New Trends in Mechanism Science: Analysis and Design (Mechanisms and Machine Science, 5) 9048196884, 9789048196883

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New Trends in Mechanism Science

MECHANISMS AND MACHINE SCIENCE Volume 5 Series Editor MARCO CECCARELLI

For other titles published in this series, go to www.springer.com/series/8779

Doina Pisla • Marco Ceccarelli • Manfred Husty Burkhard Corves Editors

New Trends in Mechanism Science Analysis and Design

Editors Prof. Dr. -Ing. Doina PISLA Technical University of Cluj-Napoca Memorandumului 28 400114 Cluj-Napoca Romania [email protected]

Univ. Prof. Dr. Manfred HUSTY University Innsbruck Technikerstr.13 6020 Innsbruck Austria [email protected]

Prof. Marco CECCARELLI University of Cassino Via Di Biasio 43 03043 Cassino (Fr) Italy [email protected] Prof. Dr. -Ing. Burkhard J. CORVES RWTH Aachen University 52056 Aachen Germany [email protected]

ISBN 978-90-481-9688-3 e-ISBN 978-90-481-9689-0 DOI 10.1007/978-90-481-9689-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010933197 © Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Table of Contents

Preface

xiii COMPUTATIONAL KINEMATICS

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage C. Huang, R. Tseng and X. Kong

3

Synthesis of Spatial RPRP Loops for a Given Screw System A. Perez-Gracia

11

Contributions to Four-Positions Theory with Relative Rotations H.-P. Schr¨ocker

21

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators G. Moroz, D. Chablat, P. Wenger and F. Rouiller

29

Kinematics and Design of a Simple 2-DOF Parallel Mechanism Used for Orientation T. Itul, D. Pisla and A. Stoica

39

Special Cases of Sch¨onflies-Singular Planar Stewart Gough Platforms G. Nawratil

47

The Motion of a Small Part on the Helical Track of a Vibratory Hopper D.I. Popescu

55

Kinematic Analysis of Screw Surface Contact ˘ ıgler J. Sv´

63

v

vi

Table of Contents

Aspects Concerning VRML Simulation of Calibration for Parallel Mechanisms A. Capustiac and C. Brisan Protein Kinematic Motion Simulation Including Potential Energy Feedback M. Diez, V. Petuya, E. Macho and A. Herm´andez Implementation of a New and Efficient Algorithm for the Inverse Kinematics of Serial 6R Chains M. Pfurner and M.L. Husty Composition of Spherical Four-Bar-Mechanisms G. Nawratil and H. Stachel

73

83

91

99

MICRO-MECHANISMS Simulation and Measurements of Stick-Slip-Microdrives for Nanorobots C. Edeler, I. Meyer and S. Fatikow

109

Analysis and Inverse Dynamic Model of a Miniaturized Robot Structure A. Burisch, S. Drewenings, R.J. Ellwood, A. Raatz and D. Pisla

117

Design and Modelling a Mini-System with Piezoelectric Actuation S. Noveanu, V.I. Csibi, A.I. Ivan and D. Mˆandru

125

LINKAGES AND MANIPULATORS Some Properties of Jitterbug-Like Polyhedral Linkages G. Kiper

137

Servo Drives, Mechanism Simulation and Motion Profiles B. Corves

147

Azimuth Tracking Linkage Influence on the Efficiency of a Low CPV System I. Visa, I. Hermenean, D. Diaconescu and A. Duta

157

Multi-Objective Optimization of a Symmetric Sch¨onflies Motion Generator B. Sandru, C. Pinto, O. Altuzarra and A. Hern´andez

165

Cyclic Test of Textile-Reinforced Composites in Compliant Hinge Mechanisms N. Modler, K.-H. Modler, W. Hufenbach, M. Gude, J. Jaschinski, M. Zichner, E.-C. Lovasz, D. M˘argineanu and D. Perju

173

Table of Contents

vii

The Optimization of a Bi-Axial Adjustable Mono-Actuator PV Tracking Spatial Linkage D. Diaconescu, I. Visa, M. Vatasescu, R. Saulescu and B. Burduhos

181

MECHANICAL TRANSMISSIONS Defect Simulation in a Spur Gear Transmission Model A. Fernandez del Rincon, M. Iglesias, A. de-Juan and F. Viadero

191

On a New Planetary Speed Increaser Drive Used in Small Hydros. Part I. Conceptual Design C. Jaliu, D. Diaconescu, R. S˘aulescu and O. Climescu

199

On a New Planetary Speed Increaser Drive Used in Small Hydros. Part II. Dynamic Model R. S˘aulescu, C. Jaliu, D. Diaconescu and O. Climescu

209

Simplified Calculation Method for the Efficiency of Involute Helical Gears M. Pleguezuelos, J.J. Pedrero and M. S´anchez Cam Size Optimization of Disc Cam-Follower Mechanisms with Translating Roller Followers P. Flores

217

225

The Dynamic Effects on Serial Printers Motion Transmission Systems D. Comanescu and A. Comanescu

235

Size Minimization of the Cam Mechanisms with Translating Roll Follower D. Perju, E.-C. Lovasz, K.-H. Modler, L.M. Dehelean, E.C. Moldovan and D. M˘argineanu

245

Kinematic Analysis of the Roller Follower Motion in Translating Cam-Follower Mechanisms E. Seabra and P. Flores

253

Kinematic Analysis of Cam Mechanisms as Multibody Systems D. Ciobanu and I. Visa

261

Peculiarities of Flat Cam Measurement by Results of Digital Photo Shooting A. Potapova, A. Golovin and A. Vukolov

269

Analyzing of Vibration Measurements upon Hand-Arm System and Results Comparison with Theoretical Model A.F. Pop, R. Morariu-Gligor and M. Balcau

277

viii

Table of Contents

Elastic and Safety Clutch with Metallic Roles and Elastic Rubber Elements I. Stroe

285

MECHANISMS FOR BIOMECHANICS A New Spatial Kinematic Model of the Lower Leg Complex: A Preliminary Study B. Baldisserri and V. Parenti Castelli

295

Selected Design Problems in Walking Robots T. Zielinska and K. Mianowski

303

Numerical Simulations of the Virtual Human Knee Joint D. Tarnita, D. Popa, N. Dumitru, D.N. Tarnita, V. Marcusanu and C. Berceanu

309

Development of a Walking Assist Machine Using Crutches – Motion for Ascending and Descending Steps T. Iwaya, Y. Takeda, M. Ogata and M. Higuchi

319

Human Lower Limb Dynamic Analysis with Applications to Orthopedic Implants N. Dumitru, C. Copilusi, N. Marin and L. Rusu

327

Forward and Inverse Kinematics Calculation for an Anthropomorphic Robotic Finger C. Berceanu, D. Tarnita, S. Dumitru and D. Filip

335

EXPERIMENTAL MECHANICS Theoretical and Experimental Determination of Dynamic Friction Coefficient for a Cable-Drum System G. Bayar, E.I. Konukseven and A.B. Koku

345

Modelling and Real-Time Dynamic Simulation of the Cable-Driven Parallel Robot IPAnema P. Miermeister and A. Pott

353

Horse Gait Exploration on “Step” Allure by Results of High Speed Strobelight Photography A. Vukolov, A. Golovin and N. Umnov

361

Simulation of the Stopping Behavior of Industrial Robots T. Dietz, A. Pott and A. Verl

369

Table of Contents

ix

Mechanical and Thermal Testing of Fluidic Muscles E. Ravina

377

Structural Dynamic Analysis of Low-Mobility Parallel Manipulators J. Corral, Ch. Pinto, M. Urizar and V. Petuya

387

DYNAMICS Spatial Multibody Systems with Lubricated Spherical Joints: Modeling and Simulation M. Machado, P. Flores and H.M. Lankarani

397

Comparison of Passenger Cars with Passive and Semi-Active Suspension Systems Based on a Friction Controlled Damper S. Reich and S. Segla

405

Dynamic Balancing of a Single Crank-Double Slider Mechanism with Symmetrically Moving Couplers V. van der Wijk and J.L. Herder

413

Evaluation of Engagement Accuracy by Dynamic Transmission Error of Helical Gears V. Atanasiu and D. Leohchi

421

The Influence of the Friction Forces and the Working Cyclogram upon the Forces of a Robot I. Turcu, C. Birleanu, F. Sucala and S. Bojan

429

Dynamic Aspects in Building up a Flight Simulator A. Pisla, T. Itul and D. Pisla

437

APPLICATIONS AND TEACHING METHODS Robotic Control of the Traditional Endoscopic Instrumentation Motion M. Perrelli, P. Nudo, D. Mundo and G.A. Danieli

449

Ethics in Robotic Surgery and Telemedicine F. Graur, M. Frunza, R. Elisei, L. Furcea, L. Scurtu, C. Radu, A. Szilaghyi, H. Neagos, A. Muresan and L. Vlad

457

Optimal Control Problem in New Products Launch – Optimal Path Using a Single Command I. Blebea, C. Dobocan and R. Morariu Gligor

467

x

Table of Contents

Mechanical Constraints and Design Considerations for Polygon Scanners V.-F. Duma and J.P. Rolland Training Platform for Robotic Assisted Liver Surgery – The Surgeon’s Point of View F. Graur, L. Scurtu, L. Furcea, N. Plitea, C. Vaida, O. Detesan, A. Szilaghyi, H. Neagos, A. Muresan and L. Vlad

475

485

Teaching Mechanisms: From Classical to Hands-on-Experiments and Research-Oriented V.-F. Duma

493

Models Created by French Engineers in the Collection of Bauman Moscow State Technical University D. Bolshakova and V. Tarabarin

503

The Models of Centrifugal Governors in the Collection of Bauman Moscow State Technical University N. Manychkin, M. Sakharov and V. Tarabarin

511

Advanced Approaches Using VR Simulations for Teaching Mechanisms S. Butnariu and D. Talaba

519

NOVEL DESIGNS Preliminary Design of ANG, a Low-Cost Automated Walker for Elderly J-P. Merlet An Active Suspension System for Simulation of Ship Maneuvers in Wind Tunnels T. Bruckmann, M. Hiller and D. Schramm Mechanism Solutions for Legged Robots Overcoming Obstacles M. Ceccarelli, G. Carbone and E. Ottaviano

529

537

545

CONTROL ISSUES OF MECHANICAL SYSTEMS Dynamic Reconfiguration of Parallel Mechanisms J. Schmitt, D. Inkermann, A. Raatz, J. Hesselbach and T. Vietor

557

Development of a Voice Controlled Surgical Robot C. Vaida, D. Pisla, N. Plitea, B. Gherman, B. Gyurka, F. Graur and L. Vlad

567

Table of Contents

xi

Modeling and Simulation of the Tracking Mechanism Used for a Photovoltaic Platform C. Alexandru The Modular Robotic System for In-Pipe Inspection O. Tatar, C. Cirebea and A. Alutei

575

583

MECHANISM DESIGN Geometric and Manufacturing Issues of the 3-UPU Pure Translational Manipulator A.H. Chebbi and V. Parenti-Castelli

595

Workspace Determination and Representation of Planar Parallel Manipulators in a CAD Environment K.A. Arrouk, B.C. Bouzgarrou and G. Gogu

605

A Theoretical Improvement of a Stirling Engine PV Diagram N.M. Dehelean, L.M. Dehelean, E.-C. Lovasz and D. Perju

613

Cylindrical Worm Gears with Improved Main Parameters T.A. Antal and A. Antal

625

Multi-Objective Optimization of Parallel Manipulators A. de-Juan, J.-F. Collard, P. Fisette, P. Garcia and R. Sancibrian

633

A Design Method of Crossed Axes Helical Gears with Increase Uptime and Efficiency T.A. Antal and A. Antal Dual Axis Tracking System with a Single Motor Gh. Moldovean, B.R. Butuc and R. Velicu The Determination of the Exact Surfaces of the Spur Wheels Flank with the Unique Rack-Bar S. Bojan, C. Birleanu, F. Sucala and I. Turcu Choosing the Actuators for the TRTTR1 Modular Serial Robot R.M. Gui, V. Ispas, Vrg. Ispas and O.A. Detesan

641

649

657

665

MECHANICS OF ROBOTS Stiffness Modelling of Parallelogram-Based Parallel Manipulators A. Pashkevich, A. Klimchik, S. Caro and D. Chablat

675

xii

Table of Contents

Incorporating Flexure Hinges in the Kinematic Model of a Planar 3-PRR Parallel Robot R.J. Ellwood, D. Sch¨utz and A. Raatz On the Dynamics of a 5 DOF Parallel Hybrid Robot Used in Minimally Invasive Surgery D. Pisla, B.G. Gherman, M. Suciu, C. Vaida, D. Lese, C. Sabou and N. Plitea

683

691

Author Index

701

Subject Index

705

Preface

The European Conference on Mechanism Science (EUCOMES 2010 Conference) is the third event of a series that has been started in 2006. EUCOMES has been started as a conference initiative to be a forum mainly for the European community working in Machine and Mechanism Science in order to facilitate aggregation and sharing interests and results for a better collaboration and activity visibility. The previous EUCOMES conferences were successfully held in Innsbruck, Austria (2006) and Cassino, Italy (2008). EUCOMES 2010 is taking place in Cluj-Napoca, Romania from 14 to 18 September 2010. EUCOMES 2010 is organized by the Center for Industrial Robots Simulation and Testing (CESTER) at the Faculty of Machine Building, Technical University of Cluj-Napoca under the patronage of IFToMM, the International Federation for the Promotion of Mechanism and Machine Science. The aim of the conference is to bring together researchers, industry professionals and students from the broad ranges of disciplines referring to Mechanism Science, in an intimate, collegial and stimulating environment. The EUCOMES 2010 Conference aims to provide a special opportunity for the scientists to exchange their scientific achievements and build up national and international collaboration in the mechanism science field and its applications. This book presents the most recent research results in the mechanism science, intended to improve a variety of applications in daily life and industry. The book is published in the Springer series “Machine and Mechanism Science”. The issues addressed are: Computational Kinematics, Micro-mechanisms, Mechanism Design, Mechanical Transmissions, Linkages and Manipulators, Mechanisms for Biomechanics, Experimental Mechanics, Mechanics of Robots, Dynamics, Control Issues of Mechanical Systems, Novel Designs, Applications and Teaching Methods. EUCOMES 2010 received 100 papers and after careful review with at least two reviews for each paper, 79 papers have been accepted for publication and presentation at the conference. We would like to express grateful thanks to IFToMM, the Romania IFToMM National Committee, the members of the International Steering Committee for the EUCOMES Conference for their co-operation: Marco Ceccarelli (University of Cassino, Italy), Burkhard Corves (University of Aachen, Germany), Manfred Husty xiii

xiv

Preface

(University of Innsbruck, Austria), Jean-Pierre Merlet (INRIA, France), Doina Pisla (Technical University of Cluj-Napoca, Romania), Fernando Viadero (University of Cantabria, Spain) and Teresa Zielinska (Warsaw Technical University, Poland), the members of the International Award Committee and the members of the Honorary Committee. We thank the authors who have contributed excellent papers on different subjects, covering many fields of Mechanism Science, and we are grateful to the reviewers for the time and effort they spent evaluating the papers. We thank the Technical University of Cluj-Napoca and Faculty for Machine Building for hosting the EUCOMES 2010 Conference and we would like to thank our colleagues: Adrian Pisla, Tiberiu Itul, Tiberiu Antal, Calin Vaida, Bogdan Gherman, Marius Suciu, Dorin Lese, Ovidiu Detesan from the Local Organizing Committee, and the sponsors of this conference for their help. Furthermore, we thank the Romanian National Authority for Scientific Research – ANCS for the financial support awarded for the EUCOMES 2010 Conference and the printing of this book. We also thank the staff at Springer, including Nathalie Jacobs (editor), and Jolanda Karada (Karada Publishing Services) for their excellent technical and editorial support. Cluj-Napoca, April 2010 The editors

Computational Kinematics

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage C. Huang1, R. Tseng1 and X. Kong2 1 National

Cheng Kung University, Taiwan; e-mail: [email protected] University, United Kingdom; e-mail: [email protected]

2 Heriot-Watt

Abstract. This paper presents the design and kinematic analysis of a multiple-mode 5R2P linkage, which is a spatial closed-loop linkage built by combining two overconstrained mechanisms: the Bennett and RPRP linkages. In the paper, the construction of the multiple-mode 5R2P linkage and its variations is investigated. The analytical inverse kinematic solution of a 4R2P open chain is adopted to analyze the 5R2P linkage for a given driving joint angle. The analysis is then conducted for the whole revolution of the driving crank. The result suggests that the multiple-mode linkage consists of three operation modes: the 5R2P, Bennett, and RPRP modes. Transitional configurations between different modes are also identified in this paper. The multiple-mode linkage features the simpler motions of the Bennett and RPRP modes as well as the more complicated motion of the 5R2P mode without the need to disconnect the linkage. Key words: kinematotropic mechanism, overconstrained linkage, Bennett, RPRP, operation mode

1 Introduction Multiple-mode linkages are designed to carry out different tasks in different modes. This paper introduces a multiple-mode 5R2P linkage that possesses three operation modes by using only one actuator. The idea is to combine two overconstrained linkages, the Bennett and RPRP linkages, with a common R joint. The resulting linkage can move like a regular 5R2P linkage and features rather complicated motion. However, if one or more joints are locked, the 5R2P linkage can move like a Bennett or RPRP linkage. The simpler motions of the Bennett and RPRP modes can also be used to bridge different branches or circuits of the 5R2P mode. Multiple-mode linkages stem from kinematotropic mechanisms [2, 5, 12] and belong to a class of reconfigurable mechanisms. The degree of freedom (DOF) of a kinematotropic linkage may change during operation; however, here we focus on the cases in which a multiple-mode mechanism has the same DOF in different modes. The use of overconstrained (paradoxic) linkages in the design of multiple-mode linkages was proposed only recently [6]. Building upon a previous development on

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_1, © Springer Science+Business Media B.V. 2010

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multiple-mode 7R linkages [3], this research investigates spatial seven-link linkages with prismatic joints, with an emphasis on the 5R2P linkage. The Bennett and RPRP overconstrained linkages are used as building blocks. The proposed multiple-mode 5R2P linkage incorporates the kinematic properties of the Bennett, RPRP, and 5R2P linkages. The kinematic analysis of a closed-loop 5R2P linkage can be modeled as the inverse kinematic analysis of a 4R2P open chain, in which we are concerned with finding all possible configurations of the linkage when given the position of its outermost body. The inverse kinematic solution of a 4R2P open chain has been obtained analytically [8], and it stems from the algorithm for solving the renowned 6R inverse kinematic problem [7, 9, 10]. In this paper, we adopt the solution method described in [8] to find all configurations of the 5R2P multiple-mode linkage for a full rotation of the input joint and identify different operation modes.

2 Construction of the Multiple-Mode Linkage Figure 1a shows the schematic drawing of a Bennett linkage, whose joints are denoted by B (and 7), A (6), C (2), and D (1). The link lengths and joint twist angles designated in the figure must satisfy the following constraints [1]: a76 = a21 ; a62 = a17

α76 = α21 ; α62 = α17 a62 a76 = sin α76 sin α62 Figure 1b shows a RPRP linkage, whose joints are denoted by B (or 7), G (3), F (4), and E (5). The link lengths and joint twist angles designated in the figure must satisfy the following constraints [4, 11]: a73 = a57 ; a34 = a45 a73 // a34 ; a45 //a57

α73 = α45 ; α34 = α57 α73 + α34 = α45 + α57 = π To construct the multiple-mode linkage, we first combine the two closed-loop linkages, BACD, and BGFE, by aligning the common joint B, as shown in Figure 2a. Secondly, we remove all the links of the combined linkages and reconnect the joints in a different order to obtain a closed-loop linkage, BAGFECD. The newly constructed 5R2P linkage, as shown in Figure 2b, possesses one degree of freedom.

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage

5

Fig. 1 (a) A Bennett linkage. (b) A RPRP overconstrained linkage.

Fig. 2 (a) Combining the Bennett and RPRP linkages with a common R joint. (b) Connecting a multiple-mode 5R2P linkage.

If we take link AB of the linkage shown in Figure 2b to be the ground link and joint B to be the driving joint, the linkage moves like a regular, one-degreeof-freedom, closed-loop linkage. What makes this linkage special is that it has two more operation modes, inherited from the two original four-link linkages. In Sections 4 and 5, we will conduct the position analysis of the linkage to confirm these operation modes.

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Fig. 3 Different combinations of the Bennett and RPRP linkages.

3 Variations in the Construction of the 5R2P Linkage Before combining the two overconstrained linkages, each of the linkages, shown in Figure 1, can be in any of its feasible configurations. When combining the linkages, as shown in Figure 2a, we are free to rotate the two linkages relative to each other about joint B. Furthermore, an offset between the two linkages along joint B is allowed, though not shown in Figure 2a. Considering these variations, we will have an infinite number of different 5R2P linkages. When reconnecting joints (the process shown in Figure 2), we must retain the orders of the original linkages. For example, the orders of the original linkages, B-A-C-D and B-G-F-E, are retained in the new linkage B-A-G-F-E-C-D. In fact, we can have 20 different combinations in constructing the 5R2P linkage. The 20 combinations are shown in Figure 3, of which the one constructed in Section 2 is identified at the upper right corner.

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage

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Fig. 4 (a) The 5R2P linkage in the Bennett mode. (b) The 5R2P linkage in the RPRP mode.

We can also reverse the orders of either one or both of the two original linkages to obtain different linkages. For example, we can reverse the order of the Bennett linkage when combining it with the RPRP linkage. In other words, we will combine linkages in the orders of B-D-C-A and B-G-F-E. As a result, another 20 combinations can be obtained.

4 Operation Modes and Transitional Configurations As mentioned in Section 2, the linkage shown in Figure 2b possesses three operation modes: the Bennett, RPRP, and 5R2P modes. In addition to the regular 5R2P motion, the linkage shown in Figure 2b can be operated like a Bennett linkage or a RPRP linkage. There are several transitional configurations that allow the linkage to switch from one operation mode to another. The configuration shown in Figure 2b is referred to as the assembly configuration of the multiple-mode linkage. It is the unique transitional configuration between the Bennett and RPRP modes. In the assembly configuration, if we lock any of joints A, C, and D, the other two joints will be automatically locked as well. As a result, the linkage can move as a RPRP linkage. On the other hand, if any of joints G, F, and E is locked in the assembly configuration, the other two joints will be locked too. Then the linkage will move like a Bennett linkage. Note that the linkage cannot be operated in the 5R2P mode starting from the assembly configuration. We have to operate the linkage in either the Bennett or RPRP mode until it reaches a transitional configuration that allows the linkage to switch to the 5R2P mode. There is usually more than one transitional configuration for switching to the 5R2P mode. Figures 4a and 4b show the 5R2P linkage in the Bennett and RPRP modes, respectively.

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C. Huang et al. Table 1 D–H parameters of the 5R2P linkage.

The merit of the proposed multiple-mode linkage, as opposed a regular 5R2P linkage, is that it allows three different motion modes without increasing the number of actuators. Furthermore, some configurations of a regular 5R2P linkage may not be reachable without disconnecting and reassembling the linkage. With the help of the two additional operation modes, all configurations of the multiple-mode 5R2P linkage can be reached without having to disassemble or reconnect the linkage.

5 Kinematic Analysis and Numerical Example In this section, we will briefly summarize the kinematic analysis of the multiplemode linkage and give a numerical example to demonstrate the concepts discussed heretofore. We are concerned with finding all possible configurations of the linkage when the driving joint angle is known. Specifying the value of the input joint angle in a 5R2P linkage is equivalent to specifying the outermost body’s position in a 4R2P open chain. We adopt the inverse kinematic solution procedures described in [8] without repeating them in this paper. We also utilize the Denavit and Hartenberg (D–H) convention for describing the geometry of links and joints of the linkage. The definitions of link parameters and coordinate systems are the same as those used in [10]. To present a numerical example, we follow the procedures described in Section 2 to construct a multiple-mode 5R2P linkage. The D–H parameter of the linkage is listed in Table 1. We conduct the position analysis of the linkage for the full rotation of the driving crank, using increments of one degree. The results of the position analysis are illustrated by joint angle plots, as shown in Figure 5. Due to limited space, Figure 5 shows only the plots of angles of joints 1 and 4 against the driving joint angle. In the plots, we can see that the multiple-mode 5R2P linkage bears the three operation modes, designated by different curves. The thick solid curves correspond to the joint angles of the RPRP mode, while the thick dashed ones correspond to the joint angles of the Bennett mode. The curve of joint

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage

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Fig. 5 Plots of two joint angles (1 and 4) against the driving joint angle (7). Table 2 Transitional configurations of the 5R2P linkage.

angle 1 is a horizontal line in the RPRP mode because joint 1 (and joints 2 and 6, not shown in the figure) is locked. Similarly, the curve corresponding to joint 4 (and joints 3 and 5, not shown in the figure) is a horizontal line in the Bennett mode. The thin dotted curves correspond to joint angles of the 5R2P mode. Next, we need to identify the transitional configurations between different modes. By comparing the joint angle plots, we can see that there are two transitional configurations, indicated by points A and C, between the RPRP and 5R2P modes. There is one transitional configuration between the Bennett and 5R2P modes, indicated by point D in the plots. Finally, point B indicates the transitional configuration between the RPRP and Bennett modes. The corresponding driving joint angles of the transitional configurations are summarized in Table 2.

6 Conclusion This paper presents the construction of a multiple-mode 5R2P linkage by using two overconstrained linkages: the Bennett and RPRP linkages. In addition to the 5R2P operation mode, this linkage also inherits the kinematic characteristics from the two overconstrained linkages. The proposed multiple-mode linkage uses only one actuator, and it eliminates the need to disconnect and reassemble the linkage in order to operate in different modes. When operating in either the Bennett or RPRP

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mode, three of the joints will be automatically locked. To switch to the 5R2P mode, we need to trigger one of the locked joints at a transitional configuration. This paper then conducts the kinematic analysis of the 5R2P multiple-mode linkage by using the inverse kinematic solution of a 4R2P linkage. The result of the analysis confirms the three operation modes and identifies the transitional configurations between different modes.

Acknowledgments The financial support of the National Science Council, Taiwan and the Royal Society of UK through an international exchange program is gratefully acknowledged.

References 1. Bennett, G.T., A new mechanism. Engineering, 76, 777–778, 1903. 2. Galletti, C. and Fanghella, P., Single-loop kinematotropic mechanisms. Mechanism and Machine Theory, 36(6):743–761, 2001. 3. Huang, C., Kong, X., and Ou, T., Position analysis of a Bennett-based multiple-mode 7R linkage. In Proceedings of ASME Mechanisms and Robotics Conference, San Diego, 2009. 4. Huang, C. and Tu, H.-T., Linear property of the screw surface of the spatial RPRP linkage. ASME Journal of Mechanical Design, 128:581–586, 2006. 5. Kong, X., Gosselin, C.-M., and Richard, P.-L., Type synthesis of parallel mechanisms with multiple operation modes. ASME Journal of Mechanical Design, 129(7):595–601, 2007. 6. Kong, X. and Huang, C., Type synthesis of single-DOF single-loop mechanisms with two operation modes. In Proceedings of ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots, London, 2009. 7. Lee, H.-Y. and Liang, C.-G., Displacement Analysis of the general spatial 7-link 7R mechanism. Mechanism and Machine Theory, 23(3):219–226, 1988. 8. Mavroidis, C., Ouezdou, F.-B., and Bidaud, P., Inverse kinematics of six-degree-of-freedom general and special manipulators using symbolic computation. Robotica, 12:421–430, 1994. 9. Raghavan, M. and Roth, B., Kinematic analysis of the 6R manipulator of general geometry. In Proceedings of the 5th International Symposium on Robotics Research, H. Miura and S. Arimoto (Eds.), MIT Press, Cambridge, 1990. 10. Tsai, L.-W., Robot Analysis: The Mechanics of Serial and Parallel Manipulator. John Wiley & Sons, New York, 1999. 11. Waldron, K.J., A study of overconstrained linkage geometry by solution of closure equations – Part II. Four-bar linkages with low pair joints other than screw joints. Mechanism and Machine Theory, 8:233–247, 1973. 12. Wohlhart, K., Kinematotropic linkages. In Recent Advances in Robot Kinematics, J. Lenarcic and V. Parenti-Castelli (Eds.), Kluwer Academic Publishers, Dordrecht, the Netherlands, pp. 359–368, 1996.

Synthesis of Spatial RPRP Loops for a Given Screw System A. Perez-Gracia Institut de Robotica i Informatica Industrial (IRI) UPC/CSIC, Barcelona, Spain; and College of Engineering, Idaho State Univesity, USA; e-mail: [email protected]

Abstract. The dimensional synthesis of spatial chains for a prescribed set of positions can be used for the design of parallel robots by joining the solutions of each serial chain at the end effector. In some cases, this may yield a system with negative mobility. The synthesis of some overconstrained but movable linkages can be done if the finite screw system associated to the motion of the linkage is known. For these cases, the screw system could be related to the finite tasks positions traditionally defined in the synthesis theory. This paper presents the simplest case, that of the spatial RPRP closed chain, for which one solution exists. Key words: dimensional synthesis, overconstrained linkages, finite screw systems

1 Introduction Synthesis of parallel robots has focused mainly on type or structural synthesis, using group theory, screw theory, or geometric methods, see for instance [3]. Dimensional synthesis examples exist, which focus on optimizing performance indices [7, 10] or on reachable workspace sizing [2, 12]; see also [14]. The dimensional synthesis of spatial serial chains for a prescribed set of positions can be used for the design of parallel robots by synthesizing all supporting legs for the same set of positions. There are a few examples of finite-position dimensional synthesis of parallel robots in the literature, most of them doing partial synthesis. Wolbrecht et al. [21] perform synthesis of 3-RRS, 4-RRS and 5-RRS symmetric parallel manipulators; Kim and Tsai [11] and Rao [20] solve the partial kinematic synthesis of a 3-RPS parallel manipulator. This method yields, in some cases, a system with negative mobility. One interesting question is whether the finite-screw surfaces generated by the task positions can give any information for the synthesis of the overconstrained closed linkages. Using Parkin’s definition for pitch [15], the screws corresponding to finite displacements can form screw systems. Huang [4] showed that the single RR chain forms a finite screw system of third order; however, the set of finite displacements of the coupler of the Bennett linkage form a cylindroid, which is a gen-

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_2, © Springer Science+Business Media B.V. 2010

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eral 2-system of screws [5]. Baker [1] has also studied the motion of the Bennett linkage. Perez and McCarthy [16] used two arbitrary displacements to generate the cylindroid associated to the Bennett linkage in order to perform dimensional synthesis. Husty et al. [9] use the geometry of the Study quadric to obtain simpler equations for the synthesis and analysis. Following this approach, Pfurner and Husty [19] present the constraint manifold of overconstrained 2-3R parallel robots as 6R closed chains. In this paper, the focus is on the simplest of the overconstrained linkages, the closed spatial RPRP linkage. Recently, Huang [6] has shown that the set of screws corresponding to displacements of this linkage forms a 2-screw system. We use this result in order to synthesize RPRP linkages with positive mobility and for a given shape of the screw system of the relative displacements. In order to do so, we state the design equations using the Clifford algebra of dual quaternions [18], which has a direct relation to the screw system. The design yields a single RPRP linkage.

2 Clifford Algebra Equations for the Synthesis 2.1 Forward Kinematics The approach used in this paper for stating design equations is based on the method of Lee and Mavroidis [13]. They equate the forward kinematics of a serial chain to a set of goal displacements and consider the Denavit-Hartenberg parameters as variables. A more efficient formulation for our purpose consists of stating the forward kinematics of relative displacements using the even Clifford subalgebra C+ (P3 ), also known as dual quaternions. In this section, we follow the approach presented in [18]. The Pl¨ucker coordinates S = (s, c × s) of a line can be identified with the Clifford algebra element S = s + ε c × s. Similarly, the screw J = (s, v) becomes the element J = s + ε v. Using the Clifford product we can compute the exponential of the screw θ 2 J, 

θ θ d θ θ d θ θˆ θˆ e 2 J = cos − sin ε + sin + cos ε S = cos + sin S. 2 2 2 2 2 2 2 2

(1)

The exponential of a screw defines a unit dual quaternion, which can be identified with a relative displacement from an initial position to a final position in terms of a rotation around and slide along an axis. For a serial chain with n joints, in which each joint can rotate an angle θi around, and slide the distance di along, the axis Si , for i = 1, . . . , n, the forward kinematics of relative displacements (with respect to a reference position) can be expressed as the composition of Clifford algebra elements. Let θ 0 and d0 be the joint parameters of this chain when in the reference configuration, so we have ∆ θˆ = (θ − θ0 + (d − d0 )ε ). Then, the movement from this reference configuration is defined by

Synthesis of Spatial RPRP Loops for a Given Screw System

13

Fig. 1 The RP serial chain, left; the RPRP closed linkage, right.

∆ θˆ

∆ θˆ

∆ θˆ

ˆ ∆ θˆ ) = e 2 1 S1 e 2 2 S2 · · · e 2 n Sn , Q( 


∆ θˆ1 ∆ θˆ1 ∆ θˆ2 ∆ θˆ2 ∆ θˆn ∆ θˆn +s S +s S ··· c +s Sn . (2) c = c 2 2 1 2 2 2 2 2 Note that s and c denote the sine and cosine functions, respectively. The RPRP linkage has a mobility M = −2 using the Kutzbach–Groebler formula; however, for certain dimensions of the links, it moves with one degree of freedom. The RPRP linkage can be seen as a serial RP chain and a serial PR chain joined at their end-effectors. The RP serial chain consists of a revolute joint followed by a prismatic joint. Figure 1 shows the RP serial chain and a sketch of the RPRP linkage with its axes. In the PR serial chain, the order of the joints in the chain is switched. For both the RP and PR serial chains, let G = g + ε g0 be the revolute joint axis, with rotation θ , and H = h + ε h0 the prismatic joint axis, with slide d. Notice that, for synthesis purposes, the location of the slider, given by h0 , is irrelevant. The Clifford algebra forward kinematics equations of the RP chain are    ∆d ∆θ ∆θ ˆ + sin G 1+ε H QRP (∆ θ , ∆ d) = cos 2 2 2     ∆θ ∆θ ∆d ∆θ ∆d ∆θ ∆θ 0 ∆d ∆θ +s g +ε − s g·h+ c h+s g + s g×h . = c 2 2 2 2 2 2 2 2 2 (3) For the PR chain, the only difference is a negative sign in the cross product.

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2.2 Design Equations and Counting The design equations are created when a set of task positions are defined. The design variables that determine the dimensions of the chain are the position of the joint axes in the reference configuration. ∆ φˆ Given a set of task positions expressed as relative displacements, Pˆ = cos 1 j + sin

∆ φˆ 1 j 2 P1 j ,

1j

2

j = 2, . . . , m, we equate them to the forward kinematics in Eq. (2), Pˆ1 j = e

∆ θˆ1 j 2

S1

e

∆ θˆ2 j 2

S2

···e

∆ θˆn j 2

Sn

,

j = 2, . . . , m.

(4)

The result is 8(m − 1) design equations. The unknowns are the n joint axes Si , i = 1, . . . , n, and the n(m − 1) pairs of joint parameters ∆ θˆi j = ∆ θi j + ∆ di j ε . For the RP (and similarly the PR) serial chains, the design equations are Qˆ RP (∆ θ j , ∆ d j ) = Pˆ1 j ,

j = 1, . . . , m.

(5)

The counting of independent equations and unknowns allows to define the maximum number of arbitrary positions m that can be reached, based only on the type and number of joints of the serial chain, see [17] for details. Consider a serial chain with r revolute and p prismatic joints. The maximum number of task positions is given by m in Eq.(6). For serial chains with less than three revolute joints, the structure of semi-direct product of the composition of displacements needs to be considered, and the maximum number of rotations mR needs to be calculated too. Assuming that the orientations are given and that both the directions of the revolute joints and the angles to reach the task orientations are known, we can count, in a similar fashion, the number of translations mT that the chain can be defined for, m=

3r + p + 6 , 6 − (r + p)

mR =

3+r , 3−r

mT =

2r + p + 3 − c . 3− p

(6)

In order to determine the maximum number of task positions for the RP and PR chains, we apply Eq. (6), to obtain m = 2.5 task positions. Additional information is obtaining when computing mR = 2 task rotations, and mT = 3 task translations. Hence, we can define one arbitrary relative displacement and a second relative displacement whose orientation is not general.

3 Screw System for the RPRP Linkage In the context of this paper, a screw surface is a ruled surface in which the lines correspond to relative displacements. A screw surface will be a screw system if it

Synthesis of Spatial RPRP Loops for a Given Screw System

15

is closed under addition and scalar multiplication, that is, if it can be written as a linear combination of screws. The linear combination of two arbitrary screws representing relative displacements form a 2-system known as the cylindroid, which is the manifold for the relative displacements of the closed 4R linkage. Huang [6], by intersecting the 3-systems associated with the RP and PR dyads, shows that the screw surface of the closed RPRP linkage forms a 2-system of a special type, the fourth special type according to Hunt [8], also known as 2-IB [22]. The screws of this system are parallel, coplanar screws whose pitches vary linearly with their distance. This screw system can be generated by two screws with same direction and finite pitches. Notice that this coincides with the results of the counting for the synthesis of the RP (or PR) serial chain. This allows us to define and use the screw system as input for the dimensional synthesis of the closed RPRP chain. We have several strategies for doing so. For instance, we can select a first relatˆ ˆ ive displacement, Sˆ12 = cos ∆2ψ + sin ∆2ψ (s12 + ε s012 ). The rotation axis of the displacement, s12 is common to both Sˆ12 and the second relative displacement. We set s12 = s13 and select a rotation angle to define the relative rotation sˆ13 . We can then set the slope of the pitch distribution in order to shape the screw system. The pitch for the finite displacement screws is [15] p1i =

∆ t1i 2 ∆ψ tan 2 1i

,

(7)

s ·s0 directly calculated from the dual quaternion using p1i = s1i ·s1i . Similarly, a point on 1i 1i the screw axis is calculated as c1i = s1i × s01i .

(8)

Define the slope of the distribution as p − p12  K =  13 c13 − c12 

(9)

If we set the value of K, we can solve for ∆ t13 in order to define the pitch of the second relative displacement, the location of its screw axis being defined. This defines the screw system; by converting to absolute displacements, we can easily check whether the trajectory of the end-effector is acceptable.

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4 Dimensional Synthesis of the RPRP Linkage for a Prescribed Screw System The solution of the RP, and similarly, PR chains is simple and yields one solution. 0 Given an arbitrary relative displacement Qˆ 12 = (qw12 + q12 ) + ε (qw0 12 + q12 ) and a ˆ second displacement Q13 such that both have same direction and a given pitch distribution, as explained in previous section, we equate them to the forward kinematics in Eq.(3). We can solve for the direction of the revolute joint g and the rotation angles,   q12 ∆ θ1i q1i      g =   , = tan , i = 2, 3. (10) 2 qw1i q12 The equations for the dual part are linear in the moment of the revolute joint, g0 ,

∆ d1i ∆ θ1i ∆ θ1i 1 0 cos h + sin g × h , i = 1, 2. (11) g0 = q − 1i ∆θ 2 2 2 sin 1i 2

Equating the solution of g0 for both relative displacements, we can solve linearly for h as a function of the slides ∆ d12 , ∆ d13 . The relation between the slides is given by the pitch condition, ∆ d12 2

qw0 12 sin

∆ θ12 2

=

∆ d13 2

qw0 13 sin

∆ θ13 2

(12)

Imposing ||h|| = 1, we can solve for the slides to obtain one solution. Using the same process, we can solve for the PR serial chain.

5 Example The dual quaternions in Table 1 were generated as explained. Sˆ12 has been randomly generated, while the rotation in Sˆ13 is such that it belongs to the workspace of the chain. We select c13 = (−1.237, 2.541, −1.601), randomly generated. Set a value of the slope of the pitch distribution, K = 0.48, to create the second displacement. These two screws generate the screw system in Figure 2, where the length of each screw is proportional to its pitch. Some of the corresponding absolute displacements of the trajectory are included in the same figure.

Table 1 Goal relative displacements for the RP and PR chains. (0.459, −0.133, −0.565, −0.672) + ε (1.660, 0.343, −0.019, 1.082) (0.135, −0.039, −0.167, 0.976) + ε (0.023, −0.570, −0.923, −0.184)

Synthesis of Spatial RPRP Loops for a Given Screw System

17

Fig. 2 Left: screw system generated by S12 and S13 (shown in corners). Right: corresponding absolute displacements. Table 2 Joint axes for the RPRP linkage at the reference configuration. Chain

Revolute joint G

Prismatic joint h

Rotations (θ12 , θ13 )

Slides (d12 , d13 )

RP

(−0.620, 0.179, 0.763) + ε (−0.436, −3.166, 0.389) (0.620, −0.179, −0.763) + ε (2.316, −1.826, 2.310)

(−0.087, 0.893, 0.442)

(-264.5,-25.2)

(5.30,-3.05)

PR

(−0.274, 0.065, −0.959) (264.5,25.2)

(-5.30,3.05)

Fig. 3 The RPRP linkage reaching the first position.

We obtain one solution for the RPRP linkage, specified in Table 2 as the Pl¨ucker coordinates of the axes and the joint variables to reach the positions. Comparing these results to the joint variable conditions in [6] we can see that our solution corresponds to the unfolded RPRP linkage. Figure 3 shows the chain reaching the three displacements, using the identity as reference displacement.

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6 Conclusions This papers presents the synthesis of an overconstrained closed linkage, the RPRP. The knowledge of the screw system that corresponds to the finite displacements of the linkage is used to ensure that the solutions of the synthesis of the RP and PR serial chains can be assembled to create a movable system. The counting of the maximum positions for the synthesis suffices to define the positions that generate the screw system. The synthesis yields a single RPRP linkage.

References 1. J.E. Baker. On the motion geometry of the bennett linkage. In Proc. 8th Internat. Conf. on Engineering Computer Graphics and Descriptive Geometry, pages 433–437, Austin, Texas, USA,, 1998. 2. D. Chablat and P. Wenger. Architecture optimization of a 3-dof parallel mechanism for machining applications, the orthoglide. IEEE Transactions on Robotics and Automation, 19(3):403–410, 2003. 3. G. Gogu. Structural Synthesis of Parallel Robots. Part 1: Methodology. Springer, first edition, 2007. 4. C. Huang. On the finite screw system of the third order associated with a revolute-revolute chain. ASME Journal of Mechanical Design, 116:875–883, 1994. 5. C. Huang. The cylindroid associated with finite motions of the bennett mechanism. In Proceedings of the ASME Design Engineering Technical Conferences, 1996, Irvine, CA, USA, 1996. 6. C. Huang. Linear property of the screw surface of the spatial rprp linkage. ASME Journal of Mechanical Design, 128:581–586, 2006. 7. T. Huang, M. Li, X.M. Zhao, J.P. Mei, D.G. Chetwynd, and S.J. Hu. Conceptual design and dimensional synthesis for a 3-dof module of the trivariant - a novel 5-dof reconfigurable hybrid robot. IEEE Transactions on Robotics, 21(3):449–456, 2005. 8. K.H. Hunt. Kinematic Geometry of Mechanisms. , Oxford University Press, 1978. 9. M.L. Husty, M. Pfurner, H.-P. Schrocker, and K. Brunnthaler. Algebraic methods in mechanism analysis and synthesis. Robotica, 25:661–675, 2007. 10. H.S. Kim and L.-W. Tsai. Design optimization of a cartesian parallel manipulator. ASME Journal of Mechanical Design, 125:43–51, 2003. 11. H.S. Kim and L.-W. Tsai. Kinematic synthesis of a spatial 3-rps parallel manipulator. ASME Journal of Mechanical Design, 125:92–97, 2003. 12. A. Kosinska, M. Galicki, and K. Kedzior. Design and optimization of parameters of delta-4 parallel manipulator for a given workspace. Journal of Robotic Systems, 20(9):539–548, 2003. 13. E. Lee and C. Mavroidis. Solving the geometric design problem of spatial 3r robot manipulators using polynomial homotopy continuation. ASME Journal of Mechanical Design, 124(4):652–661, 2002. 14. J.-P. Merlet. Optimal design of robots. In Proceedings of Robotics: Science and Systems, June, 2005, Cambridge, USA, 2005. 15. I.A. Parkin. A third conformation with the screw systems: Finite twist displacements of a directed line and point. Mechanism and Machine Theory, 27:177–188, 1992. 16. A. Perez and J. M. McCarthy. Dimensional synthesis of bennett linkages. ASME Journal of Mechanical Design, 125(1):98–104, 2003. 17. A. Perez and J. M. McCarthy. Dual quaternion synthesis of constrained robotic systems. ASME Journal of Mechanical Design, 126(3):425–435, 2004.

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18. A. Perez Gracia and J. M. McCarthy. The kinematic synthesis of spatial serial chains using clifford algebra exponentials. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, 220(7):953–968, 2006. 19. M. Pfurner and M.L. Husty. A method to determine the motion of overconstrained 6rmechanisms. In Proceedings of the 12th IFToMM World Congress, June 18-21, 2007, Besancon, France, 2007. 20. N.M. Rao and K.M. Rao. Dimensional synthesis of a 3-rps parallel manipulator for a prescribed range of motion of spherical joints. Mechanism and Machine Theory, 44:477–486, 2009. 21. E. Wolbrecht, H.-J. Su, A. Perez, and J.M. McCarthy. Geometric design of symmetric 3-rrs constrained parallel platforms. In ASME, editor, Proceedings of the 2004 ASME International Mechanical Engineering Congress and Exposition, November 13-19, 2004, Anaheim, California, USA, 2004. 22. D. Zlatanov, S. Agrawal, and C.L. Gosselin. Convex cones in screw spaces. Mechanism and Machine Theory, 40:710–727, 2005.

Contributions to Four-Positions Theory with Relative Rotations H.-P. Schr¨ocker University Innsbruck, A-6020 Innsbruck, Austria; e-mail: [email protected]

Abstract. We consider the geometry of four spatial displacements, arranged in cyclic order, such that the relative motion between neighbouring displacements is a pure rotation. We compute the locus of points whose homologous images lie on a circle, the locus of oriented planes whose homologous images are tangent to a cone of revolution, and the locus of oriented lines whose homologous images form a skew quadrilateral on a hyperboloid of revolution. Key words: four-positions theory, rotation, discrete line congruence, Study quadric

1 Introduction Four-positions theory studies the geometry of four proper Euclidean displacements, see [3, chapter 5, §8–9]. Often, it is considered as a discretization of differential properties of order three of a smooth one-parameter motion. In this article we take a different point of view and regard the four displacements as an elementary cell in a quadrilateral net of positions, that is, the discretization of a two- or more-parameter motion. Motivated by applications in the rising field of discrete differential geometry [1] we intend to study quadrilateral nets of positions such that the relative displacement between neighbouring positions is a pure rotation. The present text shall provide foundations for this project. Thus, the questions we are interested in are induced from discrete differential geometry. Nonetheless, our contributions seem to be of a certain interest in their own right. Denote the four displacements by α0 , α1 , α2 , α3 and the relative displacements of consecutive positions by τi,i+1 = αi+1 ◦ αi−1 (indices modulo four). We require that every relative displacement τi,i+1 is a pure rotation. In this case we call the quadruple (α0 , α1 , α2 , α3 ) a rotation quadrangle. Rotation quadrangles with additional properties (the displacements α2 ◦ α0−1 and α3 ◦ α1−1 are also rotations) have been studied in [3, chapter 5, § 9]. Our contribution is more general and we provide answers to questions not asked in [3]. In Section 2 we address the problem of constructing rotation quadrilaterals. Study’s kinematic mapping turns out to be a versatile tools for this task. Sub-

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_3, © Springer Science+Business Media B.V. 2010

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H.-P. Schr¨ocker

Fig. 1 Homologous points on a circle, homologous planes tangent to a cone of revolution, and homologous lines on a hyperboloid of revolution.

sequently, we study points X whose homologous images Xi = αi (X ) lie on a circle (Section 3; Figure 1, left), oriented planes ε whose homologous images εi = αi (ε ) are tangent to a cone of revolution (Section 4; Figure 1, center), and oriented lines
whose homologous images
i = αi (
) form a skew quadrilateral on a hyperboloid of revolution (Section 5; Figure 1, right).

2 Construction of Rotation Quadrilaterals One position in a rotation quadrilateral, say α0 , can be chosen arbitrarily without affecting the quadrilateral’s geometry. The remaining positions can be constructed inductively by suitable rotations but the final position α3 must be such that both τ23 and τ30 are rotations. This is certainly possible, albeit a little tedious, in Euclidean three space. The quadric model of Euclidean displacements, obtained by Study’s kinematic mapping [4, 8] seems to be a more appropriate setting for this task. It transforms two positions corresponding in a relative rotation to conjugate points on the Study quadric S . More precisely, the composition of all rotations about a fixed axis with a fixed displacement is a straight line in the Study quadric. Thus, a rotation quadrilateral corresponds, via Study’s kinematic mapping, to a skew quadrilateral Q in the Study quadric. We require here and in the following that Q spans a threespace which is not completely contained in the Study quadric S . This genericity assumption is not fulfilled in the special configuration studied in [3, chapter 5, § 9]. The quadrilateral Q is uniquely determined by three suitable chosen vertices A0 , A1 , A2 and the connecting line of A2 and the missing vertex A3 . Alternatively, Q is determined by two opposite vertices, say A0 and A2 , and two opposite edges. In any case the input data defines a three-space whose intersection with the Study quadric is a hyperboloid H containing Q. The completion of the missing data is easy. Denoting the relative revolute axes in the moving space by ri,i+1 the translation of these considerations into the language of kinematics reads: Theorem 1. A rotation quadrangle is uniquely determined by 1. a fixed displacement αi , three revolute axis ri,i+1 , ri+1,i+2 , ri+2,i+3 , and two rotation angles ωi,i+1 and ωi+1,i+2 (indices modulo four) or 2. two displacements αi , αi+2 and two relative revolute axes ri,i+1 , ri+2,i+3 (indices modulo four).

Contributions to Four-Positions Theory with Relative Rotations

23

Computing the actual rotation angles and line coordinates of the revolute axes from the quadrilateral Q is straightforward. The necessary formulas are, for example, found in [9, Satz 13]. Beware that [9] uses conventions for the Study parameters that slightly differ from modern authors, for example [4] or [8]. Therefore minor adaptions to the formulas may be necessary.

3 Homologous Points on a Circle Now we turn to the study of the locus of points in the moving space whose homologous images lie on a circle (Figure 1, left). As usual we consider straight lines as special circles of infinite radius. If the four relative displacements are general screws, the sought locus is known to be an algebraic curve of degree six [3, p. 128]. In case of rotation quadrilaterals this is still true but the curve is highly reducible: Theorem 2. The locus of points in the moving space whose homologous images lie on a circle consists of the four relative revolute axis r01 , r12 , r23 , r30 and their two transversals u and v. Proof. By the general result of [3] the sought locus can consist of six lines at most. We show that all points on the six lines mentioned in the theorem have indeed homologous images on circles. For points X ∈ ri,i+1 this is trivially true since we have Xi = Xi+1 . Since any transversal carries four points of this type, Lemma 1 below, implies the same property for all points on u and v. (At least the first part of this lemma is well-known. An old reference is [7, p. 187].)
Lemma 1. Consider a straight line
in the moving space and four affine displacements. If
contains four points whose homologous images are co-planar than this is the case for all points of
. If, in addition, the homologous images of three points lie on circle, this is the case for all points of
. Proof. We can parametrize the homologous images of
as
i : xi (t) = ai + tdi ;

ai , di ∈ R3 ;

i = 0, 1, 2, 3

(1)

such that points of the same parameter value t are homologous. The co-planarity condition reads
 1 1 1 1 det =0 (2) x0 x1 x2 x3 (we omit the argument t for sake of readability). Since it is at most of degree three in t, it vanishes identically if four zeros exist. This proves the first assertion of the lemma. Four homologous points X0 , X1 , X2 , X3 lie on a circle (or a straight line), if three independent bisector planes βi j of Xi and X j have a line, possibly at infinity, in common. This is equivalent to

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H.-P. Schr¨ocker

V2 y

d a

V3 y U23

V1 V0 x x U01

U30

c b

U12

Fig. 2 Homologous images of the transversal line u.

⎛ T ⎞ x0 · x0 + xT1 · x1 , x0 − x1 rank ⎝xT1 · x1 + xT2 · x2 , x1 − x2 ⎠ ≤ 2. xT2 · x2 + xT3 · x3 , x2 − x3

(3)

Corresponding points are assumed to be coplanar. Therefore there exist scalars λ , µ and ν , not all of them zero, such that

λ (x0 − x1 ) + µ (x1 − x2 ) + ν (x2 − x3 ) = 0.

(4)

Hence, the circularity condition becomes

λ (xT0 · x0 + xT1 · x1 ) + µ (xT1 · x1 + xT2 · x2 ) + ν (xT2 · x2 + xT3 · x3 ) = 0.

(5)

It is at most quadratic in t. Therefore, existence of three zeros implies its vanishing and the second assertion of the lemma is proved.
The set of circles through quadruples of homologous points originating from a relative revolute axis ri,i+1 is not very interesting in the context of this article. Of more relevance is the set of circles generated by points of the transversal lines u and v. Theorem 3. The circles through homologous images of points of a transversal line (u or v) lie on a hyperboloid of revolution. Proof. We consider only points on the transversal u. Any two of its successive homologous images ui , ui+1 intersect in a point Ui,i+1 , the intersection point of αi (ri,i+1 ) and αi+1 (ri,i+1 ). Therefore the lines u0 , u1 , u2 , u3 are the edges of a skew quadrilateral and lie on a pencil of hyperboloids. The intersection points with circles through homologous points form congruent ranges of points on the lines u0 , u1 , u2 , u3 and the intersection points Ui,i+1 are homologous in the correspondence between ui and ui+1 . Consider now a circle C such that every intersection point Vi with ui lies outside the segment [Ui−1,i ,Ui,i+1 ] for i ∈ {0, 1, 2, 3}. Setting a := dist(U30 ,U01 ), c := dist(U12 ,U23 ),

b := dist(U01 ,U12 ), d := dist(U30 ,U23 ),

x := dist(U01 ,V0 ) = dist(U01 ,V1 ) (6) y := dist(U23 ,V2 ) = dist(U23 ,V2 )

Contributions to Four-Positions Theory with Relative Rotations

25

(Figure 2) we see that b + x = c + y and a + x = d + y. This implies a + c = b + d. In other words, both pairs of opposite edges in the skew quadrilateral (U01 ,U12 ,U23 ,U30 ) have the same sum of lengths. This is a well-known criterion
for the existence of a hyperboloid of revolution through u0 , u1 , u2 , and u3 .

4 Homologous Planes Tangent to a Cone of Revolution In this section we study the locus of planes ε in the moving space such that its homologous images ε0 , ε1 , ε2 , ε3 are tangent to a cone of revolution (Figure 1, center). Actually, we will not discuss this in full generality but restrict ourselves to the case of oriented planes in oriented contact with a cone of revolution. This means the cone axis is the intersection line of the unique bisector planes of εi and εi+1 . Having in mind applications in discrete differential geometry (a kinematic generation of discrete principal contact element nets, see [2, 1, 6]) this additional assumption is justified. Since the offset surface of a cone of revolution is again a cone of revolution, every plane parallel to a solution plane ε is a solution plane as well. We infer that the sought locus of planes consists of pencils of parallel planes. This can also be confirmed by direct computation. Denote by e = (e0 , e1 , e2 , e3 )T the plane coordinate vector of ε . If Ai is the homogeneous transformation matrix describing αi , the coordinate vectors ei of the homologous planes are found from eTi = eT · A−1 i ,

i ∈ {0, 1, 2, 3}.

(7)

The condition that they have a point in common reads E(e0 , e1 , e2 , e3 ) = det(e0 , e1 , e2 , e3 ) = 0

(8)

and is of degree four in e0 , e1 , e2 , and e3 . Assuming that (8) holds, the condition on the conical position can be stated in terms of the oriented normal vectors ni which are obtained from ei by dropping the first (homogenizing) coordinate. The planes are tangent to a cone of revolution if the four points with coordinate vectors ni lie on a circle (note that these vectors have the same length). This condition results in a homogeneous cubic equation G(e1 , e2 , e3 ) = 0 which is nothing but the circle-cone condition known from four-position synthesis of spherical four-bar linkages (see [5, p. 179]). A closer inspection shows that there exists a homogeneous polynomial F(e1 , e2 , e3 ) of degree four such that (8) becomes E(e0 , e1 , e2 , e3 ) = F(e1 , e2 , e3 ) + e0 G(e1 , e2 , e3 ) = 0.

(9)

Hence, solution planes are characterized by the simultaneous vanishing of F and G. Since these equations are independent of e0 , the value of the homogenizing coordin-

26

H.-P. Schr¨ocker

ate can be chosen arbitrarily. This is the algebraic manifestation of our observation that the solution consists of pencils of parallel planes. Since the two homogeneous equations F and G are of respective degrees four and three there exist at most twelve pencils of parallel planes whose homologous images are tangent to a cone of revolution. But not all of them are valid: The circlecone equation G also identifies vectors n0 , n1 , n2 , n3 that span a two-dimensional sub-space. The corresponding planes ε0 , ε1 , ε2 , ε3 are all parallel to a fixed direction but do not qualify as solution planes and have to be singled out. The complete description of all solution planes is given in Theorem 4. The locus of planes in the moving space whose homologous images are tangent to a cone of revolution consists of six pencils of parallel planes: 1. The planes orthogonal to one of the four revolute axes r01 , r12 , r23 , r30 and 2. the planes orthogonal to one of the two transversals u or v of these axes. Proof. We already know that there exist at most twelve parallel pencils of solution planes, obtained by solving the quartic equation F(e1 , e2 , e3 ) = 0 and the cubic equation G(e1 , e2 , e3 ) = 0. By Lemma 2 below (which actually has been proved in a more general context in [3, pp. 131–132]), six of these pencils of planes are spurious and have to be subtracted from the totality of all solution planes. Thus, we only have to show that the planes enumerated in the Theorem are really solutions. This is obvious in case of a plane ε orthogonal to axis ri,i+1 since εi and εi+1 are identical. In case of a plane orthogonal to u or v it is an elementary consequence of Theorem 3.
Lemma 2. Consider four spherical displacements σ0 , σ1 , σ2 , and σ3 . Then there exist in general six lines (possibly complex or coinciding) through the origin of the moving coordinate frame, whose homologous lines are co-planar. Proof. The co-planarity condition applied to a vector x reads rank(x0 , x1 , x2 , x3 ) ≤ 2.

(10)

We compute classes of proportional solution vectors by solving the two cubic equations det(x0 , x1 , x2 ) = det(x0 , x1 , x3 ) = 0. (11) It has nine solutions. But the three fixed directions (one real, two conjugate imaginary) of the relative rotation σ2 ◦ σ1−1 are spurious so that only six valid solutions remain.


5 Homologous Lines on a Hyperboloid of Revolution Finally, we also investigate the locus of lines
in the moving frame whose homologous images
0 ,
1 ,
2 ,
3 form a skew quadrilateral on a hyperboloid of re-

Contributions to Four-Positions Theory with Relative Rotations

27

volution (Figure 1, right). As usual, we restrict ourselves to generic configurations. Moreover, we make an additional assumption on the lines’ orientation: Setting Li,i+1 :=
i ∩
i+1 (indices modulo four) we require that, when walking around the edges of the skew quadrilateral (L01 , L12 , L23 , L30 ) we follow the line’s orientation only on every second edge (Figure 1, right). The manifold of lines in Euclidean three-space depends on four parameters. The mentioned condition on the four homologous images
0 ,
1 ,
2 ,
3 of a line
imposes, however, five conditions: four intersection conditions for the skew quadrilateral and one further condition for the hyperboloid of revolution. Therefore, we expect no solution for a general screw quadrangle. On the other hand we already know (Theorem 3) that for rotation quadrangles the homologous images of two transversals u and v form the required configuration. We will show that these are the only solutions. Considering the skew quadrilateral condition only, it is obvious that either •
i and
i+1 intersect αi (ri,i+1 ), possibly at a point at infinity, or •
i and
i+1 are orthogonal to αi (ri,i+1 ). Only the first condition is compatible with our requirement on the orientation of the homologous lines. Therefore, it is necessary that
intersects all lines ri,i+1 (indices modulo four). Hence
= u or
= v. Combining this result with Theorem 3 we arrive at Theorem 5. The two transversals u and v of the four relative revolute axes r01 , r12 , r23 , and r30 are the only lines whose homologous images form a skew quadrilateral (with proper orientation) on a hyperboloid of revolution.

6 Conclusion and Future Research We investigated geometric structures related to four displacements αi (i ∈ {0, 1, 2, 3}) such that the relative displacements τi,i+1 = αi+1 ◦ αi−1 are pure rotations. In particular, we completely characterized the points, planes and lines whose homologous images lie on a circle, are tangent to a cone of revolution or form a skew quadrilateral on a hyperboloid of revolution (with proper orientation). In a next step we plan to study quadrilateral nets of proper Euclidean displacements such that neighbouring positions correspond in a pure rotation. Using the results of this article it is possible to show that independent pairs of principal contact element nets [2] can occur as trajectories of such nets. Since mechanically generated motions are often composed of pure rotations the theory might also be useful in more applied settings.

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H.-P. Schr¨ocker

References 1. A.I. Bobenko and Y.B. Suris. Discrete Differential Geometrie. Integrable Structure, Graduate Texts in Mathematics, Vol. 98. American Mathematical Society, 2008. 2. A.I. Bobenko and Yu.B. Suris. On organizing principles of discrete differential geometry. geometry of spheres. Russian Math. Surveys, 62(1):1–43, 2007. 3. O. Bottema and B. Roth. Theoretical Kinematics. Dover Publications, 1990. 4. M. Husty and H.-P. Schr¨ocker. Algebraic geometry and kinematics. In Ioannis Z. Emiris, Frank Sottile, and Thorsten Theobald (Eds.), Nonlinear Computational Geometry, The IMA Volumes in Mathematics and its Applications, Vol. 151, Springer, 2009. 5. J.M. McCarthy. Geometric Design of Linkages. Springer, New York, 2000. 6. H. Pottmann and J. Wallner. The focal geometry of circular and conical meshes. Adv. Comput. Math., 29(3):249–268, 2008. 7. A.M. Schoenflies. Geometrie der Bewegung in Synthetischer Darstellung. B.G. Teubner, Leipzig, 1886. 8. J. Selig. Geometric Fundamentals of Robotics. Springer, 2005. 9. E.A. Weiss. Einf¨uhrung in die Liniengeometrie und Kinematik. Teubners Mathematische Leitf¨aden. B.G. Teubner, Leipzig, Berlin, 1935.

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators G. Moroz1,2, D. Chablat1 , P. Wenger1 and F. Rouiller2 1 Institut

de Rercheche en Communications et Cybern´etique de Nantes, France; e-mail: {guillaume.moroz, damien.chablat, philippe.wenger}@irccyn.ec-nantes.fr 2 Laboratoire d’Informatique de Paris, France; e-mail: [email protected]

Abstract. This paper investigates the existence conditions of cusp points in the design parameter space of the RPR-2PRR parallel manipulators. Cusp points make possible non-singular assemblymode changing motion, which can possibly increase the size of the aspect, i.e. the maximum singularity free workspace. The method used is based on the notion of discriminant varieties and Cylindrical Algebraic Decomposition, and resorts to Gr¨obner bases for the solutions of systems of equations. Key words: kinematics, singularities, cusp, parallel manipulator, symbolic computation

1 Introduction It is well known that the workspace of a parallel manipulator is divided into singularity-free connected regions [2]. These regions are separated by the so-called parallel singular configurations, where the manipulator loses its stiffness and gets out of control. The so-called cuspidal manipulators have the ability to change their assembly-mode without running into a singularity, which thus may increase the size of the singularity-free regions [7, 9]. The word “cuspidal” stems from the notion of cuspidal configuration, defined as one configuration where three direct kinematic solutions coalesce. A cuspidal configuration in the manipulator joint space allows non-singular assembly-mode changing motions. Thus, determining cuspidal configurations is an important issue that has attracted the attention of several researchers [1,6,9,13]. In particular, [13] (resp. [1]) has analyzed the cuspidal configurations of planar 3-RPR (resp. 3-PRR) manipulators.1 More recently, Hernandez et al. [6] studied the RPR-2PRR, a simpler planar 3-DOF manipulator that lends itself to algebraic calculus [6]. In both papers, the cusp configurations were determined by looking for the triple roots of a univariate polynomial. This approach may yield spurious solutions. In this paper, the cuspidal configuration are determined directly from the Jacobian of the whole set of geometric constraints of the robot, which guar1

The underlined letter means an actuated joint.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_4, © Springer Science+Business Media B.V. 2010

29

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G. Moroz et al.

anties that only true solutions are obtained. Then, we classify the parameter space of a family of RPR-2PRR manipulators according to the number of cuspidal configurations. It is shown that these manipulators have either 0 or 16 cuspidal configurations. The proposed method is based on the notion of discriminant varieties and cylindrical algebraic decomposition, and resorts to Gr¨obner bases for the solutions of systems of equations.

2 Mechanism under Study 2.1 Kinematic Equations A RPR-2PRR parallel manipulator is A3 θ3 shown in Fig. 1. This manipulator was analyzed in [6]. It has 1 actuated prisL3 matic joint ρ1 , and 2 passive prismatic b joints ρ2 and ρ3 . The two revolute joints B2 ρ3 a B centered in A2 and A3 are actuated while 3 α B1 (x, y) the ones centered in A1 , B1 , B2 and B3 are L2 ρ1 passive. The pose of the moving platform θ1 θ2 is described by the position coordinates A ρ A 1 2 2 (x, y) of B1 and by the orientation α of the moving platform B1 B2 . The input vari- Fig. 1 A RPR-2PRR parallel manipulators ables (actuated joints values) are defined with (a = 1, b = 2, L2 = 2, L3 = 2, x = 1/2,y = by ρ1 , θ2 and θ3 . The points B1 , B2 and 1, θ = 0.2). The actuated joint symbols were filled in gray. B3 are aligned, a= (B1 , B2 ), b= (B1 , B3 ), L2 = (A2 , B2 ) and L3 = (A3 , B3 ). The geometric constraints can be expressed by the following five equations [6]: f 1 : ρ12 =x2 + y2 f 2 : x =ρ 2 + L2 cos(θ2 ) − b cos(α )

f 4 : x =L3 cos(θ3 ) − a cos(α )

f 3 : y =L2 sin(θ2 ) − b sin(α )

f 5 : y =ρ 3 + L3 sin(θ3 ) − a sin(α )

(1)

Without loss of generality, we fix a = 1 in the rest of the article.

2.2 An Algebraic Model The singular and cuspidal equations were previously computed using the three following steps [6]: 1. Reduce the equation system to a polynomial equation depending on the articular variables ρ1 , θ1 , θ2 and one pose variable α : g(ρ1 , θ1 , θ2 , α ) = 0 (this step is done by eliminating variables, either with resultants or with Gr¨obner basis )

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators

31

2. Add the constraint ∂ g/∂ α = 0 to define the parallel singularities 3. Add the constraint ∂ g/∂ α = 0 and ∂ 2 g/∂ α 2 = 0 to define the cuspidal configurations This approach has the advantage of reducing the problem of computing the cusp configurations, to the problem of analysing the triple roots of a single polynomial. However, this only gives a necessary condition for the manipulator to have cusp configurations. In particular it is possible that 3 configurations of the robot coalesce in one coordinate but not in the others. Let us come back to the theoretical definitions, using Jacobian matrices to define directly the triple roots of the original system of equations in all the input and output variables. If P is a list of polynomials and X a list of variables, let Jk (P, X ) be the union of P = {p1 , . . . , pm } and of all the k × k minors of the Jacobian matrix of the pi with respect to the Xi . For the analysis of the RPR–2PRR manipulator, we introduce: Y := [x, y, α , ρ2 , ρ3 ]

W := [b, L2 , L3 , ρ1 , θ2 , θ3 ]

S := { f1 , . . . , f5 }.

Using these notations, the parallel singularities of the manipulator are defined by {v ∈ R11 , p(v) = 0, q(v) > 0, ∀p ∈ J5 (S,Y ), ∀q ∈ {b, L2 , L3 , ρ1 } so that the cuspidal configurations are fully characterized by: S = {v ∈ R11 , p(v) = 0, q(v) > 0, ∀p ∈ J5 (J5 (S,Y ),Y ), ∀q ∈ {b, L2 , L3 , ρ1 }}

3 Main Tools from Computational Algebra The algebraic problem to be solved is basically related to the resolution of polynomial parametric systems. More specifically, one needs to solve a system of the following form: E = {v ∈ Rn , p1 (v) = 0, . . . , pm (v) = 0, q1 (v) > 0, . . . ql (v) > 0} where p1 , . . . , pm , q1 , . . . , ql are polynomials with rational coefficients depending on the unknowns X = [X1 , . . . , Xn ] and on the parameters U = [U1 , . . . ,Ud ]. There are numerous possible ways of solving parametric systems in general. Here we focus on the use of Discriminant Varieties (DV, [8]) and Cylindrical Algebraic Decomposition (CAD, [3]) for two reasons. It provides a formal decomposition of the parameter space through an exactly known algebraic variety (no approximation). It has been already successfully used for similar mathematical classes of problems (see [4]). To reduce the dimension of the parameter space to three so that it can be displayed, we set L2 = L3 . Note that the proposed method can treat the general case L2 = L3 without any problems. When L2 = L3 , the system to solve is S with the unknowns [x, y, α , ρ2 , ρ3 , θ2 , θ3 ] and the parameters [b, L2 , ρ1 ].

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G. Moroz et al.

3.1 Basic Black-Boxes First experiments are often performed for specific values of the parameters, especially singular and/or degenerated cases. Here, we mainly use exact computations, namely formal elimination of variables (resultants, Gr¨obner bases) and resolution of systems with a finite number of solutions, including univariate polynomials. Let us describe the global solver for zero-dimensional systems. It will be used as a black box in the general algorithm we describe in the sequel. Given any system of equations p1 = 0, . . . , pm = 0 of polynomials of Q[X1 , . . . , Xn ], one first computes a Gr¨obner basis of the ideal < p1 , . . . , pm > for any ordering. At this stage, one can detect easily if the system has or has not finitely many complex solutions. If yes, then compute a so called Rational Univariate Representation or RUR (see [10]) of < p1 , . . . , pm >, which is, in short, an equivalent system of the form: { f (T ) = 0, X1 =

g1 (T ) gn (T ) , . . . , Xn = }, g(T ) g(T )

T being a new variable that is independent of X1 , . . . , Xn , equipped with a so called separating element (injective on the solutions of the system) u ∈ Q[X1 , . . . , Xn ] and such that:
 u V ( p1 , . . . , pm ) − → (x1 , . . . , xn )

V(f)

u−1

−−→

→ β = u(x1 , . . . , xn ) →




V ( p1 , . . . , pm ) g1 (β ) gn (β ) ,..., g(β ) g(β )



defines a bijection between the (real) roots of the system (denoted by V (p1 , . . . , pm )) and the (real) roots of the univariate polynomial (denoted by V ( f )). We then solve the univariate polynomial f , computing so called isolating intervals for its real roots, say non-overlapping intervals with rational bounds that contain a unique real root of f (see [11]). Finally, interval arithmetic is used for getting isolating boxes of the real roots of the system (say non overlapping products of intervals with rational bounds containing a unique real root of the system), by studying the RUR over the isolating intervals of f . In practice, we use the function RootFinding[Isolate] from Maple software, which performs exactly the computations described above. For example, with (b = 2, L2 = 2, L3 = 3, ρ1 = 2), the polynomial system S defining the cuspidal configurations has 16 real solutions. One of these solutions is (ρ2 = 2.30, ρ3 = 1.86, φ = −2.36, x = 1.79, y = 0.885, θ2 = −2.88, θ3 = −0.999). We observe the three coalescing configurations around this root by calculating the direct kinematic solutions with θ2 = −2.892 and θ3 = −1.007. These solutions are shown in Figure 2.

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators A A33 A3

A3 B1 A1

33

B3 B2

A2

B1 B1 B1 B3 B3 A2A2A2 A1 B3 B2B2B2

Fig. 2 A cuspidal configuration (left), and the three converging configurations (right).

3.2 Discriminant Varieties The above method allows one to study instances of the problem and may be used together with a discretization of the parameter space to get a first idea of the complexity of the general problem to be solved. But the true issue addressed in this paper is to find criteria on the parameters that allow classifying the configurations to be studied (for example to distinguish the manipulators having cuspidal configurations from the others). This leads to a more general problem since one then has to study non zero-dimensional, semi-algebraic sets. Let p1 , . . . , pm , q1 , . . . , ql be polynomials with rational coefficients depending on the unknowns X1 , . . . , Xn and on the parameters U1 , . . . ,Ud . Let us consider the constructible set: C = {v ∈ Cn , p1 (v) = 0, . . . , pm (v) = 0, q1 (v) = 0, . . . ql (v) = 0} If we assume that C is a finite number of points for almost all the parameter values, a discriminant variety VD of C is a variety in the parameter space Cd such that, over each connected open set U satisfying U ∩VD = 0, / C defines an analytic covering. In particular, the number of points of C over any point of U is constant. Let us now consider the following semi-algebraic set: S = {v ∈ R11 , p(v) = 0, q(v) > 0, ∀p(v) ∈ J5 (J5 (S,Y ),Y ), ∀q(v) ∈ {b, L2 , L3 , ρ1 }} If we assume that S has a finite number of solutions over at least one real point that does not belong to VD , then VD ∩ Rd can be viewed as a real discriminant variety of S , with the same property : over each connected open set U ⊂ Rd such that U ∩ VD = 0, / C defines an analytic covering. In particular, the number of points of R over any point of U is constant. Discriminant varieties can be computed using basic and well known tools from computer algebra such as Gr¨obner bases (see [8]) and a full package computing such objects in a general framework is available in Maple software through the RootFinding[Parametric] package. Figure 3 represents the discriminant variety of the cuspidal configurations of the RPR-2PRR manipulator.

34

G. Moroz et al. 5 4

10 8 ρ16 4 2

3

L2 2

0 1

1 2

L2 3 4 5 5

4

3

b2

1

0

Fig. 3 Discriminant variety of the cuspidal configurations.

0

1

2

b

3

4

5

Fig. 4 V2 for the cuspidal configurations of the manipulator.

3.3 The Complementary of a Discriminant Variety At this stage, we know, by construction, that over any simply connected open set that does not intersect the discriminant variety (so-called regions), the system has a constant number of (real) roots. The goal of this part is now to provide a description of the regions for which the number of solutions of the system at hand is constant. For that, we compute an open CAD ( [3, 5]). Let Pd ⊂ Q[U1 , . . . ,Ud ] be a set of polynomials. For i = d − 1 . . .0, we introduce a set of polynomials Pi ⊂ Q[U1 , . . . ,Ud−i ] defined by a backward recursion: • Pd : the polynomials defining the discriminant variety • Pi : {Discriminant(p,Ui ),LeadingCoefficient(p,Ui ), Resultant(p, q,Ui ), p, q ∈ Pi+1 } We can associate to each Pi an algebraic variety of dimension at most i − 1: Vi = V (∏ p∈P p). Figures 3 and 4 represent respectively V3 and V2 for the manipulator at i hand. The Vi are used to define recursively a finite union of simply connected open i U / and one point ui,k subsets of Ri of dimension i: ∪nk=1 i,k such that Vi ∩ Ui,k = 0, with rational coordinates in each Ui,k . In order to define the Ui,k , we introduce the following notations. If p is a univariate polynomial with n real roots: ⎧ ⎪ ⎨ −∞ if l ≤ 0 Root(p, l) = the lth real roots of p if 1 ≤ l ≤ n ⎪ ⎩ +∞ if l > n Moreover, if p is a n-variate polynomial, and v is a n − 1-uplet, then pv denotes the univariate polynomial where the first n − 1 variables have been replaced by v. Roughly speaking, the recursive process defining the Ui,k is the following: • For i = 1, let p1 = ∏ p∈P p. Taking U1,k =]Root(p, k); Root(p, k + 1)[ for k from 1 0 to n where n is the number of real roots of p1 , one gets a partition of R that fits

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators

35

Table 1 Numerical values of the positive roots of p1 . b

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

0.0

0.533

0.564

0.617

0.656

0.707

1.0

1.41

1.52

1.62

1.77

1.88

the above definition. Moreover, one can chose arbitrarily one rational point u1,k in each U1,k . • Then, let pi = ∏ p∈P p. The regions Ui,k and the points ui,k are of the form: 1



Ui,k = (v1 , . . . , vi−1 , vi ) | v := (v1 , . . . , vi−1 ) ∈ Ui−1, j ,

ui,k

vi ∈]Root(pvi , l), Root(pvi , l + 1)[  (β1 , . . . , βi−1 ) = ui−1, j = (β1 , . . . , βi−1 , βi ), with u u βi ∈]Root(pi i−1, j , l), Root(pi i−1, j , l + 1)[

where j, l are fixed integer. For our example, we get for p3 a trivariate polynomial of degree 33, for p2 a bivariate polynomial of degree 113, and for p1 a univariate polynomial of degree 59. The zero-dimensional solver then provides the positive real roots of p1 (Table 1), from which we easily deduce the open intervals u1,k . We then use the zerodimensional solver to solve every p2 (u1,k ,U2 ) and deduce all the tests points of the U2,k in each cells of Figure 4. Finally we use the zero-dimensional solver to solve every p3 (u2,k ,U3 ) and deduce all the tests points of the U3,k , describing so the complementary of the discriminant variety.

3.4 Discussing the Number of Solutions of the Parametric System At this stage, we have a full description of the complementary of the discriminant variety of the system to be solved : a recursive process for the construction of each cell Ud,k and a test point (with rational coordinates) in each of these cells. By definition of the discriminant variety, we know that the system has a constant finite number of solutions over each of these cells and computing this number for each cell is the only remaining step. This can be done simply by solving all the systems S|U=u , k = 1 . . . nd using the zero-dimensional solver. d,k

For our example, the process described in 3.3 returns 344 cells of dimension 3 (U3,1 , . . . , U3,344 ). We solve the system S for each of the 344 associated sample points, and we get always either 0 or 16 solutions. By selecting only the cells where the manipulator has 16 cuspidal configurations, we obtain the 58 cells shown in Figure 5. Table 2 provides the different formula bounding the three dimensional cells U3,1 , . . . , U3,344 and Table 3 represents the 58 cells of Figure 5, where the manipulator has 16 cuspidal configurations.

36

G. Moroz et al. Table 2 Formula describing the boundaries of the cells in Table 3.

b1 = 0, b2 = Root(8 b6 − 11 b4 + 6 b2 − 1, 2)

L2 (b) = 1



3



2 2 2 2 4 Root b2 + 1 L6 2 − 3 b + 3b + 1 b − 3b + 1 (b − 1) (b + 1) L2 +

2 4 4 2 6 6 3 b + 1 (b − 1) (b + 1) L2 − (b − 1) (b + 1) , 2  1 − b2 , L2 (b) = 1/ 1 − b2 , L2 (b) = (1 − b2 )/b, L2 (b) = ∞ 3 4    5 b2 / 1 − b2 , L2 (b) = 1/ b2 − 1, L2 (b) = b2 / b2 − 1 7 8 2 2 (b − 1)/b, L2 (b) = b − 1 10

b3 = Root(4 b2 + b6 − 3 b4 − 1, 2), b4 = Root(b8 + 3 b6 + 3 b4 + b2 − 1, 2) √  b = Root(−2 b4 + b6 + 3 b2 − 1, 2), b = 1/ 2, b7 = 1, b8 = (2) L2 (b) = 5 6 2 b9 = Root(2 b2 +b6 −3 b4 −1, 2), b10 = Root(b8 −b6 −3 b4 −3 b2 −1, 2) L2 (b) = 6 4 6 2 b11 = Root(−4 b + b + 3 b − 1, 2) L2 (b) = 9 b12 = Root(b6 − 6 b4 + 11 b2 − 8, 2), b13 = ∞





3 2 2 2 6 6 4 4 2 2 2 2 ρ1 (b, L2 ) = Root(−ρ1 b + 3 b L2 b + 1 − L2 ρ1 − 3 b −7 L2 b + 7 L2 + L42 + L42 b4 − 2 L42 b2 + 1 ρ12 + L22 b2 + 1 − L22 , 2) 1





3 ρ1 (b, L2 ) = Root(ρ16 + −3 b4 + 3 L22 b2 − 3 L22 ρ14 + 21 L22 b6 + 3 L42 b4 − 6 L42 b2 + 3 b8 − 21 L22 b4 + 3 L42 ρ12 + L22 b2 − b4 − L22 , 2) 2 ρ1 (b, L2 ) = b2 , ρ1 (b, L2 ) = 1/b 2 3

Table 3 Cells of R3 where the manipulator has cuspidal configurations. (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 2 3 3 2 3 4 3 2 4 5 3 4 1 2 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 6 2 1 2 2 3 3 2 3 4 3 2 4 5 3 4 1 6 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 1 2 2 4 3 2 4 3 3 4 3 5 3 4 1 6 6 2 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 1 2 2 4 3 2 4 3 3 4 3 5 3 4 1 6 6 2 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 2 6 3 2 6 4 3 2 4 3 3 4 3 5 3 4 1 2 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 1 2 2 4 3 2 4 6 3 4 6 3 3 4 3 5 3 4 1 2 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 2 1 9 10 4 1 10 7 4 3 7 8 4 3 8 5 4 3 1 9 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 2 1 9 7 4 1 7 10 4 1 10 8 4 3 8 5 4 3 1 9 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 2 1 7 9 2 1 9 10 4 1 10 8 4 3 8 5 4 3 1 7 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 2 1 7 9 2 1 9 10 4 1 10 8 4 3 8 5 4 3 1 7 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [, (]L2 L2 [, ]ρ1 ρ1 [) 2 1 7 9 2 1 9 8 4 1 8 10 4 1 10 5 4 3 1 7 (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [), (]L2 L2 [, ]ρ1 ρ1 [) 2 1 9 8 4 1 8 10 4 1 10 5 4 3 1 9

]b1 b2 [ ]b2 b3 [ ]b3 b4 [ ]b4 b [ 5 ]b5 b6 [ ]b6 b7 [ ]b7 b8 [ ]b8 b9 [ ]b9 b10 [ ]b10 b11 [ ]b11 b12 [ ]b12 b13 [

8 7

ρ1

8 7 6 5 4 3 2 1 0

6 5

ρ1 4

3 2 0

1

L2

1

2

2

3

3

4 5 5

4

b

1 0 4

L2

2 0

1

2

3

b

4

5

Fig. 5 The cells of R3 where the manipulator admits cuspidal configurations, front view (left) and back view (right).

4 Conclusion We have proposed a general method to describe rigorously the design parameters for which a manipulator has cuspidal configurations. This method can be applied directly to other mechanisms, such as the ones studied in [4, 12] for example. The tools used to perform the computations were implemented in a Maple library called Siropa.2 For 3D illustration purposes, we have detailed the main computations to be performed with manipulators satisfying L2 = L3 . However, the proposed method 2

http://www.irccyn.ec-nantes.fr/ moroz/siropa/doc.

Cusp Points in the Parameter Space of RPR-2PRR Parallel Manipulators

37

allows one directly to solve the general case (L2 = L3 ) by computing a discriminant variety of the system with 4 parameters b, L2 , L3 , ρ1 , and by decomposing R4 with a CAD adapted to the discriminant variety. This description generalizes and completes the analysis done in [6]. There is still some limitations though. In particular, when the system that defines the cuspidal configurations has no solution, it may mean that there exists a manipulator with no cuspidal configurations, but it may also mean that no manipulator can be assembled with these design parameters. Thus it is essential to be able to describe precisely the set of design parameter values for which a manipulator can be assembled.

Acknowledgement The research work reported here was made possible by SiRoPa ANR Project.

References 1. Bamberger, H., Wolf, A., and Shoham, M., Assembly mode changing in parallel mechanisms. IEEE Transactions on Robotics, 24(4):765–772, 2008. 2. Chablat, D. and Wenger, P., Working modes and aspects in fully parallel manipulators. In IEEE International Conference on Robotics and Automation, pages 1964–1969. INSTITUTE OF ELECTRICAL ENGINEERS INC (IEEE), 1998. 3. Collins, G.E., Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Springer Verlag, 1975. 4. Corvez, S. and Rouillier, F., Using computer algebra tools to classify serial manipulators. In Automated Deduction in Geometry, pp. 31–43, 2002. 5. Dolzmann, A., Seidl, A., and Sturm, T., Efficient projection orders for CAD. In Jaime Gutierrez (Ed.), ISSAC, pp. 111–118. ACM, 2004. 6. Hernandez, A., Altuzarra, O., Petuya, V., and Macho, E., Defining conditions for nonsingular transitions between assembly modes. IEEE Transactions on Robotics, 25:1438–1447, Dec. 2009. 7. Husty, M.L., Non-singular assembly mode change in 3-RPR-parallel manipulators. In Computational Kinematics: Proceedings of the 5th International Workshop on Computational Kinematics, p. 51. Springer Verlag, 2009. 8. Lazard, D. and Rouillier, F. Solving parametric polynomial systems. J. Symb. Comput., 42(6):636–667, 2007. 9. McAree, P.R. and Daniel, R.W., An explanation of never-special assembly changing motions for 3-3 parallel manipulators. I. J. Robotic Res, 18(6):556–574, 1999. 10. Rouillier, F., Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Commun. Comput, 9(5):433–461, 1999. 11. Rouillier, F. and Zimmermann, P., Efficient isolation of polynomial real roots. Journal of Computational and Applied Mathematics, 162(1):33–50, 2003. 12. Wenger, P., Classification of 3R positioning manipulators. Journal of Mechanical Design, 120:327, 1998. 13. Zein, M., Wenger, P., and Chablat, D., Singular curves in the joint space and cusp points of 3-RPR parallel manipulators. Robotica, 25(6):717–724, 2007.

Kinematics and Design of a Simple 2-DOF Parallel Mechanism Used for Orientation T. Itul, D. Pisla and A. Stoica Department of Applied Mechanics and Computer Programming, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. In this paper a 2-DOF parallel mechanism used for orientation is presented. This mechanism could be applied for orientation of the TV satellite antennas, sun trackers etc. Based on the kinematic scheme of the mechanism, the algorithms for the kinematic models are solved, the workspace and singularities are described and analyzed. An optimal design of the mechanism is performed. Finally, the design of the mechanism is also described. Key words: parallel mechanism, kinematics, workspace, optimization, design

1 Introduction A generalized parallel manipulator is a closed-loop kinematic chain mechanism whose end-effector is linked to the base by several independent kinematic chains [6]. Compared with the serial mechanisms, the parallel mechanisms have some special characteristics such as: higher rigidity, potentially higher kinematical precision, stabile capacity, and suitable position of arrangement of actuators. Lately, several important technical applications of parallel mechanism such as air flight simulators, telescopes and orienting device applicable to point payloads such as solar panels, cameras, lasers, antennas etc. have been introduced. There are several papers where parallel mechanisms used for orientation are studied. Vertechy [8] presents the synthesis of 2-DOF spherical fully parallel mechanisms with legs of type US where the actuation issues and kinematics, workspace and singularity analysis are addressed (U and S are for universal and spherical joints, respectively). Gallardo [3] introduced a family of spherical parallel manipulators with a simple architecture where the analytical expressions for the forward position, velocity and acceleration of the parallel manipulators have been obtained and solved for an exemplary manipulator which has 3-DOF. Itul and Pisla [4, 5], developed the dynamic and kinematic studies regarding the possibility to use a 3-DOF parallel mechanism for the TV satellite antenna orientation or sun tracker. Dunlop et al. [2] describe a Stewart platform prototype for a radio antenna orientation that assures a wide operating range avoiding the singularities. They have found an original tech-

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Fig. 1 The kinematic scheme for parallel mechanism.

nological solution of equivalence for the upper joints in order to allow significant rotations for the mobile plate. Meschini et al. [7] make the study of a parallel mechanism with 6-DOF of type MSSM (Minimal Simplified Symmetric Manipulator), for a satellite antenna with double reflector, the mechanical system consisting of two platforms which are connected through six extensible legs and on each platform is mounted a reflector for signaling. There are also hybrid mechanisms with 2-DOF, which are used for orientation of solar panels and satellites, most of them are sun-tracking systems [1, 9]. The present paper presents the kinematics of a simple 2-DOF parallel mechanism which fulfill completely the requirements of a celestial orientation device. The paper is organized as follows: Section 2 is dedicated to the description of the studied 2-dof parallel mechanism; Section 3 deals with the geometric model. Section 4 deals with the kinematic model. Section 5 presents workspace and singularities. The optimal design are detailed in the section 6. In the sections 7 and 8 the design of a CAD model and the conclusions are described.

2 Description of the Studied 2-DOF Parallel Mechanism The kinematic scheme of the 2-DOF parallel mechanism is presented in Figure 1. It consists of a mobile platform OA1 A2 , a fixed platform OB1 B2 , which are connected through an universal joint (O) and two telescopic legs B1 A1 and B2 A2 . The mechanism is studied with respect to two reference systems: the fixed system OXYZ, with the OY axis oriented to the west and the OZ axis oriented to the south

Kinematics and Design of a Simple 2-DOF Parallel Mechanism Used for Orientation

(a)

41

(b)

Fig. 2 Devices oriented by the parallel mechanism: (a) a solar panel; (b) a satellite.

and the mobile system Oxyz. The orientation of the mobile platform is given by the angles: ϕ (azimuth) and θ (elevation). This type of parallel mechanism can be used for orientation of a solar panel (Figure 2a), a satellite (Figure 2b), mirror, camera, antenna, etc.

3 The Geometric Model The geometric model establishes the relationships between the generalized coordinates q1 and q2 of the mechanism and coordinates ϕ , θ of the mobile platform OA1 A2 . The rotation matrix has the form: ⎡ ⎤ cos θ 0 sin θ [R] = [R]x(ϕ ),y(θ ) = ⎣ sin ϕ sin θ cos ϕ − sin ϕ cos θ ⎦ (1) − cos ϕ sin θ sin ϕ cos ϕ cos θ The coordinates of guided points Ai (i = 1, 2) with respect to OXYZ reference system can be written: ⎡ ⎤ ⎡ ⎤ Xi xi ⎣Yi ⎦ = [R] ⎣yi ⎦ ; i = 1, 2 (2) Zi zi where x1 = x2 = h, y1 = −y2 = a, z1 = z2 = 0. For inverse geometric model, the generalized coordinates of the mechanism, represented by the length of the legs Bi Ai are obtained:  (3) qi = (Xi − XBi)2 + (Yi − YBi)2 + (Zi − ZBi )2 , i = 1, 2 where: XB1 =XB2 =-C, YB1 =YB2 =B, ZB1 =ZB2 =-H.

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The direct geometric model can be solved using the input-output equations (3) of the mechanism or solving in advance the following system of equations: ⎧ (Xi − XBi )2 + (Yi − YBi )2 + (Zi − ZBi )2 = q2i ; i = 1, 2 ⎪ ⎪ ⎪ ⎨(X − X )2 + (Y − Y )2 + (Z − Z )2 = 4a2 ; 2 1 2 1 2 1 (4) 2 + Y 2 + Z 2 = a2 + h2 ; i = 1, 2 ⎪ X ⎪ i i i ⎪ ⎩ X1 − X2 = 0; After the computing of the absolute coordinates for the guided points, the orientation of the mobile platform can be determined:  



Z − Z2 Y1 − Y2 a Z1 + Z2 a Y1 + Y2 ; ; θ = atan2 · . (5) ϕ = atan2 1 ; · 2a 2a h Y1 − Y2 h Z1 − Z2

4 The Kinematic Model The vectors are defined in order to establish the relations between the generalized velocities of the mechanism q˙1 , q˙2 and angular speeds ϕ˙ , θ˙ in passive point O: ⎡ ⎤ ⎡ ⎤ Xi XBi Pi = OAi = ⎣ Yi ⎦ ; PBi = ⎣YBi ⎦ (6) Zi ; ZBi The relationship (3), written in the form: (Pi − PBi)2 = q2i ;

i = 1, 2

(7)

and its derivative with respect to the time leads to the equation of kinematic model:

⎛ [A] = ⎝

(P1 − PB1) · (P2 − PB2) ·

∂ P1 ∂ϕ ∂ P2 ∂ϕ

(P1 − PB1) · (P2 − PB2) ·

[A]X˙ = [B]q˙ ⎞ ∂ P1 ∂θ ∂ P2 ∂θ

⎠ ; [B] =

(8)

q1 0 0 q2



; X˙ =



 ϕ˙ q˙1 ˙ ; q= q˙ θ˙ 2

(9) ˙ q˙ are the end-effector velocity vector, where [A], [B] are the Jacobi matrices and X, respectively actuating velocity vector. The components of the angular velocity of the mobile platform and its module are given by:  ωx = ϕ˙ cos θ ; ωy = θ˙ ; ωz = ϕ˙ sin θ ; ω = ϕ˙ 2 + θ˙ 2 ; (10)

Kinematics and Design of a Simple 2-DOF Parallel Mechanism Used for Orientation

43

5 Workspace and Singularity Analysis Considering the relationship (3), in ideal case, the shape of the workspace is a rectangle with sides π /2, π and area 4.93 rad2 , located in the plane defined by coordinates ϕ and θ . The shape of the workspace is affected by constraints imposed by: angles allowed by spherical joints; minimum transmission angles; presence of singularities; limited stroke of actuated joints. The constraints caused by spherical joints can be avoided by replacing them with revolute joints with orthogonal intersection axes. Nevertheless, the use of revolute joints with orthogonal intersection axes present also some drawbacks: a very precise manufacturing is required, the number of joints is three times higher so it is possible to have cumulations of the joint backlashes. The angles of transmission can be computed with the relationship:

Ψi = asin

| (Pi − PBi) · (P1 × P2) |

The next condition is imposed:

|Pi − PBi| · |P1 × P2|

; i = 1, 2

Ψi ≥ 20◦

(11)

(12)

The curves of singularities result from the condition: Det[A] = 0

(13)

The reachable workspace can be determined imposing the condition: Det[A] < 0

(14)

Due to the limited strokes of actuated joints, the reachable workspace is obtained from condition: qmin ≤ qi ≤ qmax ; i = 1, 2 (15) The input data are chosen: a = 0.50 m, h = 0.60 m, B = 0.15 m, H = 0.30 m, C = 0.80 m, qmin = 0.95 m, qmax = 1.6 m. The constraints (12), (14) and (15) lead to different shapes of the workspace, which are represented respectively in Figures 3a,b and 4a (light grey area). The reachable workspace is presented in Figure 4b (light grey area).

6 Optimal Design An optimal design is required in order to increase the workspace of the mechanism. The design variables of the mechanism are following: a, h, B, H, C and objective function is the workspace area taking into account the following requirements:

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(a)

(b)

Fig. 3 The reduced workspace of the mechanism: (a) due to the minimum transmission angle; (b) due to the singularities.

(a)

(b)

Fig. 4 Workspace of the mechanism: (a) reduced workspace due to the strokes; (b) reachable workspace.

1. the singularity curves must be as far as possible from the center of the ideal surface; 2. the telescopic leg must fulfill the condition (16) in order to withdraw to the bottom. No restrictions are imposed regarding: the maximum allowed stroke of actuated joints and the maximum and minimum values of the leg. 2 ∗ qimin − qimax − 0.1 ≥ 0

(16)

3. the angle of transmission could not be over to the admitted value:

Ψi ≥ 20◦ ;

i = 1, 2

(17)

As the mobile platform orientation is important, there are five design variables but only four are independent. Regarding the optimization process, one dimension is imposed and the other dimensions take values between some minimum and maximum limits, which are considered possible. The combination of dimensions that leads to the biggest area of the workspace is the solution of the problem. After the searching process, two solutions are possible: (a) a = 0.40 m, h = 0.40 m, B = 0.08 m, H = 0.51 m, C = 0.80 m, qmin = 0.802 m, qmax = 1.508 m with reachable workspace A = 4.14 rad2 . (b) a = 0.40 m, h = 0.40 m, B = 0.08 m, H = 0.28 m, C = 0.60 m, qmin = 0.660 m, qmax = 1.212 m with reachable workspace A = 3.56 rad2 . The choice for one or other of the solutions must take into account to: the size of the area of the reachable workspace; the shape of reachable workspace; total length

Kinematics and Design of a Simple 2-DOF Parallel Mechanism Used for Orientation

(a)

45

(b)

Fig. 5 The reachable workspace of the mechanism: (a) for first solution; (b) for the second solution.

Fig. 6 CAD model of the parallel mechanism.

of the elements of mechanism (L = 2a + h + 2qmax); the conditional number of the [A] matrix. In Figure 5 the reachable workspaces for the two solutions are shown by the light grey area.

7 Design of a CAD Model Based on the optimal design a CAD model of the parallel mechanism using the SolidWorks software package has been designed. The parallel mechanism was designed using the kinematic scheme from Figure 1. Figure 6 presents the CAD model and details about the universal joint and the spherical joint. The input data for the CAD model are from the first solution of optimal design: a = 0.40 m, h = 0.40 m, B = 0.08 m, H = 0.51 m, C = 0.80 m, qmin = 0.802 m, qmax = 1.508 m.

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8 Conclusions In this paper the kinematics, workspace, singularity and optimal design of a simple parallel mechanism used for orientation have been successfully carried out. The obtained results have identified the advantages of the studied parallel mechanism: it is a simple mechanism, it is inherently suited to be actuated by motors with linear motion, it has a simple design, it enables orientation of large loads like: antenna, solar panel (Figure 2a), satellite (Figure 2b). Besides its theoretically character the paper has also an applicative character in order to use the mechanism for the orientation of parabolic antennas and sun trackers. Based on the performed modeling, an experimental model will be further built. The reachable workspace represented in the azimuth-elevation coordinates shows that a significant part of Clarke Belt and the apparent trajectories of the sun are contained in this space, excepting the extreme configurations.

Acknowledgement The authors gratefully acknowledge the financial support provided by the research grants awarded by the Romanian Ministry of Education, Research, Youth and Sport.

References 1. Comsit, M. and Visa, I., Design of the linkages type tracking mechanism of the solar energy conversion systems by using MBS method. In Proceedings 12th IFToMM World Congress, Besancon, France, pp. 582–587, 2007. 2. Dunlop, G.R., Johnson, G.R. and Afzulpurkar N.V., Joint design for extended range Stewart Platform applications. In Proceedings Eighth World Congress on the Theory of Machines and Mechanisms, Prague, Czechoslovakia, pp. 449–451, 1991. 3. Gallardo, J., Rodriguez, R., Caudillo, M. and Rico, J., A family of spherical parallel manipulators with two legs. Mechanism and Machine Theory, 201–216, 2008. 4. Itul, T. and Pisla, D., Three degrees of freedom parallel structure used for the TV satellite dish orientation. In Proceedings International Conference of Inteligent Engineering Systems, INES 2004-IEEE, Cluj-Napoca, pp. 261–266, 2004. 5. Itul, T. and Pisla, D., Kinematics of a three degrees of freedom parallel strucuture with applications for the satellite antenna orientation. In Proceedings RAAD2005, Bucharest, pp. 197–202, 2005. 6. Merlet, J.-P., Parallel Robots, Kluwer Academic Publisher, 2000. 7. Meschini, A., Sinatra, R. and Pirrotta, S., A parallel mechanism for a satellite antenna with double reflector. In Proceedings of Workshop on Fundamentals Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Quebec City, Quebec, Canada, pp. 120– 130, 2002. 8. Vertechy, V.R. and Parenti-Castelli, V., Synthesis of 2-dof spherical fully parallel mechanism. In Advances in Robot Kinematics: Mechanisms and Motion, pp. 385–394, 2006. 9. Visa, I. and Comsit, M., Tracking systems for solar energy conversion devices. In Proceedings of the 14th ISES Conference EUROSUN, Freiburg, pp. 783–788, 2004.

Special Cases of Sch¨onflies-Singular Planar Stewart Gough Platforms G. Nawratil Institute of Discrete Mathematics and Geometry, Vienna University of Technology, A-1040 Vienna, Austria; e-mail: [email protected]

Abstract. Parallel manipulators which are singular with respect to the Sch¨onflies motion group X(a) are called Sch¨onflies-singular, or more precisely X(a)-singular, where a denotes the rotary axis. A special class of such manipulators are architecturally singular ones because they are singular with respect to any Sch¨onflies group. Another remarkable set of Sch¨onflies-singular planar Stewart Gough platforms was already presented by the author in [5]. Moreover the main theorem on these manipulators was given in [6]. In this paper we give a complete discussion of the remaining special cases which also include so-called Cartesian-singular planar manipulators as byproduct. Key words: Sch¨onflies-singular, Sch¨onflies motion group, Stewart Gough platform, singularities

1 Introduction The Sch¨onflies motion group X(a) consists of three linearly independent translations and all rotations about a fixed axis a. This 4-dimensional group is of importance in practice because it is well adapted for pick-and-place operations. The geometry of a planar parallel manipulator of Stewart Gough type (SG type) is given by the six base anchor points Mi ∈ Σ0 with coordinates Mi := (Ai , Bi , 0)T and by the six platform anchor points mi ∈ Σ with coordinates mi := (ai , bi , 0)T . By using Euler Parameters (e0 , e1 , e2 , e3 ) for the parametrization of the spherical motion group SO(3) the coordinates mi of the platform anchor points with respect to the fixed space can be written as mi = K −1 R·mi + t with ⎛ 2 ⎞ e0 + e21 − e22 − e23 2(e1 e2 − e0 e3 ) 2(e1 e3 + e0e2 ) R := (ri j ) = ⎝ 2(e1 e2 + e0e3 ) e20 − e21 + e22 − e23 2(e2 e3 − e0e1 ) ⎠ , (1) 2(e1 e3 − e0e2 ) 2(e2 e3 + e0 e1 ) e20 − e21 − e22 + e23 the translation vector t := (t1 ,t2 ,t3 )T and K := e20 + e21 + e22 + e23 . It is well known that a SG platform is singular if and only if Q := det(Q) = 0 holds, where the ith row of the 6× 6 matrix Q equals the Pl¨ucker coordinates li := (li ,li ) := (mi − Mi , Mi × li ) of the ith carrier line.

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1.1 Related Work and Notation For the determination of X(a)-singular planar parallel manipulators we distinguish the following cases depending on the angle α ∈ [0, π /2] between the axis a and the carrier plane Φ of the base anchor points and the angle β ∈ [0, π /2] between a and the carrier plane ϕ of the platform anchor points. Every X(a)-singular manipulator belongs to one of the following 5 cases (after exchanging platform and base): 1. α = β : 2. α = β :

(a) α = π /2, β ∈ [0, π /2[ (b) α , β ∈ [0, π /2[ (a) α = π /2 (b) α =]0, π /2[ (c) α = 0

Due to the theorem given in [6] the manipulators of the solution set of case (1a) presented in [5] are the only X(a)-singular ones with α = β which are not architecturally singular. In this paper we discuss the remaining special cases with α = β . In the following we use the notation introduced in [5]. We denote the determinant of certain j× j matrices as follows: |X, y, . . . , Xy|(i

1 ,i2 ,...,i j )

:= det(X(i

1 ,i2 ,...,i j )

, y(i

1 ,i2 ,...,i j )

, . . . , Xy(i

1 ,i2 ,...,i j )

)

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ Xi yi Xi yi 1 1 1 1 ⎢X ⎥ ⎢y ⎥ ⎢X y ⎥ ⎢ i2 ⎥ ⎢ i2 ⎥ ⎢ i2 i2 ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ with X(i ,i ,...,i ) = ⎢ . ⎥ , y(i ,i ,...,i ) = ⎢ . ⎥ , Xy(i ,i ,...,i ) = ⎢ ⎢ .. ⎥ . . j j j 1 2 1 2 1 2 ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ Xi yi Xi yi j

j

j

(2)

(3)

j

and (i1 , i2 , . . . , i j ) ∈ {1, . . . , 6} with i1 < i2 < . . . < i j . Moreover it should be noted i

that we write |X, y, . . . , Xy|i j if ik+1 = ik + 1 for k = 1, . . . , j − 1 hold. 1

It should also be said that in the latter done case study we always factor out the homogenizing factor K if possible. Moreover we give the number n of terms of not explicitly given polynomials F in square brackets, i.e. F[n].

2 Case (2a) Theorem 1. A non-architecturally singular planar SG platform, where the axis a is orthogonal to ϕ and Φ , is X(a)-singular if and only if |1, A, B, a, b, Ab − Ba|61 = 0 and |1, A, B, a, b, Aa + Bb|61 = 0 are fulfilled. Proof. Without loss of generality (w.l.o.g.) we can choose Cartesian coordinate systems in Σ and Σ0 such that A1 = B1 = B2 = a1 = b1 = b2 = 0 hold. As both carrier planes are orthogonal to the axis we set e1 = e2 = 0 and compute the condition Q := det(Q) = 0 as given in Section 1. Q splits up into z3 K 2 [F1 (e20 − e23 ) + 2F2e0 e3 ] where F1 and F2 are the two conditions given in Theorem 1.
Remark 1. A geometric interpretation of these 2 conditions is still missing. Moreover, it should be noted that the manipulators of the solution set of case (1a) also fulfill F1 = F2 = 0 due to their property rk(1, A, B, a, b)61 = 4 (cf. [5]). 

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3 Case (2b) Theorem 2. A non-architecturally singular planar manipulator with 0 = α = β = π /2 is X(a)-singular if and only if in a configuration with coinciding carrier planes the anchor points Mi and mi are within an indirect similarity, which is the product of a dilation and the reflection on the orthogonal projection of a onto Φ = ϕ . Proof. The proof of this theorem is given in the following two parts: Part [A] No four anchor points are collinear: Due to [2, 5] we can always choose coordinate systems in the platform and the base such that a2 A2 B3 B4 B5 c(3,4,5) (a3 − a4 )(b3 − b4 ) = 0 hold. 1 Now we must distinguish again two cases, depending on whether γ > α or γ = α holds with γ := ∠([M1 , M2 ], a) ∈ [0, π /2]. Case γ > α : Under this assumption we can rotate the platform about a such that the common line s of Φ and ϕ is parallel to [M1 , M2 ]. Therefore we can use the same coordinatisation as in the proof of Theorem 1 of [6], namely: Mi = (Ai , Bi , 0) and mi = (ai , bi cos δ , bi sin δ ) with A1 = B1 = B2 = a1 = b1 = 0 and sin δ = 0. We set e1 = e4 cos µ , e3 = e4 sin µ and e2 = e4 n where e4 is the homogenizing factor. Therefore we only have to consider those cases in the proof of Theorem 1 of [6] which yield the contradiction α = β . There is exactly one such case which will be discussed here in more detail. As given in [6] this case is characterized by K1 = K2 = K4 = 0, b2 = 0, e2 = 0 and bi = b3 Bi /B3 for i = 4, 5 with K1 = |A, B, Ba, Bb, a|62 ,

K2 = |A, B, Ba, Bb, b|62 ,

K4 = |A, B, Ba, Bb, Ab|62 . (4)

uv i j k u v We proceed by computing Q51 020 = 0 where Qi jk denotes the coefficient of t1 t2 t3 e0 e4 of Q. Its only non-contradicting factor can be solved for A4 w.l.o.g.. Then Q42 020 = 0 = 0 we must distinguish two cases: implies an expression for A5 . Due to Q71 010

1. A2 = a2 : Solving the only non-contradicting factor of Q62 100 = 0 for n yields b3 sin µ sin δ /(B3 − b3 cos δ ). Note that for B3 = b3 cos δ the coefficient Q62 100 cannot vanish without contradiction (w.c.). We proceed by expressing A3 from 35 26 Q44 100 = 0 w.l.o.g.. Now Q100 and Q100 can only vanish w.c. for: a. B3 = b3 : This yields a solution and the platform and the base are congruent. b. B3 = −b3: We get again a solution; but now we have an indirect congruence. c. cos µ = 0: In this case Q35 010 = 0 yields the contradiction. 2. A2 b3 (n cos δ + sin δ sin µ ) − na2B3 = 0, A2 = a2 : As Q71 010 = 0 cannot vanish w.c. for B3 = b3 A2 cos δ /a2 we can solve the above condition for n w.l.o.g.. Moreover we can express A3 w.l.o.g. from the only non-contradicting factor of Q62 010 = 0. 35 Now Q44 and Q can only vanish w.c. for: 100 100 1 The algebraic condition that Mi , M j , Mk or mi , m j , mk are collinear is denoted by C(i, j,k) := |1, A, B|(i, j,k) = 0 and c(i, j,k) := |1, a, b|(i, j,k) = 0, respectively.

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a. b3 = B3 a2 /A2 : This yields a solution and the platform and the base are similar. b. b3 = −B3 a2 /A2 : We get a solution; but now we have an indirect similarity. Note that the solutions (1a) and (1b) as well as (2a) and (2b) are identical if one takes the relative position of ϕ , Φ and a into consideration. Case γ = α : For this case we can use the same coordinatisation as in the case γ > α but now we have e2 = δ = 0. In the first step we show that K1 = K2 = 0 (cf. Eq. (4)) must hold. Therefore we compute Q[30768] in its general form. Computation of the following two linear-combinations already yields the result: K1 =

Q51 Q42 Q33 011 002 020 + + , 2ω(0,0,1) 4ω(0,0,2) 8ω(0,0,3)

K2 =

Q24 Q42 200 101 − , 8ω(0,1,3) 4ω(0,0,2)

with ω(i, j,k) := sin δ i sin µ j cos µ k . Moreover, due to 8b2 (K4 + K2 ) = Q53 100 /ω(0,0,3) we must distinguish the following two cases: 1. b2 = 0 (⇒ K4 = 0): We compute Q in dependency of K1 , . . . , K4 as given in the proof of Theorem 1 of [6] with K3 := |A, B, Ba, Bb, Aa|62 . By setting K1 = K2 = K4 = 0 we end up with Q = K3 e4 sin µ A2 F[2646]. As K1 = K2 = K3 = K4 = 0 indicate the architecturally singularity (cf. [2]) we prove that F cannot vanish 05 14 w.c.. The resultant of F200 and F200 with respect to a2 implies |a, b, B|53 = 0. a. c(1,3,4) = 0: W.l.o.g. we can express B5 from |a, b, B|53 = 0. Back-substitution 05 = H[4]G [8] and F 14 = H[4]G [8], respectively. It is not difficult yields F200 200 1 2 to verify that G1 = G2 = 0 yields a contradiction. Therefore we express b4 from H := B3 (a2 b4 − b2a4 ) − B4(a2 b3 − b2a3 ). 14 = 0 and A form F 14 . i. Assuming b3 = b5 we can compute A4 from F001 020 5 14 14 . ii. For b3 = b5 we can compute A5 from F001 = 0 and A4 form F020 In both cases we proceed by expressing A6 and b6 from K1 = K2 = 0. Now we get b2 K3 = a2 K4 . This is a contradiction as K4 = 0 yields K3 = 0. b. c(1,3,4) = 0: W.l.o.g. we can express a3 from c(1,3,4) = 0. Then |a, b, B|53 = 0 14 = 0 implies an expression for b . Moreover, yields B3 = b3 B4 /b4 and F200 5 14 = 0. Now F 14 can only vanish w.c. for A = we can compute A5 from F110 020 3 b3 A4 /b4. After computing A6 and b6 from K1 = K2 = 0, we get again the contradiction b2 K3 = a2 K4 .

2. b2 = 0: Here we distinguish again two cases: a. K4 = 0: This assumption yields Q = K3 e4 sin µ A2 F[1338]. The resultant of 05 and F 14 with respect to B can only vanish w.c. for b |b, B|4 = 0. F200 200 3 5 5 14 = 0 and F 05 = 0 yields the contradiction. For b5 = 0 we get B3 = B4 from F200 200 Therefore we can set B3 = b3 B4 /b4 because not both bi (i = 3, 4) can be 14 equal zero. Moreover we can assume b5 = 0. Then from F200 = 0 we get 14 14 = 0 B5 = b5 B4 /b4 . We proceed by computing A5 from F110 = 0. Now F020 34 implies an expression for A4 . Due to F100 = 0 we must distinguish two cases: 25 = 0, K = 0, i. B4 = −b4 A2 /a2 , A2 = a2 : In this case the 3 equations F100 1 K2 = 0 imply an indirect similarity.

Special Cases of Sch¨onflies-Singular Planar Stewart Gough Platforms

51

43 = 0, F 25 = 0, K = 0, K = 0 imply ii. For B4 = −b4 the 4 equations F010 010 1 2 an indirect congruence. 5 b. K4 = 0: We start by considering Q24 200 = 0 which implies |b, B, Bb|3 = 0. Now 5 |b, B, Bb|3 = 0 cannot be solved for any bi if and only if B3 = B4 = B5 hold. But for this special case we get the contradiction from Q15 200 = 0. Therefore we can assume w.l.o.g. that we can express b3 from |b, B, Bb|53 = 0. Now Q33 020 can only vanish w.c. for (B4 − B5 )G[16] = 0. As for B4 = B5 we get the contradiction from Q15 200 we set G[16] = 0. From this condition we can 24 5 compute a3 w.l.o.g.. Then Q15 200 and Q020 can only vanish w.c. for |b, B|4 = 0 and |a, b, A|(2,4,5) = 0, respectively. From these conditions we can express A5 and B5 w.l.o.g.. Moreover we can compute B6 from K2 = 0 which yields b6 K1 = K4 , a contradiction.

Part [B] Four anchor points are collinear: Similar considerations as for part [A] show that possible solutions of this problem must yield the contradiction α = β in the proof of Theorem 2 of [6]. But there do not exist such a contradiction in the mentioned proof. There is only one case which is not covered by the proof of Theorem 2 of [6], namely the following one: M1 , . . . , M4 collinear and m1 , . . . , m4 collinear with α = ∠([M1 , . . . , M4 ], a) = ∠([m1 , . . . , m4 ], a) = β . For this case we can use the same coordinatisation as in the case γ > α by setting B3 = B4 = b2 = b3 = b4 = e2 = δ = 0. Now Q splits up into e4 sin µ (ye4 cos µ − ze0 )H[6]F[56] where H = 0 indicates 20 = 0 we get b = b B /B . Then a architecturally singular manipulator. From F002 5 6 5 6 13 04 22 = 0 F100 = 0 and F100 = 0 imply B5 = B6 and a5 = a6 , respectively. Finally, F001 yields the contradiction.
Remark 2. The manipulators of Theorem 2 are so-called equiform platforms. The singularities and self-motions of these manipulators were extensively studied by Karger [1, 3]. In part [B] of the discussion we get no solution because an equiform manipulator with four collinear anchor points is already architecturally singular. 

4 Case (2c) Theorem 3. A non-architecturally singular planar SG platform is X(a)-singular, where a is parallel to the x-axes of the fixed and moving system, if and only if one of the following cases hold: (1) rk(1, b, B, Bb)61 = 2, (2) rk(1, b, B, Bb, A − a)61 = 3 or (3) rk(1, A, B, Ba, Bb, a, b, Ab)61 = 5. Proof. We can choose coordinate systems such that Mi = (Ai , Bi , 0) and mi = (ai , bi , 0) with A1 = B1 = a1 = b1 = 0 hold. Now we can compute Q in its general form according to Section 1 under consideration of e2 = e3 = 0. The necessity of K1 15 33 42 24 and K2 given in Eq. (4) follows directly from Q51 002 + Q002 + Q002 and Q101 + Q101 , uv i j k u v respectively, where Qi jk denotes the coefficient of t1t2 t3 e0 e1 of Q. In the following we split up the proof of the necessity into two parts:

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Part [A] rk(A, B, Ba, Bb)52 = 4 Under this assumption we can perform the generalized version of the matrix manipulation given by Karger in [2]. The 5 steps of the generalization are given by: (a) li := li − l1 (c) li := li − l3

(b) li := li − l2 Ai /A2

i = 2, . . . , 6 |A, B|(2,i)

(e) l6 := l6 − l5

|A, B|32

(d) li := li − l4

i = 4, 5, 6

|A, B, Ba, Bb|(2,3,4,6) |A, B, Ba, Bb|52

i = 3, . . . , 6

|A, B, Ba|(2,3,i) |A, B, Ba|42

i = 5, 6

.

Then l6 has the from (v1 , v2 , v3 , 0, −w3 , w2 ) with vi := ri1 K1 + ri2 K2 and w j := r j1 K3 + r j2 K4 with K3 := |A, B, Ba, Bb, Aa|62 and K4 of Eq. (4). Due to the above shown necessity of K1 = K2 = 0 we can set them equal to zero and compute Q = K4 F[1032] where F do not depend on K3 and K4 . Therefore there a two possibilities. For K4 = 0 we get solution (3). In the second case we must consider the conditions under which F is fulfilled identically. It can easily be seen that there are only 7 such conditions, namely: 5 P1 : Q53 100 = |B, Ba, Bb, b|2 = 0

5 P2 : Q40 003 = |A, B, a, b|2 = 0

26 5 P3 : Q62 001 − Q001 = |Ba, Bb, b, a − A|2 = 0

5 P4 : Q33 002 = |A, Bb, a, b|2 = 0

26 5 P5 : Q62 001 + Q001 = |Ba, Bb, B, a − A|2 = 0

5 P6 : Q42 101 = |B, Bb, a, b|2 = 0

5 5 P7 : Q42 011 = |B, Ba, b, a − A|2 − |B, Bb, A, a|2 = 0

For the discussion of this system of equations we distinguish two cases: 1. rk(b, B, Bb)52 = 3: We get a52 = λa b52 + µa B52 + νa Bb52 with (µa , νa ) = (0, 0) from P6 . As a consequence P2 and/or P4 equal/s |b, Bb, B, A|52 = 0. This yields together with P1 the relation rk(A, B, Ba, Bb, a, b)52 = 3, a contradiction. 2. rk(b, B, Bb)52 < 3: Now rk(b, B, Bb)52 = 2 must hold due to rk(A, B, Ba, Bb)52 = 4. Therefore the vectors B and Bb are linearly independent and we can set b52 = λb B52 + µb Bb52 . It can easily be seen that for any linear combination of b only the following four conditions remain, namely: |A, B, Bb, a|52 = 0,

|Ba, Bb, B, a − A|52 = 0

and K1 = K2 = 0.

As the vectors A52 , B52 and Bb52 are linearly independent due to the assumption rk(A, B, Ba, Bb)52 = 4 we can set a52 = λa B52 + µa Bb52 + νa A52 without loss of generality. Now the remaining three equations can only vanish for: a. |A, B, Ba, Bb|52 = 0 which is a contradiction or for b. νa = 1, b6 = λb B6 + µb B6 b6 and a6 = λa B6 + µa B6 b6 + νa A6 , which yield solution (2). This finishes this part. For the remaining one we can assume that there do not exist any i, j, k, l ∈ {2, . . . , 6} with |A, B, Ba, Bb|(i, j,k,l) = 0.

Special Cases of Sch¨onflies-Singular Planar Stewart Gough Platforms

53

Part [B] rk(A, B, Ba, Bb)62 < 4 A close inspection of Q shows that it vanishes independently of x, y, z, e0 , e1 for rk(A, B, Ba, Bb)62 < 4 if and only if the following 9 conditions are fulfilled: 04 6 R1 : Q40 003 − Q003 = |a, b, A, B, Ab|2 = 0

6 R2 : Q31 102 = |a, b, A, B, Bb|2 = 0

04 6 R3 : Q04 003 + Q003 = |a, b, A, B, Ba|2 = 0

6 R4 : Q33 020 = |a, b, A, Ab, Bb|2 = 0

26 6 R5 : Q62 101 + Q101 = |a, B, Ba, Bb, Ab|2 = 0

6 R6 : Q33 110 = |a, b, B, Ab, Bb|2 = 0

6 R7 : Q33 110 = |b, Ba, Bb, Ab, A − a|2 = 0

6 R8 : Q33 020 = |b, B, Ba, Bb, Ab|2 = 0

04 6 6 6 R9 : Q40 002 + Q002 = |a, b, A, Ba, Bb|2 + |a, A, B, Ab, Bb|2 + |b, B, Ab, Ba, A− a|2 = 0

As rk(B, Ba, Bb)62 = 1 implies with R1 solution (3) we are left with two more cases: 1. rk(B, Ba, Bb)62 = 3: As rk(A, B, Ba, Bb)62 = 3 must hold we can set A62 = λA B62 + µA Ba62 + νA Bb62 with (µA , νA ) = (0, 0). Then R2 and/or R3 equals the determinant of (B, Ba, Bb, a, b)62 . If the rank of this matrix is 3 we get solution (3). Therefore we can assume rank 4. For rk(B, Ba, Bb, b)62 = 4 we get solution (3) from R8 . If rk(B, Ba, Bb, a)62 = 4 holds we get solution (3) from R5 . 2. rk(B, Ba, Bb)62 = 2: We already get solution (3) if rk(U, V, a, b)62 = 2 with U, V ∈ {B, Ba, Bb} and rk(U, V)62 = 2 holds. Therefore we assume rk(U, V, a, b)62 = 4: a. (U, V) = (B, Bb): Solution (3) is implied by R2 and R6 . b. (U, V) = (B, Ba): Now R3 implies rk(A, B, Ba, Bb, a, b)62 = 4. For Bb62 = λBb B62 + µBb Ba62 with µBb = 0 we get solution (3) from R6 . For µBb = 0 and λBb = 0 we get it from R7 . For µBb = λBb = 0 we get solution (3) from R9 . In the remaining case of rk(U, V, a, b) = 3 we can set x62 = λx U62 + µx V62 + νx y62 with (λx , µx ) = (0, 0), x, y ∈ {a, b} and x = y: a. (U, V) = (B, Bb): R1 resp. R4 implies solution (3) for µx = 0 resp. λx = 0. b. (U, V) = (B, Ba): For µx = 0 we get solution (3) from R1 . For the case µx = 0, νx = 0 solution (3) is implied by R9 for Bb = 0, by R7 for Bb = RB and by R4 in all other cases. The case µx = νx = 0 is the same as the last one for x = a and y = b. But for x = b, y = a and µx = νx = 0 we get solution (1) for Bb = 0 or Bb = RB. In all other cases R4 implies solution (3). As B62 = 0 must hold the case (U, V) = (Ba, Bb) need not be discussed in item 2. This finishes the proof of the necessity of the conditions of solution (1), (2) and (3). Due to the limitation of pages we refer for the proof of the sufficiency of these conditions to the corresponding technical report [7].
It should be noted that the proof of the sufficiency of the conditions of solution (1) and (2) was done analytically in contrast to the one of solution (3), which was done geometrically according to the method of R¨oschel and Mick [8]. As a byproduct of this proof we get the following geometric characterization of solution (3): Theorem 4. Given are two sets of points Mi and mi (i = 1, . . . , 6) in two nonparallel planes Φ and ϕ , respectively. Then the non-architecturally singular planar parallel manipulator of Stewart Gough type is X(a)-singular with a := (Φ , ϕ ) if

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Mi , mi are three-fold conjugate pairs of points with respect to a 2-dimensional linear manifold of correlations, whereas the ideal point of a is self-conjugate. The geometric meaning of the condition rk(1, b, B, Bb, A − a)61 = 3 is still missing. Until now we are only able to identify a geometric meaning with the necessary condition |1, b, B, Bb|(i, j,k,l) = 0. This conditions equals DV (Gi , G j , Gk , Gl ) = DV (gi , g j , gk , gl ) with Gi := [Mi , U] and gi := [mi , U] where DV denotes the crossratio and U the ideal point of the axis a. For solution (1) this yields the following geometric interpretation (cf. [7]), where two cases must be distinguished: 1. [Mi , M j , Mk , Ml ]  [mi , m j , mk , ml ]  [Mm , Mn ]  [mm , nn ]  a, 2. [Mi , M j , Mk ]  [mi , m j , mk ]  [Ml , Mm , Mn ]  [ml , mm , nn ]  a, with (i, j, k, l, m, n) consisting of all indices from 1 to 6.

5 Conclusion We discussed the special cases (i.e. α = β ) of Sch¨onflies-singular planar Stewart Gough platforms, whereas we distinguished three cases: α = π /2 (cf. Theorem 1), α ∈]0, π /2[ (cf. Theorem 2) and α = 0 (cf. Theorem 3). As side product we also characterized all planar parallel manipulators which are singular with respect to the translational group T(3). Such manipulators are called Cartesian-singular. Due to Lemma 2.1 of Mick and R¨oschel [4] the solution set of case α = 0 equals the set of Cartesian-singular planar SG platforms, where Φ and ϕ are not parallel. A non-architecturally singular planar SG platform, where ϕ and Φ are parallel, is Cartesian-singular if and only if |1, A, B, a, b, Ab − Ba|61 = 0 holds. This follows from the proof of Theorem 1 by setting e0 = 1 and e3 = 0.

References 1. Karger, A., Singularities and self-motions of equiform platforms, Mechanism and Machine Theory, 36(7):801–815, 2001. 2. Karger, A., Architecture singular planar parallel manipulators, Mechanism and Machine Theory, 38(11):1149–1164, 2003. 3. Karger, A., Parallel manipulators with simple geometrical structure. In Proc. 2nd European Conference on Mechanism Science, M. Ceccarelli (Ed.), pp. 463–470, Springer, 2008. 4. Mick, S. and R¨oschel, O., Geometry & architecturally shaky platforms. In Advances in Robot Kinematics: Analysis and Control, J. Lenarcic and M.L. Husty (Eds.), pp. 455–464, Kluwer, 1998. 5. Nawratil, G., A remarkable set of Sch¨onflies-singular planar Stewart Gough platforms, Technical Report No. 198, Geometry Preprint Series, Vienna University of Technology, 2009. 6. Nawratil, G., Main theorem on Sch¨onflies-singular planar Stewart Gough platforms. In Advances in Robot Kinematics, J. Lenarcic and M.M. Stanisic (Eds.), Springer, 2010. 7. Nawratil, G., Special cases of Sch¨onflies-singular planar Stewart Gough platforms, Technical Report No. 202, Geometry Preprint Series, Vienna University of Technology, 2009. 8. R¨oschel, O. and Mick, S., Characterisation of architecturally shaky platforms. In Advances in Robot Kinematics: Analysis and Control, J. Lenarcic, M. Husty (Eds.), pp. 465–474, Kluwer, 1998.

The Motion of a Small Part on the Helical Track of a Vibratory Hopper D.I. Popescu Faculty of Machine Building, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. The paper presents theoretical aspects of the calculus and simulation of transportation effect of vibratory hoppers. Two mathematical models are developed, based on distinct mechanical assumptions, which describe the motion of a singular small part, assimilated with a material particle, on the vibratory track. The small part executes a complex motion, composed by slides and jumps, and follows the conical-helix shape of the track. Key words: vibrations, vibratory hopper, vibratory track, vibratory conveying, particle motion

1 Introduction Vibratory hoppers are automatic handling equipments, working on the principle of generation directed vibrations that continuously jog conveyed pieces on a helical track [1]. They can serve a wide range of industries and feed parts made in steel, plastics, ceramics, glass, etc., without danger of breakage or marring their surface finish. Their advantage over other handling equipment consists in the simple construction and gentle treatment of the handled pieces. There is a great variety of shapes and sizes of parts that can be conveyed and sorted out towards the exit of a hopper, but in great proportion they are cylindrical or flat. The helical type of vibratory motion of the bowl tends to move the parts towards the wall of the bowl and around it, following the helical track, in upward direction. The motion of parts inside a vibratory hopper is very complex. Many studies were dedicated to the kinematics and dynamics of the vibratory machine and also to the motion of material on vibratory tracks [1, 5, 10]. The relative motion of a single material particle has been studied by taking into consideration different working conditions, constructive characteristics of the vibratory machines and interactions [2–4, 8]. Experimental research has been directed to the study of the behavior of granular layers on vibratory surfaces and, due to the development of computer software, simulations, complex analysis and optimization were possible [9].

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_7, © Springer Science+Business Media B.V. 2010

55

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D.I. Popescu

Fig. 1 Vibratory hopper and the attached coordinate systems: O1 x1 y1 z1 – fixed, Oxyz – mobile with the vibratory bowl.

For the next study we consider the motion of a single small part on the helical track of a vibratory conical bowl (Figure 1). The part is assimilated with a flat material particle having a periodic motion, composed of small jumps followed by relative slides on the track.

2 The Helical Track and Its Movement The main features of vibratory hoppers can be seen in Figure 1. The bowl executes a complex helical movement, which can be mathematically modeled, taking into account the driving force and the characteristics of the elastic suspension [7]. We consider a conical bowl, with the inclination angle noted with γ , measured from the central axis (Figure 2). The lateral wall of the bowl has an inside conical helix shape, defining the track. The base of the track is inclined with an angle α to the horizontal. The conical helix of the track is defined by the following known parameters: step H and radius of the base circle noted with Rb (Figure 2). The working movement of the bowl can be described by two laws: a translation on vertical direction and a rotation around the central axis of the track (bowl). For numerical calculation we may consider harmonic laws for both movements: z0 = A · cos ω t

θ = B · cos ω t

(1)

The Motion of a Small Part on the Helical Track of a Vibratory Hopper

57

Fig. 2 Definition of the conical helix of the track and coordinates of a given point P, relative to the mobile system Oxyz.

3 The Displacement of Small Parts on the Track The aim is to describe the motion of a single small part on the helical track of a vibratory conical bowl. The small part was assimilated with a flat material particle having a periodic motion, composed of small jumps, followed by relative slides on the track. Two mechanical models have been developed, by considering the jump of the material particle described by the laws of free flight and then by taking into account the friction between the particle and the lateral wall of the track in the jump stage. Two coordinate systems were considered (Figure 1): the fixed O1 x1 y1 z1 – tied on the base-platform of the vibratory hopper, and the mobile Oxyz – in motion with the bowl and its track. The axes of the coordinate systems are parallel at the moment t =0 and the distance between the parallel planes xOy and x1 O1 y1 is noted with h. The motion of the small part may be described by the following three functions: (R, ψ , z) – in Oxyz (Figure 2) and (R1 , ψ 1 , z1 ) – in O1 x1 y1 z1 (Figure 1). For the following calculus, the track’s helix was considered right hand and its start point was placed on the axis Ox. The following reference parameters are introduced, in order to describe and calculate the motion of the material particle, consisting of small jumps followed by relative slides along the track [6, 8]: • • • •

Displacement of the particle on the track during the slide phase; Start moment of the particle jump: ts ; Displacement of the particle on the track during the jump phase; End moment of the particle jump: ta .

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D.I. Popescu

3.1 Model 1 – The Jump Is Assimilated with a Free Flight The following assumptions are considered: • During the slide motion, the material particle (associated to the small part) remains all the time on the external helix of the track, so that the track may be assimilated with this helix. • The small jump of the particle respects the laws of a free flight. • After a small jump the particle returns on the external helix of the track.

The Slide Phase The differential equation that describes the slide motion, without turning, taking in consideration the friction with the friction coefficient µ , of the material particle on the track was deduced in [6]:

ψ¨ d = R(ψ˙ + θ˙ )2 sin γ cos γ sin2 δ − (g + z¨0)sin2 δ cos2 γ − 2 d ψ˙ (ψ˙ + θ˙ ) sin γ sin δ cos δ − R θ¨ sin δ cos δ cos γ
2 − R (ψ˙ + θ˙ ) (1/ cos δ + sin δ tg δ sin γ cos γ (1 − tg γ ))   + (g + z¨0) (cos δ + sin δ tg δ sin2 γ ) (1 − tg γ ) + tg γ / cos δ

 − 2 d ψ˙ (ψ˙ + θ˙ ) sin δ sin γ (1 − tg γ ) − R θ¨ sin δ cos γ (1 − tg γ )

× µ sin δ cos γ sign(ψ˙ )

(2)

where the following notations were made: d=

H ; 2π

δ = arctg

H ; 2π R(ψ ) · cos γ

R = R(ψ ) = Rb + ψ · d · tg γ

(3)

Using the parameter ψ , the displacement of the material particle on the helical track during the slide phase results from    2  ψ 2 H 2 2 H tg γ + R2 + · dψ (4) s= 2π 2π ψ1 The particle starts the jump at the time ts , when the normal reaction of the base of the track, acting on the particle during the slide phase, becomes zero. The jump phase begins when the following condition is satisfied [6]: R (ψ˙ + θ˙ )2 sin δ tg δ sin γ cos γ − 2 d ψ˙ (ψ˙ + θ˙ ) sin δ sin γ − R θ¨ sin δ cos γ + (g + z¨0)(cos δ + sin δ tg δ sin2 γ ) = 0

(5)

The Motion of a Small Part on the Helical Track of a Vibratory Hopper

59

The Jump Phase At the beginning of jump, marked by the start time of the particle jump, ts , the position of the material particle, relative to O1 x1 y1 z1 is given by ⎧ ⎪ ⎨ ψ1 (ts ) = ψ (ts ) + θ (ts ) R1 (ts ) = Rb + ψ (ts ) d tg γ = R(ts ) ⎪ ⎩ z1 (ts ) = ψ (ts ) d + z0 (ts ) + h

(6)

The jump trajectory of the material particle, in the assumption of free flight, is calculated in the fix coordinate system O1 x1 y1 z1 (Figure 1) and then transposed in the coordinate system Oxyz, tied on the track, being described by the following functions: ⎧ ˙ ⎪ ⎨ ψs = ψ1 s − θ = ψ1 (ts )(t −ts ) + ψ1 (ts ) − θ (t)  Rs = Rb + ψs d tg γ = Rb + ψ˙ 1 (ts )(t − ts ) + ψ1 (ts ) − θ (t) d tg γ (7) ⎪ ⎩ 1 2 zs = z1 s − z0 − h = − 2 g (t − ts ) + z˙1 (ts )(t − ts ) + z1 (ts ) − z0 − h The jump ends at the time ta , when the particle returns on the conical helix of the track: (8) zs (ta ) = ψs (ta ) · d After the time ta , the material particle remains on the track and continues its motion by sliding, until the next jump (next time ts ). We assume that the collision between the particle and the track is inelastic, so the following initial conditions are respected for the slide motion of the particle after the jump, related to the coordinate system Oxyz: (9) t = ta ; ψ (ta ) = ψs (ta ); ψ˙ (ta ) = 0 The motion of the small part on the helical vibratory track, described by this model, was simulated on the computer. The solutions of the differential equation were obtained by using Runge–Kutta–Gill numerical method. Moments of startjump and end-jump were obtained by verifying conditions (5) and (8) at every iteration step. For example, considering the following values as input data: A = 0.8 mm; B = 0.8 mm, µ = 0.2; γ = 60◦ , and a time interval of 2 seconds, the displacement s of the material particle on the helical track depends on the frequency of the tack motion, as follows: s = 0.005 mm ( f = 18 Hz), s = 0.01 mm ( f = 20 Hz), s = 0.02 mm ( f = 25 Hz), s = 0.025 mm ( f = 30 Hz).

3.2 Model 2 – Contact with Friction between Particle and Track during the Jump Stage The following assumptions are considered:

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• During the slide motion, the material particle (associated to the small part) remains all the time on the external helix of the track, so that the track may be assimilated with this helix. • The material particle has two phases of motion: a slide phase and a jump phase. The slide motion of the particle is described by equation (2) and its displacement along the helical track by equation (4). • The time in which the particle starts the jump is calculated from condition (5). As follows, it is considered that the particle remains in contact with the lateral wall of the track, during the jump phase, so that its motion does not respect any more the laws of free flight. At the start time of particle jump, ts , the position of the material particle on the track is given by the functions (10), where ψ (ts ) and its derivative are obtained by numerical solving of the differential equation of slide motion (2): R(ts ) = Rb + ψ (ts ) d tg γ z(ts ) = ψ (ts ) d z˙(ts ) = ψ˙ (ts ) d

(10)

Figure 3 represents the material particle on its jump trajectory and the forces acting on it [6]: N¯ s is the normal reaction of the lateral wall of the track, during the jump; T¯s is the friction force between the particle and the lateral wall of the track; F¯ jt s is the inertia force due to transport acceleration; F¯jc s is the inertia force due to Coriolis acceleration; m g¯ is the force due to gravity; a¯rs is the relative acceleration of the particle in the jump stage. The equation of relative motion of the particle on the lateral wall of the track, during the jump phase is m a¯r s = m g¯ + N¯ s + T¯s + F¯ jt s + F¯ jc s

(11)

After replacing each vector in this equation and make appropriate simplifications, we obtain a system of two non-linear differential equations, which describe the slide motion, with friction, of the material particle on the lateral wall of the track, during the jump phase (12):   1 µ ψ˙ s d tg, γ 2 2 Rs (ψ˙ s + θ˙ ) + Rsθ¨ tg δs sin γ Rs ψ˙ s − cos δs + ψ¨ s = d tg γ mod   1 2 ˙ ˙ ˙ + 2 d ψ˙ s (ψ˙ s + θ ) tg γ tg δs sin γ + Rs θ + 2 θ ψ˙ s Rs cos δs + sin δs tg δs sin2 γ
µ z˙s 1 2 Rs (ψ˙ s + θ˙ ) + Rs θ¨ tg δs sin γ z¨s = − g − z¨0 − mod cos δs + sin δs tg δs sin2 γ  (12) + 2 d ψ˙ s (ψ˙ s + θ˙ ) tg γ tg δs sin γ

The Motion of a Small Part on the Helical Track of a Vibratory Hopper

61

Fig. 3 The material particle P on its jump trajectory, in contact with the lateral wall, and the forces acting on it.

where d=

H ; 2π

Rs = Rb + ψs d tg γ ;

δs = arctg

 mod = (ψ˙ s d tg γ )2 + R2s ψ˙ s2 + z˙2s

d Rs cos γ

The particle remains in contact with the lateral wall of the track if the reaction force Ns of this wall is positive: Ns =


1 2 m Rs (ψ˙ s + θ˙ ) + m Rs θ¨ tg δs sin γ 2 cos δs + sin δs tg δs sin γ  + 2 m d ψ˙ s(ψ˙ s + θ˙ ) tg γ tg δs sin γ > 0

(13)

When Ns becomes zero, the material particle leaves the lateral wall of the track, continuing the jump with a free flight. Similar with Model 1, the jump ends at the moment ta , when the material particle returns on the conical helix of the track. Due to the complexity of the mathematical model, phases of slide and jump may be detected only as part of the numerical solving and simulation process of the particle motion on the track.

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4 Conclusions Numerical simulations for the displacement of a singular small part on the track, based on the mathematical models presented in this paper, allowed the calculation of the mean velocity of the part, and the approximation of the output capacity of the vibratory hopper. Results serve to the analysis of the influence of different constructive and driving parameters on the upward motion of the small part on the track. The relative displacement of the material particle on the track is due to the jumps and not so much to the slides and it is direct proportional with the amplitude and frequency of bowl movement (1), which has a main influence on the conveying process. Such an analysis must also take into consideration experimental results, in order to reduce errors.

References 1. Bachschmid, N. and Rovetta, A., Theoretical and experimental research on vibromachines for the transport and handling of material. Meccanica, 11(3):172–179, 1976. 2. Cronin, K., Catak, M., Dempsey, S. and Ollagnier, R., Stochastic modeling of particle motion along a sliding conveyor. AIChE Journal, 56(1):114–124, 2010. 3. Dyr, T. and Wodzinski, P., Model particle velocity on a vibrating surface. Physicochemical Problems of Mineral Precessing, 36:147–157, 2002. 4. Han, I., Vibratory orienting and separation of small polygonal parts. Journal of Engineering Manufacture, 221(12):1743–1754, 2007. 5. Parameswaran, M.A. and Ganapathy, S., Vibratory conveying – Analysis and design: A review. Mechanism and Machine Theory, 14(2):89–97, 1979. 6. Popescu, D.I., The vibro-transportation effect of the conic helical troughs. Scientific Bulletin of Politehnica University from Timisoara, Transaction of Mechanics, Tom. 44(58), 1:73–78, 1999. 7. Popescu, D.I., Dynamic modeling of a vibratory bowl feeder, WSEAS Transactions on Mathematics, 3(1):188–191, 2004. 8. Popescu, D.I., Periodic free flight motion of particles on vibratory hoppers. In Proceeding of International Conference MicroCAD2004, Machine and Construction Design, Miskolc, pp. 85–90, 2004. 9. Soto-Yarittu, G.R. and Andres Martinez, A., Computer simulation of granular material: Vibrating feeders. Powder Handling & Processing, 13(2), 2001. 10. Winkler, G., Analysing the vibrating conveyor. International Journal of Mechanical Sciences, 20(9):561–570, 1978.

Kinematic Analysis of Screw Surface Contact ˘ ıgler J. Sv´ University of West Bohemia in Pilsen, 30614 Pilsen, Czech Republic; e-mail: [email protected]

Abstract. An analysis of the three member mechanism with higher kinematic pair, that is created by conjugate screw surfaces, is solved using the theory of screws. In a theoretical position the surface axes are in parallel position and the surfaces have a contact at the curve. Under acting force and temperature fields the correct parallel position of the surface axes changes into a skew position. The curve contact changes into contact at the point and the originally relative rolling motion changes into a spatial motion. Instantaneous state of the surfaces is expressed with the force screw, wrench and with kinematic screw, twist of their relative motion. Key words: screw surfaces, incorrect position, contact, kinematic and force screws

1 Introduction Important part of mechanic systems create the three-element mechanism with higher kinematic pair where two bodies have a contact at a point or curve. One of the most important technical application creates pair of gears [2, 4] which are created by conjugate surfaces. Let us consider two conjugate, surfaces, Figure 1, where σ2 , which is defined by parametric equation rσL 3 = rσL 3 (p2L , χ ), is a creating surface and σ3 is its envelope. The computation is performed in the basic space R ≡ (i, j, k) and therefore a transformation T : R3 → R is necessary to use. The transformation equation is given with expression σ3 σ3 (1) R rL = TRπ R (π )TR Rπ (−aw )TR R (ϕ2 )R rL , 20

2 20

2

where TR R is the transformation matrix from Ri to R j . The surface σ3 = σ3 (p2L , χ , i j

ϕ [χ ]) is created as envelope of the surface σ2 according to equations σ3 R rL

= TR R (ϕ3KO )TRR (ϕ3 )R rσL 2 , 3

3

σ2 R nL · R vL32

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_8, © Springer Science+Business Media B.V. 2010

= 0.

(2)

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˘ ıgler J. Sv´

64

Fig. 1 Creating of conjugate screw surfaces σ2 and σ3 .

The meshing surfaces are created by several separated surfaces which continuously connect each other. In described theoretical case both surfaces touch mutually in a curve.

2 Kinematic Screw Let us consider two rigid bodies 2 and 3 with a spatial moving. Kinematical state each of them is determined with instantaneous twist η3 : ω3 , p3 , η2 : ω2 , p2 around skew axes o2 , o3 , Figure 2, where pi , i = 2, 3, marks parameter of twist. Relative motion both bodies is given by relative twist η32 : ω32 , p32 around an axis o32 which position is determined by the shortest distance a of axes o2 , o3 and the angle Σ = γ2 + γ3 . Then the position of the axis of relative twist o32 can be determined [5] with the transversal a2 and angle γ2 a2 =

sin γ2 [a cos γ3 + (p3 − p2 ) sin γ3 ], sin Σ

tgγ2 =

Both quantities we get, Figure 2, from system of equations a = a2 + a3, Σ = γ2 + γ3 ,

i32 sin Σ . 1 + i32 cos Σ

(3)

Kinematic Analysis of Screw Surface Contact

65

Fig. 2 Configuration of twist axes.

ω2 sin γ2 = ω3 sin γ3 ,

ω3 = i32 ω2 ,

(4)

a2 ω2 sin γ2 + v2 sin γ2 = a3 ω3 sin γ3 + v3 sin γ3 that can be obtained using classical kinematical way. Kinematical quantities ω32 , v32 of the relative twist are given, Figure 2, with expressions

ω32 = ω3 sin γ3 + ω2 cos γ2 , v32 = a2 ω2 sin γ2 + a3 ω3 sin γ3 − v2 cos γ2 − v3 cos γ3

(5)

and the parameter of the twist is v sin γ2 sin γ3 p32 = 32 = ω32 sin Σ


 p3 p2 a− . − tgγ3 tgγ2

(6)

The axis o32 creates a line of a skew straight line surface of the third degree, Pl¨ucker conoid, Figure 3. This surface creates a locus of twists resulting from composition of twists with varying ratio on two given skew straight line o2 , o3 , Figure 2. There are two principal intersecting straight line oα , oβ which are perpendicular each other. Their point of intersection Ωk , Figure 4, lies at a half of the conoid height p. These lines together with the straight line d create the fundamental coordinate system [2] of the Pl¨ucker conoid Rk = (e1k , e2k , e3k ). In the fundamental

˘ ıgler J. Sv´

66

Fig. 3 Pl¨ucker conoid.

Fig. 4 Principal lines oα , oβ and general line of Pl¨ucker conoid.

coordinate system some line o of this surface can be expressed yk = p sin ϑ cos ϑ ,

zk = xk tgϑ

(7)

where ϑ is an angle, which the line o contains with the axis xk . Let us consider, Figure 4, two screws Θα : ωα , pα , Θβ : ωβ , pβ on oα , oβ . By composition of these screws we obtain the screw Θ : ωΘ , pΘ , which is determined with relations   (8) pΘ = pα cos2 ϑ + pβ sin2 ϑ , yΘ = pα − pβ sin ϑ cos ϑ . Let be remarked that it is identical if we discussed about the twist or wrench. Now we consider a reciprocal problem. For two given screws Θ2 (ω2 , p2 ) and Θ3 (ω3 , p3 ) we want to determine principal screws Θα , Θβ . Applying of Equation (8) to each

Kinematic Analysis of Screw Surface Contact

67

screw Θ2 , Θ3 , using relations Σ = ϑ2 + ϑ3 and a = y3 − y2 , we obtain [1] after ordering, the following relations:    a2 + p3 − p2 pα − pβ = , pα + pβ = p3 + p2 − a cotgΣ , sin Σ
  p3 − p2 1 1  , y3 = p2 − p3 cot gΣ + a v3 = Σ + arctg (9) 2 a 2 which determine the principal screws Θα , Θβ appertaining to screws Θ2 , Θ3 .

3 Force Screw As was said Equations (7), (8), (9) are valid both twists and wrenches. Let us now consider a rigid body loaded by force field that is caused by medium pressure on the surface of the body. The force field which consist from forces and couples in nodes Lk of selected coordinate axis, Figure 5, is determined with relations n

F=

s

∑ ∑ Fik ,

k=1 i=1

n

M=

s

s

∑ ∑ Mik + ∑ rk × Fk

k=1 i=1

(10)

k=1

where Fik , Mik are results of the pressure on an elementary area of surface transformed into node k, k = 1 . . . n, of the axis. Both resultants F, M can be replaced with wrench that is determined by direction of forces and a position vector rE = F × M/|F|. Let us suppose a body performing a spatial motion which is given by the twist η : η  , pη where η denotes kinematical

Fig. 5 Force effects on surface element.

˘ ıgler J. Sv´

68

screw with the axis η , amplitude of the twist η  and the pitch of the screw pη . The body is loaded by force field represented by the wrench ρ : ρ  , pρ where ρ denotes force screw with the axis ρ , intensity of the wrench, force, ρ  and the pitch of the screw pρ . Then the instantaneous virtual work [1] done by the given twist against the given wrench is W = 2ρ  η  ω˜ ρη = ρ  η  [(pρ + pη ) cos φ − d sin φ ]

(11)

where ω˜ ρη = 12 [(pρ + pη ) cos φ − d sin φ ] is the virtual coefficient in which d is the shortest distance between ρ , η and φ is the angle between ρ and η . Using Equation (11) the wrench ρ can be replaced with six wrenches ρi : ρi , pρi , i = 1–6 in six given screw axes ρi . Let us consider six arbitrarily twists η j , j = 1–6, appertaining to the body. The six wrenches ρi can be obtained under the condition that virtual work of the wrench ρ , acting against one of the six arbitrarily selected twist η j , must do the same quantity of work as the sum of the six wrenches ρi acting against the same twist η j . After replacing wrenches ρi with reciprocal wrenches ρiR , which expressing reaction force effects, it is possible to speak about equilibrium of wrenches ρ and ρiR . The determination of six screw ρi , or ρiR , on arbitrarily chosen axes ρi , denotes the determination of six intensities ρi or ρ  Ri laying on these axes. Hence pitches of these wrenches ρi , or ρiR , and the wrench ρ must be the same.

4 Contact Point of Surfaces In consequence of the force and temperature loading the originally parallel position of axes o2 , o3 changes into a skew position o∆2 , o∆3 . Surfaces σ2 and σ3 are shifted into new position σ2∆ , σ3∆ and the curve contact changes into contact in the point. The contact point of these surfaces was determined in the following way. The axis o3 was fixed and new positions of both axes σ2∆ and σ3∆ were inserted in the axis o2 . So we obtain a deformed position σ2∆ which represents the deformation of both surfaces. The surface σ2∆ is separated, by cross sections τ2∆ , Figure 6, into single profiles p∆2 . Each of these profiles rotates about the axis o∆2 through an angle

ϕ2∆ till it takes a contact with the profile p∆3 = σ3 τ2∆ . In the contact point L the ∆

∆

normal of both surfaces must fulfil the condition nσL 2 × nσL 3 = 0. From the set of contact points of profile p∆3 and p∆2 the contact point of surfaces σ3 and σ2∆ creates only this one, which fulfils the condition ϕ2p = min{ϕ2ip }, where i denotes the cross section.

Kinematic Analysis of Screw Surface Contact

69

Fig. 6 Searching of the contact point by incorrect contact σ2∆ and σ3∆ .

5 Application Application is focused in screw machines, i.e. screw compressors and engines, with liquid injection [4] in which tooth surface of rotors beside of creation of a workspace and its sealing ensure the kinematic and force accouplement both rotors, Figure 7. The position of rotors in the system R, Figure 1, is determined with relations T T T R rO = [0, 0, 0] ,R rO = [0, aw , 0] and R ν o3 = R ν o2 = [0, 0, 1] . After a bearing dis3

2

placements [3] the rotor axes o2 , o3 translate into new positions o∆2 , o∆3 . The position of the axis o∆32 in the fundamental coordinate system Rk is defined, Equation (7), with expressions

Rk

 T rO∆ = 0, yO∆ , 0 , 32

R k ν o∆

32

32

T  = cos ϑ32 , 0, sin ϑ32

(12)

∆ = ϑ ∆ + γ and tgγ is determined with the second equation of the where ϑ = ϑ32 2 2 2 system (3). In places of rotor seating and in the contact point of surfaces, Figure 8,

˘ ıgler J. Sv´

70

Fig. 7 Screw compessor with injected liquid.

Fig. 8 Wrenches effect on rotors.

wrenches ρim = −ρiRm , i = 1–6, m = 2, 3, for simplifying ρim will be used, are introduced. Likewise an auxiliary system of twists η j , j = 1–6, was introduced. Using Equation (11) to all wrenches ρim we obtain a system of six equations for forces ρ  m i for each of rotors. Form one of these equations for twist η j , is

Kinematic Analysis of Screw Surface Contact

71

Table 1 Reaction effects at bindings of rotors. Female rotor

ρ  21 [N] 1288.94

ρ  22 [N] 515.83

ρ  23 [N] 48.63

ρ  24 [N] 797.33

ρ  25 [N] 4504.75

ρ  26 [N] 6700.61

ρ  35 [N] –2933.82

pρ 3 [m] 5 –0.037

Male rotor

ρ  31 [N] –2630.76

ρ  32 [N] –3509.78

ρ  33 [N] –1265.09

ρ  34 [N] 397.57

ρm η j ω˜ ρm η1 = ρ  1 η j ω˜ ρ m η + ρ 2 η j ω˜ ρ m η + ρ  3 η j ω˜ ρ m η + · · · + ρ  6 η j ω˜ ρ m η i 1 2 1 3 1 6 1 (13) For the solution of the equilibrium of the male rotor 3 Equation (13) was rewritten into the form: m

m

m

ρ3 ω˜ ρ3 η j = ρ  1 ω˜ ρ 3 η + · · · + κ cos φρ 3 η

m

3

1

3 +ρ  5




j

5

j



pη j cos φρ 3 η − d sin φρ 3 η 5

j

5

j

3 + ρ  6 ω˜ ρ 3 η 6

(14) j

where κ = ρ  35 pρ 3 is the moment of the couple around the axis z and ρ  36 = ρ  26 5

is the normal force in the point of the contact. Obtained results of reaction effects from numerical solution for ϕ3 = 0◦ , aw = 85 mm are presented in Table 1.

6 Conclusions Using the theory of screw the relative motion of two rigid bodies with skew axes was solved. The axis of the relative screw motion as well as axes of rotary motions of bodies lying on the Pl¨ucker conoid. The change of the contact character causes a significant increasing of the value of the normal force acting between both surfaces. Theory of screws is a suitable instrument for solution of spatial problems of mechanisms with the higher kinematics pairs.

Acknowledgment This work was supported by the project MSM 4977751303 of the Ministry of Education of the Czech Republic.

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References 1. Ball, R.S., A Treatise on the Theory of Screws. Cambridge University Press, Cambridge, 1998 (first published 1900). 2. B¨ar, G., Zur Optimierung der Grundgeometrie von Hypoidgetrieben. Tagung Antriebstechnik/Zahnradgetriebe TU Dresden, IMM, pp. 372–386, 2000. ˘ ´igler, J., Machulda, V. and Siegl, J., Deformation of Screw Machine Housing under Force 3. Sv Field and its Consequences. VDI Berichte Nr. 1932, Schraubenmaschinen 2006, Dortmund, pp. 377–388, 2006. ˘ ´igler, J. and Siegl, J., Contribution to modelling of screw surfaces contact. In Proceedings 4. Sv of Xth International Conference on the Theory of Machines and Mechanisms, Liberec, Czech Republic, pp. 617–622, 2008.

Aspects Concerning VRML Simulation of Calibration for Parallel Mechanisms A. Capustiac and C. Brisan Department of Mechanisms, Precision Mechanics and Mechatronics, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. This paper presents aspects of geometrical calibration for partner parallel robot with 6 DOF, based on a wire tracking system. The wire tracking system was designed as a parallel manipulator with 3-2-1 configuration. The environment Matlab/Simulink was used to connect the models developed for both calibrated robot and the robot used in calibration with a virtual environment. Also, the paper presents the results obtained after performing virtual calibration simulations. Key words: partner parallel robots, calibration, mathematical models, simulation, wire robots

1 Introduction Parallel robots are mechanisms with at least one closed loop, topologically formed by one fixed platform and at least one mobile platform, connected by at least two open loops. Parallel robots have good dynamic behavior; a high accuracy and a good ratio between total mass and manipulated mass compared with serial robots. A modular parallel robot consists of independent designed modules, which contain actuator(s), passive joints and links. These modules can be connected into a given topology in order to obtain a large variety of parallel robots. One of the main challenges of modular development of parallel robots is the accuracy of the end-effectors if the machining tolerances, misalignment of the connected modules and other sources of error are taken into consideration. Based on the topology of the parallel mechanisms, calibration algorithms for parallel robots are divided into: calibration of the closed loop mechanism and calibration of the end effector. This paper will deals with the calibration of the closed loop. From a different point of view a robot can be calibrated in two ways: external using an auxiliary mechanism and internal or auto-calibration, using sensors or imposing mechanical constraints on the links during the calibration process [16]. This paper will present how the calibration can be done if auxiliary and/or wire mechanism is used.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_9, © Springer Science+Business Media B.V. 2010

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A. Capustiac and C. Brisan

Fig. 1 Prototype of partner robot with 6 DOF (degrees of freedom).

2 Partner Robots Partner robots (PR) are parallel robots designed into a modular manner, in order to ensure full reconfigurability according to customer needs [3]. The modularity of the PR is derived from its mathematical development such as the topology, kinematics, dynamics and control by keeping in mind the efficiency of the reconfigurability. Each independent loop can be considered as a system with input and output values from topologic, kinematic and dynamic point of view. Reconfigurability of the PARTNER robots results from an appropriate design of the mechanical architecture. Thus, all types of passive joints and/or other passive components were design according to reconfigurability requirement.

3 Calibrations of Parallel Robots Calibration of a parallel robot is the process of identifying the real geometrical parameters from its structure. Geometrical calibration of a robot implies four steps: modelling the functions of the robot, measuring the generalized coordinates of the geometrical joints, identifying the differences between the real values and the ones corresponding to the mathematical model and making the necessary corrections [7]. A robot can be calibrated external, when the mobile platform or the gripper is brought in a certain number of positions using joint control and the position is measured using external measurement systems [12] or internal, using sensors or mechanical constraints [16]. There are different possibilities to measure the position: supersonic sensors, lasers, attachable video-camera systems, serial auxiliary systems, wire tracking systems. Several solutions used in calibrating the parallel robots were

Aspects Concerning VRML Simulation of Calibration for Parallel Mechanisms

75

studied. Such solutions are: a visual measurement system, an auxiliary wire mechanism and adding sensors [1, 6, 8, 9, 11, 15]. Wire mechanisms can be modelled as parallel mechanisms because the wires can be considered extensible arms, connecting the platform and the base through spherical, universal joints. These kinds of mechanisms have to operate in small workspaces. In this paper the elasticity of the cables is neglected, the wire being considered ideal.

4 Mathematical Modeling of Partner Robots In order to create a calibration algorithm, the mathematical model of a 6 DOF Partner Robot (PR6dof) was developed by solving the inverse and forward kinematics of the system. For the PR6dof the unknowns in the inverse kinematic problem are the generalized coordinates qi (i = 1 . . . 6) of the motors. The points xBi , yBi , zBi (i = 1 . . . 6) are the coordinates of the points Bi belonging to the mobile platform. They are measured with respect to the coordinate system OX0Y 0Z0 . The inverse kinematics algorithm is used for: – Finding xB
i , yB
i , zB
i , knowing rB , the radius of the circle that defines the mobile platform’s hexagon and the hexagon’s angles; – Finding xBi , yBi , zBi by multiplying the coordinate vectors of the points B
i with the transformation matrix, H from the coordinates system OX1Y 1Z1 to the coordinate system OX0Y 0Z0 : ⎛ ⎞ cψ · cθ cψ · sθ · cϕ − sψ · cϕ cψ · sθ · cϕ + sψ · sϕ x ⎜ ⎟ ⎜ sψ · cθ sψ · sθ · sϕ + cψ · cϕ cψ · sθ · cϕ − sψ · sϕ y ⎟ ⎜ ⎟ (1) H =⎜ cθ · sϕ c ϑ · cϕ z ⎟ ⎝ −sθ ⎠ 0 0 0 1 Where ‘c’ and ‘s’ denote ‘cos’ and ‘sin’ and ψ , θ , ϕ are the Euler’s angles. – Parameters qi can be found using the following sphere equations:  (2) qi = zBi ± (L2 − (xBi − xAi )2 − (yBi − yAi )2 Where xAi , yAi , zAi are the coordinates of the points Ai , belonging to the fix platform, defined with respect to OX0Y 0Z0 for i = 1 . . . 6; described as follows: A1 (95.61; −80.25; 0), A2 (0; −125; 0), A3 (−108.52, −62.5; 0), A4 (−108.52; 62.5; 0), A5 (0; 125; 0) and A6 (95.61; 80.25; 0). This algorithm for solving the inverse kinematics of the PR was implemented in Matlab/Simulink in the IKPSB (Inverse Kinematics Partner Solving Block). In order to solve the forward kinematics of the PR, a system of 18 non-linear equations with 18 unknown variables was written: 6 equations that describe the distance between points Ai and Bi :

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A. Capustiac and C. Brisan

Fig. 2 Kinematical chain of the PR6dof; r denotes the position vector making the connection between OX0Y 0Z0 fix coordinate system and O1X1Y1Z1 mobile coordinate system, h denotes the distance between Bi and B
i and rA denotes the radius of the circle that defines the fix platform’s hexagon of the robot. L is the length of the segment Ai Bi .

(xBi − xAi)2 + (yBi − yAi )2 + (zBi − zAi )2 = L2

(3)

Six equations were written to describe the distance between the points Bi : (xBi − xBi+1)2 + (yBi − yBi+1)2 + (zBi − zBi+1 )2 = Bi B2i+1

(4)

Three co-planarity equations were written for points: B1 , B2 , B3 and B4 ; B1 , B2 , B3 and B5 ; B1 , B2 , B3 and B6 ( j = 1 . . . 3) in order to describe the fact that the 6 points are always coplanar: ⎤ ⎡ xB1 yB1 zB1 1 ⎥ ⎢ ⎢ xB2 yB2 zB2 1 ⎥ ⎥ ⎢ (5) ⎢x y z 1⎥ = 0 ⎣ B3 B3 B3 ⎦ xB j yB j zB j 1 The last 3 equations of the system were written to have a unique solution for the geometry of the mobile platform: (xBi − xB j )2 + (yBi − yB j )2 + (zBi − zB j )2 = Bi B2j

(6)

The equations were written between points B3 and B5 , B3 and B6 , B2 and B6 [2]. The platforms of the PR6dof describe non regular hexagons [4]. This configuration was chosen in order to avoid singularities for any configurations for PR [10]. To implement the system in Matlab/Simulink, the Newton–Raphson solving method was used. The algorithm of solving the forward kinematics for the PR was introduced in FKSB (Forward Kinematics Partner Robot Solving Block)

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Fig. 3 (a) The calibration solution adopted for the PR6dof. (b) The inverse kinematics for the wire robot.

5 Mathematical Modeling of the Wire Robot The solution proposed for simulating the calibration process is a wire robot with configuration 3-2-1 (Figure 3a). The inverse kinematics gives the lengths of the wires, knowing the coordinates of the points C j : xC j , yC j , zC j ( j = 1 . . . 3) belonging to the mobile platform of the wire robot with respect to O
xyz coordinate system (the connection points) and the coordinates of the points Pi : xPi , yPi , zPi (i = 1 . . . 6) belonging to the fixed frame of the wire robot with respect to O

XY Z coordinates system. O

XY Z is the fix coordinate system, attached to the base of the wire robot, with centre in O

. The reason why this solution was chosen is that the forward kinematics can be easily solved, compared to the 2-2-2 configuration, using Cayley– Menger determinants [5]. In reality, the lengths of the wires are taken from the encoders. In order to create the virtual model of the wire robot and to simulate the calibration process, the inverse kinematics was solved [13]. In this case the wire lengths are obtained from the simulation model. Figure 3b presents an arbitrary vector chain of the wire robot: bi (i = 1 . . . 6) is the vector describing the distance between the centre of the coordinates system and the base’s points of the wire robot and li (i = 1 . . . 6) are the wire’s lengths vectors. Modelling the wire robot was done with the help of the points on the base, with respect to O

XY Z coordinate system: Pi = 1 ≤ i ≤ m

(7)

where m is the number of points situated on the base of the wire robot. The vector chain shown in Figure 3b gives: Pl = Pb − Pr − PR C pi , i

i

C

1≤i≤m

(8)

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Fig. 4 Virtual models. (a) Virtual model of the PR6dof. (b) Virtual model of the 3-2-1 wire robot.

where Pr is the vector r denoted in coordinate system O

XY Z and PRc is the transformation matrix from coordinates system O1X1Y 1Z1 to O

XY Z giving the orientation of the platform with respect to O

XY Z coordinates system. The wire’s lengths are:     li = Pb − Ppi  , 1 ≤ i ≤ m (9) i

2

The algorithm of solving the inverse kinematics for the wire robot was introduced in the Inverse Kinematics Wire Robot Solving Block, named IKWRSB and implemented in Simulink. The forward kinematics of the parallel wire robot is based on Cayley–Menger determinants and three consecutive triangle operations, using these described in [14], the forward kinematics was introduced in the Forward Kinematics Wire Robot Solving Block, named FKWRSB and implemented in Matlab/Simulink.

6 Simulation Results After implementing the forward and inverse kinematics for the PR6dof and for the wire robot in Matlab/Simulink, the virtual environment was built with the help of Virtual Reality Modelling Language (VRML). Calibration of a robot implies measurements of real generalized coordinates of the joints and comparison with the coordinates from the mathematical model. To be able to simulate the process of calibration two sets of coordinates for the points describing the mobile platform of PR6dof were needed. The mathematical model proposed as a calibration algorithm solves the kinematic problems for both robotic systems. The forward kinematics of the PR6dof gives the position of the platform, ‘where the platform should be’. In reality, because of errors, the real position of the platform is different from the mathematical model position (forward kinematics of the PR6dof in this paper). In order to find the ‘real position’, the wire tracking system’s forward kinematics is used. In simulation, these values are the same.

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Table 1 Coordinate values from simulation used for validation of the calibration algorithm. Coordinates of Bi taken from the FKSB Simulation time: 2 seconds B1(61.32;–69.58;–666.2) B2(–32.04;–112.1;–691.7) B3(–137.6;–51.65;–720.5) B4(–137.5;73.35;–720.5) B5(–31.92;133.7;–691.8) B6(61.39;91.12;–666.3) Simulation time: 10 seconds B1(22.16;–30.52;–806.4) B2(–71.3;–72.87;–831.8) B3(–176.7;–12.17;–860.6) B4(–176.4;112.8;–860.8) B5(–70.65;173;–832.1) B6(22.57;130.2;–806.7)

Coordinates of Bi with introduced errors Simulation time: 2 seconds B1(64.01;–70.02;–666.9) B2(–37.01;–115.67;–692.89) B3(–135.46;–50.47;–722.4) B4(–138.0;71.83;–722.4) B5(–30.15;131.98;–696.8) B6(66.78;96.78;–668.69) Simulation time: 10 seconds B1(22.86;–32.59;–810.8) B2(–78.13;–75.15;–836.82) B3(–175.9;–13.00;–862.16) B4(–176.0;114.01;–863.06) B5(–71.02;173.09;–836.81) B6(21.09;134.28;–803.7)

Coordinates of Bi from CCB Simulation time: 2 seconds B1(61.32;–69.61;–666.31) B2(–32.17;–111.95;–691.75) B3(–137.57;–51.79;–720.48) B4(–137.3;73.35;–720.5) B5(–33.02;133.8;–691.8) B6(61.45;91.12;–666.3) Simulation time: 10 seconds B1(22.17;–30.62;–806.4) B2(–73.31;–72.67;–831.69) B3(–176.56;–12.31;–860.05) B4(–176.5;113.0;–861.00) B5(–70.51;173.06;–832.1) B6(22.59;130.06;–806.5)

The errors artificially introduced are in the range [−15, 15] mm. This part of simulation wanted to present how the PR6dof behaves in real conditions, when errors appear. Because this comparison it’s made in a simulation, errors based on practical considerations had to be introduced. A Comparing and Correction Block (CCB) was designed and introduced in Matlab/Simulink in order to identify the differences between the real values and the ones corresponding to the mathematical model and makes the necessary corrections giving the set of right solutions for the PR6dof. A calibration algorithm for the PR6dof was obtained. The virtual models for the PR and for the wire robot were built in order to visualize the movements in VRML (Figure 4). The CCB from Matlab/Simulink was build with the purpose of finding the correct values for the coordinates of the points belonging to the mobile platform of the PR6dof, solving the calibration problem. This block compares the two sets of coordinates for the mobile platform of PR6dof minimizes the errors according to precision values established by the user. The results obtained from the simulation presented validated the algorithm [12] and in Table 1 values are given for the system following a trajectory with position vectors: rinitial = [0, 0, 720, 1]T , r f inal = [−50, 50, 900, 1]T r = r(t) with a linear variation and Euler angles: Ψ = 0◦ (roll angle), θ = 15◦ (pitch angle), ϕ = 0◦ (yaw angle). All data are given in mm, the sample time of the simulations is 1 ms and the solver used is Ode45 [17]. The ‘corrected’ coordinates from block CCB gave 0.2 mm error and according to precision values established by the user the results are satisfying.

7 Conclusions This paper presents the limitations, the advantages and the disadvantages of a chosen calibration solution in order to implement it on the PR6dof. The Matlab environ-

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ment was used for simulating the process and for modelling of the prototype of the robot used for calibration. The calibration method used is based on measurements made by using a wire robot with 6 wires. For a wire robot calibration, known algorithms are used; in this paper, encoders are being used instead of actuators. The reasons that sustain the choice of this method are: increased portability by having reduced weight, high processing speed and stationary encoders. To be able to simulate the calibration process, errors based on practical considerations were introduced in the forward kinematics solving block. The result was the calibration algorithm for PR6dof. The method proposed also it’s framed in a concept of the Software in the Loop (SIL).

Acknowledgment This work was financially supported by CNCSIS through grant No. 1075 (grant IDEI type).

References 1. Amirat, M.Y., Pontnau J., and Artigue, F., Six degrees of freedom parallel robot with C5 link. Robotica, 10:35–44, 1992. 2. Brisan, C. and Capustiac, A., Development of a calibration algorithm for parallel robots. IREME, 2(4):585–592, 2008. 3. Brisan, C. and Hiller, M., Partner robotic system – A new kind of parallel robots. In: Advanced Technologies Research, Development, Application, B. Lalic (Ed.), Pro Literature Verlag, 2006. 4. Capustiac, A. and Brisan, C., Aspects concerning utilisation of Matlab in calibration simulation for partner robots. In Proceedings of the 4th International Conference Robotics 08, pp. 551–557, 2008. 5. Grandon, C., Daney, D., Papegay, Y., Tavolieri, C., Ottaviano, E., and Ceccarelli, M., Certified pose determination under uncertainties. In: Proceedings 12th IFToMM World Congress, Besancon, France, 2007. 6. Hiller, M. et al., Design, analysis and realization of tendon-based parallel manipulators. Mechanism and Machine Theory, 429–445, 2005. 7. Husty, M.L. and Gosselin, C., On the singularity surface of planar 3-RPR parallel mechanisms. Mech. Based Design of Structures and Machines, 36:411–425, 2008. 8. Jeong, J.W., Kim, J., and Cho, Y.M., Kinematic calibration of redundantly actuated parallel mechanism. ASME Journal of Mechanical Design, 126(2):307–318, 2004. 9. Jeong, J.W., KimKwak, S.H., and Smith, C.C., Development of a parallel wire mechanism for measuring position and orientation of a robot end-effector. Mechatronics, 845–861, 1998. 10. Kapur, P., Ranganath, R., and Nataraju, B.S., Analysis of Stewart platform with flexural joints at singular configurations. In: Proceedings 12th IFToMM World Congress, Besancon, France, 2007. 11. Merlet, J.-P., Parallel Robots, Springer, 2006. 12. Mroz, G., Design and prototype of a parallel, wire-actuated robot. Master of Science Thesis, Queen’s University, 2003. 13. Navhi, A., Hollerbach, J.M., and Hayward, V., Calibration of a parallel robot using multiple kinematic loops. In: Proceedings of IEEE Conference in Robotics, 1994.

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14. Ottaviano, E., Ceccarelli, M., Toti, M., and Avila-Carrasco, C., CATRASYS – A wire system for experimental evaluation of robot workspace. Journal of Robotics and Mechatronics, 14(1):78–87, 2002. 15. Renaud, P., Andreff, N., Lavest, J.M., and Dhome, M., Simplifying the kinematic calibration of parallel mechanisms using vision-based metrology. IEEE Transactions on Robotics, 22(1):12– 22, 2006. 16. Yan, J. and Ming Chen, I., Effects of constraint errors on parallel manipulators with decoupled motion. Mechanism and Machine Theory, 41(8):921–928, 2006. 17. Mathworks, Numerical analysis with Matlab, Math 475/Cs 475, available from http://www.math.siu.edu/.

Protein Kinematic Motion Simulation Including Potential Energy Feedback M. Diez, V. Petuya, E. Macho and A. Herm´andez Department of Mechanical Engineering, University of the Basque Country, Bilbao, Spain; e-mail: [email protected]

Abstract. In this article a methodology for simulating proteins function movement is presented. The procedure uses a potential energy feedback algorithm that without minizing the energy obtains succesive positions of the protein. before the simulation process, structures are normalized reducing the experimental methods produced errors. The procedure presents a low computational cost in relation to the accuracy obtained. Finally, results of the simulation for a specific protein are shown. Key words: kinematics, proteins, motion simulation

1 Introduction A few years ago a new approach made its way in the proteins field. This new approach, which could be designated as Biokinematics, tries to apply techniques and concepts of robotics and kinematics to the study of proteins. In this sense, Probabilistic Roadmap Mapping has been applied to simulate the folding of proteins and certain protein-ligand interactions [4]. In [1] modal analysis methods are used to obtain information regarding the possible movement of proteins around a given position. These tests are conducted on models of proteins in which all interatomic bonds are replaced by springs. In this article the authors describe the evolution of the work presented in [3] introducing a new normalization algorithm for protein structures and the evaluation of the potential energy of the protein as feedback to guide the simulation process.

2 Protein Modelling Basically a protein is a sequence of amino acids that are joined in a serial chain. All amino acids share a main chain consisting of a nitrogen atom, two carbon atoms and one oxygen atom linked by a double bond to the last carbon atom (see Fig. 1). From

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Fig. 1 Biochemical model. (Courtesy of Accelrys DS visualizer)

the central carbon atom, also called Cα , comes a group of atoms called secondary chain (see Fig. 1). Proteins are formed joining amino acids by a bond between the last carbon C and the nitrogen N of the next amino acid. This bond, called peptide bond, has some chemical characteristic that force the relative angle, ω , between the front and back bonds in the chain of the protein to be 0◦ or 180◦ . These angles impose the main chain atoms of the protein to be placed on planes called peptide planes, composed of atoms Cαi −Ci − Oi − Ni+1 −Cαi+1 (see Fig. 1). The bonds between atoms N −Cα and Cα − C possesses certain rotation motion. These bonds are considered the degrees of freedom (DOF) of the protein. The relative angles in these aforementioned bonds are called dihedral angles, being φ and ψ respectively. When a protein folds, because of potential clashes between the secondary chains, these DOF do not have total freedom of rotation. Ramachandran et al. [8] showed how, if the values of dihedral angles are represented in a φ -ψ chart all the proteins tend to have their angular values in certain areas (see Fig. 3). This graphical representation of the dihedral angles of a protein is called a Ramachandran plot. The importance of this chart is that only the values of φ , ψ inside the preferred areas have a biological sense. Our approach is focused on the simulation of the motion of a protein between two known positions by varying its DOF, ∆ φ and ∆ ψ . Initial and final positions data are obtained experimentally. Data can be extracted from the PDB, the global database of protein structures and sequences. To calculate the values of dihedral angles in the initial and final structures a reference plane has been chosen in each amino acid, called main plane defined by the N −Cα −C atoms. The dihedral angles values are the angles between the main plane and the previous and next peptide planes. To perform our simulations, inorganic Pyrophosphatase protein (corresponding to 1k20 entry in the PDB) has been chosen.

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3 Protein Structure Normalization Before performing the simulation, in order to eliminate protein structure internal errors, a normalization algorithm is going to be applied. Currently, the algorithm is able to correct errors regarding peptide planes and interatomic bond lengths. These two normalizations can be applied individually or sequentially (first plane and then distance normalization) depending on the type of experimental errors present in the structure of the protein. Regarding peptide planes normalization, Ci and Ni+1 atoms of i and i + 1 amino acids are projected onto the midplane formed by Cαi ,Ci , Ni+1 and Cαi+1 . As in previous works in this field [6], the middle plane calculus is made using a leastsquares fit as shown in Eq. (1). S=

∑

(xi − f (zi , yi ))2

(1)

i=1..n

f (yi , zi ) = b · yi + c · zi + d

(2)

where S is the function to minimize, (xi , yi , zi ) are the coordinates of the atoms in the peptide plane, b, c, d are constants of the plane to get and n is the number of atoms in the peptide plane. For the normalization of bond lengths, the algorithm applies distance constraints to all bonds of the protein chain to ensure that the distances between atoms ni , n j are equal to those proposed by Pauling et al. in [7]. The distance constraint is defined by the following equations:



p (3)
n j − nip
= d ji (n pj − nip−1 ) × (n pj − nip ) = 0

(4)

where nip−1 is the position vector of atom ni before applying the distance constraint, nip is the new obtained position vector, n pj is the position vector of atom n j during application of the restriction and, d ji is the normalized distance of the bonds. When assessing the validity of the normalization algorithm several factors have been taken into account: local distortions generated in the internal structure of the protein, measured by the value of the protein potential energy; biological sense of the normalized structure calculated by the Ramachandran plot; and overall difference between the normalized and original structures, for which the root mean square error (rmsdl ) has been calculated between the two superimposed structures:  rmsdl =

1 N 2 ∑ di N i=1

where di represents the distance between ni atoms of both superimposed structures and N is the number of atoms in the protein.

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energy (Kcal/mol)

Ramachandran plot preferred areas

˚ RMSD (A)

67095 57458 73112

93% 93% 87%

− 0.34 0.27

Without normalization Lenghts Planes

For the evaluation of the proteins’ potential energy the AMBER potential field is being selected with the parameters proposed by Cornell et al. [2]. This potential field is governed by the following formula: E(Kcal/mol) =

∑

bonds

Kr (r − r0 )2 +

∑

Kθ (θ − θ0 )2 +

angles

  Bi j qi q j Ai j Vn [1 + cos(nϕ − γ )] + ∑ 12 − 6 + ∑ Ri j ε Ri j i< j Ri j dihedrals 2

(6)

where Kr , Kθ and Vn are experimental constants which depend on the bond being tested, n is a multiplying factor function of the torsion bond which is being evaluated. The parameters r0 , θ0 and γ are respectively the standard bond distances, angles and torsions of the protein structure obtained experimentally and r, θ and φ are the present values. Ri j is the distance between two atoms, qi and q j are the partial charges calculated experimentally between two atoms, ε is the dielectric constant and Ai j and Bi j are values dependent on Van der Waals radii and the Van der Waals parameter εi, j of atoms i, j. As can be seen, the last term of expression (6) is highly dependent on the interatomic distance. Usually it is considered that for ˚ this energy can be neglected [5]. interatomic distances greater than 9A Table 1 shows the results for two different normalizations processes: a normalization of bond lengths and a normalization of peptide levels, data for non-normalized protein is also included: Analyzing the results in Table 1 it is apparent that the normalization of peptide planes distorts the internal structure of the protein as it increases the potential energy. This energy increase occurs only in amino acids’ radicals due to the alteration of their relative positions with respect to the main chain. Another noticeable effect is the reduction of the number of amino acids in the Ramachandran plot preferred areas from 93% to 87%. It is noteworthy that no normalization process alters in ex˚ The validity cess the overall shape of the protein while keeping the rmsdl below 1A. of the results obtained in the normalization of the peptide planes is conditioned by the experimental data quality. Some peptide bonds are far from their supposed theoretical value of 180◦ causing the normalization process to alter in excess the internal structure of the protein. With more accurate experimental data, this normalization would lead to better stabilization of the protein. Current work in the algorithm of peptide plane normalization is focused on using local coordinate systems for each peptide plane normalization.

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4 Protein Function Simulation Simulations have been carried out with an under development software called GIMPRO. The software is capable of displaying graphically a protein by reading the standard *.PDB format,varying the value of the dihedral angles manually and simulating the movement of proteins through the procedure presented in this article. In the simulations only the normalization of lengths is been applied, so it is necessary to include the peptide angle ω (Fig. 1) as an additional DOF. In the analysis, the values of the dihedral angles φi and ψi and the peptide angle ωi of each amino acid, are computed, at both initial (0) and final ( f ) experimental positions. The DOF increments are calculated and applied gradually according to the following expression:

∆ φi =

φif − φi0 ; n

∆ ψi =

ψif − ψi0 ; n

∆ ωi =

ωif − ωi0 n

(7)

where n represents the number of steps or intermediate positions in the simulation. The process progresses from the first amino acid to the last one. ei axes are unit vectors whose directions are defined by the bonds Ni −Cαi for the case ∆ φi , Cαi −Ci for ∆ ψi and Ci − Ni+1 for ∆ ωi . The vectors to be rotated, defined as r j , are formed between the atom origin of the rotation axis ei and any atom to be rotated. The rotations in each DOF are made by the vector form of the Rodrigues rotation formula, used in analysis and simulation of mechanisms, and adopted today in the VRML (Virtual Reality Markup Language): rk+1 = u + v · cos ∆ φi + d · sin ∆ φi j

(8)

where k is the current position. Rotated vector rk+1 is obtained by the sum of three j vectors: v is the component in the plane defined by rkj and ei , d is the normal component of this plane and u is the projection of rkj on ei :   v = rkj − ei · rkj · ei ;

d = ei × rkj ;

  u = ei · rkj · ei

(9)

This formula presents no singularity in any position including 0◦ and 180◦ . In [3] it was seen the need to include the evaluation of the potential energy in the process of simulation. An algorithm is been developed that evaluates the energy and uses that value as feedback to guide the simulation process. The algorithm described in Algorithm 1 begins by calculating the current potential energy value of the protein (E0 ). Next all DOF are rotated sequentially and the energy increment generated by each DOF is calculated ∆ E j from j = 1 to j = m being m the number of DOF of the protein. At the end of the process Em energy is calculated corresponding to the potential energy of the new obtained position. The algorithm checks that the increase in potential energy with respect to E0 does not exceed a boundary εk value. This εk tries to distribute the allowed increase in energy ε , in each iteration by the following expression:

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Fig. 2 RMSD error of the simulations with energy check implementation.

k·ε (10) n where k is the current iteration and n is the total number of iterations of the simulation. This distribution avoids that the DOF which generate large potential energy increases consume the available energy in a few iterations. Thus, the algorithm favors rotations that cause slight energy increases, resulting in a more continuous energy distribution throughout the movement. If the value of the potential energy exceeds the allowed one, the algorithm begins to undo the rotations of those DOF that have generated a greater increase in energy, repeating the process until the energy falls below the εk value. If this happens, the position is considered correct and stored, proceeding to obtain the next position of the movement.

εk =

Algorithm 1 Energetic algorithm for a single step of the procedure 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

E0 ← Evaluate protein potential energy for DOF j in the protein do Rotate the jth DOF (∆ ψ j ||∆ φ j ||∆ ω j ) E j ← Evaluate Potential Energy ∆ E j ← E j − E j−1 end for if Em /E0 ≥ εk then while ∆ Em /E0 ≥ εk do Unrotate DOF j related to the highest ∆ E j ∆Ej = 0 Em ← Evaluate Potential Energy end while end if

The algorithm continues with the simulation process until either every DOF have reached their final values, or until the protein is not able to keep moving after consuming all the available energy for the movement. The results of simulations obtained with this algorithm are represented in Fig. 2. Three simulations with different values of available potential energy for protein movement are presented. The potential energy available is fixed as a percentage of the initial energy of the protein being fixed at 5%, 10% and 15%. In all cases the pro-

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Fig. 3 1k20 protein Ramachandran plot of the obtained final position. (Obtained with WinCoot software)

(a)

(b)

Fig. 4 Initial (a) and final (b) positions of 1K20 protein with GIMPRO software.

tein has reached its final position without reaching the maximum potential energy increment defined value. In the charts it can be observed how the simulation that obtains the more accurate results is the one related with the 15% of available energy ˚ Moreover, for the movement, being the maximum error during the process 4.29A. this simulation follows the energy distribution proposed by the algorithm, increasing the energy cycle by cycle. In the other two simulations the protein is not able to distribute properly the available energy for the movement thereby obtaining higher errors. It has also been checked that the protein does not lose its biological sense, keeping throughout the simulation process, at least 92% of the atoms within the preferred areas of Ramachandran plot. In Fig. 4 both initial and final structures of the simulation process related with 15% of available energy are shown. In Fig. 3 the Ramachandran plot of the obtained final structure is shown. Therefore, this method obtains small error results with an acceptable increase in protein potential energy with a quite low computational effort. The simulations were carried out in a 2.13 GHz PENTIUM Core Duo with 2 Gb of ram. The simulation process has taken 3 days to obtain 103 intermediate positions.

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5 Conclusions In this article, a methodology for obtaining protein motions paths is presented. It has been described a normalization process that reduces errors in the structures. A simulation algorithm that includes an assessment of the potential energy to guide the process of calculating a new position has been described. The results show how the algorithm can evenly distribute the energy throughout the simulation process obtaining all positions with a reduced error. In this method it has been chosen a compromise between the computational cost and the quality of the simulation obtained. The method is capable of obtaining 100 intermediate positions of the protein function movement within three days.

Acknowledgements The authors of this paper wish to acknowledge the finance received from the Spanish Government via the Ministerio de Educaci´on y Ciencia (Project DPI2008-00159) and the University of the Basque Country (Project GIC07/78).

References 1. Chirikjian, G.S., A methodology for determining mechanical properties of macromolecules from ensemble motion data. Trends in Analytical Chemistry 22:549–553, 2003. 2. Cornell, W.D., Cieplak, P., Byly, C.I., Gould, I.R., Merz, K.M., Ferguson, D.M., Spellmeyer, D.C., Fox, T., Caldwell, J.W., and Kollman, P.A., A second generation force field for the simulations of proteins nucleic acids and organic molecules. Journal of American Chemical Society 117:5179–5197, 1995. 3. Diez, M., Petuya, V., Urizar, M., and Hernandez, A., A biokinematic computational procedure for protein function simulation. In IEEE Conference Proceedings pp. 355–362, 2009. 4. Kavraki, L.E., Protein-ligand docking, including flexible receptor-flexible ligand docking. Technical Report, Creative Commons, 2007. 5. Kazerounian, K., Laif, K., and Alvarado, C., Protofold: A successive kinetostatic compliance method for protein conformation prediction. Journal of Mechanism Design, 127:712–717, 2005. 6. Kazerounian, K. and Subramanian, R., Residue level inverse kinematics of peptide chains in presence of observation inaccurancies and bond lenght changes. Journal of Mechanism Design, 129:312–319, 2007. 7. Pauling, L. and Corey, R.B., Atomic coordinates and structure factors for two helical configurations of polypeptide chains. Proceedings of the National Academi of Sciences, 37, 1951. 8. Ramachandran, G.N., Ramakrishnan, C., and Sasisekharan, V., Stereochemistry of polypeptide chain configurations. Journal of Molecular Biology, 7:95–99, 1963.

Implementation of a New and Efficient Algorithm for the Inverse Kinematics of Serial 6R Chains M. Pfurner and M.L. Husty Unit Geometry and CAD, Faculty of Civil Engineering, University Innsbruck, 6020 Innsbruck, Austria; e-mail: {martin.pfurner, manfred.husty}@uibk.ac.at

Abstract. The aim of the reported project was to implement a new and efficient algorithm that yields simultaneously all solutions of the inverse kinematics of general 6R chains in a fast software prototype based on a C# code. The algorithm itself was developed in the working group of the authors and was previously only running in a computer algebra system. It is well known that the inverse kinematics problem of general 6R chains is highly nonlinear and yields in general 16 solutions. Using geometric preprocessing the new algorithm reduces the initial mathematical description to several linear and only two nonlinear equations. This paper recalls the algorithm, discusses the software and shows how this tool can be used in path planning and singularity detection along a path. Key words: inverse kinematics, serial chain, software, singularity analysis

1 Introduction In the PhD thesis of the first author [6] a new and optimal algorithm for solving the inverse kinematics of serial robots with six revolute joints was developed. Inverse kinematics deals with the problem of finding all joint parameters for a given manipulator to reach a prescribed pose (position and orientation) of the end effector. It is well known that this problem is highly nonlinear. There is a large number of papers on this topic and there are different methods to solve it (see e.g. [6]). In all previous algorithms that led to a univariate polynomial (see [4] and [7]), the mathematical description consisted of 14 or more nonlinear equations. The new algorithm reduces the solution process to the solution of a linear system of 7 equations and after that only two nonlinear equations in two unknowns remain to be solved. Clearly, because there is only one elimination step in the algorithm no spurious solutions are introduced. Furthermore, because of simplicity of the initial equations, which are developed in complete generality, i.e. without specification of the design (DenavitHartenberg (DH) [1]) parameters, this algorithm can be easily adapted to all designs of manipulators having six revolute axes.

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The aim of the reported project was to implement the algorithm, that had been previously run only on a computer algebra system, in a software prototype based on a fast C# code. The implementation is such that it works for any conceivable design of a 6R manipulator. It provides a graphical user interface (GUI) but can also be used in a command line mode. The software with the GUI runs only on a Windows platform with a .NET 3.5 framework with SP1 installed. To use all the functionality one has to install MSChart which is available for free on the Microsoft web page. In command line mode the software runs on any Windows platform with the same requirements as the GUI and furthermore on Linux platforms (we tested on Fedora and Ubuntu) using a software to run .NET applications such as mono (see http:/ /mono-project.com). The paper is organized as follows: In Section 2 we recall briefly the algorithm that is used in the background of the software. Section 3 deals with the different types of input and output. It shows how to create a text file as input for the command line and graphical mode and describes the usage in pure graphical mode as well. Section 4 shows the application of the software in a reference example. Section 5 gives a promising outlook to the application in path planning and a simple graphical singularity analysis along given paths.

2 Algorithm for the Inverse Kinematics The underlying algorithm for solving the general 6R inverse problem is recalled very briefly. For details the reader is referred to [6] or [2] and especially [3]. Using algebraic methods combined with classical multidimensional geometry the mathematical description of the inverse kinematics could be reduced to a minimal set of nonlinear equations. Kinematic mapping turned out to be most suitable for the analysis of serial chains. It maps Euclidean transformations to points in a seven dimensional image space P7 , the so called kinematic image space. This space is nothing else than a geometric interpretation of the (algebraic) Study parameters [8] which are closely related to dual quaternions. The underlying idea is that the end effector coordinate system of every mechanism generates a certain set of points, curves, surfaces or higher dimensional objects in the image space P7 . In general the dimension of this object depends on the constraints of the mechanism. The corresponding object in P7 is called constraint manifold. Using this technique the mathematical description of the inverse kinematics of general serial 6R-chains could be reduced to one quadratic equation x0 y0 + x1y1 + x2y2 + x3 y3 = 0,

(1)

the so called Study quadric and eight linear equations 3

Hi (v) :

∑ ((ai v + bi)xi + (civ + di)yi ) = 0

i=0

(2)

New Inverse Kinematics Software Prototype 3

Hi+4 (w) :

∑




 (ai+4 w + bi+4)xi + (ci+4w + di+4 )yi = 0

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(3)

i=0

where a j , b j , c j and d j , j = 0 . . . 7 are functions of the design parameters (DHparameters) of the mechanism. v is the tangent half of the revolute angles of one of the first three axes and w is the tangent half of the revolute angles of one of the second three axes. (x0 , x1 , x2 , x3 , y0 , y1 , y2 , y3 ) are homogeneous coordinates of an arbitrary point in the kinematic image space.

2.1 Solving the Inverse Kinematics Problem For the solution of the inverse kinematics problem we have to intersect Eqs. (1), (2) and (3). In taking seven of the hyperplane equations H0 , . . . , H6 one can solve linearly for xi and yi . Substitution of the solutions into the two remaining equations H7 and the Study quadric (1) yields two nonlinear equations, denoted by E1 and E2 (see [6]) in the algebraic unknowns v and w. A resultant with respect to one of the remaining unknowns, say w, yields a univariate polynomial in v, which is after canceling of some known useless factors of degree 16. Solving this univariate polynomial numerically, back substitution of the solution into E1 and E2 and finding the common solutions of these equations for w one gets up to 16 real pairs of solutions for the inverse kinematics in the tangent half of two of the revolute angles of the 6R mechanism. Computation of the remaining four angles belonging to each of the pairs of v and w amounts to solving a linear equation in one unknown for each of the angles.

3 Description of the Software The user has only to specify the robot design parameters and a sequence of discrete end effector poses. Then the whole computation is handled by the software automatically, including all special cases caused by specific values of the DH parameters. The input can be made either by using a text file or directly in the graphical user interface. In case of input via a text file it is not only possible to compute the inverse kinematics for one pose of the end effector, it is also possible to do batch processing, i.e., to compute the inverse kinematics for a series of end effector poses of the same manipulator. This batch processing is possible in both modes of the software. In this case the output consists, also in the graphical mode, of a text file with all possible real solutions of the inverse kinematics for all prescribed end effector poses. This procedure can be used for fast computation of the inverse kinematics of a discretized motion of the end effector of a manipulator. Additionally, in the case of batch processing using the GUI, the software produces a graphical output such, that for each pose of the whole series all possible link parameters are plotted in a graph.

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This tool can be of great value for fast practical path planning and maybe a simple graphical singularity analysis along a path of the end effector.

3.1 Input as a Text File In the text file one has to specify the design of the mechanism using DH parameters and an end effector pose which is declared via a 4 × 4 matrix. It is also possible to specify more than one end effector pose to compute the inverse kinematics of the same manipulator for a series of end effector poses. How to use this functionality will be described in the next two subsections. A template file can always be generated using the GUI with the function Tools - Generate Template File.

3.2 Using the Graphical User Interface Using the graphical user interface the user has the possibility to enter the Denavit Hartenberg parameters as shown in Fig. 1 on the left side. At this step it is also possible to read in the data from a text file with a prescribed format (see Section 3.1) by using File - Import. If there is more than one pose for the end effector prescribed in this file only the first one is taken. Another function in the File menu is Presets. This function provides some known examples from the literature that have been used for testing the software. After filling all blank boxes by hand or reading in from a file and pushing the next button one gets to the second page of the interface shown in Fig. 1 right. If the values in the previous part had been substituted manually the user has to specify the 4 × 4 end effector matrix in this step manually too. One can choose either European or US representation by clicking the box in front of the name. If one has read in the DH parameters from a file, in which also the end effector pose was specified, then those values are filled in automatically. Clicking the next button starts the computation of the inverse kinematics. After a short time the output is displayed. It shows all possible parameter sets of the revolute angles such that the given manipulator can reach the prescribed end effector pose (see Fig. 1 in the example). The output consists of up to 16 real solutions of the inverse kinematics. The counter in front of every solution is a link, where one can see, by clicking on it, the end effector pose that can be reached in computing the forward kinematics using this particular set of revolute angles. Furthermore it is possible to check the difference of this calculated pose to the prescribed pose. Using the functions File - Export Input and File - Export Result one has the possibility to save the input and all the solutions into files. This function allows the user to reuse the design and the prescribed pose of this computation later on but, most important, to transfer the results for formatting them for import into the controller of the mechanism. When using the function Tools - Batch Processing one has to specify the input file. Then the inverse kinematics is

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Fig. 1 Input of design and pose – Output of the computation.

solved for every given pose. The output, that consists of all possible revolute angles of the joints for every given pose, is written in a text file. Additionally a graphical output format is available. A point graph is drawn such, that in one axis direction the numbers of poses is shown and in the other axis direction for each pose of the whole series all possible link parameters are plotted as discrete points. When the step-size is chosen small enough then the sets of points appear to be curves. For a picture of this graphical output the reader is referred to the example in Section 4.2.

4 Examples The following examples show how to use the GUI with some of its functionality. In both subsections the same manipulator, a puma type robot, will be used, once with a single end effector pose and once using batch processing with 1001 end effector poses.

4.1 Single End Effector Pose The DH parameters are shown in Fig. 1 left. The inverse kinematics of this manipulator having the end effector pose shown in Fig. 1 middle was presented in [5]. All eight solutions for the inverse kinematics are shown in Fig. 1 right. In this window the software displays additionally the number of real solutions and the time taken for computation. In this case the time for calculating all the solutions took 11 milliseconds.

4.2 Batch Processing To show batch processing the end effector of the manipulator is moved along a path having the translation vector (first column of the transformation matrix)

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Fig. 2 Plot of second, fourth and sixth revolute angle.

 t=

1,

T √ √ √ 65 + 79 2 1 1 13 2 3 65 + 41 2 1 2 . + t − t, − t 3 + t 2 − t, + t 200 7 200 7 5 40 4

(4)

The rotation is described by the matrix ⎛

⎞ 2t(t 2 + 1)(t 2 − 1) −2t(t 2 + 1)2 (t 2 + 1)(t 2 − 1)2 ⎝ −2t(t 2 − 1)(t 2 − 2t + 1) t 6 − t 4 + 8t 3 − t 2 + 1 2t(t 2 + 1)(t 2 − 1)⎠ . 2t(t 4 + 2t 3 − 2t 2 + 2t + 1) 2t(1 − t 2 )(t 2 − 2t + 1) (t 2 + 1)(t 2 − 1)2

(5)

For parameter value t = 0 the pose is exactly the same as in the example of the previous subsection. Using a computer algebra system 1001 steps for this motion for t ∈ [0, 1] were computed. Then these poses were written into a file and the function Tools - Batch Processing for computing all the real roots of all 1001 end effector poses was used. The whole inverse kinematics computation for all 1001 poses took 12 seconds and 31 milliseconds on a usual laptop (1.66 GHz Intel Centrino, 3.2 GB RAM, Windows XP SP3). Parts of the graphical output are displayed in Fig. 2. These plots allow a lot of interpretations of the kinematic behavior of the mechanism along the path. A detailed discussion is devoted to extensions of this work. Note the following feature in the example: It can be seen that the ”curves” in none of the solutions are continuous. This tells clearly that the chosen path has a part (between 906 < t < 926) which cannot be reached by the manipulator. This part is outside the workspace.

5 Singularity Analysis Along a Path Here a manipulator presented in [7] will serve as the example. For a prescribed end effector motion the discretized solutions for the inverse kinematics of 1001 poses along this path yields the graphs in Fig. 3. The intersection point in the middle of this figure corresponds to the home pose of the manipulator, which is a singular position (see [2]). It is suggested that this plot shows graphically the singularities the manipulator encounters when following the prescribed path. This conjecture is substantiated by a number of numerical examples, however the development of rigorous proof is in progress.

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Fig. 3 Solutions for all joint angles of a mechanism along a path.

Fig. 4 Horziontal sections of E1 and E2 at t ∈ {0.2, 0.314, 0.45} with {2, 1, 2} intersections.

Our proposal is that in this graph one can see at least two different kinds of singularities. One type of singularity occurs when the graphs of all joint angles have a self intersection in the same pose of the end effector (for a specific value of t). In this pose a double solution occurs for all joint angles when the inverse kinematics is computed (in the example this is the case in the vicinity of pose 814; note that the path parameter t has been divided into 1001 discrete values p = 0 . . . 1000). In such a kind of singularity the number of different real solutions for the inverse kinematics in this pose p0 is one less than in the previous and following pose p = ±ε p0 . Using the described algorithm for the inverse kinematics one can find singularities along a path in the following way: if the end effector of the mechanism describes a motion the equations E1 (v, w,t) = 0 and E2 (v, w,t) = 0 (see Section 2.1) can be seen as surfaces in a 3-space with coordinates v, w and t, where t is the parameter of the motion, v and w are algebraic parameters of two of the joint angles. Singularities will occur in special points of the intersection curves. The singularity around pose 814 (p0 ) corresponds to a planar section of E1 and E2 at t = 0.314 (see Fig. 4). This section is shown in the middle plot of Fig. 4. The plots left and right show the planar sections of E1 and E2 of two poses before and afterwards p = ±ε p0 . Both poses have two intersections, corresponding to two different solutions of the inverse kinematics and the singularity occurs in the transition when the two curves are tangent, which means kinematically that two solutions of the inverse kinematics coincide. The second kind of singularity is when the graphs of all joints in Fig. 3 have vertical tangents in the same pose. In these cases the number of real solutions from one pose to the next two poses decreases (or increases) (see around pose 325). One can see this behavior in the horizontal sections of E1 and E2 as well. The value t = −0.175 corresponds to pose 325 (see Fig. 5). Note that in Fig. 5 we have just two different curves. From left to right we have four intersections, a transition where

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Fig. 5 Horziontal sections of E1 and E2 at t ∈ {−0.177, −0.175, 0.17} with {4,3,2} intersections.

the two curves have a common tangent and then there are only two intersections. The middle plot corresponds to a pose having vertical tangents in the graphs of Fig. 3. Figures 4 and 5 were plotted using Maple.

6 Conclusions This work discusses a new software prototype for computing the inverse kinematics of all designs of serial 6R chains. To the best of the authors knowledge it is the first software that produces a graphical output displaying all solutions along a prescribed path using the batch processing mode. The authors believe that this tool can be of great value for practical path planning, especially for a simple visual singularity analysis along a prescribed path.

Acknowledgement The authors gratefully acknowledge the financial support of CAST and TransIT within the proIT project KinSoft.

References 1. Denavit, J. and Hartenberg, R.S., A kinematic notation for lower-pair mechanisms based on matrices, Journal of Applied Mechanics, 77:215–221, 1955. 2. Husty, M.L., Pfurner, M. and Schr¨ocker, H.P., A new and efficient algorithm for the inverse kinematics of a general 6R, Mechanism and Machine Theory, 42(1):66–81, 2007. 3. Husty, M.L., Pfurner, M., Schr¨ocker, H.-P. and Brunnthaler, K., Algebraic methods in mechanism analysis and synthesis, Robotica, 25:661–675, 2007. 4. Lee, H.Y. and Liang, C.G., Displacement analysis of the general 7-link 7R mechanism. Mechanism and Machine Theory, 23(3), 209–217, 1988. 5. Manocha, D. and Zhu, Y., A fast algorithm and system for the inverse kinematics of general serial manipulators. In Proceedings of IEEE Conference on Robotics and Automation, pp. 3348–3354, 1994. 6. Pfurner, M., Analysis of spatial serial manipulators using kinematic mapping. PhD Thesis, Innsbruck, 2006. Available at http://repository.uibk.ac.at/alo?ohjid=1015078 7. Rhagavan, M. and Roth, B., Kinematic analysis of the 6R manipulator of general geometry, In: Proceedings of the 5th Internat. Symposium on Robotics Research, Tokyo, pp. 263–269, 1990. 8. Study, E., Geometrie der Dynamen. B.G. Teubner, Leipzig, 1903.

Composition of Spherical Four-Bar-Mechanisms G. Nawratil and H. Stachel Institute of Discrete Mathematics and Geometry, Vienna University of Technology, A-1040 Vienna, Austria; e-mail: nawratil,[email protected]

Abstract. We study the transmission by two consecutive four-bar linkages with aligned frame links. The paper focusses on so-called “reducible” examples on the sphere where the 4-4-correspondance between the input angle of the first four-bar and the output-angle of the second one splits. Also the question is discussed whether the components can equal the transmission of a single four-bar. A new family of reducible compositions is the spherical analogue of compositions involved at Burmester’s focal mechanism. Key words: spherical four-bar linkage, overconstrained linkage, Kokotsakis mesh, Burmester’s focal mechanism, 4-4-correspondence

1 Introduction Let a spherical four-bar linkage be given by the quadrangle I10 A1 B1 I20 (see Fig. 1) with the frame link I10 I20 , the coupler A1 B1 and the driving arm I10 A1 . We use the output angle ϕ2 of this linkage as the input angle of a second coupler motion with vertices I20 A2 B2 I30 . The two frame links are assumed in aligned position as well as the driven arm I20 B1 of the first four-bar and the driving arm I20 A2 of the second one. This gives rise to the following Questions: (i) Can it happen that the relation between the input angle ϕ1 of the arm I10 A1 and the output angle ϕ3 of I30 B2 is reducible so that the composition admits two oneparameter motions? In this case we call the composition reducible. (ii) Can one of these components produce a transmission which equals that of a single four-bar linkage ? A complete classification of such reducible compositions is still open, but some examples are known (see Sect. 3). For almost all of them exist planar counterparts. We focus on a case where the planar analogue is involved at Burmester’s focal mechanism [2,4,5,11] (see Fig. 3a). It is not possible to transfer the complete focal mechanism onto the sphere as it is essentially based on the fact that the sum of interior angles in a planar quadrangle equals 2π , and this is no longer true in spherical geometry. Nevertheless, algebraic arguments show that the reducibility of the included four-bar compositions can be transferred.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_12, © Springer Science+Business Media B.V. 2010

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Fig. 1 Composition of the two spherical four-bars I10 A1 B1 I20 and I20 A2 B2 I30 with spherical side lengths αi , βi , γi , δi , i = 1, 2.

Remark: The problem under consideration is of importance for the classification of flexible Kokotsakis meshes [1, 7, 10]. This results from the fact that the spherical image of a flexible mesh consists of two compositions of spherical four-bars sharing the transmission ϕ1
→ ϕ3 . All the examples known up to recent [6, 10] are based on reducible compositions. The geometry on the unit sphere S2 contains some ambiguities. Therefore we introduce the following notations and conventions: 1. Each point A on S2 has a diametrically opposed point A, its antipode. For any two points A, B with B = A, A the spherical segment or bar AB stands for the shorter of the two connecting arcs on the great circle spanned by A and B. We denote this great circle by [AB]. 2. The spherical distance AB is defined as the arc length of the segment AB on S2 . We require 0 ≤ AB ≤ π thus including also the limiting cases B = A and B = A. 3. The oriented angle < ) ABC on S2 is the angle of the rotation about the axis OB which carries the segment BA into a position aligned with the segment BC. This angle is oriented in the mathematical sense, if looking from outside, and can be bounded by −π < < ) ABC ≤ π .

2 Transmission by a Spherical Four-Bar Linkage We start with the analysis of the first spherical four-bar linkage with the frame link I10 I20 and the coupler A1 B1 (Fig. 1). We set α1 = I10 A1 for the length of the driving arm, β1 = I20 B1 for the output arm, γ1 := A1 B1 , and δ1 := I10 I20 . We may suppose

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0 < α1 , β1 , γ1 , δ1 < π . The movement of the coupler remains unchanged when A1 is replaced by its antipode A1 and at the same time α1 and γ1 are substituted by π − α1 and π − γ1 , respectively. The same holds for the other vertices. When I10 is replaced by its antipode I 10 , then also the sense of orientation changes, when the rotation of the driving bar I10 A1 is inspected from outside of S2 either at I10 or at I 10 . We use a Cartesian coordinate frame with I10 on the positive x-axis and I10 I20 in the xy-plane such that I20 has a positive y-coordinate (see Fig. 1). The input angle ϕ1 is measured between I10 I20 and the driving arm I10 A1 in mathematically positive sense. The output angle ϕ2 = < ) I 10 I20 B1 is the oriented exterior angle at vertex I20 . This results in the following coordinates: ⎛

⎞ ⎛ ⎞ cα1 cβ1 cδ1 − sβ1 sδ1 cϕ2 ⎝ ⎠ ⎝ A1 = sα1 cϕ1 and B1 = cβ1 sδ1 + sβ1 cδ1 cϕ2 ⎠. sα1 sϕ1 sβ1 sϕ2

Herein s and c are abbreviations for the sine and cosine function, respectively. In these equations the lengths α1 , β1 and δ1 are signed. The coordinates would also be valid for negative lengths. The constant length γ1 of the coupler implies cα1 cβ1 cδ1 − cα1 sβ1 sδ1 cϕ2 + sα1 cβ1 sδ1 cϕ1 + sα1 sβ1 cδ1 cϕ1 cϕ2 + sα1 sβ1 sϕ1 sϕ2 = cγ1 .

(1)

In comparison to [3] we emphasize algebraic aspects of this transmission. Hence we express sϕi and cϕi in terms of ti := tan(ϕi /2) since t1 is a projective coordinate of point A1 on the circle a1 . The same is true for t2 and B1 ∈ b1 . From (1) we obtain −K1 (1 + t12)(1 − t22) + L1 (1 − t12)(1 + t22) + M1(1 − t12)(1 − t22) + 4 sα1 sβ1 t1t2 + N1 (1 + t12)(1 + t12) = 0 , K1 = cα1 sβ1 sδ1 , L1 = sα1 cβ1 sδ1 ,

M1 = sα1 sβ1 cδ1 , N1 = cα1 cβ1 cδ1 − cγ1 .

(2)

This biquadratic equation describes a 2-2-correspondence between points A1 on circle a1 = (I10 ; α1 ) and B1 on b1 = (I20 ; β1 ). It can be abbreviated by c22t12t22 + c20t12 + c02t22 + c11t1t2 + c00 = 0

(3)

setting c00 = −K1 + L1 + M1 + N1 , c11 = 4 sα1 sβ1 , c02 = K1 + L1 − M1 + N1 , c20 = −K1 − L1 − M1 + N1 , c22 = K1 − L1 + M1 + N1

(4)

under c11 = 0 . Alternative expressions can be found in [10]. Remark: Also at planar four-bar linkages mechanisms there is a 2-2-correspondence of type (3).

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Fig. 2 (a) Opposite angles ϕ2 and ψ2 at the second spherical four-bar I20 A2 B2 I30 . (b) Composition of two orthogonal four-bar linkages with I30 = I10 .

There are two particular cases: Spherical isogram: Under the conditions β1 = α1 and δ1 = γ1 opposite sides of the quadrangle I10 A1 B1 I20 have  equal lengths. In this case we have c00 = c22 = 0 in (3), and eq. (1) converts into s(α1 − γ1 )t2 − (sα1 + sγ1 )t1 s(α1 − γ1 )t2 − (sα1 − sγ1 )t1 (for details see [10]). The 2-2-correspondence splits into two projectivities1 t1
→ sα ± sγ1 t1 , provided α1 = γ1 , π − γ1 . Both projectivities keep t1 = 0 and t2 = 1 s(α1 − γ1 )

t1 = ∞ fixed. These parameters belong to the two aligned positions of coupler A1 B1 and frame link I10 I20 . In these positions a bifurcation is possible between the two one-parameter motions of the coupler against the frame link. Orthogonal case: For a given point A1 ∈ a1 the corresponding B1 , B1 ∈ b1 are the points of intersection between the circles (A1 ; γ1 ) and b1 = (I20 ; β1 ) (compare Fig. 2a). Hence, the corresponding B1 and B1 are located on a great circle perpencos α1 cos β1 = cos γ1 cos δ1 dicular to the great circle [A1 I20 ]. Under the condition  c

c

which according to [10] is equivalent to det c22 c02 = 0 , the diagonals of the 20 00 spherical quadrangle I10 A1 B1 I20 are orthogonal (Fig. 2b) as each of the products equals the products of cosines of the four segments on the two diagonals. Hence, B1  and B1 are always aligned with I10 , but also conversely, the two points A1 and A 1 corresponding to B1 are aligned with I20 . Note that the 2-2-correspondence (3) depends only on the ratio of the coefficients c22 : · · · : c00 . With the aid of a CA-system we can prove: Lemma 1 For any spherical four-bar linkage the coefficients cik defined by (4) obey 1 Since the vertices of the moving quadrangle can be replaced by their antipodes without changing the motion, this case is equivalent to β1 = π − α1 and δ1 = π − γ1 . We will not mention this in the future but only refer to an ‘appropriate choice of orientations’ of the hinges.

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   c611 + 16 K 2 + L2 − 2M 2 − 1 c411 + 256 (M 2 − K 2 )(M 2 − L2 ) + 2M 2 c211 − 4096 M 4 = 0 .

Conversely, in the complex extension any biquadratic equation of type (3) defines the spherical four-bar linkage uniquely – up to replacement of vertices by their antipodes. However, the vertices need not be real. At the end of our analysis we focus on opposite angles in the spherical quadrangle I20 A2 B2 I30 : The diagonal A2 I30 divides the quadrangle into two triangles, and we inspect the interior angles ϕ2 at I20 and ψ2 at B2 (Fig. 2a). Also for non-convex quadrangles, the spherical Cosine Theorem implies cos A2 I30 = cβ2 cγ2 + sβ2 sγ2 cψ2 = cα2 cδ2 + sα2 sδ2 cϕ2 . Hence there is a linear function cψ2 = k2 + l2 cϕ2 with k2 =

cα2 cδ2 − cβ2 cγ2 , sβ2 sγ2

l2 =

sα2 sδ2 . sβ2 sγ2

(5)

For later use it is necessary to define also ψ2 as an oriented angle, hence

ψ2 = < ) I30 B2 A2 ,

ϕ2 = < ) I30 I20 A2 under − π < ψ2 , ϕ2 ≤ π .

We note that in general for given ϕ2 there are two positions B2 and B2 on the circle b1 obeying (5) (Fig. 2a). They are placed symmetrically with respect to the diagonal A2 I30 ; the signs of the corresponding oriented angles ψ2 are different. Remark: Also Eq. (5) describes a 2-2-correspondence of type (3) between ϕ1 and ϕ2 , but with c11 = 0 . A parameter count reveals that this 2-2-correspondence does not characterize the underlying four-bar uniquely.

3 Composition of Two Spherical Four-Bar Linkages Now we use the output angle ϕ2 of the first four-bar linkage as input angle of a second coupler motion with vertices I20 A2 B2 I30 and consecutive side lengths α2 , γ2 , β2 , and δ2 (Fig. 1). The two frame links are assumed in aligned position. In the case < ) I10 I20 I30 = π the length δ2 is positive, otherwise negative. Analogously, a negative α2 expresses the fact that the aligned bars I20 B1 and I20 A2 are pointing to opposite sides. Changing the sign of β2 means replacing the output angle ϕ3 by ϕ3 − π . The sign of γ2 has no influence on the transmission. Due to (3) the transmission between the angles ϕ1 , ϕ2 and the output angle ϕ3 of the second four-bar with t3 := tan(ϕ3 /2) can be expressed by the two biquadratic equations c22t12t22 + c20t12 + c02t22 + c11t1t2 + c00 = 0 , (6) d22t22t32 + d20t22 + d02t32 + d11t2t3 + d00 = 0 . The dik are defined by equations analogue to eqs. (4) and (2). We eliminate t2 by computing the resultant of the two polynomials with respect to t2 and obtain

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Fig. 3 (a) Burmester’s focal mechanism and the second component of a four-bar composition. (b) Reducible spherical composition obeying Dixon’s angle condition for ψ1 – equally oriented. ⎛

⎞ c11 t1 c20 t12 + c00 0 c22 t12 + c02 2 2 ⎜ 0 c22t1 + c02 c11 t1 c20t1 + c00 ⎟ ⎟ = 0. det ⎜ ⎝ d22 t32 + d20 ⎠ d11 t3 d02 t32 + d00 0 2 2 0 d22t3 + d20 d11 t3 d02t3 + d00

(7)

This biquartic equation expresses a 4-4-correspondence between points A1 and B2 on the circles a1 and b2 , respectively (Fig. 1). Up to recent, to the authors’ best knowledge the following examples of reducible compositions are known. Under appropriate notation and orientation these are: 1. Isogonal type [1,7]: At each four-bar opposite sides are congruent; the transmission ϕ1 → ϕ3 is the product of two projectivities and therefore again a projectivity. Each of the 4 possibilities can be obtained by one single four-bar linkage. This is the spherical image of a flexible octahedron of Type 3 (see, e.g., [8]): 2. Orthogonal type [10]: We combine two orthogonal four-bars such that they have one diagonal in common (see Fig. 2b), i.e., under α2 = β1 and δ2 = −δ1 , hence I30 = I10 . Then the 4-4-correspondence between A1 and B2 is the square of a 2-2-correspondence. 3. Symmetric type [10]: We specify the second four-bar linkage as mirror of the first one after reflection in an angle bisector at I20 (see [10, Fig. 5b]). Thus ϕ3 is congruent to the angle opposite to ϕ1 in the first quadrangle. Hence the 4-4correspondence is reducible; the components are expressed by the linear relation cϕ3 = ±(k1 + l1 cϕ1 ) in analogy to (5). At the end we present a new family of reducible compositions: In Fig. 3a Burmester’s focal mechanism is displayed, an overconstrained planar linkage (see [2, 4, 5, 11]). The full lines in this figure show a planar composition of two four-bar linkages with the additional property that the transmission ϕ1 → ϕ3 equals that of one single four-bar linkage with the coupler KL. Due to Dixon and Wunderlich this

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composition is characterized by congruent angles ψ1 = < ) I10 A1 B1 and < ) LB2 A2 which is adjacent to ψ2 = < ) I30 B2 A2 .2 However, this defines only one component of the full motion of this composition. The second component is defined by ψ1 = < ) I10 A1 B1 = − < ) LB2 A2 (see Fig. 3a). For the sake of brevity, we call the overall condition < ) I10 A1 B1 = ± < ) LB2 A2 Dixon’s angle condition and prove in the sequel that also at the spherical analogue this defines reducible compositions. Lemma 2 For the composition of two spherical four-bars Dixon’s angle condition ) I 30 B2 A2 is equivalent to < ) I10 A1 B1 = ± < sα1 sγ1 : sβ1 sδ1 : (cα1 cγ1 − cβ1 cδ1 ) = ±sβ2 sγ2 : sα2 sδ2 : (cα2 cδ2 − cβ2 cγ2 ).

In terms of cik and dik it is equivalent to proportional polynomials D1 = (c11 t2 )2 − 4(c22 t22 + c20 )(c02 t22 + c00 ),

D2 = (d11 t2 )2 − 4(d22 t22 + d02 )(d20 t22 + d00 ).

Proof. In the notation of Fig. 3b Dixon’s angle condition is equivalent to cψ1 = c(π − ψ2 ) = −cψ2 = −k2 − l2 cϕ2 by (5). At the first four-bar we have analogously cψ1 = −k1 − l1 cϕ2 ,

k1 =

cα1 cγ1 − cβ1 cδ1 , sα1 sγ1

l1 =

sβ1 sδ1 . sα1 sγ1

(8)

Hence, cψ1 = −cψ2 for all cϕ2 is equivalent to k1 = k2 and l1 = l2 . This gives the first statement in Lemma 2. The ± results from the fact that changing the sign of γ2 has no influence on the 2-2-correspondence ϕ2
→ ϕ3 , but replaces ψ2 by ψ2 − π . If the angle condition holds and ψ1 = 0 or π , the distances I10 B1 and I30 A2 are extremal. For the corresponding angles ϕ2 there is just one corresponding ϕ1 and one ϕ3 . Hence, when for any t2 the corresponding t1 -values by (3) coincide, then also the corresponding t3 -values by (6) are coincident. Hence, the discriminants D1 and D2 of the two equations in (6) – when solved for t2 – have the same real or pairwise complex conjugate roots. Conversely, proportional polynomials D1 and D2 have equal zeros. Hence the linear functions in (5) and (8) give the same cϕ2 for cψ1 = −cψ2 = ±1 . Therefore cψ1 = −cψ2 is true in all positions, and the composition of the two four-bars fulfills Dixon’s angle condition. The second characterization in Lemma 1 is also valid in the planar case. So, the algebraic essence is the same on the sphere and in the plane. Since in the plane the reducibility is guaranteed, the same must hold on the sphere. This can also be confirmed with the aid of a CA-system: The resultant splits into two biquadratic polynomials like the left hand side in (3). By Lemma 1 each component equals the transmission by a spherical four-bar, but the length of the frame link differs from the distance I10 I30 because otherwise this would contradict the classification of flexible octahedra. General results on conditions guaranteeing real four-bars have not yet been found. We summarize:

2 This condition is invariant against exchanging the input and the output link. The compositions along the other sides of the four-bar I10 KLI30 in Fig. 3a obey analogous angle conditions.

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Theorem 3 Any composition of two spherical four-bar linkages obeying Dixon’s angle condition ψ1 = < ) I10 A1 B1 = ± < ) I 30 B2 A2 (see Fig. 3b) is reducible. Each component equals the transmission ϕ1 → ϕ3 of a single, but not necessarily real spherical four-bar linkage. Example: The data α1 = 38.00◦ , β1 = 26.00◦ , γ1 = 41.50◦ , δ1 = 58.00◦ , α2 = −40.0400◦ , β2 = 123.1481◦ , γ2 = −123.3729◦ , δ2 = 82.0736◦ yield a reducible 4-4-correspondence according to Theorem 3. The components define spherical four-bars with lengths α3 = 60.2053◦ , β3 = 53.5319◦ , γ3 = 8.6648◦ , δ3 = 14.5330◦ or α4 = 24.7792◦ , β4 = 157.1453◦ , γ4 = 160.4852◦ , δ4 = 33.8081◦ .

4 Conclusions We studied compositions of two spherical four-bar linkages where the 4-4-correspondance between the input angle ϕ1 and output angle ϕ3 is reducible. We presented a new family of reducible compositions. However, a complete classification is still open. It should also be interesting to apply the principle of transference (e.g., [9]) in order to study dual extensions of these spherical mechanisms.

Acknowledgement This research is partly supported by Grant No. I 408-N13 of the Austrian Science Fund FWF within the project “Flexible polyhedra and frameworks in different spaces”, an international cooperation between FWF and RFBR, the Russian Foundation for Basic Research.

References 1. Bobenko, A.I., Hoffmann, T. and Schief, W.K., On the integrability of infinitesimal and finite deformations of polyhedral surfaces. In: Bobenko et al. (Eds.), Discrete Differential Geometry, Oberwolfach Seminars Series, Vol. 38, pp. 67–93, 2008. 2. Burmester, L., Die Brennpunktmechanismen. Z. Math. Phys., 38:193–223, 1893 and Tafeln III–V. 3. Chiang, C.H., Kinematics of Spherical Mechanisms. Cambridge Univ. Press, 1988. 4. Dijksman, E., On the history of focal mechanisms and their derivatives. In: Ceccarelli, M. (Ed.), International Symposium on History of Machines and Mechanisms Proceedings HMM2004, Springer, pp. 303–314, 2004. 5. Dixon, A.C., On certain deformable frame-works. Mess. Math., 29:1–21, 1899/1900. 6. Karpenkov, O.N., On the flexibility of Kokotsakis meshes. arXiv:0812.3050v1[mathDG], 16 December 2008. ¨ 7. Kokotsakis, A., Uber bewegliche Polyeder. Math. Ann., 107:627–647, 1932. 8. Stachel, H., Zur Einzigkeit der Bricardschen Oktaeder. J. Geom., 28:41–56, 1987. 9. Stachel, H., Euclidean line geometry and kinematics in the 3-space. In: Art´emiadis, N.K. and Stephanidis, N.K. (Eds.), Proc. 4th Internat. Congress of Geometry, Thessaloniki, pp. 380– 391, 1996. 10. Stachel, H., A kinematic approach to Kokotsakis meshes. Comput. Aided Geom. Des., to appear. 11. Wunderlich, W., On Burmester’s focal mechanism and Hart’s straight-line motion. J. Mechanism, 3:79–86, 1968.

Micro-Mechanisms

Simulation and Measurements of Stick-Slip-Microdrives for Nanorobots C. Edeler, I. Meyer and S. Fatikow Department Microrobotics and Control Engineering, University of Oldenburg, D-26111 Oldenburg, Germany; e-mail: {christoph.edeler, ingo.meyer, fatikow}@uni-oldenburg.de

Abstract. This paper presents the results of measurements and simulations of a linear stick-slip axis for a mobile nanorobot for the height adjustment of end-effectors, e.g. microgrippers. The function is dependent on surface condition and friction characteristics. The axis is driven by laserstructured piezoceramics, who generate a dynamic stick-slip motion. It is shown, that the dependency to surface etc. can be partly described by a phenomenon that will be denominated 0(zero)-step amplitude. It is the amplitude, where an actuator displacement is generated, but the final step length remains precisely zero. To investigate several influences on the 0-step amplitude, a test stand was build up. For simulation of friction according to literature dedicated to micro- and nanorobots, friction contacts can be modeled using the LuGre-model of friction. However, the 0-step amplitude is not covered by the model. Furthermore, the dependency to surface condition is not part of actual friction models. Thus, this effect is systematically measured and finally it is discussed how to integrate it into available friction models. Key words: nanorobot, stick-slip, piezoactuator, simulation

1 Introduction In recent years mobile nanorobots with small step sizes and high dynamics were developed in our division [6, 7, 13]. These robots combine nanometer accuracy with a large travel. The robots can carry any end-effector, such as tips, injection needles, sensors oder microgrippers [10, 11, 17]. Probes are positioned with four degrees of freedom x, y, z and rotation in the x-y plane. The drive principle used here is the stick-slip principle, where a phase of “stick” is followed up by a short phase of “slip”. This principle is well known for several decades and will not be explained further [12, 14, 16]. However, there is still a lack in understanding the details specially for micro- and nanorobots. Very few publications are dealing with this issue [1–3]. Maybe the reason is that stick-slip is understood well enough for building up drives quickly. Nevertheless, our measurements show, that literature does not cover all observed phenomena. Furthermore, actual friction models do not cover the measured effects, either. Thus, there is in fact a lack of knowledge as far as the quantitative simulation and the dependency to surface conditions and preload are

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Fig. 1 Image of (a) the Z-piezoactuators, (b) the robot with actuators and (c) the slider driven by the actuators with end-effector module.

concerned. Thanks to smaller and cheaper control electronics, stick-slip drives are more and more a competition to convential geardrives. This paper for the investigation of stick-slip drives is based on empirical laws (see section 2) and simulations (see section 3). Finally, if these details can be understood better, stick-slip drives can be designed much more effective. Figure 1a shows the piezoactuators used for the measurements. Their design is similar to that for the robot’s drive, which is described in detail in [8]. The piezoactuators are located in the robots housing (Figure 1b). The ruby hemispheres can be rotated dependent on the control voltage. The rotation is transformed into a linear motion of the slider (Figure 1c). Lastly, the axis can be moved with a resolution of 20nm within a travel of 5mm. A critical element of the slider is the preload spring based on a flexure hinge, which generates the preload for the axis. The preload is necessary for the normal force in the contact zone between the ruby hemispheres and the surface of the slider. If preload is too low, the axis is not hold in position or cannot move upwards. If preload is too high, motion is reduced or even impossible. These facts are strongly connected to the 0-step amplitude, which also depends on preload. This will be described after the introduction of the test stand which was build. The test stand (Figure 2) contains the slider similar

Fig. 2 Test stand for investigation of the stick-slip axis.

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to Figure 1c, but fabricated without preload spring. This is because preload is generated externally. The displacement of the slider is measured by laserinterferometry. The slider is hold in position by six ruby hemisheres (as it is in the robot). The piezoactuators including the hemispheres are mounted to the housing and the preload carriage. Preload is generated by the preload spring, this allows a smooth control of the preload widely independent on fabrication tolerances. The test stand is mounted on a vibration isolation table and is surrounded by a box to keep away airflow which causes temperature drift and carries acoustic vibrations. The interferometer (SIOS, Germany) allows a measurement accuracy of less than 1nm. Due to several noise sources, the test stand accuracy is around 6nm. To define some basic terms, Figure 3 shows a reference step (continuous lines). A typical control signal (thin continuous line) starts from 0nm, it moves the actuator slowly within 5ms to the positive displacement of 80nm, which corresponds to +150V. From the mechanical point of view, this is a quasistatic movement, friction is always “sticky” and the slider follows the movement properly (thick line). After that the crucial phase comes: the slip. The control signal changes from +150 to −150V in 2µs. Due to the measured response, the slider does not follow this movement, sliding occurs. What defines the slip-phase is the slewrate of the control signal (which can also be described by the slip-time and the slip-displacement). From the sliders point of view, during the slip a back-step of approximately 50nm oocurs. This back-step is induced by sliding friction and would be zero, if there was no frictional sliding force. The back-step’s size is strongly dependent on the slip-time; the shorter, the smaller. In the following stick-phase a vibration of the slider with an amplitude of about 20nm can be observed. Finally, a permanent displacement, a stick-slip step, with a size of 120nm is performed.

Fig. 3 Control signal and measured stick-slip step (continuous lines). For the simulation (dashed lines) refer to Section 3.

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2 Measurements In this section it will be shown that under certain conditions, the back-step is at its maximum and the final displacement is zero: No sliding occurs. Firstly, it is explained how the 0-step amplitude is measured. The amplitude of the control signal can be varied from 0 to 300Vpp (Volts Peak-Peak). Thus, different step sizes dependent on the amplitude can be measured, Figure 4 shows the coherency. Each point represents the average step size of five single step measurements. The amplitude is varied from 100 to –100%, which means in both directions. It is obvious, that most of the points follow a line, but the step size reaches zero at approximately 20% amplitude in this constellation (the linear response was there in all measurements). Thus, best fit of the stepsizes greater than 10nm and also the intersection points with the x-axis is calculated, which represent the 0-step amplitudes. The definition of the 0-step amplitude is as follows: A step is performed with typical slewrate, but the final step-size is zero due to the low amplitude only. That means the back-step equals the forward-step, so that there is zero displacement at the end of the slip. The 0-step amplitude does not mean, that there is no motion or vibration at all. Its only dedicated to the final zero offset between actuator (ruby hemispheres) and slider. As mentioned in the introduction, it is measured how several parameters influence the value of the 0-amplitude. Furthermore, the maximum step size (generated by 100% amplitude) as an average between the forward and backward maximum step size is calculated (here on the basis of ten measurements each). With the teststand, it is possible to vary the following parameters indepentdently: amplitude, frequency, slewrate, preload, friction (in such a way that we can change material and surface condition), slider’s mass and direction of gravity.

Fig. 4 Measurement of the 0-Amplitude. Firstly, the step lengths generated by several amplitudes in both directions are averaged. Then, a best-fit line is calculated. The intersection with the x-axis is the measured 0-Amplitude.

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Fig. 5 Measurement of the 0-step amplitude for polished steel against slewrate. There is a minimum 0-step amplitude for middle slewrates. For the simulated line refer to Section 3.

Figure 5 shows the 0-step amplitudes for polished steel with different preloads. Two facts can be obtained: The 0-step amplitude raises widely proportional to preload; and the 0-step amplitude is not constant to different slewrates. In fact, a minimum 0-step amplitude can be observed at a slewrate of about 100 to 150 V/µs. With a preload of 0.75N, 0-step amplitude reaches nealy one quarter of its entire range. If preload is more than 2N the 0-step amplitude becomes 100%, the resulting step size is zero. However, repeatability of the latter measurements is very poor. Figure 6 shows the maximum step sizes of several materials with different slewrates and constant preload at 0.3N. It can be seen that the dependency of several materials

Fig. 6 Measurement of the maximum step size for different materials against slewrate. Polished surface leads to higher sizes than rough surface, steel also than aluminum and brass. The step sizes decrease with the slewrate. For the simulated line refer to Section 3.

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C. Edeler et al. Table 1 Parameters of the LuGre model used for simulations.

σ0

σ1

σ2

vstribeck

µstatic

µkinetic

7e6

5

0

0.001

0.3

0.2

is related to and decreases proportionally with the slewrate. At very low slewrates, slip does not really occur (maybe presliding) and step sizes are rapidly decreasing. And, the smoothest surface finish (polished) leads to larger steps and harder materials are better than soft ones. The difference between the measured materials is about 20nm, which is 20% of the maximum.

3 Simulation Very few publications take into account the effect of what is introduced in this paper as 0-step amplitude. Brequet and Bergander mentioned a similar effect [2, 3]. Brequet derived it from the tangential stiffness of the surface asperities and calculates values of 20 to 40nm, which is verified in measurements. Brequet also simulates a stick-slip drive, but apparently 0-step amplitude is not an issue. From this point of view, it is difficult to evaluate the relevance of the 0-step amplitude. Based on the fact, that 0-step amplitude is a matter of friction or surface mechanics, respectively, friction models should reflect the effect. Actually, numerous friction models exist. The most sigificant for technical terms are the “classical” model of Amonton, the model of Coulomb and the LuGre-model of Canudas [4]. The model takes into account the second-order motion generated by the interconnected surface asperities. Brequet also used this model for simulation of microdrives. Based on the model of amonton, the LuGre model also takes into account friction effects like hysteresis, velocity-dependent sliding (Stribeck characteristics) and presliding. For further information on this issues the reader is referenced to [12, 14, 15]. The LuGre model was enhanced by elastic and plastic deformation characteristics to avoid quasistatic drift [5]. For the first step the LuGre model and not the elastoplastic model was chosen for the investigations, because implementation is easier. Thus, the simulation results of this paper are based on the LuGre model. Its parameters were derived from a “standard” step run similar to that in Figure 3,(measured response). The parameters can be gathered from table 1 (for the meaning of the parameters see [4]). Effects such as piezoelectric hysteresis and nonlinearity were neglected (Brequet also did). Furthermore, it is ensured that the piezoelectric actuator is fast enough to reproduce the high slewrates and frequencies up to 1MHz. This is due to extensive Finite-Element-Simulations [9]. Thus, the piezo is modeled as a linear voltage-displacement transducer. In contrast to the measured step (Figure 3, continuous lines), the slip in the simulation is located initial, then followed up by a longer stick phase (Figure 3, dashed lines). This is due to technical reasons, the

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slip itself is the same, so the results are comparable. As a matter of fact, simulation of quasistatic drift is not of interest at the moment. As a matter of fact, an axis which is characterised by six frictional contacts (hemispheres) is under investigation. Thus, the preload which is applied, divides up into six normal forces, which generate six frictional forces on their part. In contrast to this the simulation is based on a single friction contact with the full amount of preload. Theoretically the two scenarios are the same, but geometric errors and fabrication tolerances can also have an effect. The simulated 0-step amplitude is depicted in Figure 5, “simulation”. The 0-step amplitude is always exactly zero; obviously the LuGre model does not contain this particular effect. According to [4], the LuGre-model exhibits effects like breakawayforce, which is likely connected to the 0-step amplitude. However, the interconnection is unknown at this moment. The simulated maximum step sizes dependent on the slewrate are shown in Figure 6. Contradictory to the measurements, maximum step size in simulation is constant until sliding effect disappears.

4 Conclusions In this paper it is questioned, how the simulation of the 0-step amplitude does meet the measurements. As can be seen in Figure 6, the resulting step length is dependent on material and surface condition. Different parameters for the LuGre model dependent on material and surface condition can be an empirical solution. However, this is more a technical issue and there is still the question, how to deal this effect in theory. In calculations of Brequet, the comparable value to the 0-step amplitude depends on the stiffness of the surface asperities. This is in contrast to the fact, that the LuGre-model should cover all effects caused by the surface asperities. As a result of Figure 5, a slewrate can be found, where the 0-step amplitude has a minimum. This could be useful to control a drive with the lowest possible amplitude and level of vibration. Figure 6 indicates that material and surface condition influence the practicable step size. The drive could be optimized to increased velocity (larger steps) or to less vibrations (smaller steps). Finally, for the design of nanorobots a trade-off between the low 0-step amplitude and the rising back-step at decreasing slewrates has to be found.

5 Outlook It will be a challenge of the future to model the effect of the 0-step amplitude in combination with an adequate friction model. The extension of the LuGre-model, the elastoplastic model, could be a solution. Another topic of interest is the vibration, which occurs after the slip (Figure 3). Until now there is a lack of data, but

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preliminary measurements point out that it is dependent on material, surface condition and to the slider’s mass. According to this phenomenon a question comes up: How can the friction model describe not only the resulting step length, but also the shape of the measured step-signal including vibrations accurately? This could lead to different model parameters in dependency to the slewrate. Maybe this is owed to material properties as time-dependent ductility (rate of elongation).

References 1. Brequet, J.-M. and Bergander, A., A testing mechanism and testing procedure for materials in inertial drives. In Proc. Int. Symposium on Micromechatronics and Human Science MHS2002, 2002. 2. Bergander, A., Control, wear and integration of stick-slip micropositioning. PhD Thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 2003. 3. Brequet, J.-M., Stick and slip actuators. PhD Thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland,1998. 4. Canudas de Wit, C., A new model for control of systems with friction. IEEE Transactions on Automatic Control, 1995. 5. Dupont, P., Armstrong, B. and Hayward, V., Elasto-plastic friction model: Contact compliance and stiction. IEEE Transactions on Automatic Control, 2000. 6. Edeler, C., Simulation and experimental evaluation of laser-structured actuators for a mobile microrobot. In Proceedings IEEE International Conference on Robotics and Automation ICRA2008, Pasadena, 2008. 7. Edeler, C., Jasper, D. and Fatikow, S., Development, control and evaluation of a mobile platform for microrobots. In: International Federation of Automatic Control, 2008. 8. Edeler, C., Laser-based structuring of piezoceramics for mobile microrobots. In: Proceedings European Conference on Mechanism Science, Cassino, Italy, 2008. 9. Edeler, C., Dynamic-mechanical analysis of piezoactuators for mobile nanorobots. In: Proceedings International Conference and Exhibition on New Actuators and Drive Systems, ACTUATOR2010, Bremen, Germany, 2010, accepted. 10. Eichhorn, V., Fatikow, S., Wich, T., Dahmen, C., Sievers, T., Nordstroem-Andersen, K., Carlson, K. and Boeggild, P., Depth-detection methods for microgripper-based CNTmanipulation in a scanning electron microscope. Journal of Micro- and Nanomechanics, 4:27–36, 2008. 11. Guo, S., Pan, Q. and Khamesee, B., Development of a novel type of microrobot for biomedical application. In: Microsystems Technology, 2008. 12. Heslot, F., Baumberger, T., Perrin, B. and Caroli, C., Creep, stick-slip, and dry-friction dynamics: Experiments and a heuristic model. Physical Review E, 49:4973–4988, 1994. 13. Jasper, D. and Edeler, C., Characterization, optimization and control of a mobile platform. In: Proceedings International Workshop on Microfactories, IWMF2008, Chicago, 2008. 14. Persson, B.N.J., Theory and simulation of sliding friction. Physical Review Letters, 71:1212– 1215, 1993. 15. Rabinowicz, E., The intrinsic variables affecting the stick-slip process. Master Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1957. 16. Zesch, W., Multi-degree-of-freedom micropositioning using stepping principles. PhD Thesis, Swiss Federal Institute of Technology ETH Zurich, Zurich, Switzerland, 1997. 17. Zhao, C., Zhang, J., Zhang, J. and Jin, J., Development and application prospects of piezoelectric precision driving technology. Frontier of Mechanical Engineering, 3:119–132, 2008.

Analysis and Inverse Dynamic Model of a Miniaturized Robot Structure A. Burisch1 , S. Drewenings1, R.J. Ellwood1 , A. Raatz1 and D. Pisla2 1 Institute

of Machine Tools and Production Technology, Technische Universit¨at Braunschweig, D-38106 Braunschweig, Germany; e-mail: [email protected] 2 Mechanics and Programming Department, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. The Parvus is a miniaturized precision robot driven by micro drives. The robot is well suited for pick-and-place applications with a repeatability in the range of a few micrometers. Due to the miniaturization of the machine parts used in the Parvus, disturbing effects in the robot drives in combination with load torque cause dynamic vibrations of the robot structure. To optimize the dynamic behavior of the robot, the common approach of a computed torque feed forward control (CTFF) strategy is investigated. The results of the inverse dynamic model show why this control strategy is not applicable in this case, but gives insight for other approaches. Key words: miniaturized robot, inverse dynamic model, micro gear, torque control

1 Introduction In micro assembly an increasing gap concerning dimensions and costs of micro products compared to the production systems used can be observed. Assembly lines and clean rooms for millimeter-sized products often measure some tens of meters and are mostly too expensive for small- and medium-sized businesses. A solution to prevent this could be the cost-optimized, flexible desktop factory for micro production. In the last couple of years, the development of such miniaturized machine equipment is possible due to the availability of new miniaturized drives and gears. A simple scaled down version of large drives and gear concepts is not possible in all cases. The behavior of these miniaturized machine concepts can be completely different from the larger ones. Thus for miniaturized machines it has to be observed if typical structural behavior or standard control strategies can be used or new concepts have to be found.

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Fig. 1 The robot Parvus equipped with a suction gripper.

2 The Miniaturized Robot Parvus The robot Parvus, as shown in Figure 1, is a size-adapted handling device using miniaturized machine parts. This handling device, in the range of several centimeters, is well prepared for the use in easily scalable and highly flexible production technology such as desktop factories. The challenge of the functional model Parvus was to develop a miniaturized precision industrial robot with the full functional range of larger models. The robot consists of a typical parallel structure, driven by Micro Harmonic Drive gears [3, 5] combined with Maxon electrical miniaturized motors. This plane parallel structure offers two translational DOF in the x-y-plane. The z-axis is integrated as a serial axis in the base frame of the robot. The easy handling of the whole plane parallel structure driven in z-direction is possible due to the minimized drive components and light aluminum alloy structure. The rotational hand axis covers the Tool Center Point (TCP) of the robot and can be equipped with several types of vacuum grippers. The development of the Parvus, its fundamentals, the miniaturized drive systems, and the robot design approaches have already been described in previous papers, for example [1].

3 Analysis of the Parallel Robot Structure As a handling device for high precision assembly the Parvus has been investigated concerning its precision and structural behavior. The repeatability of the robot has been measured to 5.7 µm (worst value, with 3 Sigma). This value is in the range of the expected repeatability of 2–6 µm, which has been simulated based on the angular repeatability of 0.0027◦ of the micro gears, given by the manufacturer [3, 5]. As described in previous papers [1], this value has been achieved by optimizing the stiffness of the micro gears. The robot is thus well suited for typical pick-and-place applications.

y-direction [mm]

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actual path

set path

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actual path robot kinematics

set path

motors with micro harmonic drive gear

x-direction [mm]

Fig. 2 Path accuracy and dynamic behavior of the Parvus.

As stiffness of the micro gears is important for the repeatability of the robot, it is also very important for the path accuracy and the dynamic behavior of the Parvus. To characterize the path accuracy of the robot, a sample path has been measured with a 3D laser tracker. Figure 2 shows the results of the measured path accuracy at very low speed. The behavior of the plane parallel structure, visible in the path accuracy plot (Figure 2), demonstrates the vibrational deviation from the set path during the movement of the robot. Obviously there are disturbing effects within the micro drives which initiate this behavior. An investigation of whether the load torque during the movement of the robot arms or other effects such as translation errors mainly influence this behavior is necessary. It can be observed that faster movement of the robot causes higher deviations from the set path. Therefore the load torque, especially when starting and stopping the robot’s end effector, play an important role. The chosen approach to reduce the vibration of the robot arms, is optimizing the control strategy of the robot drives. At present, each drive is controlled by a cascade control and does not consider the load torque of the whole robot structure or the interaction with the drive of the other arm.

4 Approach to Optimize the Dynamic Behavior A common approach to optimize the dynamic behavior of a robot is the computed torque feed forward control strategy [2]. Thereby the necessary driving torque for the robot arms is precalculated by the inverse dynamic model (IDM) of the robot structure. This information is used within the control algorithm as a feed forward controller to optimize the drive control current and the behavior of the robot structure. However it has to be considered that miniaturized motors produce their maximum power output at high revolution speed. Therefore these motors are used in combination with high transmitting gears [5]. Thus for an adequate use of the

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mTCP mB2,2

TCP l2,1

mP2

mB1

mB2

mA1

mA2

θ1

θ2

l2,2

B1 l1,1

B2 q1

A1

l1,2

y 0

x

A2

q2

Fig. 3 Nomenclature for the plane parallel structure of the Parvus.

computed torque feed forward control strategy, it is investigated if the load torque influences the robot motors or only the micro gears.

4.1 The Analytical Inverse Dynamic Model of the Parvus In order to get more information about load torque the IDM of the plane parallel structure of the Parvus has to be analytically calculated. With the IDM and an exemplary path within the workspace, the load torque in the robot drives (gears and motors) can be calculated. Figure 3 shows the nomenclature with the point masses m, the inertia θ , the length of the links li, j , the joints Ai , Bi , TCP and the driving angles qi for the plane parallel robot structure. To calculate the torques τi at the active joints Ai for a given path of the tool center point (TCP), the dynamic equations of the robot have to be developed. First of all, the inverse kinematic problem has to be solved, followed by the derivation of the constraints and kinematic equations. Afterwards the torques at the active joints Ai can be calculated by the Lagrange equations of the first kind. With Eq. (1) the angles for a given position of the TCP (TCPx ,TCPy ) are defined, where t1 and t2 describe the trigonometric function for transforming the position of the TCP into the angle of the active joints. q = (q1 , q2 )T = (t1 (TCPx , TCPy ),t2 (TCPx , TCPy ))T

(1)

The constraints fi Eq. (2), (3) base on the fact that the links li, j are defined as rigid bodies. −→ −−−→ 2 ! f1 = |0B1 − 0TCP|2 − l2,1 =0 (2) −→ −−−→ 2 ! f2 = |0B2 − 0TCP|2 − l2,2 =0 With d2 f dt 2

i

d fi dt

(3)

= 0 (i = 1, 2) and transposing to q˙i the velocity is defined and with

= 0 and transposing to q¨i the acceleration is obtained.

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For developing the dynamic equation with the Lagrange formalism [4], the generalized coordinates are defined ξ = (x, y, q1 , q2 ). Due to the fact that only the plane parallel structure of the robot is considered, there is no potential energy (E pot = 0) and the Lagrange equation can be written as: 4

L = Ekin − E pot = ∑ Ei

(4)

i=1

In Eq. (4) the kinetic energy Ekin of the components of the Parvus ∑4i=1 Ei is given as: • Kinetic energy of the active joints Ai : 1 E1 = (θ1 q˙21 + θ2 q˙22 ) 2

(5)

• Kinetic energy of the point masses mAi,Bi at the joints Bi of the active cranks: 1 2 2 2 2 q˙1 + (mB2 + mA2)l1,2 q˙2 ) E2 = ((mB1 + mA1)l1,1 2

(6)

−→ v→ • Kinetic energy of the center of mass of each link li, j in which − P1 and vP2 describe the velocities of the point masses mP1 and mP2 . The velocities are calculated via the inverse kinematic problem by the exemplary path of the TCP: 1 −→ −→ E3 = (mP1 |vP1 |2 + mP2|vP2 |2 ) 2

(7)

• Kinetic energy at the TCP, where x˙ and y˙ are given by the exemplary path: 1 E4 = (mTCP + mB2,1 + mB2,2)(x˙2 + y˙2 ) 2

(8)

Under the condition that no external forces affect the TCP, the Lagrange equations of the first kind for the generalized coordinates are:
 ∂f ∂f d ∂L ∂L − = 0 + λ1 1 + λ2 2 (9) Λx = dt ∂ x˙ ∂x ∂x ∂x
 ∂f ∂f d ∂L ∂L − = 0 + λ1 1 + λ2 2 (10) Λy = dt ∂ y˙ ∂y ∂y ∂y
 ∂f ∂f d ∂L ∂L − Λq 1 = = τ1 + λ1 1 + λ2 2 (11) dt ∂ q˙1 ∂ q1 ∂ q1 ∂ q1
 ∂f ∂f d ∂L ∂L − Λq 2 = = τ2 + λ1 1 + λ2 2 (12) dt ∂ q˙2 ∂ q2 ∂ q2 ∂ q2 This is a linear system of equations, where λi are the Lagrange multiplicators and can be calculated by solving Eq. (9) and Eq. (10). With the obtained Lagrange

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Micro-Gears Joint

z

x y Joint Joint

TCP

Fig. 4 The multi-body model of the parallel robot structure.

multiplicators Eq. (11) and (12) reveal the required torques τi , which are shown in Eqs. (13), (14):

∂ f1 ∂f − λ2 2 ∂ q1 ∂ q1 ∂f ∂f τ2 = Λq2 − λ1 1 − λ2 2 ∂ q2 ∂ q2 τ1 = Λq1 − λ1

(13) (14)

4.2 Verification of the IDM by a Multi-Body Simulation To verify the analytic IDM a multi-body simulation (MBS) of the parallel robot structure has been developed in corporation with Technical University of ClujNapoca. The multi-body model has been created in the software ADAMS and covers the micro gears, the robot arms, and the equivalent mass of the joints as well as the tool center point, see Figure 4. The multi-body simulation of an exemplary path of the TCP gives the driving angle as well as the driving torque over time. The comparability between the analytic and the multi-body model is given under the restrictions that both models traverse the same path through the workspace, the same equivalent inertia of the drives, the same sample time, and properly defined boundary conditions such as no friction and rigid bodies. To analyze the whole spectrum of driving speed and torque, a path of the TCP has been calculated that cause a sinusoidal movement and torque of the robot drives. This path, seen in Figure 5, serves as sample path for the comparison of the IDM and the MBS. Due to the different simulation algorithms several inconsistencies occur during the first sample steps. Thus the time period from 0–8 ms is not included in the verification process. Nevertheless the acceleration from the rest position of the structure can be investigated during the simulation at the inflection points of the path, see Figure 5.

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Fig. 6 Comparison of analytic IDM and MBS (torque at left and right gear output).

Figure 6 shows the data calculated by the two different simulations with the aim to verify the analytical IDM. The curves of the simulated torque at the output of the gears show very good correlations. Due to the error lower than 1% between these curves, the analytical solution of the IDM is verified by the multi-body simulation. Furthermore, the multi-body model can be used for testing new control algorithms before their implementation in the robot.

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5 Conclusions and Outlook An inverse dynamic model has been developed for a possible feed forward control to optimize the dynamic behavior of the Parvus. The IDM has been verified using a multi-body simulation. Using the chosen sample path the IDM gives a maximum load torque at the gear output of a few µNm. The load torque at the motor shaft is calculated by dividing the simulation data by the gear transmission ratio. Thus the simulated load torque at the motor shaft, during the movement of the robot, is in the range of a few nNm. Unfortunately the friction torque of the micro gears of 45 nNm [3] is higher than the necessary driving torque at the motor shaft in the order of a few nNm. This points out that a feed forward control signal gives a torque command to the motor controller which is always lower than the breakaway torque of the robot drives. Thus in contrast to larger robots a computed torque feed forward control is not suitable for optimizing the dynamic behavior of the Parvus. Due to the miniaturized motors and high transmission gear ratio the load torque of the robot structure does not influence the motor but only the elastic micro gears. Nevertheless the load torque information calculated by the IDM, can be used for predicting resulting deflections in the elastic micro gears. This leads to the conclusion that the only approach to optimize the dynamic behavior of the robot is getting more information about the disturbing effects within the micro gears and their interaction with load torque of the robot structure. These investigations point out, that only a model of the transmission errors and elasticity characteristics of the micro gears can help to optimize the path accuracy and dynamic behavior of the Parvus.

Acknowledgement The authors gratefully acknowledge the funding of the reported work by the German Research Center within the Collaborative Research Center 516 “Design and Manufacturing of Active Micro Systems”.

References 1. Burisch, A., Wrege, J., Raatz, A., Hesselbach, J., and Degen, R., PARVUS – Miniaturised robot for improved flexibility in micro production. Journal of Assembly Automation, Emerald, 27(1):65–73, 2007. 2. Hesselbach, J., Pietsch, I., Bier, C., and Becker, O., Model-based control of plane parallel robots – How to choose the appropriate approach? In: Parallel Kinematics Seminar – PKS 2004, Chemnitz, pp. 211–232, 2004. 3. Micromotion GmbH, Catalogue: Precision Microactuators. Vol. 04/2005 MM 90 01 18, 2009, http://www.micromotion-gmbh.de. 4. Sciavicco, L. and Siciliano, B., Modelling and Control of Robot Manipulators. Springer, London, pp. 131–132, 2005. 5. Slatter, R. and Degen, R., Miniature zero-backlash gears and actuators for precision positioning applications. In: Proceedings of 11th European Space Mechanisms and Tribology Symposium ESMATS 2005, Lucerne, Switzerland, pp. 9–16, 2005.

Design and Modelling a Mini-System with Piezoelectric Actuation S. Noveanu1, V.I. Csibi1 , A.I. Ivan2 and D. Mˆandru1 1 Department

of Mechanisms, Precision Mechanics and Mechatronics, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania; e-mail: [email protected], [email protected], [email protected] 2 AS2M Department, FEMTO-ST Institute, 25044 Besanc¸on Cedex, France; e-mail: [email protected] Abstract. In this paper a new mini-system with piezoelectric actuation is presented. The piezoelectric actuation principle it is described and also an original mini-system is designed and analyzed. An experimental prototype of the compliant mini-mechanism is developed, in compact design, which amplifies the input displacement, with facile connection with the piezoelectric actuator. The piezoelectric actuator is commanded with different frequencies impulse signals. Key words: mini-system, compliant mechanism, piezoelectric actuator

1 Introduction The field of micromechanical devices is extremely broad. It encompasses all of the traditional sciences and engineering disciplines, only on a smaller scale. A distinctive feature of micro-electro-mechanical systems technology is miniaturization. Mini-systems are diverse and complex and the trends indicate that it is appropriate to configure small-scale systems somewhat differently from those which we are familiar with. Simply downsizing from dimensions is not possible because of the scaling effect. This is due to the fact that the types of forces that are dominant in miniaturized devices are different from those that are dominant in macro-scale devices. For large-scale systems, inertial effects have a greater influence while surface effects influence smaller-scale systems. Being small, micro-systems are able to produce small forces only [2, 7]. In particular, there are used materials with special properties. The size of a system and the physical parameters will be modified with a scale factor S. When the scale size changes, all the dimensions of the object change by exactly the same amount S such that 1 : S. This scale factor S can be used to describe how physical phenomena change. Knowing how a physical phenomenon scales, whether it scales as S1 or S2 or S3 , or S4 or some other powers of S, guides our understanding of how to design small mechanical systems [3].

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Fig. 1 The modules of the mini-system.

Miniaturization of the mechanical structure alone is insufficient so that many devices may work together, so microelectronics needs to be incorporated in the system. The reduced dimensions imply different manufacturing technologies, measuring and control techniques and specific actuation methods based on new types of actuators. Their actuation effect is achievable through three different means: fields interaction, mechanical interaction and induced limited strain. The actuators in the last category include elements made of so called intelligent or smart materials: piezoelectric and magnetostrictive materials, electro/magnetorheological fluids, electroactive polymers, shape memory alloys [8].

2 The Structure of the Mini-System The researched mini-system contains a mechanical part, represented by a compliant monoblock-type minigripper which is actuated by a piezo actuator. In order to command and control the mini-system with DSpace data acquisition board, it is made a Simulink model that generates the command signals using the Signal generator blocks. The signals type can be: step, sinusoidal, ramp, impulse, and so on. Viewer in real-time behaviour of the system will be through the Control Desk application. The modules included in the proposed mini-system [9] are shown in Fig. 1. The control is achieved in closed loop, with a reaction by position; it is necessary to measure the actuators displacements using suitable position sensors; the control of supply voltage of actuators is such that can obtain the exact position required by the mechanism action.

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Fig. 2 Piezoelectric actuation through the reversed piezoelectric effect.

2.1 Piezoelectric Actuators Regarding the piezoelectric materials (PZT) a tensile mechanical stress is induced that elongates and contracts the material due to the inverse piezoelectric effect, when they are placed in an alternative electric field. In order to amplify the actuators stroke, the stack and bimorph configurations are frequently used [13]. This “induced strain” or change in length occurs as electrical dipoles in the material rotate to align with an orientation that more closely aligns with the direction of the applied electric field. The change in length is generally proportional to the field strength as applied via the device actuation voltage [7]. A typical value for length change might be 0.1% of the total material length in the direction of the applied field. Application of an external field Em (m stands for motor) in opposition to the poling field E p will generate an expansion of the piezoelectric sample by a mechanical deformation ∆ u (Fig. 2). Driving a piezoelectric actuator requires sourcing sufficient current from the driver to produce the desired electrical field or voltage level in the PZT stack. Because PZT stacks have relatively high capacitance, only a small amount of the energy delivered by the driver is used to move the load. The majority of the current applied is in the form of reactive power. Thus, the driving electronics must be able to move a relatively large amount of charge in and out of the PZT stack. This is slightly analogous to a four-quadrant drive for a DC motor with the added complication that the load is capacitive rather than resistive. Shortly, the drive electronics for a piezoelectric actuator are specialized for driving capacitive loads [13].

2.2 Mini-Mechanisms Conventional joints cannot be easily miniaturized, but using compliant mechanisms this problem can be solved. Compliant mechanisms can be miniaturized for use in simple microstructures, actuators and sensors. The monolithic construction also simplifies production, enabling low-cost fabrication [9].

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The compliant mechanisms are different from the traditional rigid body mechanisms in the sense that the last ones are rigid links connected by movable joints. They are used to transfer or transform energy and motions across themselves as desired by the users. The compliant mechanism does the same work except that their usability is also dependent on flexibility of some members. Compliant mechanisms have a smaller number of movable joints materialized as rotation joints. The result is the reduced wear and reduced need for lubrication. They also result in the reduction of noise and vibration due to reduced number of components and friction movements. Backlash reduction due to a decreased number of joints increases the mechanism precision [6]. Several approaches have previously been or are currently being taken to the design of small-scale precision mini/microgrippers [1, 4, 5, 10, 11]: reducing the size of macro-gripping devices; using physical phenomena; the cantilever approach; the tweezers approach; distributed actuators; linkage-free actuators; using adhesion forces. The characteristics of an ideal multipurpose minigripper are: ability to manipulate micro-objects with various sizes and shapes, ability to work in various environments (in clean rooms, in liquids, etc.), ability to apply high forces, high accuracy, low response time, isolation between the control signals and the working area, simplicity of fabrication. Are characterized by a small opening span, a saw-like tip in order to reduce adhesion forces and are equipped with position and force sensors to provide quantitative information during grasping. The mini and micro compliant mechanisms are fabricated by exploiting different technologies (Laser Micromachining, Electro Discharge Machining, LIGA, Stereo Lithography, precision machining).

3 Modelling and Simulation of the Mini-System The developed mini-system contains the modules presented in Fig. 1: a minigripper designed as a device with flexible joints, actuated by a stack piezoactuator, an amplifier for the actuator, a displacement sensor and command and control modules. The piezoelectric actuator used, is stack type [12] and it has a maximum stroke of 15 µm, and overall dimensions are 18 × 6.5 × 6.5 mm. The analytical equations for displacements piezoelectric actuator [9] are:

∆ L = E · di j · L0 +

F cT

(1)

where E is electric field; di j is the piezoelectric coupling coefficient; L0 is the actuator length; F is the axial force and cT is rigidity.

∆ L = ∆ L0 +

F cT

(2)

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Fig. 3 The solid model and mesh realized for FEM analysis.

∆L =

qU − F cT

(3)

where q is the factor for force and U is voltage.

∆ L0 =

F0 cT

F0 = qUmax

(4) (5)

The time for response is: tr =

1 2 fr

(6)

where fr is the resonant frequency. To compensate position change, one must drive the piezoelectric actuator to a slightly different voltage in the return movement to get back to the same starting position. The value of the stroke hysteresis is a percentage of the entire commanded stroke. The overall dimensions of the mini-gripper in the structure of the studied minisystem are 54 mm × 35 mm × 2.21 mm [9]. The task for design is to find the geometry, with relevant dimensions in order to satisfy desired characteristics under specified constraints. These are: the range of motion for fingers, maximal external dimensions, available driving force and displacement and characteristics of material. The solid model (Fig. 3a) and FEM analysis are realized in the linear domain, using Solid Works and ANSYS software. The meshed body is presented in Fig. 3b. The study was performed for the following applied input voltage: 0–120 V and the material used for analyzing the mini-gripper was alloy steel with E = 2.1 × 105 N/mm2 , the Poisson’s coefficient being 0.31. One functioning characteristic of the compliant minigripper is so-called mechanical advantage (m.a.), which is the output-to-input amplification (or deamplification) ratio [6, 7]. The mechanical advantage determined through the FEM analysis for the studied mini-gripper is 6.50.

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Fig. 4 The developed mini-gripper.

Fig. 5 The command scheme in (a) Simulink and (b) ControlDesk application.

4 Experimental Results Our experimental study was focussed on the time response for different input signals. In the next figure is presented a constructive variant of a compliant mini-gripper in the structure of the studied mini-system (Fig. 4): For data acquisition at the piezoelectric actuator and compliant mini-gripper, we realized the command scheme in Simulink (Fig. 5a) and the interface using ControlDesk application (Fig. 5b). The structure of the test bench for measurements contains a Laser displacement sensor LC-2420 with resolution 0.01 µm and response time 100 µs. The stack piezoelectric actuators AE0505D18 have the following characteristics: maximum drive voltage 150 volts; operating temperature 0–45◦C; capacitance 1600–320 nF; clamping force 853 N; resonant frequency 261 KHz; Young’s modulus 4.4 × 1010 N/m2 ; recommended preload less than 100 N. The study was performed for the following inputs: voltage 150 V and impulse signal 0.1–100 Hz (Fig. 6).

Design and Modelling a Mini-System with Piezoelectric Actuation

Fig. 6 The experimental results for impulse signal and different frequencies.

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Figures 6a and b show that for a proper functioning, the mini-system has to be ac-tuated with a frequency below 50 Hz. Figure 6c shows that the mini-system oscillates for a frequency equal with 100 Hz.

5 Conclusions In this paper we pointed out some specific problems in functioning of minimechanical systems built on compliant mechanisms. In order to test the performance of the minigripper with piezo actuators an experimental test bench was developed. The mini-system starts to oscillate for frequency values higher than 50 Hz. In conclusion, for a proper functioning, the system has to be actuated with a frequency below that value. The control is achieved in closed loop, with a reaction by position; it is necessary to measure the actuators displacements using suitable position sensors; the control of supply voltage of actuators is such that can obtain the exact position required by the mechanism action. The mini-system can be successfully applied to the fields that need high precision in small workspace, such as: optics, precision machine tools and mini component fabrication. In the future the aim of our work will be to downscale the mini-system analyzed in this paper.

Acknowledgments The research work reported here was made possible by AS2M Department, FEMTO-ST Institute, Besanc¸on, France; PNII – IDEI Project, ID 221: Modelling, Simulation and Control of the Compliant Micro mechanisms and PNII – ID 1076 Development of a Modular Family of Linear and Rotary Actuators Based on Shape Memory Alloys.

References 1. Bassan, H., et al., Control of a rigid manipulator mounted on a compliant base, Robotica, 23:197–206, 2005. 2. Fatikow, S., Microsystems Technology and Microrobotics, The Mechanical Engineering Handbook Series, Springer, 1997. 3. Gad-el-Hak, M., The MEMS Handbook, The Mechanical Engineering Handbook Series, CRC Press, 2001. 4. Havlik, S., Analysis and modeling flexible robotic (micro) mechanisms. In Proc. of the 11th World Congress in Mechanism and Machine Science, Tianjin, Vol. 3, pp. 1390–1395, 2004. 5. Kota, S., Design and application of compliant mechanisms for surgical tools, Journal of Biomechanical Engineering, 127:981–989, 2005.

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6. Lobontiu, N., Compliant Mechanisms: Design of Flexure Hinges, CRC Press, 2002. 7. Lobontiu, N. and Garcia, E., Mechanics of Microelectromechanical Systems, The Mechanical Engineering Handbook Series, Springer, 2004. 8. Mˆandru, D. et al., Actuating Systems in Precision Mechanics and Mechatronics, AlmaMater, 2004. 9. Noveanu, S., Contributions concerning the study of compliant mechanisms specific to mechatronic systems, Ph.D. Thesis, 2009. 10. Refaat, M.H. and Meguid, S.A., Accurate modelling of compliant grippers using a new method, Robotica, 16:219–225, 1998. 11. Voyles, R.M. and Hulst, S., Micro/macro force-servoed gripper for precision photonics assembly and analysis, Robotica, 23:401–408, 2005. 12. http://www.thorlabs.com 13. http://www.dynamic-structures.com

Linkages and Manipulators

Some Properties of Jitterbug-Like Polyhedral Linkages G. Kiper Middle East Technical University, 06531 Ankara Turkey; e-mail: [email protected]

Abstract. A formal definition for Jitterbug-like polyhedral linkages is presented. It is shown that the supporting polyhedral shapes cannot be arbitrary and some topological properties are derived. Also it is demonstrated that the link lengths of the spatial loops comprising the linkage cannot be arbitrary. The spherical indicatrices of spatial loops are examined and are shown to be immobile. Key words: deployable structures, polyhedral linkages, Jitterbug, spherical indicatrix

1 Introduction Just as a thorough understanding in plane kinematics necessitates a good knowledge on polygons and circles, in order to study spatial kinematics, one needs to master the geometry of polyhedra and the sphere. The passage from spatial rigid structures to mobile ones lies in the question of which polyhedra are rigid and which are movable. First results on this matter are due to Bricard in 1897 [3], Bennett in 1911 [1] and Goldberg in 1942 [6]. These studies address the answer to the question “Do there exist polyhedra with invariant faces that are susceptible to an infinite family of transformations that only alter solid angles and dihedrals?” [3]. Recent studies show that there are also mobile assemblies resembling polyhedral shapes where the dihedral and planar angles are preserved. Altogether, these assemblies are called polyhedral linkages. More precisely, polyhedral linkages are used for spatial deployment, where a certain polyhedral shape is to be preserved or a transformation between some polyhedral shapes is required. That is, the links and joints of the linkage enclose a finite volume with planar boundaries – a supporting polyhedron. The supporting polyhedron continuously changes its shape via an n-parameter transformation for an n-degree-of-freedom (dof) polyhedral linkage. This study deals with linkages for which dihedral and planar angles are preserved during the shape transformation of the supporting polyhedron. This kind of linkages will be called conformal polyhedral linkages.

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Fig. 1 (a) Stachel’s Heureka Polyhedron with tetrahedral motion (4 double faces) [14] and (b) one of Verheyen’s dipolygonids (6 double faces) [15] with overlapping link pairs.

The first example of conformal polyhedral linkages, the Jitterbug, is due to Fuller in 1975 [14]. Elaborate studies on Jitterbug-like linkages were given by Verheyen [15] and R¨oschel [10–13]. Some other polyhedral linkages of this kind are synthesized by Wohlhart [16] and Kov´acs et al. [9]. In all these linkages only revolute joints are used. Verheyen and R¨oschel’s approach to synthesize Jitterbug-like polyhedral linkages is to consider the in-plane motion of a polygonal element and transfer this motion spatially by transforming them into neighboring planes. Once the motions are interrelated, the whole structure is mobile bound to certain criteria. As oppose to this approach, in this study, the spatial loops that comprise the linkage are examined to reveal some properties of Jitterbug-like linkages. This study constitutes a first step for synthesizing Jitterbug-like linkages and metric considerations such as link lengths are the subject of further studies. In the next section, some observations are stated and a definition of a Jitterbuglike linkage is given. In the third section some properties of spatial loops for a Jitterbug-like linkage are sought. In Section 4, the spherical indicatrices of spatial loops are examined. Finally the results are summarized and some further studies are mentioned in the last section.

2 Observations and a Formal Definition For the linkages in [10–13,15,16], in general, there corresponds single link for each face of the polyhedral shape. In some cases though, there are overlapping pairs of links on faces (Fig. 1). In some of Wohlhart’s linkages there are planar chains on the faces [16, 17]. As illustrated in [8], these linkages are in general degenerate cases of a more general family of linkages, where some of the faces of the polyhedral shape coincide. In all these linkages, a separate face will be considered for each link even if some planes coincide.

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(a)

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Fig. 2 (a) An octahedral linkage [12], (b) a cuboctahedral linkage [11, 15] and their supporting polyhedra.

Taking a closer look at these linkages, it is seen that adjacent joint axes are either parallel, or intersecting. The triangular links between the intersecting joint pairs are dihedral angle preserving (dap for short) elements. The links between the parallel ones are links on the faces, which can be referred to as the polygonal links. In general each loop is used to obtain expansion/contraction around a vertex. However, for some cases, the supporting faces do not intersect all at a point, such as the ones demonstrated in Fig. 2. In this study, such cases are not considered; the focus is on linkages for which each vertex is surrounded by a spatial loop. By constructing the supporting polyhedron for many configurations of the linkage in Fig. 2a, it is seen that not only the dihedral and planar angles are preserved, but also the conformal transformation is a proper dilation due to the existence of an inscribing sphere [10]. However, for the linkage in Fig. 2b, the square faces and the triangular faces dilate in different proportions, hence the total transformation of the supporting polyhedron is not a dilation. As a consequence, when all planes are intersected, the corners of the triangles are chopped off and hexagons are formed. The hexagons have variable side length ratios. That is to say, the supporting polyhedral shape transformation is not dilative for all conformal polyhedral linkages. With these observations, a Jitterbug-like linkage can be defined as follows: Let E, F, V, ni denote the number of edges, faces, vertices and valency of ith vertex for a polyhedral shape P. A Jitterbug-like linkage associated with P is a mobile linkage with E many binary (dap) links, F many polygonal links and V many loops comprising 2ni revolute joints such that the polygonal links remain parallel to faces, binary links remain perpendicular to the edges of P with the joint axes intersection on the corresponding edge and to each vertex there corresponds a spatial loop. The joint axes of polygonal links are all parallel to each other, while the axes of a dap

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link intersect. Every polygonal link is connected to another polygonal link via a dap link. The joint intersections of the dap links around a polygonal link are coplanar. This definition generally applies to, but not limited to the Jitterbug-like linkages handled in previous studies. The definition may further be broadened to include linkages, loops of which are not necessarily around vertices, like the ones in Fig. 2. The reason to this restriction is that it is easier to analyze and synthesize a spatial loop around a single vertex figure.

3 Some Properties of Suitable Spatial Loops In this section, some conditions for suitable polyhedral geometries and some properties of suitable loops are investigated. Since a vertex of a polyhedron is at least 3-valent, the simplest loop shall involve 3 pairs of intersecting or parallel joint axes. However in most cases odd-valent vertices cannot be used around an odd-valent vertex as mentioned in [18]. The motivation in claiming this is the fact that instantaneous rotation axes are fixed and so neighboring polygonal links rotate in opposite senses at all times in [18]. The links rotating in opposite senses is generally true; at least there is no counterexample so far. However, when one just concentrates on a spatial loop around a vertex, it is seen that it may be the case where two neighboring polygonal links rotate in the same sense. This is due to the Cardanic motion of a polygonal link in its plane of motion. Again, there is no example of such a linkage so far, hence for the time being, it will be assumed that the valency of all vertices are even. There are also some tricks to design a Jitterbug-like linkage for a polyhedral shape with odd-valent vertices, such as the ones for the cube in Fig. 1b and the cube in [8], where the 3-valent vertices are converted to 6-valent ones. Another trick is embedding offsets between dap links as demonstrated by Verheyen (Fig. 13a of [15]) and Kov´acs et al. [9]. In this case, the dap links are doubled in number and the dihedral angles are halved. These offset links are nothing but digons and by embedding them the number of edges is doubled, and so are the vertex valencies. In each case, eventually, the supporting polyhedral shape is altered. The two joints of a dap link are generally considered as a single joint and the link is not counted as a link in literature. Wohlhart uses the terms “double rotary joint” or “gusset” [16], while R¨oschel [13] prefers “spherical double hinges”. Here, for brevity and due to the shape of the links, this 2-dof joint will be called a “V joint”. Now, some properties of the loops will be given. For most of the linkages in literature, it is seen that generally they contain at least one 4V loop. However, as some counterexamples are already seen (Fig. 1 and [9]), there are some linkages with no 4V loops. Still, it can be showed that for the linkages with polyhedral geometries obeying Euler’s formula V − E + F = 2 there should be at least one 4V loop (V, E, F: number of vertices, edges and faces, respectively). For the proof, assume that the valencies of the vertices are all greater than or equal to six. Then F ≤ 6V/3 (equal when all vertices are 6-valent and all faces are triangular) and E ≥ 3V (equal when

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all vertices are 6-valent). So V + F ≤ 3V, while E + 2 ≥ 3V + 2 > 3V, hence Euler’s formula is not satisfied and the assumption about the vertex valencies is not valid. As noted above, polyhedral shapes involving concurrent faces and digons all vertex valencies may be greater than four. In both of these cases, the Euler characteristic is less than two: addition of a concurrent face increases F by one, E by more than one while V remains and addition of a digon increases E while F and V do not change. Still, in the generic case 4-valent vertices should be expected.

4 The Spherical Indicatrix Associated With the Loops In its most general meaning, the spherical indicatrix of a ruled surface is obtained by a cone of lines with its vertex at the center of a sphere, each line having a corresponding parallel generator on the surface. The indicatrix is the spherical curve of intersection between this cone and the sphere. It is used to separate rotations and translations in spatial linkages [7]. Consider a spatial loop around a vertex such that adjacent joint axes are either parallel, or intersecting. The spherical indicatrix of such a loop does not distinguish between parallel axes. If the sphere is unit, the link lengths of this spherical linkage are “π -dihedral angles’ and the angles between the links are “π -plane angles”. Since the joint axes are normal to the faces (n1 , n2 , n3 in Fig. 3), the associated link lengths (δ12 , δ23 in Fig. 3) being “π -dihedral angle” is obvious. For the angles between the links of the spherical linkage, consider three adjacent faces meeting at a common vertex as shown in Fig. 3. First move the tails of the face normals to the vertex. The angle between δ12 and δ23 is the angle between the normals b12 and b23 to n2 that are in the planes defined by n1 & n2 and n2 - n3 , respectively. Then b12 and b23 are also normal to the edges e12 and e23 , respectively. The angle between b12 and b23 , say β2 , is then supplementary to the plane angle α2 . For a conformal polyhedral linkage, neither the dihedral nor the planar angles should change. So the spherical indicatrix is dictated by the corresponding vertex figure and should remain unchanged during the motion. Formally, a vertex figure is the spherical polygon formed by the intersection of the faces surrounding a vertex with a small sphere centered at that vertex [4]. The side lengths and interior angles of a vertex figure are the planar angles and dihedral angles of the associated vertex, respectively and these angles fully define the spherical indicatrix that is dealt with here. The converse is also true: if the spherical indicatrices of all loops of a polyhedral linkage are fixed during the motion, the linkage should be a conformal polyhedral linkage. But, an assembly constructed for a specific polyhedral shape, such that the spherical indicatrices remain unchanged, may or may not be mobile. If a loop has an immobile spherical indicatrix, it means that the dap links are constrained to move in pure translation with respect to each other. Now, the problem is how to get these constraints. Consider the relative motion of two polygonal links connected to each other via a dap link. The dap link ensures that the angle between the planes of the polygonal links remain constant during the motion. When all links

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Fig. 3 Part of the neighborhood of a vertex (three adjacent faces) and the part of the spherical indicatrix (two spherical links) associated with the spatial loop (dashed regions) around this vertex.

are assembled together to enclose a polyhedral region, one can fix a coordinate system in space such that the instantaneous rotation axes of polygonal links do not change their directions with respect to this frame for all times, i.e. the motion is a Schoenflies motion, as noted in [17] for a specific linkage. This fixed frame can be chosen to be attached to any of the links. In [10] R¨oschel investigates the case where the motions of the polygonal links are Darboux motions. To illustrate, take two such normal directions, say n1 , n2 in Fig. 3. As the directions of n1 and n2 do not change throughout the motion, the dap link in between realizes a translational motion along the perpendicular to n1 and n2 , i.e. along edge e12 in Fig. 3. Of course e12 is not fixed in space and with respect to the fixed frame, the translation is along an ellipse [2, 15]. Several problems may occur in assembling the links. First of all, given a polyhedral shape, even if the polygonal and dap links are selected properly, the assembly may be immobile. As explained below, the spatial loops cannot be constructed arbitrarily, and even all the loops are mobile individually, the assembly of the loops, which is a loop of loops, may be immobile. The conditions for mobility are subject to further studies. If the assembly is mobile, still it may not be a conformal polyhedral linkage. There is no problem if the dof is one, but a multi-dof conformal transformation of the supporting polyhedron would be a high hope. These multi dof assemblies arise, for instance if diagonal links are used: If 12 digons are added to a regular octahedron and proper dap links are inserted, the resulting linkage has multi-dof (in Verheyen’s notation 12{2} + 8{3}|35◦15 52 – Fig. 14a in [15]). Next, the overconstrainedness of the spatial loops will be demonstrated. In an nV loop, there are 2n revolute joints and 2n links (disregarding the degenerate case where some axes are coincident). According to Gr¨ubler’s formula, in general such a loop possesses 6(2n − 1) + 52n = 2n − 6 dofs. If the special condition that the joint axes are pairwise parallel is not imposed, the spherical indicatrix of such a loop comprises 2n revolute joints and 2n spherical links and has 3(2n − 1) + 22n =

Some Properties of Jitterbug-Like Polyhedral Linkages

(a)

143

(b)

Fig. 4 (a) An octahedron with the spherical indicatrices of its vertices. (b) The assembly of the indicatrices.

2n − 3 dofs. Constraining the 2n − 3 dofs of the spherical indicatrix, the loop has 2n − 6 − 2n + 3 = −3 dofs; hence it is a structure unless some special geometric conditions are imposed. The special conditions on the axes are that adjacent joint axes are either parallel, or intersecting. These conditions make the indicatrix an nlink spherical linkage, instead of a 2n-link one. When the indicatrix has n links, it has n − 3 dof(s), and so the loop has 2n − 6 − n + 3 = n − 3 dof(s). For the 4V loop, with these conditions on the axes only, the loop has single dof, however, not all such one dof loops can be used to move about a vertex figure. Also not all vertex figures may be suitable for such loops around them. Finally, it will be showed that the spherical indicatrices of the vertices of a polyhedral shape constitute a spherical polyhedron. Consider the convex octahedron and the six spherical indicatrices of its vertices shown in Fig. 4a. Each indicatrix defines a part of a ball with four dap disk fractions. To every edge, there corresponds two identical dap disk fractions, like the dark ones in Fig. 4a. In total, there are twelve pairs of these disk fractions. Since these identical pairs are parallel to each other, if the ball fractions are assembled by mating the parallel disk fractions, a whole ball is obtained (Fig. 4b). The octahedron is just an example, but the facts mentioned are generally true for all polyhedral shapes that can be projected to a sphere, because the spherical indicatrix for a vertex figure is the spherical projection of the dual facial figure. The projection does not tell much about the metric properties of the dual face, but just gives combinatorial information (see, e.g. [5]).

5 Conclusions and Further Studies This study is a first step to investigate the properties of and hence obtain a general synthesis method for Jitterbug-like polyhedral linkages. First, a precise definition for Jitterbug-like linkages is given. Then the following properties and conditions are conceived for these linkages: (i) The valencies of vertices of a polyhedral structure to be dilated are generally even. (ii) Polyhedral geometries that obey Euler’s

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formula contain at least one 4V loop. (iii) The spherical indicatrix of a spatial loop is immobile and the dimensions of the spherical indicatrix of a loop around a vertex are uniquely given by the dihedral angles associated with the edges and planar angles of the faces meeting at this vertex. The immobility of the spherical indicatrix demands that the polygonal links go through a Schoenflies motion and the dap links are in pure translation. (iv) If an nV loop is constrained such that the spherical indicatrix is immobile, the loop is in general overconstrained. (v) The assembly of spherical indicatrices associated the with vertex figures results in a spherical polyhedron. While (i) and (ii) already provide limitations on possible supporting polyhedral shape geometries, (iii)–(v) are properties to be used in constructing suitable spatial loops and assembling these loops. Next step in the ongoing study is to extract the special geometric conditions mentioned in (iv) for an nV loop. Then the assembly conditions of these loops will be investigated. However, the motions of suitable nV loops may not comply and more special conditions are required for the supporting polyhedral shape and the link lengths of the polygonal linkages. At this stage, (v) is expected to be useful in that this property may allow the designer work on a spherical polyhedron, which should ease the work. The main novel approach in this study is that the spatial loops of Jitterbug-like linkages are issued to reveal several properties of these linkages. Hopefully, this approach will prove its usefulness in obtaining further understanding about Jitterbuglike linkages and possibly give birth to new linkages.

Acknowledgements The financial support supplied by the The Scientific and Technological Research ¨ ˙ITAK) during this study is acknowledged. Also the author Council of Turkey (TUB thanks Professor Dr. Eres S¨oylemez (Middle East Technical University, Turkey) for his valuable advice.

References 1. Bennett, G.T., Deformable octahedra. Proc. Lond. Math. Soc., 2:309–343, 1911. 2. Bottema, O. and Roth, B., Theoretical Kinematics. North-Holland, pp. 305–307, 1979. 3. Bricard, R., M´emoire sur la th´eorie de l’octa`edre articul´e. J. Math. Pures Appl., 3:113–150, 1897. 4. Cromwell, P.R., Polyhedra. Cambridge University Press, p. 77, 1997. 5. Galiliunas, P. and Sharp, J., Duality of polyhedra. Int. J. Math. Educ. Sci. Tech., 36:617–642, 2005. 6. Goldberg, M., Polyhedral linkages. Nat. Math. Mag., 16:323–332, 1942. 7. Hunt, K.H., Kinematic Geometry of Mechanisms. Clarendon Press, p. 287, 1978. 8. Kiper, G., Fulleroid-like linkages. In Proceedings 2nd EUCOMES, Springer, pp. 423–430, 2008.

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9. Kov´acs, F., Tarnai, T., Fowler, P.W., and Guest, S.D., A class of expandable polyhedral linkages. Int. J. Solids Struct., 41:1119–1137, 2004. 10. R¨oschel, O., Linked Darboux motions. Math. Pannonica, 7:291–301, 1996 11. R¨oschel, O., Zwangl¨aufig Bewegliche Polyedermodelle I. Math. Pannonica, 6:267–284, 1995. 12. R¨oschel, O., Zwangl¨aufig Bewegliche Polyedermodelle II. Studia Sci. Math. Hung., 32:383– 393, 1996. 13. R¨oschel, O., Zwangl¨aufig Bewegliche Polyedermodelle III. Math. Pannonica, 12:55–68, 2001. 14. Stachel, H., The Heureka Polyhedron. In Fejes T´oth, G. (Ed.), Intuitive Geometry, Coll. Math. Soc. J. Bolyai, Vol. 63, North-Holland, pp. 447–459, 1994. 15. Verheyen, H.F., The complete set of Jitterbug transformers and the analysis of their motion. Int. J. Comput. Math. Appl., 17:203–250, 1989. 16. Wohlhart, K., New overconstrained spheroidal linkages. In Proceedings 9th World Congress on Mechanism and Machine Science, Milano, Vol. 1, pp. 149–154, 1995. 17. Wohlhart, K., Kinematics and dynamics of the fulleroid. Multibody System Dynam., 1:241– 258, 1997. 18. Wohlhart, K., The kinematics of R¨oschel polyhedra. In Lenarcic, J. and Husty, M. (Eds.), Advances in Robot Kinematics, Kluwer Academic Publishers, pp. 277–286, 1998.

Servo Drives, Mechanism Simulation and Motion Profiles B. Corves Institut f¨ur Getriebetechnik und Maschinendynamik, RWTH Aachen University, 52062 Aachen, Germany; e-mail: [email protected]

Abstract. Mechanisms are used especially in automatic process, production and packaging machinery where a high degree of automation is required, especially because they are fast and precise when it comes to coordination of dependant motion. Due to the restricted flexibility of such motion solutions they are not optimal when the motion requirements are changing. Then often a completely new mechanism has to be installed or it is completely replaced by a servo electric drive that allows a non-uniform input motion. Whether the complete mechanism can be replaced by a servo-electric drive or if a simple mechanism with servo-drive is more favorable can be determined using proper simulation models. Using a typical motion task in process machinery a comparison of different cases will be presented, leading to a typical simulation procedure for the layout and design of suitable motion devices. Key words: mechanism simulation, motion profiles, servo drives

1 Introduction The widespread use of servo electric direct drives can often be explained by the fact that the kinematic and dynamic non-linearities of mechanisms can be avoided and thus the dimensioning process at first glance becomes much easier. Nevertheless this approach often leads to oversized motors especially for high dynamic requirements with regards to accelerations and inertia dependant torque requirements. Therefore the combination of a simple mechanism with a servo electric drive could be advantageous even though more thought must be invested during the design process. For such a case the result could be either a cheaper drive unit or shorter process times due to a more dynamic motion solution [1, 2]. It is of utmost importance to identify the correct procedures and tools for the proper design of the motion device. Especially in the development stage a sensible simulation model is important which should be as simple as possible and as detailed as necessary, without neglecting the nonlinear behavior of the mechanism. Figure 1 shows the general topology of a drive system aimed at producing non uniform motion using a servo drive. The lower row in this picture shows the electro-

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_17, © Springer Science+Business Media B.V. 2010

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Fig. 1 Topology of a typical complete servo motion system.

mechanical drive system consisting of a electric motor, a gear box acting as a torque converter or speed reducer, a mechanism and a work tool, which finally acts on the process, and the process itself. The output motion of the work tool often acts on the process such that a motion dependant force or torque may occur. Therefore in many cases a simulation model should also reflect this situation [3, 5]. The upward pointing arrows in Fig. 1 indicate different possibilities of measuring the actual situation of the system and its feedback towards the control system. Generally only one of the possibilities shown will be used in a typical system. The most common possibility is to measure the position directly on the motor shaft. If the drive system contains a gear box, a given encoder system at this position will give the best resolution, but any deviations due to elasticity in the gear box will not be detectable.

2 Modeling A sensible design of a servo electric drive system as shown in Fig. 1 is only possible if all components especially the electric drive, the mechanical systems and the control system are orchestrated in a sensible manner. This requires that suitable simulation models of each component are available in order to allow the simulation of the complete system. This paper does not only show the relevant equations but also the realization within the widespread Matlab/Simulink
simulation system.

2.1 Servo Drive and Current Controller The major blocks of the servo drive simulation model are the current controller and the electric drive model as implemented in Matlab/Simulink
[4] (Fig. 2). Two input data are required: the current demand from the motion control system and the angular velocity of the motor in order to simulate the field suppression through self induction when the motor reaches its velocity limit expressed by the back EMF constant KEMF . The dynamic behavior of the servo motor itself is

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Fig. 2 Model of servo drive and current controller.

modeled by a DC motor model using the coil resistance R and the inductivity L:

with

R 1 dI = − · I + · (U − UEMF ) dt L L

(1)

UEMF = KEMF · ω

(2)

Using the normalized variables i = I/IR and u = U/UR the parameters KR =

UR IR · R

(3)

and

L (4) R reflect the PT1 -characteristic of the model. The parameters required for this simulation model can easily be gathered from supplier catalogues and data sheets without bothering about detailed coil design and the electromagnetic characteristics. This model is as simple as possible and as detailed as necessary. The model of the current controller is a simple analog model of a PI-controller described by the PT1 parameters K p and TI . For proper consideration of the limited maximum current of the current controller an anti windup loop is part of the model. In the development phase it is often impossible to gather sufficient data about the parameters of the current controller. The pole placement theory allows to estimate sensible values by setting up the system matrix which is the matrix representation of the servo drive and current controller model. The characteristic polynomial derived from the determinant of this system matrix can also be expressed by a desired resonance frequency Ω of the system assuming a damping ratio of one and placing the roots of the polynomial such that only one pole is present: TR =

λ2 +λ

(1 + KRK p ) K ! + R = (λ + Ω )2 = λ 2 + 2λ Ω + Ω 2 TR Ti · TR

(5)

A comparison of the polynomial coefficients on left and right side of the above equation leads to the following equations for the parameters of the current controller

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Fig. 3 Physical model of servo drive system with (a) screw and nut, (b) gear box and slider crank mechanism.

KR Ω 2 · TR

(6a)

2Ω · TR − 1 KR

(6b)

Ti = and KP =

2.2 Mechanical Motion System For the comparison of a servo drive system with and without linkage a linear reciprocating motion of a slider with given mass mslider is considered. 2.2.1 Servo Drive System with Screw and Nut The servo drive system with screw and nut can be modeled as a simple motion system with two rotary inertias. Both rotational inertias JM and Jred are connected by an elastic shaft as shown in Fig. 3a. Given the inertia Jscrew and the pitch h together with the slider mass mslider the reduced rotational inertia Jred can be determined as follows:
Jred = Jscrew + mslider ·

h 2·π

2 (7)

With the torsional stiffness cK and damping kK the following differential equation system can be formulated:   JM · ϕ¨ 1 = − cK · (ϕ1 − ϕ2 ) + kK · (ϕ˙ 1 − ϕ˙ 2 ) − Tmotor (8a)   Jred · ϕ¨ 2 = cK · (ϕ1 − ϕ2 ) + kK · (ϕ˙ 1 − ϕ˙ 2 ) + Tprocess (8b)

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The motor torque is given by the motor constant kmot and the normalized current: Tmotor = kmot ·i·IR . The process torque Tprocess is the reduced torque from the process force Fprocess that acts on the screw and nut combination: Tprocess = Fprocess ·

h 2·π

(9)

Reformulating the differential equation system for the relative co-ordinate ϕr = ϕ1 − ϕ2 leads to the following resonance frequency:   cK Jred + JM 2 ω0 = (10) Jred · JM As mentioned in Section 2.1 an estimate for the desired resonance frequency Ω of the servo drive and current controller system is required to determine sensible values for the current controller parameters K p and TI . This can now be based on the calculation of the resonance frequency according to Eq. (10) and the estimate Ω ≈ 3 · ω0 . 2.2.2 Servo Drive System Slider Crank Mechanism As an alternative to the servo drive system with screw and nut, a slider crank mechanism with a gear box will be considered. The physical model will be similar to the previous with two rotational inertias coupled by an elastic shaft. But in order to ease the modeling process the model shaft axis is the output axis of the gear box which simultaneously is the input shaft of the slider crank mechanism (Fig. 3b). The most ∗ (t) is time dependent. Thereimportant difference is that the rotational inertia Jred fore for the formulation of the differential equation not only the reduced torsional stiffness damping cK = i2G · cK and kK = i2G · kK have to be considered but also red red the differential equation of the linkage with variable inertia:   ∗ JM · ϕ¨ 1 = − cK · (ϕ1 − ϕ2 ) + kK · (ϕ˙ 1 − ϕ˙ 2 ) − i · Tmotor red red   ∗ ∗ ∗ (11) · ϕ¨ 2 + 1/2J˙red · ϕ˙ 2 = cK · (ϕ1 − ϕ2 ) + kK · (ϕ˙ 1 − ϕ˙ 2 ) + Tprocess Jred red

red

Since the co-ordinates ϕ 1 , ϕ 2 and above differential equation system are based on the output shaft of the gear box the new reduced inertia on the motor side must ∗ be calculated by JM = i2G · JM . Formulating equivalence regarding mass, rotational inertia and center of gravity the equivalent masses and rotational inertias can be calculated. Next formulating the equivalence of kinetic energy for the original and ∗ can be calculated: the equivalent system the time variant rotational inertia Jred ∗ = Jcrank + m21 l12 + J2 Jred




dψ d ϕ2

2


dxB 2 + (mslider + m23 ) d ϕ2

(12)

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The differentials in the above equation can be derived from the kinematics of the ∗ slider crank mechanism. Finally the reduced process torque Tprocess is given by ∗ = Fprocess Tprocess

dxB d ϕ2

(13)

2.3 Position Control The position control is the core element of the control system, feeding the power amplifiers and the current controller with the current demand signal. This current demand signal is generated by a PID-controller. Since the PID-controller also incorporates an integrator, a saturation with an anti-windup loop is part of the system.

3 Motion Profiles Most of the building blocks shown in Fig. 1 have been discussed apart from the motion profile generation. A motion task will be considered where the slider moves for a given distance at constant speed followed by a quick return motion. Both servo drive systems will be simulated with such a motion task which is quite common in process and packaging machines. Commercial programs for motion profile generation often only allow a limited number of different motion segments such as dwell-to-dwell, dwell-to-constant-velocity or constant-velocity-to-dwell. Therefore the motion task will be programmed such, that there will be a dwell in the lower and upper dead center positions with zero velocity but also zero acceleration. This leads to high accelerations because two acceleration peaks are required for each dead center position, one for deceleration prior to reaching the dead center position and one for acceleration when starting again from the dead center position. A more sophisticated approach is to program one single motion segment from the end of constant velocity back to the start of constant velocity using a fifth order polynomial x = a0 + a1t + a2t 2 + a3t 3 + a4t 4 + a5t 5 . Considering that the boundary conditions regarding zero acceleration at the beginning and end together with the synchronous speed vsync over the length of the constant speed area ∆ xsync and a given return time ∆ treturn must be properly fulfilled, the polynomial coefficients can be calculated easily from a linear equation system. Figure 4 shows a comparison between a return motion with dwell in the dead center positions and a direct return motion using the above described polynomial. Obviously the latter requires less absolute acceleration at the start and end of the return motion. The motion profile itself can be calculated a priori within Matlab
and be used within the Simulink
model using a look-up table. For the servo drive system with slider crank mechanism and a gear box the motion profile required for the slider motion cannot directly be fed into the position controller. Instead the re-

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Fig. 4 Comparison between return motion with double dwell and return motion without dwell. Table 1 Motor parameters of the servo motors. Motor parameters

Unit

Motor 1

Motor 2

Phase resistance Electr. inductance Motor constant Back EMF Peak current Max. l. to l. Voltage Rotor inertia

Ohm H Nm/A V/(rad/s) A V kg m2

0.82 11.0 × 10−3 0.644 0.360 33.0 250 0.25 × 10−3

0.34 8.4 × 10−3 0.664 0.448 55.9 250 0.65 × 10−3

quired crank position must be calculated from the desired slider position using well known kinematic analysis. Again a priori calculation with Matlab
and Simulink
look-up table.

4 Simulation Results and Conclusions Using the simulation models described in Section 2 together with the profile generation described in Section 3 simulations have been carried out for different configurations. Two different brushless DC motors have been simulated (see Table 1). The slider mass has been assumed as 8 kg and in case of the screw and nut combination the mass of the screw was estimated as 2 kg. Configuration 1 shows the minimum achievable cycle time of 0.95 s using motor 1 for a prescribed maximum positional deviation of 0.1 mm for the return motion with double dwell, a constant

154

B. Corves Table 2 Different simulation configurations. Configuration

Screw & Nut

Slider crank

Motor

Profile

Min. cycle time

1a 1b 1c 2a 2b 2c 3a 3b

h = 6 mm h = 6 mm h = 12 mm h = 6 mm h = 12 mm h = 12 mm -/-/-

-/-/-/-/-/-/i = 65 i = 40

1 2 1 1 2 1 1 2

trapezoidal trapezoidal trapezoidal polynomial polynomial polynomial polynomial polynomial

0.90 s 0.91 s 0.82 s 0.82 s 0.64 s 0.73 s 0.64 s 0.62 s

speed distance of ∆ xsync = 100 mm and a constant speed of vsync = 0.25 m/s. The slider crank dimension are 100 mm crank length and 200 mm rod length. A list of all configuration is shown in Table 2. Configuration 1a with motor 1 shows that the return motion with dwell plus screw and nut combination gives the slowest motion time. The edgy trapezoidal acceleration profile is bad for good control behavior. Using the stronger motor 2 with higher torque but also higher back EMF does not give any advantage as long as the same pitch is used (configuration 1b). Cycle time is reduced by about 10%, with a pitch of 12 mm (configuration 1c). In fact the simulation shows very well, that using a stronger motor with higher rotational inertia and higher back EMF always requires a new balancing of torque and maximum velocity, which must be done regarding the transmission ratio, which in case of the screw and nut combination is ruled by the pitch. Instead of using a stronger motor it is also possible to reduce the cycle time by avoiding the double dwell during return and instead using the fifth order polynomial described in the previous chapter. With configuration 2a this leads to exactly the same cycle time as configuration 1c. Of course the combination of the more intelligent motion profile with a stronger motor and an increased pitch of 12 mm as shown in configuration 2b further reduces the minimum cycle time to 0.64 s. But even with the smaller motor together with the better balanced pitch of 12 mm, see configuration 2c, the achievable minimum cycle time of 0.73 s is far better than the minimum cycle time which could be achieved with configuration 1c. Configuration 3a includes a gear box with a transmission ratio of i = 65. The complete rotational inertia of the gearbox plus mechanism crank relative to the crank axis was assumed at a value of 5 kg m2 . In fact with this combination it is possible to realize the same short motion time as configuration 2b even though only the smaller motor is installed. As configuration 3b reveals, even the installation of a larger motor and new balancing of torque and velocity as can be seen from the new transmission ratio i = 40 does not improve the situation. As a conclusion it can be stated that thorough modeling of each component is a prerequisite for the design of a servo drive system with or without a mechanism. Ongoing investigations are on the influence of A/D and D/A converters and discretisation effects of encoders.

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References 1. Braune, R., Koppelgetriebe mit Servo-Antrieb in schnellen Verarbeitungsmaschinen: Nutzungspotenziale und Konzipierungsaspekte. Fachtagung Verarbeitungsmaschinen und Verpackungstechnik VVD2006, Dresden, 2006. ¨ 2. Brosch, P.F., Automatisieren von Industrie-Anlagen mit Motion Control – Eine Ubersicht. Antriebstechnik, 1:48–53, 2003. 3. Corves, B., Mechatronic in container glass forming. In: Proceedings Mechatronic 2006, 4th IFAC-Symposium on Mechatronic Systems, Heidelberg, September 12–14, 2006. 4. Corves, B., Modelling of mechanisms in container glass forming machines. In Proceedings Conference on Interdisciplinary Applications of Kinematics, Lima, Peru, 9–11 January 2008. 5. Gr¨uninger, W., K¨ummerle, M., Anheyer, W. and Corves, B., Modelling, design and implementation of a servoelectric plunger mechanism for glass container forming machines. Glass Technology, 43(3):89–94, 2002.

Azimuth Tracking Linkage Influence on the Efficiency of a Low CPV System I. Visa, I. Hermenean, D. Diaconescu and A. Duta Department of Product Design for Sustainable Development, Transilvania University of Brasov, 500036 Bras¸ov, Romania; e-mail: {ionvisa, a.duta, ioana.hermenean, dvdiaconescu}@unitbv.ro

Abstract. The paper presents research on modeling the influence of an azimuth tracking linkage on the energetic response of a low CPV system build up of a PV module and two mirrors: one on the left side and symmetrically on the right side along the length of the PV module. This paper analyses the influence of the azimuth open tracking linkage, its tracking accuracy and the mirror-PV relative geometry on the tracking efficiency and on the size of a low CPV system with mirrors. This analysis has two main objectives: maximizing the tracking efficiency of the direct solar radiation and minimizing the overall size of the CPV system. The geometric modeling of this CPV system and the numerical simulations allow the identification of the optimal solutions. Key words: azimuth tracking linkage, low CPV (concentration photovoltaic) system, incidence angle, tracking efficiency

1 Introduction The maximum direct solar radiation that falls on a PV module is theoretically obtained when the solar rays are normal on the PV module surface [10]. This can be achieved by using mechanical orientation systems of the modules, tracking the path of the sun; these tracking systems are open tracking linkages with one or two mobility degrees [3]. Novel research is focusing on the mechanisms used for Concentrating PV systems (CPV) tracking [8], which require a high accuracy, implicit a complex functional structure. A significant advantage of a concentrated photovoltaic or hybrid (PV/Thermal) systems is the increase in the overall system’s efficiency, even beyond 30% [5], thus using less PV or hybrid surface. There are three main types of concentrating systems, with low, medium and high concentrating ratio [6]. One limitation of the medium- and high-concentration systems is the requirement for highly accurate tracking, able to insure the focus of the light on the solar cells as the sun moves throughout the day, with an accuracy of ≈ 1◦ [1]. The low concentrating PV systems does not need such a high tracking accuracy [8], therefore the complexity and the costs are lower. Our research focuses on a low concentration system; our goal is to establish the influences of the tracking

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linkage, the tracking accuracy and the CPV geometric parameters on the tracking efficiency (the ratio between the direct solar radiation that falls normal on the PV module surface and the available direct radiation). This work presents the geometric modeling followed by the numerical simulations and by the identification of the optimal working parameters in the case of the azimuth open tracking linkage, as compared with the pseudo-equatorial open tracking linkage [2], discussed in a previous paper [4]. The simulations were developed considering the geographical location of the Brasov–Romania area, using specific regional parameters, considering the ideal and the in-field conditions. The low CPV system design includes a PV module and two mirrors (see Fig. 1a), symmetrically disposed on the left and right side along the length of the PV module, as presented in the literature [9]. This paper reports on the pre-dimensioning of the mirrors, according to the PV/Hybrid dimensions when using the azimuth open tracking linkage. Through numerical simulations the parameters that influence the assembly functionality are identified, making optimization possible.

2 Geometric Modeling When optimizing the dimensions of a low CPV system, the target is to identify the lowest overall size of the system with a given PV module that gains the maximum direct radiation, thus increasing the efficiency of the system. Considering a given PV module, the parameters that influence the dimensions of the CPV system are, according to Figs. 1 and 2: the type of open tracking linkage, the predefined tracking accuracy (the maximum incidence angle, υ M , between the sunray and the normal of the PV module), the inclination angle between the PVmodule and the mirror (θ ), the ratio between the mirror width and PV module width (kw ), the longitudinal deflection of the reflected sun ray on the PV module (C E) and the corresponding ratio between the additional length of the mirror and PV module width (kl ); the factors kw and kl depend on the values of υ M and θ . The tracking program accuracy is designed (and indirectly, the maximum incidence angle υ M ); in the tracking program, a step consists of a rotation time (under 1 s) and a break, till the next rotation. In the simulations the following steps were considered: 15 min, 30 min, 1 h and 1 h 30 min (with the corresponding υ M values: 2.27◦, 3.58◦, 6.88◦ and 10.46◦ respectively). Unlike the sunray angles α and ψ (Fig. 1a) that vary continuously, the azimuth linkage angles (with discreet variations) are noted α ∗ and ψ ∗ . The longitudinal deflection, C E, of the reflected sunray is due to the tracking deviation of the elevation angle (∆ α = α ∗ − α ) as Fig. 2 presents. Based on Figs. 1b and 2, the sunray-mirror incidence angles υ O1,2 were formulated as:

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Fig. 1 (a) The azimuth sunray angles (α , ψ ) and the corresponding tracking angles (α ∗ , ψ ∗ ); (b) scheme of the azimuth open tracking linkage used in a low CPV system with mirrors.

cos υo 1 = sin θ · cos α · sin(ψ − ψ ∗) + cos θ · (cos α ∗ · cos α · cos(ψ − ψ ∗) + sin α ∗ · sin α )

(1)

cos υo 2 = − sin θ · cos α · sin(ψ − ψ ∗) + cos θ · (cos α ∗ · cos α · cos(ψ − ψ ∗) + sin α ∗ · sin α )

(1’)

In the calculations we considered that the sun ray falls on an extreme point of the mirror A and is than reflected on the PV surface, reaching point E. If all the direct reflected radiation has to fall on the surface of the PV module, then the size of the mirror must be increased (see Fig. 2). The increase in the mirror size can be calculated using the longitudinal deviation C E and the ratio kl : C E = CD + E E,

(2)

CD = AC · tan ε

(2’)

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Fig. 2 Geometrical schemes of a low CPV system and the longitudinal deflections E1C 1 and E2C 2 of the reflected sunray (a) from the left mirror and (b) from the right mirror, respectively.

tan ε =

− sin α ∗ · cos α · cos(ψ − ψ ∗) + cos α ∗ · sin α cos θ · cos α · sin(ψ − ψ ∗) + sin θ · (cos α ∗ · cos α · cos(ψ − ψ ∗) + sin α ∗ · sin α )

tan χ = tan ε · cos θ 1 tan θ · cos ε The factors kl and kW can be defined as (see Fig. 2): tan µ =

(5)

2 ·C E CC

(6)

AC cos ε · sin(υo − µ ) = CC cos υo · cos χ

(6’)

kl = kw =

(3) (4)

and represents the ratios between the supplementary length of the mirror (related to the PV-module length) and the width of the PV-module, respectively between the width of the mirror and the width of the PV-module. Based on these relations, the curve families presented in Fig. 3 are developed for the azimuth open tracking linkage (continuous lines) compared with the pseudoequatorial tracking linkage (dashed lines), [4]. While the variations of the factor kw are the same for the both linkage types (Fig. 3a) the variations of the kl factor has

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Fig. 3 The variations of the deflection factors: (a) width, (b) longitudinal.

lower values for the azimuth linkage (Fig. 3b). The results also show that while the width of the mirror increases due to the θ angle (Fig. 3a), the additional length of the mirror decreases (Fig. 3b); finally, both factors increase when the step duration is longer (Figs. 3a, 3b).

3 Numerical Simulations Numerical simulations were done to evaluate the adequate tracking accuracy and the optimal relative geometry of a low CPV system, tracked by an azimuth open linkage. In these simulations, the following values where considered for the stepdurations and for the θ angles: 15 min, 30 min, 60 min, 90 min and respectively θ = 50◦ , 55◦ , 60◦ , 65◦ . The simulations were developed considering the Brasov– Romania location (with the latitude ϕ = 45.6◦N and the turbidity factor [7], Tr ≈ 3) during three significant days in the year: summer solstice, fall or spring equinox (equivalent in terms of solar radiation geometry) and Winter Solstice, using an azimuth open tracking linkage; all the calculations are done considering the clear sky as prerequisite. The variations of the sun angles (α , ψ ) and the PV angles (α ∗ , ψ ∗ ) are presented in Fig. 4; the sunray angles represented have a continuous variation, while the PV angles have a stepwise variation for the extreme values of the step durations: 15 min (with large number of steps) and 90 min (with the lowest number of steps considered). Using these angles, the variations of the incidence angles are as given in Fig. 5. In Fig. 5a there are presented the variations of the incidence angles of the sunrays that fall on the PV- surface (i.e. the angles of the direct sunray and of the reflected rays from the left mirror, M1 and from the right mirror, M2) with the normal to the PV module, considering the extreme step durations: 15 min (till noon) and 90min (after noon). The results presented in Fig. 5b show the variations of the maximum incidence angles with the value of the θ angle and outline that to large values of the maximum incidence angles corresponds small values of the angle θ . Using the in-

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(b)

(c) Fig. 4 The variations of the solar ray and PV angles for azimuth tracking (a) summer solstice, (b) fall equinox and (c) winter solstice.

(a)

(b)

Fig. 5 (a) The variation of the incidence angle for the azimuth tracking during the Summer Solstice at different θ values and the extreme values of the step duration: 15 min (till noon) and 90 min (after noon); (b) The maximum values of the incidence angles related to the θ angle.

cidence angles, the partial direct and reflected radiation and the total direct radiation received on the PV module where evaluated. The tracking efficiency was further calculated for each case. Part of the results are presented in Fig. 6, where the variations of the total direct radiation received normal on the PV module, without (Bpv) and with mirrors (BpvM) are represented at different values of the θ angle and for the extreme val-

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Fig. 6 Total direct solar radiation variations that fall on the PV-module, at different values of angle θ and for the extreme values of the step-duration, during the summer solstice (compared to the tracking efficiency of a PV system with the pseudo-equatorial tracking linkage).

ues of the steps (15 min till noon and 90 min after noon); two values are written above each curve, representing the tracking efficiency of the PV system tracked by the azimuth open linkage (first value) and of the PV system tracked by the pseudoequatorial open linkage (second value, see [4]). According to Fig. 6, the highest tracking efficiencies are reached at high θ values. It can be observed that by using pseudo-equatorial tracking the efficiency is slightly higher comparing to the azimuth tracking system; the drawback of the pseudo-equatorial tracking system is its overall size.

4 Conclusions (a) The parameters that influence the tracking efficiency and the overall size of this CPV system are: the azimuth sunray angles, the tracking program accuracy (υ M ) and the inclination angle between the mirror and PV module (θ ); the overall size of the CPV system is described by the width deflection ratio kw and the length deflection ratio kl . (b) The simulation was done considering a system located in Brasov, Romania. The results prove that, for low overall sizes, the tracking program must be accurate (under 30 min) and the value of the θ angle must be small (under 55◦ ). Continuous tracking can be avoided by using step intervals between 15 and 20 minutes. (c) The overall size described by the width factor kw increases with the angles θ and υ M , while the overall size described by the length factor kl decreases with θ angle and increases with incidence angle υ M , i.e. with tracking inaccuracy. (d) The tracking efficiency, of a CPV system fitted with azimuth open tracking linkage, increases with the θ angle, while the tracking accuracy has negligible influence. Compared to tracking efficiency of a CPV system fitted with the pseudoequatorial tracking linkage, the azimuth one is 1% lower, but the overall size of

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the system is significantly smaller, which explains the use of such a system on the concentrating photovoltaic market. (e) Based on these results, an optimized solution is the use of the azimuth tracking linkage, for maintaining the θ values between 55◦ and 60◦ , and the maximum sunray-PV incidence angle under 4◦ (i.e. the step duration lower than 30 min). Another optimized version can be obtained by combining a low accurate tracking with a fine and discreet adjustment of the mirror’s angle θ . (f) Further research will focus on the tracking efficiency, using other types of open tracking linkages and on the different positioning of the reflective components related to the PV module.

Acknowledgment This paper is supported by the CNMP Programme, PNII-Parterneriate, number 21003/2007.

References 1. Bett, A.W. and Lerchenm¨uller, H., The FLATCON System from Concentrix Solar. Concentrator Photovoltaics. Springer Series in Optical Science, Springer, 2007. 2. Burduhos, B., et al., Comparison between single-axis PV trackers of pseudo-equatorial and azimuthal type. Bulletin of the Transilvania University of Bras¸ov, Romania, 81–86, 2008. 3. Diaconescu D., et al., Synthesis of a bi-axial tracking spatial linkage with a single actuator. In Proceedings of SYROM 2009, pp. 632–617, Springer, 2009. 4. Hermenean, I.S., et al., Modelling and optimization of a concentrated hybrid photovoltaic system with pseudo-equatorial tracking. Accepted for the EVER Conference, Monaco, 2010. 5. Klotz, F.H., European photovoltaic V-trough concentrator system with gravitational tracking. In Proceedings of the 16th European PV Solar Energy Conference, Glasgow, UK, p. 2229, 2000. 6. Luque, A.L. and Andreev, V.M., Concentrator Photovoltaics. Springer, Berlin, 2007. 7. Meliss, M., Regenerative Energie-quellen Praktikum, Springer, Berlin, pp. 5–7, 1997. 8. Palmer, T., et al., Tracking systems for CPV: Challenges and opportunities, Solar Power International, 2007. 9. Reis, F., et al., Power generation and energy yield using doublesun photovoltaic solar concentration. In 24th European Photovoltaic Solar Energy Conference, Hamburg, Germany, pp. 803–806, 2009. 10. Visa, I., et al., On the incidence angle optimization of the dual-axis solar tracker. On Proceedings 11th International Research/Expert Conference TMT, Hamammet, Tunisia, pp. 1111– 1114, 2007.

Multi-Objective Optimization of a Symmetric Sch¨onflies Motion Generator B. Sandru, C. Pinto, O. Altuzarra and A. Hern´andez Mechanical Engineering Department, University of the Basque Country UPV/EHU, Bilbao, Spain; e-mail: {sandru.bogdan, charles.pinto, oscar.altuzarra, a.hernandez}@ehu.es

Abstract. The multiobjective optimal design of a four-degree-of-freedom symmetric parallel manipulator is the subject of this paper. The design has rotary actuators and generates SCARA motion. Kinematic functions are implemented and used in a multiobjective optimization. First, the manipulator description is made, and kinematic problems are solved. Then the Circular Trajectory pattern and Singularity-free-volume concepts is implemented and an operational dexterous workspace along with his volume is defined. Similarly, the manipulator’s kinematic dexterity based on the Frobenius norm is found. Finally, the kinematic results are proposed as objective functions in a Design Parameters Space approach. Key words: parallel manipulators, dexterity, workspace, multiobjective optimization

1 Introduction Parallel manipulators with fewer than 6 degree of freedom (dof), are said to be with lower mobility [6]. An increasing number of researchers have been involved in the study of such robots [5]. In industrial applications where operations such as pickand-place are essential, 3 or 4-dof parallel manipulators are easier to implement. In fact, the ones which best correspond with this type of operation are those with Sch¨onflies or SCARA motion [3,4,10], i.e., three translations and one rotation about a normal axis to the moving platform. Optimum design of such parallel manipulators is a great concern for many researchers. The majority is focused on trying to find performance indices in order to optimize the design, i.e., workspace volume, dexterity, isotropy [7, 8, 12]. In order to optimize various functions at the same time, the optimal design problem becomes a mixed multiobjective optimization problem. Often, no single solution can be achieved. Instead, a set of solutions in the design parameters space are obtained that contains the best compromise. In this paper, the optimization counts with kinematic criteria of volume and dexterity workspace. First, the manipulator and its kinematics is briefly described. Next, the singularity free volume within the workspace is computed according to various circular trajectories, which is the first objective function to optimize. Then, the

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Fig. 1 Symmetric Sch¨onflies motion generator with rotational inputs.

global dexterity analysis, using the weighted local index, is obtained which is also subject of the optimization. Then, the two objective functions are scaled and used in a global optimization process. Finally, the results are obtained as a set of Paretooptimum solutions of the design parameters.

2 Manipulator’s Description The robot under study is supplied with four identical limbs of the RRΠ RR type and has a symmetry plane π at x = −y, as shown in Figure 1(a). Each limb has five joints: R, the actuated revolute joint, of axis of rotation Ai and passing through Ai parallel either to j for i = 1, 3 or to i for i = 2, 4, where {i, j, k} form an orthonormal triplet fixed to the base platform (BP)R, of axis Bi parallel to Ai , as in Figure 1(b), and passing through Bi ; Π , a parallelogram joint of plane of normal ni and allowing for relative translations around circles of radius BiCi ; R, of axis Ci parallel to Bi and passing through Ci ; and R, of axis Di passing through Di and parallel to k. Thus, the intersection of all subsets of limb-displacements is a subgroup called Sch¨onflies subgroup X (k) [1] and, hence, the manipulator has 4-dof, i.e., 3 translations and 1 rotation upon a normal axis with mobile platform. The position of the operation point (OP) P can be expressed for limb i as p = ai + (bi − ai) + (ci − bi ) + (di − ci ) + (p − di), i = 1, . . . 4

(1)

where the position vector of a given point Xi is denoted by the corresponding subscripted lower-case boldface vector, xi . Moreover, the position vectors of points Ai are given in the frame (O, i, j, k) that is fixed to the BP.

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Vectors bi − ai are expressed in terms of the actuated-joint variables qi and the length b corresponds to the limbs Ai Bi . Further, vectors di − ci are constrained to move under spatial translations and, thus, are constant throughout the robot workspace. Moreover, vectors pi − di are expressed in a local frame M (Puvw). The transformation of the two foregoing vectors triplets is given by the matrix R which rotates frame M into B, being θ the MP orientation angle with respect to the BP. Vectors ci − bi must satisfy the constraints imposed by rigid links BiCi


c − b
2 = r2 i i

i = 1, . . . 4

(2)

being r the common length BiCi . the values of the actuated-joints qi are found upon Substituting the Eq. (1) into Eq. (2) and making use of the trigonometric identity  q  ti = tan 2i , the actuated-joints variable are obtained in terms of ti    4bz ± (4bz)2 − 4 Ti2 − 4b2 (δi − ∆i )2 ti = where



δi = and being

2 [Ti + 2kb (δi − ∆i )]

x, i = 1, 3 , y, i = 2, 4



∆i =

Xi , i = 1, 3 , Yi , i = 2, 4

,

i = 1, . . . , 4

 k=

1, i = 1, 2 −1, i = 3, 4

  Ti = (x − Xi)2 + (y − Yi )2 + z2 + b2 − r2 Xi = k (−d cos θz − c + h) Yi = k (−d sin θz + a) , i = 1, 3

Xi = k (d sin θz + a) Yi = k (−d cos θz − c + h) i = 2, 4

(3)

(4)

(5)

(6)

Then it is possible to obtain the actuated-joints as qi = 2 arctanti

(7)

The existence of the 24 different solutions of q, depends on the square root in Eq. (3). Differentiating Eq. (1) with respect to time, and canceling the passive-joint-rate terms, the velocity equation is obtained as Jx x˙ = Jq q˙

(8)

˙ T is the four-dimensional MP twist, q˙ = [q˙i ], for i = 1, . . . , 4, where x˙ = [ωz p] is the actuated-joint-rate velocity vector and the Jacobian matrices Jx and Jq are defined as

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⎡ T T⎤ s1 k r1 ⎢sT k rT ⎥ 2 2⎥ Jx = ⎢ ⎣sT3 k rT3 ⎦ , sT4 k rT4

⎡

−nT1 j ⎢ 0 Jq = ⎢ ⎣ 0 0

0 nT2 i 0 0

⎤

0 0 0 0 ⎥ ⎥ T n3 j 0 ⎦ 0 −nT4 i

(9)

where si =(di − p) × ri ni =(bi − ai ) × ri 1 ri = (ci − bi ), i = 1, . . . 4 r

(10)

3 Circular Trajectories Based Criteria In the evaluation of the pick-and-place manipulator’s workspace, the idea of an operational workspace is used to define better the trajectories that the robot has to complete during the tasks. Some researchers have used the so-called Adept cycle for pick-and-place manipulators. However, this type of trajectory is specific to a reduced part of the operational workspace and do not evaluate the global manipulator’s performance. In our case, several circular trajectories Ck centered in z axis have been computed in the plane xy for various values of z. Hence, the operational workspace V have been defined as    (∃) Ck (z) = (x(z), y(z)) ∈ ℜ2 : x2 (z) + y2 (z) = ri (z) such that: V: (11) qT ∈ ℜ, qT ∈ W , ∀i = 0, . . . , m, ∀k = 1, . . . , l points from the trajectory Ck (z) with ri varying to the last m for which all the discrete  verifies the inverse position problem qT ∈ ℜ ; l represents the maximum number of trajectories within V and being W the workspace constraints defined as  qmin ≤ q ≤ qmax W: (12) αmin ≤ α ≤ αmax with the above specified parameters representing the minimal and maximal values of the active-input, passive torque and MP orientation angle, respectively, as shown in Figure 1(b). Thus, we evaluate which of the mentioned trajectories are completely included in the workspace. On the other hand, the geometrical parameters h, a, c, d, b, r shown in Figure 1(b) are taken into account for the optimal design. However, r acts as a scale factor and, thus, the following design parameters are used in the optimal design of the manipulator a b d n A= , B= , D= , N= r r r r being n = h − c.

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Fig. 2 Complete and SFV workspace (A = 0.125, B = 0.4, D = 0.07, N = 0.173, θ = π6 ).

(a)

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Fig. 3 Volume function for A = 0.133 and N = −0.5.

3.1 Singularity Free Volume In order to avoid the singularities, the sign of the Jacobian matrix in each point of each trajectory have been computed. Hence, we have chosen only those trajectories whose points do not change the Jacobian sign, forming in this way a so-called singularity-free volume (SFV) workspace. In this way, the robot can operate without the risk to lose the control. The complete workspace and the SFV dexterous workspace for a specific set of design parameters is presented in Figure 2. In Figure 3 can be observed the convergence of the maximum volume to the superior limit of B and to the inferior one of D. Moreover, the volume is maximum for a low value of A. In Figure 3(b) a sliced view in the BD plane, is presented for a value of N corresponding to the maximum volume.

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3.2 Dexterity The dexterity of the manipulators is often connected with their kinematic singularities. The direct singularities pertaining to Jx are related for the robot at hand. However, Jx contain mixed translation and rotation terms and, thus, we dimensionally homogenized the direct Jacobian [11] using the characteristic length [2]. Thus, the condition number κF , based on weighted Frobenius norm, is defined as  


 


1 T
−1




tr Jx Jx (13) κF Jx = Jx F
Jx
, Jx F = F 4 where Jx is the Jacobian Jx with a characteristic length introduced to make terms homogeneous, and being Jx the weighted Frobenius norm. However, this performance index is a local one. As the discrete points number,p, per each circle is constant, the local dexterity κF has more influence as the circle radius is smaller. Then, as the workspace is orientation angle dependent, a global index with a mean value of κF is defined as  n  k k p 1 κF n ( θ p ) ∑ rk ∑ κ θ ∑ k  i i=1 j=1 F nk i=1 θ , θi = εg = , κ = θmax − θmin (14) F nk k p p ∑ nk rk j=1

where rk represents the radius whose value is k, nk is the number of points pertaining on the circular trajectory of radius rk , κF θn is the θ dependent local dexterity, k associated with the points pertaining on a specific radius rk and being θmax , θmin the variation limits of the orientation angle in each discrete point p. As one can see from Figure 4, the maximum dexterity occurs for values of D closer to the maximum and for values of c and h closer to each other, with a slowly bigger value of c. A sliced view in BD plane, is presented in Figure 4(b), for a value of N corresponding to εg max .

4 Multiobjective Optimization Design In the design optimization using several objective functions, we have used the design parameters space approach [9]. In this way, the objective functions must fulfill some input requirements Ck . Then, the intersection I of the regions that meet the specified conditions, is related as the set of all possible designs that accomplish with all Ck . Thus, the designer acts as a decision-maker, choosing one set of the design parameters from I as convenience. In Figure 5 have been kept only the parts that meet the requirements Ck , as shown for the volume and dexterity functions in Figure 5(a) and Figure 5(b). Then,

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Fig. 4 εg function for A = 0.133 and N = −0.166.

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(c)

Fig. 5 Design parameters space with Ck ≥ 0.45.

a volume is obtained from the intersection of the above mentioned functions as is shown in Figure 5(c).

5 Conclusions In this paper a 4-dof parallel manipulator has been proposed for the multiobjective optimization design with two objective functions. The design parameters number have been reduced in order to obtain a meaningful graphical representation. Maximal values for the objective functions, i.e., volume and dexterity of the SFV workspace, have been computed and represented in the design space formed by the parameters B, D, N for various A. These objective functions are conflicted and, thus, no single solution can be achieved in the multiobjective optimization. Hence, the Design Parameters Space approach has been used, the result being represented as a volume from which the designer is free to choose upon considerations.

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Acknowledgements The authors wish to acknowledge the financial support received from the Spanish Government via the Ministerio de Educaci´on y Ciencia (Project DPI2008-00159), the FEDER funds of the European Union and the Departamento de Educaci´on, Universidades e Investigaci´on of the Basque Government (Project GIC07/78.IT-186-07).

References 1. Altuzarra, O., Hern´andez, A., Salgado, O., and Angeles, J., Multiobjective optimum design of a symmetric parallel Sch¨onflies-motion generator. Journal of Mechanical Design, 131(3):1– 11, 2009. 2. Angeles, J. and L´opez-Caj´un, C.S., Kinematic isotropy and the conditioning index of serial robotic manipulators. The International Journal of Robotics Research, 11(6):560–571, 1992. 3. Angeles, J., Caro, S., Khan, W. and Morozov, A., Kinetostatic design of an innovative Sch¨onflies-motion generator. Proceedings of the IMECHE Part C, Journal of Mechanical Engineering Science, 220(7):935–943, 2006. 4. Company, O., Marquet, F. and Pierrot, F., A new high-speed 4-dof parallel robot synthesis and modelling issues. IEEE Transactions on Robotics and Automation, 19(3):411–420, 2003. 5. Fattah, A. and Kasai, G., Kinematics and dynamics of a parallel manipulator with a new architecture. Robotica, 18:535–543, 2000. 6. Huang, Z. and Li, Q., General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. The International Journal of Robotics Research, 21(2):131–145, 2002. 7. Kosinska, A., Galicki, M. and Kedzior, K., Designing and optimization of parameters of Delta-4 parallel manipulator for a given workspace. Journal of Robotic Systems, 20(9):539– 548, 2003. 8. Lou, Y., Liu, G. and Li, Z., Randomized optimal design of parallel manipulators. IEEE Transactions on Automation Science and Engineering, 5(2), 2008. 9. Merlet, J.-P., Parallel Robots, Springer, 2006. 10. Salgado, O., Altuzarra, O., Amezua, E. and Hern´andez, A., A parallelogram-based parallel manipulator for Sch¨onflies motion. ASME Journal of Mechanical Design, 129(12):1243– 1250, 2007. 11. Schwartz, E., Manseur, R. and Doty, K., Noncomensurate systems in robotics. International Journal in Robotics and Automation, 17(2):1–6, 2002. 12. Stamper, R.E., Tsai, L.W. and Walsh, G.C., Optimization of the three dof translational platform for well-conditioned workspace. In Proceedings of the 1997 IEEE International Conference on Robotics and Automation, pp. 3250–3255, 1997.

Cyclic Test of Textile-Reinforced Composites in Compliant Hinge Mechanisms N. Modler1 , K.-H. Modler1, W. Hufenbach1, M. Gude1 , J. Jaschinski1 , M. Zichner1, E.-C. Lovasz2, D. M˘argineanu2 and D. Perju2 1 Institut

f¨ur Leichtbau und Kunststofftechnik, Technische Universit¨at Dresden, 01062 Dresden, Germany; e-mail: [email protected] 2 Mechanical Engineering Faculty, Politehnica University of Timis¸oara, 300222 Timis¸oara, Romania; e-mail: [email protected]

Abstract. The use of so-called compliant elements with specifically adjustable compliances offers the possibility to transmit motions just by structural deformations. This paper makes a contribution to the efficient cyclic test of textile-reinforced compliant structures by developing a novel kinematic test stand realizing pure bending. Key words: compliant hinge mechanisms, textile-reinforced composites, cyclic test

1 Introduction For the experimental determination of the deformation behavior of strip shaped compliant mechanisms links, and for the analysis of the anisotropy induced coupler effects in multi-layered fiber composites, pure bending can be used. Therefore a kinematic test stand was developed, which can be used for static and dynamic bending tests. The centerpiece is a six bar mechanism that injects a pure moment or torque, devoid of any shear force. By means of this multiply actuated linkage the specimen loading under pure bending is guaranteed. The emphasis in the mechanism synthesis was on the adaptation of a suitable mechanism to the trajectory of the moving clamping point. A negligible difference between the exact mathematical description and the mechanically possible trajectory ensures a nearly pure (shear-force-free) bending moment installation. In the bending test of thin plate structures, plate deflections will be realized onto the range of the yield strength of the coating material, which necessarily requires a proper tracking of the rotation points. These result, due to the constant plate bending moments introduced, close to the ideal case, in the form of a circular arc. The aim of the special testing device development considered here is therefore to realize the most possible accurate reproduction of the circular trajectory while simultaneously tracking the force input.

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1,u

MB

h wend

3,w

Fig. 1 The model for bending deformations of a mechanism’s link under moment load. Glass Fibre/Polypropylene (GF/PP) - Knitted Fabric GF/PP - Woven Fabric GF/PP - unidirectional GF/PP - Short Fibre PP GF/PA6 - Short Fibre PA6 Carbon Fibre/Epoxy (CF/EP) - unidirectional Aramid Fibre/Epoxy (AF/EP) - unidirectional GF/EP - unidirectional EP Aluminium Steel Titanium 0

10

20

30

40

50

LM

Fig. 2 A comparison of the deformability of some materials chosen as examples.

2 Selection of Materials A key objective in the design of more compliant hinge mechanisms is to obtain the most compact size. Moreover, the elastic potential of the hinge’s material can be advantageously used to achieve minimal energy consumption when opening or closing. Suitable materials must allow high, nonlinear elastic deformation of the compliant hinge members and endure the resulting high strains without injury. Therefore it is useful to define appropriate performance indices to aid in the selection of materials, while here a slim cantilever is used as a representative link and will be adopted to facilitate small deformations (see Figure 1). The maximal displacement at cantilever’s end: wmax end =

R
2 E1 h

(1)

is in this case obtained, when the deformability (ratio R/E1 of the tensile strength to the axial Young’s modulus) is high. Figure 2 shows a comparison between the values of this index LM = 1000 R/E1 for lightweight mechanisms construction for various materials. The graph shows that, due to their high stiffness, the usefulness of metals as compliant hinge element is rather limited. In comparison non-reinforced polymers such as EP or PP-systems can sustain much greater strain. The thermoplastics seem to be the best choice for compliant links. A further increase of the deformability can be achieved through textile reinforcements, while woven or knitted fabrics provide higher values as unidirectional reinforcements.

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Fig. 3 Variation of yarn properties and architecture for the design of various compliances.

The textile reinforcement allows not only the classical geometric design parameters but also some textile-specific parameters such as bond type and thread architecture for the targeted adjustment of the compliant joints. In addition, the textile technical manufacturing processes produces these additional design options directly on the part without further processing steps (Figure 3). By the choice of reinforcing fibers for the hinge system elements with higher deformations (coupler and driven element) glass fibers are preferable to the carbon fibers as glass fibers have relatively high strength and low stiffness.

3 Experimental Analysis of Pure Bending (without Shear Force) For the experimental determination of the deformation behavior of stripe shaped compliant mechanisms links, and for the analysis of the anisotropy induced coupler effects in multi-layered fiber composites, the pure bending can be used. Therefore a kinematic test stand was developed, which can be used for static and dynamic bending tests. The centerpiece is a six bar mechanism that injects a pure moment or torque, devoid of any shear force. By means of this multiply actuated linkage the specimen loading under pure bending is guaranteed. The emphasis in the mechanism synthesis was on the adaptation of a suitable mechanism to the trajectory of the moving clamping point. A negligible difference between the exact mathematical description and the mechanically possible trajectory ensures a nearly pure (shearforce-free) bending moment installation.

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Fig. 4 Schematic of the projected bending test.

3.1 Mechanisms Technical Task for the Realization a Testing Device 3.1.1 Obtaining the Bending Angle With l as plate length, θ as bend angle and a constant curvature radius r over the plate length for pure bending follows from Figure 4(a) the relationships l(θ ) = l

sin θ l(θ ) l , r(θ ) = = θ 2 sin θ 2θ

(2)

and for the moving clamping points B, the trajectory in parametric form: xb (θ ) = l(θ ) cos θ , yb (θ ) = − l(θ ) sin θ .

(3)

◦ For a plate length l = 300 mm and θmax = 50◦ the trajectory k reaches in the interval 0 ≤ θ ≤ θmax maximum 0.4 % deviation with respect to a circular trajectory, i.e. the point B moves on a circular arc with an adequate approximation. In Figure 4(b), the geometrical relationships are represented. From these, it is found:

b0x =

l 2 − xb (θmax )2 − yb(θmax )2 , 2(l − xb (θmax )

rb = l − b0x , 
sin θmax . ψ0 = arcsin l(θmax ) rb

(4) (5) (6)

With a crank rocker mechanism A0 ABB0 that realizes an oscillation angle ψ0 , a bend angle θmax can be generated. A centric crank rocker, as ϕ0 = π is an inexpensive ◦ = 50◦ and the length l = r solution. With a minimum transmission angle µmin 4 b

Cyclic Test of Textile-Reinforced Compliant Links y

E b 0x

bent tangent test plate t (0) B(0)

B0

θ l(θ)

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t (θ)

ψ

0

x

l4 tracking trajectory

γ

θ

B(θ)

Fig. 5 Angular relationships for the bending moment input.

of the driven member follows the other links lengths l1 , l2 and l3 as well as the angle ϕ1 and ψ1 for the initial position (see [3]). Figure 6(b) shows the crank rocker mechanism in the outer dead-point positions.

3.1.2 Bending Moment Input For a pure (shear force free) bending moment imposing it is necessary that the bending moment Mb is held at any time in the same direction in the testing structure. According to Figure 5 for the movable clamping of the test plate at point B, the angle γ is obtained as being. The relationship follows 
b (7) γ (θ ) = θ − arcsin 0x sin θ . l4 The function (7) can be considered as linear and has a maximum deviation of 0.27% at θmax . The function (7) can be done with sufficient approximation by a Assur group C0CB (Figure 6), where C must be placed always on the tangent t(θ ) in Figure 5. It is enough to consider the start and end position, therefore, θ = 0 and θ = θmax . Thus, an angular relation occurs: 
b (8) γ (0) = 0 and γ (θmax ) = θmax − arcsin 0x sin θmax . l4 The link length l5 = BC will be freely chosen. Thus follow C(0) und C(θmax ) from the tangent t(0) and t(θmax ). The frame point C0 must lay on the bisector of C(0)C(θmax ). It exist 2 free parameters, l5 and l6 = C0C, that are available for finding a favorable solution.

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E A0 B0 C0

Point coordinates x

y

0 394.69 mm 78.98 mm 108.49 mm

0 146.13 mm 0 –74.73 mm

Link length

Angular dimensions

l2 = 120.193 mm l3 = 294.323mm l4 = 221.022 mm l5 = 145.580 mm l6 = 86.640 mm

0◦ ≤ θ ◦ ≤ 50◦ 0◦ ≤ γ ◦ ≤ 5◦ ψ0◦ = 65.89◦ ϕ1◦ = 32.22◦ ψ1◦ = 24.84◦

Fig. 6 Test stand mechanism design. Belt drive Tracking links Frame Torsional wave with strain gauges

Driving links

Base plate

Fig. 7 Kinematic testing stand for determining the flexural properties.

3.1.3 Kinematik Test Stand An overview of the most important dimensions is presented in Table 1. The design layout of the test stand mechanism is shown in the initial and final position in Figure 6. Figure 7 illustrates the final design. The mechanism is driven by a stepper motor with specially adapted control technology.

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Fig. 8 Surface topology of a compliant textile reinforced mechanism link before and after load, and moment characteristics of an exemplary cyclic bending test.

3.1.4 Elementary Cyclic Tests The forces and moments resulting from the investigation of textile-reinforced compliant links are very low compared with the test rig’s strength. The aim conflict emerging from this between maximizing the specimen clamping rigidity and the largest possible angle of rotation in the field of measuring areas leads to increased demands on measuring technology. In order to meet these measuring challenges, custom-made torsion bars are used, which also assume the function of the specimenclamping and torque measurement. Using highly sensitive strain gauges the occurring moments and forces can in this way be reliably detected. The totality of the collected measurements allows one to draw conclusions on the bending stiffness. Figure 8 shows the results of an exemplary endurance test carried out at 0.55 Hz. At approximately 3 million load cycles there has been no significant drop of the resultant bending moment (Figure 8(b)). Hereby the advantages to use glass fiber fabric reinforced polypropylene for hinges in compliant mechanisms can be confirmed. The texture relief developed after about 2 million load cycles (Figure 8(c)) is due to the creep of the thermoplastic matrix and requires closer investigation in further works.

4 Conclusions In the frame of the research, a new concept for shear-force-free bending test rig was developed and the first cyclic test on the selected compliant mechanisms links was successfully carried out, in which the certification of a fatigue-free operation by using textile reinforced GF-PP up to about 3 million load cycles is provided. The present work deals with the particular design and implementation of an appropriate kinematic, which realizes the before calculated trajectory of the clamping point in order to impose a pure bending moment. In this way, the input of nearly shear-force-

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free bending moments can be assured, without giving away the rotary drive, which is very advantageous for endurance tests.

Acknowledgement The authors would like to express their gratitude towards the Deutsche Forschungsgemeinschaft (DFG), which supports this research within the scope of the subproject D2 of the Collaborative Research Centre SFB 639 “Textile-Reinforced Composite Components in Function-Integrating Multi-Material Design for Complex Lightweight Applications”.

References 1. Hufenbach, W. and Gude, M., Analysis and optimisation of multi-stable composites under residual stresses. Composite Structures, 55:319-327, 2002. 2. Hufenbach, W., Jaschinski, J., Weber, T. and Weck, D., Numerical and experimental investigations on HYLITE sandwich sheets as an alternative sheet metal. In Archives of Civil and Mechanical Engineering, No. 2, pp. 67–80, Vol. VIII, Wroclaw, 2008. 3. Luck, K. and Modler K.-H., Getriebetechnik - Analyse, Synthese, Optimierung, AkademieVerlag, Berlin, 1990. 4. Modler, N., Nachgiebigkeitsmechanismen aus Textilverbunden mit integrierten aktorischen Elementen. Dissertation TU Dresden, 2008.

The Optimization of a Bi-Axial Adjustable Mono-Actuator PV Tracking Spatial Linkage D. Diaconescu, I. Visa, M. Vatasescu, R. Saulescu and B. Burduhos Department of Product Design for Sustainable Development, Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: {dvdiaconescu, visaion, maria.vatasescu, rsaulescu, bogdan.burduhos}@unitbv.ro

Abstract. Continuing a previous research on the optimization of a bi-axial mono-mobile azimuth tracking linkage, in which the elevation blocking effect was avoided, the present paper presents two new working angles, along with their particular restrictions, imposed to avoid the azimuth blocking effect. The synthesis algorithm, proposed to obtain the optimum dimensions for the considered tracking linkage is developed for to the specific geographic and climatic conditions in Brasov, Romania; the design principles of this algorithm can be adapted to any location on the Earth’s surface. Key words: bi-axial mono-mobile tracking system, spatial linkage, azimuth PV tracking system, direct tracking efficiency, functional angles

1 Introduction Following the need of increasing and improving the use of green energy devices [1, 4, 5], the paper continues the studies previously presented [2, 6] on the kinematical optimization of a biaxial azimuth tracking linkage with a single actuator and one adjustment, Figs. 1, 2a. The initial approach imposes a single working restriction regarding the limitation of β 1 elevation pressure angle (Fig. 2c) between the swing arm and the PV platform; further research highlighted the necessity to introduce other two working restrictions regarding the limitation of the β 2 and β 3 working angles (Fig. 3). Based on these new working restrictions, the present paper proposes a broad synthesis algorithm and establishes a new optimized solution for the adjustable tracking linkage, through numerical simulations, considering that the tracked PV system is implemented in the Brasov/Romania area. The dimensional optimization considers the geographic and climatic conditions in Brasov/Romania, but the algorithm is adaptable to any location on the Globe.

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Fig. 1 (a) The azimuth tracking system [7]; (b) the kinematical scheme of the proposed PV system fitted with an azimuth biaxial adjustable tracking linkage with one actuator.

2 Linkage Synthesis Optimization The proposed azimuth adjustable solar tracking linkage using a single actuator for a bi-axial orientation, presented in Fig. 1b consists of a vertical pole (0), a fork (1), a PV platform (2), a swing (3), a vertical shaft (4), a slide (5) and their links: the rotating joint (0, 1) describing the system azimuth axis (driven by a rotating actuator), the rotating joint (1, 2) defining the system elevation axis, the swing connections (2, 3) and (3, 4) consisting of two Hook joints, the rotating joint (4, 5) – parallel with the azimuth axis and the prismatic joint (5, 0) driven by a screw mechanism that adjusts manually or motorized the relative position between pole (0) and slide (5). In our approach, the azimuth movement ψ ∗ in Fig. 1a is considered independent and the PV elevation movement α ∗ and the swing movements are dependent in relation to ψ ∗ . The proposed bi-axial, mono-mobile adjustable tracking linkage has a spatial contour, Fig. 2b, in which segments: AB = h, BC = l0 , CD = l, AD = r represent the unknown dimensions of the linkage. To simplify the calculus, three relative dimensions have been adopted: H = h/l0 , R = r/l0 , L = l/l0 . During previous research [2], the extreme angular values were determined which had to be performed by the linkage at sunset/sunrise α ∗m and ψ ∗m , and at noon α ∗M and ψ ∗M , for every season of the year (see Table 1). The calculus was proceeded for the spring equinox day and the H, R, L parameters were calculated [2, 3]. Keeping R and L parameters (as evaluated for the spring equinox) another two H values are calculated for summer and winter seasons. With this goal, the displacement law, in the implicit form (1, 1 ) and then in the explicit form (2, 2 ) were established, by development of the geometrical connection CD2 = l 2 = ct (Fig. 1): 2r(l0 · sin α ∗ · cos ψ ∗ + h · cos α ∗ ) = (r2 + h2 + l0 ) − l 2

(1)

(sin α ∗ · cos ψ ∗ + H · cos α ∗ ) = [(R2 + H 2 + 1) − L2 ] · (2R)−1 = ct.

(1’)

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Fig. 2 PV system fitted with an azimuth biaxial adjustable tracking linkage with one actuator: (a) geometrical scheme at the noon position; (b) constructive scheme; (c) working angle 1 (β 1 ).

α ∗ = 2 tan−1




a = 2R · cos ψ ∗ ;

a±

   a2 + b2 − c2 (b + c)−1

(2)

c = R2 + H 2 + 1

(2’)

b = 2RH;

Starting from the form (1’), the Hi parameter can be established for spring/fall and winter seasons: ∗ ∗ ∗ ∗ ∗ ∗ · cos ψMi + Hi · cos αMi = sin αmi · cos ψmi + Hi · cos αmi sin αMi

(3)

Hi = (sin α ∗ Mi · cos ψ ∗ Mi − sin α ∗ mi · cos ψ ∗ mi )(cos α ∗ mi − cos α ∗ Mi )−1

(4)

where m indicates the sunset/sunrise value, M indicates the noon value and i indicates the season: i = {spring/fall, summer, winter}. For the summer season H is calculated according to (see Fig. 2a) [5]:   (4’) H = L · cos µ + R · cos α ∗M , where µ = sin−1 (R · sin α ∗M − 1) · L−1 For the noon linkage position, in the equinox day in which α ∗M = 45◦ and ψ ∗M = 0◦ , the R parameter can be analytically established from the null elevation pressure angle condition [2] (β 1 = 0◦ ) and L parameter, from relation (1’): R = cos α ∗M · (H + 1)  ∗ L = H 2 + R2 + 1 − 2RH cos αM

(5) (6)

According to Fig. 2c and relation (7), where eL and ePV are unit vectors, the following working restriction can be stated for the β 1 working angle: cos β1 = eL · ePV ⇒

(7)

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Fig. 3 (a) Working angle 2 (β 2 ) and (b) working angle 3 (β 3 ).

β1 = cos−1



1 H sin α ∗ − cos α ∗ · cos ψ ∗ L L



≤ 55◦

(7’)

The R and L parameters calculated in the spring equinox day were kept for the entire year and another two values for H were calculated, for the summer and winter seasons, according to the specific α ∗m , ψ ∗m , α ∗M , ψ ∗M established values (see Table 1). Further research highlighted the necessity to introduce other two working restrictions for the β 2 and β 3 working angles; according to Figs. 3a and 3b, where eH , eR and e = (eH × eR )/|eH × eR | are the unit vectors, these restrictions can be analytically stated by relations (9) and (11) respectively: ⎤ ⎤T ⎡ ⎡ cos ψ R sin α sin ψ 1⎣ sin ψ 1 − R sin α cos ψ ⎦ · ⎣ sin ψ ⎦ = eL · e = cos β2 = L L H − R cos α 0   β2 = cos−1 L−1 sin ψ ∗ ≥ 35◦ ⎤T ⎡ ⎤ ⎡ 0 R sin α ∗ sin ψ ∗ H − R cos α ∗ 1⎣ R sin α ∗ cos ψ ∗ −1 ⎦ · ⎣ 0 ⎦ = eL · eH = cos β3 = L L 1 H − R cos α ∗   β3 = cos−1 L−1 (H − R cos α ∗ ≤ 65◦ ◦

◦

(8)

(9) (10) (11)

Both restrictions on last working angles β 2 ≥ 35 and β 3 ≤ 65 are assigned to avoid the blocking effect of the azimuth movement, at sunset/sunrise, when the swing arm (l ≡ CD, see Figs. 1b and 2b) reaches the closest position to the PV platform. Because the relative dimensions obtained in the anterior research, Table 1, do not fulfill the new restrictions: β 2 ≥ 35◦ and β 3 ≤ 65◦ , the previous synthesis algorithm must be extended as follows:

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1. The R value is varied below and above its former value R = 3.4 considering the following values: R = {1.8; 2; 2.2; 2.4; 2.6; 2.8; 3; 3.2; 3.4; 3.6; 3.8; 4} and is kept equal in each season of the year. 2. . The following values are adopted for the Brasov spring/fall season: α ∗M = 45◦ , ψ ∗m = 90◦ , ψ ∗M = 0◦ ; for the summer season: α ∗M = 62◦ , ψ ∗m = 90◦ , ψ ∗M = 0◦ ; for the winter season: ψ ∗m = 52◦ , ψ ∗M = 0◦ (see Table 1). For the spring and winter seasons, α ∗m is varied close to the previous values [2]. 3. For each pair (R; spring α ∗m ), the spring H and winter H are calculated using Eq. (4), the summer H with Eq. (4’) and the spring L (valid for whole year) with Eq. (6). 4. Replacing the calculated H, R and L in Eq. (2), α ∗m is evaluated for the summer season and is recalculated for spring and winter seasons. 5. The spring/fall H solution is developed for the spring minimum recalculated α ∗m for which the three working angles restrictions are fulfilled (β 1 ≤ 55◦ , β 2 ≥ 35◦ , β 3 ≤ 65◦ ). For Brasov/Romania, the following (R, spring α ∗m ) pairs are obtained: (1.8; 19.6◦), (2; 17.6◦ ), (2.2; 16.2◦), (2.4; 18.4◦), (2.6; 21◦ ), (2.8; 23.2◦), (3; 25◦), (3.2; 26.6◦), (3.4; 28◦ ), (3.6; 29.1◦ ), (3.8; 30.2◦), (4; 31◦ ). The same procedure is applied for winter α ∗m . Finally, for each R a correspondent set (Hspring/fall , Hsummer , Hwinter , L) is obtained, resulting 12 dimensional solutions (Hspring/fall , Hsummer , Hwinter , R, L), Fig. 4. 6. Numerical simulations are developed for each solutions, the energetic calculus is done and the direct tracking efficiency is determined the for one year period, for Brasov/Romania geographic and climatic conditions [3]. 7. The solution (Hspring/fall , Hsummer , Hwinter , R, L) which encounters the highest direct tracking efficiency1 represents the optimal dimensional solution.

3 Numerical Simulations and Comparative Analysis Prior to the direct tracking efficiency calculus, numerical simulations are done for the solar angles (α , ψ ) and PV platform angles (α ∗ , ψ ∗ ), for the correspondent incidence angles, for the available direct solar radiation and for the direct solar radiation received by the optimized PV tracking system, considering the real climatic conditions for Brasov, Romania [3]. As example, graphic results of the numerical simulations are presented in Figs. 4, 5 and 6: Fig. 4 shows the variations of the parameters H, L and of the working angles β 1 and β 2 relative to R; Fig. 5 outlines the fulfillment of the working restrictions regarding the angles β 1 and β 2 during some representative days of the year while Fig. 6 exemplifies the energetic response of the tracked PV system.

1

The direct tracking efficiency represents a ratio between the energy of the direct solar radiation received normal by a particular solar tracking PV system and the energy of the available direct solar radiation. This parameter measures the energetic performance of a PV tracker.

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Fig. 4 Seasonal (Sp./Fl. = spring/fall, Sum. = summer, Win. = winter) variations obtained by applying the improved algorithm: (a) dimensional parameters H, R, L; (b) morning values for the two new working angles β 2 and β 3 .

Fig. 5 Daily variations of the β 1 working angle (a) and of the β 2 working angle (b), during the annual seasons.

Fig. 6 Daily variations, during the extreme and average (equinox) days of the spring season, in Brasov, Romania, for (a) solar angles: elevation (α ) and azimuth (ψ ) – continuous curves – and PV homologues angles (α ∗ and ψ ∗ ) – step curves; (b) the available direct solar radiation (AVAILABLE), the direct solar radiation received normal by the PV system fitted with an adjustable tracking linkage (Adjustable Trk.Link Rec) and the direct solar radiation received normal by a fix tilted PV system (FixT. Rec).

Considering the clear sky condition, the available direct solar radiation (BS ) is modeled by Eq. (12) [5], and the direct solar radiation received normal by the tracked PV system (BPV ) can be calculated using the Lambert Low (13) [3]:

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◦ BS = 1367 · [1 + 0.0334 · cos(0.9856 N − 2.72)] exp −

Tr 0.9 + 9.4 sin α



BPV = Bs · [sin α sin α ∗ + cos α cos α ∗ cos(ψ − ψ ∗ )]

(12) (13)

Based on the data in Fig. 6, the tracking program for an entire season is developed in the medium day of the season (in this case, the Spring Equinox). Even though the step curves are following only the seasonal medium day solar curves, the received direct radiation curves are still very close to their correspondent available direct radiation curves. The direct solar radiation received by a fix tilt PV system was simulated as well (Fig. 6b), to evaluate the energetic opportunity of implementing the proposed adjustable tracking linkage as opposed to the lowest cost and to the easiest in usage tracking system. Based on Eqs. (12), (13) and on the Factor of Cloud Crossing (FCC ) from the Brasov area (that describes the sky covering degree with clouds) [3], the direct tracking efficiency of the PV tracking linkage (η track ) can be calculated as the ratio between the energy of the direct radiation received normal by the tracked PV system (EB·PV ) and the energy of the available direct solar radiation (EB·S ):  



E BB·PV dt ; EB·S = FCC BB·S dt ; ηtrack = B·PV . (14) EB·PV = FCC EB·S The optimum dimensional solution and its tracking efficiency, obtained using the extended algorithm, are systematized in Table 1 comparatively with the previously developed solution.

Table 1 Optimized seasonal values of the azimuth adjustable spatial tracking linkage: the dimensional parameters H, R and L, the PV elevation extreme angles α ∗m , α ∗M , the PV azimuth extreme angles ψ ∗m , ψ ∗M and the tracking efficiency. H

R

L

α ∗m

α ∗M

ψ ∗m

ψ ∗M

Track. Ef.

45.5◦ 63◦ 28.5◦

90◦ 90◦ 56◦

0◦ 0◦ 0◦

97.96%

0◦ 0◦ 0◦

97.44%

Previous values [4] Spring/Autumn Summer Winter

3.8 2.2 4.7

3.4 3.4 3.4

2.02 2.02 2.02

27.5◦ 31◦ 20◦

Optimized values through the new algorithm Spring/Autumn Summer Winter

2.91 2.33 3.03

1.94 1.94 1.94

1.58 1.58 1.58

18.2◦ 31.9◦ 29.2◦

45◦ 62◦ 40.9◦

90◦ 90◦ 52◦

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4 Conclusions • Newly imposed working conditions are set for tracking mechanisms of PV platforms: at sunset/sunrise β 2 ≥ 35◦ and β 3 ≤ 65◦ and the characteristic functional dimensions are set for the bi-axial mono-mobile azimuth tracking mechanism. • According to the proposed kinematical restrictions an improved algorithm is developed and the functional dimensions are set at: Hspring/fall = 2.91; Hsummer = 2.33; Hwinter = 3.03; R = 1.94; L = 1.58. The new solution conducts to reduce the over all size mechanism and allows obtaining high tracking efficiency of 97.44%. • Further research will focus on implementing a similar algorithm for the nonadjustable variant of the bi-axial, mono-mobile, azimuth tracking linkage.

Acknowledgments This paper is supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and by the Romanian Government under the contract number POSDRU/6/1.5/S/6”.

References 1. Boyle, G., Renewable Energy. Oxford University Press in Association with the Open University, Glasgow, 2004. 2. Diaconescu, D., Vis¸a, I., V˘at˘as¸escu, M.M., Hermenean, I., and S˘aulescu, R., Synthesis of a bi-axial tracking spatial linkage with a single actuator. In: Proceedings of SYROM 2009, The 10th IFToMM International Symposium on Science of Mechanisms and Machines, pp. 632–617, Springer, 2009. 3. Diaconescu, D. et al., Clouds influence of the solar radiation for a mountain location, Environmental Engineering and Management Journal, 8(4):849–853, 2009. 4. Jesus, A. et al., Solar tracker with movement in two axes and actuation in only one of them, Patent WO2008/000867 A1., publication date 2008-06-18. 5. Meliss, M., Regenerative Energie-Quellen Praktikum, Springer, Berlin, pp. 5–7, 1997. 6. V˘at˘as¸escu, M.M., Diaconescu, D. and Vis¸a, I., On the simulation of a mono-actuator bi-axial azimuth PV system, Bulletin of the Transilvania University of Bras¸ov, 2(51):97–104, 2009. 7. Visa, I. et al., On the received direct solar radiance of the PV panel orientated by azimuthal trackers. InProceedings of COMEC – The 2nd International Conference on Computational Mechanics and Virtual Engineering, Bras¸ov, Romania, October, pp. 25–30, 2007.

Mechanical Transmissions

Defect Simulation in a Spur Gear Transmission Model A. Fernandez del Rincon, M. Iglesias, A. de-Juan and F. Viadero Dpto. Ingenieria Estructural y Mecanica, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain; e-mail: [email protected]

Abstract. This article describes a two-dimensional model for the analysis of the transmission error and dynamic performance of a spur gear transmission supported by bearings including defects. Knowing the position of each wheel and profiles geometry, including errors and roundings, contact points are determined. Meshing forces and associated deformations are then calculated, simplifying the deformation computation by means of its subdivision in structural and local deformations. The application of this procedure improves the results and reduces the computational effort. The model supports the possibility of contact on both flanks of the teeth simultaneously and it also takes into account the variable number of bearing elements. Finally, an application example of this method is presented, highlighting the effect of cracks, surface defects and tooth gear profile errors on the transmission error. Key words: gear dynamics, transmission error, bearings, simulation, meshing stiffness

1 Introduction The actual demand of high performance gear transmissions, in terms of torque and speed, rises the interest in the development of analytical models capable of provide a better understanding of the dynamics involved in gear transmissions, and also more effective design and maintenance tools based for example on on-condition vibratory measurements [4]. Thus, the development of physical models for gear dynamics could increase the prognosis performance of a maintenance method when trying to determine for example the depth of a tooth crack or its future growth. Most works involving gear dynamics modeling employs the so-called Transmission Error (T E) as an important parameter to describe the inaccuracies of the transmission (due to manufacturing, kinematic, static and dynamic sources), defined as the the difference between the position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output [3]. The present work mainly deals with the determination of the Loaded static Transmission Error or LT E, which can be used as an input for dynamic analysis [10] or as a useful measurement related with noise and other aspects of gear transmissions. To calculate the LT E it

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is first necessary to obtain the contact forces. There are several approaches to this end, from simplified analytical formalizing [8] to complex contact finite element modeling [1]. When obtaining the deformations, the variable meshing stiffness (due to the changeable number of teeth pairs and rolling elements in contact) also plays an important role [2, 5]. An aspect of special interest in the field of gear transmissions predictive maintenance based on vibration measurement is the inclusion of defect modeling [6]. Most authors address this issue using simple models representing the dynamic vibratory signal of a gear transmission. These models typically consider the existence of a frequency response dominated by harmonics of the fundamental frequency of tooth meshing, presenting side bands due to the modulation caused by the shaft rotation speed. Indeed, the meshing frequency is related to the existence of a variable contact stiffness which will affect the dynamic behavior of the transmission. When a local defect exists in one tooth, the value of the contact stiffness will change in different ways depending on the location, type and magnitude of the defect. The development of new strategies of maintenance and transmission prognostics (such as prediction of the defect growth) requires the use of more advanced mathematical models that those merely based on the waveform response of the system. These models should be related to the actual physical system to be a reliable source of information of the status of the transmission and to provide a better understanding of the dynamic behavior in the presence of defects.

2 General Model of Gear Transmission The general model used in this work is described in details in [11] and only the highlight of the model definition, contact forces and deformation calculation is exposed next. Initially, the tooth profiles are defined. This lets to solve the other problems. Gear generation in this work will be based on a rack-type tool following the Litvin’s vector approach [7]. This allows for the possibility of tool displacements and also undercutting conditions. In addition, a rounding profile is added in the tooth tip to handle corner contact. From the analytical properties of involute profiles and tip rounding arcs, contact points and their corresponding separation distance are obtained. Thus, positive values for separation distance denote that the points should be in contact. On the other hand, negative values indicate a non contact condition. The following types of contacting profiles have been considered: involute-involute and involute-circle arc. Finally, gear contacting forces are achieved from the model proposed by Andersson and Vedmar [9]. They divide into two groups tooth deformations; one closed to the contact which is computed by an analytical non-linear formulation of Hertzian type(local), and another which is of elastic nature and determined by means of a finite element model (structural). The displacements of the considered nodes in this method results from adding both local and structural components. Then, the

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load distribution of gear teeth can be found solving a non-linear system of equations which takes in account elastic deflections, the compatibility of initial separations, non-existence of negative loads and profile errors. This approach achieves an accuracy modeling of gear teeth contacts. It do not need a extremely refined mesh at the contact surfaces. For this reason, the computational cost is reduced when it is compared with conventional finite element models that allows the analysis of dynamic problems.

3 Defect Modeling The model described has incorporated the existence of three different types of gear defects: • Errors in the generation of gear profiles • Existence of a crack • Existence of pitting Errors in the generation of gears have been implemented as a modification of the overlap for each one of the potential contact points. It was considered that this modification is identical for all teeth. This same procedure can be used for the implementation of changes in the profiles as the fillet generation in the root or the tip rounding of the teeth. The presence of a crack will affect the overall behavior of the gear wheel so to implement this defect a finite element analysis model that includes a crack has been developed. This simplified model considers a crack located at the root of the tooth with a certain orientation and size. Figure 1 provides a representative model which shows the presence of this defect and the magnified deformation that presents a tooth when a load is applied on its flank. It is revealed that the presence of the crack affects not only the defected tooth but also adjacent teeth thus modifying the flexibility

42 40 38 36 34 32 30 28 26 24 −10

−5

0

5

10

Fig. 1 Detail of finite element model with crack.

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coefficients βi j used in [11]. In the case of surface defects such as pitting only the local behavior will be affected. Therefore, the implementation of this kind of defect will be simpler that the crack implementation. It is only necessary to change the tooth contact length in the axial direction, which can be done by simply altering the value of the load per unit length q (less contact length due to the pitting).

4 Application Example The model described above will be applied next to the simulation of a spur gear pair defined by the parameters shown in Table 1. Support bearing arrangement is identical for both wheels, so that all the rolling elements involved are in phase. Infinite torsional and bending stiffness of the shafts are considered, only taking in account the finite stiffness of the wheels. The transmission error estimation is done considering that the driving wheel (wheel 1) has a known angular position θ1 and is subject to a certain external torque Text . The objective of the simulation is to locate the position of the center of the driving wheel (x1 , y1 ) and the position (both cartesian and angular positions) of the driven wheel (x2 , y2 , θ2 ) so as to verify that:  

FBearing = x1 , y1 , θ1 , x2 , y2 , θ2 + FContact = x1 , y1 , θ1 , x2 , y2 , θ2 = FExterior (1) The solution of the resultant set of equations is carried out through an iterative procedure, because of the nonlinear character related to bearings and contact forces. Once the angular positions of both wheels are known (θ1 is given and θ2 is obtained by solving Eq. 1), the LT E corresponding to a given position and load inputs can be obtained applying the equation:
 z LT E θ1 , T = θ2 − θ1 1 z2

(2)

Once the LT E is known it is possible to deduce the torsional stiffness KT . It is necessary to clarify that due to the flexibility of the supports, the center position of the wheels will vary depending on the amount of torque applied. This fact implies

Table 1 Transmission parameters. Parameter

Value

Parameter

Value

Number of teeth (z1 = z2 ) Module Elasticity Modulus Poisson’s ratio Pressure angle Rack addendum Rack deddendum Rack tip rounding

23 3 mm 210 GPa 0.3 20 (degree) 1.25 m 1m 0.25 m

Gear tip rounding Gear face width Gear shaft radius Ball stiffness Number of balls Radial clearance Outer radius

0.05 m 15 mm 9 mm 7.055 · 109 N/m3/2 9 20 µm 14.13 mm

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−3

−1.5

x 10

10 Nm 30 Nm 50 Nm 70 Nm 90 Nm 110 Nm 130 Nm 150 Nm 170 Nm 190 Nm

−2

LTE base (rad)

−2.5 −3 −3.5 −4 −4.5 −5 0

0.1

0.2

0.3 0.4 0.5 Wheel 1 angle (rad)

0.6

0.7

0.8

0.6

0.7

0.8

Fig. 2 LTE base. −5

7

x 10

LTE base−LTE crack (rad)

6 5 4 3 2 1 0 −1 0

0.1

0.2

0.3 0.4 0.5 Wheel 1 angle (rad)

Fig. 3 LTEbase-LTEcrack.

that the working distance between wheels will also experiment a change and therefore the teeth clearance will change too. Thus the interpretation of the LT E obtained by the above procedure must be done with some caution. Figure 2 shows the LT E in the absence of defects (hereinafter LT E base) for different load levels (from 10 Nm to 190 Nm). In this figure it can be appreciated how the increasing of the applied load modifies the waveform of the resulting LT E, in terms of both its magnitude and shape. Figure 3 presents the difference in the LT E simulated measurements before and after implementing a crack of 1 mm length in the tooth root as shown in Fig. 1 (radial position r = 32.2875 mm and angular orientation −30◦). The defect has different consequences depending on the angular position of the gear. It is actually possible to distinguish three stages. At first, when the cracked tooth begins contact. This occurs at the point of maximum distance between the contact point and the crack, so the loss of stiffness due to the presence of the defect (greater LT E) is more significant. As the contact point travels on the profile and therefore the distance between the contact point and the crack is reduced, the loss of stiffness is smaller and the LT E decreases. The second stage is initiated when the cracked tooth supports the entire load, which causes the LT E sharp increase (peak in the LTE signal in Fig. 3). Gradually, as the contact point continues its displacement

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−4.4

x 10

−4.45

LTE (rad)

−4.5 −4.55 −4.6 −4.65 With Pitting 190 Nm Base 190 Nm

−4.7 −4.75 0

0.05

0.1 0.15 Wheel 1 angle (rad)

0.2

0.25

Fig. 4 LTE with pitting for 190 Nm load.

on the profile, the loss of stiffness is smaller (the same happened in the first stage), and so it is the LT E. The final stage is the natural continuation of the first one, with another tooth starting contact and the contact point of the defected tooth still gradually getting closer to the crack and subsequently LT E still decreasing. The implementation of surface defects such as pitting will only affect the LT E during the contact through the defected area. Thus, it is only necessary to represent one gearing cycle (corresponding with the defected tooth) to show the LT E affection. In the case shown in Fig. 4 the defect is located between the contact radius 34.8329 mm and 35.1329 mm with a maximum width of 3 mm in the axial direction (with a parabolic variation in the range of the contact length defined in the model [11]). For better visualization only the extreme value of the applied load (190 Nm) is represented in the figure, for both the LT E base and the LT E of the pitted gear. The inclusion of errors in the generation of gear profiles has been implemented through a model representing the deviation e(s) of the actual profile with respect to the ideal profile in accordance with the expression: ⎛

 ⎞ ff s − s0 s − s0  + sin ⎝2π fr  ⎠ e (s) = fHa  (3) 2 s −s s −s f

0

f

0

where s is the radius of curvature of the involute gear profile, s0 the lower radius and s f the upper radius. The error considered follows a sine waveform with fr cycles along the profile. Parameters defining the magnitude of the error are fHa and f f that correspond to the quantities identified in Fig. 5(b). In this figure are also shown the values adopted in the example. Positive profile error values indicate increments in the theoretical radius of curvature, and negative values will therefore result in decrements. The introduction of profile errors completely modifies the LT E in comparison with gears without this kind of errors. Figure 6(a) presents the LT E for the full range of torque values applied, and Fig. 6(b) has a detail of the LT E for the maximum

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−3

3

x 10

Profile error (mm)

2

1

fHa

ff

0

−1 0

5

s0

10 s

15

sf 20

Fig. 5 Profile error definition. −3

−3

−4.2

x 10

10 Nm 30 Nm 50 Nm 70 Nm 90 Nm 110 Nm 130 Nm 150 Nm 170 Nm 190 Nm

LTE with error (rad)

−2 −2.5 −3 −3.5

x 10

LTE with error LTE base

−4.3 −4.4 LTE (rad)

−1.5

−4.5 −4.6

−4

−4.7

−4.5 −5 0

0.1

0.2

0.3 0.4 0.5 Wheel 1 angle (rad)

(a)

0.6

0.7

0.8

−4.8 0

0.2

0.6 0.4 Wheel 1 angle (rad)

0.8

(b)

Fig. 6 (a) LTE with profile error for every load. (b) LTE with profile error for 10 Nm.

torque applied (190 Nm). For lower load levels the effect of profile error is most noticeable while increasing the load level mitigates the effect on the LT E.

5 Conclusions In this paper a procedure for determining the transmission error under load (LT E) of a spur gear transmission supported by ball bearings has been presented. The model takes in account the deformation considering that is composed as a combination of the so-called structural deformations (mainly due to bending of the tooth) and the contact deformations (of local character). As a result of this consideration, the nonlinear aspects are linked exclusively to the local contact model, while the structural behavior presents only linear character. Therefore the solution is simpler and the computational effort reduced. The model also considers the possibility of variable contact ratios due to the different loads. In addition the procedure allows a better representation of the singularities of the load transfer caused by the changing number of teeth pairs in contact. The proposed formulation has been suitably adapted so that

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it is possible to include various types of defects such as errors in the generation of gear profiles, existence of a crack and surface defects (pitting). Each of these defects has been simulated under various levels of load to monitor the influence that each of these defects has in the LT E. The compromise adopted between accuracy and computational effort provides a solution which can serve as base for future model developments, and as a tool for the study of other aspects related to gear transmissions such as wear, dynamic performance, optimization, efficiency, noise and very especially condition monitoring.

Acknowledgement The research work reported here has been developed in the framework of the project DPI2006-14348 funded by the Spanish Ministry of Science and Technology.

References 1. Argyris, J., Fuentes, A. and Litvin, F.L., Computerized integrated approach for design and stress analysis of spiral bevel gears. Computer Methods in Applied Mechanics and Engineering, 191:1057–1095, 2002. 2. Cai, Y. and Hayashi, T., The linear approximated equation of vibration of a pair of spur gears (theory and experiment). Transactions of the ASME, Journal of Mechanical Design, 116:558– 564, 1994. 3. Derek Smith, J., Gear Noise and Vibration. Marcel Dekker, 1999. 4. Heng, A., Zhang, S., Tan, A.C.C. and Mathew, J., Rotating machinery prognostics: State of the art, challenges and opportunities. Mech. Sys. and Signal Processing, 23:724–739, 2009. 5. Kuang, J.H. and Yang, Y.T., An estimate of mesh stiffness and load sharing ratio of a spur gear pair. In: Proceedings of the ASME International Power Transmission and Gearing Conference, 1992. 6. Li, C.J. and Lee, H., Gear fatigue crack prognosis using embedded model, gear dynamic model and fracture mechanics. Mech. Sys. and Signal Processing, 19:836–846, 2005. 7. Litvin, F.L. and Fuentes, A., Gear Geometry and Applied Theory. Cambridge University Press, 2004. 8. Pintz, A., Kasuba, R., Frater, J.L. and August, R., Dynamic effects of internal spur gear drives. In NASA Contractor Reports, 1983. 9. Vedmar, L. and Andersson, A., TA method to determine dynamic loads on spur gear teeth and on bearings. Journal of Sound and Vibration, 267:1065–1084, 2003. 10. Velex, P. and Ajmi, M., On the modelling of excitations in geared systems by transmission errors. Journal of Sound and Vibration, 290:882–909, 2006. 11. Viadero, F., Fernandez del Rincon, A., Sancibrian, R., Garcia Fernandez, P. and De Juan, A., A model of spur gears supported by ball bearings. In: Proceedings 13th International Conference on Computational Methods and Experimental Measurements CMEM, pp. 711– 722, 2007.

On a New Planetary Speed Increaser Drive Used in Small Hydros. Part I. Conceptual Design C. Jaliu, D. Diaconescu, R. S˘aulescu and O. Climescu Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: [email protected]

Abstract. The concept of a new planetary speed increaser drive with deformable element (chain, cog belt) is developed in the paper. The variant is obtained by applying a conceptual design algorithm proposed by the authors. The algorithm contains several steps, including the establisment of the requirements list, the development of the solving structures and, finally, the identification of the planetary transmission to be used as speed increaser drive in renewable energy systems. Afterwards, the concept kinematical and dynamic features are established in order to evaluate the solving structures. Key words: conceptual design, speed increaser, deformable element, small hydropower plant

1 Introduction The small hydropower plants are installed on rivers, being designed for certain values of the water flow and head. When the river flow falls below some predetermined value, the energy generation ceases. The small schemes may not always be able to supply energy, unless their size is such that they can operate whatever the flow in the river is. These limits given by the water flow can be overstepped by the design of the small hydro electromechanical equipment. A way to increase angular velocity is to use a gearbox between the turbine and the generator. Thus, the hydro turbine shaft can turn at a much lower speed (due to a small water flow) than is required by most electric generators. The multiplication in speed can range from 3 to 5 times [5, 6]. But using a transmission to obtain the necessary speed multiplication involves some disadvantages, related to the system efficiency, overall dimensions and cost. Therefore, objectives of requirement type (compulsory objectives) and of desirable type are set in the concept development. These objectives form the requirements list, on which the conceptual design [2, 3] is based. New solutions of planetary speed increaser drives were proposed by the authors in the previous papers [6, 7, 8]. In order to increase the base of speed increasers for hydropower applications, the concept of an innovative planetary transmission with

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deformable element (chain, cog belt) is proposed in this paper. The algorithm used to generate the concept is proposed by the authors in [3, 4] and it generalizes the VDI algorithm [9]. The algorithm that is applied below, starts from a requirements list and contains: (1) global function establishment; (2) structure of sub-functions; (3) solving structures generation by: solving the sub-functions, sub-solutions composition and elimination of inadequate solution; and (4) the best solving structure selection by evaluation. The kinematical and dynamic features (internal transmission ratio, multiplication ratio, internal efficiency and the speed increaser efficiency) are further established for different values of the the sprockets teeth number. The numerical simulations for the transmission ratio and efficiency form a data base that can be used in choosing the proper solution for given values of the multiplication ratio, efficiency and overall dimensions. Thus, the concept of a new planetary speed increaser drive is developed. The principle solution is an innovative concept of the speed increaser with deformable element and represents the input data for the embodiment design phase. The second part of the paper presents the speed increaser dynamic modeling as the first step in the design of the small hydropower plant control system.

2 The Requirements List The first step in the design process is to establish the requirements list, which contains two types of requisites: compulsory objectives (main objectives) that are needed in the elimination of inadequate solutions, and objectives of desirable type (secondary objectives) that are needed in ordering the solving structures and selection of the concept. Thus, from the requirements list of the hydropower station [5, 7], the following requirements (main objectives) for the speed increaser can be identified: 1. 2. 3. 4. 5. 6. 7.

the input and output shafts should be coaxial; the multiplication ratio should range from 3 to 5 times; it should have an average efficiency of 60%; it should not require a particular manufacturing process; the components should be available on the market and not be expensive; the transmission should contain a deformable element (chain, cog belt); the connecting elements to be easily manufactured in any workshop.

The following secondary objectives (the technical-economic evaluation criteria), listed in the order of importance, are associated with the seven main requirements: minimization of radial overall dimension; minimization of friction losses; minimization of complexity degree; minimization of manufacturing costs.

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Fig. 1 The structure of sub-functions in symbolic variant [3].

Fig. 2 The morphological matrix for the development of the solving structures variants. Notations: def – the deformable element, H – the carrier, 1 – the satellite gear and 2 – the sun gear.

3 The Development of the Solving Structures The speed increaser function comes out from the small hydropower plant global function; the function is illustrated in Fig. 1, in which FE1 denotes the transmission of mechanical energy with modification of mechanical energy parameters (speed multiplication). Based on the structure in Fig. 1, the speed increaser solving structures will be generated. The planetary solutions found in the technical literature [3, 5–8] for solving the sub-function FE1 (see Fig. 1) are graphically systematized in the morphological matrix in Fig. 2. Only the variants with deformable element were considered in the morphological matrix to develop the speed increaser drive. By means of the morphological matrix, several solving structural variants can be generated, through a compatible composition of the potential solutions (Table 1). The qualitative schemes of the structural solving variants are represented in Fig. 3, while three qualitative schemes, derived from the previous ones (Figs. 3c, d, e, g, h, i) by eliminating the tangential over-twist of the deformable element are illustrated in Fig. 4. The solving variants developed by means of the morphological matrix fulfill in qualitative terms the main objectives set in the requirements list. The technical characteristics of the solving structures are established in the next phase, in order to check the quantitative accomplishment of the requirements. Obviously, the solving structures of function FE1 will consist in the variants that meet the requirements in quantitative terms.

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C. Jaliu et al. Table 1 Examples of structural solving variants. Solving variant

Speed increaser solution

SR1 SR2 SR3 SR4 SR5 SR6 SR7 SR8 SR9

1.1+1.1 1.1+1.2 1.1+1.3 1.2+1.1 1.2+1.2 1.2+1.3 1.3+1.1 1.3+1.2 1.3+1.3

4 Kinematical and Dynamic Features In order to identify the concept of speed increaser, the secondary objectives are used to eliminate the inadequate solutions. Thus, the solving variants SR1, SR2 and SR3 are eliminated due to their large radial dimension. The kinematical and dynamic features for the rest of the solving variants are established in order to check the fulfillment of the main objectives from the requirements list. The simulations of the multiplication ratio and the efficiency for the solving variant from Fig. 4i (SR9) are presented in the paper. The simulations are performed for different combinations of sprockets teeth numbers. Then, on the basis of the known values of transmission efficiency, the speed increasers efficiency are calculated. The following relations [3] are used in the establishment of the transmissions features: • for the multiplication ratio (i) i=

1 , i41H

i41H =

ω14 ω − ω4H = 1H = 1 − i0, ωH4 −ω4H

i0 =

z2 z4 · z1 z3

(1)

• for the transmission efficiency (η )
1 − i0 · η0−1 −TH T 1 −ωH4 · TH η= = = ω14 · T1 i41H 1 − i0

(2)

It can be observed that the same multiplication ratio can be obtained for several combinations of sprockets teeth numbers; by imposing some correlations among zi , z1 = z2 + (0, 1, 2, . . . ); z2 = z3 + (. . . − 2, −1, 0, 1, 2, . . .); z4 = z3 + (0, 1, 2, . . . ), the function i can be stated as i = i(z1 ); this form is needed in the teeth numbers synthesis. Knowing from the technical literature the efficiencies of each component mechanism (η12 = η34 = 95%) [1–3], the efficiency of the speed increaser (η ) can be obtained according to relation (2). Representations of the multiplication ratio and the efficiency as functions of z1 are exemplified in Fig. 5.

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Fig. 3 Qualitative schemes of the structural solving variants. Notations: 1, 4 – sun gears, 2, 3 – satellite gears, H – carrier.

It can be observed that part of the combinations of teeth numbers is not feasible, due to the negative value of the efficiency (see Fig. 5). Considering the secondary objectives, it can be noticed that the best efficiency for the smallest overall dimension is obtained for z2 < z3 (Fig. 5b). The optimum combination of teeth numbers

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Fig. 4 Three qualitative schemes derived from Fig. 3 after elimination of the tangential over-twist of the deformable element.

Fig. 5 Numerical simulations of multiplication ratio and efficiency for the solving structure SR9, in terms of z1 and (a) z2 = z3 = 20 teeth, (b) z3 = 22 and z2 = 20 teeth, (c) z3 = 20 teeth and z2 =22 teeth.

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Fig. 6 The principle solution of the speed increaser: (a) the developed variant, (b) the variant solving the over-twisting, (c) the variant with higher durability.

can be obtained from the condition of fulfilling the requirements i = 3, . . . , 5, an average efficiency of 60% and a smaller overall dimension. For instance, in order to obtain a multiplication ratio i = 4 ± 10%, the solving variant from Fig. 4i with z1 = 34, z2 = 20, z3 = 22 and z4 = 28 teeth is obtained from Fig. 5b. For this variant, the multiplication ratio is i = 3.97 and the efficiency is η = 67%. Similar to the analyzed case, optimum solutions were obtained for the rest of solving variants from Fig. 3 (SR4, SR5, SR6, SR7 and SR8). Comparing the results it outcomes that the principle solution of the speed increaser (the concept) is designated by the structure SR9. In order to eliminate the tangential over-twisting of the deformable element and to increase the transmission durability, the authors propose variants derived from the qualitative schemes of the structural solving variants. Therefore, variants of the principle solution based on the structure SR9 are illustrated in Fig. 6. Unlike the variant from Fig. 6b, the transmission from Fig. 6c has a higher durability of the deformable element and, therefore, it has a good behaviour at bigger forces, due also to its dynamic equilibration. Its disadvantage consists in the bigger radial dimensions.

5 Conclusions • The concept of a speed increaser with deformable element (chain, cog belt) is developed in the paper by means of a conceptual design algorithm that was proposed by the authors. The algorithm is based on the requirements list and a global function of a small hydropower plant. • The steps used in the concept development are: (1) the establishment of requirements list; (2) the determination of the speed increaser sub-function; (3) the generation of solving structures by solving sub-functions, composing sub-solutions

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•

•

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and eliminating inadequate solutions; and (4) the selection of the best solving structure. The requirements list contains two types of requisites: (a) compulsory objectives regarding the multiplication ratio, efficiency, manufacture, and (b) objectives of desirable type, regarding the transmission dimensions, complexity and costs. The solving structures development phase contains: (1) generation of the solving structural variants and (2) establishment of the kinematical and dynamic features for each solving structure, and elimination of the variants whose technical characteristics do not fulfill the requirements list in quantitative terms. In order to establish the solving structures by using the main objectives in the requirements list, numerical simulations of the multiplication ratio and efficiency for different combinations of sprockets teeth numbers are presented in the paper. The simulations made for six of the nine solving variants form a data base that can be used in choosing the proper solution for given values of the multiplication ratio, efficiency and overall dimensions. The concept of planetary speed increaser is found based on the condition of fulfilling the requirements. Its structure is obtained combining the potential solutions 1.3 and 1.3 (Fig. 2). The result of the conceptual design algorithm is an innovative solution of planetary speed increaser with deformable element, which is subject to patenting. The proposed concept offers the following advantages: interchangeability and an easier replacement of the active components; a reduced overall size; a simplier manufacturing technology due to the use of a deformable element (chain, cog belt) and of normalized sprockets. The developed principle solution of the speed increaser represents input data for the embodiment design phase.

Acknowledgment The research work reported here was made possible by grant ID 140 “Innovative mechatronic systems for micro hydros, meant to the efficient exploitation of hydrological potential from off-grid sites”.

References 1. American Chain Association, Standard Handbook of Chains: Chains for Power Transmission and Material Handling, 2nd Edition, Dekker Mechanical Engineering, 2005. 2. Cross, N., Engineering Design Methods, J. Wiley & Sons, New York, 1994. 3. Diaconescu, D., Products Conceptual Design, Transilvania University Publishing House, Brasov, 2005 [in Romanian]. 4. Diaconescu, D., Jaliu, C., Neagoe, M., and S˘aulescu, R., On a generalized algorithm of the technical products conceptual design. In Proc. DAAAM’08 Symposium, Trnava, Slovakia, 22– 25 October, pp. 0377–0378, 2008. 5. Harvey, A., Micro-Hydro Design Manual, TDG Publishing House, 2005.

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6. Jaliu, C., Diaconescu, D.V., Neagoe, M., and S˘aulescu R., Dynamic features of speed increasers from mechatronic wind and hydro systems. Part I. Structure. Kinematics. In Proc. of EUCOMES 08. The 2nd European Conference on Mechanism Science, Cassino, Italy, September, pp. 355–363, Springer, 2008. 7. Jaliu, C., Diaconescu, D., Neagoe, M., and S˘aulescu R., Conceptual synthesis of speed increasers for RES. In The 10th IFToMM Int. Symp. on Science of Mechanisms & Machines, SYROM 2009, Bras¸ov, Romania, September 12–15, Springer, pp. 171–183, 2009. 8. Neagoe, M., Diaconescu, D., Jaliu, C., et al., On a new cycloid planetary gear used to fit mechatronic systems of RES. In OPTIM 2008, Proc. of the 11th International Conf. on Optimization of Electrical& Electronic Equipment, Bras¸ov, Romania, May 22–23, Vol. II-B, pp. 439–449, IEEE Catalogue 08EX1996, 2008. 9. VDI (Verein Deutscher Ingenieure), Richtlinien 2221 and 2222.

On a New Planetary Speed Increaser Drive Used in Small Hydros. Part II. Dynamic Model R. S˘aulescu, C. Jaliu, D. Diaconescu and O. Climescu Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: [email protected]

Abstract. The concept of a new planetary speed increaser with deformable element is developed in the first part of the paper. The second part presents the transmission dynamic modeling, as the first step in the design of the small hydropower plant control system. The dynamic response of the aggregate composed of a Turgo turbine – speed increaser – generator is obtained by means of Matlab-Simulink software. Key words: dynamic model, planetary speed increaser, Lagrange method, Newton–Euler method

1 Introduction A way to increase the power of a small hydropower plant is to use a gearbox between the turbine and the generator. Usually, the speed increaser multiplies the input angular velocity from 3 to 5 times [3, 4, 7]. The paper analyzes the case of a planetary speed increaser with a deformable element characterized by a multiplication ratio of 4. The analyzed transmission is an innovative chain increaser, which was proposed by the authors in the first part of the paper. The second part of the paper presents the dynamic modeling of the speed increaser, as the first step in the design of the small hydropower plant control system. The paper approaches the appropriate kinematical and dynamic features of the speed increaser, based both on Lagrange and Newton– Euler methods. The dynamic response of the aggregate composed of a Turgo turbine – speed increaser – electric generator is obtained by means of Matlab-Simulink software. The speed increaser will be manufactured and included in a small hydropower station that will be implemented on Poarta river, near Brasov.

2 Kinematical and Dynamic Features The first step in dynamical modelling consists in defining the transmission structural and kinematical aspects. Thus, the proposed planetary transmission contains

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Fig. 1 The planetary speed increaser with deformable element: (a1 ) the structural scheme of the variant solving the over-twisting, (a2 ) the structural scheme of the variant with higher durability (b) the block diagram.

(Fig. 1): a sun gear (1), a fixed sun gear (4), a double satellite gear (2–3) and a carrier (H). In the first part of the paper, the authors proposed variants derived from the conceptual scheme of the speed increaser. Thus, the variant from Fig. 1, a1 solves the problem of the tangential over-twisting of the deformable element, while the variant from Fig. 1, a2 increases the transmission durability through the disposal of the satellite gears. The variant from Fig. 1, a1 will be further considered in the dynamic modelling. The block diagram with the notations in Fig. 1b is used as a simplification of the speed increaser variants. As was presented in the first part of the paper, in order to obtain a multiplication ratio i = 4 ± 10%, the following combination of teeth numbers ensures a smaller overall dimension and a higher efficiency: z1 = 34, z2 = 20, z3 = 22 and z4 = 28 teeth. Knowing from the technical literature the efficiencies of each component mechanism (η12 = η34 = 95%) [1, 2], the efficiency of the speed increaser (η ) can be obtained according to relation (4). Thus, for this variant, the transmission kinematical and dynamic parameters are obtained based on the following relations [2, 6]: • the internal transmission ratio (i0 ) H H i0 = iH 14 = i12 · i34 =

ω1H ; ω4H

iH 12 =

z2 ; z1

iH 34 =

z4 ; z3

i0 =

z2 z4 · = 0.748 (1) z1 z3

• the multiplication ratio (i) i=

1 = 3.978; i41H

i41H =

ω14 ω − ω4H = 1H = 1 − i0 = 0.2513 ωH4 −ω4H

(2)

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• the internal efficiency (η 0 ) H H H η0 = η14 = η12 · η34 = 0.95 · 0.95 = 0.9025

(3)

• the transmission efficiency (η )
−TH T 1 1 − i0 · η0−1 −ωH4 · TH −TH 1
η = = · = − = 0.67 ω14 · T1 T1 ω14 ω1H i41H 1 − i0 (4) where izxy is the transmission ratio from element x to element y, while element z is considered blocked; ωxy represents the relative speed between elements x and y. 4 = η1H

The angular velocities and angular accelerations transmission functions (5) can be established based on relation (2), while the moments transmission function (6) based on relation (4): i41H =

ω14 ω ⇒ ωH4 = 414 = i · ω14 = 3.97 · ω14 ωH4 i1H

εH4 =

ε14 = i · ε14 = 3.97 · ε14 i41H

4 η = η1H =

(5)

−ωH4 · TH 4 ⇒ TH = −T1 · i41H · η1H ; ω14 · T1

TH =

4 T1 · η1H i

(6)

In the cases of neglecting friction and considering friction, the moments transmission function is given by the following relations: TH = T1 · i41H =

T1 = 0.25 · T1; i

TH =

T1 · η = 0.17 · T1 i

(7)

3 Premises for Dynamic Modeling The speed increaser dynamic modeling relies on the following premises: • in the dynamic modeling, the inertial effects due to the satellite gears rotation are neglected (their masses being considered in the axial inertial moment of the afferent carrier shaft, Fig. 2b), while the inertial effects of the mobile central elements are considered integrated into the shafts that materialize the external links of the planetary gears; under this premise, the static correlations between the external torques of each planetary gear are valid, while the dynamic correlations interfere only for the shafts that materialize the planetary gears external links. The mechanical inertia momentums of the two shafts (see Fig. 2) are J1 = 0.035 [Kg · m2];

JH = 0.02 [Kg · m2]

(8)

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Fig. 2 The scheme of the planetary speed increaser: (a) the dynamic scheme for the input element, (b) the dynamic scheme for the output element, and (c) the block diagram of the considered machine (water turbine-speed increaser-electric generator).

• the rubbing effect is considered by means of the efficiency (η ); • the Turgo turbine and the generator have the following mechanical characteristics, which were established on experimental stands: Tt = −0.583 · ωt + 38.132 [Nm];

−Tg = 0.0089 · ωg − 4.243 [Nm]

(9)

4 Dynamic Modeling In the dynamic modeling, the main objective is to determine the transmission functions for the angular velocities, accelerations and moments, relative to time:

ωt = ωt (t) ;

εt = εt (t) ;

ω14 = ω14 (t) ;

ε14 = ε14 (t) ;

ωH4 = ωH4 (t) ; ωg = ωg (t) ;

Tt = Tt (t)

εH4 = εH4 (t) ;

εg = εg (t) ;

T14 = T14 (t) TH4 = Tt (t)

Tg = Tg (t)

(10)

The dynamic modeling is made by means of Figs. 2a, 2b and 2c, for the following cases: I. neglecting friction, and II. considering friction.

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I. In case I, the Lagrange equation is being used [5], whose quantities are resulting from Fig. 2:     d ∂ Ec ∂ Ec − =Q dt ∂ ω ∂φ Ec =

 1 J1 · ω12 + JH · ωH2 ; 2

Q = Tt · i41H + Tg

(11)

in which J1 and JH represent the mechanical inertia momentums of the elements 1 and H, respectively. Deriving Ec first relative to ω H (as independent variable) and then relative to time, it can be written as     d ∂ Ec d ∂ Ec = ε1 (1 − i0)J1 + εH JH ; =0 dt ∂ ωH dt ∂ φ

ε 1 (1 − i0)J1 + εH JH = Tt · i41H + Tg

(12)

By replacing the known parameters into relation (12), the dynamic equation without friction outcomes (analytical form (13), numerical form (14):    2   2    − bg − bt 1 − i0 = 0 (13) εH JH 1 − i0 − ω ag + at 1 − i0

εH · 0.02221 − ωH · 0.0279 + 5.35221 = 0

(14)

II. In case II, according to Figs. 2a, 2b and 2c, the following system of equations can be written using the Newton–Euler method: JH εH = Tg − TH (15) The equation used for modeling the machine dynamic system (Fig. 2c) is obtained solving system (15) and taking into account friction:    2 4   2 4    4 − ω ag + at 1 − i0 · η1H − bg − bt 1 − i0 · η1H εH JH 1 − i0 · η1H =0 (16) εH · 0.02149 + ωH · 0.0161 + 2.268 = 0 (17) T1 + TH + T4 = 0;

T1 · i41H · η + TH = 0;

J1 ε1 = Tt − T1 ;

5 Numerical Simulations The values of the output and input angular speeds in state-state regime (εH = 0) are obtained from relations (14) and (17): • when friction is neglected: ωH = 191.407 [s−1 ], ω1 = 48.107 [s−1 ]; • when friction is considered: ωH = 140.869 [s−1 ], ω1 = 35.405 [s−1 ].

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Fig. 3 Simulink scheme that models the motion equation of the planetary speed increaser.

The Simulink scheme that models the motion equation of the planetary speed increaser is presented in Fig. 3. The machine dynamic response, including the speeds, accelerations and moments as functions of time, for the turbine, speed increaser and generator is drawn in Fig. 4.

6 Conclusions • The paper presents the dynamic model of the planetary speed increaser with deformable element that was proposed in the first part of the paper. The dynamic response of the aggregate composed of a Turgo turbine – speed increaser – electric generator is obtained by means of Matlab-Simulink software. • The theoretical aspects, from which the dynamic response was obtained, are presented in the paper; the mechanical momentums of inertia were considered and the mechanical characteristics of the Turgo turbine and generator were established on experimental stands. • The considered inertia mechanical momentums (see Fig. 2) have as an input element 1, the turbine rotor, the input shaft, while for the output element H – the output shaft, generator rotor and the inertial effect of the satellite considered as a concentrated mass. • The analyzed planetary transmission with deformable element (cog belt, chain) increases the input speed 3.97 times and decreases the input moment 5.84 times. • The small hydropower plant consisting of turbine-speed increaser-generator starts practically in about 5 s (without friction) and in 7.5 s (with friction), after which enters in the steady-state regime.

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Fig. 4 Dynamic response of the aggregate: turbine/input angular speed (a), output angular/generator speed (b), turbine/input angular acceleration (c), generator/output angular acceleration (d), turbine moment (e), input moment (f), output moment (g) and generator moment (h).

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• The dynamic model is useful in the design of the control system for the small hydro stations. The system control program can be established by considering certain environmental conditions/seasons. • Based on the dynamic modeling, the authors will accomplish the design, manufacturing and testing of the speed increaser for an off-grid hydropower station in the framework of a research project.

Acknowledgment The research work reported in the paper was made possible by grant ID 140 “Innovative mechatronic systems for micro hydros, meant to the efficient exploitation of hydrological potential from off-grid sites”.

References 1. American Chain Association, Standard Handbook of Chains: Chains for Power Transmission and Material Handling, 2nd Edition, Dekker Mechanical Engineering, 2005. 2. Diaconescu, D. and Duditza, Fl., Wirkungsgradberechnung von zwangl¨aufigen Planetengetrieben. Teil I: Entwicklung einer neuen Methode. Antriebstechnik, 33(10):61–74, 1994. 3. Harvey, A., Micro-Hydro Design Manual, TDG Publishing House, 2005. 4. Jaliu, C., Diaconescu, D.V., Neagoe, M., and S˘aulescu, R., Dynamic features of speed increasers from mechatronic wind and hydro systems. Part I. Structure. Kinematics. In Proc. of EUCOMES 08. The 2nd European Conference on Mechanism Science, Cassino, Italy, September, pp. 355–363, Springer, 2008. 5. Jaliu, C., Diaconescu, D.V., Neagoe, M., and S˘aulescu, R.: Dynamic features of speed increasers from mechatronic wind and hydro systems. Part II. Dynamic aspects. In Proc. of EUCOMES 08. The 2nd European Conference on Mechanism Science, Cassino, Italy, September, pp. 365– 373, Springer, 2008. 6. Jaliu, C., Diaconescu, D.V., Neagoe, M., S˘aulescu, R., and V˘at˘as¸escu, M., Conceptual synthesis of speed increasers for renewable energy systems. In Proc. of the 10th IFToMM International Symposium on Science of Mechanisms and Machines, SYROM 2009, Bras¸ov, Romania, September 12–15, pp. 171–183, Springer, 2009. 7. Von Schon, H.A.E.C., Hydro-Electric Practice – A Practical Manual of the Development of Water, Its Conversion to Electric Energy and Its Distant Transmission, France Press, 2007.

Simplified Calculation Method for the Efficiency of Involute Helical Gears M. Pleguezuelos, J.I. Pedrero and M. S´anchez Department of Mechanics, Universidad Nacional de Educaci´on a Distancia (UNED), Madrid, Spain; e-mail: [email protected]

Abstract. The traditional methods for the calculation of the efficiency of cylindrical gear transmissions are based on the hypotheses of constant friction coefficient and uniform load distribution along the line of contact. However, the changing rigidity of the pair of teeth along the path of contact produces a non–uniform load distribution, which has significant influence on the friction losses, due to the different relative sliding at any point of the line of contact. In previous works, the authors obtained a non-uniform model of load distribution based on the minimum elastic potential criterion, which was applied to preliminary calculations of the efficiency of spur and helical gears. Recently, a new analytic formulation of the load distribution has been found. In this work, this new model of load distribution is applied to study the efficiency of involute spur gears with transverse contact ratio between 1 and 2, assuming the friction coefficient to be constant along the path of contact. Analytical expressions for the power losses due to friction, for the transmitted power and for the efficiency are presented. Key words: helical gears, load distribution, elastic potential energy, efficiency

1 Introduction The study of the efficiency of gear transmissions is an important aspect of the design process due to the significant cost and the environmental impact associated to power loss. The efficiency of gear transmissions is high; but the heat produced by power losses may result in surface defects arising after operating periods shorter than the expected ones. Classic models of efficiency of spur gears available in technical literature [1, 4, 9] are based on the hypotheses of constant friction coefficient and uniform load distribution along the line of contact. Neither of them is accurate, but the efficiency of spur gears is high, and very accurate calculations were not required in the past. But the transmitted power to size ratio is growing quickly nowadays and more accurate models may be suitable. Calculation models are based on power losses due to the friction loads along the meshing cycle. And obviously power losses depend on the load, the relative sliding and the friction coefficient at any contact point.

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Empirical models for load distribution and a constant friction coefficient have been used in some studies, as the ones of Michlin [10] and H¨ohn [5]. A preliminary authors’ study using the load distribution model of minimum elastic potential can be found in [12]. Other models using non–uniform friction coefficient models can be found in [3, 6, 20]. More complex models using the EHL theory to formulate the variation of the friction coefficient have been developed by Martin [8], Wu [21] and Li [7]. A similar model combined with load distribution models, including experimental validation, was presented by Xu [22]. Other studies to compute the power loss based on experimental data can be found in the literature [2, 18]. The authors developed a model of load distribution along the line of contact based on the minimum elastic potential criterion [11, 19] and applied it to the determination of the efficiency of spur gears [14–16], resulting in slightly greater values of the efficiency comparing with those obtained from the traditional models [1, 4, 9]. Preliminary studies of the efficiency for helical gears have been also reported [13], but results were given as tabulated values for specific studies obtained from numerical integration of the losses along the meshing cycle. Recently, a new analytic formulation of the load distribution has been provided [17], which has been used in this work to develop an accurate, analytic equation for the efficiency of helical gears with transverse contact ratio between 1 and 2.

2 Model of Load Distribution Pedrero et al. [17] present in detail the model of load distribution of minimum elastic potential energy. In general terms, that model was obtained by computing the total elastic potential energy from the equations of the theory of elasticity, considering all pairs of teeth in simultaneous contact, with an unknown fraction of the load acting on each one, and minimizing its value by means of variational techniques (Lagrange’s method). It has been demonstrated that the load per unit of length depends on the inverse unitary potential v(ξ ), which is defined as the inverse of the elastic potential for unitary load and face width. Obviously, the inverse unitary potential depends on the contact point, which is described by the ξ parameter of the contact point at the pinion profile as:
2 z1 rC1 ξ= −1 (1) 2 2π rb1 where z is the number of teeth, rC the distance from the rotation axis to the contact point, rb the base radius and subscript 1 denotes the pinion (subscript 2 will denote the wheel). For helical gears, the load per unit of length at a point of the line of contact described by ξ , at the meshing position corresponding to a reference transverse section contacting at a point described by ξ 0 , is given by [17]:

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Fig. 1 Typical shapes of the graphs of the inverse unitary potential v(ξ ) and its integral Iv (ξ0 ).

 εβ cos βb v (ξ )   F f ξ , ξ0 = b Iv ξ0

(2)

where εβ is the axial contact ratio, βb the base helix angle, b the face width, F the total load and function Iv (ξ 0 ) can be derived from: z1 −1     Iv ξ0 = v (ξ ) d ξ = ∑ j=0

lc

ξ0 + j

ξ0 + j−εβ

v (ξ ) d ξ

(3)

where lc denotes the line of contact. The reference transverse section can be any arbitrary transverse section of the helical tooth, however the expression for Iv (ξ 0 ) depends on the chosen section. The reference transverse section corresponding to Eq. (3) is the end section of the tooth with higher contact point. The inverse unitary potential v(ξ ) is described very accurately by [17]:   v (ξ ) = cos b0 (ξ − ξm ) (4) where

ξm = ξinn +

εα ; 2

b0 =

 

−1/2 ε 2 1 1+ α −1 2 2

(5)

being ξ inn the involute parameter of the inner point of contact of the pinion and εα the transverse contact ratio. Function v(ξ ) is equal to 0 outside the interval of contact [ξinn , ξinn + εα ]. Figure 1 left shows the typical aspect of function v(ξ ) for non-undercut teeth. For helical gears, the load per unit of length at any contact point (described by ξ ) at any meshing position (described by ξ 0 ) was given by Eq. (2), in which v(ξ ) and Iv (ξ 0 ) are given by Eqs. (4) and (3), respectively. The shape of function Iv (ξ 0 ) is similar to the shape of the evolution of the length of contact along the mesh cycle. Depending on whether the sum of the fractional parts of both transverse and axial contact ratios (dα and dβ , respectively) is less than 1 or not, function Iv (ξ 0 ) takes different shapes (Figure 1 right represents the first case). Note that the influence of dα + dβ is the same as that on the length of contact.

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3 Model of Efficiency The transmitted energy from contact at the inner point of contact of a pinion tooth to contact at the same point of the next tooth (i.e., ∆ ξ 0 = 1) is given by Wu = Frb1

2π z1

(6)

Similarly, the total mechanical (load-dependent) energy lost during a small rotation of the pinion can be expressed as the friction coefficient µ , multiplied by the differential normal load f dl and by the relative sliding. All these parameters depend on the contact point and can be expressed as a function of ξ as [12]:      2π rb1  2
 2π  d Ws = µ ( f dl) rb1 + rb2  ξ − tgα t  d ξ0 (7) rb2 z1 z1 where αt
is the operating transverse pressure angle. We can define a new parameter ζ of the profile points as ζ = ξ − ξinn , in such a way that the interval between contact at the inner point of contact and the next tooth contacting at the inner point of contact is given by 0 ≤ ζ ≤ 1, and the complete meshing interval of a tooth is given by 0 ≤ ζ ≤ εα , being εα the transverse contact ratio. Then, integrating the losses along the complete line of contact and the complete meshing cycle, the efficiency can be expressed as [13]:

Ws 1 1 Iη η = 1− = 1 − 2π µ + (8) Wu z1 z2 where Iη =

   1 Iη v ζ0 0

  d ζ0 Iv ζ0

(9)

being: Iη v (ζ0 ) =

z1 −1  ζ +i 0

∑

i=0

ζ0 +i−εβ

v(ζ ) |ζ − λ εα | d ζ ;

Iv (ξ0 ) =

z1 −1  ζ + j 0

∑

j=0

ζ0 + j−εβ

v(ζ )d ζ

(10)

where λ is the ratio between εα 1 and εα , with εα 1 being the contribution to the contact ratio of the approach interval:

εα 1 =

z1 − ξinn ; 2π

λ=

εα 1 εα

(11)

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Fig. 2 Function Iη (εα , εβ , λ ) and coefficients C1 (λ ) and C2 (λ ).

4 Approximate Equation for the Efficiency From Eqs. (9) and (10) we can conclude that function Iη depends on three dimensionless parameters: εα , εβ and λ . The problem is to find an analytic expression of Iη as a function of these three parameters. A lot of calculations have been carried out, considering the following ranges of variation of the parameters: 1 < εα < 2;

0 < εβ < 2;

0.4 < λ < 0.6

(12)

Results are shown in Figure 2. It can be observed that all the points of equal λ are placed at well defined planes, all of them parallel to εβ axis. Consequently, function Iη will not depend on ε β , and will vary linearly with εα , i.e.: Iη = C1 (λ ) + C2 (λ ) εα

(13)

It can also be observed that planes corresponding to λ = λ0 and λ = 1 − λ0 are coincident, as easily understandable. Coefficients C1 (λ ) and C2 (λ ) were obtained by linear regression for several values of λ , resulting in the values shown in Figure 2. Then, both coefficients were fitted to a second degree parabola, which give the result: C1 (λ ) = −0.0236 + 0.1666(λ − 0.5)2 C2 (λ ) = 0.2368 + 1.1340(λ − 0.5)2

(14)

Correlation factors obtained for these expressions were 0.9968 and 0.9999, respectively. Also, error in the estimation of the efficiency with Eqs. (8), (13) and (14) with respect to the value obtained by numerical integration of Eq. (9) was never higher than 0.1%, which means an absolute error in the estimation of the efficiency less than 10−3 .

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Fig. 3 Relative errors in the estimation of function Iη and the efficiency.

Fig. 4 Influence of the helix angle and the number of teeth on pinion and wheel.

5 Results Some studies have been carried out to find design recommendations. The most interesting results are the following: 1. The estimation of the efficiency through a similar approach but considering the load uniformly distributed along the line of contact, yields conservative values. Differences are not significant in terms of the efficiency (higher in all the cases than 0.98, Figure 4 left), but more important when expressed in terms of the losses (up to 50% higher, Figure 4 right). 2. Higher values of the efficiency are obtained for gears designed with balanced specific sliding on pinion and wheel, and values of the face width and the rack shift coefficients resulting in a maximum difference between the maximum and minimum length of contact along the meshing cycle. 3. Higher values of the efficiency are also obtained for higher values of the pressure and helix angles and the operating center distance.

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6 Conclusions A model of efficiency for involute helical gears has been developed from a non uniform load distribution model, based on the minimum elastic potential criterion, and assuming the friction coefficient to be constant along the path of contact. The efficiency has been expressed by a very simple, analytic equation, as a function of the friction coefficient, the number of teeth on pinion and wheel, the transverse contact ratio and the approach ratio. This expression allows to compute the efficiency with very small errors, always lower than 0.001 (0.1% relative error), if compared with numerical calculations. In spite of its simplicity, the proposed equation provides values of the power loss and the efficiency very similar to those obtained by experimental techniques, for a wide range of geometric and operating parameters. Some studies of the influence on the efficiency of several design parameters have been carried out. It can be affirmed that the efficiency increases for bigger pressure and helix angles and operating center distance, for balanced specific sliding on pinion and wheel, and also for values of the face width and the rack shift coefficients providing a maximum difference between the maximum and minimum length of contact along the meshing cycle.

Acknowledgment Thanks are expressed to the Spanish Council for Scientific and Technological Research for their support of the project DPI2008–05787.

References 1. Buckingham E., Analytical Mechanics of Gears, McGraw-Hill, 1949. 2. Diab Y., Ville F., Velex P. and Changenet C., Windage losses in high speed gears – Preliminary experimental and theoretical results, Journal of Mechanical Design, 126, 2004. 3. Diab Y. and Velex F., Prediction of power losses due to tooth friction in gears, Trybology Transactions, 29, 2006. 4. Henriot G., Engrenages. Conception, Fabrication, Mise en Oeuvre, 7th ed., Dunod, Paris, 1999. 5. H¨ohn B.-R., Michaelis K. and Wimmer A., Low loss gears, AGMA Technical Paper 05FTM11, 2005. 6. Lehtovaara A., Calculation of sliding power loss in spur gear contacts, Tribotest Journal, 9(1), 2002. 7. Li S., Vaidyanathan A., Harianto J. and Kahraman A., Influence of design parameters on mechanical power losses of helical gears pairs, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 3, 2009. 8. Martin K.F., The efficiency of involute spur gears, Journal of Mechanical Design, 103, 1981. 9. Merritt H.E., Gears, Sir Isaac Pitman & Sons, 1946.

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10. Michlin Y. and Myunster V., Determination of power losses in gear transmissions with rolling and sliding friction incorporated, Mechanism and Machine Theory, 37, 2002. 11. Pedrero J.I., Art´es M. and Fuentes A., Modelo de distribuci´on de carga en engranajes cil´ındricos de perfil de evolvente, Revista Iberoamericana de Ingenier´ıa Mec´anica, 3(1):31– 43, 1999. 12. Pedrero J.I., Estrems M. and Fuentes A., Determination of the efficiency of cylindric gear sets. In Proc. IV World Congress on Gearing and Power Transmissions, Paris, France, Vol. 1, pp. 297–302, 1999. 13. Pedrero J.I. and Pleguezuelos M., Modelo de c´alculo del rendimiento de transmisiones por engranajes helicoidales. In Actas VII Congreso Iberoamericano de Ingenier´ıa Mec´anica, M´exico D.C., M´exico, 2006. 14. Pedrero J.I. and Pleguezuelos M., Modelo de c´alculo del rendimiento de transmisiones por engranajes rectos. In Actas VIII Congreso Iberoamericano de Ingenier´ıa Mec´anica, Cuzco, Per´u, 2008. 15. Pedrero J.I., Pleguezuelos M. and Sanchez M., Model of efficiency of high transverse contact ratio spur gears. In Proc. JSME International Conference on Motion and Power Transmissions, Sendai, Japan, 2009. 16. Pedrero J.I., Pleguezuelos M. and Mu˜noz M., Simplified calculation method for the efficiency of involute spur gears, ASME Technical Paper DETC2009/PTG-87179, 2009. 17. Pedrero J.I., Pleguezuelos M., Art´es M. and Antona J.A., Load distribution model along the line of contact for involute external gears, Mechanism and Machine Theory, doi:10.1016/j.mechmachtheory.2009.12.009, 2010. 18. Petry-Johnson T.T., Experimental investigation of spur gear efficiency, MS Thesis, The Ohio State University, 2007. 19. Pleguezuelos M., Modelo de distribuci´on de carga en engranajes cil´ındricos de perfil de evolvente, Capitulo 6: Modelo de rendimiento, Tesis Doctoral, UNED, Madrid, 2006. 20. Vaishya M. and Houser D.R., Modeling and measurement of sliding friction for gear analysis, AGMA Technical Paper 99FTMS1, 1999. 21. Wu S. and Cheng H.S., A friction model of partial EHL contacts and its application to power loss in spur gears, Tribology Transactions, 34, 1991. 22. Xu H., Development of a generalized mechanical efficiency prediction methodology for gear pairs. MS Thesis, The Ohio State University, 2005.

Cam Size Optimization of Disc Cam-Follower Mechanisms with Translating Roller Followers P. Flores Mechanical Engineering Department, University of Minho, 4800-058 Guimar˜aes, Portugal; e-mail: [email protected]

Abstract. The main purpose of this work is to present a general approach for cam size optimization of disc cam mechanisms with eccentric translating roller followers. The objective function considered in the present study takes into account the three major parameters that influence the final cam size, namely the base circle radius of the cam, the radius of the roller and the offset of the follower. Furthermore, geometric constraints related to the maximum pressure angle and minimum radius of curvature are included to ensure good working conditions of the system. Finally, an application example is offered with the intent to discuss the main procedures and assumptions adopted in this work. Key words: cams, roller followers, synthesis, optimization

1 Introduction The cam synthesis procedure, which deals with the cam profile determination to ensure a desired follower motion, plays a key role in the cam design process. Moreover, the cam size optimization refers to the evaluation of the cam profile for a given follower motion taking into account the minimization or maximization of a specific cost or profit function. In general, minimization cost functions are related to material volume and inertia of moving parts, while maximization cost functions are associated with the cams performance, such as for example, the force transmission efficiency [1, 5, 6, 12, 14–17]. The subject of cam design process has been an intensive topic of investigation during the last years. Cardona and G´eradin [2] studied the kinematics and dynamics of mechanisms with cams. This work was then extended by Fisette et al. [9] to model cam systems using the multibody systems methodologies. Cardona et al. [3] also based on the multibody systems, proposed a methodology to design cams for motor engine valve trains using a constrained optimization algorithm. Demeulenaere and De Schutter [8] developed a design procedure to perform the synthesis of cams that compensate the fluctuation in the camshaft speed. Mandal and Naskar [11] introduced the concept of control points in the synthesis of optimized cam mechanism.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_26, © Springer Science+Business Media B.V. 2010

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They also considered different spline types to define the cam profile. Carra et al. [4] studied the synthesis of cams with negative radius roller-follower. The purpose of this work is to present a general procedure for cam size optimization of disc cam mechanisms with translating roller followers. The objective function used here accounts for the influence of the base circle radius (Rb ), offset of the follower (e) and radius of the roller (Rr ) as design variables. Furthermore, additional constraints related to the performance of the cam mechanism are considered, namely those associated with the maximum admissible pressure angles and minimum curvature radius of the cam surface. The resulting constrained optimization multivariable problem is highly nonlinear and even non-differentiable.

2 Disc Cams with Translating Roller Followers Figure 1 shows a representation of a cam disc mechanism with an offset translating roller follower. By analyzing the geometric configuration of Fig. 1, it can be observed that the normal to the common tangent between roller and cam intersects the horizontal axis at point O24 , which represents the instantaneous center of rotation between cam and follower. Thus, since the follower describes a translational motion, all points of the follower have velocities equal to that one of point O24 . So, the velocity of point O24 can be written as [7] vO = y˙ = ωC O12 O24 24

(1)

Dividing Eq. (1) by ωC and knowing that y
= yields

y˙ dy = ωC dθ

y
= O12 O24

(2)

(3)

which is a strictly geometric relation. Therefore, the term O12 O24 represents the converted velocity y/ ˙ ωC of the follower. Now, based on Fig. 1, the following relation can also be written as O12 O24 = e + (a + y) tan φ (4) Solving Eq. (4) into Eq. (3) for φ yields

φ = tan−1

y
− e a+y

where the geometric parameter a is given by [2]
a = (Rb + Rr )2 − e2

(5)

(6)

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227

Fig. 1 Schematic representation of a disc cam mechanism with offset translating roller follower.

From the analysis of Eq. (5), it can be concluded that for a given follower motion, that is, by knowing y and y
, the three geometric parameters Rb , Rr and e can be adjusted to obtain a suitable pressure angle. The radius of curvature of the cam surface, ρ p , is another important factor that influences the cam size and performance of the cam mechanisms. For a cam mechanism with translating roller follower the radius of the pitch curve can be written as a parametric expression as [10]  3  2 Rb + Rr + y + y
2 ρp =  (7) 2   Rb + Rr + y + 2y
2 − Rb + Rr + y y



3 Cam Profile Generation In this study, the analytical approach based on the theory of envelopes is used to generate the cam profile [7], in which the desired positions of the follower are determined from an inversion of the cam-follower system in which the cam is held stationary. In what regards to Fig. 2, the family of circles in the XY plane describing the roller follower is given by [10] F(X ,Y, θ ) = (X − XR )2 + (Y − YR )2 − R2r = 0

(8)

where Rr is the roller radius and (XR , YR ) are the coordinates of the center roller R XR = (Rb + Rr ) sin(θ + δ ) + y sin θ

(9)

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Fig. 2 Theory of envelopes to generate cam profile.

YR = (Rb + Rr ) cos(θ + δ ) + y cos θ

(10)

in which the parameter δ is given by the following relation [2]

δ = sin−1

e Rb + Rr

(11)

Differentiating Eq. (8) with respect to parameter θ , yields that dX dY ∂F = −2(X − XR ) R − 2(Y − YR ) R = 0 ∂θ dθ dθ

(12)

and from Eqs. (9) and (10) results in dXR = (Rb + Rr ) cos(θ + δ ) + y cos θ + y
sin θ dθ dYR = −(Rb + Rr ) sin(θ + δ ) − y sin θ + y
cos θ dθ Solving Eqs. (8) and (12) gives the Cartesian coordinates of the cam profile  X = XR ± Rr  Y = YR ∓ Rr

dYR dθ

dXR dθ

 

 

dXR dθ dXR dθ

2

2



dYR + dθ 

dYR + dθ

(13) (14)

2 −1/2 (15) 2 −1/2 (16)

It is should be noted that the plus and minus signs in Eqs. (15) and (16) reflect the two envelopes, an inner profile and an outer profile, as shown in Fig. 2.

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229

4 Cam Synthesis and Objective Function In what follows some of the most fundamental constraints and an objective function are presented in order to define a general procedure that can help in the definition of the optimum cam size for a prescribed follower motion. The two first constraints associated with the cam synthesis procedure are the maximum allowed values of the pressure angle for the rise and return follower motions, which can be stated as   (17) C1 : φRise−max = max φ (h, β1 , Rb , e, Rr , y, y
) ≤ 30   (18) C2 : φReturn−max = max φ (h, β2 , Rb , e, Rr , y, y
) ≤ 45 where h is the stroke of the follower, which is considered to be equal for the rise and return phases, and β1 and β2 represent the amplitudes of the cam angle rotation for the rise and return follower motions, respectively. The maximum value of the pressure angle can be evaluated by employing Eq. (5) if the follower motion is prescribed, which is commonly known as the cam law [1]. Another important constraint related to the pressure angle directly results from the domain analysis of the function represented by Eq. (5), which yields that C3 : Rb + Rr ≥ e

(19)

The constraints imposed to the radius of curvature of the pitch curve, ρ p , for the convex and concave cam surfaces of a disc cam with an eccentric translating roller follower can be written as, respectively C4 : ρ p (Rb , Rr , y, y
, y

) > Rr

(20)

C5 : ρ p (Rb , Rr , y, y
, y

) < 0

(21)

In short, unacceptable cam profiles are within the interval 0 ≤ ρ p ≤ Rr . The value of the radius of curvature can be computed by using Eq. (7) for both convex and concave cases, when the follower motion is known. From a practical engineering point of view, two linear inequality constraints should be used in order to guarantee that the system can be easily assembled C6 : Rr ≤ e

(22)

C7 : e ≤ Rb

(23)

Finally, in order to ensure that the obtained results, in the cam synthesis procedure, are reasonable, some simple bounds are imposed to variables Rb , e and Rr ub C8 : Rlb b ≤ Rb ≤ Rb

(24)

C9 : elb ≤ e ≤ eub

(25)

230

P. Flores ub C10 : Rlb r ≤ Rr ≤ Rr

(26)

where the superscripts lb and ub denote the lower and upper bounds, respectively. The objective function, as defined in the present study aims to minimize cam size and the maximum absolute pressure angles in rise and return motions. The design variables considered are the base circle radius of the cam, the eccentricity and the radius of roller follower. The objective function considered here includes three different components. The first one influences the mass of the cam via the size of the base circle radius. The other two terms are related to the performance of the system, that is, they are associated with the pressure angles for the rise and return follower motion. In this work, all of the three components of the objective function have the same weight, that is, they are equally penalized. Thus, to conclude, the objective function can be stated as o f (Rb , e, Rr ) = Rb + tan−1 φRise−max + tan−1 φReturn−max

(27)

As a result, the cam size optimization procedure presented above leads to a problem that can be formulated as follows. Considering that the user specifies the values of ub lb ub lb ub the parameters h, β1 , β2 , Rlb b , Rb , e , e , Rr and Rr , the problem is min o f

Rb , e, Rr

(28)

subjected to constraints C1 to C10 . In the present work, this optimization problem is solved by employing the in-built function of Matlab code named fmincon [13].

5 Results and Discussion In this section, a disc cam-follower mechanism with a translating roller follower is considered for a demonstrative application example. The follower motion is of type RDR, that is, Rise-Dwell-Return. The follower describes a rise of 30 mm during the cam rotation from 0 to 100◦. Then, the follower remains stationary for amplitude of cam rotation angle equal to 110◦. Finally, the follower returns to its initial position during the remaining cam rotation. The base circle radius of the cam is 20 mm. The radius of the roller is equal to 10 mm and the eccentricity of the follower is null. These geometric data are considered as standard in the present study and are used as reference for comparative purpose. In order to keep the analysis simple, the angular velocity of the cam is assumed to be equal to 1 rad/s. Figure 3a shows the evolution of the pressure angle for a complete cam rotation. A cicloidal motion of the follower was considered. By observing Fig. 3a it is visible that the value of the pressure in the lift phase is higher than 30◦ , which leads to problematic working conditions. Traditionally, this problem can be solved by employing a trial-and-error approach, i.e., changing the values of Rb and/or e and analyzing the consequences in terms of the pressure angle. In the present example, if the value of Rb is incremented to 30 mm, then the maximum pressure will be reduced to 28.4◦,

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Table 1 Global obtained results when Rb ∈ [20, 60], e ∈ [0, 20] and Rr ∈ [0, 20]. Rb0 e0 Rr0

20.0 0.0 10.0

60.0 20.0 20.0

40.0 10.0 10.0

20.0 10.0 20.0

Rbopt eopt Rropt of

28.0 14.6 14.6 28.0+20.1+35.7

28.0 14.6 14.6 28.0+20.1+35.7

28.0 14.6 14.6 28.0+20.1+35.7

28.0 14.6 14.6 28.0+20.1+35.7

Fig. 3 (a) Pressure angle for a complete cam rotation; (b) Standard cam profile and optimized cam profile obtained with the proposed approach.

which solves the problem. Another possibility is to include, for instance, an eccentricity of 15 mm, which results in a maximum pressure angle in the lift phase of 25.3◦ although leading, in turn, to an increase of the pressure angle to 45.6◦ in the return motion. It is obvious that different combinations among the design parameters Rb , e and Rr lead in different results. Therefore, the question naturally asked by the designer is how to overcome this difficulty, and how to determine the optimum cam size value? With the intent to find out the optimum cam size, the methodology presented in the previous section is used. Table 1 shows the obtained results when all of the three design variables, Rb , e and Rr , are taken into account in the optimization problem within the intervals [20, 60], [0, 20], [0, 20], respectively. The obtained solution is the same for the four different sets of initial guesses. In the present example, operating conditions were improved by increasing the base circle radius of the cam from 20 to 28.0 mm, by introducing an eccentricity equal to 14.6 mm and by specifying the nominal radius of roller equal to 14.6 mm. It should be highlighted that the maximum number of iterations required in all numerical solutions is less than 26. Finally, Fig. 3b presents the two cam profiles obtained for the standard and optimized cases when the numerical approach is used.

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6 Conclusions A methodology to deal with the cam size optimization of disc cam systems with offset translating roller followers has been presented in this paper. In the sequel of this process the most fundamental issues associated with the cam theory analysis and synthesis were revisited. It was shown that the shape of the cam profile is conditioned by the cam law selected, together with the geometric parameters that characterize the cam and the roller follower of a specific system. The methodology developed for optimization of the cam size was based on the most suitable cam operating conditions, namely in what concerns to the maximum allowed pressure angle, radius of curvature, base circle radius, eccentricity and roller radius. In addition, some relevant constraints related to the cam design process were also considered in the present work. The resulting problem corresponds to the standard form of a minimum finding problem for a constrained nonlinear multifunction. The proposed methodology is simple and can be easily extended by including other constraints, such as those associated with the velocity and acceleration limits, as well as to other types of cam laws.

References 1. Angeles, J. and Lopez-Cajun, C.S., Optimization of Cam Mechanisms, Kluwer Academic Publishers, Dordrecht, 1991. 2. Cardona, A. and G´eradin, M., Kinematic and dynamic analysis of mechanisms with cams. Computer Methods in Applied Mechanics and Engineering, 103:115–134, 1993. 3. Cardona, D., Lens, E., and Nigro, N., Optimal design of cams. Multibody System Dynamics, 7:285–305, 2002. 4. Carra, S., Garziera, R., and Pellegrini, M., Synthesis of cams with negative radius follower and evaluation of the pressure angle. Mechanism and Machine Theory, 39:1017–1032, 2004. 5. Chang, W-T. and Wu, L-I., A Simplified method for examining profile deviations of conjugate disk cams. Journal of Mechanical Design, 130:052601-11, 2008. 6. Chen, C. and Angeles, J., Optimum kinematics design of drives for wheeled mobile robots based on cam-roller pairs. Journal of Mechanical Design, 129:7–16, 2007. 7. Chen, F.Y. Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982. 8. Demeulenaere, B. and De Schutter, J., Synthesis of inertially compensated variable-speed cams. Journal of Mechanical Design, 125:593–601, 2003. 9. Fisette, P., P´eterkenne, J.M., Vaneghem, B., and Samin, J.C., A multibody loop constraints approach for modelling cam/follower devices. Nonlinear Dynamics, 22:335–359, 2000. 10. Flores, P., Projecto de Mecanismos Came-Seguidor, Publindustria, 2009. 11. Mandal, M. and Naskar, T.K., Introduction of control points in splines for synthesis of optimized cam motion program. Mechanism and Machine Theory, 44:255–271, 2009. 12. Marcinkevicius, A.H., Synthesis of a cam system for the measurement and follow rest of multi-step shaft necks at machining. Mechanism and Machine Theory, 42:1029–1042, 2007. 13. MatLab, 2004, Version 7. http://www.mathworks.com. 14. Mundo, D., Liu, J.Y., and Yan, H.S., Optimal synthesis of cam-linkage mechanisms for precise path generation. Journal of Mechanical Design, 128:1253–1260, 2006. 15. Ottaviano, E., Mundo, D., Danieli, G.A., and Ceccarelli, M., Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators. Mechanism and Machine Theory, 43:996–1068, 2008.

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16. Ye, Z. and Smith, M.R., Synthesis of constant-breath cam mechanisms. Mechanism and Machine Theory, 37:941–953, 2002. 17. Zhang, Y. and Shin, J-H., A computational approach to profile generation of planar cam mechanisms. Journal of Mechanical Design, 126:183–188, 2004.

The Dynamic Effects on Serial Printers Motion Transmission Systems D. Comanescu and A. Comanescu Chair of Mechatronics and Precision Mechanics, University Politehnica Bucharest, 060042 Bucharest, Romania; e-mail: [email protected], [email protected]

Abstract. The paper deals with the aspects of chaotic regimes generally applied to peripheral devices or particularly serial printers motion transmission systems. On the basis of Duffing mathematical model a wire transmission is analyzed and the optimal domain for various motion laws is determined. Key words: ink jet printer, laser jet printer, peripheral devices, chaotic regime, mathematical model

1 Introduction By studying a non-linear oscillator with a cubical rigidity term, which describes the spring “hardening” effect observed in many mechanical problems Duffing obtained the differential equation x¨ + δ x˙ − β x + α x3 = γ cos ω t

(1)

with δ > 0, α > 0 and γ > 0. Such differential equations are used to solve partial derivative equations. For example, by means of the Galerkin approximate method Moon and Holmes managed to obtain the flutter phenomenon for an elastic beam in an irregular magnetic field, and also Dowell for a plane plate placed in a fluid current [1–4]. Firstly, by ignoring the disturbing force γ cosω t, the motion equation becomes x¨ + δ x˙ − β x + α x3 = 0

(2)

which is equivalent to the first degree differential equations system x˙ = 0; y˙ = β x − α x3 − δ y

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_27, © Springer Science+Business Media B.V. 2010

(3)

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By analyzing the stability of both critical points and the equilibrium positions the solutions of the equation system are y = 0; β x − α x3 − δ y = 0

(4)

There are possible two cases for β > 0and β < 0. In the case β < 0 it is only one equilibrium position (x = 0, y = 0). For this position the characteristic values of the Jacobi array are given by the equation


−λ 1

= λ2 +δλ −β = 0 ∆1 =

(5) β −δ − λ
Since δ > 0 and β < 0 one may conclude that the equilibrium position is stable with the following characteristics: a stable node for δ > −4β , a stable furnace for δ < −4β and a centre for δ = 0. In the case β > 0 there are three equilibrium positions       (x = 0, y = 0); x = β /α , y = 0 ; x = − β /α , y = 0 (6) For the equilibrium position (x = 0, y = 0), the characteristic values of the Jacobi array are still given in the equation mentioned above, in fact the equilibrium position is unstable and therefore a saddle [5, 8]. The other two equilibrium positions are concerned, the characteristic values of the Jacobi array being given by the equation


−λ 1

= λ 2 + δ λ + 2β = 0 ∆1 =

(7) −2β −δ − λ
Since δ > 0 and β > 0 one may deduce that the two equilibrium positions are stable as follows: for δ
= 0 a stable furnace with δ < 8β and stable nodes with δ ≥ 8β and for δ = 0 centers. In order to have a global information on the phase portrait, firstly one may consider the case δ = 0 in which the differential equation system x˙ = 0; y˙ = β x − α x3 − δ y

(8)

x˙ = y; y˙ = β x − α x3

(9)

becomes

By multiplying the first differential equation by y˙ and the second one by x˙ and subtracting them the following equation is obtained yy˙ − β xx˙ + α x3 x˙ = 0

(10)

y2 /2 − β x2/2 + α x4/4 = C

(11)

By integrating it is deduced

The Dynamic Effects on Serial Printers Motion Transmission Systems

237

Fig. 1 Curves in the case β < 0 and curves in the case β > 0.

Fig. 2 Curve for δ > 0 , β < 0 and curve for δ > 0 and β > 0.

Fig. 3 The stable variety divides the Oxy plane in two regions.

a curves family depending to the C parameter. For δ
= 0 and δ > 0 one may note that the centers become stable focal points. The curves deduced from yy− ˙ β xx+ ˙ α x3 x˙ = 0 for δ > 0 are presented in Figure 2 in the cases β < 0 and β > 0. The phase portrait in the second figure of Figure 2 is the image of a shell with very sinuous channels and when it grows, it tightens very much. A trajectory that starts from a random point of the Oxy plane can either reach the stable focal point from the left side, or that of the right side. If these channels are very narrow, a certain sensibility comes up on the initial conditions, in the sense that, at the slightest variation of these conditions, the trajectory can either reach the stable focal point on the left, or that of the right side [4, 7]. This feature characterizes the chaotic motion. Despite all these, the motion is not chaotic. By eliminating the unstable invariant variety the stable variety divides the Oxy plane in two distinct regions one marked and the other unmarked (Figure 3). Due to this sensibility of the initial condition especially in the presence of the damping phenomenon, when the channels tighten very much, it is easily understood why in case of a disturbing force, a chaotic behavior of the dynamic system may show up. It is easily comprehended why complicated trajectories can be generated when a force F(t) acts continuously, trajectories that are characteristic to the chaotic movements.

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Fig. 4 a) Linear coordinates and angular coordinate, b) the stroboscopic image equivalent to a number of 4000 time tk intervals.

2 Duffing’s Particular Case The particular case of Duffing’s complete equation x¨ +

1 8 2 1 x˙ − x + x3 = cos ω t 25 5 15 5

(12)

studied by Seydel [6] has the differential equation x¨ +

1 8 1 2 x˙ − x + x3 = cos ω t 25 5 15 5

its equivalent the system being the following x˙1 = x2 ;

1 8 1 2 x˙2 = x1 − x31 − x2 + cos x3 ; 5 15 25 5

x˙3 = ω

(13)

By using the Poincar´e application in the stroboscopic form, in the sense that the values of x1 , x2 , x3 are evaluated in certain moments (k = 1, 2, 3, . . .) conveniently selected, the expression of the disturbing force from the previous differential equation suggests the period T = 2π /ω , so that tk = k · 2π /ω . The x1 and x2 may be linear coordinates and the x3 angular one (Figure 4). The aspect of the multitude of dots suggests lines “nested” in a very complicated structure. The impression that the “lines” in this diagram offer may mislead a deep increase of any “line”, which emphasizes a bunch of “sub lines”, of the same structure as the initial “lines”. Each new increase leads to the same observation: the “lines” open up in bunches of sub lines while keeping the same structure, so the auto similarity feature characteristic to fractions comparable to that of the Cantor multitude. In practice, the Duffing differential equations may show up when there are solved physics problems with partial derivative equations by using the separating variables method assuming that the general solution is a sum of products of coordinate functions, conveniently chosen and satisfying the limit conditions. By applying the Galerkin method for example, an infinite number of ordinary differential equations,

The Dynamic Effects on Serial Printers Motion Transmission Systems

239

having the time functions unknown are obtained [5, 7, 8]. Mapping the system, thus keeping a finite number of ordinary differential equations, the system such obtained may be integrated.

3 The Case of a Wire Mechanical Transmission In the case of a wire transmission for the ink-jet matrix printer carriage the wire stretching force has the expression H≥

H≥

(1 + µ  ρr ) · (µ g + x¨) · m + rJ2 x¨ eµ0 π −1 eµ0 π +1

− µ  ρr

or

x¨ · [(1 + µ  ρr ) · m + rJ2 ] + m · (1 + µ  ρr ) · µ g eµ0 π −1 eµ0 π +1

− µ  ρr

(14)

where: µ  – is the friction coefficient from the wire wheels bearing, ρ – the radius of the wire wheel shaft, m – the mass of the mobile carriage, r – the radius of the wire wheel, µ0 – the coefficient of friction between the wire and the wire wheels, J – the mass inertial moment relative to the Oz axis ( the rotation axis) of the wire wheels, µ – the coefficient of friction in the carriage translation guidance. In order to ensure the tension within the wire an elastic system of compensation is used. This is presumed to be a non-linear oscillator with a cubical rigidity term, which has the spring “hardening” phenomenon. This phenomenon is described by the differential equation x¨ + δ x˙ − β x + α x3 = γ cos ω t

(15)

with δ > 0, α > 0, γ > 0. In the particular case of the Duffing equations x¨ +

1 1 8 2 x˙ − x + x3 = cos ω t 25 5 15 5

(16)

it is adopted that x = A sint;

x˙ = A cost;

x¨ = −A sint

(17)

According to Equation (16) x¨ =

2 1 8 1 cos ω t − x˙ + x − x3 3 25 5 15

(18)

a third degree equation is obtained C1 · A3 + C2 · A + C3 = 0

(19)

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where C1 =

8 cos3 t, 15

C2 =

1 1 sint − cost + sint, 5 25

C3 =

2 cos ω t 3

(20)

By applying numerical methods to determine the wire stretching force Hk and for the following values

ω = 2π · 200;

p = 0.01;

8 1 (cos(tk ))3 ; C2k = sin(tk ); 15 15 so that there are obtained

C1k =

k = 0 . . . 100; C3k =

tk = 0.5 + pk

2 cos(ω tk ) 3

and x = −0.0005

C1k · x3 + C2k · x + C3k = 0 so1k := Find(x); Ak := so1k ; ak := −Ak · sin(tk )

µ p = 0.2;

ρ = 0.003;

µ z = 0.2;

J = 0.032;

µ = 0.4; m = 0.08; r = 0.02; g = 9.81

       aK . m 1 + µ p · ρr + rJ2 + µ · mg · 1 + µ p · ρr HK := ρ  (eµ z·π −1) eµ z +1 + µ p · r For the adopted parameters mentioned on the figures some curve variations are presented in Figures 5 and 6. For x = α t 3 + β t 2 + γ t + δ ; x = 3α t 2 + 2β t + γ ; x¨ = 6α t + 2β

(21)

taking into account Equation (16) one may obtain  2 1  cos ω t − 3α t 2 + 2β + γ + 3 25  3 8  3 1 3 αt + β t2 + γ + δ − αt + β t2 + γt + δ + 5 15 6α t + 2β =

(22)

The four equations system with four unknowns α , β , γ , δ is deduced for the limit conditions as follows: 8 2 γ δ − + − δ3 3 25 5 15  3π 1 3π 2 π π α + 2β = − β t= ⇔ α + + γ 2ω ω 25 4ω 2 ω  3  3 1 π 8 π3 π2 π π2 γ +δ − + α+ β+ α+ β +δ 5 8ω 3 4ω 2 2ω 15 8ω 3 4ω 2

t = 0 ⇔ 2β =

The Dynamic Effects on Serial Printers Motion Transmission Systems

241

Fig. 5 µ := 0.4; m := 0.08; r := 0.02.

 1 3π 2 6π 2 2π π α + 2β = − − t= ⇔ α + β +γ + ω ω 3 25 ω 2 ω  3  3 1 π 8 π3 π2 π π2 π + α + 2β + γ +δ − α + 2β + γ +δ 5 ω3 ω ω 15 ω 3 ω ω 3π 9π α + 2β ⇔ 2ω 2ω   1 27π 3 1 27π 2 3π 9π 2 3π γ +δ =− α + β +γ + α+ β+ 25 4ω 2 ω 5 8ω 3 4ω 2 2ω  3 8 27π 3 9π 2 3π γ +δ − α+ β+ 15 8ω 3 4ω 2 2ω

t=

(23)

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Fig. 6 µ := 0.24; m := 0.05; r := 0.02.

By means of numerical methods and for ω := 2 · π · 10 one may obtain the following unknown values [α β γ δ ]t = [−0.666 0.291 − 16.499 1.057]t   By adopting k := 0 . . . 100, p := 0.01, tk := 0.5 + p · k and ak := 6 · α · tk + 2 · β with the same previous characteristics, the H wire stretching force may be determined as follows

    ρ ak · m · 1 + µ p·r ρ + rJ2 + µ · m · g · 1 + µ p· r Hk := (eµ z·π −1) µ p·ρ eµ z·π +1 − r In the previous circumstances the time variations of Hk is presented in Figure 7. For an elaborate research it is possible to modify the ω parameter.

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Fig. 7 The H wire stretching force dependence on time.

Fig. 8 The αi , βi , γi , δi dependence on ω .

If i = 1 . . . 100 and p = 1, then υi = pi and ωi = 2πυi from Equation (23) the solution assembly is determined as (αi βi γi δi )t = soli their variations in function of the i parameter being presented in Figure 8. Having in view the time parameter it is possible to obtain for k := 0 . . . 100, p := 0.01 respectively tk := 0.5 + p · k, so that ak,i = 6αitk + 2βi

Hk,i = {ak,i [m(1 + µ p · ρ /r) + J/r2] + µ mg(1 + µ p · ρ /r)}  µp·ρ

(eµ z·π − 1)/(eµ z·π + 1) − r and for the previously characteristics the Hk,i three-dimensional representation is given in Figure 9.

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Fig. 9 The H wire stretching force dependence to time and ω .

4 Conclusions By applying the sin or linear law the functional optimal domains, in which the chaotic effects does not occur are only determined by the constructive characteristics of the mechanical system. The paper reveals the major importance of mechanical system constructive characteristics.

References 1. Baker, J., McRobie, F.A., and Thompson, J.M.T., Implication of chaos theory for engineering sciences, J. Mech. Eng. Sci., 211(5):349–363, 1997. 2. Comanescu, D., Echipamente de calcul si birotica, Edit. Printech, Bucharest, 1998. 3. Comanescu, D., Birotica – Concepte si structuri, Edit. Printech, Bucharest, 2000. 4. Comanescu, D. and Comanescu, A., About vibrations in the storage devices, The Romanian review of precision mechanics, Optics & Mechatronics, 5:17–25, 2001. 5. Kim, J.H. and Stringer, J., Applied Chaos, John Wiley, New York, 1992. 6. Seydel, R., From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis, Elsevier, 1988. 7. Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos: Geometric Methods for Engineers and Scientists, John Wiley, New York, 1986. 8. Voinea, R. and Stroe, I., Mechanical Structures Dynamics, Edit. Academiei, Bucharest, 2000.

Size Minimization of the Cam Mechanisms with Translating Roll Follower D. Perju1 , E.-C. Lovasz1, K.-H. Modler2, L.M. Dehelean1 , E.C. Moldovan1 and D. M˘argineanu1 1 Mechanical Engineering

Faculty, “Politehnica” University of Timis¸oara, 300222 Timisoara, Romania; e-mail: dan.perju, erwin.lovasz, liana.dehelean, cristi.moldovan, [email protected] 2 Faculty of Mechanical Engineering, Technische Universit¨at Dresden, 01062 Dresden, Germany; e-mail: [email protected] Abstract. An important problem in the cam mechanisms synthesis is their total size as a minimization condition (criteria). Usual this condition is reduced to the cam size only, which is not correct because the cam does not function alone. For a cam mechanism with translating follower, the length of follower’s guiding is a very important contributor to the total size of the cam mechanism. In this paper the case of cam mechanism with roll edge follower is taken into consideration. Key words: cam mechanism, translating follower, minimum size of the cam mechanism

1 Introduction The minimum size of the cam is usually the first condition in the designing process of a cam mechanism. The cam’s size is decided by its minimum/basic radius and the stroke. But the cams do not function alone and the total size of cam mechanism will be influenced also by the follower’s guiding length. The guiding length is also an contributor of cam mechanism size on follower’s translation direction. An important parameter for cam mechanism’s size establishment is the maximum admissible pressure angle but not only. The same problem appears even if the pressure angle is zero like on cam mechanism with flat follower. In this paper first cases will be treated, namely with roll follower. Many researches treated exclusively the cam-size minimization through graphical (Hodograph’s and Flocke’s method [4, 10, 11, 13]) and analytical optimization methods. The analytical optimization procedures developed for the cam-size minimization consider as constraints the pressure angle or/and contact-stress [1–3, 5–7, 12, 14]. The authors present at first in [8, 9] the problem of the determining the total size of the cam mechanism with translating follower.

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Fig. 1 The geometry, kinematics and forces equilibrium on a current position of a cam mechanism with roll translating follower.

2 The Pressure Angle In Fig. 1 a current position of cam mechanism is presented. The follower’s velocity (vT ) can be written as [4, 10, 11, 13]: vT = vC3 = vC2 + vC3C2

(1)

where vC2 = ω × r is the velocity of the follower’s roll (theoretical) center point belong to cam, and vC3C2 is the relative velocity between cam and follower at center point level. If relationship (1) is divided to cam’s angular velocity (ω ) and it is represented 90◦ clockwise rotated, it becomes:

where OT =

OT = OC2 + C3 T

(2)


ds dt vT ds
= = = s ω dφ d φ dt

(3)

The pressure angle α can be calculated from triangle OCT:

Size Minimization of the Cam Mechanisms with Translating Roll Follower

tan α =

s
− e ET = , EC rb · cos β + s

β = arcsin

e rb

247

(4)

where rb is theoretical minimum/basic radius, s is the current position of follower’s center roll, e is the eccentricity, which is considered positive for follower’s translation line on right hand side of the cam’s rotation centre O.

3 The Basic Circle’s Radius As usual, the basic circle’s radius is found imposing the non-blocking condition, i.e., α ≤ αa or tan α ≤ tan αa (5) where αa is maximum admissible pressure angle for active or passive stroke, respectively. From relationship (4) with condition (5) and β ≤ 10◦ the basic radius it results in the form: s
− e −s (6) rb = tan αa or in non-dimensional expression (in respect with the stroke h):

 rb 1 e s s = − − tan αa h tan αa h h h

(7)

The minimum value for rb is obtained for the cam position φ = φM for which ∂ rb /∂ φ = 0 resulting in the condition: s

(φM ) = s
(φM ) · tan αa

(8)

The maximum value of the pressure angle (αmax ) can be found for the cam’s position where ∂ α /∂ φ = 0. From relationship (6), the following is obtained: s

(rb + s) − s
(s
− e) 1 dα d (tan α ) = · = dφ cos2 α d φ (rb + s)2

(9)

from which the following results: s

(rb + s) − s
(s
− e) dα = dφ (rb + s)2 + (s
− e)2

(10)

The condition ∂ α /∂ φ = 0, taking into account the relationships (6) and (10), becomes: 


s

(s − e) (11) −s = 0 tan αa or:

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s (φM ) = s (φM ) tan αa

(12)

which is identical to condition (8), i.e. the basic radius has a minimum value if the maximum value of pressure angle satisfies the condition (5), in cam’s position where condition (8) or (12) is fulfilled. Observation: The solution s = e in equation (11) is inacceptable because follower’s velocity cannot be a constant during the active/passive stroke.

4 The Guiding Size The guiding length results from forces equilibrium on the follower: F · cos α ≥ Q + µ · (R1 + R2 )

(13)

where Q is the total force acting on the follower; F is the acting force from cam to follower; R1 and R2 are the reaction forces in guideway,  x x , R2 = F · sin α · (14) R1 = F · sin α · 1 + b b calculated by neglecting transversal follower’s dimension. With these values the condition (13) becomes:  x F · cos α ≥ Q + µ · F · sin α · 1 + 2 b

(15)

where x is the current distance of roll’s center C to the first edge of guiding. Using the so called loading coefficient Φ = Q/F ∈ (0, 1) from relationship (15), in the critical position of the follower, the guiding length can be found: b ≥ 2 · xcr

µ · sin αa cos αa − µ · sin αa − Φ

(16)

The critical position is reached at α = αa = α (φM ), φ = φM , see condition (8) or (12), (Fig. 1): (17) xcr = h + rr − s(φM ) The relationship (16) with condition (17), in non-dimensional form, becomes:   2 · µ · sin αa rr s(φM ) b ≥ (18) 1+ − h cos αa − µ · sin αa − Φ h h

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5 The Total Size of a Cam Mechanism with Translating Roll Follower Taking into account the dimensions indicated in Fig. 1, the total size in the follower’s translation direction is:
H ≥ 2 · h + rb · cos β + b + (rb · cos β + h)2 + e2 (19) or in non-dimensional expression: r · cos β b H ≥ 2+ b + + h h h

  rb · cos β 2  e 2 1+ + h h

(20)

6 Example Problems In order to compare different usual transmission functions from the mechanism’s size point of view, the analyzed type of cam mechanisms with translating follower was taking into consideration, having the following main parameters:

φa/p = 90◦ ,

e = 0,

rr = 0.1 · h,

Φ = 0.5,

µ = 0.1,

αa = 40◦

(21)

The transmission functions taken into account are [10, 13]: 1. Cosine: 2. Sine:

3. Parabolic:

4. Polynomial 3-4-5:

1 s = (1 − cos2φ ) h 2 1 s = (4φ − sin 4φ ) h 2π ⎧ s  8 2 ⎪ for φ ∈ (0, φa 2) ⎪ ⎨ h = π2 φ    ⎪ ⎪ ⎩ s = 1 − 2 1 − 4 φ2 for φ ∈ (φa 2, φa ) 2 h π   8 30 24 s = 3 φ 3 10 − φ + 2 φ 2 . h π π π

(22)

Applying the presented procedures the main parameters implied in the size establishment of cam mechanism are presented in Table 1. For the considered cam mechanisms the variation of the pressure angle on the active stroke, according to relationship (4), are presented in Fig. 2.

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Table 1 The main relative dimensions of the cam mechanisms with design parameters (21). Transmission function (cam type)

rb /h

b/h

H/h

φM

1. Cosine 2. Sine 3. Parabolic 4. Polynomial 3-4-5

0.785 1.04 1.014 1.0

0.5 0.48 0.38 0.47

5.07 5.56 5.41 5.47

33◦ 35 39◦ 05 45◦ 37◦ 46

Fig. 2 The variation of pressure angle on active stroke.

7 Conclusions The proposed method allows to minimize a certain cam mechanism with translating roll follower taking into account both cam and follower’s guide way as well. The most important parameter in the mechanism’s size establishment it is the follower’s stroke, follows by the basic circle’s radius and guiding length. The cam mechanism’s size, from transmission functions point of view, indicates that the “smooth” function (like sine or polynomial) have a size greater than the “dour” functions (like cosine or parabolic), as can be seen in the in presented example problems. That is to confirm the universal valuable rule according to in any technical problem cannot have only advantages.

References 1. Angeles, J. and Lopez-Cajun, C.S., Optimization of Cam Mechanisms. Kluwer Academic Publishers, Dordrecht, 1991. 2. Carra, S., Garziera, R., and Pellegrini M., Synthesis of cams with negative radius follower and evaluation of the pressure angle. Mech. Mach. Theory, 39:1017–1032, 2004.

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3. Chan, Y.W. and Kok, S.S., Optimum cam design. Int. J. Comput. Appl. Technol., 9(1):34–47, 1996. 4. Duca, C., Popovici, A., and Opris¸an, C., Bazele proiect˘arii mecanismelor cu came. Ed. Ghe. Asachi Ias¸i, 1999. 5. Fenton, R.G., Optimum design of disc cams. In Proceedings of the Fourth World Congress on Theory of Machines and Mechanisms, Newcastle-upon-Tyne, 8–12 September, Vol. 4, pp. 781–784, 1975. 6. Golovin, A., Borisov, A., and Os’kin, A., Geometrical analogues of contact stresses for cams design. In Proccedings of the 1st Conference EUCOMES, Obergurgl, 21–26 February, 8, pp. 1–11, 2006. 7. Loeff, L. and Soni, A.H., Optimum sizing of planar cams. In Proceedings of the Fourth World Congress on Theory of Machines and Mechanisms, Newcastle-upon-Tyne, 8–12 September, Vol. 4, pp. 777–780, 1975. 8. Lovasz, E.-Ch., Perju, D., Modler, K.-H., Z˘abav˘a, E.S., and Hotea, A., On the size of cam mechanism with translation follower. Mechanisms and Manipulators, 6(1):39–44, 2007. 9. Lovasz, E.-C., Perju, D., Modler, K.-H., M˘argineanu, D.T., V˘ac˘arescu, V., and Z˘abav˘a, E.S., Cam mechanism with flat/tangential translating follower and its size. In Proceedings of the Tenth International Symposium on Theory of Machines and Mechanisms, SYROM’09, Brasov, 10–12 October, pp. 645–654, Springer Verlag, 2009. 10. Lovasz, E.-Ch. and C˘ar˘abas¸, I., Principii de sinteza a mecanismelor cu rot¸i dint¸ate s¸i came. Ed. Politehnica, Timis¸oara, 2006. 11. Luck, K. and Modler, K.-H.: Getriebetechnik – Analyse, Synthese, Optimierung. Springer Verlag, Wien/New York, 1990. 12. Navarro, O., Wu, C.-J., and Angeles, J., The size-minimization of planar cam mechanisms. Mechanism and Machine Theory, 36:371–386, 2001. 13. Perju, D., Mechanisms for Precision Mechanics, Vol. 1. Ed. Univ Politehnica Timisoara, 1990 [in Romanian]. 14. Terauchi, Y. and El-Shakery, S.A., A computer-aided method for optimum design of plate cam size avoiding undercutting and separation phenomena – I. Mechanism and Machine Theory, 18:157–163, 1983.

Kinematic Analysis of the Roller Follower Motion in Translating Cam-Follower Mechanisms E. Seabra and P. Flores Mechanical Engineering Department, University of Minho, 4800-058 Guimar˜aes, Portugal; e-mail: {eseabra,pflores}@dem.uminho.pt

Abstract. The main purpose of this work is to present a complete kinematic analysis of the roller motion in translating cam-follower mechanisms with translating roller followers. The main kinematic variables of the roller are the angular velocity and angular acceleration. These quantities are deduced in a general form, in such a way they can be easily used for different cam laws. Furthermore, the influence of the functioning cam-follower parameters can also be included. Finally, an example of application is offered to analyze and discuss the main procedures considered in this work. Key words: roller motion, cam-follower mechanisms, kinematic analysis

1 Introduction Cam-follower mechanisms constitute a simple, versatile, compact and economic way by which a part can be given a prescribed motion. Other important features associated with this class of mechanical devices are the ability to obtain an arbitrarily specified motion for the follower with a close control over its kinematics characteristics and a strong mechanical liability, mainly due to the reduced number of moving parts involved. The spectrum of engineering applications of the cam-follower mechanisms is wide and comprises, for example, systems of open and close valves in internal combustion engines, cutting and forming presses, textile machinery, medical devices, just to mention a few. Over the last decades the fundamental theories of the cam-follower systems have been quite well treated in several standard textbooks [2, 7, 8]. Due to its capability to ensure complex output motion, the disc cam mechanism with translating roller follower is indubitably one of the most popular and used systems among the different cam-follower types. In general, the studies of the cam follower mechanisms with translating roller follower do not include the analysis of the relative motion between the cam and roller. However, this motion is inevitable in the system due to the rolling effect between these two mechanical components. Moreover, due to the variation of the distance from center of the cam rotation to

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the cam profile, the roller has changes in the angular motion about its center and, consequently, the roller is subjected to angular accelerations. These angular accelerations associated with the friction force developed between the roller and cam surface causes a resistant torque, which magnitude depends on the system’s inertia and on the pressure angle [3]. Therefore, based on these assumptions some criteria can be defined in order to eliminate, or at least minimize, the slip between the roller and cam surfaces. Moreover, geometric parameters such as the base circle of the cam, the offset of the follower and the radius of the roller follower influence the cam design process. In addition, when the change of the two main performance cam parameters, that is, the pressure angle and the curvature radius of the cam, is combined with the above factors, the cam design becomes a nontrivial task. The degree of complexity can become even higher when the designer requires the optimization of the cam size. The cam size is one of the main factors to be considered when selecting the base circle radius, especially if there are space limitations [1, 6, 9]. It is known that, the pressure angle and the curvature radius are inversely influenced by the base circle radius of the cam. The problem of rotational motion of the roller elements has been studied by few researchers. Yudin and Petrokas [10] are the few who did this type of analysis, however, this publication is in Russian [5]. Thus, the purpose of this work is to present the complete kinematic analysis of the roller motion cam mechanisms with translating roller followers. In the sequel of this process, some fundamental aspects related to the cam-follower systems are revisited. Finally, some results obtained from computational simulations are presented with the intent to understand and discuss the main assumptions and procedures adopted throughout this work.

2 Kinematics of the Follower Motion Figure 1 shows a schematic representation of a generic cam disc mechanism with an offset translating roller follower. A translating roller follower consists of an arm, constrained to move in a straight line, with a roller attached to its extremity by means of a pin. Thus, as the cam rotates, the roller rolls on the cam surface and causes the arm to translate. This relative rolling motion helps to reduce wear, and, for this reason, the roller follower is often preferred over followers that have sliding contact [2]. It is known that in most of the cases a cam follower mechanism is required to be displaced through a specific rise and fall. The shape of the displacement curve plays a key role in the synthesis of a cam mechanism. Therefore, the velocity, acceleration and, in many cases, further derivatives of the follower displacement are of great importance. According to Norton [7], the fundamental tenet of the cam design process can be stated as “the cam-follower function must be continuous through the first and second derivatives of the displacement (i.e. velocity and acceleration) across the entire interval”. Besides, the jerk must be finite across entire interval as well. In short the displacement, velocity and acceleration characteristics of the follower

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255

[t] Fig. 1 Representation of a generic disc cam mechanism with offset translating roller follower.

motion must be continuous functions. In general, these functions are expressed in terms of the lift, the angle of the lift and as function of the angle of rotation of the cam characterizing some specific cam laws. The most general form of the follower displacement for a translating roller follower can be written as [2]
 θ (1) y = hf β in which h is the lift or stroke of the follower, β represents the angle of lift and θ is the angle of rotation of the cam. Now, by successive derivatives of Eq. (1) with respect to the angle of cam rotation yields
 h θ y
= f
(2) β β
 h θ y

= 2 f

(3) β β
 h


θ


(4) y = 3f β β which represent the follower velocity, acceleration and jerk, respectively. These functions can easily be converted to a time base, multiplying each of the equations by ω , ω 2 and ω 3 .

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3 Analysis of the Roller Follower Motion From a kinematic point of view, the mathematical condition that represents the non slip condition between the roller and cam surfaces states that the velocity of the contact point between the cam profile and the circumference of the roller tangent to this profile must be the equal, as illustrated in Fig. 1. The tangential velocity of the contact point, when evaluated from the roller motion point of view, is given by vCt = ωr Rr

(5)

in which ωr is the angular velocity of the roller about its center and Rr represents the radius of the roller. On the other hand, this tangential velocity when associated with the cam motion is (6) vCt = ωC BC where ωC is the angular velocity of the cam and BC represents the distance between the contact point and pole B, which varies continuously during the cam rotation. With regard to Fig. 1, the distance BC is given by BC =

a+y − e sin ϕ − (a + y) tan ϕ sin ϕ − Rr cos ϕ

(7)

in which all the geometric parameters are the same as presented in the previous section. Since the velocities represented by Eqs. (5) and (6) are equal, and assuming that there is no slip between roller and cam surfaces, the angular velocity of the roller can be written as   ω a+y (8) ωr = C − e sin ϕ − (a + y) tan ϕ sin ϕ − Rr Rr cos ϕ If there is no eccentricity between the direction of the follower motion and the axis of the cam rotation, that is, e = 0, then Eq. (8) can be reduced to

ωr =

ωC [(a + y) cos ϕ − Rr ] Rr

(9)

Analyzing Eqs. (8) and (9) it is evident that the roller angular velocity varies with the cam rotation angle. The angular acceleration of the roller about its center can be obtained by differentiating Eq. (9) with respect to time, yielding

αr =

ω C2 cos ϕ
(y − ϕ
y
) Rr

(10)

where y
is the first derivative of the follower displacement with respect to cam, being the remaining parameters the same as previously. The dimensionless parameter ϕ
represents the first derivative of the pressure angle with respect to cam angle. Thus, using Eq. (5) the parameter ϕ
can be expressed as

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257

Fig. 2 Displacement diagram of the follower motion – RDR.

ϕ
=




cos ϕ a+y

2



y

(a + y) − (y
− e)y




(11)

in which y
represents the second derivative of the follower displacement with respect to cam angle. In a similar way, the second derivative of the angular velocity of the roller represents the angular jerk of the roller about its center and can be expressed as

ϑr =

 ωC3 cos φ 

y + y
tan φ φ
2 − y
(tan φ φ
+ φ

) − φ
y

Rr

(12)

where ϕ

represents the second derivative of the pressure angle with respect to cam angle. The remaining parameters have the same meaning as it was presented previously. The parameter ϕ

can be obtained by differentiating Eq. (18) with respect to cam angle, yielding



ϕ =




cos ϕ a+y

2











y (a + y) + (y − e)y








− 2ϕ ϕ
+


 y
tan ϕ y
− e

(13)

where y


is the third derivative of the follower displacement with respect to θ .

4 Results and Discussion In this section, a disc cam-follower mechanism with a translating roller follower is considered for a demonstrative application example. The follower motion is of type RDR, that is, Rise-Dwell-Return, as it is illustrated in Fig. 2. The follower describes a rise of 30 mm during the cam rotation from 0 to 100◦. Then, the follower remains stationary for amplitude of cam rotation angle equal to 110◦. Finally, the follower returns to its initial position during the remaining cam rotation. The base circle radius of the cam is 20 mm. The radius of the roller is equal to 10 mm and the eccentricity of the follower is null. In order to keep the analysis simple, the angular velocity of the cam is assumed to be equal to 1 rad/s.

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Fig. 3 Roller angular velocity and acceleration, considering four different cam laws of the follower motion.

Fig. 4 Roller angular velocity and acceleration for cicloidal motion, considering four different follower strokes.

The behavior of the roller can be influenced by three main factors, namely the cam law employed, the follower stroke and the amplitude of the cam rotation angle to reach the desired stroke. Thus, in what follows, the roller angular velocity and acceleration are plotted against the ratio θ /β taking into account these three variables. The plots depicted in Figs. 3–5 are relative to the rise motion only, because this is in fact the most critical and relevant part of follower motion and, consequently, plays the key role in the cam design process. From the global obtained results it can be observed how the roller kinematics is affected by the law for the follower movement. It is also possible to understand the influence of the geometrical cam mechanism parameters on the roller motion, as it is illustrated in Figs. 4 and 5, having, therefore, consequences in terms of the systems design and optimization [4].

5 Conclusions In this work, the motion analysis of the roller follower was studied by considering an application example in which different cam laws were used. Based on these res-

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Fig. 5 Roller angular velocity and acceleration for cicloidal motion, considering four different cam angle of rotation.

ults it can be observed that among all the standard follower motions, the cicloidal or sinusoidal acceleration is indubitably one of the most appropriate types due to very good operating conditions that it can assure. In fact, the cicloidal motion presents continuity in all of the kinematic characteristics, which means that acceleration varies smoothly and, consequently, this solution causes lesser wear, shocks, stress, vibration and noise than other basic follower motion curves. In summary, it can be concluded that the plots of roller kinematic characteristics are quite useful tools in the selection of the most suitable cam law for a specific cam-follower mechanism.

References 1. Angeles, J. and Lopez-Cajun, C.S., Optimization of Cam Mechanisms, Kluwer Academic Publishers, Dordrecht, 1991. 2. Chen, F.Y., Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982. 3. Flores, P., Projecto de Mecanismos Came-Seguidor, Publindustria, Guimar˜aes, Portugal, 2009. 4. Flores, P., Projecto de Mecanismos Came-Seguidor, Publind´ustria, Portugal, 2009. 5. Golovin, A., Lafitsky, A., and Simuskhin, A., Experimental and theoretical research of cams wearing of cams mechanism. In Proceedings of the EUCOMES08, The Second European Conference on Mechanism Science, pp. 343–350, Springer, 2008. 6. Mundo, D., Liu, J.Y., and Yan, H.S., Optimal synthesis of cam-linkage mechanisms for precise path generation. Journal of Mechanical Design, 128:1253–1260, 2006. 7. Norton, R.L., Cam Design and Manufacturing Handbook, Industrial Press, New York, 2002. 8. Shigley, J.E. and Uicker, J.J.: Theory of Machines and Mechanisms, McGraw Hill, New York, 1995. 9. Ye, Z. and Smith, M.R.: Synthesis of constant-breath cam mechanisms. Mechanism and Machine Theory, 37:941–953, 2002. 10. Yudin, V.A. and Petrokas, L.V.: Theory of Mechanisms and Machines. Textbook for Universities, 2nd edition, M.: Higher School, 1977 [in Russian].

Kinematic Analysis of Cam Mechanisms as Multibody Systems D. Ciobanu and I. Visa Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: {daniela.ciobanu, visaion}@unitbv.ro

Abstract. Developing the virtual prototypes of novel products using commercial software leads to a unitary modeling of all the component mechanism types (linkages, cams, gears, etc.). The paper presents the analytic modeling of the bodies (cam, follower), considering the geometrical restrictions from the joints of the planar cam mechanisms, using the Multibody System Theory. Based on this, a unitary analytic model for all the types of the planar cam-linkage mechanisms can be obtained. Key words: cam follower mechanisms, geometrical restriction, multibody system theory

1 Introduction Real time simulation of the product dynamic behavior is a modern way of performance analysis and optimization in the product design process. For a company, it saves time in product developing, reduces number of physical prototypes and experiments, reduces the manufacturing costs and the products price, and increases the quality of the product. To perform this task, involving large computers and a unitary method of modeling for all the subsystems of a product represents a must. The defining elements of the cam and linkage mechanisms as multibody systems have been mentioned in literature during the past 20 years: in 1989 the monograph Computer Aided Kinematics and Dynamics of Mechanical Systems [4] is issued, where the linkage and cam mechanisms are defined using the multibody system method, while in 1998, Shabana defines the dynamic model for the multibody systems and in 2001 in [5] is presented an algorithm for structural synthesis of the linkage mechanisms. Since 2002, the multibody systems method is reported as a powerful tool to analyze different types of mechanisms [1, 5]. However, there is no unitary method based on a concrete algorithm to define and especially to concept the cam and linkage mechanisms as multibody systems. The paper proposes a kinematic modeling of the planar cam-linkage mechanisms by using the multibody systems method, specialized software being used to

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automatically generate the kinematic-dynamic functions and to solve those, using numerical methods.

2 Theoretical Aspects The aim of the kinematic analysis by the multibody systems method is the study of the bodies’ motions (bodies’ displacements, velocities, accelerations), without considering the forces, bodies’ masses and their distribution [5]. In this study, a kinematical model is used consisting of the structural and the geometrical model. In kinematic modeling using the multibody systems method it is necessary: (a) to choose the reference systems (ground reference frame, bodies reference frames); (b) to define the geometrical model of the multibody system; (c) to choose the generalized coordinates and to define the bodies’ position in the ground reference frame; (d) to analytically define the geometrical restrictions; (e) to analytically define the driving constraints, in order to perform the numerical simulation; (f) to establish the displacements, velocities and accelerations functions.

2.1 Geometrical Model of the Cams In modeling a complex cam-linkage mechanisms as multibody systems the cams are considered as bodies of the multibody. The geometrical model of a cam i is defined in the reference frame attached to cam (Oi XiYi ), independent of the cam motion (rotation, translation, general plan motion) and consists of interesting points (ex. location of cam center, translation orientation, etc.) followed by defining the cam profile (real, theoretical). The reference frame attached to the cam is orientated so that OiYi axis passes through the initial cam position which is considered where the theoretical follower point (Pi )0 reaches the tangent point between the inferior pause zone and follower far zone (Fig. 1). In the case of a rotational cam, Fig. 1a, the geometrical model consists of the real/theoretical profile and point Ai defined by the coordinates (x(i) , y(i) ), and for Ai Ai the translational cam, Fig. 1b, the geometrical model consists of the real/theoretical , y(i) ) and Bi (x(i) , y(i) ). profile and interesting points Ai (x(i) Bi Bi Ai Ai The theoretical and real profiles of a cam are usually described by one of the following [2, 3]: • default function: Fi (x(i) , y(i) ) = 0; i i • polar parametrical equations: ρ = ρ (ϕ i ), θ = θ (ϕ i ), where ϕ i – represents the rotational angle of the cam; = xi (φi ), y(i) = yi (φi ); • parametrical equations in Cartesian coordinates: x(i) i i

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Fig. 1 The cam theoretical and real profile.

• points Pi (x(i) , y(i) ). P P i

i

In order to develop the geometrical restriction equations of Curve-Curve type, the normal and the slope to the cam profile are required. When the real cam profile is defined by points the following methods can be used to determine the normal/slope: a. the circle method; b. the secant method.

3 Geometrical Model of the Follower Defining the follower geometrical model is made in the reference frame attached to it, where the interest points, lines and curves are described depending of the follower type. The paper presents the geometrical model of the oscillating and the translational follower with roller. The geometrical model of the oscillating follower with roller (Fig. 2), consist of: point Pi for describing the Rotational geometrical restrictions (R), point Qi where the roller is mounted and the roller radius rr , required to described the Curve-Curve (CC) geometrical restriction. The geometrical model of the translational follower with roller (Fig. 3) is described by interest points (points Pi and Ri , for describing the direction towards the geometrical restriction type Translational (T), point Qi , where the roller is mounted) in the reference frame attached and the roller radius rr . Based on the parameters previously defined on the geometrical model of the bodies, the geometrical restrictions in the cam mechanism are analytically modeled.

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Fig. 2 Geometrical model of the oscillating follower with roller.

Fig. 3 Geometrical model of the translational follower with roller

3.1 Cam-Follower Geometrical Restriction In the case of planar came-linkage mechanisms the following restriction types are met: (a) simple geometrical restriction type Rotation (R) and Translation (T); (b) composed geometrical restriction type Rotation-Rotation (RR), TranslationTranslation (TT) and Rotation-Translation (RT); and (c) geometrical restriction type Curve-Curve (CC) [3]. Considering the geometrical constraint of Curve-Curve (CC) type, two bodies i (cam) and j (curve follower) are considered in planar motion. Their location is defined in the fixed reference frame OXY using the attached reference frame location: Oi XiYi (defined by XO , YO , ϕ i ) and O j X jY j (defined by XO , YO , ϕ j ), as Fig. 4 i i j j presents. In order to generate the analytical model of Curve-Curve geometrical restriction (CC), the following cases are studied: (1) cam and follower profiles are defined by functions; and (2) cam profile is defined by points and follower profile by functions.

3.1.1 Cam and Follower Profiles Defined by Functions Initial data: , y(i) ) = 0 equation of the curve that describes the real cam profile; • Fi (x(i) i i

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Fig. 4 Analytical model of geometrical restriction CC type.

• Fj (x(j j) , y(j j) ) = 0 equation of the curve that describes the follower profile. In order to describe the analytical equation of CC restriction type, the equation of the tangent to the cam profile in the contact point Pi (defined in the ground reference frame) and equation of the tangent to the follower profile, in the contact point Pj (defined in the ground reference frame) are required. The scalar equation of the geometrical restriction of CC type, Fig. 4b, is governed by the condition that the two slopes mi and m j in the contact point (defined in ground reference frame) are equal [3]: FCC : mi (Pi ) = m j (Pj )

(1)

Besides the generalized coordinates xO , yO , φ i , xO , yO , φ j , the following addii

i

j

j

tional unknown point coordinates are considered: x(i) , y(i) for cam and x(Pj) , y(Pj) for P P i

i

j

j

follower. The coordinates of these points are determined from the following conditions: • the contact point Pj of the follower, defined in the ground reference frame, belongs to the follower profile: Fj (xP , yP ) = 0 j

j

(2)

• the contact point Pi of the cam, defined in the ground reference frame, belongs to the cam profile: (3) Fi (xP , yP ) = 0 i

i

• the contact points must be superposed in the ground reference frame, Pi ≡ Pj : x P − xP = 0

(4)

yP − yP = 0

(5)

i

i

j

j

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Fig. 5 Geometrical model of the cam defined by points.

3.1.2 Cam Profile Defined by Points and Follower Profile by Functions When the real cam profile is defined by points (equally spaced as possible), in order to write the CC type geometrical restriction equation, the circle method and the secant method can be used, as presented in Fig. 5: • Circle Method Using this method, the equation of CC type geometrical restriction is defined by starting from the coordinates of the points that determine the real cam profile, defined in the reference frame attached to the cam, and then the algorithm presented below is followed: – the circle center coordinates are determined O(x(i) , y(i) ), defined by the points O O Pi−1 , Pi , Pi+1 ; – the equation of the normal that passes trough the circle center O(x0 , y0 ) and point Pi (xP , y pi ), in the ground reference frame is developed. i For a curved follower profile (Fig. 6) defined by an implicit function Fj (x(j j) , y(j j) ) = 0, the scalar equation of the geometrical restriction of CC type is given by equation (1), with the following unknowns: the generalized coordinates xO , yO , φi , xO , yO , φ j and the coordinates of the contact point i i j j from the follower profile Pj .

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Fig. 6 Analytical model of geometrical restriction CC type.

Fig. 7 Describing of cam profile using the secant method.

The coordinates of point Pj result from the supplementary equations (4) and (5). • Secant Method For defining the equation of the geometrical restriction of CC type using the scant (i) method, the coordinates of the points (x(i) r , yr ) are chosen, equally spaced on the real cam profile, and the following algorithm is used: – the line between points Pi−1 and Pi+1 is drawn (Fig. 7); – the slope of line Pi−1 Pi+1 is determined and marked as mi ; – the through point Pi is built the unit vector perpendicular to the unit vector Pi−1 Pi+1 ; – the condition of perpendicularity of the two unit vectors is imposed. The scalar equation of the geometrical restriction of CC type, for a curved follower given by an implicit function Fj (x(j j) , y(j j) ) = 0, is given by equation (1). The unknowns in this case are the generalized coordinates and the coordinates of the contact point on the follower profile Pj . The coordinates of point Pj are determined by the supplementary equations (4) and (5). The secant method can be used for a large number of equally spaced points (as possible) that define the real cam profile. For a reduced number of points, when the arcs defined by points are not equal, this method generates larger errors comparing to the circle method and is thus not recommended [3].

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4 Conclusions The method of kinematical modeling developed in this paper allows the users to perform the kinematical analysis, either by developing the whole model or by using commercial software. In this case it is important the geometrical model of the bodies and the choice of generalized coordinates. Using this method the whole range of cam-linkage mechanisms is described by six types of geometrical restrictions (R, T, RR, TT, RT, CC), that can be unitary approached in a single analytical model. Modeling the cam profile by points assures a good accuracy of the kinematical analyses, comparing to the model by functions, for an angular step properly chosen. Modeling the geometrical constraints of the cam-follower using the circle and the secant methods, leads to a number of equations reduced with two, comparing to the model based on functions, thus simplifying the analytical model. For a reduced number of points used to describe the cam profile the secant method generates larger errors comparing to the circle method, and for a large number of points used to describe the cam profile, the circle and secant method can be comparable, considering the accuracy of the kinematical analysis.

Acknowledgments This paper is supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and by the Romanian Government under contract number POSDRU ID59323.

References 1. Chang, D. and Li, H., Kinematics simulation of air shearing mechanism based on ADAMS, In Proceedings 2nd International Conference on Intelligent Computing Technology and Automation, 2009. 2. Ciobanu, D. and Visa, I., Modeling and kinematic analysis of cam mechanisms as multibody systems. In Proceedings of SYROM 2005, The 9th IFToMM International Symposium on Theory of Machines and Mechanisms, Bucharest, Romania, 2005. 3. Ciobanu, D., Modeling cam-linkage mechanism using multibody system method, PhD Thesis, Transilvania University of Brasov, 2009. 4. Haug, J.E., Computer Aided Kinematics and Dynamics of mechanical Systems, Allyn and Bacon, USA, 1989. 5. Vis¸a, I. and Gavrila C., Structural synthesis of transversal coupling by MBS. In Proceedings of the IFToMM International Symposium, Besancon, France, 2007.

Peculiarities of Flat Cam Measurement by Results of Digital Photo Shooting A. Potapova, A. Golovin and A. Vukolov RK-2 Department, Bauman Moscow State Technical University, 2nd Baumanskaya Street, 105005 Moscow, Russia; e-mail: [email protected], [email protected], [email protected]

Abstract. During technical diagnosis it is often needed to determine object dimensions remotely by results of some optical data. The digital photography is the perspective method for non-invasive measurement of industrial machinery objects. But the peculiarities of equipment and digital photo process bring some important restrictions and constraints to measurement process. We tried to explain the peculiarities and unite the advantages of different methods of photographical non-invasive measurement for easy flat cam mechanism to recover the cam profile and estimate precision of measurements. Key words: digital photography, measurement, cam mechanism, profile, remote diagnostic

1 Introduction There is an actual line in mechanisms theory experimental methods – the noninvasive measurement of industrial machinery objects. The family of such opticalbased methods is called ‘technical vision’. There are two main types of technical vision: • Target-based technical vision. The method consists of measurement of distances between optical targets and/or contrast boundaries coordinates on the frame field. The traditional technique of technical vision, it is useful for stationary diagnostic systems. Usage of this technique for non-invasive measurement of machinery objects is described in [2, 3]. • Scale-based technical vision. This technique uses optical focusing parameters and/or metric standard objects on frame field to measure the true scale of the digital image and calculate dimensions of the selected object. This method is particularly useful for robotics because of highly developed digital photography technology and computing. The main advantages of digital photography as the technical measurement technique are the equipment cheapness and possibility of measurement ‘on location’ from working machine. To improve the methodology and prepare the obvious ma-

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terial we selected the flat cam mechanism as the object of exploration. This work is the extension of [1], where the cam dimensions and profile structure were recovered using digital photographs and cam manufacturing process hypothesis. Also it is needed to pay an attention to shadowing boundary blur, reference points positioning and true scale recognition according to digital imaging peculiarities.

2 Target Setting The bending press cam mechanism was selected as the exploration object because of its easy cam shape and good accessibility. This machine is one of demonstration objects in BMSTU technological laboratory. The cam is threadbare that allows us to make estimates for method precision and wearing quantity (if possible). Because of radial dimension supremacy in cam mechanisms theory the polar coordinate system was selected for profile recovery. In accordance with aforesaid, it is possible to set the main tasks of experimental exploration: 1. Record the digital photographs of bending press cam mechanism. 2. Determine the true scale of the image in accordance with shooting properties and metric standard object. 3. Define the reference point of measurement. 4. Recover the cam profile in polar coordinates relative to reference point. 5. Indicate the error-making factors. 6. Make the estimates for measurement error values to provide a confidence interval width for the cam profile at every point.

3 Experiment on Location: Planning and Providing The registration device was Canon EOS 400D digital photo camera with CCD 10.5 MPix sensor and 100× ISO speed. Optical system had various focal distance (18–50 mm) with angular aperture near 150 degrees. Camera was placed on distance about 0.5 m from cam and centered by indicated focusing points. There are no external lighting devices used. Aperture value was forced to 16.0 for maximum field depth. In case of absence of the special computer vision systems with possibility of shape recognition, it is needed to use a metric standard object for scaling. The special metrological-verified linear metering plate with 30 mm of length was used as the metric standard object. It was placed just above the cam axis. The main photograph that was used for treatment is shown in Fig. 1. Main peculiarity of this photograph is large shadow amount. Also it is needed to strictly control metering plate parallelism to frame field edges. There were three practically identical photographs used for cam profile exploration. The treatment technique was the same for every photograph.

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Fig. 1 Bending press cam mechanism with metering plate placed above cam axis.

4 Data Analysis Methodology Description The first operation of data treatment was scale determining. To avoid perspective distortion the cam shape check was performed. On this step no geometrical distortion had exposed. Next the photograph was aligned to make metering plate and frame field edges parallel.

4.1 True Scale Determination The process of true scale determination is described in Fig. 2. The several scan lines (with 1 pixel width) were selected on the 100% scaled photograph fragment that contains an image of metering plate. In consideration of plate length is 30 mm; several values of scale were obtained. All values are represented in Table 1. According to [2] the true scale value can be obtained as an average of all measurement results with statistic treatment. As the model of results dispersion the standard Gaussian model was selected. Full error amount, standard deviation and repeating error probability were estimated using standard model and Student criteria.

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Fig. 2 Scan lines measurement for scale determination. Table 1 Results of true scale determination. Number of scanlines Average on selection Standard deviation Error amount Student’s Coefficient True scale value (average) Confidence interval width for imaging Confidence interval width

14 Pixels Pixels Pixels

264.054 0.4302 0.1145

2.1604 (for 95% of confidential probability) Pixels per mm 8.80175 Pixels 0.4947 mm 0.0562

4.2 Reference Point Definition It is important for profile recovery precision that the reference point is determined as precise as possible. Because of no visible centered parts of cam mechanism, the reference point was determined on photograph (Fig. 1) within four operations: 1. The reference point locates on the crossing of the central screw head diagonals.

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2. According to bending machine technical passport, the large radius of cam is manufactured as part of true circle, so it is possible to find the center of the circle by two chords. 3. By analogue of #2 for the small radius of cam the reference point can be located. 4. According to [1] the working sides of cam can be approximately considered as straight tangents to small circle. The reference point can be located on the crossing of perpendiculars to them in tangency points. The listed operations resulted in four possible locations of reference point. After the statistical treatment the average coordinates of reference point are determined with maximum possible deviation about 0.001 mm for 90% of confidential probability.

4.3 Cam Profile Recovering The coordinates of cam profile were obtained from photograph by point selection on the profile, and then the scanlines are selected to determine the profile point displacement relative to reference point. Length of cam radius in selected point obtained as result of calculation by Pythagorean Theorem. For each point the an-gular coordinate, radius-vector length, expectation value, confidence interval width and standard deviation were calculated. In case that all errors in image are considered as random, density of error amount distribution on [a, b] interval is circumscribed by equation (1). The expectation value can be calculated by equation (2) fX (x) =

1 1 (x) b − a [a,b]

(1)


b

a+b x dx = (2) b − a 2 a After all calculations the array of 73 cam profile points defined in polar coordinates was constructed. The evolvent of cam profile presented in Fig. 3 was calculated using MathCAD array processor. M[X] =

5 Error-Making Factors and Compensation of Errors There are several error-making factors can influence the result. The factors with most influence were discovered, error amounts are calculated, and all inaccuracies are compensated if possible. Some hardware (CCD)-specific error-making factors and its compensation operations are also described in [3, 4]

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Fig. 3 Cam profile evolvent.

Fig. 4 Pixelation errors origin.

5.1 Pixelation Errors As we can see in Fig. 4 the pixels of digital image are square. Because of limited camera sensor resolution it is possible that the real object bound can differ with the same visible one on the digital photograph. Also if the boundaries of an object are not parallel to frame edges some pixels can not be accounted during measurement because they cover object boundary by one corner only. So, the only way to compensate the pixelation errors is to measure by large amount of pixels with data averaging and statistical treatment. During metering plate measurement the pixelation errors are compensated by themselves because the metering plate edges are parallel to frame boundaries. As the result of compensation of this error type, the fractional numbers of pixels have already considered above.

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Fig. 5 Shadow blurring zones.

5.2 Shadow Blurring As we can see from Fig. 5, there are several zones on the picture where the cam edge is practically indiscernible because of lighting conditions. It is very difficult to measure the true cam profile point coordinates in these zones. The shadow zones create doubled ‘illusion edges’ of the profile with high difference between real radiusvector length and the measured one. In particular, on the photograph (Fig. 1) the minimum difference is about 0.7 mm (6.15 pixels). Shadow blurring makes it impossible to precisely determine the tangency points of cam working sides. To compensate this type of errors the following decision was made: the profile points from the shadow zones were considered as the points of small cam circle (Fig. 1).

5.3 Final Model After all error-making factors were eliminated; the final model of the cam mechanism was developed. It is presented in Fig. 6.

6 Conclusions • The possibility of digital photo equipment usage for non-invasive measurement of industrial machinery objects is shown. • Method of fast and precise scaling with small error amount is proposed. • The described methods are tested on real machine (bending press), the obtained results can be approved as precise and fast measurement.

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Fig. 6 Final model of the cam mechanism with elimination of all errors those are possible to compensate.

• There are large perspectives to use proposed method for remote diagnostics on real industrial systems.

References 1. A. Golovin, A. Lafitsky, and A. Simuskhin, Experimental and theoretical research of cams wearing of cams mechanism. In Proceedings of 2nd International Conference EuCoMeS-2008, M. Ceccarelli (Ed.), pp. 107–119, Springer, 2008. 2. V. Zarubin and A. Krischenko (Eds.), Math Statistics, BMSTU, 2002 [in Russian]. 3. A. Vukolov, A. Golovin, and N. Umnov, Horse gait exploration on “step” allure by results of high speed strobelight photography. In Proceedings of EuCoMeS-2010 Conference, 2010. 4. A. Vukolov and A. Kharitonov, Kinematical analysis of mechanical systems by results of digital video recording. In Proceedings of the 10th IFToMM International Symposium on Science of Mechanisms and Machines SYROM 2009, I. Visa (Ed.), pp. 457–464, Springer, 2009.

Analyzing of Vibration Measurements upon Hand-Arm System and Results Comparison with Theoretical Model A.F. Pop1 , R. Morariu-Gligor1 and M. Balc˘au2 1

Mechanical and Computer Programming Department, 2 Descriptive Geometry and Engineering Graphics Department, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania; e-mail: cristea [email protected], [email protected], [email protected]

Abstract. The work analyses the results of measurements for vibrations made for the system handarm and their comparison to the mechanical theoretical model. The theoretical results have been obtained by the integration of the system of equations corresponding to the hand-arm system. The integration of the equations has been done taking into account the signal at the entrance in the system (forced excitation) coming from the machine-tool. In these conditions were measured the displacements due to vibrations; vibrations produced during a mechanical process on the machinetool (cutting process) and then it was analyzed how they are transmitted through the hand-arm system up to the organism. The signal of entrance in the system was applied to the hand and fingers and the measurements were done for several standard rotation of the machine-tool, the exciting source (lathe). Key words: hand-arm system, mechanical vibration, measurements machine-tool

1 Introduction It is useful for the measurement of the mechanical vibrations to know the human response to vibrations as well as knowing the variation of real systems to excitations (for instance the control interface of the hand). This data offer the possibility of anticipation of forces and movements causing vibrations, the interpretation of those data in the purpose of minimizing the transmission of vibrations, respectively the design in the same purpose regarding tools and striker devices [5]. Not at least these studies led to assure the healthy measures regarding the equipments of the persons exposed to vibrations during work. In the literature of specialty [3,4,7] there are studies regarding the transmission of vibrations from striker tools (e.g. saws in wood industry, striker hammers in mining etc.) to the human body but few studies were done on the machine-tool (analyzed for longer periods 5–20 years) and the effect of the vibrations transmitted by it on the exposed person’s state of health. Because most often the vibrations are transmitted from tools to human body through the hand it has been chosen the analysis of their transmission through the hand-arm system up to the shoulder. Also, for this reason it has been studied and

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Fig. 1 Components of a mono-axial piezo-electric transducer of compression.

analyzed a mechanical model close to the real hand-arm system whose mechanic characteristics (mass, elasticity, dumping) have been taken closest to the real data.

2 Experimental Considerations Vibration measurements presented in this work were done on a volunteer person (man around 40 years) who made a cutting process on a SNA 580X lathe, the power of lathe P = 11 kW, operator who kept his hand fixed on the crank of the longitudinal slide of the lathe [2]. The study has been done for seven rotations of the machine-tool (lathe) 250, 315, 400, 500, 630, 800 and 1000 rpm but for reasons of space in this work will be presented the data obtained for the smallest and biggest rotation taken into study, namely 250 and 1000 rpm respectively. Displacements, velocities and accelerations due to mechanic vibrations transmitted from the machine-tool to the hand-arm system were measured by a mono-axial transducer of compression, piezo-electric KD 42. It was fixed on a light aluminum support of 45 g glued with wax on the direction of the anatomic system of coordinates Ozh (along the arm). On its turn the support was fixed on the surface of the skin by an elastic strip stretched on the studied anatomic location (wrist, elbow and shoulder). Exist signals consequently to the measurements were captured on an acquisition board mounted in the computer and given back through the programming medium Matlab [4,6] under the form of graphics, on the display or they could be saved in files.txt.

3 Mechanical Model of the Hand-Arm System It has been tried the approach of a mechanical model closest to the real model of the hand-arm system having 5 degrees of freedom. So it has been done the mechanical model presented in Figure 2 whose mechanic characteristics (mass, spring, damper).

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Fig. 2 Mechanical model of the system hand-arm. Table 1 Initial conditions taken into account for the study of the movement. Name

Initial conditions values and positions

Vibrating direction Excitation Elbow angle Subject position Handle diameter Frequencies scale Gripping force Griping place Shoulder rotation Subject

Axe zh (in conformity with anatomical coordinate frame) 5,297 (m/s2 ) 90◦ Stand up 38 (mm) 4–16.66 (Hz) 25 (N) Palm 0◦ men

The excitation of the system has been analyzed only for the direction of transmission Ozh of the anatomic system of coordinates (given by the specialty literature [9]) STAS 5349/2003 and namely on the long of the arm. The forced excitation has been produced by the machine-tool (lathe) and studied for the rotations of machine-tool at 250 and 1000 rpm. In order to realize the system of equations, system presented under matrix form by the relation [1] there were taken into account the initial conditions presented in Table 1. Here [M], [C] and [K] represent the matrix of the masses, of dumping and elasticity having dimensions (5 × 5); {Ki } and {Ci } are matrix of dimensions (5 × 1) and

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Table 2 Researches of T. Cherian, S. Rakheja, R.B. Bhat [1] corresponding to elasticity and dumping mechanical characteristics to system hand-arm. Vasco-elasticity parameters to system hand-arm k0 k1 k2 k3 k4

= 155.8 ×10 = 23.6 ×103 = 444.6 ×103 = 415.4 ×103 = 50.25 ×103 3

N/m N/m N/m N/m N/m

kt1 = 2 Nm/rad kt2 = 2 Nm/rad

c0 c1 c2 c3 c4

= 30 Ns/m = 202.8 Ns/m = 500 Ns/m = 164.6 Ns/m = 50 Ns/m

ct1 = 4,9 Nms/rad ct2 = 6,14 Nms/rad

Table 3 Anthropometric parameters to system hand-arm. Anthropometric parameters m1 = 0.45 kg m2 = 1.15 kg m3 = 1.9 kg

l3 = 0.298 m l3 = 0.178 m l3 /l3 = 0.6

Jc3 = 0.0149 kgm2

{U} is a vector matrix corresponding to the generalized coordinates of the movement of dimension (5 × 1){U}T = {z1 , z2 , z3 , x3 , θ3 } and z(t) = z0 sin(ω t)) forced excitation. The differential equations model is matrix representation in the following mode: ¨ + [C]{U} ˙ + [K]{U} = {Ki }z + {Ci}˙z [M]{U} 0 0

(1)

where z1 represents the displacement of the hand on the direction Ozh , z2 represents the displacement of the forearm on the directions Ozh , z3 , x3 represents the displacements of the arm on the directions Ozh and Oxh and θ3 represents the rotation in the joint of the elbow on the direction Oyh , generalized coordinates zi , xi (i = 1, 2, 3) measured in [meter] and θ [radian]. The mechanical model presented in Figure 2 is a model with 5 degrees of freedom (4 translations and a rotation in the joint of the elbow). Mechanical characteristics were taken according to the specialty literature [1, 2] and they are presented in Table 2, and the anthropometric data were determined in Table 3 so: anthropometric measurements were done on a group of 5 subjects (men); taking in study a mean of those measurements needed to determine the values of the masses m1 – of hand, m2 – of forearm and m3 – of the arm; of the centers of weight corresponding to those and of the mechanic moment of inertia axial of the arm. Masses m1 , m2 and m3 were determined by the rules of three simple (mathematically) referring to the volume and weight of the analyzed person and to the volumes of the hand, forearm and arm estimated to conoidals. The ratio l3 /l3 = 0.6 from Table 3 defines the ratio of the distances from the elbow to the mass center of the arm and from the joint of the elbow to the shoulder, value determined by approximating the arm to a conoidal and finding the center of weight of the conoidal. The calculus of the mechanic moment of inertia axial Jc3 (Table 3) was made approxim-

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ating the arm to a rigid bar supported at the two supports namely at the joint of the elbow and of the shoulder. The system of differential equations corresponding to the dynamics of the assembly hand-arm is obtained by applying the theorem of movement of the mass centers and of the theorem of kinetic moment reported to the point θ3 for mechanical model presented in Figure 2. The system of equations that corresponds to the mechanic model presented in relation 1 (matrix form) will be a system of 5 differential equations of order 2, nonlinear, non-homogenous and transcendent. In order to solve it there were taken in calculus some simplifying hypothesis: it has been studied the model with masses distributed as a model of masses concentrated for the mass m1 (hand) and m2 (forearm) and the arm was assimilated to a rigid bar, homogenous articulated at supports; it was neglected the joint of the hand (wrist) on the reason that the movement of the system hand-arm, hand fixed on the machine-tool does only very little rotations from the joint of the hand (wrist). The elbow was considered folded at 90◦ and the angle at the joint of the shoulder was considered 0◦ ; also the wrist (joint of the hand) was simplified to a spring and a dump; it was imposed the direction of transmission of the vibrations from the machine-tool to operator’s hand on the direction Ozh construction from dampers (c1 , c2 and c3 ) and on the direction Oxh for the anatomic system of coordinates of the hand Oxh yh zh being 5349/2003 [9]. Solving the system of differential equations corresponding to the mechanic model (relation 1) of the Figure 2 was done using the method Runge Kutta-Gill of order 4 and the medium of programming Matlab [6]. The integration of the system of equations was done for 2s. The solutions resulted in graphic form were for displacements and velocities: z1 , z˙1 – displacement and velocity corresponding to the hand (movement of the mass center m1 ); z2 , z˙2 – displacement and velocity corresponding to the forearm (movement of the mass center m2 ); z3 , z˙3 displacement and velocity corresponding to the arm (movement of the mass center m3 ). All these are done on the direction of transmission Ozh ; x˙3 – displacement and velocity corresponding to the arm (movement of the mass center m3 on the direction Oxh ); θ3 , θ˙3 – displacement (rotation) and angular velocity corresponding to the arm, rotation made in the joint of the elbow on the direction Oyh . The results obtained from solving the system, corresponding to the rotation in the joint of elbow θ3 , for the rotation of the machine-tool of 250 and 1000 rpm are very small in value, this framing in the interval of 3–4◦. For this reason the graphic representation of the other values of the displacements obtained z1 , z2 , z3 , x3 did not take into account these, from the point of view of the scale of representation. The values obtained for the displacements z1 , z2 , z3 , x3 are acceptable from the point of view of the reality; they framing according to the Figure 3 for all anatomic locations studied in the interval 10−3 or even 10−4 m.

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Fig. 3 Displacements corresponding to the hand-arm system to rotation of 250, 1000 rpm.

4 Transmissions of Mechanic Vibrations In order to verify the data obtained on a theoretical way in Section 3, they were analyzed the experimental results [7,9] obtained in Section 2 and the values were compared. It is reminded that the measurements of vibrations were done for the wrist, joint of elbow and shoulder at an excitation of the machine-tool of 250 and 1000 rpm (Figures 4a, 4b, 4c). The graphic presented in Figure 4a shows that the displacements due to the vibrations on the wrist for the rotations of the machine-tool of 1000 rpm presents the biggest values compared to the other anatomic locations studied (elbow and shoulder). For the wrist the littlest values were reported at the wrist for 250 rpm on the axe Ozh and for this reason it has been reported an auxiliary axe on the right of the graphic. Figure 4b represents the displacements corresponding to the arm, respectively the displacements made by the joint of the elbow on the directions Ozh and Oxh . It is noticed that the biggest values were obtained for the elbow at the rotation of 1000 rpm on the axe Ozh , and the littlest values were obtained for the rotation of 250 rpm on the axe Oxh . In Figure 4c it is shown that the biggest displacements on shoulder were obtained on the direction Oyh of the anatomic system, direction neglected in the theoretical study at the rotation of the machine-tool of 1000 rpm. These values were followed by the values obtained on the axe Ozh and Oxh at 1000 rpm. The littlest values were obtained regardless the direction of measurement on the shoulder for the rotation of the machine-tool of 250 rpm. In conclusion, it may be said that for all anatomic locations studied: wrist, joint of the forearm (elbow) and arm, on the direction of excitation Ozh the biggest values of the transmissions of the vibrations (displacements) were obtained for the

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Fig. 4 Measurements effectuated to the system hand-arm (wrist, elbow, shoulder) corresponding to machine-tool rotation of 250 and 1000 rpm.

biggest rotation of the machine-tool taken in study of 1000 rpm, and the littlest values of transmission of the vibrations were obtained for the smallest rotation of the machine-tool of 250 rpm for the wrist.

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5 Conclusions The present work proposes to bring contributions regarding the study of transmissibility of mechanic vibrations from the machine-tool to the human operator’s hand through the hand-arm system. By comparing the results obtained in paragraphs 3 and 4, namely those theoretic and experimental it is noticed that these are comparable from the point of view of the state of size 10−3 ÷ 10−4 meters for all anatomic locations taken in study (hand, forearm, arm) and for the direction of transmission Ozh of the anatomic system of coordinates, namely along the arm. Future researches in the field propose to realize devices of diminution or dissipation of the transmission of the vibrations through the hand-arm system and the choice of the anatomic location suitable for obtaining the desired effect of minimizing the transmission of the vibrations.

References 1. Cherian, T., Rakheja S. and Bhat, R.B., An analytical investigation of an energy flow divider to attenuate hand-transmitted vibration. International Journal of Industrial Ergonomics, 17:455– 467, 1996. 2. Cristea, A.F., Trut¸a, A., Sas, A. and Arghir, M., Studiu privind influent¸a vibrat¸iilor asupra corpului uman datorate prelucr˘arii pe ma s¸ina de frezat universal˘a pentru scule FUS-22 – Analele Universit˘a¸tii din Oradea, Fascicula Inginerie Managerial˘a s¸i Tehnologic˘a, Oradea, Romˆania, pp. 55–60, 2002. 3. De Mester, M., De Muynck, W., De Bacquer, D., and De Loof, P., Reproducibility and value of hand-arm vibration measurements using the ISO 5349 method and compared to a recently developed method. In Proceedings of the Eight International Conference on Hand-Arm Vibration, Ume˚a, Sweden, pp. 11–13, 1998. 4. Dobry, M.W., Energy flow in Human-Tool-Base System (HTBS) and its experimental verification. In Proceedings of the Eight International Conference on Hand-Arm Vibration, Ume˚a, Sweden, pp. 7–9, 1998. 5. Doel, K. and Pai, D.K., The sounds of physical shapes, University of British Columbia, Vancouver, Canada, vol. 7, pp. 382–395, 1998. 6. Matlab: Virtual Reality Toolbox for Use with Matlab and Simulink, User’s Guide, Vol. 3, The MathWorks, USA, www.Mathworks.com, 2009. 7. Nishiyama, K., Taoda, K., Yamashita, H. and Watanabe, S., The temporary threshold shift of vibratory sensation induced by hand-arm vibration composed of four one-third octave band vibrations, Journal of Sound and Vibration, 200(5), 1997. 8. Nortman, S., Aroyo, A., Nechyba, M., and Schwartz, E., A Humanoid Robot Implementation, University of Florida, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.8767, 2003. 9. Palmer, K.T., Coggon, D.N., Griffin, M.J. et al., Hand-transmitted vibration: Occupational exposure and their health effects in Great Britain, University of Southampton, UK, 1991.

Elastic and Safety Clutch with Metallic Roles and Elastic Rubber Elements I. Stroe Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: [email protected]

Abstract. The modular design imposes finding the optimal solutions from constructive and functional point of view. The constructive design must be correlated with the technological one. Thus, it is possible to obtain mechanical components with reduce building limit and weight, with high durability and small price. In this context, the present paper presents the conceiving and the design of a new clutch with multiple functions, the elastic and safety clutch. This type of clutch combines the functions of elastic and safety clutches, and it will be denoted as elastic and safety clutch with metallic roles and elastic rubber elements. Key words: clutch, elastic, safety, multiple functions

1 Introduction The clutches – machine parts or equivalent mechanical systems of those – are absolutely necessary on mechanical transmissions. The clutch realises the connection between two successive elements of a kinematic chain or between two units of a complex transmission. There exists a large variety of the clutches conceiving shapes. The use of the clutches is determined by the characteristic of the motoring machine, the kinematic chain which the clutch is equipping it and the working pattern of the driven machine. The main function of the clutches is characteristic to all of them and is the function of transmitting the motion and the torque moment. The other functions, specific to each clutch type are: the motion commanding, the load limitation (with or without interrupting the kinematic flux), the protection against shocks and loads; the compensation of assembling errors; the compensation of the errors which can appear during working; the limiting of revolution; the one-sense transmission of the motion. All of these functions can appear singularly or concomitantly. These clutches are denoted as simple clutches – which constitutes simple units from structural point of view and which cannot be divided in more simple units [1, 3]. In many cases, the mechanical transmissions need multiple functions in order to work in optimal conditions and parameters. To obtain multiple functions, combined

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Fig. 1 The classification of the mechanical clutches.

clutches are used. These clutches are obtained by connecting (usual, series connection) two or more clutches which – in this constructive shape – will adequately fulfil the complex functional role imposed by the transmission. By analyzing the speciality literature it is discovered that in the machine manufacturing domain, the use of the combined functions of elastic clutches and safety clutches is frequently required and/or needed. Because combining the two types of clutches leads to a high complexity degree (from both technical and economical point of view), the necessity of conceiving a new type of clutch which joints the two groups of functions (safety and elasticity) is required. The new clutch will have from the constructive point of view a reduced complexity, similarly to a simple one [1, 3, 4]. The clutch combines the functions of an elastic clutch with intermediary rubber elements with the functions of a safety clutch. In Fig. 1, a classification of the mechanical clutches which includes also the new type of the elastic and safety clutch is presented [4–6].

2 Elastic and Safety Clutch The paper presents a new type of clutch named “Elastic and Safety Clutch”, that can accomplish the functions of an elastic and of a safety clutch, but which is not a combined clutch [4, 5, 8]. The angle of relative rotation between the two semiclutches depends on the number of rolls. Figure 1 presents the strict systematization of the mechanical clutches with simple functions as well as the approach of finding the new type of clutch.

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Fig. 2 Structural scheme.

2.1 The Structural Schemes of Clutch The elastic and safety clutches are characterized by the following functions [4, 5, 8] (functional and technical criteria): • they make the bundle between two shafts (with relatively fix variable position) and ensure the moment transmission and the rotation motion between the shafts (according to the general definition); • the power transmission is braked off when the resistive moment outruns an imposed limit value; • the braking off of the energy flux is realized on the basis of an elastic element deformation. Out of the analysis of the properties corresponding to the elastic and safety clutches, a distinctive importance goes to the elastic element modeling, in order to ensure the automated braking off every flux, at the torque limit value. From their use within technique, the critical analysis [2–4] of the mechanism leads to the conclusion that the cam mechanism (Figure 2) lends oneself (the best) to the demands previously formulated. Figure 3 presents the structural schemes of now the elastic and safety clutch. The clutch is part from a new family of clutches [4–8] which combines the functions of the elastic clutches with the ones of the safety clutches, Figure 1. The clutch has in its structure a cam which is degenerate in an element with multiple metallic roles (3 in Figure 3), radial disposed, slipping mounted on the bolts; the followers are degenerate in multiple rubber roles (2 in Figure 3), equiangular disposed.

2.2 Construction of the Clutch Starting from the structural schemes and from the representative functions and proprieties, of the elastic and safety clutch, the next criterions of constructive generation can be formulated: • the clutch must absorb radial and angular tilts;

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Fig. 3 The structural scheme of the new type of clutch.

• the relative movement between the semi clutches, as well as the releasing must be made without shocks; • the clutch must have a reduce rigidity; it is suggested a characteristic Mt (ϕ ) with a rising inclination and a big damping capacity; • the elasticity of the clutch could be modified, by changing or adding constructive elastic elements; • when the clutch is turning around, big axial forces does not appear; • the clutch must not break down when an elastic element is destroyed; • the elastic constructive elements, that can be destroyed fast, must be replaced fast; if it is possible without demount the clutch; • the changing of the rotation sense must be permitted without duty cycle; for the safety enlargement in running the component elements of the clutch must not have protuberances. Based on these criterions, for each structural representative scheme, there is generated one constructive variant [4, 5]. The constructive solution allows the pretension adjustment without taking to pieces the clutch, Figure 4 [4, 5]. The connection between the two semi-clutches 1 and 2 is realized through the rubber roles 6, which are fixed on bolts 7 using antifriction mechanical sleeve 8. The charge is transmitted from semi-clutch 1 to semi-clutch 2 through (throughout) the rubber roles 6 and metallic roles 5. During the working of the clutch, two important states can be distinguished: • in the first phase, which corresponds to the well (normal) working of the mechanical transmission, the metallic roles together with the elastic ones will mutual coil, the relative motion between the semi-clutches tacking place; • the second phase corresponds to a load higher to the one permitted by the transmission; in this moment, the relative motion between the clutches is amplified and the elastic elements are strongly distort. Thus, this fact will lead to the interruption of the torque moment transmission.

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Fig. 4 The design of the clutch.

3 The Torque Moment The elastic and safety clutches are characterized by a variable rigidity (nonlinear characteristic) – Eq. (1); the security condition of the mechanic transmission is presented in Eq. (2) [1, 3]: dMt (ϕ ) , (1) k(ϕ ) = d(ϕ ) Mt lim (1 + ∆ ) ≤ Mt max a ,

(2)

where k(ϕ ) represents the tangent to the curve of the torsion moment, which is written depending on the relative rotation; ϕ is the relative rotation angle, between the semi-clutches; Mt (ϕ ) is the torsion moment corresponding to the clutch deformation with the angle ϕ ; Mt lim is the torsion moment when the uncoupling produces or ends; Mt max a is the maximum torsion moment admitted by the strength of the most weak clutch element; ∆ is the relative error reset inputs in function of the clutch. Figure 5 presents the geometrical model for the torque determination moment. The torsion moment that can be transmitted by the clutch is given by Eq. (3), while the relative torsion angle between the two semi-clutches is given by Eq. (4):

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Fig. 5 Geometrical model for the torque moment determination.

1 Mtcap = D0 zA0 E1 2




ϕmax 2h D − ϕmax

 ≥ Mtc ,

(3)

0

ϕ=

2Mt h  , D0 Mt + 12 D0 zA0 E1

(4)

where D1 is the arrangement diameter of the iron rolls; D2 is the arrangement diameter of the rubber rolls; D0 is the winding diameter of the rolls in the relative motion between the semi-clutches; z is the number of rolls equiangular disposed; A0 is the initial section surface of the rubber elastic elements; E1 is the elastic modules of the elastic element in pre-compressed state; and h is the thickness of the elastic element in post-compressed state.

4 Determination of the Elastic Characteristic of the Clutch Figure 6 presents the elastic prototype of the elastic and safety clutch with metallic roles. The characteristic of the clutch is determined statically by fixing the semiclutch 1 and applying different loads to the semi-clutch 2. The obtained results were graphically represented and also approximated using second and third order equations. These graphs are presented in Figure 7. Analyzing the above presented graphs it results that the closest characteristic to the ideal one is the curve described by the second order equation. The clutch has a progressive characteristic and depends upon the elasticity of the rubber elements. The relative torsion angle between the semi-clutches depends on the number of roles. The presented clutch can take over axial deviations, radial deviations and angular deviations depending on the dimensioning of the clutch.

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Fig. 6 Elastic and safety clutch with metallic roles and elastic rubber elements.

Fig. 7 The elastic characteristic of the clutch.

5 Conclusion The elastic and safety clutches with metallic roles and elastic rubber elements present the following advantages: • the clutches have a simple construction, cabaret reduced dimension, cheap price; • the clutches ensure the compensation of axial, radial and angular deviation in relatively large limits; • the clutches ensure the a relative movement between the semi-clutches, according to the nature and to the disposing mode of the component elements; above the accepted limits, the elastic clothe becomes a safety one; • the clutches ensure the limitation and the adjustment of the transmitted moment;

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• the clutch can take over technological and assembling deviations; • the clutch allows the damp of the torsion shocks transmitted in different transient regimes of the mechanical transmission; • the clutch has a progressive characteristic and depends on elasticity of the rubber elements; • the relative torsion angle between the semi-clutches depends on the number of roles.

References 1. 2. 3. 4.

5. 6.

7.

8.

Dr˘aghici, I., et al., Calculul s¸i construct¸ia cuplajelor. Editura Tehnic˘a, Bucharest, 1978. Dudita, Fl., et al., Optimizarea structural a mecanismelor, Editura Tehnic˘a, Bucharest, 1982. Pampel, W., Kupplungen, Band I, VEB Verlag Technik, Berlin, 1958. Stroe, I., Theoretical and experimental contribution regarding the conceiving and modulations of a new class of clutches with multiples functions – Elastic and safety clutches. Ph.D. Thesis, Transilvania University of Bras¸ov, 1999. Stroe, I. and Eftimie, E., Elastic and Safety Clutch, Editura Ecran Magazin, Bras¸ov, 2001. Stroe, I., Design procedure of elastic and safety clutches using cam mechanisms. In Proceedings on CD-ROM of the Twelfth World Congress on Mechanism and Machine Science, Besancon, France, June 17–21, 2007. Stroe, I., Elastic and safety clutch with radial disposed elastic lamellas. In Proceedings of EUCOMES 08, The Second European Conference on Mechanism Science, M. Cecarelli (Ed.), pp. 133–138, Springer, 2009. Stroe, I., Simple mechanical clutch with multiple functions. In Proceedings of SYROM 2009, I. Visa (Ed.), pp. 433–438, Springer, 2009.

Mechanisms for Biomechanics

A New Spatial Kinematic Model of the Lower Leg Complex: A Preliminary Study B. Baldisserri and V. Parenti Castelli DIEM-Department of Mechanical Engineering, University of Bologna, 40126 Bologna, Italy; e-mail: {benedetta.baldisserri, vincenzo.parenticastelli}@mail.ing.unibo.it

Abstract. The importance of the human joint passive motion, i.e., the articulation motion in virtually unloaded conditions, for the study of human diarthrodial joints has been widely recognized. Recently, it has been shown that equivalent mechanisms make it possible to obtain physicalmathematical models that can replicate the articular passive motion well. These models also represent a useful tool for both pre-operation planning and prosthesis design. Although the human ankle joint has been extensively investigated, studies that examine the kinematic behaviour of the tibio-talar joint and also outline the motion of the fibula bone are still lacking. This paper focuses on the 3D kinematic model of the articulation that involves four bones: the tibia, fibula, talus and calcaneus. In particular, a new spatial equivalent mechanism with one degree of freedom is proposed for the passive motion simulation of this anatomical complex. The proposed mechanism is believed to play an important role for future developments of models of the entire human lower limb. Key words: tibia-fibula-ankle complex, passive motion, kinematic model

1 Introduction The term ‘ankle complex’ is used to identify the anatomical structure composed of the ankle (that refers to the tibio-talar joint) and the subtalar joints. The static and dynamic behaviours of the human ankle complex have been extensively investigated in order to improve surgical planning and prosthetic design. In particular, the importance of the passive motion, i.e., the motion under virtually unloaded conditions, has been extensively demonstrated [1, 3, 7]. Indeed, this makes it possible, for instance, to clarify the role played by the principal anatomical structures of the articulation (ligaments, articular surfaces, . . . ) and to allow a more reliable dynamic analysis. A large number of studies in the literature have tackled the analysis of the ankle complex by means of mechanical models that can describe the relative motion of the bones under different loading conditions. Frequently, these models approximate the motion of the human ankle joint by a planar motion in the sagittal plane, by modelling the entire articulation with a hinge joint [4, 5, 8, 9]. Moreover, a number of models simplify the anatomical structure of the ankle complex involving only

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a few anatomical elements chosen among those that play an important role in the mechanical behaviour of the ankle (such as bones, ligaments, tendons, cartilage, etc.). For example, papers aimed at studying the behaviour of the ankle joint and at also investigating the motion of the fibula bone are lacking in the literature, although the kinematic behaviour of the fibula is directly involved in the kinematic analysis of the ankle. The aim of this study is to develop a 3D kinematic model for the passive motion simulation of the articulation that involves the four bones, the tibia, fibula, talus and calcaneus (henceforth the entire articulation is called TFC for brevity). In particular, starting from experimental data and previous studies, a new equivalent spatial mechanism with one degree of freedom (DoF) is proposed, in which every rigid link corresponds to a specific anatomical element. This makes the geometry of the mechanism fit the anatomical structures of the natural joint. This model of the TFC complex passive motion is believed to represent a useful tool for studying the fibula’s role during the ankle motion and thus to gain a better understanding of the ankle complex mechanical behaviour. Moreover, by involving most of the anatomical structures that affect the ankle motion, the TFC model can play an important role for future development of models that will describe the motion of the entire leg.

2 Modelling Basic Assumptions Passive motion models of human joints involve only passive structures, such as ligaments and bones, and can be developed by means of equivalent mechanisms. Indeed, the use of equivalent mechanisms has made it possible to achieve good results [1, 2, 7]. The efficiency of this approach for the modelling of the passive motion shows that if a close correspondence between articulation anatomical structures and mechanism elements can be recognized, then a useful equivalent mechanism can be devised, which satisfactorily simulates the passive motion of the articulation. More precisely, the following correspondences can be assumed: 1. bones with rigid bodies; 2. ligament isometric fibres with rigid rods; 3. ligament-to-bone insertions with spherical pairs (or universal joints for also considering the ligaments twisting around their own axes); 4. bone contact points with higher pairs which have 5-DoFs. For instance, sphereon-sphere contact higher pairs are equivalent to two rigid bodies linked by a rigid rod connected to the two bodies by spherical pairs. In fact, during the relative motion of two conjugated spherical surfaces, the two surfaces maintain the contact at a single point and the distance between the centres of the two spheres does not change. For the modelling of the TFC complex passive motion, four main bones, namely the tibia, fibula, talus and calcaneus, and some ligaments are taken into account. In particular, the talus and calcaneus are considered as a single rigid body because

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the relative motion between these two bones is negligible with respect to the relative motion of the other bones. Hence the four bones, the tibia, fibula, talus and calcaneus are modelled as three rigid segments; namely, the tibia, fibula and talus/calcaneus. Among all the ligaments of the lower leg, two ligaments have been experimentally proved to have some isometric fibres during ankle passive motion, namely, the calcaneofibular, CaFiL, and tibiocalcaneal, TiCaL, ligaments [6]. For the other ligaments, the isometry of some fibres is taken as an assumption since it allows the modelling of these ligament fibres with rigid rods. Furthermore, experimental studies showed that during passive joint flexion, the ankle complex behaves as a one-DoF system [1, 2, 6]. Based on this observation, it is assumed that the TFC complex under virtually unloaded conditions can also be considered as a one-DoF system. Feasible one-DoF spatial equivalent mechanisms can thus be developed. If, for instance, all contacts between the bones are modelled as sphere-on-sphere higher pairs, they can be represented with rigid rods linked to the bones by spherical pairs. Ligament isometric fibres can also be modelled by rods with ending spherical pairs. It can be easily shown by the Kutzbach formula that a mechanism with three rigid bodies (tibia, fibula and talus/calcaneus) interconnected by binary links through spherical pairs requires 11 binary links connecting the three segments in order to have one-DoF. The 11 rods and the three rigid bodies can be arranged to form different mechanisms according to how the bone contacts are modelled and consequently how many ligaments are considered. Indeed, the adopted model for the articular surfaces of the bones involved in the articulation determines how many binary links have to be used to represent bone contact points and consequently how many ligament isometric fibres should be used. These considerations can be the starting point for the development of different one-DoF spatial mechanisms to simulate the passive motion of the TFC complex. For instance, two possible mechanisms are shown in Figures 1a and 1b. For the first mechanism (Figure 1a) five bone contact points are considered (black dots depicted in Figure 1a) and modelled with sphere-on-sphere higher pairs. Therefore, six ligaments are used and modelled with rods connected with the bone segments by spherical pairs that are centred at points representing the ligament insertions on the bones. The second mechanism depicted in Figure 1b is different: six bone contact points are considered (black dots depicted in Figure 1b) and modelled with sphere-onsphere higher pairs. Therefore, only five ligaments are used and modelled with rods connected with the bone segments by spherical pairs that are centred at points representing the ligament insertions on the bones. Both models shown in Figures 1a and 1b could be adopted thanks to the presented method. However, in a previous study on the ankle joint [2], the contacts of the tibia with the talus and fibula were successfully modelled as two sphere-on-sphere higher pairs and as one sphere-on-sphere higher pair respectively. Therefore, by using similar assumptions for the tibia-talus and fibula-talus contacts, a further new model of the TFC complex can be devised, which is simpler than the two previous

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Fig. 1 Mechanisms for the TFC complex passive motion.

ones. The new one-DoF mechanism, which is reported in the next section, is the one proposed for modelling the TFC complex passive motion.

3 The Proposed Model of the TFC Joint In the new mechanism (see Figure 2) six rods represent six ligament isometric fibres connected to the bones by spherical pairs centred at points Ai and Bi , i = 1, 2, 3, Ci and Di , i = 6, 7, 8, A10 and C10 : points Ai and Bi , i = 1, 2, 3, represent the insertion points respectively on the tibia and talus-calcaneus segments of the ligaments TiCaL, TaTiL-ant (anterior tibiotalar ligament) and TaTiL-post (posterior tibiotalar ligament); points Ci and Di , i = 6, 7, 8, represent the insertion points respectively on the fibula and talus-calcaneus segments of the ligaments CaFiL, TaFiL-ant (anterior talofibular ligament) and TaFiL-post (posterior talofibular ligament); A10 and C10 represent the insertion points respectively on the tibia and fibula segments of a fibre of the interosseus membrane. Moreover, points A4 and A5 , B4 and B5 and D9 , C9 , represent the centres of the mating spherical surfaces fixed to the tibia, talus/calcaneus and fibula respectively. For the definition of the sphere-on-sphere higher pairs, the following approach can be used: the areas of the surfaces of the mating bones which come into contact during passive motion can be digitized and approximated by their best fit spherical or plane surfaces. Moreover, based on a careful inspection of the proximal part of the tibia and fibula, the contact point between the two bones is modelled by a plane-on-sphere higher pair, that better approxim-

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Fig. 2 The proposed mechanism for the TFC complex passive motion.

ates the articular contact between the two bones; point C represents the centre of the mating spherical surface fixed to the tibia proximal end and in contact with the plane surface fixed to the fibula proximal end. For the equivalence between sphere-on-sphere contact higher pairs and rigid rods linked to two rigid bodies by spherical pairs, the equivalent mechanism depicted in Figure 2 can be more synthetically represented by the one-DoF mechanism shown in Figure 3. Here the meaning of the points Ai , Bi , C j , D j , Ak , Ck (i = 1, 2, . . . , 5, j = 6, 7, . . . , 9 and k = 10) and C is the same as in Figure 2. This one DoF mechanism will be considered as the equivalent mechanism of the TFC complex for the passive motion. The mathematical model of the equivalent mechanism shown in Figure 3, i.e., the model that provides the relationship between the independent variable of motion and the dependent ones, which define the configuration of the mechanism, is provided by the closure equations of the mechanism. The equations make it possible to find the relative position of the three bones (tibia, fibula and talus/calcaneus) during the ankle passive flexion. In particular, they are obtained by a suitable exploitation of some kinematic constraints of the mechanism, which make the mathematical model (i.e., the equations) simpler than the one obtained based on the link-to-link DenavitHartenberg transformation matrices.

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Fig. 3 The equivalent mechanism for the TFC complex.

With reference to Figure 3, it can be noted that each pair of points (Ai , Bi ), i = 1, 2, . . . , 5, (C j , D j ), j = 6, 7, 8, 9, and (A10 , C10 ), is constrained to maintain a constant mutual distance, Li , L j and L10 respectively, during motion. This makes it possible to write: (i = 1, . . . , 5)
Ai − Rtc · Bi − ptc
2 = L2i ( j = 6, . . . , 9)
D j − Rc f · C j − pc f
2 = L2j 2 2
A10 − Rt f · C10 − pt f
= L10

(1)

where Ai is the position vector of the point Ai measured in the reference system St , Bi and D j are the position vectors respectively of the points Bi and D j measured in the reference system Sc , and C j is the position vector of the point C j measured in the reference system S f . The Cartesian reference systems St , Sc and S f are embedded in the tibia, talus/calcaneus and fibula segments respectively (see Figures 2 and 3). The generic vector pi j represents the position of the origin O j of the generic reference system S j with respect to the generic reference system Si ; the generic matrix Ri j is the orthogonal rotation matrix 3x3 that transforms the components of a vector measured in the generic reference frame S j into the components of the same vector measured in the generic reference frame Si (the indices c, t and f refer to Sc , St and S f reference systems respectively). The matrix Ri j can be expressed as a function of three parameters that represent the orientation of the reference system S j with respect to Si . The plane-to-sphere articular contact between the tibia and fibula at the proximal end can be represented by constraining the centre of the sphere to move in a plane

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parallel to the articulating plane that approximates the fibula surface at the proximal end. Hereafter, n denotes the unit vector perpendicular to the plane in contact with the sphere, C the centre of the sphere, and H the point of the plane that the point C belongs to. The plane-to-sphere contact is expressed as: t

with

n · (C −t H) = 0

(2)

= Rt f · n = Rt f · H + pt f

(3)

tn tH

where the vector n is measured in the reference system S f , H is the position vector of the point H measured in the reference system S f , C is the position vector of the point C measured in the reference system St , and the matrix Rt f and the vector pt f have the same meaning as explained above. The system of Eqs. (1) and (2) represents the closure equations of the mechanism. When considering the tibia as a fixed body, for a given geometry, this system can be regarded as a system of eleven non-linear equations in twelve variables, that are the three components of vector pt f , the three orientation parameters that define the rotation matrix Rt f , the three components of vector ptc and the three orientation parameters that define the rotation matrix Rtc . By simple matrix operations, pc f and Rc f can be easy calculated using the vectors pt f and ptc , and the matrices Rt f and Rtc . Given the angle that measures the ankle flexion – i.e. the rotation between talus and tibia in the sagittal plane – the remaining 11 variables can be found by solving the system of Eqs. (1) and (2). The equivalent mechanism proposed here is believed to provide good results both to simulate the TFC complex passive motion and to fit the anatomical structures. However, some limits can be pointed out. The assumption of considering an isometric fibre for every ligament is not yet fully justified by experimental data, except for CaFiL and TiCaL; on the other hand, representing the ligaments as rigid rods makes the mechanism less complex and has led to good results in previous studies. If the presented model fits the experimental data of the TFC passive motion well, it will be necessary to experimentally prove the real existence of isometric fibres in the selected ligaments. Moreover, there is a single DoF in the mechanism between the tibia and the talus/calcaneus: this means that the two sphere-on-sphere higher pairs between the talus and tibia and the three rigid rods representing the three ligaments, TiCaL, TaTiL-ant and TaTiL-post, constrain the relative motion between the tibia and the talus/calcaneus to one-DoF motion. Therefore the fibula is driven only by this motion.

4 Conclusion The modelling of the passive motion of the TFC complex (tibia, fibula, talus/calcaneus) by a spatial equivalent mechanism has been proposed in this paper. The

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particular approach presented here allows the use of different valid solutions for the development of the sought-for equivalent mechanism. The proposed new model of the TFC complex passive motion is believed to replicate both the passive motion and to provide a good representation of the main anatomical structures of the TFC complex. It can thus be a useful tool to acquire better knowledge of the ankle complex mechanical behaviour and for future development of a model that could describe the motion of the entire leg. Experimental sessions are in progress to validate the efficiency of the proposed model.

Acknowledgement The financial support of Aer-Tech Lab is gratefully acknowledged.

References 1. Di Gregorio, R., Parenti Castelli, V., O’Connor, J.J. and Leardini, A., Equivalent spatial parallel mechanisms for the modelling of the ankle passive motion. In: ASME DETC 2004, 28th Biennial Mechanisms and Robotics Conference, 2004. 2. Franci, R. and Parenti Castelli, V., A 5-5 one degree of freedom fully-parallel mechanism for the modelling of passive motion at the human ankle joint. In: Proceedings of ASME-IDETC/CIE 2007, International Design Engineering Technical Conferences and Computers and Information in Engineering, Las Vegas, Nevada, USA, pp. 1–8, 2007. 3. Franci, R., Parenti Castelli, V. and Sancisi, N., A three-step procedure for the modelling of human diarthrodial joints. In: Proceedings of the RAAD2008, 17th International Workshop on Robotics in Alpe-Adria-Danube Region, Ancona, Italy, 15–17 September 2008. 4. Hansen, A.H., Childress, D.S., Miff, S.C., Gard, S.A. and Mesplay, K.P., The human ankle during walking: Implications for design of biomimetic ankle prostheses. Journal of Biomechanics, 37:1467–1474, 2004. 5. Ji, Z., Findley, T., Chaudhry, H. and Bukiet, B., Computational method to evaluate ankle postural stiffness with ground reaction forces. Journal of Rehabilitation Research and Development, 41:207–214, 2004. 6. Leardini, A., O’Connor, J.J., Catani, F. and Giannini, S., Kinematics of the human ankle complex in passive flexion; a single degree of freedom system. Journal of Biomechanics, 32:111– 118, 1999. 7. Leardini, A., O’Connor, J.J., Catani, F. and Giannini, S., A geometric model of the human ankle joint. Journal of Biomechanics, 32:585–591, 1999. 8. Pilkar, R.B., Moosbrugger, J.C., Bhatkar, V.V., Schilling, R.J., Storey, C.M. and Robinson, C.J., A biomechanical model of human ankle angle changes arising from short peri-threshold anterior translations of platform on which a subject stands. In: Proceedings 29th Annual International Conference of IEEE-EMBS, pp. 4308–4311, 2007. 9. Schepers, H.M. and Veltink, P.H., Estimation of ankle moment using ambulatory measurement of ground reaction force and movement of foot and ankle. In: First IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, pp. 399–401, 2006.

Selected Design Problems in Walking Robots Teresa Zielinska and Krzysztof Mianowski Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, ul. Nowowiejska 24, 00-665 Warsaw, Poland; e-mail: [email protected]

Abstract. The development and the usability of the developed machines has been constantly improved, aided by the improvements in the computer technology. There have been many publications underlying that legged locomotion is the best form of locomotion through varying terrain, as compared to using wheels. The surface of a terrain may be uneven, soft, muddy and generally unstructured. Legs in the biological environment demonstrate significant advantage in such situations. The proper design solution can improve the robot performances increasing the mobility and decreasing the energy spending. In this manuscript we summarise the fot design concepts what is important but rather neglected problem especially when develipong the muli-legged walking machine. Key words: foot design, walking machines, robot design

1 Introduction: Role of Adjusted Foot Design Biologically inspired symmetric gaits produce the fastest displacements of walking machines. In hexapod robots those gaits are statically stable, but in quadrupeds and bipeds the dynamical effects decide about postural stability. The role of the foot during fast, efficient gaits cannot be neglected but till our days it was not much discussed. The legs of multi-legged walking machines have usually 2 or 3 active degrees of freedom. The additional degrees of freedom (if introduced) are passive. The foot compliance is typically obtained using springs. Many multi-legged robots have feet shaped as ball or as a rotating plate. The feet are attached to the shank by passive prismatic joints (Fig. 1). More complex designs consist of 3 passive DOFs [1]. The potentiometers are sometimes utilized as sensors for monitoring the joint positions (Fig. 2). The biologically inspired foot with three fingers and 2 active DOFs (Fig. 3), is an unique example of a more complex structure [2]. In gait synthesis the attention is paid to the positioning of active joints

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Fig. 1 Foot shaped as a ball or as a plate.

Fig. 2 Leg-end joint with 3 DOFs.

Fig. 3 Biologically inspired foot.

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Fig. 4 View of the foot: left – in the design software, right – in one of our prototypes.

Fig. 5 Masses distribution and proportions of human body.

2 Spring Loaded Foot We proposed and analysed the spring loaded foot. The foot (or foot sole) and shank (or foot upper part) is connected by the spring (Fig. 4). This endows the leg with compliance – the spring length changes proportionally to the vertical force. This change is small, but produces postural equilibrium. The detailed theoretical analysis of contribution to postural stabilisation is given in [4]. Recently the spring loaded foot was also tested in small biped [5] developed by our research group. Robot prototype was 0.3295m tall, with length of thigh equal to 0.10 m and shank with foot – 0.145 m. The mass of the robot was 1.61 kg, with thigh mass equal to 0.16 kg, shank with foot –0.34 kg, and in that foot –0.10 kg. The distance between the ground projection of the robot ankle and rear of the foot was 0.041 m. The point mass of thigh was located below the hip joint In the distance

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Fig. 6 Masses distribution and proportions of robot body.

Fig. 7 One of the features of human gait.

equal to half of thigh length. Point mass of the shank considered together with foot was located below the knee in the distance equal to 83.7% of the shank segment length. Legs were about 3.7 times shorter than in the human. The scheme of robot and human bodies are given in Figs. 5 and 6. The human based gait pattern was implemented in the biped, details and results are discussed in [6]. The robot walk in some extend it is similar to detailed human motion performance where the pelvis traces a sinusoidal curve with an amplitude of approximately 0.06 m [3]. The pick of this shift occurs at the end of mid stance (Fig. 7). As is illustrated in Fig. 8 in single support phase the robot is inclined to the side for about 10o the pelvis central point side shift is about 0.04 m. Figure 9 shows the sequence of picture taken during robot walk. The detailed analysis of the stabilising role of the foot is given in [4]. Due to the limitied length of this manuscript and the amount of sequenced calculations it is not possible to incude it in this paper. The description of the synthesis of the presented gait is given in [5, 6].

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Fig. 8 Robot rolling during walk.

Fig. 9 Robot during the walk (selected frames from recorded movie).

3 Conclusions Our target is to use biological inspiration for robot design and motion controlling.. The proper design solutions can improve the robot performances increasing the mobility and decreasing the energy spending. This can be obtained by introducing the

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foot or leg compliance what is the recent and very promising trend in robotics. The confirmation that the feet play an important role in postural stabilization of a forlegged walking machine using has been formally obtained [4] and experimentaly tested. In the considered prototype (as in the majority of quadrupeds) the feet rotate freely around the vertical axis passing through the point of their attachment to the shank. The knowledge of leg-end force application point displacements towards the feet mounting points has an important influence on the synthesis of dynamical stable gaits. The knowledge of those displacements is needed to assess whether the foot supporting area will assure postural stability. The change of the foot area or the change of the leg configuration can be used to adjust the stable posture during the walk.

Acknowledgement The research on dynamic stability of symmetric gaits is supported by the Ministry of Scientific Research and Information Technology Grant N N514 297935.

References 1. Garcia E., Galvez J.A., and Gonzalez de Santos P., On finding the relevant dynamics for modelbased controling walking robots, Journal of Intelligent and Robotic Systems, 37(4):375–398, 2003. 2. Spenneberg D., Albrecht M., Backhaus T., Hilljegerdes J., Kirchner F., Strack A., and Schenker H., Aramies: A four-legged climbing and walking Robot. In Proc. of 8th International Symposium iSAIRAS, Munich, September 2005, CD ROM. 3. Thompson D.M., Introduction to the study of human walking, kinesiology and biomechanics. Department of Biostatistics and Epidemiology, University of Oklahoma Health Sciences Center, 2006. 4. Zielinska T., Trojnacki, M., Postural stability in symmetrical gaits. Acta of Bioengineering and Biomechanics, 11(2):1–8, 2009. 5. Zielinska T., Biological inspiration used for robots motion synthesis. Journal of PhysiologyParis, 103(3–5):133–140, 2009. 6. Zielinska T., Chew Ch.M., Kryczka P., and Jargilo T., Robot gait synthesis using the scheme of human motion skills development. Mechanism and Machine Theory, 44(3):541–558, 2009.

Numerical Simulations of the Virtual Human Knee Joint D. Tarnita1 , D. Popa1 , N. Dumitru1 , D.N. Tarnita2 , V. Marcusanu3 and C. Berceanu1 1 Department

of Applied Mechanics, University of Craiova, 200512 Craiova, Romania; e-mail: [email protected], popa [email protected], nicolae [email protected], berceanu [email protected] 2 Department of Anatomy, U.M.F., 200349 Craiova, Romania; e-mail: dan [email protected] 3 Department of Orthopedics, Emergency Hospital, 200642 Craiova, Romania; e-mail: [email protected] Abstract. In this paper we are presenting the results of the research based on dynamic simulation of human knee virtual model. Starting from the parts realized with computer tomography help, there were realized virtual models for the femur, tibia, meniscus and ligaments that form the complex human knee joint, and their meshing in finite elements. There are presented comparative stress diagrams for the normal walking and for the case of sectioned anterior crisscross ligament. Maps of stress are presented for the whole joint and for its components, to different movement moments. 3DP rapid prototyping technology is used to obtain the components of the human knee joint necessary for future experimental research. Key words: dynamic simulation, finite element method, knee joint, ligament, 3D printing

1 Introduction The knee is one of the most important joints in the human body and it is composed of bones, ligaments, tendons and cartilages. The geometrical aspects of the bone systems modeling are dominated by the necessity of using complex spatial models because most of the bone elements have complicated geometrical shapes in space. Many researchers [1, 3, 6, 8, 11, 14] have developed several studies using programs of analysis with finite element in various situations (flexion, extension), as well as other attempts. An interesting study has been made in [14] regarding theoretical constitutive laws for large deformations which appear in ligaments and tendons in the joint composition. In this study, a virtual model of complex human knee joint has developed. A comprehensive review was presented in [9] to summarize the state-ofthe-art of knee models. Two types of models were identified: phenomenological and anatomical. The mechanics of the knee joint in relation to normal walking is studied in [15]. In vivo kinematics for four designs of knee prosthesis during level walking, stair climbing and non weight-bearing flexion-extension has been studied in [13]. In [10] the authors have determined the three-dimensional tibiofemoral articular con-

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Fig. 1 The significant images of the tibia bone.

tact patterns of a posterior cruciate ligament-retaining total knee replacement during in vivo weight-bearing flexion. In [12] the guidelines of using such a formulation in performing dynamic simulations of knee joint activities were established.

2 The Modeling of the Virtual Knee Joint The process of static behavior simulation of the knee joint components requires: (a) A three-dimensional modeling of the joint components. In order to obtain the transversal sections of the bones we employed a Philips AURA CT scanner from the Clinical Emergency Hospital of Craiova. For the femur and tibia extremities we used scanning distances of 1 mm, while for the diaphisys areas distances of 3 mm [16]. In Figure 1 we present the upper tibia images, the tibial diaphysis and lower tibia images, which outline the changes in shape of the tibia bone. In total, 135 images from upper tibia, 62 images from tibial diaphysis and 92 from lower tibia have been analysed. In order to properly define the virtual components of the bones we used SolidWorks software. The 3D virtual models are completely parameterised and can be imported in order to perform kinematic of FEA simulation with dedicated software. In Figures 2 and 3 two details from the upper and lower areas of the femur bone the virtual models for the femur and tibia bones are presented. (b) A mesh generation of the geometrical model in finite elements. In this stage we used ANSYS software, a well-known, performant software for finite element analysis (FEA). The splitting is made using a tetraedrical 3D element which takes account from the composite like structure of the natural bones (compact and spongy), by introducing the corresponding elastic constants for both materials. The finite element model of the knee joint consists of two bony structures (tibia, and femur), their menisci and two principal ligaments: anterior/posterior cruciate. The mechanical functions of ligaments are to guide joint motion within a normal range and to restrict abnormal joint movement. The 3D model of the human knee is presented in Figure 4. (c) The establishment of the contour conditions. On the tibial plateau the two menisci have been attached, afterwards the femur has been disposed knowing the correct position between the two menisci. A virtual rotation axis has been defined over the two femoral condylis. The two

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Fig. 2 The sections which define the femur shape.

Fig. 3 The 3D virtual model for femur and tibia.

Fig. 4 The 3D model of the anatomical knee joint.

crisscross ligaments have been defined with the insertion points on the femur and tibia. (d) Gathering of the results. By solving the system of equations one can find the nodal displacements over three directions in each nod of the finite elements. The displacements are used to determine the mechanical deformations and stress found in the analysed structure. Taking into account the obtained results, one can make a correct analysis regarding the behavior of the bones and joints subjected to external forces, can determine the areas predisposed to fracture and allow kinematical and dynamical analyses.

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3 The Kinematic and Dynamic Analysis of the Human Knee Joint The biomechanical system of the knee joint has been exported as ACIS (.sat) file in VisualNastran in order to study this joint from the kinematical and dynamical point of view. This study takes into account the following hypotheses: • •

A full locomotion cycle (walking) during 0.8 seconds with an average speed of 2.4 km/h was considered. A vertical force having the functional variation over the femoral head: Fz = F0 · sin(3.963 ∗ t)

• • • •

(1)

where: Fz is the vertical component of the force which acts on the femoral head; F0 is the amplitude of the force which for walking is considered 800 N; t is time (from 0 to 0.8 seconds). Between femur and tibia a motor joint was considered having as movement

α = −90 · sin(3.963 ∗ t)

(2)

Two rigid joints were defined between the menisci and tibia. Two rigid joints were defined between the crisscross ligaments, tibia and femur.

3.1 The Results from the Kinematic and Dynamic Analysis Case 1. Normal walking A first important result is the simulation movie in .avi format from which in Figure 5 we present four significant frames. In Figure 6 we present the laws of variation in time for force and moment in the virtual knee. In Figure 7 the stress maps found in the knee joint (general view of the tibial plateau with the menisci and crisscross ligaments) are presented. At t = 0 sec, the angle between tibia and femur is α = 0◦ . At t = 0.12 sec, the angle between tibia and femur is al pha = 30◦ . At t = 0.28 sec, the angle between

Fig. 5 Four significant frames taken from the simulation movie.

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Fig. 6 The components of the force and the moment from the virtual knee joint.

Fig. 7 The stress map for tibial plateau at t = 0.12 sec, and at t = 0.4 sec.

Fig. 8 The stress map at t = 0 sec, t = 0.12 sec, t = 0.4 sec (general lateral view).

tibia and femur is α = 60◦ . At t = 0.4 sec, the angle between tibia and femur is α = 90◦ . Case 2. The anterior crisscross ligament is sectioned; tibia is internally rotated by 20◦ and translated with 15 mm. In Figures 9 and 10 one can observe the stress maps found in the knee joint (general view and special views). The dynamic maps obtained can give important data for the analysis of the behavior of the knee joint in two cases: first, for the normal walking and second, for the sectioned ligament. These diagrams show that the less stressed surfaces are determined in the first case for the normal walking and the most stressed areas are obtained for the second case. These surfaces are located on menisci, tibial plateau and posterior crisscross ligament. Using these dynamic maps decomposed in frames

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Fig. 9 The stress map at t = 0.2 sec, t = 0.4 sec and t = 0.8 sec (general view).

Fig. 10 The stress map at t = 0.4 sec and t = 0.8 sec (tibial plateau view with the menisci and the posterior crisscross ligament).

Fig. 11 The comparative stress diagram for the two analyzed cases.

(static maps), the information can be structured in the different ways, such comparative diagrams that shows the differences between the analyzed cases. In Figure 11 an example was presented for these kinds of analysis which shows the major differences of the maximal stress between the two studied cases. In this figure it was determined that the maximal stress for the second case is 3.54 times bigger than the walking studied case.

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Fig. 12 Zcorp 3D printer and the methodology used to obtain an experimental assembly using RP.

Fig. 13 Tibia bone, radius bone and humerus bone, virtual models and prototiped.

4 3DP Technology Used to Prototype the Knee Joint Components The 3DP (three-dimensional printing) is a rapid prototyping technology, used to create complex three-dimensional parts directly from a computer model of the part, with no need for tooling [4, 5]. This method (Figure 12) combines a 3D printer, CAD development software and special materials from which the prototype will be created.Computer software splits the 3-D CAD data into a series of thin horizontal cross-sections (slices). Each new layer is fabricated through lowering of the piston by a layer thickness and filling the resulting gap with a thin distribution of powder. An inkjet printing head then selectively prints a binder solution onto this layer of powder to form a slice of the 3-D CAD file. This method can produce high accuracy filler structures for the fabrication of complex 3D prototypes [17]. Using the Rapid Prototyping 3D Zcorp 310 Printer system, we manufactured the prototypes for human bones (Figure 13) and one can finally obtain functional assemblies which can be used in the future work in different experiments [2, 16].

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5 Conclusions Using finite element analysis we have studied the mechanical parameters of the human knee joint in the case of normal walking and in the case of sectioned anterior crisscross ligament, tibia having a translation of 15 mm and an internal rotation of 20◦ . Using the results from the kinematic and dynamic simulation one can outline the following conclusions: • • • • • • • • •

The forces from the virtual knee joint are maximal at t = 0 sec, 0.4 sec, 0.8 sec in the case of normal walking, while in the second studied case those forces are maximal at the end of the time interval. The moments which appear in the virtual knee joint are maximal at the beginning, the middle and the end of the studied time interval in the case of normal walking and at t = 0.8 sec for the second case. In the case of a healthy knee subjected to loads, the tension in the components that form the joint is uniform distributed, having areas with maximal loads both for flexion and extension movements. During walking the stresses in the bones, ligaments and menisci augment, generally, though in flexion they maintain their distribution. The stress maps in the components of the virtual knee joint show different values as function of the degree of flexion during normal walking, being, in general, larger at the debut and the finale of the studied period of time. The maximal values for stresses during normal walking are bigger just in the time interval between the initial moment and 0.1 sec, in the rest of the interval the stresses are bigger for the second case. The maximal values for the second studied case are approximately 14 times bigger than the maximal values recorded for normal walking. Using RP as a fabrication method one can finally obtain functional assemblies which can be used in the future work in cinematic or dynamic studies. The complexity of the shapes and dimensions of the fabricated parts recommends RP method as a future fabrication method in an increasingly number of applications and scientific fields of research.

Acknowledgements This research activity was supported by CNCSIS-UEFISCSU, Grant No. 86/2007, Idei 92- PNCDI 2.

References 1. Amiel, D., et al., Ligament structure, chemistry and physiology. In Daniel, D.D., Akeson, W.H. and O’Connor, J.J. (Eds.), Knee Ligaments: Structure, Function, 1998.

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2. Berceanu, C. and Tarnita, D., Rapid prototyping and embedded control for an anthropomorphic robotic hand. In Proceedings of 3rd International Conference on Computational Mechanics and Virtual Engineering COMEC 2009, Brasov, October 2009; 3. Bendjaballah, M.Z., et al., Biomechanics of the human knee joint in compression: Reconstruction, mesh generation, and finite element analysis. Knee, 2:69–79, 1995. 4. Chua, C.K. et al., Rapid Prototyping – Principles and Applications, 2nd ed., World Scientific Publishing, NJ, USA, 1997. 5. Dimitrov, D. et al., Advances in three dimensional printing – State of the art and future perspectives, Rapid Prototyping Journal, 12(3):136–147, 2006. 6. Donahue, L.H., et al., A finite element model of the human knee joint for the study of tibiofemoral contact, J. Biomech. Eng., ASME, 124(3):273–280, 2002. 7. Frank, C.B., Woo, et al., Medical collateral ligament healing: a multi-disciplinary assessment in rabbits, American Journal of Sports Medicine, 11(6):379–389, 1983. 8. Gil, J., et al., Development of a 3D computational human knee joint model. In Proc. 1998 ASME Int. Mech. Eng. Cong. Expo., BED 39, ASME, pp. 1–2, 1998. 9. Hefzy, M.S. and Cooke, T.D.V., Review of knee models, Applied Mechanics Review, 49(10part 2):1–7, 1996. 10. Li, G., Suggs, J., Hanson, G., et al., Three-dimensional tibio-femoral articular contact kinematics of a cruciate-retaining total knee, J. Bone Joint Surg. Am., 88(2):395–402, 2006. 11. Li, G., et al., A validated three-dimensional computational model of a human knee joint, J. Biomech. Eng., ASME, 121(6):657–662, 1999. 12. Moeinzadeh, M.H., Engin, A.E. and Akkas, N., Two-dimensional dynamic modelling of human knee joint, J. Biomech., 16:253–264, 1983. 13. Migaud, H., et al., Cinematic in vivo analysis of the knee: A comparative study of 4 types of total knee prostheses, Rev. Chir. Orthop. Reparatrice Appar. Mot., 81(3):198–210, 1995. 14. Moglo, K. and Shirazi-Adl, A., On a refined finite element model of the knee joint, Adv. Bioengrg., 43:203–204, 1999. 15. Morrison, J.B., The mechanics of the knee joint in relation to normal walking, J. Biomech., 3(5):1–61, 1970. 16. Tarnita, D. et al., Considerations on the dynamic simulation of the virtual model of the human knee joint, Materialwissenschaft und Werkstofftechnik, Materials Science and Engineering Technology, 40(1/2):73–81, 2009. 17. Thilmany, J., Printing in three dimensions, Mech. Engrg., periodical article, 1 May 2001. 18. Tumer, S.T. and Engin, A.E., Three-body segment dynamic model of the human knee, J. Biomech. Engrg., 115:350–356, 1993.

Development of a Walking Assist Machine Using Crutches – Motion for Ascending and Descending Steps Tatsuro Iwaya, Yukio Takeda, Makoto Ogata and Masaru Higuchi Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo, Japan; e-mail: [email protected]

Abstract. This paper discusses the motion of a walking assist machine using crutches when it is used to assist people with walking difficulties to ascend or descend steps. First, the basic composition of the machine, which is under development in the Mechanical Systems Design Laboratory in Tokyo Institute of Technology, is shown. Based on the features of motions for ascending and descending steps by a healthy person using crutches that were measured using a motion capture system, a parameterized velocity profile as the reference trajectory of the actuator in the machine is proposed. Values of the parameters are determined through dynamic simulation taking into consideration stability, consumed energy and user’s comfort. The effectiveness of the motion is discussed with experimental results. Key words: robotics, life support, biomechanics, walking assist, motion synthesis

1 Introduction Many paraplegics use wheelchairs for locomotion. Though the energy efficiency and stability of a wheelchair are good enough, it requires a large space and its useful range is limited due to its difficulty in handling steps. Since social environments have been designed with walking in mind, walking assist devices which utilize users’ lower limbs or use biped locomotion strategies are expected to help make users’ lives more independent. Levels of ability and disability of paraplegics should be taken into consideration when walking assist devices are developed. The goal of our work is to develop a walking assist machine with actuators for persons who cannot move their lower limbs voluntarily but can move their upper body, including their arms. Especially, young people are potential users of such a machine if their upper body can have sufficient power even in the case when they have disability in their lower limbs. We took self mobility, economy, safety, and health enhancement into consideration in the conceptual design of the walking assist machine. Then, we determined the following conditions for the machine: 1. It should be wearable for space efficiency and should be light weight.

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Fig. 1 Basic concept of a WAMC.

2. The user can move while keeping the standing position, which realizes the eye sight height at the same level of healthy person. 3. The user can utilize his/her own remaining function and ability, such as upper limbs. 4. It should operate in accordance with user’s intention. 5. It must not exert excessive load on user’s joints 6. It should not require excessive force/power supplied by user’s upper body during motion and support. 7. It should use few actuators and be energy efficient. 8. It may contribute to rehabilitation of lower limbs. Studies on walking assist machines with actuators have been done. A walking assist machine which supports the whole legs by the soles of the user’s feet has been developed [5]. It is intended to assist the walking of persons who have less leg strength. Functional electrical stimulation is a method considered useful in assisting walking of paraplegics. It is used together with actuators [1, 3] and it may contribute to a reduction in energy consumed by actuators and avoidance of loss of muscle strength and joint solidification. However, it has not been put into practical use since muscle fatigue occurs and it requires many actuators. We started to develop a walking assist machine using crutches (WAMC) in 2007. A wearable device with actuators generates motion in the lower limb to create a swing-through crutch gait while the user operates axillary crutches with both arms. The basic composition of the WAMC is presented in Figure 1. It satisfies conditions (1)–(8) mentioned above. In order to make a practical WAMC, the mechanism, energy source, control system, and operating system should be determined taking into account several walking patterns in daily life, such as straight, ascending/descending steps, and turning motions, safety measures for external disturbances, etc. In our previous papers [2, 4], a basic composition for the WAMC was proposed and motion for flat floor was determined. In the present paper, motions for ascending and descending steps are discussed.

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Fig. 2 Hardware composition of WAMC.

2 Walking Assist Machine Using Crutches The hardware composition of WAMC is shown in Figure 2. The telescopic links (c), each of which is located on the outer side of each lower limb and driven by a linear actuator (f), are connected with the harness (e) and the axillary crutches (a) by spherical joints (b) at their upper ends, and are connected with the soleplate (i) by spherical joints (j) at their lower ends. The user’s feet are constrained on the soleplate by the toe band (h). The ankle spring (g) between the soleplate and the link is used for assisting ankle torque. The waistband (d), which connects both links at the user’s waist, prevents the hip and knee joints from flexing backward when the link shortens its length. Using this composition, all forces applied to the user’s body are basically supported by the links and the crutches through the harness, not by the user. The user can walk with the WAMC by operating the crutches. The actuators are controlled to follow the reference velocity profile that will be discussed in the following sections. Grip switches (k) and contact sensors (l) are used for deducing the user’s intention. Sensors (m) are equipped with to measure the position of the crutch tips, height of the steps and so on. Selection of walking mode such as flat floor mode, step ascending/descending mode can be done with the sensors’ data and user’s judgment. An energy source (o), such as a battery, and controller (p) are attached to the user’s body. Sensors (n) such as accelerometers are attached to the user and a brake is attached to the spherical joint between the soleplate and the links for safety from external disturbances.

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Fig. 3 Analysis model of walking using crutches (double support phase).

Fig. 4 Parameterized profile of the reference velocity of the linear actuator (DS: double support phase, FHCS: first half of crutch support phase, LHCS: latter half of crutch support phase, LS: leg support phase).

3 Motion Synthesis for Ascending and Descending Steps 3.1 Reference Trajectory and Parameters In order to clarify the motion which should be generated by WAMC, we measured swing-through crutch locomotion by a healthy person to ascend and descend steps. We modeled a human using crutches as a planar serial mechanism with six links and six revolute joints during the single support phase by the legs, and as a planar closed-loop mechanism with seven links and seven revolute joints during the double support phase (Figure 3a). In the figure, geometrical parameters of the step and positions of the foot and the crutch tips are defined. We put markers on the body of the subject and the crutches. Their positions were measured by a motion capture system. In order to investigate the planar motion in the sagittal plane, we transformed 3D measurement data to 2D data. A planar model with three moving links, three revolute joints and one prismatic joint shown in Figure 3b is used in dynamic simulation, in which the effect of user’s body is included. In the figure, R corresponds to the center of the spherical joint (j) in Figure 2. We conducted experiments under conditions listed in Table 1. From the experimental data, we found that the change of velocity between a virtual point corresponding to R and a point S (shoulder), which correspond to the displacement and velocity of the linear actuator of WAMC, can be modeled by the parameterized

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Table 1 Experimental conditions. Terrain

LSC [mm]

Lstep [mm]

hs [mm]

Flat floor Ascending (1) Ascending (2) Descending

– 200, 250, 300 200, 250, 300 300, 350, 400

550, 600, 650 550, 600, 650 550, 600, 650 –

– 80 180 80, 180

Table 2 Parameter values used in dynamic simulation.

Mass [kg] Moment of inertia [kg m2 ] Length [m] Center of gravity [m]

Link CS

Link SG2

Link RG3

8.42 1.23 1.30 SG1 = 0.468

60 1.53 0.50 SG2 = 0.165

15 1.12 0.77 RG3 = 0.70

Fig. 5 Relationships between parameters for ascending a step (hs = 80 mm).

curve shown in Figure 4. This curve has four parameters Vex , Vin , Vout and θex to be determined according to the walking conditions. We investigated the applicability of this curve as the reference velocity of the linear actuator through dynamic simulation. As the result, we found that walkings for ascending and descending steps as well as on flat floor are made possible when appropriate values of parameters of this velocity profile are used. Though the parameters to be determined according to the walking conditions are Vex , Vin , θex and Vout , we considered Vex , Vin and θex as variables and Vin as a constant (Vin = 0.3 m/s) in the following.

3.2 Dynamic Simulation and Determination of the Motion Dynamic simulations were carried out to determine values of parameters of the reference trajectory. Table 2 shows the values of the dynamic parameters of the model in Figure 3b used in simulation. First, we investigated the area of set of parameters, by which stable walking for ascending a step can be achieved. Figure 5a shows the minimum value of Vex vs. θex for three cases of Vin for ascending a step of

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Fig. 6 Relationships between parameters for descending a step (hs = 80 mm).

Fig. 7 Overview of the experimental apparatus.

hS = 80 mm. As shown in this figure, it is known that the minimum value of Vex is dependent on θex , and almost independent of Vin . Since Vex affects the body of the user during the DS phase, smaller Vex is expected preferable to improve user’s comfort. Figure 5b shows the relationships between energy consumption of the linear actuators during a step E and θex when the minimum Vex shown in Figure 5a is used. From this figure, it is known that smaller value of Vin is preferable in order to improve energy efficiency. Then, we obtained the relationship between the minimum value of Vin and θex , when the minimum value of Vex shown in Figure 5a is used, as shown in Figure 5c. On the other hand, we found through preliminary experiments that users prefer to constant θex even if the terrain shape changes. They said that they felt fear at the beginning of crutch support phase when θex changes according to the terrain shape because they could not prepare for this instance. Based on the results mentioned above, we developed the determination procedure of parameter values, in which walking stability, energy consumption and user’s comfort are taken into consideration, as follows: 1. θex is determined based on the user’s comfort through preliminary experiments. 2. The minimum value of Vin is selected using Figure 5c. 3. The minimum value of Vex is selected using Figure 5a. As for the motions for descending steps, an example of results is shown in Figure 6. From this figure, it is known that the same procedure as that for ascending steps can be applied to determination of reference velocity for this case.

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Table 3 Properties of the experimental apparatus. Mass of worn parts Mass of waist harness Mass of crutches Length of telescopic link

Max. Min.

17.8 kg 0.55 kg 2.0 kg 1.44 m 1.04 m

Fig. 8 Photos of experiments in ascending a step (hs = 80 mm).

Fig. 9 Photos of experiments in descending a step (hs = 80 mm).

4 Experimental Investigations Figure 7 shows the experimental apparatus of the WAMC that we designed and built. Properties of the experimental apparatus are listed in Table 3. Each linear actuator is composed of a DC servo motor, which is located in front of the waist of the user, a reduction gear, and a rack-pinion. Each linear actuator is controlled by a velocity feedback controller. The axillary pad of the crutch is connected with the harness, that the user wears, and the upper side of the telescopic link by a lateral strap. By these connections of the user’s body to the telescopic link and the crutch, the user’s supporting strength can be greatly reduced. The user’s body is lifted up from the bottom of the crutch when the linear actuator extends. When the user’s leg is in a swing phase (in the air), the WAMC, together with the user’s body, hangs from the crutch through the harness. We carried out walking experiments based on the results in Figures 5 and 6. In the experiments, we modified and used the values of parameters taking into account walking stability and user’s comfort. Modification was experimentally done by trial and error. Figures 8 and 9 show the photos during ascending and descending a step with our experimental WAMC. User’s impression after experiments was, “I could walk with WAMC without strengthening lower limb until the beginning of LS phase. At the beginning of LS phase, I might unconsciously strengthen my

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lower limb”. Through these experimental results, we confirmed the effectiveness of the proposed method to determine the reference motion for ascending/descending steps by WAMC. However, we need to improve accuracy of the dynamic simulation model by which modification at the experiment by trial and error is not required, and we need to consider motion during the LHCS and LS phases by investigating appropriate value of Vout that was not considered as a design parameter.

5 Conclusions In the present paper, we discussed the motions for ascending and descending steps by WAMC. Our conclusions are summarized as follows: 1. We proposed a parameterized velocity profile as the reference trajectory of the actuator of the WAMC for ascending/descending steps. 2. We determined the values of the parameters of the reference velocity profile in (1) through dynamic simulation taking into account stability, energy consumption and user’s comfort. 3. We confirmed the effectiveness of the reference trajectory in (1) and (2) with experiments.

References 1. Goldfarb, M., Korkowski, K., Harrold, B., and Durfee, W., Preliminary evaluation of a controlled-brake orthosis for FES-aided gait. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11:241–248, 2003. 2. Higuchi, M., Ogata, M., Sato, S., and Takeda, Y., Development of a walking assist machine using crutches (Proposition of its composition and basic experiments). In Proceedings of the 3rd International Conference on Manufacturing, Machine Design and Tribology, p. 197, 2009. 3. Popovic, D., Tomovic, R., and Schwirtlich, L., Hybrid assistive system – The motor neuroprosthesis. IEEE Transactions on Biomedical Engineering, 36:729–737, 1989. 4. Takeda, Y., Sato, S., and Higuchi, M., Development of a walking assist machine using crutches. In Proceedings of Mechanical Engineering Congress Japan, pp. 207–208, 2008. 5. Tanaka, E., Ikehara, T., Omata, T., Owada, T., Nagamura, K, Ikejo, K., Sakamoto, K., and Inoue, Y., Development of a walking assist machine supporting whole legs from their soles. Transactions of JSME, Series C, 72:3871–3877, 2007.

Human Lower Limb Dynamic Analysis with Applications to Orthopedic Implants N. Dumitru1 , C. Copilusi1 , M. Marin1 and L. Rusu2 1 Faculty

of Mechanics, University of Craiova, 200585 Craiova, Romania; e-mail: dumitru [email protected] 2 Faculty of Educational Physics and Sport, University of Craiova, 200585 Craiova, Romania; e-mail: [email protected]

Abstract. In this paper we present the human lower limb inverse dynamic analysis in the sight of some useful dynamic parameters determination for orthopedic implants design. We elaborate the human lower limb mathematic model based on some cinematic and dynamic considerations obtained on experimental and analytical way. The inverse dynamic analysis has on its base the Newton–Euler formalism completed with Lagrange’s multipliers method. Through this analysis we obtain the connection forces which appear at the human lower limb’s joints level for walking activity. With these results we will study through finite element method an orthopedic implant which will partially replace the human lower limb’s ankle joint. Key words: implant, arthroplasty, dynamic model, human lower limb, connection forces

1 Introduction Many research centers have developed various models, but the most common implants are the one offered by Tucson and RAMSES orthopedic centers [12, 13]. An implant provided in [12], for implantation is presented in Figure 1. The implant elements and the design are identified, finding a resemblance with a timber which posses a single degree of mobility [13]. This implant has a single degree of mobility, so as it is known in the ankle joint can be performed more movements such as plantar flexion, dorsal flexion, internal and external rotation, the main being the dorsal and plantar flexion [3]. The talus component’s virtual model is different from a standard lower component, by the existence of micro-channels in order to ensure fluid penetration of the synovial capsule joint for friction decrease and respectively increase the joints reliability. Some researchers develop dynamic analyses but only for specific human lower limb joints and they did not develop a dynamic analysis by taking in account the whole human lower limb [11].

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Fig. 1 Implant structure and elements identification [13].

2 Human Lower Limb Dynamic Analysis The dynamic analysis aim is to identify the connection forces variation laws depending on time. This interface will allow excitation of each joint in normal operating conditions or critical ones appropriate with those existent in reality. The correct finality of this objective is guaranteed by the design of an integrated dynamic model – experiment, finite element modeling and simulations. We present the results of a dynamic model numerical calculations for which we design the ankle joint implant model and experimental tests. The inverse dynamic analysis method will use the Newton–Euler formalism completed with Lagrange’s multipliers method [2, 5, 6]. The mathematical model [4], necessary to achieve the human lower limb inverse dynamic analysis, will be developed based on kinematic analysis carried out previously. This was possible by taking into account the kinematic scheme that emphasizes the analysis and contact with ground (Figure 2). The kinematic constraint equations are Φ (q,t) = 0, (1) where qr is the generalized coordinates vector when the elements are considered rigid, that means non-deformable; t is time. Or customizing for i elements, we obtain: T
qi = {rCTicx }T , ϕi ,

i = 1, . . . , 9,

(2)

[rCT cx ]T = {XCT cx ,Y XCT cx , ZCT cx }.

(3)

Xi = XCT cx , Yi = YCT cx , Zi = ZCT cx , i i i ⎧ Tcx X − XC = 0, ⎪ ⎨ i i Yi − YCT cx = 0, i ⎪ ⎩ Zi − ZCT cx = 0.

(4)

i

i

i

i

We note that

i

From this we obtain

(5)

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Fig. 2 The human lower limb mathematical model for inverse dynamic analysis [4].

⎧ T cx x − Exi = 0, ⎪ ⎨ Ci yCT cx − Eyi = 0, i ⎪ ⎩ T cx zC − Ezi = 0,

with i = 1, . . . , 9.

(6)

i

We differentiate relation (1) in rapport with time, and we obtain Jq q˙ + or relations (6):

∂φ =0 ∂t

(7)

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⎧ ∂ T cx ⎪ T cx ⎪ ⎪ ⎪ x˙Ci − ∂ t XCi = 0, ⎪ ⎪ ⎪ ⎨ ∂ y˙CT cx − YCT cx = 0, i ⎪ ∂ t i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T cx ∂ T cx ⎩ z˙C − ZC = 0. i ∂t i We differentiate relation (7) in rapport with time, and we obtain: ˙ q q˙ + Jq q¨ + (Jq q)

∂ 2φ + 2J˙qq˙ = 0. ∂ t2

(8)

(9)

By differentiating relations (8) in rapport with time, we obtain ⎧ ⎪ ∂2 ⎪ ⎪ x¨CT cx − 2 Exi = 0, ⎪ i ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂2 T cx y ¨ − Ey = 0, C ⎪ i ∂ t2 i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ⎪ ⎩ z¨CT cx − Ez = 0. i ∂ t2 i

(10)

Equations (7) and (9) can be written in a compact form by introducing the notations: ∂Φ = v, (11) Jq q˙ = − ∂t Jq q¨ = −2J˙qq˙ − (Jqq) ˙ q q˙ −

∂ 2φ = a. ∂ t2

(12)

The mechanical work of mass forces is 8

T cx δ L = δ rCT cx G1 + δ rCT cx G2 + · · · + δ rC8 G8 = ∑ δ rCT cx Gi . i 1

2

(13)

i=1

The equation of motion has the following form:      M JqT q¨ Qa , = Jq 0 λ a

(14)

where M represents the mass matrix with M = diag(mi , mi , mi , Ji ), i = 1, . . . , 9; λ is the Lagrange multipliers vector, with λ = λ1 , λ2 , . . . , λ36 ]T ; and Qa is the vector of active forces. We obtain the following Lagrange’s multipliers:

λ = [Jq ]−1 [Qa − M q], ¨ not q¨ = δ q = {X¨CT0 , Y¨CT0 , Z¨CT0 , ϕ˙ i }T , i

i

i

(15) i = 1, . . . , 9.

(16)

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Fig. 3 Variation of the connection force components for G joint equivalent to the human lower limb ankle joint mathematical model (for a single step in the z axis direction).

We know the motion laws ϕi (t), ϕ˙ i (t) and ϕ¨ i (t), obtained on experimental way, in a dynamic mode analysis. From relation (20) we determine Lagrange’s multipliers λ , by creating an algorithm in the Maple soft frame. In this way we compute the inverse dynamic analysis from which we obtain the connection forces existent at the kinematic joints level. Thus, through the algorithm developed above, we obtained ankle joint connection forces variations related to the dynamic model structure previously in the three xyz directions. In Figure 3 the connection force component variation is presented. This force actuates in the z direction for G joint equivalent to the human lower limb ankle joint mathematical model (for a single step).

3 Dynamic Results Application on a Human Ankle Joint Virtual Model with Orthopedic Implant For this application we use a human bony virtual model of a shank which has an orthopedic implant. The bony models were taken from a male through a CT scan. This human subject is 53 years old, has a height of 1.80 meters and a weight of 88 kg. By using a CT scan we design the bony virtual models with CATIA’s aid and the orthopedic implant design. In Figure 4, the virtual model and the boundary conditions were presented. The material properties were obtained from databases existent in the specialty engineering and biomechanics literature for: titanium, bony models and UHMWPE [7]. We consider the human ankle joint law obtained on experimental way by using SIMI Motion software, in order to study the orthopedic implant’s behavior [3]. The dynamic behavior was studied by using MSC Nastran software, and the results are presented in diagrams in Figures 5, 6 and 7. In these figures the variations of von Misses maximum internal stress, von Misses efforts and nodal displacements depending on time are represented.

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Fig. 4 Shank virtual model with orthopedic ankle implant and the boundary conditions.

Fig. 5 Internal von Misses stress [MPa], depending on time [sec].

Fig. 6 Von Misses efforts, depending on time [sec].

4 Conclusions The research purpose was to obtain an implant’s optimum model in terms of dynamic behavior which is intended to be implemented in the human ankle joint’s level. Mathematical modeling preceded by identification of the lower limb kinematic behavior on a real human subject, completed through CAD simulation models, was based on Newton–Euler algorithm completed with Lagrange’s multipliers

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Fig. 7 Nodal displacements variation [mm], depending on time [sec].

method [6]. We identified in dynamic mode the connection forces variation laws from each joint for normal and critical locomotion situations. With these results, by finite element modeling, an analysis of ankle joint dynamic response was possible, implicitly for implant with similar loadings and boundary conditions with the ones existing in reality. The results are important, because the modeling was designed in a parametric system with the possibility of considering some optimal solutions for implants, which are different by shape and size.

Acknowledgments The research work reported here was made possible by Grant CNCSIS – UEFISCSU, project number PNII – IDEI code 401/2008.

References 1. Aubart, F., Alexandre G. and Saidani N., Robot-assisted implantation of articular prostheses: The French experience in a homogeneous series of 50 cases. J. Bone Joint Surg., 83-B:56–67, 2001. 2. Amirouche, F., Computational Methods in Multibody Dynamics. Prentice-Hall, 1992. 3. Buzescu, A. and Scurtu L., Anatomy and Biomechanics. A.N. E.F.S. Publishing House, 1999. 4. Copilusi, C., Research regarding mechanical systems applicable in medicine. PhD Thesis, University of Craiova, 2009. 5. De Jalon, J.G. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems, The Real-Time Challenge Series. Springer, 1994. 6. Dumitru, N., Nanu, Ghe. and Vintil˘a, D., Mechanisms and Mechanical Transmissions. Classic and Modern Modeling Techniques. E.D.P. Bucharest Publishing House, 2008. 7. Eichmiller, C., Tesk, A. and Croarkin, M., Mechanical properties of ultra high molecular weight polyethylene NIST. Polymers Division Reference Material, Paffenbarger Research Center, Statistical Engineering Division, National Institute of Standards and Technology, Gaithersburg, 2008.

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8. Karol, G., The effect of design variations on stresses in total ankle arthroplasty. PhD Thesis, University of Pittsburgh, 2002. 9. Murnaghan, J.M., Warnock, D.S. and Henderson, S.A., Total ankle replacement. Early experiences with STAR prosthesis. Ulster Medical Society Journal, 64–72, 2005. 10. McGeer, T., Passive dynamic walking. International Journal of Robotics Research, 9(2):62– 82, 1990. 11. SouthWest Orthotic Catalog, http://www.southwest-ortho.html, 2008. 12. Maitrise Orthotic Catalog, http://www.maitrise-orthop.html, 2008.

Forward and Inverse Kinematics Calculation for an Anthropomorphic Robotic Finger C. Berceanu, D. Tarnita, S. Dumitru and D. Filip University of Craiova, 200585 Craiova, Romania; e-mail: berceanu [email protected], [email protected], [email protected], emil [email protected]

Abstract. This paper proposes a theoretical approach related to the kinematic issues of dexterous robotic hands. A new anthropomorphic robotic hand is presented in the first part, while in the second part of this paper the authors present a general method which allows fast and easy computations in order to find the solution(s) of the direct and inverse kinematic problem for this hand. Key words: kinematics, mechanical, robotic hand

1 Introduction Robotic end-effectors have been developed in two directions: as industrial grippers or as articulated hands. While the former have presented a great number of shortcomings (simplified mechanical model with a reduced number of degrees of freedom which leads to a limited range of motion and forces exerted on a grasped object, degree of anthropomorphism etc.), the latter have augmented the flexibility (the hand can adapt to objects of different shapes, can grasp objects more stably due to a larger number of points of contact or gripping surfaces, can perform fine motion without moving the entire arm and can reorient the grasped object without regrasping) of the grasping process of a robot, not considering the anthropomorphic features that recently developed mechanical articulated hands have achieved [13]. Although the first developed mechanical hands were aimed for prosthetic applications, nowadays the focus has shifted toward research in robotics laboratories. In the past 20 years many robotic articulated hands have been developed all around the world, some of these reaching performances close to those of a human hand [3, 5, 7–9]. Biagiotti et al. [1], Bicchi [2] and Sotto Martell et al. [12] make an exhaustive survey of robotic hands with emphasis on the mechanical model, actuation, sensors, control, degree of anthropomorphism etc. In the first part of this paper is presented a new developed robotic hand, while in the second part is described in detail a method of solving the direct and inverse kinematic problem of this artificial hand. The solution of the direct kinematic problem is important due to the fact that most of

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Fig. 1 A picture which illustrates the placement of the actuators, electronic control board and proximity sensor.

the modern control strategies of these robotic hands are based either on position or force algorithms [11]. Knowing the positions of the finger tips one can approximate the position of the grasped object, in the hypothesis that the hand grasps the object only with the finger tips. The direct kinematic problem in the case of an artificial hand can be formulated as follows: given the joint angles between the articulated links (phalanxes), what are the Cartesian coordinates of the finger tips? Conversely, for the inverse kinematic problem one shall compute the joint angles as a function of the Cartesian coordinates of the finger tips.

2 The Anthropomorphic Hand and Finger The already developed mechanical hand (at the Faculty of Mechanics, University of Craiova, Romania) consists of a palm and five fingers, each finger having three phalanxes (proximal, medial and distal), while the thumb finger can be in opposition with the other fingers, exactly like the human hand thumb. This feature is very important for the precision and power grasps postures of the hand [4]. Simulations have been made in order to verify the correct closing of the fingers, as shown in Figure 2. The actuation system is based on seven electric actuators (DC servomotors) placed in an additional component which substitutes the forearm of the human body. From these seven actuators, five (dimensions 40.7 × 19.7 × 42.9 mm, torque 130 Ncm at 6 V, mass 55 g) are used to drive the fingers in flexion or extension, one (placed in the palm: dimensions 29.8 × 12 × 29.6 mm, torque 28 Ncm at 6 V, mass 16.8 g) is used to rotate the thumb with respect to the palm, while another DC servomotor (di-

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Fig. 2 Power and precision grasps simulations with the new artificial hand.

mensions 66 × 30.2 × 64.4 mm, torque 250 Ncm at 6 V, mass 160 g) will power the hand in flexion-extension movements with respect to the forearm. The thumb mechanism allows thumb opposition over 0-90 degrees range. The output shaft of the five DC servomotors placed in the forearm will be mechanically connected with cable transmissions routed on the pulleys of each finger and from here linked to the distal phalanx. The kinematic scheme of the artificial hand shows 16 degrees of freedom (D.O.F.) for the mechanical model of which 4 D.O.F. for the thumb and 3 D.O.F. for the other four fingers (see Figure 3). Therefore, we can state that the artificial hand presented in this paper has an underactuated mechanism (a mechanism with fewer actuators than degrees of freedom) [10]. Some advantages of underactuation in robotic hands: reduced fabrication costs (by adopting one actuator per finger vs. four actuators per finger as in the human finger anatomical model), simplified control algorithms and, therefore, low cost microcontrollers, reduced power consumption, reduced mass of the whole assembly etc. The exteroceptive sensor system of this artificial hand is based on a ultrasonic proximity (distance) sensor placed in the palm and five force sensors disposed on the distal phalanxes of each finger.

3 A Method to Compute the Solution of the Direct and Inverse Kinematics Problems of the Robotic Fingers Forward kinematics is the computation of the position and orientation of the endpoint (end effector) of a chain of joints (kinematic chain). Inverse kinematics is the reverse process, computing the required joint trajectories to move an end-effector to the required position. The problem of inverse kinematics is nontrivial, as there are often multiple solutions to any given position. An efficient and standard method of calculating the forward kinematics is the Denavit–Hartenberg (D–H) convention

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Fig. 3 The kinematic layout of the hand and the drive system of a finger.

Fig. 4 The Denavit–Hartenberg parameterisation.

[6]. The D–H convention is used to parameterise each of the links in a chain into a set of two rotations and two translations. Each matrix associated with the rotation and translation is multiplied to give the 4 × 4 homogeneous transformation matrix for each link. Multiplying each of these link matrices produces the end-to-end transformation matrix for the chain. From this final matrix the position and orientation of the end-effector can be obtained. Figure 4 shows the four parameters that define the i-th link in a chain: the link length, ai ; the link twist, αi ; the link offset di and the link rotation ϕi . ⎛ ⎛ ⎞ ⎞ 1 0 0 0 1 0 0 ai ⎜ 0 cos(α ) − sin(α ) 0 ⎟ ⎜ ⎟ i i ⎟ ; Trans = ⎜ 0 1 0 0 ⎟ Rotxi = ⎜ (1) xi ⎝ 0 sin(αi ) cos(αi ) ⎝ 0 0 1 0 ⎠ ⎠ 0 0 0 1 0 0 0 1

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Table 1 The Denavit–Hartenberg kinematic parameters for a 3 D.O.F. finger. Link

αi

ai [mm]

ϕi

di

1 2 3

0 0 0

28 21.5 26

ϕ1 ϕ2 ϕ3

0 0 0

⎛

cos(ϕi ) − sin(ϕi ) ⎜ sin(ϕ ) cos(ϕ ) i i Rotzi = ⎜ ⎝ 0 0 0 0

0 0 1 0

⎞ 0 0 ⎟ ⎟; 0 ⎠ 1

⎛

1 ⎜ 0 Transzi = ⎜ ⎝ 0 0

0 1 0 0

0 0 1 0

⎞ 0 0 ⎟ ⎟ di ⎠ 1

The transformation for a single link is the multiplication of these matrices: ⎛ ⎞ cos(ϕi ) − sin(ϕi ) cos(αi ) sin(ϕi ) sin(αi ) ai cos(ϕi ) ⎜ sin(ϕ ) cos(ϕ ) cos(α ) − cos(ϕ ) sin(α ) a sin(ϕ ) ⎟ i i i i i i i ⎟ Ti = ⎜ ⎝ ⎠ 0 sin(ϕi ) cos(ϕi ) di 0 0 0 1

(2)

(3)

The end-to-end transformation of a kinematic chain with n links is the matrix multiplication of each link matrix: n

T = ∏ Ti

(4)

i=1

The position and orientation of the end-effector can be extracted easily from this matrix.

4 The Direct Kinematic Problem According to D–H formalism it is possible to link two different coordinate frames by calculating the transformation matrix between them. This matrix is always a function of four parameters called D–H parameters. The finger movement is in a single plane (the joint axes are always parallel during the flexion-extension motion) so the twists will be nule (αi = 0, i = 1, . . . , 3). In addition, because the finger has only revolute joints, the offsets are also nule (di = 0, i = 1, . . . , 3). Therefore, the remaining D–H parameters are the angles between phalanxes axes (ϕi ) and the link (phalanxes) lengths ai . For the finger in Fig. 5 the D–H parameters are shown in Table 1. The coordinates of P with respect to the reference frame O0 x0 y0 z0 can be computed with: [r0 ] = [T ][r2 ] (5)

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Fig. 5 The case of an anthropomorphic finger with three phalanxes (links).

where [T ] is the global transformation matrix from the reference frame O0 x0 y0 z0 to the local frame O2 x2 y2 z2 and [r2 ] is the position vector of the P point in O2 x2 y2 z2 . Applying the method described above one can find the solution of the direct kinematic problem:  xP0 = [a2 + a3 cos(ϕ3 )] cos(ϕ1 + ϕ2 ) − a3 sin(ϕ3 ) sin(ϕ1 + ϕ2 ) + a1 cos(ϕ1 ) yP0 = [a2 + a3 cos(ϕ3 )] sin(ϕ1 + ϕ2 ) + a3 sin(ϕ3 ) cos(ϕ1 + ϕ2 ) + a1 sin(ϕ1 ) (6) In addition, to the equations obtained above one can introduce the joints kinematic constraints: ⎧ ⎪ ⎨0 ≤ ϕ1 ≤ π /2 (7) 0 ≤ ϕ2 ≤ π /2 ⎪ ⎩ 0 ≤ ϕ3 ≤ π /4

5 The Inverse Kinematic Problem To solve the inverse kinematic problem one can start from the equations which expresses the solution of the direct kinematic problem. The unknowns of equations (6) are ϕ1 , ϕ2 and ϕ3 . One can conclude that the system is undetermined (contains three unknowns in two equations). To make this system solvable the following condition is introduced ϕ3 = 0 which means that the distal and medial phalanxes have

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the same orientation given by ϕ2 angle, or the distal phalanx can be considered just a part of the medial phalanx. The system (6) becomes:  xP0 = (a2 + a3 ) cos(ϕ1 + ϕ2 ) + a1 cos(ϕ1 ) (8) yP0 = (a2 + a3 ) sin(ϕ1 + ϕ2 ) + a1 sin(ϕ1 ) Solving this system one can obtain the solutions of the inverse kinematic problem: ⎧ ⎪ ⎨ϕ1 = ± arccosm + 2kπ , k ∈ Z (9) ϕ2 = ± arccosn + 2kπ , k ∈ Z ⎪ ⎩ ϕ3 = 0 where the arguments m and n are a function of D–H parameters (the phalanxes lengths ai , i = 1, . . . , 3) and of the Cartesian coordinates of the finger tip xP0 and yP0 . x2 + y2P0 − [a21 + (a22 + a23)] (10) n = P0 2a1 (a2 + a3) m= where



xP0 λ + yP0γ λ 2 + γ2

λ = a1 + (a2 + a3) cos(ϕ2 ) γ = (a2 + a3) sin(ϕ2 )

(11)

(12)

6 Conclusions This paper presents a newly developed mechanical hand. The applications of this anthropomorphic hand can be found in robotics as a robot grasping device or in the medical field as human hand prosthesis. The analytical method of solving the direct and inverse kinematic problem, based on a renowned formulation (Denavit– Hartenberg convention) can be addressed to any other artificial hand. The mathematical relations obtained as solution(s) for the direct and inverse kinematic problem can be succesfully used to implement positional control algorithms for robotic hands.

Acknowledgement This research activity was supported by CNCSIS-UEFISCSU, grant 86/2007, Idei 92PNCDI 2.

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References 1. Biagiotti, L., Lotti, F., Melchiorri, C., and Vassura, G., How far is the human hand? A review on anthropomorphic robotic end-effectors, DEIS–DIEM, University of Bologna. 2. Bicchi, A., Hands for dextrous manipulation and robust grasping: A difficult road towards simplicity, IEEE Transactions on Robotics and Automation, 16(6), December 2000. 3. Butterfass, J., Grebenstein, M., Liu, H., and Hirzinger, G., DLR hand II: Next generation of a dextrous robot hand. In Proc. IEEE Int. Conf. on Robotics and Automation ’01, ICRA 01, Seoul, Korea, May 21–26, 2001. 4. Cutkosky, M.R., On grasp choice, grasp model and the design of hands for manufacturing tasks. IEEE Transactions on Robotics and Automation, 5(3):269–279, 1989. 5. Dario, P., Micera, S., and Menciassi, A., Cyberhand – A consortium project for enhanced control of powered artificial hand based on direct neural interfaces. In Proc. 33rd Neural Prosthesis Workshop, Bethesda, MD, USA, October 16–18, 2002. 6. Denavit, J. and Hartenberg, R.S., A kinematic notation for lower-pair mechanisms based on matrices, Journal of Applied Mechanics, Transactions of the ASME, Ser. E, 77:215–221, June 1955. 7. Jacobsen, S. C., Iversen, E. K., Knutti, D.F., Johnson, R.T., and Biggers, K.B., Design of the Utah-MIT dexterous hand. In Proc. IEEE International Conference on Robotics and Automation, April 7–10, 1986. 8. Kargov, C. Pylatiuk, H. Klosek, R. Oberle, S. Schulz, G. Bretthauer: Modularly designed lightweight anthropomorphic robot hand, In: Proc. of the 2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems September 3-6, 2006, Heidelberg, Germany 9. Kawasaki, H., Shimomura, H., and Shimizu, Y., Educational-industrial complex development of an anthropomorphic robot hand ‘Gifu hand’, Advanced Robotics, 15(3):357–363, 2001. 10. Montambault, S., Analyse et optimisation de prehenseurs mecaniques sous-actionnes, Ph.D. Thesis, Departement de genie mecanique, Universite Laval, Canada, 1997. 11. Salisbury, J.K., Kinematic and force analysis of articulated hands, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1982. 12. Soto Martell, J.W. and Gini, G., Robotic hands: Design review and proposal of new design process, World Academy of Science, Engineering and Technology, 26, 2007. 13. Sundar, N., Dexterous robotic hands: Kinematics and control, Ph.D. Thesis, MIT Artificial Intelligence Laboratory.

Experimental Mechanics

Theoretical and Experimental Determination of Dynamic Friction Coefficient for a Cable-Drum System G. Bayar, E.I. Konukseven and A.B. Koku Middle East Technical University, 06531 Ankara, Turkey; e-mail: {bayar,konuk,kbugra}@metu.edu.tr

Abstract. Cable-drum systems are utilized to convert the rotary motion of a drum into a translational motion of a linear stage connected to the cable. These systems are preferred where low backlash and high stiffness is expected. They are commonly employed in machines like elevators, photocopy machines, printers, plotters etc. For machines having long working ranges, cable-drum systems employing a high resolution encoder offers a practical low cost alternative in position sensing. In most traditional machines and equipments, to get linear position information; potentiometers, linear encoders, laser range finders etc. are commonly used. However, these alternatives are expensive and their installation is not straight forward. Cable-drum systems are not problem free either. The problem coming from using cable-drum system as a linear position sensor grows out of dynamic friction. In this study, the change in the dynamic friction coefficient of the cabledrum system is modeled by using Euler and LuGre friction approaches. In order to see the change of the friction values, the developed model is simulated. To verify of the theoretical results, an experimental set-up is constructed. Both results are presented. It is concluded that for better positioning control change in dynamic friction coefficient during the motion should be accounted for. Key words: dynamic friction coefficient, cable, drum, linear encoder, positioning system

1 Introduction Cable-drive and cable-drum systems are used in various machines where rotary motion obtained from actuators are converted into translational motion. Cable-drum systems are preferred due to several reasons. They decrease the mass and the inertia of the moving assembly hence preferred in many robotic applications. High stiffness of cable-drum systems make them suitable for positioning applications requiring high precision. Apart from the conventional CNC machines and industrial robots, printers, plotters, scanners, tape recorders are some of the examples that use cabledrum systems. Speed of the tape of a tape recorder is provided by using cable-drive [6]. Paper feeding mechanism of the printers is actuated by a cable-drum system [3]. Ink-jet heads are also actuated by cable driven system in printers [4]. Cable-drum systems are mainly used for two objectives: power transmission and sensing linear

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position [2]. In the power transmission systems, cable is wrapped around the input and the output drums. On the other hand, in the linear positioning systems, drum is mounted to a high resolution rotary encoder. The latter system is an alternative solution for the position control of a moving table. Considering the fact that, linear scale (encoder) is commonly used for this purpose, and realizing that the cost of the linear scale increases as the stroke of the axis of measurement increases, the cost-effectiveness of drum system is easily seen. Moreover installation of traditional position sensors is problematic both in the sense of securing suitable space during design and providing proper alignment during installation. Cable-drum systems on the other hand are easy to install, and the encoder and the drum can be conveniently placed almost anywhere in the system as required. Despite the advantage of a cable-drum based position measurement system having low cost and ease of assembly, the problem arising from dynamic friction has to be dealt with. In [2], it has been shown that a cable-drum system can be used as a linear positioning sensor despite dynamic friction causes some problems. In [8], friction attributes to the cable-drive system have been investigated. Stiffness of a cable-drive system has been ascertained by theoretically and experimentally in [7]. In [5] it was shown that the friction between the fiber and the drum is affected by lubricant. In this study, a linear position sensor employing a cable-drum system is developed. Dynamic friction coefficient occurring between the drum and the cable is modeled via Euler and LuGre friction models. In order to observe the characteristics of the friction dynamics, a simulation model has been constructed. In order to verify the derived analytical model, an experimental set-up has been constructed. In this experimental positioning system set-up, carriage is driven by a DC motor through a power screw and position of the carriage is measured via a cable-drum attached to this system. As a result, this work highlights the fact that the theoretical and experimental results suggest that cable and drum setup can be used as a position sensing device especially when the travel is long and a low cost alternative is sought. The organization of the paper is as follows: the next section explains the design of the cable-drum system. Section 3 presents the details of the dynamic model of the proposed system. Section 4 illustrates the simulation studies. Section 5 gives the details of the set-up constructed and experimental studies. The last section, Section 6, relates to the discussion on the theoretical and experimental results.

2 Design of a Cable-Drum System The way cable-drum system used as a position sensor is illustrated in Figure 1. A circular drum with radius R is mounted to a high resolution incremental encoder (i.e. Siemens 6FX2001) which has a resolution of 2500 pulses per revolution. As seen in Figure 1, the preload is adjusted by two set screws. Cable used in this system is a steel wire rope with 0.5 mm diameter and does not elongate under the current loading pattern. It should be emphasized that cable is subjected to an alternating

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Fig. 1 Cable-drum system designed to measure linear position.

load during the motion. Moreover, according to the direction of travel, alternating load on the cable changes its direction, and yet, there is no external load on the cable. Note that preload on the cable has been properly adjusted so that it is higher than the alternating load.

3 Modeling The mathematical representation of the cable-drum system is derived based on the model illustrated in Figure 2. In this model, the cable is wrapped around the drum for 360 degrees. Point C, the center of the drum, is coincident with the rotation center of the high resolution encoder. It is assumed that there is an eccentricity at the drum of which center is specified by C0 . The eccentricity distance is denoted by R0 . The motion of the cable-drum system is expressed into two directions. Through the right and left directions of the travel are denominated by (a) and (b) in Figure 2. The equation of motion of the system described above can be given as: J θ¨ + Td ε = (FL − FR) R

(1)

where Td is the disturbance torque on the shaft of the encoder. J is the inertia of the drum. ε is the sign of the angular velocity of the drum. θ¨ is the angular acceleration of the drum. FR and FL are the cable forces on both directions and expressed as given in Equation (2) for situation (a), which is specified in Figure 2. FR = F0 − ∆ FR ,

FL = F0 + ∆ FL

(2)

where F0 denotes the preload on the cable. Perturbation forces, ∆ FL and ∆ FR , are all external forces on the cable-drum system that may not be foreseen. It should be noted that these two quantities are nonnegative.

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Fig. 2 Cable-drum system model.

Combining Equations (1) and (2), the equation of motion takes the following form: (3) J θ¨ + Td ε = R (∆ FL + ∆ FR) As mentioned above, cable is wound on the drum. While drum is moving the relationship between the dynamic friction coefficient and the cable forces can be represented by using Euler equation [2] in the following form:

γ = eµ (u)α =

FL FR

(4)

where µ (u) is the dynamic friction coefficient which effects the relative velocity between the drum and the cable. α is the wraparound angle of the cable which is 360◦ in our configuration as shown in Figure 2. Cable extension on both sides is assumed to be equal to each other. That is, during the motion, length of the cable does not change. This can be given in a general form as follows:

  ∆ FR L − L0 − x ∆ FL L0 + x FL ⇒ = (5) δ= AE AE AE where A and E are the cross sectional area and the modulus of the elasticity of the cable respectively. L is the overall length of the cable. L0 is distance between the starting point of the sensing and the fix point of the cable and x is the translation of the center of the drum as shown in Figure 1. Combining equations (4) and (5), the following definitions for ∆ FR and ∆ FL are obtained.
 F0 (γ − 1) L0 + x F (γ − 1)(L − L0 − x)
 , ∆ FL = 0 (6) ∆ FR = L − L0 − x + γ (L0 + x) L − L0 − x + γ L0 + x

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For situation (b) expressed in Figure 2, cable forces are defined in Equation (7), and the perturbation forces for this case are predicated as given in Equation (8). FR = F0 + ∆ FR , FL = F0 − ∆ FL
 F (γ − 1) L0 + x F (γ − 1)(L − L0 − x)
 , ∆ FL = 0 ∆ FR = 0 γ L + (1 − γ )(L0 + x) γ L + (1 − γ ) L0 + x

(7) (8)

To solve Euler approach given in Equation (4), LuGre model can be utilized as given in Equation (9) and the details of this model and the specifications of parameters can be found in [1]. µ (u) = σ0 z + σ1z˙ + σ2 u (9) The relative velocity between the drum and the cable can be described in the following form [2]: 
(10) u = x˙ − Rθ˙ + R0 θ˙ cos θ + θ0 where R0 and θ0 are the eccentricity and the initial angle of the drum, respectively.

4 Simulation Studies The model derived in the previous section for observing the behavior of the dynamic friction coefficient of the cable-drum system is simulated in Matlab/Simulink environment. The translational velocity of the center of the drum is given as reference input shown in Figure 3a. Velocity has a trapezoidal profile. The maximum value of the velocity is about 15 mm/s and the travel takes 35 seconds. With this velocity profile, the moving system travels forward direction first and then returns back. According to the reference input shown in Figure 3a, position of the developed system in time is expected to be as indicated in Figure 3b. This simulated position displays that system travels about 250 mm forward and comes back to the starting position. Difference between the reference position and the simulated output posi-

Fig. 3 (a) Reference velocity input (x). ˙ (b) Simulated position output.

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Fig. 4 Difference between the reference and the output position values.

tion is indicated in Figure 4. As seen from this figure, the maximum difference is about 1.1 mm. This difference comes from the eccentricity of the drum’s center position, slip to occurring between the drum and the cable, and dynamically changing coefficient of friction. Parameters used in this study are σ 0 = 10000, σ 1 = σ 2 = 0.001, L = 490 mm, L0 = 10 mm, F0 = 10 N, R = 7.5 mm, R0 = 0.2 mm, θ0 = 10◦ , α = 2π , J = 1 kg·mm2, AE = 8000 N.

5 Experimental Set-up In order to verify the results obtained based on the derived model an experimental set-up has been constructed the schematic of which is shown in Figure 5. As shown in this figure, cable is fixed by set-screws to two stationary supporters located on both sides of the setup. Drum is located on the carriage which is driven through a power-screw mechanism having 4 mm pitch. Guides are installed to both sides of the system in order to provide the stability for the carriage during the motion. Drum, on which cable is wound, is attached to a high resolution encoder. In order to verify the position estimate obtained from the cable-drum system, a linear encoder (manufactured by Heidenhain) having resolution of half micrometer is attached to the carriage as well. Power-screw is driven by using a DC-motor, rated to work in 12–24 V range and has 18:1 gearbox. DC-motor is controlled by a motion control card manufactured by Faulhaber. The motion and the direction control inputs are provided by a DAQ card manufactured by Humusoft (MF-624). Furthermore DAQ card is utilized for obtaining position information coming from both encoders. All of the aforementioned hardware is interfaced as shown in Figure 5 and controlled by xPC target toolbox of Matlab in real time. The constructed experimental set-up is also shown in Figure 6a. The reference velocity input used for the simulation studies shown in Figure 3a is used in the experiment as well. Using the reference input, the carriage travels in forward direction and comes back to the starting location. During this motion, dynamic friction coefficient has been computed. In addition to friction coefficient

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Fig. 5 Schematic view of the experimental set-up.

Fig. 6 (a) Constructed set-up. (b) Simulation and experimental results of friction coefficient.

results of the simulation study, the results of the experimental work are represented in Figure 6b. Both results are similar and show parallelism that dynamic friction coefficient of the cable-drum system does not have a constant value during this motion. It shows a characteristic that changes according to the motion event, direction of travel, cable forces, cable preload, and slippage occurred between the cable and the drum.

6 Conclusions Due to their ease of installation and control, low backlash and high stiffness properties, cable-drum mechanism can be used as a low-cost alternative to linear encoders

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for position measurement. However, dynamic friction coefficient changes during the motion of the system and in order to get more accurate position estimates this change has to be accounted for. Hence, use of constant friction coefficient is not recommended if a cable-drum pair will be replaced with a linear encoder and used for position measurement.

References 1. Canudas, W., Olsson, H., Astrom, K.J. and Lischinsky, P., A new model for control of systems with friction, IEEE Transactions on Automatic Control, 40:419–425, 1995. 2. Kılıc¸, E., Novel position measurement and estimation methods for CNC machine systems, MS Thesis, METU, 2007. 3. Murcia, et al. US Patent 6,565,173, 20 May 2003. 4. Seu, US Patent 5,779,376, 14 July 1998. 5. Smith, D.P., Tribology of the belt driven data tape cartridge, Tribology International, 31:465– 477, 1998. 6. Stephens et al., US Patent 5,878,934, 9 March 1999. 7. Tu, C.F. and Fort, T., A study of fiber capstan friction 1: Stribeck curves, Tribology International, 37:701–710, 2004. 8. Yang, C., Chiou, Y. and Lee, R., Tribology behavior of reciprocating friction drive system under lubricant, Tribology International, 32:443–453, 1999.

Modelling and Real-Time Dynamic Simulation of the Cable-Driven Parallel Robot IPAnema Philipp Miermeister and Andreas Pott Fraunhofer Institute for Manufacturing Engineering and Automation IPA, 70569 Stuttgart, Germany; e-mail: [email protected]

Abstract. In this paper, the mechatronic model of the cable-driven parallel robot IPAnema is presented. The dynamic equations are established including the dynamic behavior of the mobile platform, the pulley kinematics of the winches, and a cable model based on linear springs. The model of the actuation systems consists of the electro-dynamic behavior of the power train as well as the dynamics of the servo controller. The presented model is feasible for real-time simulation, controller design, as well as case studies for high-dynamic or large-scale robots. Simulation results and experimental measurements with the cable-driven parallel robot IPAnema are presented and compared. Key words: cable-driven parallel robot, dynamics simulation, hardware-in-the-loop

1 Introduction Cable-driven parallel robots are able to generate high velocities and accelerations due to very small inertia. On the other hand, large workspaces and high payloads are possible due to the efficient force transmission through the cables. In the last decade, a lot of research has been carried out to study both theory (see e.g. [1, 5, 6, 12]) and implementation [8, 10] of cable robots. A dynamic model and fuzzy control for a cable robot with six cables was presented as model for radio telescopes [13]. A dynamic model using techniques from multi-body system dynamics of the cable robot Segesta was introduced [2, 3]. The dynamics of under-constrained cable robots was used for simulation and control [7, 9]. A simulation model allows to study the wide variety of possible scenarios quickly and in a cost-efficient way. This paper proposes a mechatronic model of cable robots, which can be used for virtually prototyping of large-scale or high-dynamic robots. Further applications are design and optimization of force control as well as motion planning.

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Fig. 1 Mechatronic model of a cable driven parallel robot.

2 Dynamic Model of a Cable-Driven Parallel Robot The mechatronic system of the cable robot is divided into a mechanical part including the platform with n cables and winches on the one side and the electrical part on the other side as illustrated in Figure 1. The electrical part consists of n servo motors and position controllers. The governing numerical control is not further modeled. Its generated position signal θi with i = 1, . . . , n is used as reference signal for the cascaded controller. The dynamic behavior of the individual subsystems is described primarily by ordinary differential equations (ODEs) of first or second order. For simulation and numerical integration the equivalent state space representation is obtained by transforming the high order ODEs into first order ODE-systems. The overall system structure with its forward dynamics, inverse kinematics and the modeled subsystems with their associated input and output quantities is shown in Figure 1. On the electrical side the servo motors are described by their electrodynamics equations with the supply voltage udq,i as input quantities and the measured motor currents idq,i and torque TSM,i as output quantities. The measured motor currents are fed back to the inner current control of the cascaded control, while the torque TSM,i acts on the drum inside the winch together with the cable force fi and therefore both are considered as input quantities for the winch mechanics. The drum angle θeff,i and rotary velocity θ˙eff,i are used as output quantities of the winch subsystem. The drum angle correlates with the rotor angle, that is needed for the outer position and velocity control loop. Describing the platform pose by generalized coordinates x allows to determine the cable lengths q and the structure matrix AT [12] by inverse kinematics. The platform motion is determined by the cable forces f as well as by the applied force fP and torque τP , which act on the tool center point.

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Fig. 2 Advanced kinematic loop including a winch pulley

3 Modelling of Robot Mechanics For parallel kinematics the computational effort for solving the forward kinematics is significant and the solution is not unique. The proposed simulation model uses inverse kinematics and forward dynamics, which are fast and easy to solve and can be done in real-time. To avoid forward kinematics the generalized coordinates of
T the platform pose are chosen as x = r T T , where r is the position vector with respect to the inertial frame K0 and  is the platform rotation described by quaternions in order to avoid singularities and to provide numerical stability. It should be noted, that quaternions have to fulfill the normalization constraint 2 − 1 = 0 if used to parameterize rotations. The associated rotation matrix R is derived from the quaternions as shown in [11]. The cable vector l is determined by a vector loop as shown in Figure 2 with the closure-constraint   l(x) = (0) rC + R0C (C) r1 (x) + RCM (α1 (x))(M) rP2 (α2 (x)) − r − Rb. (1) Vector rC refers to the fixed pivot point C, vector r1 describes the center point of the pulley with respect to KC , and vector rP2 describes the cable contact point A with respect to KM . R0C and RCM denote the transformation from KC into K0 and from KM into KC , respectively. The mobile platform is described as a free floating rigid body without constraints, for which the stationary state follows from the force equilibrium AT f + w = 0 with the applied wrench w = [ fP τP ]T . Using the Newton–Euler formulation for the platform dynamics yields a differential algebraic equation system (DAE) with six second-order differential equations, one algebraic

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equation, and seven unknowns         0 mP E 0 E 0 E 0 Dlin 0 x¨ − x˙ + ˙ + ωI 0 IP 0 Drot 0 P 0 P ˜ Pω IP P˙   

 

 

T

T

gC

= w + AT f

(2)

2 − 1 = 0

(3)

Thereby mP and IP describe the platform mass and tensor of inertia, while the damping matrices Dlin and Drot regard the friction of the environmental medium and cable attachment points. Matrix P describes the relation between quaternions and ˙ [11]. Deriving Eq. (3) twice with respect to time yields the rotary velocity ω = P  an ODE-system of the form ⎡ ⎡ ⎤ ⎤   Dlin 0 mP E 0 T C ⎣ 0 IP P ⎦ x¨ = ⎣ 0 Drot P ⎦ x˙ + w + A f − g . (4) 0 ˙ 0  0  

  

M

k

Solving of the linear second-order equation M x¨ = k and subsequent integration of x¨ involves drift effects caused by differentiation of Eq. (3) which can be dealt ˙ onto the constraint manifold, respectively. The cables with by projecting  and  are modeled as parallel spring damper system with a variable spring rate ci and damping rate di , due to the changing effective cable length qeff,i . Cables absorb only tractive forces yielding a piecewise function that reads  c (q (θ ))qi (x, θ ) + di (qeff,i (θ ))q˙i (x, θ ) for fi > 0 fi (x, θ ) = i eff,i (5) 0 for fi
0 . The effective initial cable length qeff,0,i = qN,i (x0 ) is calculated by the nominal cable length qN,i (x) =
li (x)
2 + qP,i (x) + qG,i whereas qP,i describes the cable length resting on the pulley and qG,i describes the cable length inside the winch as shown in Figure 2. With the unwound cable length qD,i = ri θi the effective cable length reads qeff,i = qeff,0i + qD,i and the difference between the nominal cable length and the unwound cable lengths can be written as qi (x, θi ) = qN,i (x) − qN,i (x0 ) − qD,i .

(6)

Deriving Eq. (6) with respect to time for i = 1, . . . , n yields q˙ = AT x˙ − q˙D .

(7)

The winch dynamics is primarily determined by the winch mechanic’s moment of inertia JW,i and the frictional torque TF,i (fi ), which itself depends on the cable force fi . With the cable force on one side and the motor torque on the other side,

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the drum acceleration follows with rDrum,i fi + TRot,i + TF,i (fi ) . θ¨eff,i = JW,i

(8)

4 Modelling of Robot Electrics The robot’s electrical system includes the drives with embedded position controllers for the servo motors, the amplifier, and the servo motors. The amplifiers use pulsewidth modulation to provide the motors with the necessary rotating three-phase voltage. For the simulation model the amplifiers are assumed as ideal devices and have not been modeled while the permanent magnet synchronous motors (PMSM) are model as simplified motors without damping windings, iron loss, and with symmetrical star-connected motor windings. To realize field orientated control, the electrodynamics differential equations are transformed from the stator’s three phase system into a two phase rotor fix frame using Clarke and Park transformation [4]. Introducing the winding resistance R12,i and the flux linkage ψdq,i , the voltage differential equation with respect to the dq-frame reads ψ˙ dq,i = udq,i − R12,i idq,i − θ˙eff,i TI,i ψdq,i ,

(9)

where udq,i describes the input voltage
controlled by the upstream cascaded control  and the matrix TI,i is defined as TI,i = 01 −1 . Motor current 0   idq,i = L12,i −1 ψdq,i − ψR,dq,i

(10)

is used for the inner current control loop. L12,i describes the winding inductance and ψR,dq,i the rotor flux linkage caused by the permanent magnets. Considering the pole pair number ZP,i , the motor torque is obtained by TRot,i =

  3 ZP,i TI,i ψdq,i idq,i . 2

(11)

The position control calculates the reference value for the downstream velocity control by the set point θi and the effective angle θeff,i using the controller amplification kθ,i   (12) θ˙ref,i = kθ,i θi − θeff,i . For the velocity control loop a proportional integral controller with an amplification of kθ,i ˙ is used. Calculation of the control deviation θ˙i = ˙ and a time constant kT θ,i ˙ ˙ θref,i − θeff,i leads to the desired motor current   0   . (13) idq,ref,i = −1 θ˙i + kT θ,i θi kθ,i ˙ ˙

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The reference value for the d-axis is set to zero, since a motor current along the d-axis has no influence on the motor torque. As with the velocity, the current is controlled by a proportional integral controller with an amplification of kdq,i , a time constant kTdq,i , and the control deviation idq,i = idq,ref,i − idq,eff,i yielding the supply voltage     −1  id,i dt kd,i id,i + kTd,i   . (14) udq,i = −1  kq,i iq,i + kTq,i iq,i dt

5 Implementation and Validation The cable-driven parallel robot IPAnema (Fig. 3) provides a six degrees-of-freedom end-effector with seven or eight cables and focuses on industrial applications in the field of material handling. The winches are equipped with the permanent magnet synchronous motors IndraDyn S by Bosch-Rexroth, which contain multi-turn absolute encoders. The control system is based on an industrial PC with the real-time extension RTX and an adopted NC-controller by ISG (Stuttgart, Germany). The robot can be programmed by G-Code (DIN 66025) similar to machine tools. The model is implemented by the use of Simulink and Matlab, whereas each subsystem is modeled individually and connected as indicated in Fig. 1. For realtime simulation the Real-Time Workshop of Simulink is used to generate source and header files, which provide public functions for accessing the simulation model. The programming languages C and C++ can be chosen. Embedding the simulation model in the RTX real-time environment is straight forward and can be done by including the header files in a main program which calls the simulation function. Validation of the robot model is done by comparison of the actual platform trajectory with a simulated trajectory using the Leica laser tracker system AT901 for position determination with a sample rate of 1 ms. Figure 4 shows the measured and simulated circular trajectory respectively in the xy- and xz-plane with a maximum diameter of 600 mm and a velocity of 0.9 m/s while Fig. 5 shows the platform movement in direction of the x-axis with respect to time. The maximum positional deviation of the simulated trajectory versus the measured trajectory amounts to 5 mm for the movement in the xy-plane and 9 mm for the xz-plane. The motor’s internal current sensor is used to obtain the motor torque for comparison against the simulated torque with a sample rate of 1 ms (Fig. 5). The difference between simulation and measurement result from the uncertain initial cable tension, the not modeled compliance of the robot frame, and the subsidence of the cables and winch mechanics. In contrast to the platform position, the motor torque is closely related to the cable forces and more sensitive to subsidence and differing cable force distribution, which results in a higher deviation of the simulated motor torque.

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Fig. 3 3D model and experimental setup of the cable-driven parallel robot IPAnema with eight cables.

Fig. 4 Two-dimensional view of platform trajectories obtained by simulation (dotted line) and measurement (solid line).

Fig. 5 The diagram at the top shows the comparison of the simulated (dotted line) and measured (solid line) platform motion in direction of the x-axis. The diagram in the middle shows the associated positional deviation of the x-axis with a peak value of –9 mm. The diagram at the bottom shows the characteristics of the associated simulated and measured motor torque for one of the eight servo motors.

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6 Conclusions In this paper a mechatronic model of the cable robot IPAnema was presented. The model consists of the mechanical submodels for platform dynamics, cable behavior, and winch dynamics. The characteristics of the motors and controllers were introduced. The resulting system model was generated using Matlab-Simulink and implemented into the real-time controller environment of the cable robot IPAnema. The comparison between simulation results and measurements with the IPAnema prototype show good agreement. Thus, the model can be used in the future to plan and validate new geometries for robots as well as to design the force-control.

References 1. Bruckmann, T., Mikelsons, L., Brandt, T., Hiller, M., and Schramm, D., Wire robots Part I – Kinematics, analysis and design. In Parallel Manipulators – New Developments, ARS Robotic Books, Vienna, Austria, I-Tech Education and Publishing, 2008. 2. Bruckmann, T., Mikelsons, L., Brandt, T., Hiller, M., and Schramm, D., Wire robots Part II – Dynamics, control and application. In Parallel Manipulators – New Developments, ARS Robotic Books, Vienna, Austria, I-Tech Education and Publishing, 2008. 3. Fang, S., Design, Modeling and Motion Control of Tendon-Based Parallel Manipulators. Fortschritt-Berichte VDI, Reihe 8, Nr. 1076. VDI Verlag, D¨usseldorf, 2005. 4. Fischer, R., Elektrische Maschinen. Springer-Verlag, 2003. 5. Gosselin, C., On the determination of the force distribution in overconstrained cable-driven parallel mechanisms. In Proceedings of the Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, pp. 9–17, 2008. 6. Gouttefarde, M., Merlet, J.-P., and Daney, D., Wrench-feasible workspace of parallel cabledriven mechanisms. In Proceedings of IEEE International Conference on Robotics and Automation, Roma, Italy, pp. 1492–1497, 2007. 7. Heyden, T., Bahnregelung eines seilgefhrten Handhabungssystems mit kinematisch unbestimmter Lastfhrung. Fortschritt-Berichte VDI, Reihe 8, Nr. 1100. VDI Verlag, D¨usseldorf, 2006. 8. Hiller, M., Fang, S., Mielczarek, S., Verhoeven, R., and Franitza, D., Design, analysis and realization of tendon-based parallel manipulators. Mechanism and Machine Theory, 40(4):429–445, 2005. 9. Maier, T. (2004). Bahnsteuerung eines seilgefhrten Handhabungssystems. FortschrittBerichte VDI, Reihe 8, Nr. 1047. VDI Verlag, D¨usseldorf, 2004. 10. Merlet, J.-P. and Daney, D., A new design for wire-driven parallel robot. In Proceedings 2nd International Congress on Design and Modelling of Mechanical Systems, 2007. 11. Schielen, W., Technische Dynamik. B.G. Teubner, Stuttgart, 1986. 12. Verhoeven, R., Analysis of the workspace of tendon-based Stewart platforms. PhD Thesis, University of Duisburg-Essen, 2004. 13. Zi, B., Duan, B., Du, J., and Bao, H., Dynamic modeling and active control of a cablesuspended parallel robot. Mechatronics, 18(1):1–12, 2008.

Horse Gait Exploration on “Step” Allure by Results of High Speed Strobelight Photography A. Vukolov, A. Golovin and N. Umnov RK-2 Department, Bauman Moscow State Technical University, 2nd Baumanskaya Street, 105005 Moscow, Russia; e-mail: [email protected], [email protected], n [email protected]

Abstract. Most walking machines cannot operate within the typical natural environment, including ground relief especially. In that case, the main tasks are adaptive gait control realization and perceptual computing system developing. There are no paths to solve this problem without new experimental exploration of animals’ gaits. Usage of high speed strobelight photography made its possible to observe some hidden aspects of horse gait and leg control processes, like definition of hoof stepping point location. Key words: horse, gait, strobelight, leg control, probability

1 Introduction On natural ground relief, usage of walking chassis among wheels or crawler is the way to increase mobile robots cross-country ability. The control of such systems requires not only realization of adaptive single-leg stepping cycle, but the gait selection also. Gait selection is the complex task because of large amount of possible gaits and selection criteria set indeterminateness. Animal gaits experimental exploration may effectively help to solve the problem. The investigations on the theme of animal gaits were eventually conducted through long time (for example, by Muybridge [2]). Very large amount of visual data was obtained. But it is not possible to understand the gait selection or leg control criteria using these works. Animals produce leg control signals in accordance with ground surface imperfections allocation. Thus, the problem of animal sensors’ ground surface asperity analysis process requires fundamental exploration. However, it is possible to obtain any significant result of such investigation only when experimental “on location” technique or special ground surface pattern is used. Very fine timing and high geometric precision are required for the experimental methods, with imperative presence of minimal impact over object. Now, when information technologies are highly developed, some non-invasive experimental techniques for biological systems kinematics exploration are popular, such as:

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1. Synchronized photography (the modern name is Bullet-time). This method requires the set of registration devices (photo cameras, Schlieren cameras, etc.) that are placed in specified points along the object motion trajectory. The registration devices also must be united through overall synchronization system. The timing resolution of the method is highly limited because of synchronization constraints and limited quantity of registration devices on selected trajectory part. Geometrical resolution is defined by type of registration device and not limited. The synchronized photography was used at the first time near end of the XIXth century by Muybridge [2] for general horse gait exploration. 2. Video recording. Timing of this method is very fine because of fast frame exchanging (at least from 24 to 30 frames per second in standard camcorder systems, up to 50,000 frames per second in case of modern rapid-class videography systems usage). However, the geometrical resolution of video recording is strongly limited by equipment and input signal treatment process peculiarities. After timing resolution of 500 frames per second, this method is applicable in special studio environment only. The technique of kinematical analysis of mechanical systems by results of video recording was described in [6]. 3. Motion capture [4]. The method consists in optical markers placement on parts of the selected object. Motion of the markers is controlled by special measurement equipment. Motion capture is applicable in studio environment only, but it can obtain extremely fine timing and high geometrical resolution. For specific purposes of mechanical and biological systems kinematics exploration the method is practically out of usage because of high experiment cost. Also, practically all of results interpretation techniques are closed by commercial reasons (these techniques are used in cinema craft and computer graphics). 4. Strobelight photography. This method consists of repeated exposure of the object’s selected positions sequence on single frame. Repeated exposure effect achieves through special lighting equipment (photographic stroboscopes) usage. Strobelight photography has strictly defined high (up to 200–600 Hz) timing frequency, geometrical resolution restrictions are defined only by type of camera. The trajectory control precision is extremely high as the result of single frame repeated exposure. It is not required to provide studio environment for this method usage. In accordance to aforesaid, the Strobelight photography method was selected for exploration of horse gait on “step” allure.

2 Target Setting The “step” allure was selected from the main row of horse gaits as easy gait that close to be reproduced by mechanic or cybernetic systems. Another factor of this selection is the high repeatability of horse gaits.

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Fig. 1 Step allure cycle diagram (by Sukhanov). Digits represent the position indexes on the graph; black points and stripes represent hoofs’ ground contact periods: outer stripes for rear legs and inner stripes for front ones.

2.1 Selected Gait Description Using the Existing System As the main system of gait description Sukhanov’s cycled-table system [5] was chosen. In Fig. 1 the cycle diagram of step allure is represented.

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In accordance to Fig. 1 and [1, 3] it is visible that the simple leg positions sequence relatively to ground surface can not obtain full representation about single points trajectories and stability providing.

2.2 Tasks of Experimental Exploration 1. Record the strobelight photographs of horse legs motion on “step” allure. 2. Take qualitative analysis of selected leg points and joints trajectories. 3. Examine the leg motion within peculiarities of gait automation and ground surface relief accordance process.

3 Experiment on Location: Planning and Providing The photo session was made in All-Russian Exhibition Center horse farm training arena. The riding track has cinder coating with characteristic obstacle height is proportionate to hoof height. The registration device was Canon EOS 400D digital photo camera with CCD 10.5 MPix sensor and 100× ISO speed. Optical system had various focal distance (18–50 mm) with angular aperture near 150 degrees. Strobelight was provided by photographic stroboscope Sigma EF-530DGS with impulse duration near 2 × 10−4 s, 100 Hz of maximum frequency and 64 impulses of maximum batch length. Camera was placed at distance of 0.5–0.7 m from track edge, transversely to main horse run trajectory. The light grey protection bands were placed on the front horse legs for trajectory analysis easing. Stroboscope was placed on the camera’s body and it produced direct light stream with vertical aperture 10–45 deg, horizontal aperture 25–100 deg. Shutter speed was snapped to full stroboscope batch length. Synchronization of exposure beginnings was manual. There was no light adsorbing background material is used because of dirt on track coating. About 100 photographs were obtained in two sessions. Shoots were taken during experiment on straight part of track after 5–7 min of warming. It made its possible to obtain good reproducible results. Automated measurements were not being provided for the photographs. So, there is no need to calculate and take into consideration some unimportant (from a qualitative viewpoint) error-making factors, like manual synchronization errors, small geometrical distortion and CCD sensor noise. Final treatment of the photographs consisted of white balance and contrast correction.

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Fig. 2 Step allure strobelight photograph with painted joints trajectories. Batch length is 6 impulses, strobe frequency is 30 Hz.

4 Automatism of the Gait 4.1 Ground Surface Feedback While examinations of leg sensor feedback with ground surface the photograph (Fig. 2) was used. There are joints trajectories painted on the photograph. Main attention was given to front legs during analysis of the photograph. Empirical data is evidence of front legs are main elements of mass center trajectory definition. In accordance to zoological experience the rear legs are practically not concerned in allure cycle realization process. They provide force transfer from gravity and motive powers to ground surface, and also instantaneous balanced state of

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Fig. 3 Step allure strobelight photograph with painted leg parts trajectories at fraught state transfer phase. Batch length is 4 impulses, strobe frequency is 25 Hz.

the system. There are three- and two-pointed supporting states alternate during step cycle (see Fig. 1). The imprinted moment in Fig. 2 corresponds to cycle phase that precede phase of fraught state transfer between front legs (positions 10–11 in Fig. 1). As we can see the trajectories of cubital and knee joints are practically rectilinear. Simultaneously, the hoof implements complex motion. Hoof ending trajectory has a specific form with the curvature radius that is droningly increasing in proportion to hoof approaching to ground surface. This kind of motion allows inertial braking of the leg. The inertial braking makes hoof speed as closer to zero as it is possible near contact point. Second front leg works at that time as rigid part with elastic element near top of the hoof. The work of elastic element can clearly be seen in Fig. 3, where the knee ending trajectory painted during fraught state transfer phase. Due to the contact point is displaced to front from leg axis, the hoof and knee joint work like a hydraulic buffer.

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In accordance to worded, we can make a summary that on “step”allure the front leg that is not in contact with ground works as threshold sensor. This sensor allows the leg to change state immediately on contact and set joints abrupt to rigid position (practically full straight leg with cubital joint in 180+3 dergees position with angle reference point in the center of cubital joint and CCW direction). In that way the fraught state transfer between legs occurs smoothly, and there are no kicks when the step cycle repeats. The imprinted moment in Fig. 3 corresponds directly (compared with Fig. 2) to cycle phase of fraught state transfer (positions 11–12 on Fig. 1).

4.2 Transversal Control of Legs Transversal control of the horse front legs strongly depends of ground surface characteristics. Zoological experience testifies that the contact point location for front hoof calculated by the animal’s nervous system with accounting of ground surface obstacles allocation. Moreover, the influence of obstacles allocation to probably mass center displacement relative to static steady position is also component of the calculation. For rear legs the contact point location calculation is based on current step cycle time – the parameter that controls overall allure tempo. Thus, the tempo is controllable within wide range (up to 30% of current step cycle time) which makes it possible to overpass the lengthy obstacles by step-over motion (on low speed allures like “step”). The overall probability of allure instability or fall is not more than 1% (for horse without rider). The time of new hoof contact point location calculation is not more than 1/15 to 1/10 of the full step cycle time. The detailed investigation of probability calculation and concerned leg control mechanisms for horses and other biological systems are tasks for future works.

5 Conclusions 1. Strobelight photography technique allows us to obtain the large amount of experimental data on location using widespread equipment and without a studio environment. 2. When hoof contact points are calculated, the instantaneous balanced state of the system is provided by rear legs. The front legs practically not concerned in process. 3. The main control of system mass center motion is realized by front legs. The structure of front legs and more adaptable element control allows it. 4. Allocation of the ground surface obstacles is taken into consideration by the animal’s nervous system through bringing of probabilistic constituent into the leg control process.

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5. Exploration of transversal leg control, animal walking cycles and hoof stepping point placement open a new way to construct new four-leg highspeed walking robots for any purpose, e.a. with adaptive (predictive) leg control and different front/rear leg motion programs.

Acknowledgements We would like to express our great gratitude to the personnel of All-Russian Exhibition Center horse farm. Long discussions with them helped us to make clear some darkened aspects of horse biology and gaits.

References 1. Brem, A., The Life of Animals. Vol. 1, Mammals, Moscow, 1939 [in Russian]. 2. Muybridge, E., Available via Wikipedia English. http://en.wikipedia.org/wiki/Edward Muybridge. 3. Hildebrand, M., Motions of the running cheetah and horse. J. Mammal 40(4), 481–495, 1959. 4. Motion Capture. Available via Wikipedia English. http://en.wikipedia.org/wiki/Motion capture. 5. Sukhanov, V.B., The General System of Symmetrical Locomotion of Ground Vertebrates. Peculiarities of Inferior Tetrapodes Motion, “Science” Publishing House, Leningrad, 1986 [in Russian]. 6. Vukolov, A. and Kharitonov, A., Kinematical analysis of mechanical systems by results of digital video recording. In Proceeding of the 10th IFToMM International Symposium on Science of Mechanisms and Machines SYROM 2009, I. Visa (Ed.), pp. 457–464, Springer, 2009.

Simulation of the Stopping Behavior of Industrial Robots T. Dietz, A. Pott and A. Verl Fraunhofer Institute for Manufacturing Engineering and Automation (IPA), 70569 Stuttgart, Germany; e-mail: {thomas.dietz, andreas.pott, alexander.verl}@ipa.fraunhofer.de

Abstract. Ensuring safety for human robot interaction requires knowledge of the dynamic stopping behavior of the robot. This paper presents a multibody model developed to simulate the movement of industrial robots during the braking process. The main contribution is the suggested state-based braking model to overcome numerical problems of the friction behavior upon transitions between sliding and sticking. An analysis of the model upon ocurrance of these transitions is presented. Concluding this paper the simulation model is evaluated using measurement data from a KUKA KR16 industrial robot. Key words: friction, robot model, brake model, stopping behavior

1 Introduction Situations involving direct human robot interaction both in industrial environments as well as in everyday life will considerably increase in the near future. This will allow to apply robots for tasks requiring direct human instruction and collaboration. Further direct interaction scenarios are required to use robots to provide services to humans in everyday life. Common to these applications is that protective equipment separating between human and robot cannot be installed. One of the main concerns for the realization of interactive scenarios is safety for the human operator. The frequently proposed approach of limiting the robot’s energy by reducing speed and forces [11] leads to significant deterioration of robot performance and limits possible applications. Operating the robot with high dynamic energy close to humans requires knowledge of the braking behavior of the robot to ensure that no collision occurs. Therefore, a simulation of the robot brakes and hence a friction model apt for the combination with a robot model is required. The modelling of robots borrows many methods from multibody mechanics and a wide array of literature can be found on this topic, e.g. [3, 15]. Experimental analysis of friction reveals that friction interfaces exhibit a wide range of complex behavior, e.g. velocity dependence of friction force [16] and hysteresis behavior [8], mixed elastic and plastic microscopic displacement in the sticking regime [2] and dependency on the rise time of external forces and influence of

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dwell time [14]. Due to the wide range of mathematically challenging phenomena many friction models exist. Friction models can be categorized as static and dynamic friction models. One of the most commonly known types of static friction models are Coulomb type models [1], which distinguish between two friction regimes. In the sliding regime the friction force depends on the relative velocity of the friction surfaces and the normal force. In the sticking regime the friction force is multivalued and lies inside an interval bounded by the product of normal force and friction coefficient. Upon violation of this interval a transition to the sliding regime occurs. The transition between both regimes poses numeric problems since it relates to a change in the number of degrees of freedom. The main problem is the exact detection of the moment of transition which is difficult due to the integration in discrete steps. This usually either leads to increased computation effort due to the permanent detection of zero-crossings or to oscillating, hunting-like behavior. One approach to overcome these difficulties is the regularization of the Coulomb model around zero velocity [7]. This however violates one of the very core properties of friction since real sticking is no longer possible. [5] suggested to solve the related linear complementarity problem (LCP) to overcome structural problems related to friction. However, numerically solving LCPs is very difficult for general multibody systems. Aim of dynamic friction models, e.g. [4, 13], is to capture more elaborate effects of friction such as hysteresis and presliding displacement. These models introduce additional state variables for each brake increasing the order of the dynamic system. Dynamic friction models are not closer examined in this paper due to their increased complexity and the focus of this paper on simulating gross motion sliding behavior. This paper proposes a model for the description of the stopping motion of serial kinematic chains. Core aspect is the formulation of a brake model satisfying both formal criteria for friction behavior as well as adequate numeric performance and stability. The proposed friction model lies on the boundary of static and dynamic friction models. On the one hand the general properties of the friction model resemble Coulomb type friction models. On the other hand the model is able to take into account the motion state of the friction interface due to its state-based nature. This paper starts with a short description of the robot model in Section 2. Section 3 outlines the derived brake model and presents the analysis of the general behavior of the model for a simple mechanism. The behavior of the model is compared with measurements of an industrial robot in Section 4. Finally, Section 5 presents conclusions and gives an outlook.

2 Robot Model As a first step a multibody model of the robot is set up. To ensure high reconfigurability for different kinematics the model is build of standardized elements consisting of a rigid body and optionally a prismatic or revolute joint as shown in Figure 1.

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Fig. 1 Single robot joint with symbol definitions on the example of a revolute joint.

The single elements are described with KIi and KOi denoting the input and output coordinate frames of element i. K rIO is the displacement induced by element i i i measured in frame KIi , while K rS is the position of the related centre of gravity Si . i i The minimal coordinate qi specifies rotation or displacement of element i + 1 with respect to element i. mi is the mass and KI θS the inertia tensor of element i with i i+1 respect to Si in coordinate frame KIi . The axis of rotation or translation of element i + 1 with respect to i is expressed by KO eri or KO eti . i i The equations of motion of the multibody system under the influence of friction are the starting point for the robot model ˙ =0 ˙ − f(q, q˙ ) − WT (q)λT (q, q) M(q)q¨ − g(q, q)

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where q is the set of minimal coordinates of the multibody system, M(q) denotes ˙ contains the gyroscopic acceleration terms and f(q, q) ˙ dethe mass matrix, g(q, q) notes the vector of generalized forces except for the generalized friction forces. The ˙ analog generalized friction forces are captured in the separate term WT (q)λT (q, q) to [6], but here only for clearer distinction. The columns of matrix WT (q) contain ˙ the generalized force vectors associated to the friction forces contained in λT (q, q) WT (q) = {wT i }, and wTi (q) = (Ja i − Jbi )ni

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where Ja i and Jbi denote the Jacobian matrices of the bodies involved in the friction interface and n is the normalized force direction. It is assumed that the multibody system is time invariant which usually holds for robotic manipulators. Time dependency of the minimal coordinates and their derivatives is omitted in the notation for brevity. The composition of the equations of motion for numeric purposes follows the kinetostatic approach using the recursive Newton-Euler method and kinematic differentials [9].

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3 Brake Model The core problem with finding a suitable model is to address numerical problems due to the structural variance described above. Yet the model should not impose excessive computational burden for example by modelling microscopic friction behavior without significant influence on the gross movement of the multibody system.

3.1 Model Purpose and Assumptions The brake model was developed to simulate the stopping behavior of multibody systems. Starting from an initial state with sliding brakes the motion of the multibody system is simulated. During the simulated motion the axes will switch to the sticking state by and by. Therefore in general a space of time exists where some of the axes are sliding while others are already sticking. Eventually all axes end up in the sticking state and the simulation is terminated. Due to the targeted application it appears reasonable to focus on a stringent correctness of the implementation with respect to energy conservation for the use of standard solver methods. The focus lies on the large scale behavior rather than on the ability to capture small scale effects. Avoidance of structural variance of the overall model is pursued to facilitate model implementation. It is assumed that the friction torque originates mostly from the brakes. The friction torque of the bearings and therefore dependency on the posture of the robot is neglected. This assumption prevents that dynamic bearing loads appear on the right side of the equations of motion, but needs to be checked in experiment.

3.2 A State-Based Brake Model The proposed friction model originates from a Coulomb type friction model. This allows to capture velocity dependency of friction forces while neglecting dynamic friction effects with supposedly minor impact on the gross braking motion. To overcome numeric problems due to structural variance a state-based approach is used for the friction model. The friction curve is regularized in the sticking regime, but on the position level rather than on the velocity level. This retains the core property of real stiction while avoiding problems due to the non-differentiable Coulomb friction curve at zero velocity. The regularization is equivalent to introducing flexibility in the sticking state. This allows treating the multibody system as a structural invariant system since the set of minimal coordinates does not change. A damper element resembling internal friction is added in the sticking state. In reality flexibility of joints and drive system also exists in the sliding phase. However, in the sliding phase the drive system is taut, while in the sticking phase the influence of the joint flexibility is assumed to rise due to the alternating load moment. The damping

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coefficient is computed from an eigenfrequency estimate and a damping ratio of 3 to ensure overcritical damping. The brakes have two states representing the sticking and the sliding regime of the brake. A look-up table with linear interpolation λT,lookup i (q˙i ) yielding positive brake values for positive velocities is used in the simulation to compute the braking torque in the sliding regime. In the sticking regime the brake torque is computed by evaluating a spring damper element:
in sliding state −λT,lookup i (q˙i ), (3) λT i (qi , q˙i ) = −cjoint i (qi − qi0 ) − djoint i q˙i , in sticking state where cjoint denotes the joint stiffness and djoint the damping coefficient in stiction. All joints are initialized in the sliding state. The equations of motion are integrated using the Livermore Solver for Ordinary Differential equations, with Automatic method switching for stiff and nonstiff problems, and with Root-finding (LSODAR) [10]. LSODAR supports automatic switching between solution methods for stiff and nonstiff problems and root-finding to detect zero-crossings in the joint velocities. Upon detection of a zero-crossing the brake associated with this joint switches to the sticking state. The automatic switching to stiff solution methods is beneficial since sticking of at least one joint will render the set of differential equations of motion to be a stiff problem. This behavior is typical for friction due to the fast modes close to stiction and the relatively slow modes in the sliding regime. As mentioned before it is vital for the model to accurately keep track of the kinetic energy of the multibody system. This allows to assess the robot’s braking performance by analysing the energy flow in the model and prevents numeric problems in the solver. Energy conservation during state transitions is achieved by introducing an offset qi0 in equation (3). a a The robot model was implemented using the M a aBILE library [12]. For the brake model an additional kinetostatic transmission element has been created. The brake element is inserted into the model’s kinematic chain for all brake-equipped joints.

3.3 General Model Behavior To assess the simulation model a sliding to sticking transition for a single axis with constant braking torque is analysed (see Figure 2). The brake is activated at around 0.5 s. The setup without gravity shows a linear decline of velocity as expected from the Coulomb friction model. Due to the state space nature of the model no oscillating behavior is observed upon reaching zero velocity. If gravity is introduced an additional load acts on the brake. After a phase of acceleration due to the influence of gravity the activated brake leads to a rapid decrease of velocity. The spring damper element activated upon transition to sticking leads to a small overshoot in position since gravity will lead to a reversal of torque in the joint upon entering sticking. There is no creepage of the axis position as expected for real stiction.

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4 Experimental Model Validation For evaluation of the simulation results the stopping behavior of a KUKA KR16 industrial robot equipped with a 10 kg load is analyzed. First braking torques of axes 1 to 3 are determined through single axis movements by triggering an emergency stop (Stop 0) upon reaching a defined position and speed. The resulting braking ramps are measured in different postures. The brake torque is determined by fitting a simplified brake model assuming constant brake torque to the measurements, resulting in a scalar, velocity independent value for the brake torque λT,lookup i . After triggering of the emergency stop a time delay of approximately 12 ms is observed that corresponds to the interpolation cycle of the robot. This delay is not taken into account closer since the delay is not related to the mechanics of the robot. Table 1 lists the torque values measured for different poses of the robot. The measurements reveal that the assumption of the brake torque being independent of

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Fig. 3 Comparison of measurement and simulation of the joint position during braking for axes 1 to 3 of a KUKA KR16 robot (cjoint = 1.1 · 106 N m rad-1 , djoint = 1 · 105 N m s rad-1 for all axes). The reaction time of the robot is shaded in grey.

the bearing load does not hold for axis 1. For further analysis the values measured closest to the analysed posture are used. Figure 3 shows the comparison of the measured and simulated robot position during braking from a linear path. The behavior of model and measurement show general agreement. The relatively strong disagreement in axis 2 is assumed to partly originate from the low velocity of this axis at the beginning of the braking process due to workspace restrictions. I.e. the mechanical restrictions of the respective axes prohibit reaching higher velocity. Another important factor is the delay due to the interpolation cycle of the robot which is not constantly 12 ms. This shows the demand to analyze the brake behavior while excluding at least some of the error sources that might lead to the observed deviations. E.g. deviations could originate from the neglected bearing load and velocity dependency of the friction force, the inaccuracies due to the roughly estimated parameter set of the robot, particularly inertia and mass, or from uncertainties caused by the limited data set due to robot wear and in particular workspace restrictions. It is assumed that the influence of these inaccuracies is particularly strong on axes 2 and 3 since the mass of joints closer to the base is higher. Therefore the influence of simulation inaccuracies of axes closer to the robot base on axes closer to the TCP is stronger than vice versa.

5 Conclusions and Outlook This paper presented a state-based brake model focussed on the simulation of the stopping behavior of multibody systems. The model captures the structural variance introduced by friction in an internal state and regularizes the sticking behavior on the position level. This allows to overcome numeric problems for transitions between sliding and sticking usually present for Coulomb type models. The brake model accurately keeps track of the kinetic energy of the multibody system. Transition between sticking and sliding match the expected gross behavior of the multibody system without creepage effects resulting from regularisation.

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The combined robot and brake model is used to simulate the stopping behavior of robots. Although general agreement is achieved experimental results show that more work is required to capture the dependency of friction torque and bearing load. Research is under way to assess the model with a more generic experimental setup allowing to test a wider range of state space regions under more standardized conditions. Findings from such experiments will be used to perform a more detailed analysis of safety critical configurations in the robot’s workspace and to assess the sensitivity of the model with respect to parameter uncertainties.

Acknowledgement The research work reported here has been mostly funded by Dynamic Structures Ltd., Vancouver, British Columbia, Canada.

References 1. Canudas, C., Astrom, K. and Braun, K., Adaptive friction compensation in DC-motor drives. IEEE Journal of Robotics and Automation, 3:681–685, 1987. 2. Courtney-Pratt, J. and Eisner, E., The effect of a tangential force on the contact of metallic bodies. Proceedings of the Royal Society, 238:529–550, 1957. 3. Craig, J.J., Introduction to Robotics: Mechanics and Control. Addison-Wesley Longman, 1989. 4. Dahl, P., A solid friction model. Technical Report tor-0158(3107-18)-1, The Aerospace Corporation, 1968. 5. Glocker, Ch., Complementarity problems in multibody systems with planar friction. Archive of Applied Mechanics, 63:452–463, 1993. 6. Glocker, Ch., Set-Valued Force Laws: Dynamics of Non-Smooth Systems. Springer, 2001. 7. Gogoussis, A. and Donath, M., Coulomb friction joint and drive effects in robot mechanisms. In: Proceedings of the 1987 IEEE International Conference on Robotics and Automation ICRA, pp. 828–836, 1987. 8. Hess, D.P. and Soom, A., Friction at a lubricated line contact operating at oscillating sliding velocities. Journal of Tribology, 112:147–152, 1990. 9. Hiller, M. and Kecskem´ethy, A., Equations of motion of complex multibody systems using kinematical differentials. Transactions of the Canadian Society of Mechanical Engineering, 13:113–121, 1989. 10. Hindmarsh, A.C., Odepack – A systematized collection of ode solvers. Scientific Computing, 1:55–64, 1983. 11. ISO 10218-1:2006: Robots for industrial environments – Safety requirements – Part 1: Robot. International Organization a a for Standardization, 2006. 12. Kecskem´ethy, A., M a aBILE – User’s guide and reference manual. University Duisburg-Essen, 1994. 13. Olsson, H., Control systems with friction. Lund Institute of Technology, 1996. 14. Rabinowicz, E., The nature of the static and kinetic coefficients. Journal of Applied Physics, 11:1373–1379, 1951. 15. Sciavicco, L. and Siciliano, B., Modelling and Control of Robot Manipulators. Springer, 2000. 16. Stribeck, R., Die wesentlichen Eigenschaften der Gleit- und Rollenlager – The key quantities of sliding and roller bearings. Zeitschrift des Vereins Deutscher Ingenieure, 46:1342–1348, 1432–1437, 1902.

Mechanical and Thermal Testing of Fluidic Muscles E. Ravina DIMEC, University of Genoa, 16145 Genoa, Italy; e-mail: [email protected]

Abstract. The paper attempts to give a contribution about the mechanical and thermal behavior of fluidic muscles, through theoretical modeling and experimental tests. Fluidic muscles are innovative mechanical components particularly attractive for their flexibility and performances. In particular, braided pneumatic muscle actuators are a very significant development of the pneumatics, being able to operate in a wide range of environments and applied in a wide spectrum of industrial and non-industrial applications, providing a good control of force and position and showing excellent power/weight performances. But, on the contrary of conventional pneumatic actuators, artificial muscles operate under significant thermal gradients. Internal temperature distribution is conditioned by the size and is strictly related to the actual mechanical performances. Integrated mechanical and experimental testing implemented on an original test bench and are focused and discussed. Key words: experimental mechanics, fluidic muscles, testing, pneumatics, actuators

1 Introduction Fluidic muscles are innovative and promising actuators developed to address the control and compliance issues of conventional pneumatic actuators. Among the most interesting of these actuators is the family of braided fluidic muscles. Braided pneumatic muscle gives possibility to achieve accurate and repeatable linear displacement that is realized because of material deformation and flexure. Braided fluidic muscles have a membrane-contraction system able to emulate a human muscle (Fig. 1). Large initial strengths as well as big accelerations make them suitable to industrial or non-industrial processes where quick cycles and high levels of dynamics are required. Several advantages and disadvantages can be listed, with respect to a conventional cylinder. Typical advantages of this mechanical component are: • considerably higher (initial) maximum force: strength produced is ten times grater than the strength of a conventional pneumatic cylinder having the same size;

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Fig. 1 Braided fluidic muscle.

• considerable reduced mass per force unit; • reduced costs respect to comparative products; • easy positioning, by means of pressure regulation, and possibility to achieve intermediate positions; • reduced compressed air consumption for many applications; • possibility to generate highly dynamic operations, thanks to high accelerations; • no stick-slip characteristics; • suitable for clean rooms and contaminated environment; • silent positioning; • possibility to be operated by means different fluids; • no need to use lubricants. Regarding disadvantages, the following must be considered: • • • • •

considerable increased assembly lengths for required stroke; maximum force reduced down to zero; double-acting function is not possible; guidance of load not possible; aging of rubber material: service life depends on the degree of contraction and operating temperature; • vulnerable with regard to sharp edged external damage and welding splashes; • risk of aneurysm or cracks forming if overloaded. Several domains successfully apply artificial muscles: robotics, mechatronics, healthcare, rehabilitation automotive industry, construction engineering, industrial automation, and food industry. Having human muscle-like model this linear actuator is based on a contracting braided membrane. In particular it is a vulcanised rubber hose strengthened by aramid fibers: increasing the inner muscle pressure results in a longitudinal contraction combined with a concomitant expansion in the radial direction. Contraction causes tensile forces along longitudinal direction. A flexible closed membrane, reinforced with a double-helical net of inextensible fibers, represents their essential element. At the two ends, the muscle is closed with fittings by means of which: at one end the compressed air is sucked and at the other the pulling force is exerted on its load. Under the action of the compressed air introduced inside, braided muscle expands radially and contracts axially, producing

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Fig. 2 Experimental test bench.

the contractile force. This one produces the relative approaching of the connection flanges situated at both endings. Muscles can operate at different working pressures, depending on which different values of the contractile force. The developed forces, the contractions and their shape (volume) in expanded state, depend on the muscle’s geometry in rest position and inlet pressure. Braided muscle has been studied and modelled compared with other geometries of muscles by different researches [1–5], and used in advanced applications [6, 7]. But an aspect less considered is its internal temperature distribution: during operating phases the inner temperature increases, following laws depending on the length and on the motion frequency. Hereafter experimental analyses based on of mechanical and thermal tests are described.

2 Experimental Setup The experimental investigation is developed through an original test bench (Fig. 2), expressly designed and realized for monitoring and diagnosis of fluidic muscles. Different sizes of muscles can be tested, simulating variable external loads by means an antagonist pneumatic actuator (Fig. 2a). Muscle can be actuated with timed on-off pneumatic valve or by proportional flow control valve (Fig. 2b). The test bench is equipped with a load cell located between muscle and antagonist cylinder, while pressure transducers detect the inside pneumatic conditions. Muscle terminals can be misaligned in different way, simulating actual operating conditions (Figs. 2c and 2d). The variable geometry of the muscle surface, continuously submitted to contractions and stretching, does not allow the temperature measurement by means contact conventional sensors (thermocouples or thermometers): for this reason infrared thermography approach is successfully applied. The

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heat produced within the muscle is conducted, through the composite layers forming the muscle, on the outside surface, where can be detected by thermo camera: in this experience IR Flex Cam T camera (by Infrared Solution, Inc.) is used. Test bench is managed by acquisition and active modular control unit (PXI and SCXI modules, by National Instruments, Fig. 2e) and operates under flexible virtual instrument implemented through LabView code.

3 Mechanical Testing Muscle contraction is a significant mechanical parameter: its maximum value is related to the structural geometry of the muscle and cannot exceed 25% of the muscle length. Its actual value influences the position error. Methodical tests oriented to detect this influence are implemented, varying in particular sizes of muscles (diameters and lengths), working frequency, number of cycles and air pressure. Figure 3 collects results on position error (mm) vs. number of working cycles for muscles having diameter 40 mm, length 360 mm and operating with pneumatic supply at 4 bar on 500 cycles at 0.5 Hz of cycle frequency. Contraction factors are 10, 15 and 20% respectively (Figs. 3a, 3b and 3c). Position error increases increasing contraction for “long” muscles (length/diameter ratio > 8) but decreases increasing contraction for “short” muscles (length/diameter ratio < 4). Also pneumatic pressure influences the mechanical behavior of muscles: assuming the position error as significant reference on the motion accuracy, specific results are detected. Figure 4 collects results of the position error vs. the number of cycles imposed to long muscles (LM) and short muscles (SM) at low (2 bar) and high (6 bar) pressure, under three different contraction factors (10, 15 and 20 %). The range of error variation decreases for long muscles increasing pressure, while short muscles seem to be less sensitive to this physical variable. In addition, short muscles show always-positive offset, while long muscles are affected by positive or negative errors, particularly at low pressures. As previously cited, fluidic muscles operate under gradients of temperature: in fact cyclical working conditions increase the inner air temperature, but its distribution is not uniform, depending by several geometrical and physical parameters. Non-uniform temperatures produce relaxation effects, influencing the position error. This effect could be compensated by fluid proportional control, but a lot of industrial and non-industrial applications use on-off pneumatic circuits: furthermore, muscles are often used closing with one on the heads. Without “washing” gradients of temperature increase. Influence of temperature has been tested and correlations between mechanical and thermal phenomena are attempted.

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Fig. 3 Position error vs. number of cycles under different contraction factors (long muscles).

4 Thermal Testing Inner longitudinal distribution is strictly related, being equal other parameters, to the muscle length. Examples of this effect are shown in Fig. 5: two 40 mm short and long muscles, working at 0.5 Hz frequency with 20% contraction factor and inlet air pressure of 4 bar, are compared. Axial temperature distribution is more uniformly distributed in short than in long muscles. In long muscle the heat is concentrated at the opposite side with respect to the pneumatic connection. Testing has shown that this phenomenon increases reducing the contraction factor and the internal air pressure.

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Fig. 4 Influence of air pressure to position error, for long and short muscles.

Fig. 5 Axial temperature distributions in short (a) and long (b) muscles.

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Fig. 6 Minimum (a) and maximum (b) temperatures vs. pressure and contraction factor.

A systematic analysis of minimum and maximum temperatures as functions of mechanical and geometrical parameters has been developed. Figure 6 reports results for 120 and 360 mm length muscles. Codes in abscissa identify the contraction factor (c10, c15 and c20 corresponding to 10, 15 and 20%) and air pressure level (p20 and p60, corresponding to 2 and 6 bar). All pneumatic parts of the muscle are monitored, including metallic terminals, fittings and hoses. Also the effect of the working frequency has been tested: fluidic muscles are able to cover a very wide range of working frequencies, satisfying very slow and very fast motion conditions. Synthesis of these tests is collected in Fig. 7: codes Lxx corresponds to the muscle length and Fyyy to frequency. Contraction factor is selected to 10% and air pressure at 6 bar. Long muscles reach higher temperatures: this trend is confirmed also to higher working frequencies, but with more temperatures stabilization.

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Fig. 7 Minimum and maximum temperatures as functions of muscle length and frequency.

Fig. 8 Temperatures vs. average position error for long muscles.

5 Relationships Fluidic muscles are mechanical devices working under variable temperature. The final goal of the research is the correlation between thermal behaviour and mechanical performances. Figure 8 reports the trend of maximum, minimum and average temperatures vs. average position error e (standard deviation 0.3) for long muscles tested under 10% of contraction and 500 working cycles. The fourth order polynomial relationships (T = Ae4 + Be3 + Ce2 + De + E) approximate the experimental temperature functions well: for long muscles the equation parameters are A = 1; B = −5; C = 10; D = −9; E = 3. Short muscles show a very different behavior (Fig. 9): the equation parameters able to approximate the

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Fig. 9 Temperatures vs. average position error for short muscles.

maximum and average temperature distributions are A = −1; B = 2; C = −1.5; D = 0.5; E = 0.05. The minimum temperature distribution is nearly constant. The definition of mathematical correlation between temperatures and position errors is subject of deepening: these results seem to be a good reference for the generation of oriented models.

6 Conclusions A methodical analysis of the mechanical and thermal behaviour of fluidic muscles has been developed, applying not intrusive detection of temperatures. A specific experimental set-up has been used to detect correlations between temperatures and position errors, taking into account air inlet pressure, number of operation cycles, working frequency and muscles size. A significant data base of mechanical performance and on temperature distribution is now available: it will be the reference for further deepening, oriented to compare standard and anomalous working conditions, in order to attempt predictions on equipment troubleshooting and anomalous or failure conditions.

References 1. Ahn, K.K., Anh, H.H. and Kiet, N.T., A comparative study of modeling and identification of the pneumatic artificial muscle (PAM) manipulator based on recurrent neural networks. In Proc. Int. Symp. on Electrical & Electronics Eng., HCM City, Vietnam, 2007. 2. Chou, C.P. and Hannaford, B., Measurement and modeling of McKibben pneumatic artificial muscles. IEEE Transactions of Robotics and Automation, 12(1):90–102, 1996.

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3. Davis, S. and Caldwell, D.G., Braid effect on contractile range and friction modeling in pneumatic muscle actuators. Int. J. Robotic Research, 25(4):359–369, 2006. 4. Davis, S., Tsagarakis, N., Canderle, J. and Caldwell, D.G., Enhanced modeling and performance in braided pneumatic muscle actuators. Int. J. Robotic Research, 22:213–227, 2003. 5. Feja, K., Kaczmarski, M. and Riabcew, P., Manipulator driven by pneumatic muscles. In Proc. 8th Int. Conf. on Climbing and Walking Robots, London, UK, 2005. 6. Ramasamy, R., Juhari, M.R., Mamat, M.R., Yaacob, S., Mohd Nasir, N.F. and Sugisaka, M., An application of finite element modeling to pneumatic artificial muscle. Amer. J. Appl. Sci., 2(119):1504–1508, 2005. 7. Van der Linde, R.Q., Design, analysis and control of a low power joint for walking robots, by phasic activation of McKibben muscles. IEEE Trans. Robotics and Automation, 15(4):599–604, 1999.

Structural Dynamic Analysis of Low-Mobility Parallel Manipulators J. Corral, Ch. Pinto, M. Urizar and V. Petuya Department of Mechanical Engineering, University of the Basque Country, 48013 Bilbao, Spain; e-mail: {j.corral, charles.pinto, monica.urizar, victor.petuya}@ehu.es

Abstract. This paper focuses on the study of static and dynamic stiffness (normal modes and natural frequencies) by a method which enables to evaluate the vibration behaviour of low mobility parallel manipulators, i.e., less than 6 degrees-of-freedom. The purpose of this study is to improve the structural behaviour of this type of manipulators under the effect of static and dynamic external loads. As a starting point, we use a previously developed methodology for static rigidity obtaining for the aforementioned manipulators characteristics, using as case study DAEDALUS I parallel robot. The method is applied to a simplified model to analyze the evolution of the static stiffness, the frequencies and normal modes in the whole workspace. The method involves the use of one analytical model, other numerical, based on finite elements and a comparison with an experimental campaign on a prototype. From this study, the results of the natural frequencies trends of the manipulator are represented in tridimensional maps. Key words: parallel manipulators, low-mobility, stiffness, natural frequencies

1 Introduction In the last decade, the Parallel Manipulators (PM) are being widely studied. On the one hand, kinematic characteristics such as workspace, velocity and acceleration analysis have been studied [4, 7]. On the other hand, dynamic capabilities have also been a widely researched field in order to size the actuators, know the secondary dimensions and also for control strategies. But there is another fundamental aspect in the operation of the PM related to the mechanical characteristics and with a strong influence on the precision: the static and dynamic rigidity. The first source of inaccuracy error of the tool mounted on the mobile platform is that associated with assembly errors. In [6], the influence of assembly geometric errors and manufacturing tolerances in the final position and orientation of the mobile platform is evaluated. Sensitivity functions showing which of the considered factors (orientation of the legs and actuators, parallelism between active and passive axes, among others) predominate in the correct positioning are shown. From the static point of view there is an extensive bibliography in which models of stiffness based on different approaches can be found [7]. Some models take into account only the

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Fig. 1 DAEDALUS I prototype.

stiffness of the actuators, considering the legs and the kinematic joints as rigid solids [2, 3]. More complex models that take into account the stiffness of elements and the kinematic joints have been developed [1, 9]. Regarding the dynamics, the most commonly models found in the literature are those which consider the elements as rigid bodies. These models are used to solve dynamic problems, direct and the inverse problems, in order to determine which elements have a larger influence in the dynamic characteristics of the system. Furthermore, these approaches are used to design the manipulator’s control system when the driver has underdamped response [5]. In addition, models that take into account the flexibility of the elements, which are based on analytical models of constrained Lagrange equations and also finite element models, can be found. Finally, some works present an experimental study of the dynamic problem. Implementation and guidelines for experimental analysis of frequency response using analyzers, accelerometers and vibrational excitation tools like impact hammers and shakers are developed.

2 Case Study: DAEDALUS I Manipulator Throughout this section the so called DAEDALUS I (Figure 1) low-mobility manipulator is presented. It is a spatial 4-dof manipulator and its mobility is usually represented by the acronym 3T1R. The main applications of this type of manipulators are associated with pick & place operations in which the external loads are generally in vertical direction. This fact justifies a structural study based on the behaviour in this direction. The actuated inputs are four prismatic joints implemented as ball screws which are connected to electric motors. The motion pattern of the mobile platform consists of three translations and a rotation around an axis through the centre of the mobile platform that is perpendicular to the plane Y Z. The manipulator comprises a fixed frame, a mobile platform and four identical legs connecting them. Each leg consists of the 2R kinematical chain which is connected to the mo-

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Fig. 2 CAD model of the manipulator and maximal workspace.

bile platform by means of another revolute joint. In addition, all the revolute axes of each leg are parallel. For the manipulator’s workspace obtaining the inverse kinematic problem must be solved (see [8] for details). The knowledge of the workspace is critical in deciding the distribution and number of experimental measurements and simulations, as well as the order in which they are made. In Figure 2 DAEDALUS I maximal workspace is shown.

3 Structural Behaviour Analysis 3.1 Static Stiffness of the Manipulator Next, the static behaviour in vertical direction and, in particular, under a vertical load will be analyzed. Thus, from all the elements in stiffness matrix K, the study is limited to kzz element. Under these conditions the equivalent model is reduced to those elements of the manipulator that, for its configuration and resistant features, are capable of bearing vertical loads. In the DAEDALUS I manipulator these are the Z and U legs (see Figure 1) and the mobile platform. These assumptions lead to the following equation: (1) fz = kzz · δz where fz is the vertical force, δz the vertical displacement and kzz the stiffness matrix element that relates both statically. Analytical stiffness behaviour is represented by the two legs mounted in parallel and then serially connected with the mobile platform, and the expression is:

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Young Modulus Poisson ratio Density

Aluminum

Steel

71 GPa 0.33 2770 kg/m3

210 GPa 0.3 7850 kg/m3

Fig. 3 Vertical displacements field for the simplified model of stiffness.

1 1 1 = U + MP Z kzz kzz + kzz kzz

(2)

U and k Z the stiffness of the legs and where kzz is the stiffness of the model, kzz zz MP kzz the mobile platform stiffness. To obtain the stiffness of each element of the analytical model, Classical Theory of Elasticity has been used. By means of this model the total displacement of the Tool Centre Point (TCP), under a unit load is calculated as a function of position and orientation. Thus, a function of stiffness kzzANA = kzzANA (x, y, z, θ ) is obtained. Secondly, a FEM model of the manipulator to obtain a comparison with the analytical model results has been developed. This model consists of approximately 152,000 parabolic tetrahedral elements and the boundary conditions with the fixed element are formed by two clampings. For all the rotational joints, kinematic restraints have been imposed so that the static behaviour can be analyzed. The resulting model is shown in Figure 3. The properties of materials and components used are summarized in Table 1. From this model the numerical stiffness kzzMEF = kzzMEF (x, y, z, θ ) is obtained. Finally, to establish a reference, experimental tests with the manipulator’s prototype have been carried out by means of a set of lumped loads and measuring the resulting vertical displacements. From the experiments, the value of kzzEXP = kzzEXP (x, y, z, θ ) as a function of position and orientation is obtained. For both models a number of representative points and uniformly distributed through the whole workspace of the manipulator have been measured.

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Fig. 4 Lumped masses distribution and reduced model.

3.2 Dynamics: Normal Modes and Natural Frequencies To obtain the equation of motion of the chain, any of the methods discussed in the introduction can be applied resulting in the following equation: K ·
= ω2 · M ·


(3)

which represents a system of n dof where the vector
represents the amplitudes of the motion of the system and ω the natural frequencies. The system of equations (3) represents a generalized problem of eigenvalues and eigenvectors in which the square roots of the eigenvalues represent the system’s natural frequencies and eigenvectors associated with each the normal modes or vibration modes. In this work, the structural dynamic approach represented by equation (3) has been specifically addressed by means of an analytical model and other numerical based in FEM. Through a suitable discretization, the continuous system decomposition in a specific number of dof can be achieved and then, get the model solution to obtain the natural frequencies of the system and the corresponding normal modes. The analytical model is based on a lumped parameters approach in order to obtain an equivalent dynamic model consisting of a lumped mass, an equivalent stiffness in the TCP and its displacement (meq , keq , xeq ). In this model, keq is the stiffness of the model obtained in (2). This lumped model considers the masses located in 9 points (centres of mass of each element and the axes of the pins). In Figure 4 the lumped masses have been depicted on the reduced dynamic model. Based on the CAD model of the manipulator, the corresponding finite element model is obtained (see Figure 3). Regarding the boundary conditions remain the same as those from the static analysis. However, additional restrictions introduced in some rotational joints that were previously required to solve the static case without the stiffness matrix singularity are no longer needed. In this case, the results of constant height (Z = 150 mm) map of the manipulator workspace is presented; hereinafter this plane will be referred as working plane. The orientation of the manipulator has remained constant: θ = 0◦ . Each of these frequencies corresponds to a vibration mode of the system. For the pattern mode definition increasing order of natural frequencies in the central position of the work-

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Fig. 5 Representation of the first four normal modes.

ing plane (x, y) = (570, 663) mm was chosen. In Figure 5 the normal modes (mi ) for the analyzed frequencies are presented. As shown in Figure 5, vibration modes include bending and torsional deformations of the legs. For the manipulator’s application task and the excitation frequencies of the external dynamic loads, the corresponding vibration mode effect can be evaluated. Furthermore, as later shown it also depends on the point in the workspace where the TCP is. An experimental campaign on the manipulator’s prototype to compare the results obtained from the analytical models is carried out. “LMS
SCADAS Mobile Multi Channels” FFT (Fast Fourier Transform) analyzer together with a series of accelerometers that record the signal response of the prototype has been used. The excitement in the system has been generated by means of an impulse hammer and the response is collected in four different points of the system. In addition, three different impulse directions have been considered to adequately excite the modes predicted by the numerical model.

4 Analytical, Numerical and Experimental Results From the application of this method we obtain a series of results that allow representing natural frequencies maps. In these maps, trends and extrema of the natural frequencies of the manipulator can be deduced. Figures 6a and 6b show the first natural frequency maps in the working plane obtained for the two models and for the experimental prototype in Figure 6c. These maps present analogous trends of f1 in the working plane. The maximum error between prototype’s reference results and models is 16% and the average error in the z-constant plane is smaller than 7%. Another interesting result is the evolution of the first four frequencies in the working plane. Relative separation between frequencies can be analyzed (Figure 7).

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Fig. 6 Analytical (a), FEM (b) and experimental (c) f1 maps.

Fig. 7 First four frequencies evolution in the working plane.

5 Conclusions Based on these results, conclusions about the dependence of static and dynamic structural behaviour with the configuration of the manipulator and its position in the workspace are established. The main added value of this work are the graphs that represent a practical tool for the end user so the areas of the workspace that are most appropriate for optimal use of the manipulator in each specific application can be chosen. Another key issue is the computational cost, whereas numerical and experimental approaches need large computational times a well-validated analytical model leads to a significant time saving. As future lines the following ones are suggested: (1) A detailed study of the dependence of structural and dynamic properties depending on the configuration. (2) Determination of possible crossings often motivated by changes in the frequency with position.

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Acknowledgments The authors of this paper gratefully acknowledge the funding received by the Ministry of Education and Science of Spain (Project DPI2008-00159), FEDER funds of the European Union and the Basque Government (Project GIC07/78).

References 1. Company, O., Pierrot, F. and Fauroux, J-Ch., A method for modelling analytical stiffness of a lower mobility parallel manipulator. In Proceedings International Conference on Robotics and Automotion (ICRA05), Barcelona, Spain, 2005. 2. El-Khasawneh, B.S. and Ferreira, P.M., Computation of stiffness and stiffness bounds for parallel link manipulators. International Journal of Machine Tools & Manufacture, 39:321–342, 1999. 3. Gosselin C., Stiffness mapping for parallel manipulators. IEEE Transactions on Robotics and Automation, 6(3):377–382, 1990. 4. Hern´andez, A., Altuzarra, O., Pinto, Ch. and Amezua, E., Transitions in the velocity pattern of lower mobility parallel manipulators. Mechanism and Machine Theory, 43(6):738–753, 2008. 5. Kozak, K., Ebert-Ophoff, I., and Singhose, W., Analysis of varying natural frequencies and damping ratios of a sample parallel manipulator throughout its workspace using linearized equations of motion. In Proceedings of the 2001 ASME Design Engineering Technical Conference, Pennsylvania, USA, 2001. 6. Merlet, J.P., Parallel Robots, 2nd ed. Springer, 2006. 7. Pinto, Ch., Corral, J., Altuzarra, O., and Hern´andez, A., A methodology for static stiffness mapping in lower mobility parallel manipulators with decoupled motions. Robotica, DOI: 10.1017/S0263574709990403, 2009. 8. Salgado, O., Altuzarra, O., Petuya, V., and Hernandez, A., Synthesis and design of a novel 3T1R fully-parallel manipulator, Journal of Mechanical Design, 130:042305, DOI: 10.1115/1.2839005, 2008. 9. Wang, Y. Liu, H., Huang, T., and Chetwynd, D.G., Tricept stiffness modeling of the robot using the overall Jacobian matrix. Journal of Mechanisms and Robotics, 1(2):021002-1-021002-8, 2009.

Dynamics

Spatial Multibody Systems with Lubricated Spherical Joints: Modeling and Simulation M. Machado1, P. Flores1 and H.M. Lankarani2 1 Mechanical Engineering

Department, University of Minho, 4800-058 Guimar˜aes, Portugal; e-mail: {margarida, pflores}@dem.uminho.pt 2 Mechanical Engineering, Wichita State University, Wichita, KS 67260-0133, USA; e-mail: [email protected]

Abstract. The dynamic modeling and simulation of spatial multibody systems with lubricated spherical joints is the main purpose of the present work. When the spherical clearance joint is modeled as dry contact; i.e., when there is no lubricant between the mechanical elements which constitute the joint, a body-to-body (typically metal-to-metal) occurs. The joint reaction forces in this case are evaluated through a Hertzian-based contact law. The presence of a fluid lubricant avoids the direct metal-to-metal contact. In this situation, the squeeze film action, due to the relative approaching motion between the mechanical joint elements, is considered utilizing the lubrication theory associated with the spherical bearings. In both cases, the intra-joint reaction forces are evaluated as functions of the geometrical, kinematical and physical characteristics of the spherical joint. These forces are then incorporated into a standard formulation of the system’s governing equations of motion as generalized external forces. A spatial four bar mechanism that includes a spherical clearance joint is considered here as example. Key words: lubricated spherical clearance joints, dry contact, spatial multibody dynamics

1 Introduction The main objective of this study is to present a general methodology for modeling and simulating of realistic spherical joints with clearances under the framework of multibody systems methodologies. The motivation for this research work comes from current interest in developing mathematical and computational tools for the dynamics of multibody systems in which the effects of clearance, surface compliance, friction and lubrication in real joints are taken into account [4]. These phenomena can be included in a multibody system with a spherical joint, and in many applications the function of the mechanical systems strongly depends on them. Typical examples are the spherical joints of a vehicle steering suspension and the artificial hip implants. In the vehicle example, either due to the loads carried by the system or misalignments that are required for their operations, real spherical joints must be lubricated or include bushing elements, generally made with metals or polymers. By using rubber bushings, a conventional mechanical joint is transformed into a joint with clearance allowing for the mobility of the over-constrained system in which it

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is used [1, 8]. In the example of artificial hip articulations, some amount of clearance is always necessary to allow the assembly of the femoral head and acetabular cup and to facilitate the system’s operation. Hip implants are in fact the most typical application of lubricated spherical joints in which the knowledge of the clearance size and external load are of great importance for their dynamic performances. Most of the available literature on this subject is mostly formulated on the tribological level, in the measure that the geometrical properties (e.g., clearance size) of the hip implants are not considered as variables. The kinematic (e.g., angular velocities) and dynamic (e.g., applied forces) characteristics of the hip elements are also assumed to be constant in most of the studies [2, 10]. Yet, these quantities vary during any daily human activity. Liu et al. [7] developed a simple contact force formulation of the spherical clearance joints in multibody mechanical systems, using the distributed elastic forces to model the compliant of the surfaces in contact. Flores et al. [3] also presented a methodology to assess the influence of the spherical joint clearances in spatial multibody systems. Both of these approaches are only valid for the case of dry contact between the socket and the ball. The present work deals with the dynamic modeling and analysis of spatial multibody systems with lubricated spherical joints. This study extends previous authors’ work [3] to include the lubrication action into the spherical joints. In a simple way, the intra-joint forces developed at the dry and lubricated spherical joints are evaluated as functions of geometrical, kinematical and physical properties of the joint elements. For these purposes, two different approaches are employed. For the dry contact situation, the joint contact-impact forces are computed by using a continuous constitutive contact force law based on the Hertz theory. The energy-dissipative effect associated with the contact-impact process is included by a hysteresis-type damping term. For the case in which a fluid lubricant exists between the joint elements, the squeeze lubrication action is modeled employing the hydrodynamic lubrication theory. The aim here is to evaluate the resulting forces from the given state of position and velocity of the spherical bearings. For both cases, the joint reaction forces are introduced into the equations of motion as external applied forces. Finally, results for a spatial four bar mechanism with a lubricated spherical clearance joint are presented to discuss the main assumptions and procedures adopted throughout this work.

2 Proposed Methodology Figures 1a depicts two bodies i and j connected by a spherical joint with clearance. When there is no lubricant in the joint, an internal impact takes place in the system and the corresponding impulse is transmitted throughout the multibody system. Contact-impacts of such occurring within a spherical clearance joint, are one of the most common types of dynamic loading conditions which give rise to impulsive forces, and in turn excite higher vibration modes and affect the dynamic characteristics of the mechanical system. For a spherical joint with clearance, the contact

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Fig. 1 (a) General configuration of a spherical clearance joint in a multibody system; (b) A generic representation of a spherical lubricated joint with squeeze-film.

between the socket and the ball can be modeled by the well known Hertzian contact law [6]
 3(1 − c2r ) δ˙ n FN = K δ 1 + (1) 4 δ˙ (−) where K is the generalized stiffness constant and δ is the relative normal deformation between the spheres, cr is the restitution coefficient, δ˙ is the relative penetration velocity and δ˙ (−) is the initial impact velocity. The exponent n is set to 1.5 for the cases where there is a parabolic distribution of contact stresses, as in the original work by Hertz. When the space between the ball and socket is filled with a lubricant, the joint becomes a spherical joint with lubrication action, whose kinematic aspects are similar to those for the spherical joint with clearance. The only difference is that the relative radial velocity leads the lubricant squeeze action when there is no contact between the ball and socket walls. One possible way to evaluate the lubrication forces developed in a spherical joint with lubricant is to equate the pressure flow to the pressure that results from displacement associated with the geometric configuration of the spherical joint elements. The configuration considered here is of a hemispherical seat which, owing to symmetry, has a cross section identical to that of Fig. 1b. Based on fundamental principles, Pinkus and Sternlicht [9] demonstrated that, for a spherical joint with lubricant, the amount of fluid that passes a conical element surface is given by Q=−

π Ri sin θ h3 d p 6 µ Ri d θ

(2)

where Ri is the ball radius, h represents the thickness of the film lubricant, µ is the dynamic lubricant viscosity, θ is the angular coordinate and p is the pressure. On the other hand, based on the geometry of Fig. 1b, the flow at any angle θ due to the radial velocity, e, ˙ is expressed as

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Q = π eR ˙ 2i sin2 θ

(3)

where e˙ is the velocity of the ball center that is responsible for the squeeze-film action, i.e., e˙ = dh/dt. Equation (3) corresponds the total flow rate through the spherical joint, and hence, it represents the amount of fluid that flows through the joint and is required to ensure a desired fluid thickness equal to h. Thus, for the flow continuity, combining Eqs. (2) and (3) results the general expression for pressure gradient dp as follows dp =

6µ e˙ sin θ (c/Ri ) Ri (1 − ε cos θ )3 3

dθ

(4)

in which c is the radial clearance and ε = e/c. Then, the pressure distribution in the lubricated spherical joint can be obtained if Eq. (4) is integrated with an appropriate set of boundary conditions. Thus, the resulting squeeze FS film force on the ball, which should be applied to equilibrate the fluid pressure, can be evaluated as the integral of the pressure field over a hemisphere, yielding   6π µ eR ˙ i 1 1 1 − (5) ln(1 − ε ) + 2 FS = ε (1 − ε ) 2ε (c/Ri )3 ε 3 In short, this illustrates that the squeeze lubricant forces at any instant of time can be evaluated in terms of the instantaneous eccentricity and relative radial velocity, as well as the constant parameters of the fluid viscosity, joint clearance and the ball radius. The direction of the squeeze force is collinear with the line of centers of the socket and ball, which is described by the eccentricity vector in Fig. 1a. Thus the squeeze force can be introduced into the equation of motion of a multibody system as generalized forces, with the socket and ball centers as points of action for the force and reaction force, respectively. It is important to note that for high speed rotating machinery the lubricant viscous effect due to rotation also needs to be considered and included in the analysis. One possible methodology that must be followed to account for both wedge-film and squeeze-film effect is the one proposed by Goenka [5]. This, work presents the details of the Reynold’s equation for the more general lubricated circumstances of a spherical joint with lubrication. Then, in a similar manner the resulting lubrication force in both radial and tangential directions developed should be introduced into the equations of motion of the mechanical system under analysis.

3 Results and Discussion In this section, an application example is used to assess the effectiveness of the proposed methodologies for modeling spherical joints in a multibody mechanical system in spatial motion. A four bar mechanism, depicted in Fig. 2, is selected for the study and for which a non-ideal spherical joint exists between the coupler and

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Fig. 2 Schematic representation of the spatial four bar mechanism, which includes a non ideal spherical joint between the coupler and rocker. Table 1 Geometrical and inertial properties of the spatial four bar mechanism. Moment of inertia [kg m2 ] Body No. 2 3 4

Length [m]

Mass [kg]

Iξ ξ

Iηη

Iζ ζ

0.020 0.122 0.074

0.0196 0.1416 0.0316

0.0000392 0.0017743 0.0001456

0.0000197 0.0000351 0.0000029

0.0000197 0.0017743 0.0001456

rocker bodies. In the present study, this particular joint is modeled as clearance joint with and without lubricant. The remaining revolute and spherical joints are considered to be ideal joints. Due to the existence of a joint clearance, the system has a total of five degrees of freedom. In order to keep the analysis simple, the bodies are assumed to be rigid. The numbers of the bodies and their corresponding local coordinate systems are illustrated in Fig. 2. The geometrical and inertial properties of the bodies that constitute the four bar mechanism are listed in Table 1. In the dynamic simulation of this four bar mechanism, the system is released from rest corresponding to the initial configuration illustrated in Fig. 2. Furthermore, initially the socket and the ball centers are coincident. The acceleration due to gravity is taken as acting in the negative Z-direction. Hence, the dynamic behavior of the system is affected by the potential energy associated with the heights of centers of mass of all bodies. The main parameters used for the computational simulations of the system with ideal, dry and lubricated spherical joints and for the numerical methods required to solve the system dynamics are presented in Table 2. In what follows, some obtained results are presented and analyzed with the intent to demonstrate the performance of the four bar mechanism described above. This performance is quantified by plotting the reaction forces that develop at the spherical clearance joint and the relative motion between the socket and ball produced during the overall system’s motion. When the spherical joint is modeled as dry joint, the

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M. Machado et al. Table 2 Parameters used in the dynamic simulation of the spatial four bar mechanism. Nominal socket radius Nominal ball radius Radial clearance Restitution coefficient Poisson’s ration Young’s modulus Lubricant viscosity Baumgarte stabilization parameters – α , β Integrator algorithm Integration time step

10.0 mm 9.8 mm 0.2 mm 0.9 0.3 207 GPa 400 cP 5 Gear 0.00001 s

Fig. 3 Joint reaction forces developed at the spherical clearance joint: (a) dry joint model; (b) lubricated joint model.

intra-joint forces are evaluated by using Eq. (1), while for the case of lubricated model the reaction forces are computed by employing Eq. (5). The global results are compared with those obtained when the system is modeled with ideal joints only. The joint reaction forces that are developed at the spherical joints for the dry and lubricated models are plotted in Fig. 3, from which it can be observed that the dry joint model produces higher peaks. This is due to the fact that the socket and ball surfaces undergo successive direct collisions with each other and the impulsive forces associated with this type of contact are transmitted throughout the system. The presence of fluid lubricant between the socket and ball surfaces avoids this situation and acts like a damper element, which ultimately causes lower peaks in the joint reaction forces. In this case, the joint elements are apart from each other owing to the squeeze-film action. The difference in the relative positions of the socket and ball surfaces for the dry and lubricated spherical models is quite visible in the charts of Fig. 4. For the dry model, a period of impacts is immediately followed by rebounds, after which the ball tends to be in a continuous or permanent contact with the socket surface. In this phase, the relative deformation varies in the circumferential direction. In turn, the case of lubricated spherical model, the ball surface never reaches the socket due to the opposed action of the fluid, as observed in Fig. 4(b).

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Fig. 4 Eccentricity evolution with time: (a) dry joint model; (b) lubricated joint model.

4 Conclusions The main objective of the work presented in this paper was to contribute to the improvement of the methodologies dealing with the analysis of realistic spatial multibody systems, which include imperfect or real joints. These are joints in which the effects of clearance and lubrication are taken into account in the multibody systems formulation. The methodology developed here incorporates the contact-impact forces due to the collisions of the bodies that constitute the spatial spherical clearance joints into the equations of the motion that govern the dynamic response of the multibody mechanical systems. A continuous contact force model is utilized which provides the intra-joint due to dry contact and lubrication forces that develop during the normal operations of the mechanisms. The effect of lubrication in spherical clearance joint is considered by the squeeze film providing pressure or force in the radial joint direction. Thus, a suitable model for spherical clearance joints is embedded into the multibody systems methodology. The methodology is easy and straightforward to implement in a computational code because resultant forces due to the fluid action are in explicit forms. A spatial four bar mechanism was considered here with a spherical clearance joint. The system was driven by the gravitational effects only. The existence of a non-ideal joint in the system significantly increases the amount of dissipated energy. Furthermore, the fluid lubricant acts as a nonlinear spring-damper in so far as lubricated spherical bearing absorbs some of the energy produced by the rocker. The lubricant introduces effective stiffness and damping to the four bar mechanism, which plays an important role in the stability of the mechanical components. Finally, it should be highlighted that the results presented in this paper represent an upper bound of the joint reaction forces and rocker kinematics due to the existence of realistic joints, since the elasticity of the links was not included in the analysis. This effect tends to reduce the joint reaction forces and rocker kinemat-

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ics. Using the methodologies presented in this paper, the effects of clearance and lubrication in spherical joints can be effectively studied.

Acknowledgments This work is supported by the Portuguese Foundation for the Science and Technology (FCT) under the research project BIOJOINTS (PTDC/EMEPME/099764/2008). The first author expresses her gratitude to FCT for the PhD grant SFRH/BD/40164/2007.

References 1. Ambr´osio, J. and Ver´ıssimo, P., Sensitivity of a vehicle ride to the suspension bushing characteristics. Journal of Mechanical Science and Technology, 23:1075–1082, 2009. 2. Dowson, D., Hardaker, C., Flett, M., and Isaac, G.H., A hip joint simulator study of the performance of metal-on-metal joints: Part I: The role of materials. The Journal of Arthroplasty, 19(8):118–123, 2004. 3. Flores, P., Ambr´osio, J., Claro, J.C.P., and Lankarani, H.M., Dynamics of multibody systems with spherical clearance joints. Journal of Computational and Nonlinear Dynamics, 1(3):240– 247, 2006. 4. Flores, P., Ambr´osio, J., Claro, J.C.P., and Lankarani, H.M., Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies, Lecture Notes in Applied and Computational Mechanics, Vol. 34, Springer-Verlag, Berlin/Heidelberg/New York, 2008. 5. Goenka, P.K., Effect of surface ellipticity on dynamically loaded spherical and cylindrical joints and bearings. PhD Dissertation, Cornell University, Ithaca, New York, 1980. 6. Lankarani, H.M. and Nikravesh, P.E., A contact force model with hysteresis damping for impact analysis of multibody systems. Journal of Mechanical Design, 112:369–376, 1990. 7. Liu, C., Zhang, K. and Yang, L., Normal force-displacement relationship of spherical joints with clearances. Journal of Computational and Nonlinear Dynamics, 1(2):160–167, 2006. 8. Park, J. and Nikravesh, P.E., Effect of steering-housing rubber bushings on the handling of a vehicle. SAE Paper 970103, SAE, Warrendale, Pennsylvania, 1997. 9. Pinkus, O. and Sternlicht, S.A.: Theory of Hydrodynamic Lubrication. McGraw Hill, New York, 1961. 10. Smith, S.L., Dowson, D. and Goldsmith, A.A.J., The effect of diametral clearance, motion and loading cycles upon lubrication of metal-to-metal total hip replacements. Proc. Inst. Mech. Eng. J. Mech. Eng. Sci., 215:1–5, 2001.

Comparison of Passenger Cars with Passive and Semi-Active Suspension Systems Based on a Friction Controlled Damper S. Reich1 and S. Segla2 1Department of Applied Mechanics, Technical University of Koˇsice, 042 00 Koˇsice, Slovakia; e-mail: [email protected] 2 Department of Engineering Mechanics, Technical University in Liberec, 461 17 Liberec, Czech Republic; e-mail: [email protected]

Abstract. The fundamental purpose of suspension system in various kinds of vehicles is the improvement of the ride comfort and reduction of dynamic road-tyre forces leading to better ride comfort and handling properties. In this paper the ride comfort and dynamic road-tyre forces of a passenger car (segment D) with passive suspension system are compared to the same passenger car with semi-active suspension system based on a friction controlled damper. The passenger car is modelled in the general multi-body design program CarSim. For creating the control system of the semi-active friction damper another general program Matlab/Simulink was used. The control strategy is based on the reduced feedback. Parameters of the control algorithm of the semi-active friction damper were optimized for a given road profile using genetic algorithms. The results of numerical simulations show that implementation of the semi-active friction dampers could produce a significant improvement of the ride comfort and handling properties of passenger cars. Key words: passenger car, passive suspension, semi-active suspension, controlled friction damper, ride comfort

1 Introduction Designing vehicle suspension systems is a complex task owing to the necessity to meet several conflicting demands. The passenger comfort should be as high as possible, whilst the dynamic road-tyre forces should be as low as possible to improve handling (road holding) performance and minimize road damage. The same holds for the working space or relative motion between the body of the vehicle (the sprung mass) and the axle with the wheels (the unsprung mass) – it must be minimized or at least limited to a reasonable value. Most suspension units are still passive ones which do not require any external source of power. The vibration isolation characteristics of most passive suspensions are rather limited. Low damping gives good vibration isolation at high frequencies but poor resonance characteristics. Higher damping results in good resonance characteristics but the high frequency performance is poor. By using various kinds of sensors, controllers and active actuators controlled by feedback signals, it is possible to create active suspension systems that can produce

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ideal vibration control for different running conditions. In this case the optimum transmissibility has no resonance amplification and the suspension system performance is superior to any passive suspension system. But they are still more complex, expensive and less reliable than the passive suspensions. Further problems are caused by their high energy consumption (an external source of power is needed) and possible instability. Despite all these disadvantages such suspension systems are already employed in the most expensive and luxurious passenger cars. A compromise between passive and active suspensions are semi-active suspensions. The damping characteristics are controlled for example by modulation of fluid-flow orifices or of dry friction forces based on a control scheme involving feedback variables. For practical reason it is important that the feedback signals be relative displacement and relative velocity across the suspension, since these quantities can be measured directly even for a moving vehicle. The most important control methods are the skyhook control, LQG control, and sliding mode control. These methods are based on varying the system’s damping ratio without changing the system stiffness. There are only a few papers dealing with semi-active vibration control by controlling system stiffness. Magnetorheological (MR) dampers have attracted interest in using controllable actuators for their quick time response and low energy consumption. They belong to the most promising semi-active devices used nowadays in automotive engineering and are employed in a number of passenger cars and other types of vehicles. The key feature of an MR damper is the magnetorheological oil whose rheological properties can be substantially altered by applying a magnetic field. Variable damping can be produced by controlling the magnetic field by the electrical current in a solenoid. This paper deals with semi-active suspension system of a passenger car (segment D) employing a friction controlled damper. A damper of this type is nonconventional in the field of semi-active suspensions as opposed to the MR damper which is more widespread.

2 Friction Model and Semi-Active Control For modelling dynamic friction in the friction semi-active dampers the LuGre model is used. Extensive analysis of the model and its application can be found in [1, 2, 4]. The model is related to the bristle interpretation of friction. The friction force for the friction model with velocity dynamics can be expressed in the form [1] FT = FN µ (v, y) , y˙ = f (v, y) , (1) where FT is the friction force, FN is its normal component (control input), y denotes the average bristle deflection, and v is the relative sliding velocity at the friction interface. Using the control algorithm based on the control strategy with reduced feedback [5], the normal component of the friction force can be expressed in the following

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Fig. 1 Passenger car of segment D in the program CarSim.

⎧ ⎪ ⎨

form FN∗

=

⎪ ⎩

Fact µ (v, y)

if

µ (v, y) · Fact > 0 ,

0

if

µ (v, y) · Fact < 0 ,

(2)

where Fact = K1 z˙1 + K2 z˙2 ,

(3)

z1 is the unsprung mass displacement, z2 is the sprung mass displacement, and K1 , K2 are the feedback gains. However, this control strategy is prone to chatter and sticking. For this reason the control variable FN∗ serves only as input to the next step of control in the form ⎧ ∗ if sgn (µ ) · v > ε , ⎪ ⎨ FN ∗ (FN /ε ) · sgn( µ ) · v if 0 < sgn ( µ ) · v < ε , (4) FN = ⎪ ⎩ 0 if sgn (µ ) · v < 0 . where the boundary layer thickness ε should be appropriately selected. It determines the effective viscous damping coefficient during operation within the boundary layer and, besides, it must be large enough to prevent chatter [5].

3 Passenger Car Models The general purpose multi-body design program CarSim (Mechanical Simulation Corp.) was used to model and simulate a full-passenger car of segment D, Figure 1. Typical representatives of this passenger car segment are e.g. Audi A4, Opel Vectra, and Renault Laguna. The advantage of the program CarSim is its open source capability which enables replacement of built-in passive dampers by dry friction semi-active dampers developed in Matlab/Simulink. The basic mass parameters of the passenger car are as follows: total mass mT = 1370 kg, inertia moment Ix = 606.1 kg·m2 , inertia moment Iy = 4192 kg·m2 , inertia

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Fig. 2 Basic geometrical dimensions of the passanger car (in milimeters).

Fig. 3 Road unevenness “Big Jump Smooth”.

moment Iz = 4058 kg·m2 , where x is the longitudinal axis of the car, y is its lateral axis, and z is the vertical axis. All these axes are passing through the car centroid. The basic geometrical dimensions of the car are presented in Figure 2. Parameters of the car model with passive suspension were obtained experimentally. As for the second car model, its semi-active suspension systems with friction dampers were modelled in Matlab/Simulink. Their time delays were 0.005 s. The vehicle speed is v = 8.3 m·s−1 (30 km/hour), the road unevenness is so called “Big Jump Smooth” (the height of 0.5 m and length of 20 m), Figure 3.

4 Optimization and Results The multicriterial objective function expressing minimization of the vertical centroid acceleration, angular acceleration about the car lateral axis, and road-tyre dynamic forces of the front and rear axles has the following form: fop = 2ρA ( fop,A1 + fop,A2) + ρF ( fop,F1 + fop,F2 ) , where

(5)

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Table 1 Search intervals for design variables. Variable K1F K2F K1R K2R

Lower bound 0 0 0 0

Upper bound 10 000 10 000 10 000 10 000

Table 2 Results of optimization.

 T

fop,A1 =

0

 T

fop,F1 =

0

Variable

Passive suspension

Semi-active suspension

K1F K2F K1R K2R f op

– – – – 850.15

500 9712 2484 9724 321.76

z¨2T (t) dt ,

 T

fop,A2 =

0

(FF (t) − Fstat,F (t))2 dt ,

φ¨y2 (t) dt ,  T

fop,F2 =

0

(FR (t) − Fstat,R (t))2 dt , (6)

zT is the vertical centroid displacement, ϕ y is the angular displacement about the lateral axis of the vehicle, FF is the front road-tyre force, FR is the rear road-tyre force, Fstat,F is the static front road-tyre force, and Fstat,R is the static rear road-tyre force. In order to have the same influence of both parts of the objective function (5), with exception of the coefficient 2 which means that we want greater emphasis on the ride comfort, the value of the weighting coefficient ρ F in the first optimization iteration is fop,A1 + fop,A2 ρF = ρA , (7) fop,F1 + fop,F2 The search intervals for the design variables (the feedback gains of the limited state feedback control) are given in Table 1. Results for minimization of the objective function fop are given in Table 2. In Figure 4 acceleration of the sprung mass centre of gravity is shown for both passive and semi-active dry friction suspension systems. It is apparent that the semiactive friction system offers significantly better ride comfort characteristics. In Figures 5 and 6 road-tyre forces for the front and rear axle are presented. Both these figures show that not only ride comfort, but also handling (road holding) characteristics were significantly improved compared to the passive suspension system.

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Fig. 4 Acceleration of the sprung mass centre of gravity.

Fig. 5 Tyre forces of the front axle.

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Fig. 6 Tyre forces of the rear axle.

5 Conclusions Controlling friction involves revisiting an early-day damping technology and enhancing it in the light of the latest mechatronic advances. In the paper automotive semiactive friction dampers were used and for modelling dynamic friction the LuGre model was used. The passenger car was modelled in the general multi-body program CarSim. Its open source capability enabled replacement of the built-in passive dampers by dry friction semi-active dampers modelled in Matlab/Simulink. The control strategy based on reduced feedback was modified to avoid chatter and sticking. The time delay was 0.005 s. It was shown that the semi-active friction dampers can significantly improve not only the ride comfort but also handling properties of passenger cars (compared with passive suspension systems) by reduction of the sprung mass accelerations and dynamic road-tyre forces.

References 1. Dupont, P., Kasturi, P., and Stokes, A., Semi-active control of friction dampers. Journal of Sound and Vibration, 202(2):203–218, 1997. 2. Gaul., L. and Nitsche, R., Friction control for vibration suppression. Mechanical Systems and Signal Processing, 14(2):139–150, 2000.

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3. Genetic Algorithm and Direct Search Toolbox – User’s Guide. The Mathworks Inc., Natick, 2004. 4. Guglielmino, E. et al., Semi-Active Suspension Control. Improved Vehicle Ride and Road Friendliness. Springer. 2008. 5. Reich, S., Optimization of semi-active dampers in suspension systems. PhD Thesis, Technical University of Koˇsice, 2009 [in Slovak]. ˇ 6. S´aga, M. et al., Selected Methods of Analysis and Synthesis of Mechanical Systems. VTS ZU, ˇ Zilina, 2009 [in Slovak]. 7. Segla, S. and Reich, S., Optimization and comparison of passive, active, and semi-active vehicle suspension systems. In Proceedings of the 12th IFToMM World Congress, Besanc¸on, France, 2007.

Dynamic Balancing of a Single Crank-Double Slider Mechanism with Symmetrically Moving Couplers V. van der Wijk and J.L. Herder University of Twente, Enschede, The Netherlands, and Delft University of Technology, Delft, The Netherlands; e-mail: [email protected], [email protected]

Abstract. This article presents a systematic investigation of the dynamic balancing of a single crank-double slider mechanism of which the coupler links have symmetric motion. From the equations of the linear momentum and angular momentum the force balance conditions and moment balance conditions are derived. Both general and specific force balanced and moment balanced configurations are obtained step by step, are put in perspective, and are illustrated to gain a systematic overview of the balancing possibilities of this type of mechanism. Also the addition of a counter-rotating element is considered. Key words: dynamic balancing, shaking force, shaking moment, crank-slider mechanism

1 Introduction Dynamic balancing has been an issue for many decades [1, 2, 9]. Due to the motion of mechanism elements, shaking forces and shaking moments are exerted to the base, imposing vibrations. In addition to reduced wear, noise, and fatigue [5], with dynamic balancing the accuracy [3, 7] and the payload capacity [4] are improved, and other common advantages due to the reduction of mechanism vibrations apply. Crank-slider mechanisms are often considered for dynamic balancing since their reciprocating motion is a considerable source of shaking forces and shaking moments. Of a wide variety of crank-slider mechanisms a multitude of individual balancing solutions is available [1, 2]. However a systematic investigation of the balancing solutions that covers the full potential of a specific mechanism and that puts the various balancing solutions in perspective, appears to be missing. This article considers a planar single crank-double slider mechanism, meaning a mechanism with a single crank pivoted to the base and with two couplers that are individually jointed to the crank, that are moving symmetrically, and with each having a slider at their extremity. First the force balance conditions are derived and the possible force balancing solutions are shown. Then the moment balance conditions are derived and possible

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l 3A l3 h2 l4

l1 q1

g

h1

m*3

m4

y

-q2 x

l1

h2

g

l4

O

l3 O

*

l2

*

l4 B

a)

b)

l2 l 1

*

l2

q1

-q2 m 3 h1

m*1

m*4

Fig. 1 (a) Initial single crank-double slider mechanism; (b) representation with balancing masses.

moment balancing solutions are illustrated. The research is limited to slider trajectories that are straight.

2 Force Balance Figure 1a shows the initial mechanism. The crank is pivoted to the base at O and driven by θ1 . The couplers with lengths l3 and l4 have joints with the crank at distances l1 and l2 , respectively. The sliders move along a straight line with an offset hi with the origin. The angle between l1 and l2 is determined by γ . The two couplers move symmetrically if l h l1 = 3= 1 l2 l4 h2

(1)

Figure 1b shows the mass of each element. The crank has a mass m∗1 , the couplers have masses m∗3 and m∗4 , and the sliders have masses m3 and m4 . The masses of the crank and couplers are presented as counter-masses (CMs), drawn lumped and at realistic locations, and their positions represent the center-of-mass (CoM) of each element. The positions of the masses can be written in [x, y]T notation as 
∗
 −l1 cos θ1 + l2∗ cos(θ1 − γ ) l1 cos θ1 + l3 cos θ2 ∗ r3 = r1 = −l1∗ sin θ1 + l2∗ sin(θ1 − γ ) l1 sin θ1 + l3 sin θ2 r∗3 r∗4




l cos θ1 − l3∗ cos θ2 = 1 l1 sin θ1 − l3∗ sin θ2




−l2 cos(θ1 − γ ) + l4∗ cos(θ2 − γ ) = −l2 sin(θ1 − γ ) + l4∗ sin(θ2 − γ )




−l2 cos(θ1 − γ ) − l4 cos(θ2 − γ ) r4 = −l2 sin(θ1 − γ ) − l4 sin(θ2 − γ ) 



Dynamic Balancing of a Single Crank-Double Slider Mechanism

415

For force balance, the linear momentum PO of the moving masses must be zero [9]. Without considering the slider constraints, the linear momentum of the mechanism can be written as PO = m∗1 r˙ ∗1 + m3 r˙ 3 + m∗3r˙ ∗3 + m4r˙ 4 + m∗4 r˙ ∗4 (2) ⎤ ⎡ ∗∗ ∗ ∗ ∗ ∗ ˙ ˙ (m1 l1 − m3 l1 − m3 l1 )θ1 sin θ1 + (−m1 l2 + m4l2 + m4 l2 )θ1 sin(θ1 − γ )+ ⎥ ⎢ (−m3 l3 + m∗3 l3∗ )θ˙2 sin θ2 + (m4 l4 − m∗4 l4∗ )θ˙2 sin(θ2 − γ ) ⎥ =⎢ ⎣ (−m∗ l ∗ + m l + m∗ l )θ˙ cos θ + (m∗ l ∗ − m l − m∗ l )θ˙ cos(θ − γ )+ ⎦ 1 1 3 1 1 1 2 4 2 1 3 1 1 4 2 1 (m3 l3 − m∗3 l3∗ )θ˙2 cos θ2 + (−m4l4 + m∗4 l4∗ )θ˙2 cos(θ2 − γ ) PO is zero when each of the following four general force balancing conditions holds. m∗1 l1∗ − m3 l1 − m∗3 l1 = 0 −m∗1 l2∗ + m4 l2 + m∗4 l2 = 0

(3)

−m3 l3 + m∗3 l3∗ = 0 m4 l4 − m∗4 l4∗ = 0

The motion constraints of the sliders make θ2 and θ˙2 dependent on θ1 and θ˙1 by

−1 h1 − l1 sin(θ1 ) (4) θ2 = sin l3 −l cos(θ1 )θ˙1 (5) θ˙2 = 1 l3 cos(θ2 ) Also by including these motion constraints each of the four force balance conditions of Eq. (3) must hold. This is also true for the specific situation when h1 = 0 and l3 = l1 , which make Eqs. (4) and (5) linear with θ2 = −θ1 and θ˙2 = −θ˙1 . Still each of the four conditions of Eq. (3) must hold to have a force balanced mechanism. When γ = 0 or γ = π , terms in Eq. (2) can be combined and as a result the four force balance conditions of Eq. (3) are reduced to just two conditions. For γ = 0 these two force balance conditions are m∗1 (l1∗ − l2∗ ) − (m3 + m∗3)l1 + (m4 + m∗4 )l2 = 0

−m3 l3 + m∗3 l3∗ + m4 l4 − m∗4 l4∗ = 0

(6)

and for γ = π these two force balance conditions are m∗1 (l1∗ + l2∗ ) − (m3 + m∗3)l1 − (m4 + m∗4 )l2 = 0

−m3 l3 + m∗3 l3∗ − m4 l4 + m∗4 l4∗ = 0

(7)

Figure 3 shows the configurations for both γ = 0 and γ = π . In this case the position or value of only two masses are determined for force balance, as opposed to the need of the position or value of three masses for the general configuration of Fig. 1b.

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V. van der Wijk and J.L. Herder

3 Moment Balance For a moment balanced mechanism, the angular momentum of all moving elements is zero [9]. The angular momentum of the mechanism of Fig. 1b can be written as HO = I1 θ˙1 + (I3 + I4 )θ˙2 + r1∗ × m∗1 r˙1∗ + r3 × m3 r˙3

+r3∗ × m∗3 r˙3∗ + r4 × m4 r˙4 + r4∗ × m∗4 r˙4∗  = I1 + m∗1 (l1∗2 + l2∗2 − 2l1∗ l2∗ cos(γ )) + m3 l12 + m∗3 l12 + m4 l22 + m∗4l22 θ˙1 +  I3 + I4 + m3 l32 + m∗3l3∗2 + m4 l42 + m∗4 l4∗2 θ˙2 +   m3 l1 l3 − m∗3 l1 l3∗ + m4 l2 l4 − m∗4 l2 l4∗ θ˙1 + θ˙2 cos(θ1 − θ2 ) = 0 (8)

with I1 , I3 , and I4 being the inertia of the crank and the two couplers, respectively. Under the general force balance conditions of Eq. (3), the third term of HO is zero. However the first two terms cannot become zero individually, meaning that the general configuration cannot be moment balanced. Moment balancing is possible when θ1 and θ2 are linearly dependent, for which there was found one solution when θ2 = −θ1 and θ˙2 = −θ˙1 . Then the mechanism is moment balanced with I1 − I3 − I4 + m∗1 (l1∗2 + l2∗2 − 2l1∗ l2∗ cos(γ )) + m3 (l12 − l32 ) + m∗3 (l12 − l3∗2 ) + m4(l22 − l42 ) + m∗4 (l22 − l4∗2 ) I1 − I3 − I4 + m∗1 (l1∗2 + l2∗2 − 2l1∗ l2∗ cos(γ )) + m∗3 (l12 − l3∗2) + m∗4 (l22 − l4∗2 )

(9) = =0

in which l3 = l1 and consequently from Eq. (1) l4 = l2 were substituted. Figure 4 shows the moment balanced configurations for an arbitrary γ , for γ = 0, and for γ = π. Additional counter-rotation For the moment balance of the general configuration a counter-rotating element, or counter-rotation (CR), can be added. It is proposed to use one of the CMs as CR as shown in Fig. 2, in which m∗4 has become a counter-rotary counter-mass (CRCM) with inertia ICR . It is also possible to add separate CRs at the base, however for low mass and low inertia addition CRCMs are advantageous [8, 9]. The rotation of the CRCM is accomplished with belt (or chain) transmissions. A pulley at O, mounted to the base, drives a pulley at B which is mounted to another pulley at B which drives the CRCM. The diameters of the pulleys are dO , dB,1 , dB,2 , and dCR , respectively. The velocity of the CRCM is determined by the motion of the crank and couplers with

θ˙CR = (1 − k2)ϕ˙ + k2θ˙2 = k1 (1 − k2)θ˙1 + k2θ˙2

(10)

Dynamic Balancing of a Single Crank-Double Slider Mechanism

417

*

l 3A m*3

m4 -q2

dB,1 dB,2

*

dO

g

l4

O

l2

B

l4 qCR

l1

h2

j k2

l3

k1

dCR

*

l1

*

l2

q1

-q2 m 3 h1

m*1

m*4 I*CR

Fig. 2 Configuration with m∗4 being a counter-rotating element for moment balancing.

with transmission ratios k1 = 1 − dO /dB,1 and k2 = 1 − dB,2/dCR and with ϕ˙ = k1 θ˙1 being the rotation of the pulleys at B. For counter-rotations the transmission ratios are negative. The angular momentum of the mechanism including the CRCM is written as HO = I1 θ˙1 + (I3 + I4 )θ˙2 + ICRθ˙CR + r1∗ × m∗1 r˙1∗ + r3 × m3 r˙3 +r3∗ × m∗3 r˙3∗ + r4 × m4 r˙4 + r4∗ × m∗4 r˙4∗   = U + k1 (1 − k2)ICR θ˙1 + V + k2 ICR θ˙2 +   m3 l1 l3 − m∗3 l1 l3∗ + m4 l2 l4 − m∗4 l2 l4∗ θ˙1 + θ˙2 cos(θ1 − θ2 ) = 0

(11)

with U = I1 + m∗1 (l1∗2 + l2∗2 − 2l1∗ l2∗ cos(γ )) + m3 l12 + m∗3 l12 + m4 l22 + m∗4 l22 and V = I3 + I4 + m3 l32 + m∗3 l3∗2 + m4 l42 + m∗4 l4∗2 . The moment balance conditions then become ICR =

−V k2

k2 U −U = ICR (1 − k2) V (1 − k2) 0 = (m3 l3 − m∗3 l3∗ )l1 + (m4 l4 − m∗4l4∗ )l2

k1 =

(12)

For the situation that θ˙2 = −θ˙1 the moment balance condition becomes U − V + (k1 (1 − k2) − k2 )ICR = 0

(13)

or ICR =

I1 − I3 − I4 + m∗1 (l1∗2 + l2∗2 − 2l1∗ l2∗ cos(γ )) + m∗3 (l12 − l3∗2) + m∗4 (l22 − l4∗2 ) (14) k2 − k1 (1 − k2)

The CRCM is not rotating when k2 − k1(1 − k2) = 0, which is for k2 = k1 /(1 + k1). Then the moment balance condition of Eq. (9) is obtained and the CRCM is not

418

V. van der Wijk and J.L. Herder A

a)

b)

l3 *

-l 3 l1 *

q1

*

l 2-l 1

m3

m*3

A l*4

h1

m*4

m*1

l3 *

-l 3 B

l1

l2

O h2

m4

*

l4

-l 4 m*4

l2

m3

m*3

l4

q1

h1

h2 m4

O

l*1+l*2 m*1

B

Fig. 3 Force balanced configurations when: (a) γ = 0; and (b) γ = π ; The position or value of two masses are determined for force balance.

necessary. An advantage of more independent parameters is that the design is more free. Also one of the transmission ratios is superfluous since the angular momentum solely depends on θ˙1 . Figures 5a and 5b show how in this case mass m∗1 is used as CRCM. Depending on the required direction of counter-rotation, a pair of gears as in Fig. 5a or a belt transmission as in Fig. 5b is used. The counter-rotation can be expressed as

θ˙CR = gθ˙1

(15)

with g = dO /dCR for the pair of gears and g = 1 − dO /dCR for the belt transmission. The angular momentum of the mechanism then can be written as  HO = U + gICR θ˙1 + (V ) θ˙2 +   m l l − m∗3 l l3∗ + m l l − m∗4 l l4∗ θ˙ + θ˙ cos(θ − θ ) = 0 (16) 3 1 3

1

4 2 4

2

1

2

1

2

and with θ˙2 = −θ˙1 the moment balance condition becomes ICR =

I1 − I3 − I4 + m∗1(l1∗2 + l2∗2 − 2l1∗l2∗ cos(γ )) + m∗3 (l12 − l3∗2 ) + m∗4 (l22 − l4∗2 ) (17) −g

4 Discussion When the CRCM of the configuration of Fig. 2 is located in joint B or close to joint B, it may become difficult to apply the pulleys. A solution for this is presented in Fig. 5c. The difference with the configuration of Fig. 2 is that from the pulley at B the rotation is one-to-one transferred to, e.g., a pulley at slider m4 (the pulleys have equal diameter). The pulley at the slider drives the CRCM at or nearby B with transmission ratio k2 . For this configuration the balance conditions do not change but are equal to the configuration of Fig. 2.

Dynamic Balancing of a Single Crank-Double Slider Mechanism m*3

a)

l*3

419

b)

c)

A

A

m4

l1

q1

g

l4

l3

l1 m3

O

l2 l 1

B

q1

m*1

* 2

l

l4

m*4

l*4

m*4

l3

q1

m3

O

*

l*4

l l1 B l2

m*1

-l*3

* 4

l3

m*3

l*2-l*1

m4

A

m*4 *

-l 3

m*3 l4

O

m4

m3

* * m*1 l 1+l 2

l2 B

Fig. 4 Moment balanced configurations when θ˙2 = −θ˙1 with: (a) arbitrary γ ; (b) γ = 0; and (c) γ = π. m*3

a)

m*3

b) *

m4

l4 l*4

B

l1

m4

l3

q1

g * l2 l 1

l3

A

O

* 2

l

m*4

m*1 I*CR

m3

l4

l*4

B m*4

* l2 l 1

A

l1

l2

m*1 I*CR

h2 m3

k2

g

l4

m3

l3

l1

l3

O

*

m*3

m4

q1

g

l*3 A

dB,2

*

l3

dO

q1

h1

O *

dB,1

qCR

l2 B dCR

k1

l1 *

l2

m*1

c)

m*4 I*CR

Fig. 5 Moment balanced configurations with an additional counter-rotating element; (a) m∗1 as CRCM driven with a pair of gears for rotation with crank; (b) m∗1 as CRCM driven with a belt for rotation opposite to crank; (c) transmission solution when CRCM m∗4 is near or in B.

The investigation was limited to straight slider trajectories by which the mass moment of inertia of the sliders is not of importance. When slider trajectories are curved, the sliders will rotate and cause shaking moments. In addition, when the CoM of the slider is not at the joint with the coupler, also shaking forces are imposed. In specific situations, curved slider trajectories can be such that the sliders balance each other. Slider trajectories may also be such that the relation between θ1 and θ2 becomes linear, by which specific balancing solutions exist for which l3 = l1 and h1 = 0. These trajectories are called reaction-free paths, which are often included with motion planning of a robot [6].

5 Conclusion The dynamic balancing of a single crank-double slider mechanism of which the coupler links have symmetric motion was investigated systematically. The force balance conditions and moment balance conditions were derived from the equations of

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the linear momentum and angular momentum. For the general configuration only force balance configurations are possible while moment balancing configurations were found for specific design parameters of the mechanism. With an additional counter-rotating element moment balance solutions of the general configuration are possible and for specific specific design parameters of the mechanism, the number of independent design parameters increases for which the design of the mechanisms is more free. All balancing solutions were illustrated, resulting in a systematic overview which is a representation of the balancing potential of this type of mechanisms.

References 1. Arakelian, V.H. and Smith, M.R., Erratum: Shaking force and shaking moment balancing of mechanisms: A historical review with new examples. Journal of Mechanical Design, 127:1035, 2005. 2. Arakelian, V.H. and Smith, M.R., Shaking force and shaking moment balancing of mechanisms: A historical review with new examples. Journal of Mechanical Design, 127:334–339, 2005. 3. Foucault, S. and Gosselin, C.M., On the development of a planar 3-dof reactionless parallel mechanism. In Proceedings of DETC 2002, ASME, 2002. 4. Lim, T.G., Cho, H.S., and Chung, W.K., Payload capacity of balanced robotic manipulators. Robotica, 8:117–123, 1990. 5. Lowen, G.G. and Berkof, R.S., Survey of investigations into the balancing of linkages. Journal of Mechanisms, 3:221–231, 1968. 6. Papadopoulos, E. and Abu-Abed, A., Design and motion planning for a zero-reaction manipulator. In Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 1554–1559, 1994. 7. Raaijmakers, R., Besi zoekt snelheidslimiet pakken en plaatsen op (besi attacks the speedlimit for pick and place motion). Mechatronica Nieuws, 26–31, 2007 [in Dutch]. 8. Van der Wijk, V. and J. L. Herder, J.L., Guidelines for low mass and low inertia dynamic balancing of mechanisms and robotics. In Kr¨oger and Wahl (Eds.), Advances in Robotics Research, Proc. of the German Workshop on Robotics, pp. 21–30, Springer, 2009. 9. Van der Wijk, V., Herder, J.L., and Demeulenaere, B., Comparison of various dynamic balancing principles regarding additional mass and additional inertia. Journal of Mechanisms and Robotics, 1(4):04 1006, 2009.

Evaluation of Engagement Accuracy by Dynamic Transmission Error of Helical Gears V. Atanasiu and D. Leohchi Department of Theory of Mechanisms and Robotics, Technical University “Gheorghe Asachi” of Iasi, 700050 Iasi, Romania; e-mail: {vatanasi, dleohchi}@mail.tuiasi.ro

Abstract. The paper presents an analytical and computer-aided procedure for the prediction of the dynamic transmission error of helical gear pairs by using variable mesh stiffness. An improved model of the mesh stiffness is developed by including specific meshing characteristics of helical gear pairs. The effects of addendum modification coefficients and face-width of gear pairs on the time varying mesh stiffness are investigated. A comparative study has been performed to analyze the effect of the total contact ratio and its components on the dynamic transmission error of helical gear pairs. Key words: helical gears, dynamic transmission error, mesh stiffness, contact ratio

1 Introduction The dynamics characteristics of gear pairs are significant for the design and motion control of transmissions used in the structure of specific servomechanisms. The dynamic transmission error (DTE) can be described as the deviation of the motion of the driven gear from the otherwise nominal motion and is considered as a parameter that is employed to indicate the engagement accuracy of the gear drive [3, 7]. The time-varying mesh stiffness represents the main cause of undesired vibrations in the case of gear transmissions with high manufacturing precision. Many studies have been performed on the mesh stiffness of spur gear pairs, but there are few studies on the helical gear pairs. Cai [2] presented an approximate stiffness function of helical gears, but this model can be used only for a specified range of geometrical parameters of gear pairs. Other reported studies [3–5] have assumed a constant mesh stiffness in the dynamic analysis of helical gears. In this paper, a general model to calculate the cyclic meshing stiffness of helical gears is developed. The time-variation of contact length is included in this model. The effects of some design parameters, as tooth face-width and addendum modification coefficients on the mesh stiffness were investigated.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_49, © Springer Science+Business Media B.V. 2010

421

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V. Atanasiu and D. Leohchi

Fig. 1 The equivalent line of action of a helical gear.

The engagement accuracy under dynamic conditions is a result of the instantaneous contact conditions between active tooth profiles. The paper presents an analytical and computer-aided analysis of the dynamic transmission error of helical gear mechanisms. In order to use the contact ratio as a parameter in gear design, its effect on the dynamic transmission error of helical gear pairs is investigated.

2 Meshing Characteristics The characteristic of helical gears is mainly involved in the inclination of the contact lines. Both, the position and the length of the contact line at time t have a significant influence on the amount of the mesh stiffness of gears. Therefore, it is necessary to establish these parameters with high accuracy. Figure 1 shows some of the relevant features of the meshing plane of action for a pair of helical gears. The meshing starts at the point A, passes through point Et and finishes at point Ex . In the analysis it is useful to consider an equivalent line of action AE as shown in Figure 1. The position of the line of contact is indicated with one of the coordinate X or t of the equivalent line of action, where X shows the meshing position and t is the meshing time: X = Xo pbt , t = Xotz . (1) In the above equations, Xo = 0 ∼ εγ , and tz is the meshing time period of passing a transverse base pitch pbt of the helical gears. At helical gears, the contact length of a tooth pair is not a constant during the engagement cycle. The time-varying of the length of a contact line depends on the ratio between εα and εβ , where εα represents the transverse contact ratio and εβ is the overlap contact ratio. The instantaneous length of a contact line can be expressed as lci = (εα /εβ )(b/ cos βb )Cx , where b is the face-width of a gear pair, βb is the helix

Evaluation of Engagement Accuracy by Dynamic Transmission Error of Helical Gears

423

Fig. 2 Nonlinear vibration model.

angle on the base cylinder, and the parameter Cx = f (εα , εβ ) is shown in [1]. The total contact length Lc at time t is the sum of the contact lengths of each of the tooth pairs.

3 Dynamic Model of a Gear Pair The dynamic model for a gear pair in mesh is shown in Figure 2. In this model, the teeth are considered as springs and the gear blanks as inertia masses. The gear mesh interface is represented by the time-varying mesh stiffness ki (t), the viscous damper c and the composite tooth profile error ei (t). The differential equations of motion can be expressed as J1 θ¨1 + c(θ˙1 rb1 − θ˙2 rb2 )rb1 + ki (t) · (θ1 rb1 − θ2 rb2 )rb1 = T1

(2)

J2 θ¨2 − c(θ˙1 rb1 − θ˙2 rb2 )rb2 − ki (t) · (θ1 rb1 − θ2 rb2 )rb2 = −T2

(3)

where θ1 , θ2 are the rotation angle of the pinion and the driven gear, respectively. J1 and J2 are the mass moments of inertia of the gears, T1 and T2 denote the external torques applied on the gear system, rb1 , rb2 are the base circle radii of the gears, c represents the damping coefficient, and ki (t) is the instantaneous mesh stiffness of a tooth pair and t shows the meshing time. The dynamic transmission error (DTE) along the line of action is defined as δd = rb1 θ1 − rb2 θ2 . The meshing resonance frequency of a gear pair expressed as
1 km fn = (4) 2 π me where me represents the equivalent inertia mass and km is the average mesh stiffness of the gear pair.

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V. Atanasiu and D. Leohchi

4 Time-Varying Mesh Stiffness The deflection f j of the tooth is defined as the displacement of the point of applied load in the direction of the load [1, 6]. The individual tooth stiffness k j can be denoted as k j = F/ f j , where F is the normal tooth load per unit length F = Fn /lci . The mesh stiffness of a gear pair includes the deformation associated with the tooth bending deflection and the contact Hertzian deformation. The time-varying mesh stiffness is mainly caused by the following factors: (i) the variation of the single mesh stiffness along the line of action; (ii) the fluctuation of the total number of total pairs in contact during the engagement cycle. The effects of bending, shear and Hertzian contact deformation are taking into account in the analytical method to calculate the tooth deformation [1, 6]. The mesh stiffness ki corresponding to a contact line is calculated by using an iterative procedure. Thus, the contact line of a pair of teeth is divided into many equal intervals and ki is computed as the integral value of the mesh stiffness k j as follows:  lc

ki =

0

k j · dl

(5)

The teeth pairs in contact act like parallel springs. Therefore, the sum of individual mesh stiffness for all pairs in contact at time t represents the variable mesh stiffness kt (t) and can be written as N

kt (t) = ∑ k ji (t)

(6)

i=1

where N is the total number of teeth pairs in mesh in the same time. The average value of the mesh stiffness km can be expressed as km =

1 tz

 tz N

∑ ki (t) dt

(7)

0 i=1

where ki (t) is the stiffness of the i-th meshing tooth pair.

5 Simulation Results In the analysis, the main categories of the helical gear pairs based on the contact ratio are the following: (a) total contact ratio is less than 2.0; (b) total contact ratio is over than 2.0. The contact ratio of helical gear pairs can be influenced by parameters such addendum modification coefficients and tooth face-width. The addendum modification coefficient is defined as the ratio between the distance from the datum line on the generating rack to the reference diameter of gear and tooth module. For the given center distance and gear ratio results the sum of addendum modification

Evaluation of Engagement Accuracy by Dynamic Transmission Error of Helical Gears

425

Table 1 Specifications of the helical gear pairs. Gear pairs

x1

b [mm]

εα

εβ

εγ

km [N/µm

xm [µm]

fn [Hz]

GP1 GP2 GP4 GP5

0.0 1.2 0.0 1.2

12 12 35 35

1.52 1.14 1.52 1.14

0.39 0.39 1.15 1.15

1.91 1.53 2.67 2.29

258.2 167.4 641.3 377.6

9.2 14.1 3.7 6.3

2712 2203 2991 2311

coefficients as xs = x1 + x2 . The distribution of the sum xs between pinion and gear permits to change the gear diameters and the tip and root thickness of teeth. Specifications of the gear pairs which are selected in the analysis are shown in Table 1, where x1 is the addendum modification coefficient of the pinion and εγ represents the total contact ratio. These characteristics are for helical gear pairs having tooth module mn = 2.5 [mm], center distance a = 90 [mm], number of pinion teeth z1 = 18, number of gear teeth z2 = 51. The nominal torque is T1 = 50 [Nm], the pinion speeds are ω1 = 150 ∼ 500 [s−1 ], and the equivalent error is considered ei = 0. Results given in Table 1 show that the transverse contact ratio is modified by addendum modification coefficients, while the overlap contact ratio is changed by the tooth face-width. A computer program was developed for simulating the dynamic characteristics of helical gear pairs. The equations of motion are solved by the fourth-order Runge– Kutta method. Results given in Figure 3 show the effect of the total contact ratio and its components on the mesh stiffness variation during a mesh cycle. An increase of the transverse contact ratio has as effect the increase of the average mesh stiffness km . The mesh stiffness km is larger with increasing the overlap contact ratio. The relation between them is nonlinear. The engagement accuracy of helical gear pairs is expressed by the dynamic transmission error. Figure 4 shows the variation of this dynamic parameter expressed in a dimensionless form by T E = DT E/xm , where the static displacement xm due to the static load Fn is calculated by xm = Fn /km . In this figure, the dynamic transmission error T E is shown for different values of the pinion speed as a function of the normalized time t/tz , where dotted line is for the pinion speed ω1 = 150 [s−1 ] and solid line is for ω1 = 500 [s−1 ]. In the analysis, even if the meshing frequency fz is constant, the ratio fz / fn is not a constant for the gear pairs GP1 − GP4, because the natural frequency fn varies with the equivalent inertia mass and average mesh stiffness of the gear pairs. The gear mesh stiffness is dependant on the total contact ratio, while the equivalent inertia mass varies with the overlap contact ratio. At the same time, by varying the tooth face-width on a constant torque, the normal load on the length of line of contact is not a constant. These aspects are specific ones in the analysis of helical gear pairs. The effect of transverse contact ratio upon dynamic behaviour of helical gears is more evident

426

V. Atanasiu and D. Leohchi

Fig. 3 Examples of meshing stiffness of helical gears.

when the contact ratio εγ < 2. When εγ > 2, the effect of transverse contact ratio upon dynamic behaviour of helical gears is smaller. The mesh stiffness has a significant effect upon the amount and amplitude of the dynamic transmission error. As shown in Table 1 and Figure 3, the gear pairs having a higher average stiffness km have a lower dynamic transmission error. Generally, the dynamic behaviour of helical gear pairs is improved by increasing the total contact ratio.

6 Conclusions An analytical procedure for calculating dynamic transmission error of helical gears is presented. The amplitude variation of the transmission error under dynamic condition is considered as a major indicator for the engagement accuracy. The following observations should be noted: (1) The influence of contact ratio on dynamic behaviour of helical gear pairs was examined by using the mesh stiffness parameter. The complicated variation of the time-varying mesh stiffness along the line of action is studied by an improved analytical model. The effects of addendum modification coefficients and tooth facewith of gear on the both, contact ratio and mesh stiffness are considered in the analysis.

Evaluation of Engagement Accuracy by Dynamic Transmission Error of Helical Gears

427

Fig. 4 Dynamic transmission errors under different pinion speeds.

(2) Helical gear pairs which have a higher value of the mesh stiffness have a lower level of variation of the dynamic transmission error. (3) The increase of the overlap contact ratio has a favorable effect on the dynamic characteristics of helical gear pairs, especially when this contact ratio is over 1.0. The results presented here can be integrated into the early design stages of a particular helical gear pairs to improve its engagement accuracy under dynamic conditions. The dynamic model takes into account the realistic mesh stiffness and might be used for more accurate control of geared servomechanisms.

Acknowledgement This work was supported by CNCSIS–UEFISCSU, project number 76 PNII–IDEI code 296/2007.

References 1. Atanasiu, V., An analytical investigation of the time-varying mesh stiffness of helical gears. Bul. Inst. Polit. Iasi, XLIV(XLVIII):1–2, S.V., 7–17, 1998.

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2. Cai, Y., Simulation on the rotational vibration of helical gears in consideration of the tooth separation phenomenon. Journal of Mechanical Design, 117:460–469, 1995. 3. He, S. and Singh, R., Dynamic transmission error prediction of helical gear pair under sliding friction using Floquet theory. Journal of Mechanical Design, Transactions of the ASME, 30:052603/1–9, 2008. 4. Kar, Ch. and Mohanty, A.R., An algorithm for determination of time-varying frictional force and torque in a helical gear system. Mechanism and Machine Theory, 42:482–498, 2007. 5. Velex, P. and Ajmi, M., Dynamic tooth loads and quasi-static transmission errors in helical gear – Approximate dynamic factor formulae. Mechanism and Machine Theory, 42:1512–1526, 2007. 6. Weber, C. and Banaschek, K., Form¨anderung und Profilr¨ucknahme bei gerad- und schragverzahten Radern. Schriftenreihe Antriebstechnik, 11, Vieweg Verlag, Braunschweig, 1955. 7. Zhang, Y. and Fang, Z., Analysis of transmissions errors under load of helical gears with modified tooth surfaces. Trans. of the ASME, Journal of Mechanical Design, 119:120–125, 1997.

The Influence of the Friction Forces and the Working Cyclogram upon the Forces of a Robot I. Turcu, C. Birleanu, F. Sucala and S. Bojan Machine Elements and Tribology Department, Technical University of Cluj-Napoca, 40020 Cluj-Napoca, Romania; e-mail: {ioan.turcu, corina.birleanu, felicia.sucala, stefan.bojan}@omt.utcluj.ro

Abstract. The paper presents a calculation method of the forces and the moments that operate in the rotation and translation couplings from the serial industrial robots structure. Through using some methods and calculation programs, the forces and moments, used for dimensioning the mechanisms and driving motors from couplings, can be accurately determined. The theoretical relations and the calculations programs also take into consideration the friction forces acting in couplings and the working cyclogram. Key words: friction force, working cyclogram, driving axe, friction coefficient, driving force

1 Introduction Through the working cyclogram of a robot the authors of the paper define precisely the trajectory of the end efector in the workspace together with the time parameters, velocities and accelerations. The cyclogram can be transposed graphically or as a spreadsheet. An important problem that must be solved with designing an industrial robot is the correct dimensioning of motors and acting systems. No mater the motor type, its size is established on the basis of the maximum power needed during the working process. With the translation journals the axial force and the linear velocity determine the needed power, and with the rotation ones the torque and the angular speed. For a certain working regime imposed to the robot, the accurate knowledge of the forces and moments from the driving couplings becomes the main unknown for the motors and afferent acting systems dimensioning and checking. The specialist literature of the robots defines these parameters (the force and moment) “generalized driving forces” and presents a series of their calculation methods. It can be established that the majority of papers [3, 4], present methods and calculation programs, which consider the translation and rotation couplings as “ideal” couplings without friction or they determine the friction forces with approximate calculation relations. Taking into consideration the relations presented in [1, 3], together with the modeling methods recommended in the case of the serial type robots structure,

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more accurate calculation of the generalized driving forces, respectively of powers, can be made. On the basis of these forces and moments, the afferent motors and mechanisms will be chosen. Which are the needed data for this kind of calculation? The cinematic scheme of the robot structure; the constructive dimensions and the masses of the component elements from the mechanical structure; the referring data to the final efector load; the working cyclogram (or working cyclograms if it is an optimization problem); the friction coefficient.

2 Theoretical Considerations The main stages of the calculation algorithm are: (1) The cinematic scheme is made (Figure 1), represented at zero configuration. (2) The parameters of type DM are calculated for each element of the respective structure. (3) Through the application of the MGD (direct geometrical modeling) and MCD (direct kinematical modeling) algorithms from [3], the matrixes of the generalized parameters are determined. (PG) and (DH) of the omogen transformations, cinematic parameters (linear velocities and accelerations, respectively angular speeds and accelerations), differential matrixes, Jacobean matrix together with the transposed and the reverse ones. (4) Through iterations the forces and the moments of the connection forces, as well as the driving generalized forces are determined. In conformity with the designations from [3] the resultant force and the moment from each coupling are established as follows: the Newton equation or the theorem of the masses center motion and the Euler dynamic equation or theorem of the kinetic moment with respect to the masses center, applied in this iteration method is:
 i ¯ Fi = ∆θ Mi i v˙¯i +i ω˙¯ i xi r¯C +i ω¯ i xi ω¯ i xi r¯C + Mii [R]0 v˙¯0 ; i i r¯ Ci

N¯ i = ∆θ ·

i

i

Ii∗ · iω˙¯ i +i ω¯ i xi Ii∗ ·i ω¯ i



i

0

.

(1)

is the position vector of the mass center with respect to {i}; Mi is the element’s

mass and i Ii∗ is the inertial tensor of the link (i) related to {i} originating in the mass center;   j n 2 1 − ∆ ∆ m m i ¯ ·i [R]n+1 f¯n+1 + fi = (−1)∆m ∏ ik [R] j F¯j (2) 1 + 3.∆m n+1 2 − ∆m1 ∑ j=i k=i i

n¯ i = (−1)∆ m

1 − ∆m i { [R]n+1 n¯ n+1 +i r¯n1 xin xin+1 [R]n+1 f¯n+1 } 1 1 + 3∆m n+1

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Fig. 1 Cinematically scheme of the robot.

+

∆m2 2 − ∆m2

n

∑

j j ¯ j ¯ i j [R]( r¯C ji x Fj + N j ).

(3)

j=1

With these relations the reactions from the driving journals of the robots are determined, using a calculation program that considers the couplings without friction, “ideals”.

3 The Robot Presentation For a better understanding of the study method, one considers a certain structure of a robot destinated to effect a technological process. The proposed structure, presented in Figure 1 contains four driving journals, three translations and one of rotation (3TR or TTTR). The proposed technological process, materialized through the working cyclogram, needed the displacement of a load (a certain weight) from a point in the other point in space and the rotation in a certain plane with a given angle. The displacement is made by the three translation couplings on the distances l1 = 2600 mm, l2 = 1300 mm and l3 = 1040 mm and the rotation in the vertical plane is effected by the rotation journal with the angle θ4 = 180◦ . The value of the load (an important parameter referring to the designing data) was considered equal to 300 N. Variable is considered the time in which the whole process must be realized, as well as the working cyclograms for each driving coupling.

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4 The Method and the Calculation Stages Considering less friction couplings for the robot structure presented in Figure 1 we can determine the forces and moments that operate inside the couplings with the help of the next relations [3]: • the components of the forces on the Cartesian system axes: ⎤ ⎡4 ⎤ ⎡ f4x M4 · [(q¨1 + g) · sq4 + q¨2 · cq4 − 4 xC · q˙24 ] 4 ⎥ ⎢4 ⎥ ⎢ ⎣ f4y ⎦ = ⎣ M4 · [(q¨1 + g) · cq4 − q¨2 · sq4 + 4 xC4 · q˙24 ] ⎦ 4

(4)

M4 · q¨3

f4z

  ⎤

M3 + M4 · q¨2 − M4 · 4 xC · q¨4 · sq4 + q˙24 · cq4 4  ⎥ 

⎢3 ⎥ ⎢ ⎣ f3y ⎦ = ⎣ M3 + M4 · q¨1 + g + M4 · 4 xC4 · q¨4 · cq4 − q˙24 · sq4 ⎦ (5) 

3f M3 + M4 · q¨3 3z

 ⎤ ⎡2 ⎤ ⎡ − M3 + M4 · q¨3 f2x



 

⎢ 2 ⎥ ⎢ M + M + M · q¨ + g + M · 4 x · q¨ · cq − q˙2 · sq ⎥ ⎦ (6) ⎣ f2y ⎦ = ⎣ 2 4 3 4 4 C4 4 4

1 4

4 2 2f M2 + M3 + M4 · q¨2 − M4 · xC · q¨4 · sq4 + q˙4 · cq4 2z 4

 ⎡1 ⎤ ⎡ ⎤ M3 + M4 · q¨3 f1x

 

⎢1 ⎥ ⎢ ⎥ M2 + M3 + M4 · q¨2 − M4 · 4 xC · q¨4 · sq4 + q˙24 · cq4 ⎣ f1y ⎦ = ⎣ ⎦ 4

  

4 x · q¨ · cq − q˙ 2 · sq 1f M · q ¨ + M + M + M + g + M · 4 1 2 3 4 1 4 C4 4 4 4 1z (7) • the components of the moments on the Cartesian system axes: ⎡3

f3x

⎤

⎡

⎡

⎤

⎡

⎤ 0 ⎢ ....... ⎥ ⎢ .......................... ⎥ ⎢4 ⎥ ⎢ ⎥ 4x · 4 f ⎢ n ⎥ ⎢ ⎥ C4 4z ⎢ 4y ⎥ = ⎢ ⎥ ⎣ ........ ⎦ ⎣ ........................... ⎦ 4 I ∗ · q¨ + 4 x · 4 f 4n z 4 C 4y 4z 4n

4x

(8)

4

⎡





 ⎤ − 3zC · M3 · q¨1 + g 3 3x ⎢ ⎥ ⎢ ....... ⎥ ⎢ ............................................................................................... ⎥   ⎢3 ⎥ ⎥ ⎢ ⎢ n ⎥ = ⎢ −4 x · 4 f · cq + l · 4 f · cq − 4 f · sq + M · 3 z · q¨ ⎥ ⎢ 3y ⎥ ⎢ ⎥ C4 4z 4 3 4x 4 4y 4 3 C3 2 ⎥ ⎣ ........ ⎦ ⎢ ⎣ ................................................................................................ ⎦ 3 n3z 4 I ∗ · q¨ + 4 x · 4 f z 4 C4 4y (9) ⎡

3n

⎤

4 x · 4 f · sq − l 4 f · sq + 4 f · cq C4 4z 4 3 4x 4 4y 4

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 ⎤ −4 xC · 4 f4y − 4 Iz∗ · q¨4 − l2 · 3 f3y − M2 · 2 zC · q¨1 + g ⎡2 ⎤ 4 2 n2x ⎢ ............................................................................ ⎥ ⎥   ⎢ ......... ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 4 x · 4 f · cq + l · 4 f · cq − 4 f · sq ⎢ ⎥ + − ⎢ ⎥ ⎢ C4 4z 4 3 4x 4 4y 4 ⎥ ⎢ 2 ⎥ ⎢ ⎥ 3 3 3 ⎢ n ⎥ +M3 · zC · q¨2 + f3x · q3 − f3z · l2 ⎥ (10) ⎢ 2y ⎥ = ⎢ 3 ⎢ ⎢ ........ ⎥ ⎢ .............................................................................. ⎥ ⎥ ⎢ ⎥ ⎢   ⎥ ⎣ ⎦ ⎢ ⎥ 4 x · 4 f · sq − l · 4 f · sq + 4 f · cq − ⎣ ⎦ C 4z 4 3 4x 4 4y 4 2n 4

 2z −M3 · 3 zC · q¨1 + g − 3 f3y · q3 3 ⎡   

 ⎤ 4 4 4 ∗ ⎡1 ⎤ xC · f4y + Iz · q¨4 + l2 + q2 · 3 f3y + M2 2 zC + q2 · q¨1 + g n1x 4 2 ⎢ ⎥ ⎥ ................................................................................................ ⎢ ........ ⎥ ⎢ ⎢ ⎥   ⎢ ⎥ ⎢ 4

 ⎢ ⎥ ⎢ x · 4 f · sq − l · 4 f · sq + 4 f · cq + l − q · 3 f − ⎥ ⎥ C 4z 4 3 4x 4 4y 4 1 3 3y ⎢ 1 ⎥ ⎥  4 ⎢ n ⎥ ⎢  ⎢ ⎥ 1 3 1y ⎢ ⎥ = ⎢ − M · x −M ·l +M · z · q ¨ + g ⎥ 1 C1 2 1 3 C3 1 ⎢ ........ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ................................................................................................ ⎥   ⎢ ⎥ ⎢ 4

 ⎣ 1 n ⎦ ⎢ − x · 4 f · cq + l 4 f · cq − 4 f · sq − l − q · 3 f − ⎥ ⎥ C 4z 4 3 4x 4 4y 4 1 3 3x 1z ⎣ ⎦    3 

4 3 − l2 + q2 · f3z + −M2 · l1 + M3 · zC · q¨1 + g 3 (11) ⎡

In conformity with the theories applied in the robots field, the Cartesian coordinates systems are thereby oriented that the Oz axe is the axe across it overlaps the velocity vector respectively the force’s vector or the acting moment vector. If we take into consideration the friction in the coupling, the relations of the forces and acting moments are [4, 5]:   if + µ r i ixy   r Qim = i niz + sgn(ϖ i ) (12) +µiz ρi f i  fiz  + 3µi l i nixy i

 

 Qim = i fiz z  + sgn viz { µx

        i f + fix  + a3 i niz  +  iy   i   + µi y   + li3 i niy  + l3 i nix 

 i

(13)

i

with a, r and l denoting the nominal constructive dimensions; µ is the friction coefficient which corresponds with the cylindrical surface; µz is the friction coefficient, which corresponds with the frontal plane surface; nz is the driving torque in driving axes without friction; and µx , , µy are the friction coefficients which correspond with the plane surface.

5 Results and Conclusions To demonstrate the influence of the working cyclogram on the forces and the acting moments inside the robot’s couplings in a dynamic regime we take into consideration two working variants: tttr and 3tra. The two variants differ through: the traject-

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I. Turcu et al. Table 1 The parameters type DM of robot 3TR at zero configuration θ¯ (0) . i

1 2 3 4

Mass

Masses center

Mi [kg]

0

41.09 207.8 108.5 32.46

–74.5 –190 480.4 1210

xCi

0 yCi [mm]

0.00 0.00 1500 1393

Mechanical inertial moments 0

zCi

95 95 95 95

i ∗ Ix

i ∗ Iy

0.57 48. 18.9 0.014

0.978 47.93 18.87 0.347

i ∗ i ∗ Iz Ixy ×106 [kg·mm2 ]

0.012 0.623 0.186 0.347

0.0 0.0 0.0 0.0

i ∗ Iyz

i ∗ Izx

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

Fig. 2 Version tttr – the driving forces variation in the four couplings.

ory made by the end efector, velocity (q), ˙ acceleration (q) ¨ and the total time to run over the trajectory. So, for tttr variant, the total time is 29 seconds and for the 3tra variant the total time is 19,5 seconds. For the considered robot we determined the DM parameters taking into consideration the zero configurations, which are presented in Table 1. For the calculation of the forces and moments inside the robot’s couplings a calculation program, called SimMecRob, has been utilized presented in [1, 2]. The results are presented in two ways: • graphical (direct diagrams designed by computer, where it is indicated only the minimal and maximal value) (Figures 2 and 3): • numerical (tables with values at different time intervals or only maximal and minimal values).

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Fig. 3 Version 3tra – the driving forces variation in the four couplings. Table 2 The parameters type DM of robot 3TR at zero configuration θ¯ (0) . Cyclogram tttr

3 tra i = 1 → 4i

f iz [N] niz [N.m]

Min Max Dif Min Max Dif

1

2

3

4

1

2

3

4

3782 3903 121 –5.9 12.0 17.9

–36.0 17.2 53.2 –1194 284.0 1479

–11.7 5.6 17.3 –34.0 34.4 68.4

–2.6 1.2 3.8 –34.0 34.4 68.4

2810 4837 2027 –288 568 856

–604 362 966 –1182 296 1478

–385 195.4 580.7 –43.0 36.0 79.0

–88.7 44.9 133.6 –43.0 36.0 79.0

Taking into consideration a value of the friction coefficient recommended by technical literature, µ = 0.1235 [2], and with the relations (12, 13) it has been calculated the force respectively the acting moment. From the analyses of the results we can draw conclusions and make some recommendations: (a) for the same mechanical structure and the same technological process the forces and moments essentially differs function the cinematically and dynamic parameters, as well as the imposed trajectories; (b) it also results considerable differences for the two situations, that in which the coupling is considered without friction and the other when the coupling is considered with friction,

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(c) an accurate dimensioning of the acting motors can be made only if friction from couplings is also considered. (d) the acting motor must be chosen according to the relation (12 or 13) where we take into consideration the maximum values of the niz and fiz from Figures 2 and 3 or Table 2.

References 1. Husty, M.L. and Gosselin C., On the singularity surface of planar 3-RPR parallel mechanisms. Mech. Based Design of Structures and Machines, 36:411–425, 2008. 2. Minami, M., Ikeda, T. and Takeuchi, M., Dynamical model of mobile robot including sliping of carryng object, International Journal of Inovative Computing, Information and Control, 3(2):353–369, April 2007. 3. Negrean, I., Vuscan, I. and Haiduc, N., Robotics – Kinematic and Dynamic Modelling, Editura Didactica si Pedagogica R.A., Bucuresti, 1998. 4. Turcu, I., Belcin, O. and Bojan, S., Frecarea in cuplele de rotat¸ie s¸i translat¸ie la robot¸i, Editura Risoprint, Cluj-Napoca, 2003. 5. Turcu, I., Birleanu, C. and Bojan, S., Studies concerning the influence of the friction forces upon the acting forces to a serial robot with four mobility degrees. In Proc. Fourth Intern. Congres on Mechanical Engineering Technologies ’04, Varna, Bulgaria, September, Vol. 5, pp. 200–203, 2004.

Dynamic Aspects in Building up a Flight Simulator A. Pisla, T. Itul and D. Pisla Technical University of Cluj-Napoca, 4001141 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. A very tempting and demanding application within the robotic field concerns the flight simulators. The definition of a flight simulator given by Collins Dictionary is: “A ground-training device that reproduces exactly the conditions experienced on the flight deck of an aircraft” [1]. The computer based flight simulators still take advantage of the human organ illusion. When the pilots turn any direction in flight simulator, the motion base supports the real bank to some extent. The dynamic analysis of the flight and the dynamic behavior of the simulator are crucial for identifying the differences between fly and simulation stress imposed on the structure and to determine the adequate generalized force value in the control. Therefore in this paper the generalized forces to induce a controlled acceleration in the cockpit frame are determined and the fitting diagrams are depicted. Key words: helicopter flight simulator, dynamics, parallel structure, simulation

1 Introduction The research for improvement of the flight simulators envisaged the D class simulators concerning the dynamic behavior imposed by: • aerodynamics development using revolutionary concepts, • redesign engines – with reduced fuel consumptions, noise and air pollution (possible diesel engine and jet solutions), • performance limits – for good and safety piloting, • excellent ride control – with application of active and adaptive control methods. In extreme situation, the “pilot” may suffer over 6 g accelerations (in our case 2 g). The acceleration conditions are created in a large masses environment (in our case almost 2000 Kg), the inner structures must resist to the inertial forces and moments imposed by the accelerated motion. Each structural element and each connection must be modeled, analyzed and tested to fully correspond to the stress conditions. Moreover it is considered that on the motion platform can be placed up to seven persons that are “manipulated” with high speed and high accelerations.

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2 Functional Modeling The dynamic modeling is not only used for the motion platform control but also to test the structural frame of the cockpit. The “hanging” of the structure in the real case must be replaced by the platform forces transmission, therefore a totally different distribution of forces and stress is achieved. The well known simulators with motion are using a special made frame that replicate only inside the cockpit interior considering that the real spacecraft structure is not capable to transmit the forces in this way without material fractures. The motion restrictions of mobile platform and the absence of simulated of gravity (G) are the current limitations of a simulator. For commercial flights the pilots do not feel the need of simulated G, since a 60 degree inclination angle, which causes 2G acceleration, the maximum bank to minimize the inconvenience of passengers in commercial flights [12]. In case of the helicopter flight the situation is different; the rapid change in direction may create easily accelerations over 2G. This situation must be replicated on the flight simulator. The force determination considering friction was made in [9, 11]. The force-motion simulation, to generate the motion commands requires the inverse kinematics, according to the orientation given in Figure 1. Generally the real cockpit structure is only used when static flight simulators are designed (Figure 2, left), in motion always a special cabin with the reconstruction of the inner elements is designed (Figure 2, right).

Fig. 1 Motion orientation.

Fig. 2 Simulator cockpit.

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Fig. 3 The 3D model of the helicopter resistance structure and the concentrated masses.

It is necessary to make the stress analysis of the studied helicopter skeleton (Figure 3, left), in flight and for the simulated conditions, considering Stewart platform as a parallel manipulator whose 6-DOF end-effector is connected to a base of the skeleton. The deformations range and evolution are visualized by interfacing the dynamic model with a generated solid model, where the concentrated masses are equally distributed in the frame nodes (Figure 3, right). The dynamic modeling is important to determine the forces that must be induced in order to obtain the dynamically desired conditions.

3 Dynamic Modeling Helicopter Flight Simulators (HFS) could be used by the aviation, the military for training, for disaster simulation and helicopters components development. The different types of flight simulators range from video games up to full-size cockpit replicas mounted on hydraulic, electric or electromechanical actuators [1, 14]. The modern simulators are used to normal and emergency procedures, for situations that are unable to safely do in a real aircraft like loss of flight surfaces and complete power loss. In all cases dynamics behavior plays a very important role. In [3] an analytical study of the kinematics and dynamics for the Stewart platform-based machine tool structures is presented. It is estimated that the forces due to the friction could represent a certain part from the forces/torques which are necessary for the manipulator movement in typical situations. In this paper the inverse dynamical model with friction of a 6-DOF parallel structure for helicopter flight simulation is used starting from the Newton–Euler equations [4–7]. Also Riebe and Ulbrich [13] used a dynamic model for a Stewart platform with six DOF, based on Newton–Euler equations including the frictional behavior optimizing the parameters describing the friction model. The kinematic model yields through the differentiating of the input-output equations of the mechanism [8]:

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Fig. 4 One kinematic chain.

[A] · X˙¯ = [B] · q˙¯

(1)

where  T ⎤  P¯1 − P¯ × e¯1 ⎥ ⎢  ⎢ e¯T P¯ − P¯  × e¯ T ⎥ ⎥ ⎢ 2   2 2 ⎥ ; [B] = I , =⎢ 6 ⎥ ⎢ .. .. ⎥ ⎢ . . ⎦ ⎣  T  e¯T6 P¯6 − P¯ × e¯6     = v¯ ω¯ T = X˙ Y˙ Z˙ ωX ωY ω Z T ,   = q˙1 q˙2 . . . q˙6 T , ⎡

[A]

X˙¯ q˙¯

e¯T1

and e¯i is the unit vector of the Ai Bi direction leg (Figure 4). The relation (1) could be written in the form: ˙¯ v¯ = [Jv ] · q;

ω¯ = [Jω ] · q˙¯

(2)

The dynamic model that includes the friction forces or torques evaluation start from the frictionless model [10] upgraded with the model of the viscous friction together with the model of Coulomb friction selected used in [2], considering that the necessary torque from a rotation passive joint to win the friction has the relative velocity

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Fig. 5 Dynamic modeling of the parallel platform.

sense (Figure 4). Using the Newton–Euler equations the drive generalized forces yields (Figure 5): ⎡ ⎤ ⎤ ⎡ Q1 ma¯ + M a¯c − (m + M) g¯ ⎢ ⎥ ⎥ ⎢ ⎢ Q2 ⎥ ˙¯ ⎥ ⎢ ⎢ ⎥ −T ⎢ ([Io ] + [Ic ]) · ω + ⎥ (3) ⎢ . ⎥ = [A] · ⎢ ⎢ .. ⎥ ¯ × (([Io ] + [Ic ]) · ω¯ ) + ⎥ ω + ⎦ ⎣ ⎣ ⎦ ¯ + ([R] · p¯c ) × M (a¯c − g) Q6 where Qi is the drive generalized force from the Bi Ai leg; m, M are the masses of the mobile plate and the helicopter cockpit & pilots; g¯ = [0 0 −9.806]T is the gravitational acceleration; p¯c = oc is the position vector of the cockpit mass centre; a¯c is the cockpit mass center acceleration, [Io ] , [Ic ] are the the inertia tensors. The necessary torque to win the friction in the passive joints is: M f i j = c j ωi j + µ j

dj   Q  sgn(ω ); ij 2 i

i = 1, 2, . . . , 6; j = 1, 2, . . . , 5

(4)

where c j is the viscous coefficient from rotational joint j; µ j is the Coulomb friction coefficient from rotational joint j; d j is the diameter of the j joint pin. In the active prismatic joint the necessary force for win the friction could be given through the relation:   Q fim = ct q˙i + µt Qi  sgn(q˙i ); i = 1, 2, . . . , 6 (5) where ct is the viscous coefficient from active joint; µt is the Coulomb friction coefficient from the same joint. In order to get the generalized drive forces which are necessary to overcome the friction from the passive joints, the principle of virtual power is applied:

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Q f p · q˙¯ = ∑

5

∑ M f i j · ωi j

(6)

i=1 j=1

This leads to the force expressions for each leg: ⎤ Q f1p

p ⎢Qf ⎥ 6 5 6 5 ⎢ 2⎥ T T ¯ ⎢ . ⎥ = [Jv ] · ∑ ∑ M fi j · a¯i j + [Jω ] · ∑ ∑ M fi j · bi j ⎣ .. ⎦ i=1 j=1 i=1 j=1 Q f6p ⎡

(7)

Finally the total active generalized forces are: m p Qm i = Qi + Q f i + Q f i + ms q¨i

(8)

where ms is the leg upper part mass.

4 Results For the simulation three simple motions along the reference system axes with the input harmonic laws have been considered: X −X:

X = 0.5 sin(2π t); Y = 0; Z = 1.8; α = β = γ = 0

Y −Y :

Y = 0; Y = 0.5 sin(2π t); Z = 1.8; α = β = γ = 0

Z−Z:

√ X = 0; Y = 0; Z = 0.5 sin(2π t); Z = 1.8 + 0.5 sin( 2π t); α = β = γ = 0

Fig. 6 The Gough–Stewart platform.

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Fig. 7 The resulted generalized forces for the control.

The input data, considering also the inertia and the friction data are corresponding to the real platform located in the Systems Dynamics Simulation Laboratory within the Technical University of Cluj-Napoca [14] (Figure 6):

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m = 300 kg;

M = 1500 kg;

ms = 25 kg;

d j = 0.024 m;

Ix = Iy = 37.5 kg m2 ;

Iz = 75 kg m2 ;

Ixc = 384 kg m2 ;

Iyc = Izc = 624 kg m2 ;

µt = tan(3◦ ) = 0.05241;

ct = 0.005 Ns/m; c j = 0.003 Nms/rad;

µj =

4 µ π t

In Figure 7 the diagrams of the total generalized forces from the legs 1, 3 and 5 for the harmonic motions on the three axes are presented. In the Z–Z diagram is revealed the good superposition of the identified curves.

5 Conclusions The paper presents some aspects concerning the dynamics of a Stewart–Gough parallel platform for helicopter simulation in order to induce controlled forces in the monitorized testing of the stresses in the simulator structure. Using a numerical and graphical simulation, the diagrams for the dynamics representation are computed. The identified algorithms offer the possibility of more dynamic studies. The identified forces expressions will be used directly in the simulation control in order to avoid critical situations and to ensure the safeties of the device.

Acknowledgements The authors gratefully acknowledge the financial support provided by the research grants awarded by the Romanian Ministry of Education, Research, Youth and Sport.

References 1. Andreev, A.N. and Danilov, A.M., Information models for designing, conceptual broad-profile flight simulators. Measurement Techniques, 43(8), 2000. 2. Cobuild, C., Advanced Learner’s English Dictionary: Paperback with CD-ROM, Amazon Publishing House, 2006. 3. Craig, J.J., Introduction to Robotics. Mechanics and Control, Addison-Wesley, 1989. 4. Harib, K. and Srinivasan, K., Kinematic and dynamic analysis of Stewart platform-based machine tool structures. Robotica, 21(5):541–554, 2003. 5. Itul, T., Pisla, D. and Pisla, A., Dynamic model of a 6-DOF parallel robot by considering friction effects. In Proceedings of 12th IFToMM World Congress, Besanc¸on (France), June 18–21, 2007. 6. Itul, T. and Pisla, D., The influence of friction on the dynamic model for a 6-DOF parallel robot with triangular platform, Journal of Vibroengineering, 9:24–29, 2007. 7. Merlet, J.-P., Parallel Robots. Kluwer Academic Publisher, 2000.

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8. Nahon, M.A. and Gosselin, C.M., A comparison of flight simulator motion – Base architectures. Journal of Mechanical Design, 122, 2000. 9. Pisla, D.L., Kinematic and Dynamic Modeling of Parallel Robots. Dacia Publisher, 2005 [in Romanian]. 10. Pisla, D.L., Itul, T.P., Pisla, A. and Gherman, B., Dynamics of a parallel platform for helicopter flight simulation considering friction. In Proceedings of the 10th IFToMM International Symposium on Science of Mechanisms and Machines, SYROM’09, Brasov, Romania, 2009. 11. Pisla, A., Plitea, N. and Prodan, B., Modeling and simulation of parallel structures used as flight simulators. In: Proceedings of TMT2007, Tunisia, 2007. 12. Plitea, N., Pisla, A., Pisla, D. and Prodan, B., Dynamic modeling of a 6-dof parallel structure destinated to helicopter flight simulation. In Proceedings of ICINCO 2008, Madeira, 2008. 13. Reid, L.D. and Nahon, M.A., Response of airline pilots to variations in flight simulator motion algorithms. Journal of Aircraft, 25(7):639–646, 1988. 14. Riebe, S. and Ulbrich, H., Modeling and online computation of the dynamics of a parallel kinematic with six degrees-of-freedom. Archive of Applied Mechanics, 72:817–829, 2003. 15. Motion Technology. Available at: http://www.boschrexroth.com/business units/.

Applications and Teaching Methods

Robotic Control of the Traditional Endoscopic Instrumentation Motion M. Perrelli, P. Nudo, D. Mundo and G.A. Danieli Dipartimento di Meccanica, Universit`a della Calabria, 87036 Arcavacata di Rende (CS), Italy; e-mail: {michele.perrelli, paola.nudo, d.mundo, danieli}@unical.it

Abstract. This paper deals with the development of a special 3+1 Degrees of Freedom (DOFs) end effector used to control the standard laparoscopic instrumentation’s positioning and actuation during laparoscopic surgery. Nowadays, the robotic system da Vinci is the most used robot in the laparoscopic surgery field. The cost of this robot is very high, which limits its spread in hospitals. Moreover, this robot can use only customized tools, called endoWrist, to perform the laparoscopic surgery. This instruments are, however, very expensive and can be used only for few operations, increasing the required economical efforts. The aim of our research is to create an end effector, supported by a navigator arm, able to manipulate standard laparoscopic instruments in order to reduce the cost of the robotic surgery, while keeping most of the advantages introduced by the other robotic systems. The particular structure of the end effector proposed here makes it possible to create a virtual spherical wrist at the insertion point. This solution allows to have the mobility needed to perform the surgery and to avoid excessive stress to the patient skin. The main disadvantage is represented by the DOFs of the standard laparoscopic tools that are less than in the endoWrists. However, the proposed end effector allows to manipulate standard laparoscopic tools smaller than customized robotic instruments, which is a fundamental feature in neonatal and paediatric surgery. Key words: robot-assisted surgery, least invasive surgery, standard tools control, virtual spherical wrist

1 Introduction Least invasive surgery is taking more and more momentum as a good method to reduce invasiveness, recovery times and, consequently, global costs of surgical procedures, minimizing also related risks [8, 10, 14, 15]. In the case of traditional laparoscopic surgery, the laparoscopic tools control may not be easy, especially if the movements required are very fine. After robots have been introduced in the surgical rooms many problems were overcome [2, 9, 11, 16]. The robotic system allows to make finer movements than surgeons could do manually, removing problems as the physiological hands tremors [1] and reducing also the blood loss [17].

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Fig. 1 (a) Complete structure of Navi-Robot; (b) mobility degrees of a Navi-Robot’s arm.

However, the cost of surgical robots as the da Vinci is very high. Furthermore, the endoscopic instruments, called endoWrist, are very expensive and even usable for no more than ten operations. Moreover, in some surgical treatments, the benefit/cost ratio is not high enough to justify the use of surgical robots [13]. This originated the idea [5, 6] to develop an end effector allowing the same quality of motion control, using existing endoscopic instruments, which would allow a greater diffusion in hospitals. Furthermore, a robot able to manipulate standard instruments even allows to use small size tools, 2–3 mm in diameter versus 5–10 mm of endoWrist. This feature is very useful in the case of neonatal and paediatric surgery, where body structures are small. Our end effector is designed to be supported by a three arm system called Navi-Robot [3, 4, 12], shown in Figure 1. The Navi-Robot is a hybrid parallel/serial robot born to be used in orthopaedics. It is composed by three 6 DOFs self-balanced arms, see Figure 1b. Thanks to the balancing weight, the vertical motion results practically weightless and this feature makes the manual motion of the entire arm in navigation mode very comfortable. The central arm of the Navi-Robot is able to switch between navigation mode and robotic mode. Each joint of this arm is composed by two brakes, an actuator and a position sensor. One of this two brakes is used to freeze the position of the single joint when needed, while the other one is used to switch from robot to navigator mode and vice versa. The two lateral arms of the robot are only usable as navigators. The first five joints are equipped with a position sensor and a braking system to freeze the arms in the desired configuration. The sixth joint of the two navigator arms is identical to the joints of the central arm, so that the Navi-Robot can be used both in orthopaedic and in laparoscopic surgery just connecting different end effectors through an appropriate connecting system located at the end of the kinematic chain, as shown in Figure 1a. In orthopaedics, the lateral arms are used as measuring systems and then the last joint is used in navigation mode [4, 12]. When the Navi-Robot is used in laparo-

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Fig. 2 Scheme of structure for fixed point rotation.

scopic surgery, instead, this joint works in robot mode giving one DOF that, together with the 3+1 DOFs given by the end effector, allows orienting and positioning correctly the standard laparoscopic tools. In this paper, different solutions for the end effector are investigated and presented.

2 Possible Solutions for the End Effector The end effector is designed to allow moving and actuating the instrument using 4 DOFs to position and to orient the forceps. Another DOF, used to actuate the laparoscopic tools, will be described in details in Section 3. In laparoscopic surgery the instrument is introduced through a trocar into the abdomen of the patient. Therefore, the end effector must be controlled in order to fulfil the fixed point constraint at the trocar site. To satisfy this constraint and to simplify the control unit of the end-effector, different solutions were studied. In order to obtain a rotation around a fixed point, the mechanism shown in Figure 2 can be designed. To simplify the control, we have imposed that the insertion point lies on the axis of the sixth rotary joint of the lateral arms of Navi-Robot. This joint (joint 1 in Figure 2), is used to tilt the laparoscopic tools in the transversal plane. In order to obtain a rotation around a fixed point in the longitudinal plane a 4-bar linkage, in which the two centers of rotation lie on the axis of first rotary joint, can be used. The 4-bar linkage is to be dimensioned to ensure that the center of rotation of the second link and the fixed point at the trocar site are coincident. After choosing the fixed point in the workspace of the robot, the holder of the standard laparoscopic tools is connected to the rest of the structure tilted of an appropriate angle β . This way, the axis of joint 4, actuated to rotate the traditional instruments on their axis, crosses the fixed point. This part of the system is composed by one prismatic joint (joint 3), actuated to control the penetration of the instrument, one rotary joint (joint 4) and the standard tools actuator. This particular kinematic chain allows creating a virtual spherical wrist at the fixed point. Since the fixed point is located at the trocar site, it is not possible to locate a fixed hinge in that point. To overcome this problem and, at the same time, to preserve the

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Fig. 3 Scheme of pulleys-belts based end-effector.

Fig. 4 Double-Parallel-4-Bar-Linkage mechanism.

functionality of the end-effector, a 3 DOFs open kinematic chain, obtained removing joint 2.4 of the original 4-bar linkage, was designed, as shown in Figure 3. This system is actuated by pulleys and belts, allowing to reduce the DOFs from three to one, guaranteeing the necessary free space around the trocar site to perform the surgical tasks. Using this type of transmission the fixed point constraint is satisfied if: q2 = −q1 .

(1)

q3 = π + q1 .

(2)

The tilting angle of the laparoscopic tools, with respect to the horizontal plane of the end-effector, is γ = ±q1 − β . (3) Since the fixed point constraint at the trocar site is to be met, it is fundamental to use a structure as rigid as possible in order to reduce problems of bending. In fact, if some deformation happens, the effective dimension of the end-effector’s links can change, making the system unusable for the required tasks. To improve the stiffness, a Double-Parallel-4-Bar-Linkage (DP4BL), shown in Figure 4, can be used instead of the simple 4-bar linkage shown in Figure 2. A DP4BL presents three external hinges represented by joints 1, 5 and 6 in Figure 4. To use this mechanism as end effector while leaving the necessary free space around the trocar site, a mechanism based on the DP4BL concept, with a different joint configuration was designed, as shown in Figure 5. Now, the fixed point at the trocar site is coincident with the center of rotation of the laparoscopic instrument.

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Fig. 5 (a) DP4BL used in end-effector structure; (b) parameters for the dimensional synthesis.

To guarantee the intersection between the axis of joints 1 and 4 with the insertion point of the instrument, the following dimensional synthesis was performed. Since not all standard laparoscopic tools are equipped with a DOF to rotate the forceps, it is necessary to rotate it together with the handle. The maximum characteristic length of the handle in the instrument’s transversal plane is 13 cm. With reference to Figure 5b, a distance l1 = 15 cm between the axes of joints 3 and 4 was set in order to avoid collision between the handle and the prismatic joint. Moreover, since the end-effector must be located over the patient abdomen, l4 and l5 was imposed, respectively, equal to 35 and 30 cm to create a sufficient free area around the trocar site. Using these link lengths, the following value of angle β is obtained:
 −1 l1 = 25.3769◦. β = sin (4) l4 In order to make the system manufacture and assembly as easy as possible, an angle β = 25◦ was selected and, consequently, the following values of l2 , l3 and l4 were derived: l1 = 35.493 cm, (5) l4 = sin(β ) l3 = l4 · cos(β ) = 32.1676 cm,

(6)

l2 = L − l3 ,

(7)

where L is the length of the standard laparoscopic tools. Figure 7a shows a schematic view of the end effector with the embedded frames according the Denavit–Hartenberg (D–H) convention [7]. Figure 7b shows in details the frames assigned to solve the DP4BL, in order to calculate the transformation matrix between the frames 1 and 2 fixed to joints 2 and 3 in Figure 5, respectively. The D–H parameters used to calculate the kinematic equation of the end effector are shown in Table 1.

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Fig. 6 (a) D–H general frame assignment; (b) DP4BL frame assignment. Table 1 D–H parameters. 



T10

T11

T11

T11

θi

q1

q2

q2 = − π2 − q2

π q 2 = − 2 − q2

−β

0

q4

di

d1

0

0

0

0

− π2

d4

0

0

0

π 2

d3 0

0

0

a2

a2

a 2

0

a3

0

αi ai

T21



Tji

π 2

T32

T43

Since these D–H parameters, the transformation matrix representing the kinematic equation of the end effector is 

where





T40 = T10 · T11 · T11 · T11 · T21 · T32 · T34 ,

(8)

T ji = Rot(zi , θ j ) · Transl(zi , d j ) · Rot(xi , α j ) · Transl(xi , a j )

(9)

indicate the transformation from reference frame {i} to { j} fixed to joint i + 1 and j + 1, respectively. The last DOF of the end effector, is used just to actuate the laparoscopic instrument as will be explained in the following section.

3 The Traditional Instrument Actuator The actuation of the traditional instrument is controlled by a mechanism composed by five gears, gears 1 to 5 in Figure 8a. The bevel gear 1 is connected to an electrical actuator. The bevel gear 2, actuated by gear 1 through the spur gear 3, engages with gear 4 which is composed by a spur gear on the bottom and a bevel gear on the top. By rotating gear 4, the bevel gear 5 actuates the handle of the laparoscopic tool. This last gear, in fact, actuates

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Fig. 7 (a) Tools actuator internal view; (b) tools actuator external view.

the 4-bar linkage that link1 and link2 create with the handle of the instrument, as shown in Figure 8b. The laparoscopic instrument is fixed to the spur gear 9, whose axis is collinear with the axis of gear 4. Gear 9 also supports the axis of gear 5 which engages with the bevel portion of gear 4. The bevel gears 6 and 7 and the spur gears 8 and 9 are used to create the rotary joint 4 shown in Figure 5. To have a pure rotation of the instrument both gears 1 and 6 are to be rotated at the same time, while only gear 1 is to be rotated to activate the instrument. It has to be noticed that the transmission described above is omokinetic.

4 Conclusions In this paper, a mechanism able to hold and actuate the standard laparoscopic instruments was presented. The efforts necessary to build this system are justified by the money saving that the instrument can represent for the hospitals, with the consequent increasing diffusion of robotic systems in the medical centres.The main advantage of the proposed system over electronically controlled devices is the low response time. Moreover, the standard tools manipulation could make it possible to use this robotic system in neonatal and paediatric least invasive surgery. In order to verify the performance of the end-effector, in terms of resolution, accuracy and repeatability, a first prototype is going to be built and tested.

References 1. Anderson, C., Ellenhorn, J. et al., Pilot series of robot-assisted laparoscopic subtotal gastrectomy with extended lymphadenectomy for gastric cancer. Surgical Endoscopy, 21(9):1662– 1666, 2007. 2. Advincula, A.P., Song, A. et al., The role of robotic surgery in gynecology. Current Opinion in Obstetrics and Gynecology, 19(4):331–336, 2007.

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3. Cosco, F.I., Moschella, D., Gatti, G., and Danieli G.A., Kinematic error model of NaviRobot, a hybrid parallel/serial robotic structure. In Proceedings 12th IFToMM World Congress, Besanc¸on, France, June 18–21, 2007. 4. Danieli, G.A., Measuring open kinematic chain able to turn into a positioning Robot. PCT/IT05/000487 del 08/08/2005, WO2006016391 A1 of February 16th, 2006, EP 05778744.2, IPC B25J 19/00, 2006. 5. Danieli, G.A. and Moschella, D., Endo-Navi-Robot per il controllo e movimentazione micrometrica di strumentazione endoscopica tradizionale, particolarmente adatto per laparoscopia neonatale. Domanda di Brevetto No. CS2007A000046, deposited 26 October 2007. 6. Danieli, G.A. and Riccipetitoni, G., Robotized system of control and micrometric actuation of an endoscope. PCT/IT05/000486 del 08/08/2005, WO2006016390 A1 of February 16th, 2006, EP 05778903.4, IPC A61B 19/00, 2006. 7. Denavit, J. and Hartenberg, R.S., A kinematic notation for lower-pair mechanism based on matrices. Journal of Applied Mechanics, 77:215–221, 1955. 8. Dewaele, F., Caemaert, J. et al., Intradural endoscopic closure of dural breaches ion a case of post-traumatic tension pneumocephalus. Minimally Invasive Neurosurgery, 50(3):178–181, 2007. 9. Goldstraw, M.A., Patil, K. et al., A selected review and personal experience with robotic prostatectomy: Implications for adoption of this new technology in the United Kingdom. Prostate Cancer and Prostatic Diseases, 10(3):242–249, 2007. 10. Hernandez-Divers, S.J., Stahl, S.J. et al., Endoscopic orchidectomy and salpingohysterectomy of pigeons (Columba livia): An avian model for minimally invasive endosurgery. Journal of Avian Medicine and Surgery, 21(1):22–37, 2007. 11. Korets, R., Hyams, E.S. et al., Robotic associated laparoscopic ureterocalicostomy. Urology, 70(2):366–369, 2007. 12. Moschella, D., Gatti, G., Cosco, F.I., Aulicino, E., Nudo, P., and Danieli, G.A., Development of Navi-Robot, a new assistant for the orthopaedic surgical room. In Proceedings 12th IFToMM World Congress, Besanc¸on, France, June 18–21, Paper No. 713, 2007. 13. Muller-Stich, B.P., Reiter, M.A. et al., Robot assisted versus conventional laparoscopic fundoplication: Short-term outcome of a pilot randomized controlled trial. Surgical Endoscopy, 21(10):1800–1805, 2007. 14. Nappi, C., Di Spiezio Sardo, A. et al., Prevention of adhesions in gynaecological endoscopy. Human Reproduction Update, 13(4):379–394, 2007. 15. Nievas, M.N.C.Y., Haas, E., Hollerhage, H.G. et al., Combined minimal invasive techniques in deep supratentorial intracerebral haematomas. Minimally Invasive Neurosurgery, 47(5):294– 298, 2004. 16. Tayar, C., Karoui, M. et al., Robot assisted laparoscopic mesh repair of incisional hernias with exclusive intracorporeal suturing: A pilot study. Surgical Endoscopy, 21(10):1786–1789, 2007. 17. Wang, G.J., Barocas, D.A., Raman, J.D. et al., Robotic vs open radical cystectomy: prospective comparison of perioperative outcomes and pathological measures of early oncological efficacy, BJU INT, eFIRST date 25 September 2007.

Ethics in Robotic Surgery and Telemedicine F. Graur1 , M. Frunza2 , R. Elisei1 , L. Furcea1 , L. Scurtu3 , C. Radu1 , A. Szilaghy3, H. Neagos1 , A. Muresan1 and L. Vlad1 1 University

of Medicine and Pharmacy “Iuliu Hatieganu” Cluj-Napoca, 400012 Cluj-Napoca, Romania; e-mail: [email protected] 2 Babes-Bolyai University, 400084 Cluj-Napoca, Romania; e-mail: [email protected] 3 Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. Robotic surgery and telemedicine become wide spread in medical world and these techniques are used more and more in a large number of surgical interventions. The limits of all these modern technologies are pushed by surgeons, required by patients and stimulated by industry. There is a concern regarding the accelerating evolution of technology which may lead to some fantastic scenarios such as autonomous surgeon, criminal interference during a telesurgical intervention and other aspects. In this article the ethical issues of the robotic surgery and telemedicine are debated in order to initiate the development of a set of ethical rules in this field. Key words: ethics, robotic surgery, telemedicine, telesurgery

1 Introduction Robotic surgery becomes more and more part of our lives. If this is a real necessity or a pressure from the market or at last a demand from the patients is yet unknown. Robotics and telemedicine are developing very fast, without a sound legislation and ethical rules. Nowadays experts from many developed countries establish connection with surgical centers from developing countries and perform interventions using telesurgery, but the latest doesn’t have rules for these interventions. Robotic surgery and telemedicine have many interconnections therefore the ethical issues are discussed in a close relation. In this article the authors want to discuss the main problems raised by the use of robots and telemedicine technology in surgery and the future direction of research and also to initiate the development of a set of ethical rules in the field.

2 Ethical Issues In the EU legislative context, the use of robots in healthcare is assumed to be regulated by the Council Directive 93/42/EEC of 14 June 1993 concerning medical

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devices, the Council Directive 90/385/EEC on the approximation of the laws of the Member States relating to active implantable medical devices, and the amendments from the Directive 2007/47/EC of the European Parliament and of the Council of 5 September 2007 [14–16]. These directives generally aim at preserving the safety concerns by harmonizing the rules of manufacturing, selling and handling of the devices designed to assist the physician to combat a disease. In this respect, they are important in providing a uniform legal framework across Europe. However, the above mentioned documents may be deemed imprecise, by not taking into account the possibility of autonomous machines. Another shortfall is that, by imposing too strict guidelines on the producers of such devices, they could be tempted in escaping the framework by designing products that are not intended for monitoring or curing a disease, that end up by being used in a medical context [7].

2.1 Ethics of Implementing High-Costs Medical Equipment How ethical is to implement some very expensive devices as robots for patients and medical systems? In developing countries with a weak health system, there will be few centers that will afford very expensive equipments such as surgical robots, while the rest of the hospitals and clinics are underfinanced. This will lead to patient addressability imbalance and also specialists shortage in underequiped clinics. Telemedical consultations may lead to unfair competition between doctors and this can stimulate the specialists shortage in the area. High technology counterbalanced by specialists shortage in underserved regions. The main benefit of the robotic surgery and telemedicine is that in some underserved regions or countries the patients can reach some top expert-demanding interventions. In low density populated regions even for usual interventions these robots could help, for example near the poles or on an orbital spatial station. The presence of a robot and a connection with a specialist in a surgical center could save lives when a specialist is missing [3]. On the other hand the surgeons will migrate from these underdeveloped regions to the well-equipped centers which can offer a sound infrastructure even for these high-tech interventions. In these top centers they will be trained better, the new technology is affordable, and also the financial aspects stimulate the migration. Such problems may create many difficulties in a global health system especially in the underdeveloped regions. This aspect is already visible in the health systems of the Eastern Europe with an increasingly number of specialists that migrates to Western European countries [2, 3]. Cross-border care should not change usual medical ethics, for instance on confidentiality, but may mitigate or aggravate migration of specialists [3]. High costs of the equipment may prove to be unaffordable to clinics and hospitals, especially when it involves expensive robots or advanced technology such as

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videoconferencing applications and multiple ISDN lines. However, such as the case presented by Chan et al., proves telemedicine (in the format of live and taped registration of surgery broadcasted between Brisbane, Australia and Florida, USA) as the only realistic way of training novice surgeons in rare surgical procedures, such as fetal endoscopic surgery [1]. Another issue is that the sophisticated equipment required for telemedicine is (at least in the US case) for the time being more often covered from providers and/or grants, and only infrequently reimbursed by various government agencies, which endangers the long-term prospects of telemedical projects [9].

2.2 Ethics of Robot Responsibility Who is responsible for robots errors? Assigning robot responsibility is dependable on the degree of autonomy that robot has. The various degrees of autonomy of machines in general range from: kinetic autonomy (capability for moving a part from one’s structure), cognitive autonomy (capability for recognizing information, processing and manipulating it), learning autonomy, decisional autonomy (capability to take decision to act in a certain way), classificatory autonomy, first-order institutional autonomy (capability for selecting in which institution to participate in order to solve trust problems) and second-order institutional autonomy (capability to innovate institutionally) [8]. Though, when it comes to the issue of dealing with machines errors, especially if those errors are significant, the responsibility is currently attributed to owners and designers. The paradigmatic example is the existing legislation on dogs, which, in case of a serious harm (a road accident or a personal injury) is attributed to their owners [7, 8]. Since there is no legislations for the use of the surgical robots (especially for the autonomous robotic surgery) the medical instances should implement some rules for this aspect. However, the existing framework generally deals with robots in warfare, and can be applied only by analogy to medical robots. In the latter case, and especially in robots endowed with a superior degree of autonomy, it will become of critical importance to distinguish between the acts that can partially be and those that cannot be attributed to a human person. This issue is further complicated by the fact that, in a system of justice such as ours, the issue of responsibility translates (in case of a mistake) in the possibility of somebody found accountable for the mistake, and eventually being punished for it. Or, the issue of punishment clearly makes sense only for humans. How could somebody “punish” a robot and, even if one can imagine such ways, what sense would it make? There are two big frameworks for punishments. The utilitarian one attempts to define punishment by its positive consequences. The punishment in itself is something negative, that should be avoided, but if one intends to maximize the utility of a situation, such as utilitarians do, in order to be justified, a punishment should max-

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imize positive consequences, it should serve to the purpose of reducing the existing crimes. The second one is the retributive theory, that somehow resembles the “eye for an eye, tooth for a tooth” ancient principle. The criminal needs to be punished, i.e., needs to suffer, in exchange for his villain act [12]. Both definitions of punishments are missing an important point when hypothetically applied to correct a robot error. The utilitarian one is senseless, since robots are not supposed to willingly err, and therefore would not gain anything from a punishment. The retributive one is equally flawed, since a robot could not “suffer” for the consequences of a punishment. Though, for properly speaking about a robot responsibility, the current framework for sanctioning it - the current theories of punishment - need to be reconsidered. To avoid any possible surgical robot’s error which can put patients life in danger in the future, a full body scan (MRI or CT) will be made, which will provide enough information to construct a virtual 3D patient. On this 3D virtual patient the robot will make a virtual intervention and only after this test intervention will be a success the robot will be allowed to do the real one.

2.3 Cultural and Social Issues Cultural and social issues may raise some debates, because during telesurgery the surgeon and the patient cannot meet face to face. One problem raised by telemedicine is the fact that it eliminates physical proximity between doctor and patient, and in this way an important aspect of the health process – the personal connection – is disregarded [9]. Some authors even speak about the marginalization of the patient, who is reduced at the level of a mere “beneficiary” or, worse, “data source” [4]. Another worrying issue is the fact that the practice increases the workload of doctors, that are already very busy. An additional issue is given by the fact that people in disadvantaged countries are facing supplementary problems - for example, related to poor equipment or techniques, they are confined from enjoying the benefits of telemedicine [6]. One issue that could look trivial but is still highly judged by actual surgeons is the challenge of telesurgery to provide the surgeon enough data in order to “feel” the operation site, because surgery is still considered a highly specialized and trainingrequiring form of “craft” [10]. Still, an important obstacle to the development of telemedicine at a global level may be the differences among various medical bodies concerning disciplinary actions in case of malpractice, which is a very sensitive issue worldwide [9]. Another issue may be the difference between customs: for example a mangynaecologist could not treat a woman-patient from a islamic region through telemedicine because of religious rules. There may be also legal issues regarding the difference of licensing from one country to another. A doctor may not be licensed in a country but can perform interventions through telemedicine in another country. Also there may be some illegal

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procedures in a country but legal in another country (eg. human cloning or abortion) and therefore the ethical rules must respect some symmetry. Ethics and law coincide regarding encouragement of cross-border recognition of medical licensure and specialist accreditation, and of harmonization of standards of professional practice, so that jurisdictional differences will not obstruct patients’ access to telemedical and robotic surgical services. Similarly, commercial insurance and professional self-defence association arrangements should be adjusted to the realities of medical globalization, and not obstruct the upgrading of care that technology can afford patients in medically deprived settings. Spreading of benefits depends in part on the spreading of risk. Further, courts addressing conflict of laws issues should take care to avoid dysfunctional decisions that would deter practitioners from providing cross-border services in unfamiliar legal territory [3].

2.4 Confidentiality During the transmission of data there could appear leaks and loss of confidentiality especially when the data are collected by a third party between patient and doctor. The electronic media must be secured by specialized providers who will assure also the connections between “master” and “slave”. Confidentiality is at risk due to means of electronic eavesdropping, but special care is needed to prevent inadvertent copying of communications such as diagnoses. Care must also be taken to ensure that non-physician intermediaries who collect and transmit data about patients, such as medical technicians, observe confidentiality. Where patients are treated or monitored in their homes, family members may become, and perhaps need to become, involved in their care, and have access to information. Practitioners at a distance from patients should ensure, as much as they can, that local personnel have informed patients of this, and obtained their agreement [3]. There are some examples of typically daily situations which involve supplementary measures in order to maintain confidentiality: for instance, when computer systems need to be repaired or maintained by an external provider, or when patient data are sent between hospitals in telemedical consultations. In the former case, it is important to extend the confidentiality agreement to the provider; in the latter, that only partial and relevant data need to be sent, and not all the patients’ data [11]. All existing regulations (legal or ethical) put a great deal of emphasis on the issue of confidentiality; however, some authors warn that the standards and protocols that are habitual in developed countries may actually impede the developing of telemedicine in developing or disadvantaged countries. Here, hospitals may lack the advanced technology that is considered a norm in the existing ethical guidelines and patients may not be computer literates at a level that would allow them to provide an informed consent concerning their files sent at distance; however, patients may be better served by the use of telemedical practices than without them. An example provided by Mars from the rural Sub-Saharan region clearly illustrates this point: in

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this case, a junior doctor uses her personal camera and commercial email address to send the X-Ray films of a patient, in search of advice from a specialist physician. [6] Although this practice is not conform to existing regulations on telemedicine, clearly the patient benefits from the use of technology. In this case, ethical regulations and guidelines need to be adjusted to the existing level of equipment and not to impose supplementary burdens on physicians working in extremely rural settings.

2.5 Reliability of Equipment The micrometer level precision of the movements of a robots arms has been shown in the manufacturing of the microchips. The already existing advanced technology could be transferred from computer manufacturing to surgery operations. The robots will perform very accurate cuts compared with the surgeon’s hand that sometimes could be trembled by emotions or fatigue. The robots could perform ultrasonic scans and limit their movement as to not penetrate tissues that are not to touch. The surgeon would be relieved of the nerve consuming operation of scalpel and other surgical instruments and focus on the coordination of the surgery team and the progress of surgery operation. Stanberry discusses this issue via a real-life case (Therac-25) when at least six people were harmed (some of them died) due to the misuse of the software of an X-ray machine. The real dosage of radiation administered to the patients far outweighed the recommended dose, but the instrument that should have measured the correct dose was faulty. The example may be extrapolated to telesurgery, and should warn about the necessity of introducing an “off switch” for medical robots, and the possibility of letting a human surgeon to double check all decisions of a machine [10]. Another issue that concerns the use of equipment is the establishment of recognized technical and clinical standards – e.g., when using email communication between physician and patient. Authors underline that these standards are being established at local and national level, and the generalization of them at international level is still at the beginning [6, 9].

3 Rules to Be Followed “Primum non nocere, secundum succurrere” is the Latin phrase for “first do no harm, second hasten to help” and this principle should be the main rule also for newly developed techniques. Of course the main principles of general medical ethics should reflect in the new list of rules. In its 1999 statement on telemedicine, the World Medical Association emphasizes that regardless of the telemedicine system under which the physician is operat-

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ing, the principles of medical ethics globally binding upon the medical profession must never be compromised [13]. Recognizing the possible ethical limitations introduced by telemedicine and telehealth, the World Medical Association aimed at developing ethical guidelines and principles for those practicing telemedicine. Those principles strive towards:

3.1 The Physician-Patient Relationship The guidelines warn that telemedicine should not aim at replacing the face-toface encounter between physician and patient, but only to supplement it with tele-mediated encounter when the physical one is not possible. However, they acknowledge the advantages of a physician-to-physician tele-consultation, particularly when it concerns talking with a distant expert, that would benefit the patient.

3.2 Accountabilities and Responsibilities of a Physician The guidelines attempt to extend physician’s responsibility for the directions and treatment s/he recommends during telemedical consultations. The physician also bears responsibility for ensuring that non-physicians participating in telemedicine (for instance those involved in transmitting data or monitoring the patient) are properly trained to use the facilities.

3.3 Patient Consent and Confidentiality The general rules of obtaining consent and preserving confidentiality apply, with the supplement of observing the security standards for not leaking data in the process.

3.4 Quality Issues The guidelines address the quality of care and safety (a physician must be convinced that the highest standards of care and safety apply to telemedicine, as in regular consultations), but also the quality of data and information (i.e., a physician must rely on the data provided by the patient and the accompanying persons who eventually monitor the patient for the diagnose and treatment proposed).

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3.5 Authorization and Competence in Practicing Telemedicine The guidelines demand that the physician offering tele-consultations should be licensed or recognized in both places (own and patient’s location).

3.6 Patient Records The data tracking and protection should be as rigorous for telemedicine as it is for regular consultations [13]. There must be a code of ethics between doctor-patient, doctor-doctor and even doctor-institution. Good ethical practices will remove the need of the Consumer Protection Act. To inculcate ethics in ones clinical attitude must start from the beginning and it must start from Medical School. Some have questioned whether taking the Hippocratic Oath is necessary. The Hippocratic Oath is based on old concepts and a new oath should be stated but until then one should rely on ethics [5].

4 Conclusions The modern doctor is more and more dependent of new technology, therefore there is a need to implement some rules for use these devices. Ethical issues discussed in this article shows that the problem has many faces, maybe with some not shown yet. The only solution to solve these problems is to try to adapt the old rules or to implement new rules as well as specific guidelines for doctors who use the modern technology. Regarding the research in this domain there should be organized ethical boards who will study the future problems and connect with the boards from partner countries. There should be a site with FAQ (frequently asked questions) for patients who will have a robotic surgical intervention or a telemedical consult and also a hot-line for direct questioning. The public perception is not a well informed one, the patients being informed only by mass-media (stimulated by industry) or directly by the doctor. Even the way to inform the patient regarding the benefits or disadvantages of the robotic surgery or telemedicine should follow some ethical rules.

Acknowledgement The research work reported here was financed by the PNCDI-2 P4 Grant, entitled “Innovative development of a virtual e-learning system for hepatic laparoscopic surgery – HEPSIM” and the PNCDI-2 P4 Grant, entitled “Multidisciplinary develop-

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ment of surgical robots using innovative parallel structures”. They have been awarded by the Ministry of Education, Research, Youth and Sports of Romania.

References 1. Chan, F.Y., Soong, B., Taylor, A., Bornick, P., Allen, M., Cincotta, R. and Quintero, R., Fetal endoscopic telesurgery using an internet protocol connection: Clinical and technical challenges. Journal of Telemedicine and Telecare, 9(S2):12–14, 2003. 2. Chen, L.C. and Boufford, J.I., Fatal flows – Doctors on the move. N. Engl. J. Med., 353:1850– 1852, 2005. 3. Dickens, B.M. and Cook, R.J., Legal and ethical issues in telemedicine and robotics. Int. J. Gynaecol. Obstet., 94:73–78, 2006. 4. Irvine, R., Mediating telemedicine: Ethics at a distance. Internal Medicine Journal, 35:56–58, 2005. 5. Madhok, P., Medical ethics: Boon or bane? Bombay Hospital Journal, 45(03), General practitioners section, July 2003. 6. Mars, M., Does Sub-Saharan Africa need different ethical and medico-legal telemedicine guidelines. Acta Informatica Medica, 15(3):155–157, 2007. 7. Nagenborg, M., Capurro, R., Weber, J. and Pingel, C., Ethical regulations on robotics in Europe. AI-Soc., 22:349–366, 2008. 8. Perri, G., Ethics, regulation and the new artificial intelligence, Part II. Information. Communication Society, 4(3):406–434, 2001. 9. Silverman, R.D., Current legal and ethical concerns in telemedicine and e-medicine. Journal of Telemedicine and Telecare, 9(S1):67–69, 2003. 10. Stanberry, B., Telemedicine: Barriers and opportunities in the 21st century. Journal of Internal Medicine, 247:615–628, 2000. 11. Stanberry, B., The legal and ethical aspects of telemedicine. 1: Confidentiality and the patient’s rights of access. Journal of Telemedicine and Telecare, 3:179–187, 1997. 12. Ten, C.L., Crime and punishment. In P. Singer (Ed.), A Companion to Ethics, Blackwell, Oxford, pp. 366–372, 1993. 13. World Medical Association Statement on Accountability, Responsibilities and Ethical Guidelines in the Practice of Telemedicine, Adopted by the 51st World Medical Assembly Tel Aviv, Israel, October 1999 and rescinded at the WMA General Assembly, Pilanesberg, South Africa, 2006. http://www.wma.net/en/30publications/10policies/20archives/a7/index.html. 14. Council Directive 93/42/EEC of 14 June 1993 concerning medical devices. http://eurlex.europa.eu/LexUriServ/LexUriServ.do?uri=CONSLEG:1993L0042:20071011:en:PDF. 15. Council Directive 90/385/EEC on the approximation of the laws of the Member States relating to active implantable medical devices. http://eurlex.europa.eu/LexUriServ/LexUriServ.do?uri=CONSLEG:1990L0385:20071011:en:PDF. 16. Directive 2007/47/EC of the European Parliament and of the Council of 5 September 2007. http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2007:247:0021:0055:en:PDF.

Optimal Control Problem in New Products Launch – Optimal Path Using a Single Command I. Blebea, C. Dobocan and R. Morariu Gligor Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected], [email protected], [email protected]

Abstract. Equations of state that govern the process of new products launched in the market are a generalization of the Nerlowe–Arrow equations [2]. The existence of four independent parameters regulating the behavior enable the release process of finding sufficient conditions for determining an optimal trajectory using a single command. Maximizing Hamiltonian corresponding dynamic system results two control functions which can be independently or simultaneously used. In this paper we demonstrate the possibility of using two independent commands, thus obtaining two minimum times, to get a higher test time optimization of data release in the initial and final conditions taking the smaller of the two minimum times corresponding to the two control functions. Key words: optimal control, state equations, Hamiltonian, “goodwill” function, cost function

1 Introduction The advertising activity has an important role in launching a product. Here we give and discuss the problem of determining the advertising policy in order to prepare the introduction of a new product on the market. Suppose that the company has three goals: the first objective is to maximize the product image (good impression) at the moment T of the launch with the purpose to minimize the cost of advertising. The second objective requires to determine the release (or entry on the market) – release time T , when introducing the product will begin and the third goal lies in planning to launch and advertising launch campaign during the interval [0, T ]. To formulate the control problem for the market launch of new products (LPPN) we considered three unknown functions: x(t), y(t), u(t), t ∈ [0, T ], where T is time, x(t) is the “level of good impressions” presented above, about the new product to be launched, the size (value) to measuring the percentage, y(t) is the function that shows the “level of investment at time t”, and u(t) is the control function, the function that shows the intensity, increase or decrease of the cost y(t) of the investment at time t, u(t) can be interpreted as an instrument of pumping costs. The equations of state, the equations governing the state system are a generalization of Marlowe–Arrow equations. The equations of state are:

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⎧  α ⎪ ⎨ x (t) = b · [u(t)] − c · x(t), y (t) = u(t) · e−pt , p > 0 ⎪ ⎩ t ∈ [0, T ], α ∈ (0, 1)

b, c > 0 (1)

For α = 1 we obtain the very Marlowe–Arrow equations [2]. The positive constants b, c, p, α are regulated by the release process behavior. b is the coefficient representing the influence of zoom in or out of control function u(t) determined by the increase or decrease the function of “goodwill” x(t). c is the coefficient decrease (degradation) of “goodwill” x(t). p is the exponential coefficient that balances the costs of promotion variation y(t) expressed by the control function u(t) so that there is a correlation between the function x(t) of “goodwill” and the function y(t) the costs of promotion at the time t. Matching consists in: if x(t) increases it result that y(t) decreases and vice versa. α ∈ (0, 1) is a factor that defines the influence of the external market. The system solutions (1) are subject to restrictions of behavior. Baseline t = 0 “goodwill” had a value x0 ≥ 0, so x(0) = x0 , so investment value is zero, that is y(0) = 0. For the control function we have to give a finite interval [0, u0 ], thus 0 ≤ u(t) ≤ u0 , ∀t ∈ [0, T ]. At final time t = t ∗ ≤ T , it is given the state (x, ¯ y), ¯ that is ¯ In conclusion, the optimal control x, ¯ y¯ ∈ ℜ+ is given, so that x(t ∗ ) = x¯ and y(t ∗ ) = y. problem is to determine a minimum time optimal trajectory:  x = x∗ (t), t ∈ [0,t ∗ ] , Γ ∗: y = y∗ (t), so that in the given conditions, the time in which a point ring (x(t), y(t)) ∈ Γ ∗ moves from A(x(0), y(0)) to B(x, ¯ y) ¯ is minimal. This time is t ∗ ≤ T . Formally, the control problem is expressed as: ⎧  x (t) + c · x(t) = b · [u(t)]α , b, c > 0 ⎪ ⎪ ⎪ ⎪ ⎪ y (t) = u(t) · e−pt , p > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x(0) = x0 , y(0) = 0, x0 ∈ ℜ+ (2) 0 ≤ u(t) ≤ u0 , t ∈ [0, T ] ⎪ ⎪ ∗ ∗ ∗ ⎪ x(t ) = x, ¯ y(t ) = y, ¯ t ≤T ⎪ ⎪ ⎪ ∗ = min{t} ⎪ ⎪ t ⎪ ⎪ ⎩ t ∈ [o, T ] The optimal control problem (2) is solved using the “Pontreaguine Maximum Principle”. We will apply this principle to problem (2). First we introduce the Hamilton function: H(x, y, u) = [b · uα (t) − c · x(t)] ·C1 · ect + C2 · e−pt · u(t) = [b · uα (t) − c · x(t)] ·C1 · ect + C2 · e−pt · u(t)

(3)

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On the basis of Pontreaguine’s Maximum Principle we further determine the peak of function H(u), in variable u, where the function H(u) is: H(x, y, u) = [b · uα (t) − c · x(t)] ·C1 · ect + C2 · e−pt · u(t) = [b · uα (t) − c · x(t)] ·C1 · ect + C2 · e−pt · u(t) where A = b ·C1 · ect ,

B = C2 · e−pt ,

(4)

C = −c ·C1 · x(t) · ect

2 Maximizing the Hamiltonian The determination of the maximum of H(u) involves the calculation of the critical points for the function H(u), i.e. solving the equation H  (u) = 0, leading to α · A · uα −1 + B = 0, α ∈ (0, 1) it results that B · uα −1 = −α · A or uα −1 = − αB·A . There should be an issue depending on the sign of C1 and C2 . (a) If C1 and C2 are zero, it results that Ψ1 = 0, Ψ2 = 0, ∀t ∈ [0, T ]. This solution is inconsistent with Pontreaguine’s Maximum Principle. (b) If C1 > 0,C2 > 0, we obtain H  (u) > 0. In this case the optimal control function (optimal control) is: u∗ (t) = u0 ,t ∈ [0, T ]

(5)

(c) If C1 < 0,C2 < 0, the result is H  (u) < 0. It is noted that in this case: max H(u) = H(0) = C, u∈[o,u] ¯

so u∗ (t) = 0,t ∈ [0, T ]. In this case the problem of control makes no sense. (d) If C1 < 0,C2 > 0 we are led to (5) or (6). (e) If C1 > 0,C2 < 0 where u∗ (t) = λ · eµ t , λ > 0, µ > o with   C1 1/1−α λ = −α · b · , C2 and so

u∗ (t) = λ · eµ t ,

µ=

c+ p 1−α

t ∈ [0, T ]

3 Determination of the Optimal Path and the Minimum Time For case (5), the state equations are:

(6)

(7)

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x (t) + c · x(t) = b · u0α u y (t) = u0 · e−pt ⇒ y(t) = − p0 · e−pt + K2

From y(0) = 0 ⇒ K2 = u0 /p and thus we obtain the cost function: y(t) =

u0 (1 − e−pt ), p

For t = t ∗ , y(t ∗ ) = y¯ so y¯ =

t ∈ [0, T ]

u0  ∗ 1 − e−pt p

from which we obtain the minimum time relative to the cost function t∗ =

u0 1 · ln p u0 − p · y¯

In conditions t ∗ ∈ (0, T ]. From t ∗ > 0 the following condition results: u0 >1 u0 − p · y¯ and u0 − p · y¯ > 0 ⇔ p
p · y¯

We show that further conditions are imposed to parameter p > 0. Since ex function ∗ is increasing and t ∗ ≤ T ⇒ e pt ≤ e pT . ∗

e pt =

u0 u0 − p · y¯

(8)



u This equation has two solutions: p1 = 0 and p2 ∈ 0, y¯0 . p1 does not agree because p > 0.  u It is noted that the values of p which agree are within the range p0 , y¯0 , i.e. u

p0 ≤ p < y¯0 . For the function x(t) we integrate the differential equation: x + c · x = υ ,

where υ = b · uα0

(9)

The general solution of equation (9) is: x(t) = K1 · e−ct +

b ·u α c 0

The Cauchy problem solution is: 

b b x∗ (t) = x0 − · uα0 · e−ct + · uα0 c c

(10)

(11)

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∗

In the end, the final condition x(t ∗ ) = x¯ leads us to x¯ − γ = x0 − γ · e−ct , where ∗ x0 −γ γ = bc · u0 α . Of equal ect = x− ¯ γ > 1 resulting: t∗ =

x −γ 1 · ln 0 >0 c x¯ − γ

(12)

Finally, another limitation of input data is: x¯ b > α c u0

(13)

As the current point (x(t), y(t)) reaches in the same time in (x, ¯ y), ¯ the value of t ∗ : x − b ·u α u0 1 1 · ln = · ln 0 bc 0 p u0 − p · y¯ c x¯ − c · u0 α

u p ∈ p0 , 0 y¯

(14)

 (15)

For this case, the problem of new products launch on the market is more laborious and we see that it has a solution if all the conditions that have resulted over resolving ¯ y¯ and the coefficients of adjustment b, c, p, α are verified. the input data: u0 , x0 , x, Finally, the case given by (7) is treated similarly.

4 The Numerical Model and Optimal Path

∗

Γ :



x = x∗ (t), y = y∗ (t),

t ∈ [0, t˜∗ ].

Table 1 The impact of the coefficients of adjustment.

Case 1 Case 2

u0

x0

x¯

y¯

b

c

p

α

λ

µ

2 2

0.05 0.05

0.85 0.85

5000 5000

0.5 0.5

0.5 1

0.0001 0.0001

0.25 0.25

0.0625 0.0625

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Fig. 1 The representation of minimum time and the optimal path in case 1.

Fig. 2 The representation of minimum time and the optimal path in case 2.

5 Conclusions This case is important as such, but it also occurs in the situation of optimal control problems under transversatility conditions [4]. The numerical model is suggestive for the use of parameters b, c, p, α . A more detailed numerical study has to be carried out.

References 1. Buratto, A. and Viscolani, B., An optimal control student problem and a marketing counterpart, Mathematical and Computer Modelling, 20:19–33, 2002.

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2. Buratto, A. and Viscolani, B., New product introduction: goodwill, time and advertising cost, Mathematical Methods of Operations Research, 55:55–68, 2002. bibitem3. Dobocan, C., Optimal control problem in new products launch. In Annals of DAAAM for 2009 & Proceedings: “Intelligent Manufacturing & Automation: Focus on Theory, Practice & Education”, Vienna, Austria, 25–28 November, p. 1033, 2009. 3. Dobocan, C., Theoretical researches and contributions about new products launch on the market, Doctoral Thesis, Technical University of Cluj-Napoca, Faculty of Machines Building, Romania, 2010.

Mechanical Constraints and Design Considerations for Polygon Scanners V.-F. Duma1,2 and J.P. Rolland2 1 Faculty

of Engineering, Aurel Vlaicu University of Arad, Romania; e-mail: [email protected] 2 Institute of Optics, University of Rochester, Rochester, NY 14627, USA; e-mail: [email protected]

Abstract. We study the mechanical aspects of the design of optical scanners. From the most used types of scanners, we focus on those with rotating mirrors, of special importance in nowadays high end applications. A designing scheme is developed for these complex optomechatronic systems. Their kinematic aspects, i.e. scanning function (generated considering the rotating mirror as a cam and the laser beam as its follower) and scanning velocity (of the laser spot) are approached. The main mechanical issues involved are pointed out, i.e. manufacturing methods, materials and tolerances, in relationship with kinetostatic and dynamic aspects, i.e. structural integrity and facets deformation of the device. We perform, with regard to some of the main parameters of the polygon, a Finite Element Analysis (FEA) for the evaluation of these aspects, for three types of the most used polygonal mirror scanners. Key words: polygon scanners, optomechanics, kinematics, FEA, structural integrity, rotating shafts

1 Introduction Optical scanners are built in a variety of solutions [8], and there are more than 40 different designs described in the literature [1]. From the most utilized scanners (Figure 1), we highlighted the rotating mirror devices we shall approach in this paper. The necessity of this study comes from the rapid development of high performance Polygon Mirror (PM) scanners for high end applications (especially in medical imaging, where cost issues are secondary to performances) in the last decade [6, 9, 13], after another decade, in the 90s, when they seemed to have lost ground [11] to oscillatory scanners, galvanometer or resonant [8]. A lot of stress is placed on the optical design [1–7, 12] of the various setups the PM scanners are used for, while less consideration is given to the mechanical aspects. However, the latter usually limits severely the theoretically high performances of the optical systems envisaged. While we shall provide in this study an overlook of these aspects and limitations, several specific problems will be addressed in more detail, using different parts of the theory of mechanisms [10]: (i) scanning func-

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Fig. 1 Most used types of optical and laser scanners.

tions and velocities, considering the rotating PM as a cam and the laser beam as its follower [4]; (ii) structural integrity and facets deformation due to high rotation speeds (ω ); (iii) influence of manufacturing tolerances, techniques and materials to the above issues. There are numerous other aspects, e.g. dynamic balance of the PM, a rotating shaft with high ω and performances that require special studies, of importance especially for more rarely used asymmetric configurations, e.g. monogons and irregular polygons [1, 8]. In the present study we shall focus only on the most used PM scanners.

2 Opto-Mechanical Design of Scanners The ‘trajectories’ of the optical and mechanical design of scanners intersect (Figure 2). They can be considered, for a simplified approach, somehow parallel, but with the necessary feedback from the mechanical part, which actually limits the practical possibilities of achieving the usually high expectations of the optical part. At any phase of this process, not fulfilling the requirements means to go back and to reconsider the previous steps. The optical calculus provides the main geometric and kinematic parameters of scanners (which will further on determine their constructive characteristics): (i) The scan resolution, i.e. the number of N light spots that can be perceived distinctively on the scanning path imposes the type of scanner [2, 8]: for N < 1000, non-mechanical scanners, i.e. electro- or acousto-optic; for N < 10, 000, the galvoscanner is effective, but limited in speed for large apertures or angular fields; for N < 20, 000, one may use resonant scanner, but with limits regarding frequency; for N > 20, 000, the PM or the holographic scanners; for N up to 100,000, drum scanners with curved image fields are compulsory (often, with monogons). Major advantage: the PM scanners cover the most necessary (and used) domain of N. (ii) The necessary scan frequency fs gives the number of facets n of the polygon, because its rotating speed ω is technologically limited. For PMs (Figure 3b) fs = nω /2π ; for monogons (Figure 3a), n equals 1.

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Fig. 2 Scheme of the designing calculus of optical and laser scanners.

(iii) The inner radius R of the polygon results from the condition of obtaining a sufficient width (a) of a facet for a given n, where a is obtained from the shape and width of the laser beam (e.g. Gaussian or superGaussian, with Full Width (FW) at 1/e2 intensity) [8]: R = 0.5a/ tan α , with α = π /n (Figure 3b). (iv) Once the type and scanner architecture (objective, pre- or post-objective – see the position of the PM with regard to its lens) [2] is selected, the scan magnification (m) and the scanning function h(t) can be obtained (Section 3). (v) The duty cycle η , i.e. the scan efficiency (scan time versus total time) is obtained from the analysis of the total and useful angular domains of the scanner [3–5] (Section 3). In some designs it can also be an input data [8], as η is related to the facet size to beam size after truncation. The beam truncation ratio of the input beam affects the output beam profile. All the apertures have to be about 3 times FW Half Maximum (FWHM) for a purely Gaussian output beam. An essential optical aspect, related to point (iii), is to minimize with the design the FW: 1/e2 or even FW: 95% over the scan. This is related to avoiding vertical banding otherwise one would get a sinusoid looking grey on a line. This is a main source of failure for 1200 dpi or higher (dpi, dots per inch, measure of the precision

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Fig. 3 Rotating scanning heads: (a) monogon; (b) polygon. Parameters: ω , rotating speed; L, distance from the fix beam to the lens; D, lens diameter; for the PM only: R, n, α = π /n, a, e (Figure 2).

Fig. 4 Scanning function h(θ ) and velocity v(θ ), θ = ω t, ω = cst. for an off-axis polygon with R = 15 mm, e = 6 mm, L = 100 mm, ω = 2500 rad/s and D = 80 mm.

of writing – always in a trade-off with the speed of writing). From an optics design point of view, the Strehl ratio on these systems must be higher than 0.9.

3 Kinematics: Scanning Function and Velocity Two of the most utilized rotating scanners, i.e. monogons and polygons are presented in Figure 3, for one of their most common setup: with the incident (parallel, well-collimated) beam perpendicular to the optical axis of the lens (L1 ). The characteristic functions of the PM device are presented in Figure 4: the scanning function h(θ ), θ = ω t, ω = cst, that represents the current position of the collimated beam with regard to the optical axis, and the scanning velocity v(t) = dh/dt, with equations deduced in [4]. One may see the non-linearity of both functions, which imposes the use of their central (approximate linear) part or using hardware or software solutions for linearizing. For the monogon, these functions are also non-linear, but trivial [3], while for the PM the parameters e and L can be used as degrees of freedom in the design, to obtain a more linear h(θ ) for a given R [5].

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Fig. 5 Pyramidal scanner (double pass: η > 1) [1, 2].

Fig. 6 Torques on polygon scanners.

The scanners presented in Figure 3 are pre-objective (the mirror is placed before the lens), and their scan magnification m, i.e. the ratio between the optical rotational angle and the mechanical one is 2 (applying the laws of reflection). This is of particular importance in certain applications [9, 13], while in others it is convenient to have m equals 1, therefore the pyramidal scanner is used (Figure 5). The characteristic angles provide the duty cycle. For the PM setup in Figure 3b this results [3, 4]: η = n(θ2 − θ1 )/2π , where θ1 and θ2 are respectively the angles for which the inferior and the superior margins of the lens L1 (thus, of the scanning domain) are reached (Figure 3a); n equals 1 for monogons. In setups as those in Figure 3, η < 1, while when overfilling the mirror (e.g. in Figure 5, for a pyramid scanner), the beam is divided in two and one obtains η = 1 . . . 2 – in a conceptual similarity to the theory of gears. Often complicated F-theta lenses impose a displacement in the scan that is linear with theta. Optically this means that the lens has distortion to achieve F-theta mapping, i.e. uniform spot velocity in the scanned plane. These lenses are the final step of the optical design [1] – Figure 2. They may use spherical components in the simpler forms or include some aspheric surfaces to control the Strehl ratio in the highest performance scanners (with more than 3600 dpi).

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4 Kinetostatics of Polygons The practical aspects of the mechanical design (manufacturing techniques, materials and tolerances – Figure 2) have to be considered in relationship with the functional ones, i.e. kinematic, but also kinetostatic and dynamic. High end applications require a high scan frequency, therefore high rotation speeds ω (30,000 to 60,000 rpm) and a large number of facets (n). Polygons with n equals 24 to 128 became usual, in contrast to those for medium applications, with n equals 5 to 12 (with ω usually up to 4,000 rpm). For large n, the dimensions of the PM, i.e. R also has to be increased to provide the necessary facet width, i.e. beam aperture. Large PMs require motors with increased power, because of high bearings and windage torques (Figure 6). Technological effort has been directed towards reducing these two types of torques, and consequently, Tmotor . Windage is a major issue: with special solutions, e.g. caged polygon, rounded corners, or for open assemblies, with two plates to sandwich the PM, Twindage can be lowered 2 to 40 times [8]. Tmotor can be thus decreased, and therefore different, lighter bearings may be used (the cost is consequently lowered). Of special interest are aerodynamic bearings, for the highest rotation speeds, while conventional ball bearings remain for medium applications.

5 Dynamics of Polygons and Manufacturing Issues Materials and manufacturing methods of polygons are chosen together: for medium to low end applications one may use brass, copper or electroless-plated nickel, with injection molding or replication (the latter with epoxy resine). Glass is best when a high optical quality (i.e. reflectivity) is needed, and total reflections are used, e.g. for monogons. Aluminum alloys, stainless steel and even beryllium (although the latter is expensive, difficult and toxic to process) are the materials for demanding working regimes, i.e. high rotation speeds and torques. Manufacturing methods for these materials are: assembly construction (planar mirrors fastened to a structure, e.g., for irregular polygons), single-point diamond machining and conventional polishing (the most used). Two major dynamic aspects have to be studied in relationship with the above issues: structural integrity and facet deformation, the latter correlated with the tolerances imposed by the application. For asymmetric assemblies, e.g. irregular polygons, dynamic balance is an issue. Also, especially for commercial applications, a trade-off is always asked from the designer between quality (good material, precise manufacturing methods and tight tolerances) and equipment cost. Most decisive for the quality of the scanner, the tolerances of the rotating polygons have evolved, improving roughly by a factor of two in the last two decades [8]: (i) pyramidal error: average variation from the desired angle between the actual facet and the mirror datum (responsible for pointing error and scan line bow); (ii) facetto-axis variance: total variation of the pyramidal error from all facets within one

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polygon (typically 2

. . . 1

); (iii) facet-to-facet angle variance: variation of the angle between the normals of adjacent facets (usually ±10

. . . ± 30

); (iv) variation in the inner radius R (±60 µm for the facet radius average position; ±25 µm for the facet radius variation within the polygon); (v) surface figure, measured as the deviation from an ideal flat surface (λ /8 . . . λ /10, where λ is the laser wavelength); (vi) surface roughness; (vii) reflectance and uniformity; (viii) durability: typically up to 20,000 h (around 5 years) at high ω . These values are state-of-the-art for polygons, as indicated by the top few companies that produce them. The two dynamic issues mentioned result: (i) durability: the structural integrity of the PM has to be provided (the coatings also have special issues, in order to be resistant and consistent over time, but this is not the subject of this study); (ii) flatness of the facets, influenced by: initial fabrication tolerances, distortion due to mounting stress, to centrifugal stresses for high ω (the most critical as it is inherent to the device), and to stress relief over long periods of time. Except for the influence of ω , the other factors can be kept under control by proper manufacturing. The study of these two issues has to address various factors: n, ω , inner radius R, polygon thickness (b), radius of the central hole and shaft (r), and material. There are only brief recommendations in the literature, to the best of our knowledge, obtained from experiments and Finite Element Analysis (FEA) performed by companies, but kept as their expertise, with the exception of some brief rules of thumb. One has to analyze and test each particular configuration that results from a specific design, in order to verify its mechanical viability. In Figure 7 we present the results of a FEA performed using ABAQUSTM for a PM made of aluminum alloy 5052 (yield strength equals 218 MPa) with n = 6, R = 25 mm, r = 5 mm and b = 10 mm, for the indicated values of ω . A quarter of the mirror, as marked in Figure 7a was considered in order to do the analysis. Six eccentric holes of 2 mm diameter each are made to provide the mounting of the PM on the shaft. The holes are placed on the directions of the polygon points, to provide a symmetrical deformation of the facets – for an optimal optical behavior of the mirror. Figure 7b shows the positions and values of the minimum and maximum stress for ω1 equals 60 krpm. In Figure 7c the stresses are determined for ω2 equals 100 krpm (same positions of the extreme values, but higher levels). For comparison, the FEA is performed for a PM with the same parameters and with 4 facets (Figure 7d), and 8 facets (Figure 7e) – both for ω2 . For these PM parameters, the exact value of the yield stress is reached for n = 6, and it is surpassed for n = 4, while σmax is only 197 MPa for n = 8, at 120 krpm (the upper possible limit of nowadays motors of PMs).

6 Conclusions We achieved an overview of the issues one has to solve in designing a Polygonal Mirror (PM) scanning head. The links between the optical and the mechanical part of the process were highlighted in an optomechanical approach that has to be taken into account to fulfill the high requirements of the optical calculus. We have pointed

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Fig. 7 (a) FEA of the dynamic regimes of a (quarter of a) rotating polygon with n = 6 facets; (b) extreme values of the stress for ω1 = 60 krpm; (c) stress distribution for ω2 = 100 krpm; (d) FEA for ω2 for a PM with the same indicated parameters and with n = 4 facets, respectively (e) with n = 8 facets (but the same inner radius gives different facets widths and outer radiuses of the PM). The maximum stress is reached in the inner part of the PM (and in its central plane), and it is higher (at the same inner radius R) for PMs with a smaller number of facets: σmax (n = 4) = 184.8 MPa > σmax (n = 6) = 150.6 MPa > σmax (n = 8) = 136.7 MPa (values for ω2 ). The minimum stress is in the exterior of the apex of the PMs and it is higher for a larger n : σmin (n = 4) = 8.53 MPa < σmax (n = 6) = 9.68 MPa < σmax (n = 8) = 14.57 MPa.

out the kinematic aspects, determined using methods characteristic to the theory of mechanisms. The design of the PM scanner was presented, in its essential steps. As we have seen, extensive Finite Element Analysis (FEA) followed by experimental tests is required to obtain the optimum parameters of the polygon for different geometries, materials, and ranges of rotation speeds. In this study we have approached the issue of structural integrity of PMs to the very limit of rotation speeds of the motors. Future work will be focused on the facet deformations of the mirrors, with a high impact on their optical behavior within the system.

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Acknowledgments The study was made possible by PN II IDEAS Grant 1896/2008 of NURCUEFISCU, Romania, by Fulbright Scholar Senior Research Grant 474/2009 of the US Department of State, and by the NYSTAR Foundation. We thank Jonathan Young for his invaluable help with the ABAQUSTM program.

References 1. Beiser, L. and Johnson, B., Scanners. In Handbook of Optics, M. Bass (Ed.), McGraw-Hill, pp. 19.1–19.57, 1995. 2. Beiser, L., Fundamental architecture of optical scanning systems. Applied Optics, 34:7307– 7317, 1995. 3. Duma, V.F., On-line measurements with optical scanners: metrological aspects. Proc. SPIE, 5856:606–617, 2005, http://dx.doi.org/10.1117/12.612103. 4. Duma, V.F., Novel approaches in the designing of the polygon scanners. Proc. SPIE, 6785, 67851Q, 2007, http://dx.doi.org/10.1117/12.756730. 5. Duma, V.F. and Nicolov, M., Numerical and experimental study of the characteristic functions of polygon scanners. Proc. SPIE, 7390:7390-42, 2009, http://dx.doi.org/10.1117/12.827443. 6. Duma V.F., Rolland J.P. and Podoleanu A.Gh., Perspectives of scanning in OCT. Proc. SPIE, 7556:7556-10, 2010, http://dx.doi.org/10.1117/12.840718. 7. Li, Y., Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns, Applied Optics, 47:386–398, 2008. 8. Marshall, G.F. (Ed.), Handbook of Optical and Laser Scanning. Marcell Dekker, 2004. 9. Oh, W.Y., Yun, S.H., Tearney, G.J. and Bouma, B.E., 115 kHz tuning repetition rate ultrahighspeed wavelength-swept semiconductor laser. Optics Letters, 30:3159–3161, 2005. 10. Perju, D., Mecanisms of Fine Mechanics. Politehnica, 1990. 11. Sweeney, M.N., Polygon scanners revisited. Proc. SPIE, 3131:65–76, 1997, http://dx.doi.org/ 10.1117/12.277766. 12. Walters, C.T., Flat-field postobjective polygon scanner. Applied Optics, 34:2220–2225, 1995. 13. Yun, S.H., Boudoux, C., Tearney, G.J. and Bouma, B.E., High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter. Optics Letters, 28:1981– 1983, 2003.

Training Platform for Robotic Assisted Liver Surgery – The Surgeon’s Point of View F. Graur1 , L. Scurtu2 , L. Furcea1, N. Plitea2 , C. Vaida2 , O. Detesan2 , A. Szilaghy2, H. Neagos1 , A. Muresan1 and L. Vlad1 1 University

of Medicine and Pharmacy “Iuliu Hatieganu” Cluj-Napoca, 400012 Cluj-Napoca, Romania; e-mail: [email protected] 2 Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. Minimally invasive surgery was developed as a necessity to improve the quality of life after surgical interventions, to decrease the costs of hospitalization and to shorten recovery of the patient. Rapid development and the growing complexity of minimally invasive surgery are arguments for implementing a comprehensive training method to acquire special technical skills. The paper presents the development of an training platform for hepatic minimally invasive surgery using parallel robots, which focuses on two important aspects: the training of surgeons in laparoscopic liver surgery and the possibility of interactive pre-planning of the surgical procedure in a virtual environment. Key words: training platform, laparoscopic liver surgery, robots, simulation

1 Introduction A minimally invasive procedure refers to the introduction of surgical instruments inside the human body through small incisions trying to minimize the damage of healthy tissue. Minimally invasive surgery (MIS) was claimed in medical practice due to strong impact on the quality of life of the patients by decreasing the rate of complications, time and costs of hospitalization, rapid reintegration in the socioprofessional life and also by offering a better and a wider view of the operating field. Hepatic surgery is one of the most complicated procedures in abdominal surgery, due to the vascularization and internal structure of the liver. Therefore, laparoscopic liver surgery requires many hours of training and practice before performing this type of surgery. A number of simulators dedicated to laparoscopic surgery were created, which is well suited to virtualization. However, until now there is no training system focused on laparoscopic liver surgery. This research represents an important contribution in developing the surgeon’s skills for hepatic surgery because integrates the newest acquisitions of the technique in the training platform like: haptic system, robotic manipulation of laparoscope, voice command, 3D vision, virtual reconstruction of the real organ and the possibility to perform the virtual intervention before the real one [5].

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A laparoscopic liver surgery simulator intends to be a useful tool for young surgeons trying to gain new skills in laparoscopic surgery, as well as for experienced surgeons who wants to simulate a difficult intervention before the real one [3]. Laparoscopic surgery is completely different from conventional surgery which offers wide exposure, tissue contact, binocular vision and the use of traditional equipment. It, thus, demands specialized training [7]. Laparoscopy consists in performing surgery by introducing different surgical instruments in the patient abdomen through 0.5-1 cm wide incisions. The surgeon can see the abdominal anatomy with great clarity by watching a high resolution monitor connected to an endoscope introduced inside the patient abdomen. This technique bears several advantages over traditional open surgery: it decrease the trauma entailed by the surgical procedure on the patient body, it decrease the patient stay in hospitals and the cost of health care, and therefore it reduce the morbidity. If these minimally invasive techniques are clearly beneficial to the patients, they also bring new constraints on the surgical practice. They significantly reduce the surgeon’s access to the patient body. In laparoscopy, the surgical procedure becomes more complex with the decreased number of DOF (degrees of freedom) of each surgical instrument. They must enter the abdomen through fixed points of the incisions. Because the surgeon cannot see his own hand on the monitor, this technique requires specific hand-eye coordination. Therefore, an important training period is required before a surgeon acquires the skills necessary to perform minimally invasive surgery adequately [1]. Currently, surgeons are trained to perform minimally invasive surgery by using mechanical simulators, living animals or virtual reality-based surgical simulators. First method is based on ”endotrainers” representing an abdominal cavity inside which are placed plastic objects representing human organs. These systems are sufficient for acquiring basic surgical skills, but are not realistic enough to represent the complexity of the human anatomy (respiratory motion, bleeding). The second training method consists in practicing simple or complex surgical procedures on living animals (often pigs). This method has two limitations. On one hand, the similarity between the human and pig anatomy is limited and certain procedures cannot be precisely simulated with this technique. Therefore, this method of training is considered dangerous and inadequate. On the other hand, the evolution of the ethical code in most countries may forbid the use of animals for this specific training, as it already the case in different European countries [1]. In addition, the animal laboratories are very expensive to maintain and are far from being cost-effective [7]. About the third training method, there were created a number of simulators dedicated to laparoscopic surgery which is well suited to virtualization. Simulab develops a simple trainer, LapTrainer (www.simulab.com), a system that helps the surgeon to gain basic skills in laparoscopic procedures. The training on The Laparoscopy (LapVRTM) surgical simulator, developed by Medical Immersion (www.immersion.com), shortens the learning curve in laparoscopic surgery. The LapSim System is developed by Surgical Science (http://www.surgical-science.com) and it has multiple teaching programs in a completely virtual environment meant to increase the dexterity of the surgeon in minimally invasive surgery procedures. A more advanced system, The LAP MentorTM multi-disciplinary LAP surgery simulator, created by Simbionix

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(www.simbionix.com), offers training opportunities to new and experienced surgeons for everything from improving basic laparoscopic skills to performing complete laparoscopic surgical procedures. A virtual reality-based surgery simulator intends to be a useful tool for both young and experienced surgeons. However, the purchase and maintenance price of these simulators is high, so their availability is limited. Furthermore, these simulators do not allow training of surgeons in laparoscopic liver surgery. Because of the limitations of current training methods, it is necessary to develop a training platform to present a new completely integrated system of teaching, learning, evaluation, based on modern educational principles, on flexibility and allowing wide accessibility among surgeons. Robotics assisted minimally invasive surgery recently introduced in surgical practice needs a new training phase. In fact, the surgeons must become familiar with robotics systems before moving to clinical applications [2]. On a worldwide level, today only one robotic system has been approved to be used on human patients, da Vinci Surgical System, developed by Intuitive Surgical (www.intuitivesurgical.com). Despite numerous advantages (3D visualization with immersion, ergonomic comfort, reducing the surgeon’s tremor, high accuracy, greater maneuverability of intracorporeal tools), the impact of surgical robots has been very limited due several drawbacks: they are voluminous, occupying a large space in the operating room and, especially, are very expensive [6]. The surgeon’s requirements for a training system for robotic liver surgery are: the haptic system with a easy to use menu and rapid changeable instruments; 3D vision with immersion; voice command to command the robotic arm; a realistic 3D reconstruction of the real environment of the patient in order to perform an accurate simulation of the future real intervention; a good connection with the main system to avoid the delays; an easy way to upload the basic images from the CT. Therefore the strong points of this platform are: the first trainer for laparoscopic hepatic surgery; the possibility of performing of the virtual intervention before the real one; the laparoscope movements by voice command; haptic system for tactile feedback; three-dimensional vision; the use of a robotic system in the training platform for liver laparoscopic surgery; facile changing of laparoscopic instruments from the menus. The weaknesses of this platform are: the necessity of high-tech computers for uploading many 2D images and compiling the 3D structures; some factors from real environment will not be perfectly simulated: heart beats, phrenic movements, haemorrhage, practice of biliary-digestive anastomosis; the necessity of joysticks with haptic system, of the 3D vision system, pedal for electrocautery and voice command; the miss of the robotic arms for active instruments.

2 The Description of the Training Platform for Hepatic Robotic Minimally Invasive Surgery Training platform is structured in two parts (Figure 1): basic system and secondary system.

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Fig. 1 The principle scheme of the training platform.

The basic system that includes the graphic station, the surgeon console with the surgical instruments, the robot for positioning of the laparoscopic camera, the robot control unit and the video application for generating virtual environment. The graphic station is used to process information received from external devices connected and calculates the process of the virtual simulation. Generic instruments for the left and right hand can be instruments with or without force-feedback. They are designed to send data from the haptic devices to the virtual instruments in the computed environment about the movements and forces applied by the surgeon’s hand. The virtual environment is a 3D reconstruction of the internal organs such as

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Fig. 2 Three-dimensional reconstruction of the liver as a virtual environment.

liver with inner structures and surrounding organs. All the structures have assigned consistencies through finite element analysis. The 3D reconstruction is computed using slices from CT-scans of the real patient, therefore, the surgeon can perform a virtual intervention as the first step before performing the real one on that patient (Figure 2). The following steps have to be followed for the generation of such a 3D virtual liver model: CT image acquisition; visualization of CT images with specialized software, processing operation for reconstruction, selecting of interesting regions, segmentation of interesting liver regions (liver, liver vessels, liver segments, possible tumors), 3D generation of a liver. The 3D reconstructions delineating inner structures and tumors of the liver are used by surgeons to analyze the spatial relationships between these elements. The accuracy of the 3D reconstruction depends on: the characteristics of the CT devices; the used protocol by the radiologists; the spatial resolution during the main scanning; the hardware characteristics of the graphical station. The haptic instruments will interact with the virtual environment and will transmit to the surgeon tactile sensations. The experimental model of PARAMIS parallel robot, designed to position and to move the laparoscopic camera, was built in a joint collaboration between the Center for Testing and Simulation of Industrial Robots (CESTER), Technical University of Cluj-Napoca , The Institute of Machine Tools and Production Technology (IWF), Technical University of Braunschweig, Germany and the Surgical Clinic III, University of Medicine and Pharmacy, Cluj-Napoca [4]. The robot control unit serves to transmit the voice command from the surgeon, the keyboard or mouse to the robot control software. The control system of the virtual environment transmits the information from the generic instruments (left and right hand) and from the laparoscopic camera to the video application of the training platform. The 3D complex

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model of the liver is generated by three-dimensional reconstruction with the software Amira using medical images from CT Scan (Computer Tomography Scanning) and MRI (Magnetic Resonance Imaging). This three-dimensional model of the liver includes the internal structure of the liver [3]. In the computed environment, the 3D model has assigned the elasticity, plasticity and flow properties of the liver and its vascularization. The secondary system that is located away from the basic system includes the haptic devices, the 3D display with accessories, the electrocautery pedal, the voice command of the robot and the computer. The virtual modeling of the training platform for hepatic robotic minimally invasive surgery is presented in Figure 3. The haptic devices (5 and 9) represent the left and right hand of the surgeon manipulated by the surgeon which controls the virtual instruments and can interact with the virtual organs of the patient. When a virtual instrument has contact with the organs, based on the texture of the tissue, the haptic device will react and the surgeon will have the tactile feeling of resistance as of the real tissue. The 3D display (6) with its accessories (the emitter, 3D glasses, mirror reflection and the computer graphics card dedicated for 3D visualization) allows the stereo visualization in surgical virtual environment. The electrocautery pedal (12) is used to send information during the virtual operation about cauterization on the virtual tissue, when it is operated. The voice command of the robot is used for motion of the virtual laparoscopic camera and is controlled by surgeon via a Bluetooth microphone (3), using the voice recognition system. The computer (11) processes the data to create interactive virtual environment where the surgeon can work. The training platform makes it possible to simulate laparoscopic liver surgery via internet. Using the internet connection, the surgeon (2) connects to the graphic station which is situated away from him. The stereo visualization of laparoscopic intervention is performed with the 3D display (6), emitter (7), 3D glasses (4), mirror reflection (8) and the computer graphics card (11), which is dedicated for 3D visualization. Surgeons from any location can interact with the virtual environment using common computer peripherals, keyboard and mouse (13). With the training platform, surgeons can perform virtual procedures in laparoscopic liver surgery and pre-planning of the surgery, can see the different procedures and can teach new surgical procedures. Thus, the surgeons can familiarize with simpler or complicated procedures, based on a training protocol. In addition, the surgeons can test and evaluate their skills in laparoscopic liver surgery, leading to the improvement of the surgical act with an impact on the postoperative evolution. The training platform is able to simulate the surgery procedures with the help of robotic systems designed for minimally invasive surgery. For this, it was built a parallel robot designed to position the laparoscopic camera in minimally invasive surgery [4]. The addressability of this training platform is large, being represented by young surgeons who are mainly preoccupied by liver laparoscopic surgery, as well as experienced surgeons interested in obtaining a competence in the hepatic minimally invasive surgery. The teaching methodology for the training platform will include a theoretical course followed by the simulation of the laparoscopic intervention. The surgeon will be logged on the training site and will have the joystick with haptic

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Fig. 3 The virtual modeling of the training platform for hepatic robotic minimally invasive surgery.

system, the 3D vision system, the electrocautery pedal and the voice command at the training console. At the beginning there will be used some standard models, without anatomic variants or pathological lesions and the modules for simulation of complications will not be activated. Later, after some training, will be used models compiled from real cases with tumors or other lesions and the modules for complications will be active. Each simulation will evaluate the candidate regarding specific parameters in order to quantify the acquisitions on the learning curve.

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3 Conclusions The paper presents the concept of an advanced training platform using parallel robots for training and pre-planning in hepatic minimally invasive surgical procedures. This new developed system intends to help surgeons to develop their skills in laparoscopic liver surgery and to offer the possibility to determine the best approach in a procedure before actually performing it. The integration of the robots in the platform allows surgeons to get familiar with such new systems, work with them and contribute to their further development.

Acknowledgement The research work reported here was financed by the PNCDI-2 P4 Grant, entitled “Innovative development of a virtual e-learning system for hepatic laparoscopic surgery – HEPSIM” and the PNCDI-2 P4 Grant, entitled “Multidisciplinary development of surgical robots using innovative parallel structures”. They have been awarded by the Ministry of Education, Research, Youth and Sports of Romania.

References 1. Delingette, H., Simulation d’interventions chirurgicales. In: Deuxime Journes de la Recherche en Robotique JNRR 1999, Montpellier, pp. 109–118, 1999. 2. Friconneau, J.P., Karouia, M., Gosselin, F., Gravez, Ph., Bonnet, N., and Leprince P., Force feedback master arms, from telerobotics to robotics surgery training. In CARS 2002, pp. 1–6, Springer, 2002. 3. Graur, F., Pisla, D., Scurtu, L., Plitea, N., Cote, A., Lebovici, A., Draghici, S., Furcea, L., Muresan, A., Neagos, H.C., and Vlad, L., Liver 3D reconstruction modalities – The first step toward a laparoscopic liver surgery simulator, IFMBE Proceedings, 26:251–257, 2009. 4. Pisla, D., Plitea, N., and Vaida, C., Kinematic modeling and workspace generation for a new parallel robot used in minimally invasive surgery. In Advances in Robot Kinematics: Analysis and Design, Springer, pp. 459–468, 2008. 5. Plitea, N., Vlad, L., Popescu, I., Pisla, D., Graur, F., Tomulescu, V., Vaida, C., Furcea, L., and Forgo, Z., E-learning platform for hepatic robotic minimally invasive surgery using parallel structures, Acta Technica Napocensis, 51:23–28, 2008. 6. Shoham, M., Burman, M., Zehavi, E., Joskowicz, L., Batkilin, E., and Kunicher, Y., Bonemounted miniature robot for surgical procedures: Concept and clinical applications. IEEE Transactions on Robotics and Automation, 19(5):893–901, 2003. 7. Udwadia, T.E., Guidelines for laparoscopic surgery. Indian Journal of Medical Ethics, 5(2), 1997.

Teaching Mechanisms: from Classical to Hands-on-Experiments and Research-Oriented V.-F. Duma Faculty of Engineering, Aurel Vlaicu University of Arad, Romania; and Institute of Optics, University of Rochester, Rochester, NY 14627, USA; e-mail: [email protected]

Abstract. Science and engineering teaching are a challenge in the modern world, and in particular in Romania. We present the steps we have taken in the last decade and in particular in the last five years, to improve Mechanisms teaching. We have gradually passed in this process from classical to hands-on-experiments and to research-oriented activities, in order to provide the proper background for each student to find personal motivation and to develop – both as a specialist and as a more fulfilled person. Several student projects we have coordinated are pointed out, and our vision on a modern school of engineering is compared to and confirmed by some of the realities of the American academic system. Key words: higher education, students, research, mechanisms, mechatronics, Fulbright

1 Introduction A French philosopher, O.M. Aivanhov once said: the teaching children and students receive in schools and universities does not address their heart or their soul, much less their spirit; it addresses only their intellect . . .

Teaching has always proved a major challenge. Passing knowledge, shaping abilities, awaking in students a real drive for science, engineering or humanities, are issues of all professors, in any school. We are now facing throughout the globe a sort of drifting away from science and engineering of young generations in favor of humanities and social studies. On the other hand, due to the high progress of the last decades and to the globally developed technological context, science or engineering-oriented students are better prepared and motivated when they begin university studies. This is not a sociological paper, but we have to admit that, due to the accelerated technological progress all science education faces challenges it never encountered before. Is it losing ground because we do not know how to provide students with a more complete, a more human approach? Would science education not be even more efficient this way? These are questions we have to ask. Teaching Mechanism is facing its own particular challenge: it belongs to a ‘classical’ field, while students are mostly interested in novelties. Mechanisms are in

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the broad engineering knowledge but most interesting today are interdisciplinary fields, e.g. robotics and mechatronics. One has therefore to provide, when teaching, both basics and insights in rapid developing, fascinating fields (thus addressing the ‘heart’ of students, as well). This study is a contribution with practical solutions we developed to respond to the general aspects listed above and to the particular ones that address science (not only mechanisms) teaching in Romania. The entire country witnessed two decades of major changes. Education was no exception. Where are the students in all this – sometimes – turmoil? They are actually the silent mass so far for several reasons: (a) small education fees (one does seem not to have much to lose if one does not count time and energy spent); (b) lack of democratic exercise in expressing one’s opinions and interests; (c) not knowing another system and not being too aware of one’s possibilities; (d) a still autocratic behavior in schools at all levels, where the teacher/professor is perceived as a ‘master,’ not as a more experienced partner in the path to professional development, valued for his/her expertise. What can be done to fight this inertia? To get students out of this silent phase and to arise their interest for professional and personal growth – in everyone’s interest? We speak of ‘knowledge society’ at a European level, but what can be done for students in science in a country that is in a ‘transition’ that is now older than most of them?

2 Looking for Solutions How can one provide human comfort when teaching science? This is a question I have asked myself many years ago, while dissatisfied with the lack of emphasis ‘classical’ teaching put on personal development of students. It seemed that their stay in school is conceived like a period ‘in brackets,’ a suspension of real life in order to concentrate on study – aspect that they reject anyway. These two, life and study should ideally merge and students must feel both passion and accomplishment, not only from learning things but from applying them, even at an early stage of their development; from integrating university years into their life, while indeed using them for their primary purpose. For this they must find motivations strong enough to keep them both from distractions and from (too) early jobs that are both a problem in Romanian universities nowadays. In this need for personal growth, a balance seems necessary for the students to get not only information, but also (or primarily) confidence in what they can do, self respect based on the potential they have proven themselves; therefore, a balance between intellect and ‘heart.’ Another (related) aspect is giving students the possibility to interact, to accomplish team work. That is a critical aspect; something you realize you have to learn yourself first, in order to teach others, e.g. in project-based work, involving mix teams of undergraduate (UG), postgraduate (PG) students and faculty. This latter commitment exists in some places, but it is not a tradition or one of the scopes of the system. However, we have to make it one to be able to interact correctly within our institutions and in larger clusters; also to prepare graduates that

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can integrate fast in the team-work environment (of professional but also human relationships) which is a must in any company. Without the intention of being original or of providing a recipe, we shall give in the following a brief presentation of the steps we have taken and of some of our results. Our focus will be on the last 5 years, when things really began to move. Hopefully this one of the myriad of experiences will be of benefit to others. A confirmation of our approach was received in the contact with the American university system, through the Fulbright Fellowship experience of the author, during the 2009– 2010 academic year at The Institute of Optics, University of Rochester (UoR), NY. Relevant parallels will be pointed out, with regard to the activity in this top research university, rated 77th in the international ranking in 2009, but 1st in Optics in USA. Some of the next steps to follow, learned from this experience, have to be highlighted. The presentation of our efforts in improving the activity in Mechanisms (but also in related fields, e.g. Mechatronics) will be done ‘from top to bottom,’ i.e. from the most high-ranked and difficult activity, i.e. journal publications, to the most basic one, i.e. class teaching. The reason for this approach is that we have concluded that one cannot improve an activity from within its boundaries. One has to move to the ‘higher’ level, to see what the requests for its outcomes are, in order to efficiently ‘move’ things in the right direction at the ‘base.’

3 Research-Oriented Solutions (1) Journal of Mechanical Engineering (ME): A solution to the above problems, including students’ motivation, had to come by first providing a stimulating place for opinions to be expressed and for students, even UG, to have their efforts recognized. A Scientific Bulletin in ME was therefore my first step. This journal was launched in 2005 [13]. It gained national recognition from NURC in 2006. Students, both UG and PG are encouraged to submit papers: from some 7 articles/issue (4 issues per volume), an average of 1.5 has student contributions. To make a parallel, when arriving at UoR one finds a journal, JUR (Journal of Undergraduate Research) entirely focused on publishing peer-review articles, from all domains, but only from UG students (a more radical solution, one may say). (2) Student Sessions of Scientific Communications: This tradition in Engineering at ‘Aurel Vlaicu’ University of Arad (UAVA) expanded every year (mid-May). For the 2007 and 2008 editions I organized, approximately 20 papers were presented each edition (several from the Mechanism and Mechatronics Student Research Group I founded in 2003). These sessions prove a good opportunity for students to learn to communicate, prepare and make presentations. This is of course the outcome of an involvement in a research topic supervised by a faculty. It is also an indirect way of stimulating UGs in classes. It is only an apparent paradox that supplementary work (but one out of personal choice) actually enhances – through commitment – the performances of a student. It implies a supplementary effort from behalf of the

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Fig. 1 Study of the manufacturing process of a gear with z = 8 teeth and module m = 10 mm: (a) experimental stall; (b) simulation; (c) AutoCAD simulations (X = −2 mm profile movement).

faculty as well, but that is again only apparent, because faculty benefits as well from the work, time, and sometimes really good ideas of students, in developing their research topics. It is a win-win situation. In Fig. 1 such results on simulation of the manufacturing process of gears are presented [9]. To continue the parallel, at UoR there are two such sessions at the Institute of Optics, in October and April. They are much better designed: students’ oral and poster presentations and one day of employment interviews with interested companies (that contribute to the Institute) are programmed, on the University’s expenses. This is certainly a good idea for Romanian universities, in order to support students and graduates. So far, our students have only been encouraged to expand, translate and polish in English, then to submit the research as a paper to the Scientific Bulletin. A more practical approach is however necessary. (3) Student participation at Conferences has been encouraged, for the Symposium at UAVA every two years (in October), but also for other Conferences, including international. Besides the students in Mechanisms, those in Automation and Automotives are especially active in this direction – from the Faculty of Engineering. The Proceedings of UAVA [5] had in 2008, by example, 16 papers with UG and PhD students as first authors or co-authors out of a total of 39 papers. (4) Students Scientific Group in Mechanisms and Mechatronics was established in 2003, after two stimulating Students Seminars I organized in 2002. Brainstorming proved a good way to point out issues left aside in the everyday routine. Consulting the students for solutions at their lack of satisfaction (perceiving what has to be changed) remained a procedure ever since. The activity in the Students Group is organized on two steps: (i) With 2nd year students that are willing to participate. They usually conclude their activity at the end of that year, by building a mechanism model (Figs. 2, 3) [8, 11] or with an advanced study, with or without a communication at the Students Session. At least three models and experimental stalls are built every year. (ii) With the most passionate students that continue their activity and usually conclude it with a paper, at least. Some of them (although very few, yet) go further

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Fig. 2 Mechanisms with gears (a, b) and cams (c) developed by students in 2008–2009 (UAV).

Fig. 3 Linkages built by students (UAV Mechanism Laboratory) in 2008–2009: examples.

on to prepare their diploma using this research. Thus they benefit from 2 or 3 years of more relaxed, but intense and challenging work for their diploma thesis. This allows them to grow as specialists, it stimulates their entire work throughout university and it allows them to deepen a subject, while eliminating the inefficient stress of the ‘last minute’ (even last semester) diploma preparation. They are most likely to become some of the best engineers, and, why not, faculty of tomorrow. To whatever extend this research-oriented activity goes, it allows the student to develop self confidence, seeing that he or she is capable of solving more real-life problems, and to recognize (by practice) the value and utility of the information (and science & engineering education) students received. This is more stimulating when the topics approached are in the ‘first line’ of international research. Actually the professor should not give students trivial or purely didactic problems. That is the road to failure of this method. However difficult, it has to be challenging to be stimulating. Of course, a balance has to be kept, with regard to the level and capacities of the student(s). Assigning individual themes or designing compatible teams of 2 or 3 is another challenge – for the professor. Two such projects are presented here, involving (for the above reason) not only mechanisms, but optomechatronics: (a) polygon scanners [8], studied experimentally [10] and in MathCAD after an indepth theoretical study [2] (Fig. 4); (b) optical choppers, with the same research ‘trajectory’ [4, 6] (Fig. 5). Another example of complex study in 2008–2009 involved two students that completed after an Erasmus stage in Denmark their diploma with a project on a hydro-turbine on the Mures River that they built in part. In a research university like UoR, some half of the UG and most PG (Master) students are implied in research. For my Fulbright year there I worked within the group in Robert E. Hopkins Center for Optical Design an Engineering that was created to provide students with a computer lab, a fabrication lab, and a testing

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Fig. 4 (a) Experimental stall for the study of polygon scanners [2]; (b) scanning function [8].

Fig. 5 (a) Experimental stall for the study of optical choppers [4]; (b) transmitted light flux [6].

lab, that all, from UG to PG can access to develop hands on experience. Activities with the Center are integrated as part of lectures, to provide a place for experiencing hands on learning. Also several classes have a lab component to give students handson experience with knowledge taught in the classroom. In Romania, this is rather the exception than the rule. However it should become a goal here as well.

4 Hands-on-Experiments This approach, of research-orienting students, emerged after some years of gradually moving from classical teaching to hands-on-experiments. While strong theoretical background has to be provided, it is common knowledge that students are best motivated when practical work is involved. This can be done on different levels: (i) mechanism models for every lecture and seminar, so that the students may get a grasp of the real thing, even when doing theory; (ii) group contests in the lab hours; (iii) developing mechanisms of their own; (iv) deepen, even at early stages, subjects that may lead to papers and diploma preparation (Figs. 4, 5). An interesting experiment was done in 2007–2008 with a group of students in “Tools, devices and control apparatuses” [12]. After a year of labs and a onesemester project, we were able (but the project was initiated from the start) to ‘produce’ a book with the most relevant topics. Only students willing to make a supplementary effort participated (14 out of 32) and they each produced a chapter. I

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coordinated but also participated with them, adding three chapters to the final book that was published one semester after completing [3]. It was a productive experience, a great memory for the participants (each received three copies, as authors), and the feeling of doing something useful: the book is to be used by their fellows next years. One of the students expanded his chapter as a journal paper [1]. All those described here has also provided students with some instruction in technical writing, a topic that is not yet present in UAVA, and to the best of our knowledge, nor in most Romanian universities. At UoR such an instruction is provided, while the groups are much smaller (10–20 students) for each subject. This helps interactions in classes and it represents the real student-centered education.

5 Improving Classical Teaching In lectures, the use of mechanisms models every class awakes and mantain the students’ interest. Together with problem solving, doing things with a purpose is helpful in a results-oriented society. It is also efficient to point out first the ‘destination point,’ discuss what is required from a mechanism to perform the task, than move to a ‘classical expos´e’ (with models available at all times). In labs, a ‘game’ technique is applied. Students are divided, e.g. in four subgroups, each with a different mechanism (but of the same type). Their assignement is similar, but given the different mechanism, each subgroup has to work independently. The 1st subgroup to obtain correct results gets 2 points bonus at each individual grade; the 2nd gets 1 point, the 3rd 0, while the last one gets a penalty (–1 point). Each subgroup’s results are presented and discussed. The competition is stimulating, usually fun and it teaches students much more than mechanisms, actually. They learn to interact, to work as a team, to manage their time, to assign and fulfil duties. They learn more about themselves, their qualities and flaws. All grades obtained in labs do contribute to the final exam grade. As in seminars and labs, there is a free dialogue. Brilliant answers get bonuses, added to lab grades to improve L, Eq. (1). Projects in mechanisms unfortunately dissapeared at UAVA with curricula reduction to four years. However, each student receives an individual homework (HW) to submit before each test that concludes the corresponding chapter, i.e., structure, kinematics, kinetostatics and dynamics. Two chapters, gears and cams, are not included in the HW due to lack of time. They are approached at seminars and labs in more detail. The results of each part of HW are given to students before they take the test so they may see how much they have understood of each topic. Tests and HWs are in agreement with reducing the weekly hours of class teaching and ‘moving’ the weight center of the students’ activity towards individual study. For this to be effective, individual study with a feed-back must be provided. The final grade is calculated with:
 G=

∑

i=1,6

Ti + 2(L + HW)

10 + BR + BP

(1)

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(formula presented at the beginning of the semester) where Ti , i = 1 . . . 6 are test grades, L and HW are averages of lab and homework grades, BR = 1 . . . 2 is the bonus for research work (in the Students’ Group), and BP is the bonus for attending classes (0.5 for the entire semester). The system is flexible and adapted for each class level. In American universities this tests & HWs system is a tradition, while independent study is a must, in conditions that support such a demand.

6 Conclusions We presented our approach in mechanisms and engineering teaching. Although each must follow one’s personal style and beliefs, ‘moving’ from classical teaching to hands-on-experiments and to research-oriented activities has become a must. So has providing a more ‘human’ approach to subjects which get more specialized and more difficult with the amount of increasing information in the field. Last, but not least, a convergent approach of life and education, of professional and personal growth are also an increasing necessity, to my belief.

Acknowledgments The study was made possible by PN II IDEAS Grant 1896/2008 of NURCUEFISCU, Romania, and by Fulbright Scholar Senior Research Grant 474/2009 of the US Department of State, host Professor Jannick Rolland, whom I thank for the useful suggestions. Grateful thanks to all my professors and teachers who have contributed to my education – in all respects.

References 1. Berar, T. and Duma, V.F., Devices of orientation of pieces with irregular outline. Sci. and Tech. Bulletin of Aurel Vlaicu Univ. of Arad, Mech. Eng., 3:28–40, 2007. 2. Duma, V.F., Novel approaches in the designing of the polygon scanners. In: Proc. SPIE, Vol. 6785, 67851Q, 2007, http://dx.doi.org/10.1117/12.756730. 3. Duma, V.F. (Ed.), Tools, Devices and Control Apparatuses. Experimental Works. Aurel Vlaicu Univ. Arad, 2008 [in Romanian]. 4. Duma, V.F., Theoretical approach on optical choppers for top-hat light beam distributions. Journal of Optics A: Pure and Applied Optics, 10:064008, 2008. 5. Duma, V.F. (Ed.), Proceedings of the 2nd International Symposium Research and Education in an Innovation Era, Engineering Sciences, Arad, November 20–22, 2008. 6. Duma, V.F., Nicolov, M., and Kiss, M., Optical choppers: Modulators and attenuators. In: Proc. SPIE, Vol. 7469, 7469-3, 2009. 7. Eckhardt, H.D., Kinematic Design of Machines and Mechanisms. McGraw-Hill, 1998. 8. Marshall, G.F. (Ed.), Handbook of Optical and Laser Scanning. Marcel Dekker, 2004.

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9. Patrusel, V. and Duma, V.F., Simulation of the manufacturing process of gears using toothed rack tools (MAAG). Sci. and Tech. Bull. of Aurel Vlaicu Univ. of Arad, Series: Mech. Eng., 3:41–51, 2007. 10. Pelle, R. et al., Experimental stall for the study of polygonal mirror scanners. Sci. and Tech. Bull. of Aurel Vlaicu Univ. of Arad, Series: Mech. Eng., 4:37–48, 2008. 11. Perju, D., Mecanisms of Fine Mechanics. Politehnica (1990 [in Romanian]. 12. Vonica, C.-P., Design of Devices, Vols. I–IV. Aurel Vlaicu Univ. Arad, 2005 [in Romanian]. 13. www.uavsb.xhost.ro.

Models Created by French Engineers in the Collection of Bauman Moscow State Technical University D. Bolshakova and V. Tarabarin Bauman Moscow State Technical University, 105005, Moscow, Russia; e-mail: [email protected], [email protected]

Abstract. Study of the history of higher technical education, of its first stage in particular has attracted a lot of interest over recent years. One of the main constituent parts at this stage of academic training of engineers was collecting models of mechanisms and machines. Every technical school and university had such a collection. In this article we shall observe models developed and made in France in the XIXth century and collected in BMSTU. Short descriptions of certain models and background information of the constructors are given. Key words: history of science and technology, higher engineering education, models of mechanisms and machines, constructors and manufacturers of models

1 Introduction Nowadays, computer-aided technologies have reformed most fields of human activities. In such circumstances, the society faced acute problems with structural adjustments of educational system as a whole and technical or engineering education in particular. To avoid gross mistakes it is necessary to study the history of engineering education buildup starting from its very first stages. Such analysis of the ways of engineering training development allows to single out the most successful methods created during the period of its existence. One of the principles widely used in engineering training is use of aids training and practical training. Models of mechanisms and machines were exceedingly important at this stage. Being scaled down and simplified replicas of real objects they retained all the main functions of a machine and allowed to visually demonstrate the transformation of simple processes of a machine into the necessary (often complicated) end-effector movements.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_58, © Springer Science+Business Media B.V. 2010

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1.1 Teaching Models of Mechanisms and Machines Models of mechanisms and machines were used in the laboratories and assemblies of Academies of Science as early as the XVIIIth century. Robert Willis, Ferdinand Redtenbacher, and A.-J. Morin [4, 5] were among the first who began using models in teaching. The first demonstration models were developed by the professors themselves or by the engineers supervised by them. The models were made in the workshops of a school or a university, in a single copy as a rule. Most of those models are lost now. As the number of technical schools and universities grew a demand arouse for a considerable amount of teaching models and in the middle of the XIXth century small firms in Germany and France started production of such models. Thus we know models of Reuleaux made by G. Voigt and I. Schroder [4, 6, 8]. In France, models are made by Bourdon, Eugene Philippe, Alexander and Pierre Clair, Jules Digeon and his son [1, 3]. Models are developed by order of the National School of Arts and Crafts and by order of a number of technical museums, schools and universities.

2 Families of Clair and Digeon – Constructors and Teaching Model Manufacturers Pierre Clair (1804–1870) laid the foundation of a generation of French machine engineers who constructed and manufactured models of mechanisms and machines under the trademark of “Clair”. Pierre Clair was born in a rural family. He started his professional carrier as an apprentice in a handicraft workshop [1]. Then he entered the school of mines in Saint-Etienne where he was distinguished for his remarkable abilities. After finishing his studies at school Clair together with Thimonnier and Ferrando took to engineering and manufacturing of thread sewing machines. With his first sewing machines Pierre Clair comes to Paris where he tries to bring them in production. In Paris he is captured by a new idea. Rather soon he launches his own manufacture of equipment for silkworm farms and for silk thread production. P. Clair’s works received high appraisal from the specialists. At the national exposition of 1839 he wins a bronze medal and meets professor of applied mechanics at the National School of Arts and Crafts Arthur-Jules Morin. This encounter gave start to a long-term profitable and mutually beneficial cooperation. Since 1840 P. Clair is a nominated supplier of the National School of Arts and Crafts and remains so till 1855. That was the noonday of the school. By the orders of Jules Morin he manufactures models of mechanisms and machines meant for teaching mechanics. Pierre Clair creates unique models that can be safely called masterpieces. Clair’s workshops were situated at 5, Rue Duroc. Today it is the Blind association of Valentin Hauy (Fig. 1). Having mastered the manufacture of teaching models (Fig. 2) Clair started the production of measuring tools which are also distinguished for high quality. The products of “P. Clair” are becoming

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Fig. 1 Blind aid society of Valentin Hauy, which is now situated at the former Clair workshops.

Fig. 2 The models designed and manufactured by the firm of Pierre and Alexander Clair (photo from the museum “Conservatory of Arts and Meters”, Paris.)

more and more popular; the models are exhibited in international expositions and win different prizes. In 1855 Pierre Clair delegates his management authorities to his son Alexander (1831–1880), Authorized mechanic and good manager. A. Clair keeps up his father’s business successfully. But products under the trademark of “A. Clair” reflect the trends characteristic of French industry of the late XIXth century. Mechanisms and machines created by Alexander are standardized and mass produced. Their physical configuration is not so perfect as before and the material used for production is iron casting and steel instead of the more expensive copper and brass. Alexander Clair diversifies the assortment of manufactured products. He starts making training matrices that demonstrate the ways of joining up different materials (e.g. wood and metal) or geometric figures and constructions. Alexander participates in numerous expositions, wins prizes and reaches international level which allows him to expand his customer base for account of foreign educational institutions. Thus he supplies models of devices and machines for the Museumlaboratory of Aldini-Valeriani in Bologna, which now possesses 23 models by Clair and for the Museum of religious school in Quebec where 44 models are kept. The collection of BMSTU comprises six models by Clair [2,7]. The alumnus of the School of Arts and Crafts of Chalons sur Marne Jules Digeon founded his first enterprise – a horseshoe workshop – together with Lucien Dancre who had a patent for that type of activity [3]. But the enterprise soon went bankrupt, so they had to sell their personal property to settle their debts. To keep his family Jules Digeon was forced to hire himself out as an engineer draftsman to some constructor in Pontoise. His talent as a draftsman allows him to enlist to the National School of Arts and Crafts where in 1872 he becomes the teacher of perspective and shadow projections at the course of professor La Gurnerie. Later on he gets the posi-

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Fig. 3 The models designed and manufactured by the firm of Jules Digeon and his son (photo from the museum “Conservatory of Arts and Crafts”, Paris).

tion of tutor of graphics in the Polytechnical School and in 1878 he gets the position of professor of drawing at the Lycee Louis-le-Grand. By virtue of his position he entered the scientific world and the system of higher technical education. Digeon became member of Association of civil engineers, Association of national industry stimulation and Association of higher education. Later these connections will help him in signing up orders and in selling his products. In 1873 Digeon opens a workshop producing models of mechanisms and machines. At first the workshop executes orders of Conservatory of Arts and Crafts. Digeon attends to repair and modernization of models made earlier and he also designs and constructs new models (Fig. 3). Academic programmes of the Conservatory vary and the Digeon workshop creates numerous models of machines for chemical production, paper manufacturing, models of farm machinery. Besides, Digeon makes drafts of various machines for the manufacturing department of the conservatory. All together he made over 300 models for the collection of the Conservatory. Digeon’s workshops expand as the number of orders grows. In 1882 Clement Ade, an engineer, involves Digeon in development of a lightened steam engine for an airplane. In 1888 the famous Gustave Eiffel orders him a gold-plate model of his tower which was being constructed, 1/50th of the real size. At the World exposition of 1889 Digeon wins the gold medal in the category of models for higher education. At the time the workshop of Digeon providing models for higher and technical schools and museums in the country and abroad had about 40 employees. On the models of the galleries of the Conservatory Digeon creates a demonstration room at his workshop where he exhibited models for sale. In 1888 Carl, son of Jules Digeon, becomes the head of the workshop and the workshop transforms into “Jules Digeon and Son”. Jules Digeon died in Paris in April of 1901. After his death Paul-Emile Duhanot – ex-director of the workshops – purchased the firm. For some time the workshop continued manufacturing models mainly for the museum of Lille and the Conservatory. At the beginning of the XXth century there appeared several new firms engaged in manufacturing teaching models. However production of models would reduce and practically stopped in the period of World War I.

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3 Models of Mechanisms and Machines by Clair and Digeon in the Collection of the BMSTU The collection of models of mechanisms and machines of the chair of Theory of Machines and Mechanisms of BMSTU comprises 6 models by Clair and two models by Digeon [7]. Models by Alexander Clair: (1) Ratchet pawl with non-reversible step motion of the output element; (2) Overrunning friction clutch coupling of Dobo; (3) Frontal disk-type variable-speed friction drive unit; (4) Geared transmission formed by eccentric gearwheel and non-circular gear; (5) Geared transmission with non-circular gears. (6) Hyperbolic gearing. All these models were manufactured in the workshops of A. Clair in Paris, France in 1875–1880. All the models are placed upon a wooden leg have cast-iron bedding, most of the details are made of steel, some are made of bronze and brass. Let us see the models in details. We start with the ratchet gear model (Fig. 4) designed for conversion of oscillatory motion of the lever to the ratchet wheel incremental motion in same direction. The model consists of the balance arm 1, two ratchets 2 and 3 and wheel 4. During the balance arm raising the ratchet tooth 3 slides along the ratchet wheel tooth, and ratchet tooth 2 prevents clockwise rotation of wheel 4. At the end of rising motion the ratchet tooth goes down into the cavity and gears into the wheel tooth. During the falling motion of the lever 1 ratchet 3 turns the wheel 4 counterclockwise by angle increment. Ratchet tooth 2 in the meantime slides along the wheel tooth and at the end of the motion becomes engaged with it. Afterwards every oscillation of the motion lever is repeated. Free-wheeling clutch (Fig. 4b) is a free wheel mechanism. It represents a kind of self-acting clutches transmitting rotary moment in one direction only. Free-wheeling clutch is turned off upon exceeding of the driven link angular speed as related to the driving link, providing for free running of the driven link. Design of the clutch was suggested by French Engineer Dobo. The model consists of a drum with the internal cylindrical functional surface fastened onto the shaft 2, and a lever connected with shaft 1. Two pads are mounted on the lever, on the rotating pairs, that ate pressed against the functional surface of the drum with plate springs. When shaft 1 rotates counterclockwise, pads turn and slide as related to the drum. Changing direction of rotation causes jaw choking in the drum in this case the drum and the lever turn as one unit. Front frictional disc variable-speed gear model (Fig. 4c) consists of two flat discs 1 and 2 mounted to the input and output shafts. End planes of these discs contact the end planes of the two intermediate discs 3 that are pressed against them by the spring from both sides. Discs 3 on friction type bearings C are mounted on adjusting lever 4. Lever is connected to the stand by the rotating pair D. Motion of the lever and connected discs changes the unit gear ratio. The model (Fig. 5a) represents a single gearing consisting of two gear wheels: round 1 and elliptical 2. Wheel spindle 1 is offset from its geometric center by the amount of eccentricity. Wheels are located relative to each other in such a way that teeth in the major axis of one wheel would be engaged with the teeth in the minor

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Fig. 4 The models from collection BMSTU designed and manufactured of the firm of Clair.

Fig. 5 The models from collection BMSTU designed and manufactured of the firm of Clair.

axis of the other one. Gear ratio of the unit within one rotation is variable. During the input wheel rotation with constant speed the output wheel will be rotating with variable speed. Gear ratio of the unit determined by the ratios of its teeth is a constant value and makes two. Model of the gear drive with out-of-round toothed wheels (Fig. 5b) consists of two identical elliptical toothed wheels 1 and 2. Wheels are located relative to each other so that the teeth in the major axis of one wheel would be engaged with the teeth in the minor axis of the other one. Gear ratio of the unit within one rotation is variable. During the input wheel rotation with constant speed the output wheel will be rotating with variable speed. Gear ration is maximum, when teeth in the minor axis of the input wheel are engaged with the teeth in the major axis of the output wheel. Ratio of teeth in the model wheels is same, gear ratio (gear ration of the wheel teeth) makes one. The sixth model in the collection of BMSTU is a crossed-axis helical gear or a hyperbolic gearing mechanism (Fig. 5c). The model is probably designed and produced in A. Clair’s workshop. It can be said from the model design. Doubts occurred because “A. Clair” nameplate in the model is practically illegible. The model consists of the two helical or screw gears 1 and 2, with the crossing axes. Hyperboloids of rotation are axoidal surfaces of the screw gears. Cones or cylinders inscribed in the hyperboloids are normally considered as pitch surfaces of screw gears. In this case engagement is approximate. Screw gears did not obtain a wide circulation because they are difficult to make. Digeon’s models are represented in the collection of BMSTU by two units only. First one is a centrifugal friction clutch mechanism (Fig. 6a). Only part of this model

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Fig. 6 The models from collection BMSTU designed and manufactured of the firm of J. Digeon and son.

survived, with a manufacturer’s mark, serial number and place of manufacture. In the second model (Fig. 6b) the manufacturer’s mark has been lost, there is only a cavity in the place where it originally used to be. However, comparing this model with famous Digeon models indicates that by design features it can be assigned to this company’s models. The model consists of two friction mechanisms that represent pitch surfaces of tooth gears and several detachable tooth profiles. Inscriptions in the profiles are made in French, denoting the country of origin. The following inscriptions are engraved in four profiles out of five: C4 – “Rayon et Epicycloide” (Radius and epicycloid), B1 – “Corde et Epicycloide” (Chord and epicycloid), B3 – “Developpantes” (Reamers), B2 – “Hypocycloide et Epicycloide” (Hypocycloid and epicycloid). Some details have been lost in the model that is why it is difficult to get an exact idea of its purpose.

4 Conclusions From today’s point of view, gear principles and types so brightly illustrated in teaching models developed by P. and A. Clair would surprise no one. But in those days many of these mechanisms had been innovations. Before 3D-modeling appeared a new design solution should originally have been represented as a model or as a prototype. Working successfully in this field P. and A. Clair, together with other French mechanical engineers of XIXth century made an invaluable contribution towards technological development. Besides, during over a hundred years in different countries of the world several generations of engineers and technicians completed their training in the models developed by them. Nowadays products labeled “Clair” represent not only historical value, but also antique. They become interesting as collectibles and as objects for investments in the unique period pieces.

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Acknowledgement This work was made as a part of the collaboration project by and between BMSTU and LARM University of Cassino “Application of Model of Mechanisms and Machines in Teaching Machine Mechanics”.

References 1. Emptoz, G., Pierre et Alexandre Clair, constructeurs de modeles. Revue du musee des arts et metiers, 3:4–10, 1993. 2. Golovin, A. and Tarabarin, V., Russian Models from the Mechanisms Collection of Bauman University, History of Mechanism and Machine Science, Vol. 5, Springer, 2008. 3. Louis, A., Jules Digeon, l’age d’or du modle rduit. Revue du musee des arts et metiers, 45, 2006. 4. Moon, F., The Machines of Leonardo Da Vinci and Franc Reuleaux, Kinematic of Machines from Renaissance to the 20th Century, History of Mechanism and Machine Science, Vol. 2, Springer, 2007. 5. Redtenbacher F., Die Bewegungs-Mechanismen: Darstellung und Beschreibung eines Theiles der Maschinen-Modell-Sammlung der polytechnischen Schule in Carlsruhe, F. Bassermann, Heidelberg, 1866. 6. Schruder, J., Illustrationen von Unterrichts-Modellen und Apparaten, Actien-gesellschaft, Polytechnisehes Arbeits-Institut, Fabrik f. Unterrichts-Modelle, Zeichen-u. Mal-Gerathe, 2 Band, Buchdruckerei von H. Uhde, Darmstadt, August 1902. 7. Tarabarin, V., Golovin, A., and Tarabarina, Z., The historical part of collection models of machines and mechanisms of TMM’s Department of the Bauman Moscow State Technical University. In Proceedings of 12th International Symposium on Theory of Machines and Mechanisms, Besanc¸on, 2007. 8. Voigt, G., Kinematic Models after Reuleaux, Catalog, Berlin, 1907.

The Models of Centrifugal Governors in the Collection of Bauman Moscow State Technical University N. Manychkin, M. Sakharov and V. Tarabarin Robotics and Complex Automation, Bauman Moscow State Technical University, Moscow, Russia; e-mail: [email protected], [email protected], [email protected]

Abstract. In the period of development of higher technical education great importance was given to the visibility in training of engineers. This was realized with the help of models of machines and mechanisms. Collections of models of mechanisms were formed practically in every engineering school. This article describes the collection of centrifugal governors of BMSTU collected at the end of the 19th century. Some models were brought from Europe; others were designed and manufactured in the school (IMTS). The article gives a brief description of some models and basic principles of their operating (operation); it also provides information on the history of their creation and production. Key words: history of the science and mechanisms, theory of the regulation, steam engine, centrifugal governor, model of mechanism, 3D model

1 Introduction Organization of training courses that were included into the discipline of Applied Mechanics occurred in the 1970s. In different universities curriculums were different. In the Moscow University (MSU) and Imperial Moscow Technical School (IMTS) these courses were created by Orlov [7]. In IMTS at the department of Applied Mechanics future engineers were read: the theory of machines, the theory of mechanisms, hydraulics, thermodynamics, strength of materials and theory of steam engines. The course of the theory of steam engines was high demanded at that time, and theory of regulating was an important part of that course. This course provided a description of various models of centrifugal governors and the basics of their theory. Models of centrifugal governors of various designs were manufactured in the workshops of IMTS and purchased in Germany for demonstrations on lectures and study on the practical lessons. Some of these models survived to this day. This article deals with these models, provides an analysis of their design and classification, describes the results of animation of these models based on 3D modeling.

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Fig. 1 Steam engine from Munich museum. Photo was taken by V. Tarabarin.

2 Regulation of the Steam Engines It is known that the steam engine is an external combustion heat engine that converts the energy of hot steam into mechanical work (Figure 1). In the drive of steam engine driving force and external loads vary both within one cycle and for longer periods of time. Uneven rotation of shaft of the machine causes additional dynamic loads and affects the quality of products. E.g., in the textile industry this leads to uneven thickness of the thread or to its breakage, in process of cutting this leads to increase of surface roughness and to reduction of accuracy [9]. Control devices are used to reduce the unevenness of the rotation, they are: flywheels, which accumulate kinetic energy and reduce the fluctuation of the velocity inside the cycle, and centrifugal governors, which maintain constancy of the speed at longer intervals. For the first time governors have been used to regulate distances and efforts between millstones of windmills and watermills. In 1788 Scottish engineer James Watt [5, 10, 11] applied such a governor to maintain the uniformity of rotation of the shaft of the steam engine. The classic Watt’s governor is a centrifugal governor of direct action, because the movement of coupling has direct impact on the value of the driving force, decreasing or increasing the amount of steam (Figure 2).

3 Types of Centrifugal Governors According to the operating principle regulators are divided into static, astatic and pseudoastatic. The last were later developed based on astatic to eliminate their shortcomings [3, 7, 10]. Static governors are such governors, where each value of shaft speed corresponds to its equilibrium position of loads and coupling. In astatic governor loads and coupling will be in equilibrium only at one value of the velocity of

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Fig. 2 Watt’s centrifugal governor, where 1 – shaft, 2 – loads, 3 – coupling, 4 – lever arms.

the shaft in any possible position of governor elements. At a speed different from the specified value equilibrium is impossible. Let us consider classic Watt’s centrifugal governor. In the initial position of the governor the shaft rotates at some constant speed ω 0 , at the same time the moment of the centrifugal force cannot overcome the moment of gravity forces of loads 2 and the coupling 3. With the increasing speed the moment of the centrifugal force increases, exceeds the moment of gravity forces and turns levers 4, moving up loads 2 and coupling 3. Moving of the coupling leads to the displacement of control levers and reduction of steam. As a result, rotation speed of the shaft begins to decline and loads and coupling move down. In addition to the gravity forces and inertial forces there are the forces of friction, acting on the elements. Under their action coupling stops above or below the final position for a given shaft speed. The greater the difference between actual and desired position of coupling, the more the actual speed differs from the nominal value. Obviously, the speed range corresponding to equilibrium positions of the governor should be as small as possible. This reduces the regulation error and unevenness of rotation. Watt’s classic static governor has a large range of equilibrium and is highly uneven. However, it has been widely spread and successfully worked in steam engines of small capacity. Many scientists and inventors were engaged in its improvement and created its new structural varieties. One of these scientists was the great Russian mathematician and engineer Chebyshev [1, 2]. In his writings Chebyshev develops a method of determining basic sizes of Watt’s governor in order to achieve the best astatic characteristics while maintaining the original simplicity of design. For optimal results, he designs levers of complex shape with variable width and radius of curvature. Governors designed by Chebyshev were produced in IMTS workshops. Two of these models are preserved in the collection of BMSTU. There are also fully astatic governors. E.g., Garnett’s governor, in which astatic characteristics are achieved by design features [7, 11]. Loads are attached to a firm

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Fig. 3 (a) Chebyshev’s centrifugal governor, (b) another Chebyshev’s centrifugal governor from the collection of BMSTU.

rail frame, which provides a parabolic path of their movement (Figure 4a). The same principle is realized in the model of another governor, where a heavy coupling is designed in a form of a cover, the inner part of which serves as the sideways for the path of divergent loads (Figure 4b). It may seem that astatic governors, allowing only one equilibrium speed, are very convenient in practice. However, in the process of regulating their coupling always slipped equilibrium, and the governor performed, in fact, oscillatory motion, which was totally unacceptable. Thus, the desire to get complete astatic characteristic was a mistake. Speaking about inapplicability of astatic regulators it is necessary to mention another great Russian mathematician and engineer Vyshnegradsky [4, 5]. He came to a conclusion that the governor without introducing additional elastic forces and forces of friction (provided by a spring or cataract), cannot provide good movement regulation. Astatic regulators are absolutely not suitable even when using these devices. Pseudoastatic governors turned up to be the best for practical application, which replaced static ones and became popular. Pseudoastatic governors combined a sufficient stability of static governors (each configuration corresponding its equilibrium rate), and thus they allowed only minor variations of adjustable speed. The design of pseudoastatic governors provides various ways of reduction of velocity fluctuations. The first pseudoastatic governor from BMSTU collection is Kley’s governor (Figure 5a). This model is characterized by intersecting rods. Such construction realizes a simple, but effective mechanism of reduction of velocity fluctuations. The disadvantage of this model is its large size. Pr¨oll’s governor (Figure 5b) is a successful modification of Kley’s governor, providing the same small speed variations at twice smaller sizes. The upper lever arms are bent, to provide the free movement of loads, coupling has large mass and pear-shaped form.

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Fig. 4 (a) Garnett’s centrifugal governor, (b) the governor with the massive cover-like muff, (c) centrifugal governor with massive ring. Governors are from the collection of BMSTU.

Fig. 5 (a) Kley’s centrifugal governor, (b) Pr¨oll’s centrifugal governor. Governors are from the collection of BMSTU.

The collection of BMSTU contains an original design of centrifugal governor, which was produced using drawing of Redtenbacher [8]. The inertial element of this governor has a shape of a massive ring (Figure 4c). There is a groove in the upper part of the ring at the arc of approximately 160◦; this lightens one side of the ring. As a result, in the static position the ring is located with a slight slope to the vertical axis. With increasing frequency of rotation of the shaft inertia forces turn the ring around the horizontal axis and it turns into horizontal position. The model is designed and manufactured in 1862 by Ivanov Pavel, Moscow Craft School (MCS) student.

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Fig. 6 (a) Research facility, (b) theoretical diagrams, where 1 – diagram for Watt’s governor, 2 – diagram for Pr¨oll’s governor, 3 – experimental diagram.

4 Experimental Characterizations Models of mechanisms were used in the training not only for the demonstration at lectures and seminars. It served as objects for technical drawing and sketching, for experimental studies. To verify this application of models authors conducted an experimental determination of the working characteristics of one of the models of governors. Pr¨oll’s governor was used as the object of research (Figure 6a). Since it was impossible to equip a model with an engine which could produce a variable speed, then the following experimental method was used. At the end of the lever clutch control a vertical ruler was installed. To measure the frequency of the shaft controller we used a frequency chronometer F5041, a signal voltage of 1.5 volts from alkaline battery was supplied to its input. The circuit was opened with plate spring, which contacted with the tops of the teeth of a bevel wheel controller. During the experiment, the model speed was set by hand, providing the movement of the coupling from lower to upper position. In front of the facility a digital camera was installed, which was used to photograph the facility, so that you can determine the position of the lever clutch and the frequency meter readings in the photo. Photography was done randomly, until there were a sufficient number of points to build a diagram. Using the experiment results we built the working characteristics and compared them with the calculated characteristics (Figure 6b). The analysis showed a minor difference between the experimental and the theoretical characteristics [10]. Moreover, theoretical curves were compared with the Watt’s governor. Pr¨oll’s governor has a smaller range of equilibrium rates and steeper performance.

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Fig. 7 3-D models of: (a) Pr¨oll’s governor, (b) Garnett’s governor, (c) Chebyshev’s governor.

5 Computer 3D Modelling of Governors In modern world of multimedia systems of distance studying practical application of full-scaled physical models becomes impossible. Here we can successfully use their photos and video fragments. In a relatively short period of time (about 50 years) the collections of models in the majority of technical universities were lost. Interest in saving the collections appeared about 15 years ago [6], however, the problem of saving the collections of models exists today. There are large collections of models in technical museums of London, Paris, Munich and Moscow, but they are often stored in vaults and not available for visitors. Also we should not forget about their fragility and uniqueness, many models survived only in single copy. With the development of computer technologies it has become possible to create 3D models and films, based on these models. Such films are useful in lectures-presentations, as they are compact and well understood. They can demonstrate models in motion, show it in perspective, and demonstrate the assembly of model from individual parts. Storing 3D models on digital media allows you to keep the models themselves, as well as translate them into drawings and make copies. In this work some governors were recreated in 3D models. The development of this project is done via software Autodesk Inventor 2010. A library of components was created for each governor – bases, loads, rods, gears, fasteners and so on. From these elements the whole models were assembled, that fully reproduce the mechanisms (Figure 7).

6 Conclusions Centrifugal governors were among the first mechanical systems of automatic control. They are important monuments of history of technology, historical background of creation of theory of regulating. The study of their constructions, ways of their

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development, principles of regulating embodied in them is important not only for history of science and technology. Models of the governors demonstrate the level of design and manufacturing of engineering products in the mid-19th century, technological capabilities of small-batch production. They are interesting object to study engineering ideas of the era of steam engines.

Acknowledgments This work was performed under the project of cooperation between BMSTU and LARM University of Cassino, “Application of models of mechanisms for learning the mechanics of machines”.

References 1. Artobolevsky I.I. and Levitsky N.I., Mechanisms of P.L. Chebyshev. A Collection of “Scientific Heritage of P.L. Chebyshev”, USSR Academy of Science, 1945. 2. Chebyshev P.L., A Comlete Set of Works, USSR Academy of Science, 1946. 3. Maxwell J.C., On governors. Proceedings of the Royal Society of London, 16:270–283, 1868. 4. Maxwell J.C., Vyshnegradsky I.A., and Stodola A., Theory of Automatic Regulation, USSR Academy of Science, 1949. 5. Merkin D.R., Introduction to the theory of stability of motion, Science, 1976. 6. Moon, F., The Machines of Leonardo Da Vinci and Franc Reuleaux, Kinematic of Machines from Renaissance to the 20th Century, History of Mechanism and Machine Science, Vol. 2, Springer, 2007. 7. Orlov F.E., Steam Engines. Lections of Prof. Orlov F.E., IMTS, 1890. 8. Redtenbacher F., Die Bewegungs-Mechanismen: Darstellung und Beschreibung eines Theiles der Maschinen-Modell-Sammlung der polytechnischen Schule in Carlsruhe, F. Bassermann, Heidelberg, 1866. 9. Steam Engines. History, Description and Their Application. St. Petersburg, print. Eduard Prace and Co., 1838. 10. Zernov D.S., Theory of Steam Engines, IMTS, 1896. 11. Zernov D.S., The Building of Steam Engines. Lections of Prof. Zernov D.S., IMTS, 1895.

Advanced Approaches Using VR Simulations for Teaching Mechanisms S. Butnariu and D. Talaba Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: {butnariu, talaba}@unitbv.ro

Abstract. This paper describes a new methodology for teaching mechanisms at mechanical engineering courses based on Virtual Reality technology. Laboratory exercises are a fundamental part of teaching for many fields of science. Classical laboratory methods can be constraining because of time, expenses, scale, safety, or complexity of the real life exercises. Modern tools such as Simulink (part of Matlab software) and software based on Virtual Reality (VR) allow engineers and students to simplify algorithms and models of mechanisms, creating thus new teaching materials in order to increase the capacity of knowledge transmission and absorption. The proposed methodology integrates the above tools for the specific case of articulated mechanisms. The used software, hardware and methodology are described in detail, as well as the results. Key words: Virtual Reality, mechanism, teaching method

1 Introduction Rapid advances in information technology are reshaping the learning styles of students in many higher education institutions. Today, the standard desktop interface is complemented by multiuser virtual environment in which peoples interact with each other as well as augmented realities. Higher education institutions can increase the pedagogical efficiency by basing their strategic investments on using these emerging educational technologies to match the increasingly diversified learning needs of their students [3]. On the other hand, the models used in industry for the study of mechanical systems become more and more complex. Many learning situations require the student to mentally transform 2-D objects into dynamic 3-D objects. To understand the science concepts, learners may need to translate among reference frames, to describe the dynamics of a model over time, to predict how changes in one factor influence other factors, or to reason qualitatively about physics processes that are best explored in a 3-D space.

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The 3-D modeling tools available in education may allow students to engage in the kind of concept, building activities necessary to promote the development of conceptual understandings [1]. This paper presents an approach to integrate Virtual Reality tools into an innovative teaching and learning concept that reduces the need for classical experiment involving expensive artifacts. Extensive research, conducted recently in various parts in the world, dealt with this topic. Thus the approach in [8] is based on crossmodal interactions between visual and haptic perception. This study looks at a simulation on the principles of levers using both visual and haptic feedback. Grimaldi and Rapuano presents in [4] a teaching method where the virtual laboratory (VL) represents the environment in which the learning activities are performed. All the above research demonstrates the attractiveness of the proposed concept, especially the advantages of involving haptic feedback [4, 8]. Nevertheless, in all cases, commercial haptic devices are used that are customized to the application. In our approach we propose the use of dedicated haptic devices that are not generic but designed from the very beginning for the specific application. In this way the space occupied by the device itself is minimized, because less DOF are involved. The paper is structured as follows: the first part introduces the virtual reality tools used; in the second part the used resources in experimentations are presented. The third part defines the methodology, shows how the generic model is developed to integrate each different model. We finally present the conclusions of the study, results and discussions.

2 Virtual Reality for Experimentation Virtual Reality is a field of study that aims to expose the user to a synthetic experience that mimics the real life counterpart. The experience is dubbed synthetic, illusory, or virtual because the sensory stimulation to the user is generated by the system. For all practical purposes, the system usually consists of various types of displays for delivering the stimulation, sensors to detect user actions, and a computer that processes the user action and generates the display output. To simulate and generate virtual experiences, developers often build a computer model, also known as virtual worlds or virtual environments (VE) which are, for instance, spatially organized computational objects (called the virtual objects), presented to the user through various sensory display systems such as the monitor, sound speakers, and force feedback devices [5]. Virtual Reality allows the user to ignore the physical reality, in order to experience a virtual change in time, place, and/or type of interaction: interaction with an environment simulating reality or interacting with an imaginary or symbolic world. The Theory of Mechanisms and Machines is an area of science education very appropriate to employ haptic interfaces and may involve experiments that require the learner to both apply and respond to force feedback [8].

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Fig. 1 The haptic system developed.

3 Resources The course that we are reporting on this paper is delivered during 4 weeks, 2 hours a week. The creation of 3-D computational models has traditionally required advanced computer hardware and programming skills. However, recent advances in 3D modeling wysiwig editors, coupled with the declining cost and increasing power of personal computers, has opened up opportunities for students to build themselves complex 3-D models. According to these advances, we used for 3-D modeling SolidWorks software. To develop the mathematical model, we used components of Matlab/Simulink software. Simulink is a platform that has most of the same functionality of Matlab, but allows engineers to develop systems graphically, with a block diagram interface. Simulink has standard blocksets that allow the user to implement common tasks, such as I/O, summers, signal routing, scopes, etc. With SimMechanics (part of Simulink), engineers who design controlled mechanisms can simulate both their mechanical and control systems in the same Simulink environment. The engineers can now automatically build mechanical models from their SolidWorks assemblies and simulate the dynamic motion of the system. The new version of SimMechanics provides the ability to collaborate and share designs in Simulink, allowing thus iterating and improving the product design. To create the files for Virtual Reality simulations we used software modules, such as VrmlPad or VRbuild of Matlab. Nevertheless, the 3-D model and its operation must be seen into a Virtual Reality viewer program. For this reason, we used BS Contact [2]. Additionally, VRML is a platform-independent language and is easily viewed over the Web using a free plug-in and a Web browser. Once Simulink, SolidWorks, VrmlPad and BS Contact were the chosen software, hardware had to be chosen also to properly support the integration. We use a dayto-day portable PC, based on Intel Centrino processor, running Win XP. It has analog and digital I/O, three USB ports and typical PC connections such as keyboard, mouse, and VGA. To generate the torque needed for the haptic return, we developed

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Fig. 2 The methodology.

a specialized 1 DOF haptic system composed of the following components: power source 24 V DC, a positioning controller by Maxon, type EPOS 70/10, electric motor by Maxon, EC-powermax, brushless, 200 Watt, fitted with a planetary gearhead type GP 42 C, with gear ratio 1:50. The controller is connected to the PC through a serial link USB (Figure 1). On the output shaft of the gearbox, a 200 mm length crank is mounted, to deliver the torque generated by simulated virtual mechanism.

4 Methodology and Results In principle, the student task could be formulated as to cover sequentially, the following steps (Figure 2): (i) Analyzing and creating the CAD model. To analyze a mechanism, first, a CAD model is needed in order to identify the components, their position and the couplings between them. Usually it begins by the geometric modeling components. To this aim, the students will design and build a 3-D model using a specialized modeling software (SolidWorks). Geometric objects made in SolidWorks can now be saved directly in VRML format. VRML is used for developing 3D objects. It is based on the behavior concept where each object in a scene possesses certain qualities. These qualities include data array sets such as Position, Colour and Sensor. Motion in a VRML world is achieved using interpolators (nodes that take a set of values, which define a range, and generate the intermediate values automatically). Input to the interpolators is known as key and keyValue. An array for a key consists of three values. These are the intersection values and can only range from 0 to 1. keyValue also consists of three sets of values, with three numbers per set separated by commas. Each set of these values represents the position data of an object [7].

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Fig. 3 Graphical results in SimMechanics.

(ii) Creating and verifying the Simulink model. To transfer the model manually by the CAD model, without errors, is difficult and involves a series of programming knowledge. A more rational and rigorous solution is to use the software packages that automatically perform this task. On the other hand, to study the behavior of the model in a real operating context requires its dynamic modeling. The software used is Matlab (Simulink), whose module SimMechanics allows dynamic modeling of CAD models developed with SolidWorks. It is however required to export the CAD model prior to using the SimMechanics and this is performed by SimMechanics-CAD-Translator program. Next, a multi-body system model is defined, composed of bodies connected by joints. The geometrical model will include SimMechanics equivalent coresponding bodies and joints, body coordinate systems and constraints. Each body corresponds to an element block, and allows SimMechanics be connected to Simulink blocks so that the result can be saved and reused in different areas [6]. To create the SimMechanics model mass characteristics are needed for each body as well as the characteristics of the links defined in SolidWorks. The listing is saved in an XML file generated by SimMechanics. This file allows an automatic creation of an appropriate SimMechanics model, by using a specific Matlab command: mech import. After the automatic creation, the model can be verified by SimMechanics by simply running the simulation. Note that not all models can be transformed through restrictions of the elements generated by SolidWorks. Once at this point, we can study the dynamic behavior of components using the SimMechanics facilities. We use Joint Sensor blocks, connected to different joints and Scope blocks to view the behavior of some parameters: linear/angular position, velocity, acceleration, computed force/torque and/or reaction force/torque (Figure 3). (iii) Connecting with Virtual Reality programs. Visualization of the simulation in SimMechanics is not a good solution, since the quality is poor. To obtain high quality views and dynamic interactions with the model, we propose the use of Virtual Reality technology that requires making the connection between the mathematical models generated by SimMechanics and the Virtual Reality viewer, i.e. BS Contact. We can use different types of sensors (Body Sensor blocks, connected to the coordinate system of the body), which measure any combination of translational

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Fig. 4 Several proposed mechanisms.

Fig. 5 Student testing a virtual mechanism with the proposed system.

position, velocity, acceleration and rotational orientation, angular velocity, angular acceleration. These measured values have to be send to the VRML model in order to drive the mechanism motion. For this, we use the Go To blocks, one for each measured value. On the other hand, for receiving signals sent by SimMechanics model, a model with blocks of Virtual Reality (VR Sink) is built which uses connectors From. All the obtained values from the mathematical model will be sent to the controller of electric motor in order to generate a torque to haptic device handle proportional to the input data into the system. (iv) Results. Currently, at the disciplines studying the mechanisms, the practical laboratory work are conducted by studying a physical model of the mechanism (a scale model) and by using analytical methods in order to calculate its kinematics and dynamics. In order to study new mechanisms, new models are needed, but this thing lead to increasing the dedicated time. To study a wider range of mechanisms, creating new methods to unfold the lesson is involved, taking into account limited material resources. The laboratory lesson will take place in the following structure: (1) exposure of problem, included examples from industry, (2) objectives, (3) used resources, (4) methodology, (5) output data, (6) conclusions. To carry out the lesson, several types of mechanisms are presented to students, some of which are represented in Figure 4. Their construction, geometry, masses and operating characteristics will be input data for laboratory work. In this early stage there are two options to start the Laboratory: kinematic and dynamic calculation made using analytical methods or using the facilities offered by the MATLAB software. In order to reduce time, the method which uses MATLAB will be chosen; the other method will lead to resolve the problem by writing a program using C++ language. Each student will choose one type of mechanism and then, in order to solve the problem, they will follow the steps outlined in sections 4 (i), (ii) and (iii). Data entry can be modified at any time, having provided different results. Thus, geometric features can be mod-

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Fig. 6 The haptic device: EC-powermax motor and planetary gearhead.

ified in the model, characteristics of material bodies (specific gravity), constraints and loads acting on a physical model. Any change to the input data will lead to changes in kinematic or dynamic features of the studied mechanism; it may be observed in view mode and at the haptic level. Each student will work individually on own computer. After making CAD models, Simulink, VRML files and connect to the VR programs, they will perform a computer simulation, connected to the visualization device: ordinary display, Head Mounted Display HMD, 3D display or CAVE system (Figure 5) and to haptic device. We list some of the advantages of this method: short time for learning, low material resources, a lot of different types of mechanisms studied, possibility of input data modification, possibility of output data faster, 3D visualization, haptic reaction. This method can be applied to students of the departments of Mechatronics and Robotics, assuming knowledge of Matlab, Solidworks, programming controller EPOS. Simulink modules combined with Virtual Reality graphics allow control of the position, rotation and image size set in virtual reality, resulting in a stereoscopic 3D animation. 3D scenes are described in VRML language and the haptic feedback of the mechanism, depending on its characteristics, can be felt thanks to the device presented in Figure 6.

5 Conclusions We can now conclude that the use of specialized haptic device bring improvements in the methods for teaching the mechanisms science. We are able to provide a cheap and effective method to study the behavior of a wide range of mechanisms, using the facilities offered by some CAD software and Virtual Reality technology. We have presented a methodology to study kinematics and dynamics of mechanisms, including all steps for conversion the CAD model into the mathematical model and to export to VRML virtual scene to obtain quality simulations, as well as haptic feedback delivered to the user in a very similar manner as in a real experiment. Additionally, the proposed application allows to quickly evaluating the effect of

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changes in design, making thus the students able to correctly evaluate the various design decision. The haptic interface provides force feedback to the user, allowing the user to feel the mechanical resistance to a force exerted. The haptic device with 1 dof presented in the paper is sufficient for the purpose of feeling the resistance of a mechanism with 1 dof. The device is not claimed to be generic but only for mechanisms with mobility M=1. When we test a mechanism that does not exist yet as a physical prototype, overlapping the virtual repsesentation of the mechanism helps students to see the entire system in the 3D space, without the need of a physical prototype. The advantage of the method proposed is that it allows the presentation of a wide range of mechanisms and also allows the students to feel how the mechanical resistance opposed by the mechanism changes when design is modified. A haptic interface allows the students to feel the mechanical resistance opposed by a mechanism without the need to use a physical model. SimMechanics model and simulation in Virtual Reality Toolbox provides information about the dynamic behavior of mechanical components. By simulating the behavior of mechanical, control, and other dynamic systems, engineers and students improve the quality of their practical works and shorten the time required to create a new design. Also the SimMechanics model can be used in the design of control components of mechanical systems as well as modeling purpose in other areas.

Acknowledgement We would like to thank our coleagues F. Garbacia and H. Erdely for their work concerning haptic application.

References 1. Barnett, M., Yamagata-Lynch, L., Keating, T., Barab, S.A. and Hay, K.E., Using Virtual Reality computer models to support student understanding of astronomical concepts. The Journal of Computers in Mathematics and Science Teaching, 24(4):333–356, 2005. 2. Bitmanagement Software GmbH, http://www.bitmanagement.com/en/products/interactive-3dclients/bs-contact, 2010. 3. Dede, C., Planning for neomillennial learning styles. Educause Quarterly Magazine, 28(1):7– 12, 2005. 4. Grimaldi, D. and Rapuano, S., Hardware and software to design virtual laboratory for education in instrumentation and measurement. Measurement, 42:485–493, 2009. 5. Kim, G.J., Designing Virtual Reality Systems: The Structured Approach. Springer-Verlag, London, 2005. 6. Mathworks, Inc. SimMechanics References 3, http://www.mathworks.com, 2009. 7. Ong, S.K. and Mannan, M.A., Virtual reality simulations and animations in a web-based interactive manufacturing engineering module. Computers & Education, 43:361–382, 2004. 8. Wiebe, E.N., Minogue, J., Jones, M.G., Cowley, J. and Krebs, D., Haptic feedback and students learning about levers: Unraveling the effect of simulated touch. Computers & Education, 53:667–676, 2009.

Novel Designs

Preliminary Design of ANG, a Low-Cost Automated Walker for Elderly J-P. Merlet INRIA Sophia-Antipolis, 06902 Sophia Antipolis Cedex, France; e-mail: [email protected]

Abstract. We present the overall design of ANG, a new walking aid for the elderly. The objectives of this walking aid is to provide a low-cost and low-intrusivity aid for the elderly. ANG is based on a classical 4-wheel Rollator, the two rear wheels being fixed while the two front are caster wheels. The rear wheels have been motorized, a bistable clutching mechanism allowing to clutch and unclutch the motors at will. ANG will provide fall prevention, navigation assistance, walking aid and may also be used as a rehabilitation tool. We present in this paper the mechanical design of ANG and some of the modes in which it may be used. Key words: walker, elderly aid, assistance robotics

1 Introduction The ability to walk is one of the most important and fundamental function for humans. A consequence of aging societies is that the number of elderly is rapidly increasing in many countries. Such people may suffer from locomotion handicap because of the decline of physical and muscular strength. A related problem is the fall: in France in 1997 there has been 2.5 millions elderly falls with over 40% of people over 60 years old concerned by this problem, which may lead to dramatic consequences. This paper focuses on the design of a walker-type support system with the following motivations: • low cost: although assistance robotics addresses helping elderly, the cost of system such as humanoid robots is by far too high to be afforded by a vast majority of elderly. Our purpose is to design a low-cost system that is however offering a large panel of functionalities, that will be presented later on; • low intrusivity: social acceptance is a key issue for assistance robotics. Our goal is to design an assisting system that look familiar; • interface: elderly have various physical and cognitive abilities and consequently various interfaces to control the system must be available.

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Fig. 1 The ANG walker in a typical elderly flat. The two motors are located behind the solar panel while the electronic is located below the motors

Various passive and active walker have been proposed in the past: the passive walker of MARC at Virginia University [1],the active PAM-AID and its extension GUIDO [10], PAMAID [14], NURSEBOT [4], the sit to stand devices MONIMAD of LRP [12] and of Chugo [2], IWalker [9], RT-Walker [8], the sophisticated CAREo-BOT [5] and the omnidirectional walker of Chuy [3].

2 Design ANG (Assisted Navigation Guide) is based on a commercially available 4-wheels Rollator walker, the two rear wheels being fixed while the two front are caster wheels (Figure 1) and hence is similar in principle to the NURSEBOT and iWalker. A first modification was to motorize the two rear wheels with 12V cc 157W DC motors whose velocity is reduced by two levels of belted drives (with reduction 1:2.1 and 1:2) and a planetary gear (with reduction 1:15). The high reduction ratio leads to a system that is mechanically irreversible. Encoders with 2048 impulses per turn have been added directly on the rear wheels and we consider having also encoders to indicate the direction of the front wheels. There is a braking system on the handles and the elderly may seat on the walker. The originality of the motor system of ANG is that it uses a clutch system based on a computer-controlled bistable spring electro-magnet that allow to clutch and unclutch the motors through a Oldham coupling. The interest of such clutch is that it uses energy only during the transient phase (Figure 2). The motors rest on a plate that is located about 45 cm from the ground while the electronic part and batteries are located below the plate. A direct consequence is that the center of gravity (CoG) of ANG is about 20 cm below the CoG of the original walker, thus ensuring a better stability. The wheel diameter is 0.2 m and the reduction ratio is 1:63, while the motor maximal speed is 8300 rpm, leading to a maximal velocity of about 1.38 m/s (roughly 5 km/h) which is quite higher than the average velocity of elderly.

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Fig. 2 The clutch system for the motor of ANG. The electro-magnet is located below the belt pulley on the left side.

The weight of ANG is about 20 kg, to be compared to the 180 kg of Care-O-bot 3, and it is planned to be used both indoor and outdoor. Further planned equipments are: • front and lateral ultra-sound sensors, rear infra-red distance sensors located all along the handles of the walker and ultra-sound sensors looking backward and close to the ground • a GPS locator, compass, accelerometers, RFID receiver and a webcam; • force sensors in the handle of the Rollator; • a 5W solar panel at the rear of the Rollator; • tactile and joystick interfaces. The control part is based on low-cost USB building blocks Phidgets1 that allows one to manage various sensors and provides sophisticated motor control. An on-board computer (currently a micro-laptop with a wifi connection) manages the various modes of ANG.

3 Functionalities Two basic modes are available for ANG: the free mode in which the motors are unlatched and the system basically behave like the initial Rollator, although the wheels rotations are measured and the motor mode in which the motors are clutched and provide either braking power or motion help. In a normal walking mode we believe that the free-mode should be preferred as much as possible for various reasons:

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• to maintain the cognitive/physical activities of the elderly; • in the motor mode it will always be difficult to determine the intent of the elderly and even with the best control approach the user will always feel a slight delay in the answer of the walker, that will be negatively resented; • active walker present usually a larger task completion time than passive walker [14]. However the motor mode is necessary as will be illustrated in the following sections.

3.1 Fall detection and prevention As mentioned in the introduction fall is a major problem for elderly. This issue has been addressed by Hirata with the passive RT-walker that provides only braking power to the rear wheel. Using distance measurements of several laser rangefinders and a simplified 2D human kinematics model with seven links a stability region is determined for the human center of gravity (CoG). If the CoG is outside the stability region fall is assumed and a fall preventive algorithm based on the braking of the rear wheels is applied [8]. We plan to use a similar method but with several improvements. The 2D kinematic model used by Hirata may allow one to detect vertical and horizontal fall but not lateral one. Furthermore if a fall occurs in spite of the walker behavior the passive RT Walker cannot provide a support to the elderly. In our free/fall-detection mode the distance sensors, the force sensors in the handles, the acceleration of the walker and the velocities of the wheels will be used to detect a possible fall, while the motors are unlatched. Our first experiments have shown that a horizontal fall can simply be detected by monitoring the wheels velocity. To manage the lateral and vertical fall we will rely on an array of distance sensors located all along the handles of the walker, ultra-sound sensors to locate the foot position. The sensors will be used to locate specific points of the joints and links of 3D human model. An initial calibration procedure will allow one to place the sensors in the optimal position and to identify the main geometrical parameters of the 3D model for the current user. Being given the usually slow velocity of the walking of elderly we believe that the sensors measurements will allow us to locate his CoG and to detect all types of possible fall. If a fall is detected the walker moves in the fall prevention mode and the motors are clutched to try to prevent the fall. For a vertical or lateral fall the best strategy will be that the velocity of the walker become 0, providing a strong support to the elderly. This may not be the best strategy for a horizontal fall as the weight of the elderly may cause a sliding of the walker even if the wheels are blocked by the motors. It may be better to adopt an anti-skid approach in which the walker moves along the fall direction, progressively braking but avoiding the skidding of the wheels. In spite of the prevention scheme falls may still occur: the location of the elderly with respect to ANG will be determined by using the distance sensors and the walker will come close to the elderly to provide a support. Here again the walker sensors

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may be used to determine the pose of the elderly on the ground in order to present the walker in the best support position. If the elderly still cannot get up we will use a smart object (namely a Nabaztag) that can call for help using voice synthesis or even make a call phone to a rescue center. The rescuer may then even move the walker around using a web based navigator tool (see the motion section) to assert the gravity of the fall and to reassure the elderly.

3.2 Trajectory, Motion planning and Navigation assistance The purpose of motion planning for a walker is to provide a navigation help that may be local (e.g. follow a wall using distance sensors) or global (e.g. determine the trajectory to reach a given location, using, for example, a GPS). Trajectory planning addresses the safety of the walker motion (e.g. braking when the walker moves downhill, avoiding obstacles, etc.). Several trajectory algorithms and motion planning methods, directly derived from the motion planning and mobile robotics community, have been proposed in the past either for active or passive walkers [4, 6, 7, 9] but the utility of such tool has yet to be demonstrated as clinical evaluations have shown a negative opinion of the elderly [15]. Trajectory and motion planning algorithms will also be available for ANG but we plan to add several motion functionalities. In the slave mode the motors are clutched and ANG will follow or be in front of the elderly, being at a constant distance from him. This will allow the user to move temporarily away from the walker (for example to fetch a piece of grocery). Such mode will use the distance sensors to determine the location of the elderly but will require sophisticated control algorithm as soon as the elderly will turn. The distance-constraint mode is a variant of the slave mode with the user having his hands on the handles and the distance threshold being fixed to the average value of the distance to the hip when the elderly is using the walker in the free mode. This will allow to adapt the velocity of the walker to the velocity of the user. If the distance threshold is decreased below the average distance value the walker moves in the assisted mode which allow to provide a motion help, drawing the elderly when moving uphill or braking when going downhill. We plan to implement also a teleoperated mode in which a tactile interface and a simple joystick will allow the elderly to control the walker motion within a limited distance. ANG will also be able to work in an autonomous mode by using the ultra-sound and IR sensors for local navigation. This may be completed outdoor by a navigation mode using GPS information, that will allow the walker to guide the elderly to reach a specific location. An important aspect of ANG will be situation monitoring: the walker will be used on a regular basis for specific trajectories (e.g. moving from the bed to the kitchen) and its motion may be recorded. As the physical abilities of elderly change slowly

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with respect to time monitoring these trajectories may help to anticipate difficulties for the elderly.

3.3 Rehabilitation Walking rehabilitation and training is another use of walkers [11, 13]. In the rehabilitation mode the elderly will have to complete specific trajectories as determined by a doctor while ANG may exert a resistive force or introduce some planned disturbances. The originality of ANG will be to store all sensor data during a rehabilitation session for analysis by a doctor. Data will be downloaded through the wifi connection and the doctor will be able to modify on-line the training exercises during a session.

3.4 Energy management Energy management is a crucial issue for assistant robots, although energy performances is most of the time neglected. Walkers have an extensive use and it cannot be accepted that the batteries fail in the middle of this use. It is quite difficult to find the energy consumption of other walkers but we believe that the six 12V 1.2Ah lead-acid batteries of ANG will lead to at least the same motion range (10.9km) than PAMAID [14] although the battery power is much less than the Care-O-bot 3 (60Ah at 50V). As walkers are often used outdoor the 5W solar panel may extend this range. We plan however to implement a battery management system (BMS) that will use the motor energy when walking downhill to charge the batteries, prevent the user to use the walker with a low battery state, allow for a solar follower charging mode (the walker will rotate according to the sun motion using the compass and the charging current of the solar panel) and an automated docking mode in which the walker at home will autonomously dock to charge its batteries.

3.5 Interfaces ANG is planned to be used by elderly that may have various level of physical abilities and may be reticent to use a hi-tech device. Furthermore the abilities of this type of user may change on the long time. Hence interfaces is a major issue for acceptance and effective use by elderly. Sensors such as force sensors in the handle or distance sensors may be used to detect the user’s intents, manage local navigation (e.g avoiding obstacles or following a wall) but one should take care of ambiguous or even erroneous user/sensors inputs. Furthermore a fully automated walker may not solicit the elderly physical/cognitive abilities, while managing on his own the

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walker may be a good training exercise. Hence we plan to provide several simple interfaces allowing the user to manage the mode and motion of the walker. The use of these interfaces will be monitored, especially during specific trajectories of the walker that are performed every day, to detect a loss of ability to use the current interface (and possible propose a new one) and possible motion problems.

4 Conclusions We have presented the preliminary design of an assistant walking aid that combines the advantages of passive and active walker. The main ideas of this design is to propose a low-cost, low-intrusivity walker solution. Using an already accepted mechanical design is a key solution for social acceptance. We also believe that a key issue for a walker is to use an appropriate 3D human kinematic model that may be used for fall detection/prevention and determining the user intents. This model however has to be adapted to the elderly morphology and physical abilities, that may change over time. The proposed walker is also intended to be used as a calibration tool that will allow to determine the physical parameters of this human model.

References 1. Alwan M. et al., Stability margin monitoring in steering-controlled intelligent walkers for the elderly. In Proc. AAAI Fall Symposium, Arlington, 4–6 November, pp. 1509–1514, 2005. 2. Chugo D. et al., A moving control of a robotic walker for standing, walking and seating assistance. In Proc. Int. Conf. on Robotics and Biomimetics, Bangkok, 21–26 February, pp. 692–697, 2008. 3. Chuy O. et al., Motion control algorithms for a new intelligent robotic walker in emulating ambulatory device function. In Proc. IEEE Int. Conf. on Mechatronics and Automation, Niagara Falls, July, pp. 1509–1514, 2005. 4. Glover J. et al., A robotically-augmented walker for older adults. Technical Report CMU-CS03-170, CMU, Pittsburgh, August 1, 2003. 5. Graf B., An adaptive guidance system for robotic walking aids. Journal of Computing and Information Technology, 17(1):109–120, 2009. 6. Graf B. and Schraft R.D., Behavior-based path modification for shared control of robotics walking aids. In Proc. 10th Int. Conf. on Rehabilitation Robotics, Noordwijk, 12–15 June, pp. 317–322, 2007. 7. Hirata Y., Hara A., and Kosuge K., Motion control of passive intelligent walker using servo brakes. IEEE Trans. on Robotics, 23(5):981–989, October 2007. 8. Hirata Y., Komatsuda S., and Kosuge K., Fall prevention of passive intelligent walker based on human model. In IEEE Proc. Int. Conf. on Intelligent Robots and Systems (IROS), Nice, 22–26 September, pp. 1222–1228, 2008. 9. Kulyukin V. et al., iWalker: Toward a rollator mounted wayfinding system for the elderly. In Proc. IEEE Int. Conf. on RFID, Las Vegas, 16–17 April, pp. 303–311, 2008. 10. Lacey G. and MacNamara S., User involvement in the design and evaluation of a smart mobility aid. Journal of Rehabilitation Research & Development, 37(6):709–723, 2000. 11. Lee C-Y. et al., Development of rehabilitation robot systems for walking-aid. In Proc. IEEE Int. Conf. on Robotics and Automation, New-Orleans, April, pp. 2468–2473, 2004.

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12. M´ed´eric P., Pasqui V., Plumet F., Rumeau P., and Bidaud Ph., Design of an active walkingaid for elderly people. In Proc. 3rd International Advanced Robotics Program: International Workshop on Service, Assistive and Personal Robots, Madrid, 2003. 13. Mir´o J.V. et al., Robotics assistance with attitude: A mobility agent for motor function rehabilitation and ambulation support. In Proc. 11th Int. Conf. on Rehabilitation Robotics, Kyoto, 23–26 June, pp. 529–536, 2009. 14. Rentschler A.J. et al., Intelligent walkers for the elderly: Performance and safety testing of VA-PAMAID robotic walker. Journal of Rehabilitation Research & Development, 40(5):423– 432, September–October 2003. 15. Rentschler A.J. et al., Clinical evaluation of Guido robotic walker. Journal of Rehabilitation Research & Development, 45(9):1281–1294, 2008.

An Active Suspension System for Simulation of Ship Maneuvers in Wind Tunnels T. Bruckmann, M. Hiller and D. Schramm Lehrstuhl Mechatronik, University Duisburg-Essen, 47048 Duisburg, Germany; e-mail: {bruckmann, hiller, schramm}@imech.de

Abstract. Wind tunnels are an experimental tool to evaluate the air flow properties of vehicles in model scale and to optimize the design of aircrafts and aircraft components. Also the hydrodynamic properties of marine components like ship hulls or propulsion systems can be examined. For advanced optimization, it is necessary to guide the models along defined trajectories during the tests to vary the angle of attack. Due to their good aerodynamical properties, parallel wire robots were successfully used to perform these maneuvers in wind tunnels. Compared to aircraft hulls, marine models may be very heavy-weight (up to 150 kg). Thus, the suspension system must be very stiff to avoid vibrations. Additionally, fast maneuvers require powerful drives. On the other hand, the positioning system should not influence the air flow to ensure unaltered experimental results. In this paper, different designs are presented and discussed. Key words: wire robot, wind tunnel, tensed mechanism

1 Introduction Parallel kinematic machines can show major advantages compared to serial manipulators in terms of precision, load distribution and stiffness when they are adequately designed. On the other hand, classical parallel kinematics have a relatively small workspace compared to serial systems. To overcome this drawback, Landsberger [8] presented the concept of parallel wire driven robots in 1985. These robots share the basic concepts of classical parallel robots, but show some specific differences: • The long and flexible wires can be coiled on winches. This allows larger strokes in the kinematical chain and thus larger workspaces. • No complicated joints are required. Instead, wire guides are installed. • Simple and fast actuators can be used. Motors and winches can be mounted at nearly arbitrary positions, since the wires can be easily guided to any required point. Wires can transmit only tension forces, thus at least m = n + 1 wires are needed to tense a system which has n degrees of freedom [9]. This results in a kinematical

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redundancy. Accordingly, the solution space of the wire force distribution has the dimension m − n. Many applications require to follow a continuous trajectory through the workspace. In this case, the force distribution must also be continuous while the wire forces are limited by lower and upper bounds to prevent slackness and wire breaks, respectively. This makes the force computation a complicated task, especially when the computation has to be performed in realtime on a controller system. State-of-the-art for the simulation of quasi-stationary positions of models during airplane maneuvers is the application of static wire suspensions. To cover also dynamic maneuvers, the usage of active wire robots for wind tunnel maneuvers was evaluated in detail by Lafourcade [6, 7]. The SACSO (Suspension ACtive pour SOufflerie) robot of CERT-ONERA was an active wire suspension for wind tunnels. In the last years, chinese researchers presented their results, e.g. the WDPSS (Wire-driven Parallel Suspension System) [11] which was optimized for large attack angles, required for the simulation of modern jet aircrafts. These large attack angles are of special interest in the project presented here: The main focus of hydrodynamic research in the last decade was the simulation of the viscous flow around a ship at constant speed and parallel inflow to the ship longitudinal axis. Nowadays, numerical methods are able to simulate the viscous flow around a maneuvering vessel. For the validation of these methods, experimental data are now required, particularly during a predefined complex motion. The motion of the ship model can be realized through a superposition of longitudinal motion simulated through the inflow in the wind tunnel and a transverse or rotational motion of the ship generated by the suspension mechanism. Presently, mechanisms to enforce the motion of a ship model are available for towing tank applications. The design conditions for these mechanisms are totally different from wind tunnel applications: In the towing tank the weight of the ship model is compensated by the buoyancy force. On the other hand, the required forces to move the model along a certain path are much higher due to the higher density of the water in comparison with air. In wind tunnels, the weight of the model is the main factor influencing the required forces to guide the model. In an interdisplinary project between the Institute for Fluid Dynamics and Ship Theory at the Hamburg University of Technology and the Chair for Mechatronics at the University Duisburg-Essen, a dynamic wind tunnel suspension system is developed for installation at the Hamburg University of Technology. In the project described here, the system is designed for 150kg weight and a motion frequency of 0.5Hz at an amplitude of approximately 0.5m.

2 Discussion of Suspension Mechanisms Considering these points, several known approaches for a ship hull suspension system were discussed by the involved engineers:

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• Serial Kinematics: In 2002, Airbus Industries placed a KUKA KR 30/15L robot [5, 10] next to a wind tunnel to move a sensor within the air flow. The sensor was attached to the robot by a thin guidance which was aerodynamically optimized. As a drawback, the robot would introduce strong disturbances into the airflow when installed within the wind tunnel. Therefore, it can not be used for the purpose addressed here. • Classic Parallel Kinematics: Parallel kinematic manipulators can be very stiff, but use relatively slim kinematic chains. Therefore, they are an option for aerodynamically sensitive applications. Nevertheless, due to the high loads demanded and in order to be able to move a mass around 150kg, the transmission elements building the kinematic chains between the model and the wind tunnel must be considerably strong and thick. They would have a remarkable influence onto the air flow. • Parallel Wire Robot: As presented by Lafourcade [7] and other authors, parallel wire robots are applicable for wind tunnel suspensions. They have a relatively small aerodynamical footprint and allow for high loads. During the design phase, several parallel designs were proposed and compared with respect to the desired application. The next section describes the considered solutions.

Design 1: Parallel Wire Robot Due to points described above, a wire robot using eight wires (see Fig.1) was chosen as a first approach. Notheworthy, wires are already used for static suspension systems in wind tunnels. In the past also successful wind tunnel experiments using wire robots were presented (see e.g. [7]). As a preliminary design and due to architectural requirements, eight wires are considered, coiled by winched installed in the corners of a cuboid (see Fig.1). Examining the wire robot, the shape of wires is not optimal in terms of aerodynamical properties. Due to the changing length of the wires, it is not possible to cover them by aerodynamical shapes. Thus, turbulences and vibrations of the wires may occur.

Design 2: Parallel Robot using Tensed Links A promising improvement might be a tensed system based on thin, rigid links and linear actuators (see Fig. 2). Wires and winches are replaced by links of constant length and rails to guide the base points of the parallel links, respectively. This gives three advantages: • The tensed system applies only positive tensions (i.e. pulling forces) onto the links. This allows a very thin design of the links since buckling may not be taken

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Fig. 1 Wire robot for ship hull model suspension.

Fig. 2 Hybrid robot for ship hull model suspension.

into account for durability analysis. This property is inherited from the wire robot concept. It requires force control approaches, similar to wire robots. • The constant length of the links allows to give them a desirable shape design. The free formable profile allows the reduction of flow disturbances, e.g. by choosing Rankine [1] profiles. • The acutators are linear drives which can the bought off-the-shelf which reduces the development effort. This tensed design is capable for high loads, but leaves only a small aerodynamic footprint. Regarding the parameter synthesis and the geometrical design, some technical conditions had to be taken into consideration: Due to the rectangular shape of the testing area in the wind tunnel and some architectural limitations, a symmetric design using eight links was chosen for first simulations. Noteworthy, the parameters for the base and platform connection points were set to reasonable values. For first analysis steps, the linear drive axes were aligned with the cross direction of

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the wind tunnel. Their finally optimal values will be subjected to other optimization steps. To evaluate this design, a trajectory including the required motions and acceleration profiles was chosen as a benchmark and design criterion. Two questions arised at this design phase: 1. Is the workspace large enough to allow the desired maneuvers? 2. Is it possible to perform the predefined trajectories by drives of reasonable power consumption? To answer these questions, test maneuvers were performed by simulation. As a consequence of the movable base points, the system has a workspace covering large parts of the testbed volume in the wind tunnel. This workspace will be optimized in future steps for defined trajectories and loads. Along the chosen trajectories, this system with tensed links used a high peak power consumption in the upper drives (see section 3). On one hand, this is due to the mass of the mock-up and the required accelerations itself. This is defined by the application and cannot be influenced. On the other hand, the required positive pre-tension in the lower links increases the tension in the upper links additionally. This is due to this specific suspension design. Since high tensions in the upper links coincide with high velocities in the respective drives, high power peaks occur during the benchmark trajectories. Noteworthy, at the same time some of the lower drives run at low velocities while their links have advantageous angles of attack regarding the load compensation. These advantageous angles of attack could be used for an improved design. This third design described in the next section.

Design 3: Hybrid Tensed Link Approach An improved concept is very advantageous in terms of drive power distribution, maximum link forces and peak power consumption (see section 3). As opposed to a wire robot, it assumes that the lower links are capable for transmitting small and limited (!) negative (i.e. pushing) forces through the links. Having a tensed system equipped with a force control system, it is possible to predefine the force limits. These user-definable force limits can be used during the strength calculation of the links. This allows to choose a compromise between slim links, aerodynamic properties, size of the workspace and the required drive peak power. Since this design may use both unilateral and bilateral constraints, it is a hybrid mechanism between classical parallel kinematics and parallel wire robots. Formally, the constraints can be set e.g. as follows: • The number of degrees of freedom of the end effector is six. • The four lower links may support compressive forces and are thus treated as bilateral constraints, i.e. two degrees of freedom are remaining uncontrolled. • To control these degrees of freedom, at least three unilateral constraints are needed. Using e.g. four links for the upper suspension which support only tensile forces, the system has a redundancy r = 2.

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The ratio between allowable pushing and pulling forces is of special interest for the effectivity of this design compared to a conventional parallel manipulator. As an example, the links can be realized by Rankine [1] profiles which are similar to ellipses. The ratio between length LA and width LB of the ellipse should be compromise between a high geometrical moment of inertia I and an optimal aerodynamical L shape. Here, LA = 4 is chosen. A lightweight, but very tensile material is carbon B

= 2000 N/mm2 . fiber reinforced plastic. Its tensile strength in fiber direction is R+
The elasticity module is E
= 140000 N/mm2 . The ellipse profile is described by LA = 40 mm und LB = 10 mm. The length of the links is l = 2500 mm. Using buckling modes according to Euler, the smaller of the two geometrical moments of inertia of the ellipse is IA = π4 LA L3B = 31415.92 mm4 . Using a buckling length of l = 2500 mm and the described material, the critical bucking force is E IA

FK = π 2 l
2 = 6945.40 N. On the other hand, the critical tensile force FZ using the π L L = 2513274.12 N. tensile strength R and a cross section A gives FZ = RA = R+
4 A B The high ratio FZ /FK ≈ 360 shows the potential of Design 3.

3 Simulation Results for the Three Designs As already mentioned, the three designs were evaluated by simulation. This requires the calculation of force distributions for the wires. The used force computation approaches are described in [2–4]. The figures 3 to 5 show simulation results. They describe the force distribution and power consumption distribution of the eight force transmission elements and drives, respectively, for the three designs described in the second chapter, using a sinusoidal trajectory: • Design 1: Wire robot, only pulling forces f ∈ [100 . . .50000] N • Design 2: Link robot, only pulling forces f ∈ [100 . . . 50000] N • Design 3: Link robot, pulling and small pushing forces f ∈ [−200 . . .50000] N While the power consumption distribution for the example trajectory becomes more inefficient from Design 1 to Design 2, the final Design 3 is very advantageous: The simulation shows a significant reduction of peak forces and peak power consumption. In practice, using the described linear actuator concept of Design 2 and 3, it is possible to design a range of links with different positive and negative force transmission capabilities and cross-sections of flow. This allows to find the optimal compromise for every experimental setup.

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Fig. 3 Simulation results for the parallel wire robot (Design 1) on a sinusoidal trajectory, f min = 100 N, f max = 50000 N.

Fig. 4 Simulation results for the tensed parallel manipulator (Design 2) on a sinusoidal trajectory, f min = 100 N, f max = 50000 N.

Fig. 5 Simulation results for the hybrid tensed link approach (Design 3) on a sinusoidal trajectory, f min = −200 N, f max = 50000 N.

4 Conclusions and Future Work In this paper, the problem of designing an active wind tunnel suspension system is addressed. For this specific application, a minimized aerodynamical footprint is of major importance. At the same time, the suspension system must be able to carry high loads. The properties of three alternative designs are compared here. Finally, a novel hybrid system, mixing the kinematic properties of classical parallel kinematic mechanisms and wire robots, is presented. It offers the possibility of variable link shapes to optimize the aerodynamic footprint of the system. During simulation, it

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shows a good distribution of the required peak power per drive and thus, it allows to use less powerful motors. Presently, the suspension system is engineered. The geometrical parameters of the suspension system will be optimized to ensure a good system performance and large workspaces. The construction and design of the overall system and the implementation of the realtime control system is subject to current work. First experiment results will be presented in the near future.

Acknowledgement This work was supported by the German Research Council (Deutsche Forschungsgemeinschaft) under HI370/24-1 and SCHR1176/1-2.

References 1. Ashley, H., and Landahl, M., Aerodynamics of Wings and Bodies. Courier Dover Publications, 1985. 2. Bruckmann, T., Mikelsons, L., Brandt, T., Hiller, M., and Schramm, D., Wire robots part II dynamics, control & application. In Parallel Manipulators – New Developments, A. Lazinica (Ed.), ARS Robotic Books, I-Tech Education and Publishing, Vienna, Austria, pp. 133–153, 2008.. 3. Bruckmann, T., Mikelsons, L., Pott, A., Abdel-Maksoud, M., Brandt, T., and Schramm, D., A novel tensed mechanism for simulation of maneuvers in wind tunnels. In Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, CA, USA, August 30–September 2, 2009, ASME International, to appear. 4. Bruckmann, T., Pott, A., Franitza, D., and Hiller, M., A modular controller for redundantly actuated tendon-based Stewart platforms. In Proceedings of EuCoMeS, the first European Conference on Mechanism Science, Obergurgl, Austria, February, M. Husty and H.-P. Schroecker (Eds.), Innsbruck University Press, 2006. 5. Kuka Industrial Robots. Robot guides probe in wind tunnel. Available online, 2002. 6. Lafourcade, P., Contribution l’tude de manipulateurs parallles cbles. PhD Thesis, Ecole Nationale Suprieure de l’Aronautique et de l’Espace, December 9, 2004. 7. Lafourcade, P., Llibre, M., and Reboulet, C., Design of a parallel wire-driven manipulator for wind tunnels. In Proceedings Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, October 3–4, C. M. Gosselin and I. EbertUphoff (Eds.), 2002. 8. Landsberger, S., and Sheridan, T., A new design for parallel link manipulator. In Proceedings International Conference on Cybernetics and Society, Tucson, Arizona, pp. 812–814, 1985. 9. Ming, A., and Higuchi, T., Study on multiple degree of freedom positioning mechanisms using wires, Part 1 – Concept, design and control. International Journal of the Japan Society for Precision Engineering, 28:131–138, 1994. 10. Warmbold, J., Robot made to measure – Measuring air currents on an airbus model. Available online, 2002. 11. Yaqing, Z., Qi, L., and Xiongwei, L., Initial test of a wire-driven parallel suspension system for low speed wind tunnels. In Proceedings on 12th IFToMM World Congress, Besanc¸on, France, June 18–21, 2007.

Mechanism Solutions for Legged Robots Overcoming Obstacles Marco Ceccarelli, Giuseppe Carbone and Erika Ottaviano LARM: Laboratory of Robotics and Mechatronics, DIMSAT, University of Cassino, Via Di Biasio 43, 03043 Cassino (Fr), Italy; e-mail: {ceccarelli, carbone, ottaviano}@unicas.it

Abstract. Leg design for legged robots is approached by looking at mechanism solutions with features that are useful for low-cost applications in overcoming obstacles. LARM experiences have been used as background in order to outline new mechanism designs whose basic performance have been evaluated to prove the soundness of the approach and feasibility of the proposed linkage solutions. Key words: robotics, leg mechanisms, linkage design, analysis

1 Introduction Walking machines are proposed as transportation machinery with the aim to overpass the limits of wheeled systems by looking at legged solutions in nature, [8]. But only in a recent past efficient walking machines have been conceived, designed, and built with performances that are suitable for practical applications, as presented for example in [1, 2, 6, 8 ]. Biped, quadruped and hexapod robots as well as humanoid robots have been developed in the last part of 20-th century all around the world in research centres and universities as machines that can help/substitute human beings in dangerous or exhausting tasks, like in transportation of military staff, mine detection and grass cutting, in-pipe inspection or planetary exploration. Compared to wheeled/tracked vehicles, walking machines show the advantage that they can act in highly unstructured terrain. Legged robots can cross obstacles more easily; they depend less on the surface conditions and, in general, they exhibit better adaptability, [1, 2, 5–8, 11]. In this paper design solutions for leg mechanisms overpassing obstacles are reported as results of design activity for enhancing existing prototypes at LARM, [5, 7]. In particular, linkage solutions are searched as in [9, 10] for pantograph-based designs, and for modular designs in an Hexapod robot, [3], and even with new structure, [4], with features of robust mechanical design, easy-operation, low-cost construction, and proper functionality.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_63, © Springer Science+Business Media B.V. 2010

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Fig. 1 A scheme for leg design in overpassing an obstacle:(a) sagittal plane; (b) front view.

2 Leg Design Problem and Requirements Leg design problems with obstacle overcoming conditions can be discussed by referring to the schemes in Fig. 1. Looking at the instantaneous equilibrium gives the necessary conditions for a walking overcoming obstacles of the maximum step height. In a sagittal plane, Fig. 1a, the equilibrium can be expressed by a a Rh −P −PL L ≥ 0; g g

RV −P−PL ≥ 0;

RV dR−PdP−PL(dP+dL)−CinS ≥ 0,

(1) where Rh and Rv are the horizontal and vertical components of R contact reaction; P is robot weight; PL is leg weight; a is motion acceleration and g is the gravity; dL, dR, and dP are the indicated distances; CL is the torque actuating a leg; CinS is the sagittal component of the inertial torque due to waist balancing movement. Point Q is assumed as the foot contact point about which the system will rotate in a possible fall. In the front plane, Fig. 1b, the equilibrium can be expressed by Rl = P

aS ≥ 0; g

RV = P

aS ≥ 0; g

RV pR − PpP − PL(pP + pL) − Cinl ≥ 0, (2)

where Rl is the lateral component of R; aS is the lateral acceleration of robot body; Cinl is the lateral component of the inertial torque of waist balancing movement. Point S is assumed as the foot contact point about which the system will rotate in a possible fall. The components Rh and Rl refer to friction actions at the foot contact area. By using Equations (1) and (2) expressions can be deduced for design and operation features that are useful to overcome obstacles of height h. From geometric viewpoint the obstacle/step of height h can be overpassed when the leg mobility gives (3) l1 (1 − cos φ1 ) + l2 (1 − cos φ2 ) > h

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Fig. 2 Leg mechanism designs developed at LARM: (a) Chebyshev-Pantograph linkage; (b) screw-actuated binary mechanism; (c) link-cable parallel manipulator architecture.

in which l1 and l2 are the lengths of leg links, whose angles φ1 and φ2 are measured with respect to a vertical line. Thus, in general the design problem related with overpassing obstacles can be formulated by using (1) to (3) to size properly the leg links and to give required mobility ranges and actions for proper leg operation.

3 Solutions of Leg Designs for Walking Robots Basic considerations for a leg design can be considered as the following: the leg should generate an approximately straight-line trajectory for the foot with respect to the body in the propelling phase; the leg should have a robust mechanical design; the leg operation should posses the minimum number of DOFs to ensure the required motion capability. Research activity has been carried out at LARM since 1995 by developing prototypes, as reported in [5]. In Fig. 2 are shown the investigated solutions at LARM for which further enhancements are needed to give obstacle overpassing capability. In particular, in Fig. 2a a leg mechanism is shown for 1 DOF actuation by combining a Chebyshev linkage with a pantograph, whose main characteristic is that the Chebyshev linkage generates the proper trajectory that is amplified and used by the pantograph for the foot point. Alternative solution has been also experienced with Chebyshev linkage connected at the P top point of the pantograph. In Fig. 2b, a screw-actuated two-link design is shown as conceived for a hexapod robot with a powered wheel as a foot. Main feature is the compactness and robust design for a heavy payload capability. In Fig. 2c a study of feasibility is reported for a leg mechanism as a parallel manipulator with 3-1-1-1 architecture in which cables and links are used. For each of the above solutions it is needed to design improvement for having a flexible capability in overpassing obstacles by looking both at design sizes and operation features. Thus, possible solutions have been thought with the design ar-

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Fig. 3 Design solutions for obstacle overpassing capability with Chebyshev-Pantograph leg mechanism by means of: (a) lifting Chebyshev linkage frame; (b) lifting guiding cam; (c) slider-crank module; (d) adjustable leg link.

Fig. 4 Design solutions for obstacle overpassing capability with two-link mechanism by means of: (a) two-links leg; (b) adjustable leg link; (c) screw-actuated binary mechanism; (d) five-bar linkage leg.

chitectures in Figs. 3 and 4, respectively for the built prototypes with an additional DOF for increasing the step height on demand for overpassing obstacles. In particular, design solutions for leg linkage mechanisms are schematically shown in Fig. 3, as based on the 1-DOF leg solution in Fig. 2a. Figure 3a shows a design solution in which the body frame of the Chebyshev mechanism can be lifted up. Figure 3 shows a scheme in which the foot path generator roller-follower cam can be lifted up by a prismatic actuator. The design solution of Fig. 3c uses a slider-crank mechanism connected to the leg point C for providing the additional lifting action. The leg design in Fig. 3d is based on an adjustable leg link near the foot with a suitable actuated prismatic joint. For all the design solutions in Fig. 3 the additional DOF can be used to modify the leg-end-point trajectory at point A in terms of step size L and step height h, according to the scheme of Fig. 1. It is worth noting that all proposed solutions refer to an additional DOF that is obtained by a prismatic actuator. For the first three cases it can be conveniently fixed to the body of the walking robot, and for the scheme of Fig. 3d it must be designed and built within the leg link. Therefore,

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Fig. 5 Schemes for the kinematic analysis of 2-DOF leg: (a) Chebyshev and pantograph mechanism in Fig. 3a; (b) adjustable leg link; (c) actuated shoulder of screw-actuated binary mechanism in Fig. 4c.

from the mechanical point of view, the design solutions in Figs. 3a, b and c can be considered of main engineering significance, if compared to the solution in Fig. 3d. The solutions in Figs. 3a and b can be used to shift the foot-point trajectory in a vertical direction only, according to the obstacle size h. The design solution in Fig. 3c can be used to modify the foot-point trajectory in the vertical/horizontal direction. For the design solution in Fig. 3d the foot-point trajectory can be modified according to obstacle size both during flight phase and support phase. Design solutions are schematically shown in Fig. 4, as a modification of the 1DOF screw-actuated binary mechanism leg solution in Fig. 2b. In particular, the scheme in Fig. 4a shows a design solution based on 2 DOFs where two commercially available DC motors can be used for providing the necessary actuation at the two revolute joints with a modular design by integrating a motors into a link body. The scheme in Fig. 4b shows a design solution with 3 DOFs as based on an adjustable leg link. This design solution can either have a linear actuator for adjusting the height of the foot or just have a linear spring to add a passive degree of freedom for just providing adaptability on rough terrains. The scheme in Fig. 4c shows a design solution in which the original screw-actuated leg mechanism is enabled to rotate by using an additional actuator in its shoulder. The scheme of Fig. 4d proposes a completely new design solution with two DOFs as based on a five-bar mechanism by using two commercially available DC motors on the body frame. A careful coordination of the two active DOFs is required for achieving a suitable trajectory of the foot with a considerable motion capability.

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4 Operation Feasibility The above-mentioned design solutions in Figs. 3 and 4 can be considered for an implementation in the existing prototypes at LARM. In particular, among the solutions in Fig. 3, the slider-crank module and adjustable leg link can be considered convenient from engineering practical viewpoints. Similarly, among the solutions in Fig. 4 the adjustable leg link and actuated shoulder of screw-actuated binary mechanism can be considered feasible for cost-oriented and regulation requirements. Referring to Fig. 5 for the analysis of the new leg solutions in Figs. 5a and b, the mobility of point B with respect to OXY frame can be evaluated as a function of the input crank angle α (t) and kinematic parameters of the Chebyshev mechanism by the coordinates xB = −a + m cos α + (c + f ) cos θ in which

⎛

θ = 2 tan−1 ⎝

−C2 +

yB = −m sin α − (c + f ) sin θ  ⎞ C22 − 4C1C3 ⎠ 2C1

(4)

(5)

whose coefficients C1C2 and C3 can be obtained form the closure equation of the four-bar linkage in Fig. 5a, in the form C1 = a2 − d 2 + M 2 + c2 + 2ac − 2m cos α (c + a)

C2 = 4mc sin α

C3 = a2 − d 2 + m2 + c2 − 2ac + 2m cos α (c + a)

(6)

The position of foot point A with respect to the fixed frame can be obtained as XA = xB + (g + h) cos ϕ2 − (i + r) cos ϕ3 yA = yB − (g + h) sin ϕ2 − (i + r) sin ϕ3

(7)

in which the variable s is function of time because of the linear actuator, and ϕ2 and ϕ3 angles are computed from the closure equations of OBCG in Fig. 5a as  ⎛ ⎞   −K2 + K22 − 4K1K3 ⎠ , ϕ = cos−1 −xB + g cos ϕ2 , (8) ϕ2 = 2 tan−1 ⎝ 3 2K1 j where K1 = x2B − j2 + g2 + y2B + (p + s)2 + 2xB g + 2yB(p + s) K2 = −4g(yB + (p + s)) K3 = x2B − j2 + g2 + y2B + (p + s)2 − 2xB g + 2yB(p + s).

(9)

The suitable trajectory for overpassing an obstacle can be planned in such a way that during the support phase (the approximately rectilinear path) points B, C and

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Fig. 6 Operation capability of the built screw-actuated binary mechanism leg: (a) fully stretched configuration; (b) intermediate configuration; (c) fully bent configuration.

A should be always aligned (s = cost = s0 ) but in flight phase variable s(t) can be modified according to the size of the obstacle. Therefore, the leg can operate with 1 DOF during the support phase and with 2 DOFs during the flight phase. The required torque T can be conveniently computed by using Principle of Virtual Power as function of the input crank velocity ω in the form T =

mrA g (x˙B + y˙B + ϕ˙ 2 (g + h)(sin ϕ2 − cos ϕ2 ) − ϕ˙ 3(i + s)(sin ϕ3 + cos ϕ3 )) ω m g ˙ ϕ3 − sin ϕ3 )) (10) + rA (s(cos ω

x˙B and y˙B are the components of point B velocity and mrA is the overall equivalent lumped leg mass that can be conveniently reduced at point A. Angles ϕ2 and ϕ3 are obtained by the pantograph kinematics in Fig. 5a) as reported in Equation (8). For the case of adjustable leg link in Fig. 5b foot point A trajectory can be expressed through the coordinates XA = l1 cos α1 + (l2 + s) cos α2

YA = l1 sin α1 + (l2 + s) sin α2

(11)

in which the extra mobility for obstacle overpassing is achieved by the s range. A new leg as actuated shoulder of screw-actuated binary mechanism in Fig. 4c has been designed and built at LARM in Cassino as shown in Fig. 6. The new leg consists of two link modules whose total length is 2L = 540 mm and total weight is 850 grams. Maximum step size is equal to 200 mm as due to the mobility range of the screw that is limited by the configurations at 0 deg (Fig. 6a) and configuration at about 80 deg (Fig. 6c) due to the mechanical limits of the screw-nut transmission. Similarly, the step height can be computed as equal to 200 mm from h = 2L[1 − sin(θ1 ) − sin(θ2 )]. Referring to Fig. 5c the equilibrium for the leg link can be written as

(12)

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Fig. 7 Measured motor torque versus time in test with the leg prototype of Fig. 6: (a) with no load; (b) with a load of 300 grams; (c) with a load of 676 grams.

P = F(sin(α ) − cos(α ) cot g(ξ ))

(13)

with the aim to compute the evaluate the actuator force F on the screw system to carry a load P. The involved angles can be computed as

α=

θ1 − θ2 , 2

ξ = θ2 +

α . 2

(14)

If one refers to a configuration with the knee fully bent θ1 is equal to 90 deg, θ2 is equal to 10 deg so that Equation (14) gives the angle α as equal to 40 deg and ξ as equal to 30 deg. In addition, by referring to DC motors having nominal torque of 1.2 Nm, a nominal speed of 60 rpm, and a step of the screw equal to 3 mm per revolute, the principle of virtual powers and Equations (13) and (14) yield to a maximum speed of 6 mm/s, a maximum force F of about 10 N and a maximum payload of about 6.9 N. Experimental tests have been carried out on the built prototype in Fig. 6 in which the joint of the screw on the lower arm has been moved upwards to avoid interference of the screw with the ground. Preliminary experimental results have been reported in Fig. 7. All the experimental tests have shown suitable results with motor torque below a nominal motor torque of 1.2 Nm as a validation of the computed above design schemes. In addition, it is worth to note that the torque plot shows that after the initial sudden variation, due to inertias, the leg motion is related to a quasi constant action that, although depending obviously by the load, can be considered suitable for low-energy consumption low-cost easily controlled commercial actuator.

5 Conclusions Obstacle overpassing is a significant feature of legged robots but usually requires suitable powered design and controlled operation. In this paper we have proposed

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mechanism solutions for LARM prototypes whose main characteristics are linkage design and easily controlled additional DOFs.

References 1. Berns K., Walking machine catalogue, Available online at http://www.walking-machines.org/, 2010. 2. Carbone G. and Ceccarelli M., Legged Robotic Systems, Cutting Edge Robotics ARS Scientific Book, Wien, pp. 553–576, 2005. 3. Carbone G. and Ceccarelli M., A low-cost easy-operation hexapod walking machine, International Journal of Advanced Robotic Systems, 5(2):161–166, 2008. 4. Ceccarelli M. and Carbone G., A study of feasibility for a leg design with parallel mechanism architecture. In Proceedings IEEE/ASME Conference on Advanced Intelligent Mechatronics AIM’09, Singapore, Paper No. 131, 2009. 5. Ceccarelli M., Carbone G., Ottaviano, E. and Lanni, C., Leg designs for walking machines at LARM in Cassino, Workshop Robotica per esplorazione lunare unmanned, Roma, 2009. 6. Iagnemma K. and Dubowsky S., Mobile Robots in Rough Terrain. Estimation, Motion Planning, and Control with Application to Planetary Rovers, Springer-Verlag, Berlin, 2004. 7. LARM webpage: http://webuser.unicas.it/weblarm/larmindex.htm, accessed April 2010. 8. Morecki A. and Waldron K.J., Human and Machine Locomotion, Springer, New York, 1997. 9. Ottaviano E., Lanni C. and Ceccarelli M., Numerical and experimental analysis of a pantograph-leg with a fully-rotative actuating mechanism. In Proceedings of the 11th World Congress in Mechanism and Machine Science, Tianjin, Vol. 4, pp. 1537–1541, 2004. 10. Ottaviano E., Ceccarelli M. and Grande S., A biped walking mechanism for a rickshaw robot, Mechanics Based Design of Structures and Machines, 2010 (in print). 11. Song, S.M., Lee, J.K. and Waldron, K.J., Motion study of two and three dimensional pantograph mechanism, Mechanism and Machine Theory, 22(4):321–331, 1987.

Control Issues of Mechanical Systems

Dynamic Reconfiguration of Parallel Mechanisms J. Schmitt1 , D. Inkermann2, A. Raatz1 , J. Hesselbach1 and T. Vietor2 1 Institute

of Machine Tools and Production Technology, Technische Universit¨at Braunschweig, 38092 Braunschweig, Germany; e-mail: {jan.schmitt, a.raatz, j.hesselbach}@tu-bs.de 2 Institute for Engineering Design, Technische Universit¨ at Braunschweig, 38106 Braunschweig, Germany; e-mail: {inkermann, vietor}@ikt.tu-bs.de

Abstract. Shorter product life cycle as well as higher product complexity and diversity require more flexible manufacturing systems. Dynamic reconfiguration is a time efficient way to adapt system properties to rapidly changing process requirements. The potential to reconfigure parallel mechanisms depends on specific and optimized machine components, which enable modification of kinematic behaviour of the system. In this contribution influence of several component parameters on system properties and the needs to develop more suitable machine components are highlighted. Furthermore, a possibility to adapt kinematic properties of a planar RRRRR-mechanism, using an adaptive revolute joint, is introduced. Key words: reconfiguration, machine components, parallel robots, singularities

1 Introduction Development of modern manufacturing systems is faced with changes, taking place in the ecomonic, social and technological context. Competitive systems have to offer high flexibility as well as opimized process conditions to determine high output and high product quality at the same time. In general robots are designed for one specific application within the manufacturing process. Using these robots for changing application e.g. caused by different products, leads to compromises between product quality and output. Reconfigurable parallel robots are a promising approach to assure optimized process conditions in different applications. System inherent properties such as workspace dimension or workspace position as well as accuracy of the mechanism are changed with regard to present requirements. Furthermore reconfiguration helps to avoid singularities within the workspace or to pass them in an assured manner. A suitable way to reconfigure (parallel) robots, is to use adaptive machine components. These are able to adjust themselves to different and (may be conflicting) requirements, such as zero backlash and low-friction, and leads to fulfillement of present application demands [11]. In this paper general aspects of reconfigurable parallel robots are discussed and the use of adaptive machine components is introduced. The application of an ad-

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aptive joint and its functionality as well as the reconfiguration conditions and restrictions are theoretically analyzed. The kinematic structure used to validate the reconfiguration concept is a planar RRRRR-mechanism. Firstly the diversity to reconfigure the 5-bar to a 4-bar mechanism as well as the specific benefits are shown. Secondly a strategy for assured passing through singularities of type 2 is shown to enlarge available workspace.

2 Reconfiguration Approaches In recent years several approaches for the reconfiguration of parallel robots have been presented. The majority of these are based on modular concepts and enable manual changing or rearranging of machine components such as drives, struts, as well as whole subassemblies. According to Kreffts’ classification of reconfigurationstrategies [4], this approach is designated as static reconfiguration (SR). Numerous system parameters can be modified in wide ranges, but the robot has to be out of action. This results in a high rebuilding time, therefore this strategy is suitable in cases of process reorganization. With regard to time efficiency, dynamic reconfiguration strategies have to be applied. Two types of dynamic reconfiguration can be distinguished and both of them intend to modify system properties during operation mode. With dynamic reconfiguration of type 1 (DR1) physical dimensions of the robot are not changed, but different configurations are used. Following this reconfiguration strategy type 1 and/or type 2 singularities have to be passed. Extensive investigations have been conducted e.g. by Budde [1] and Helm [2]. Major advantage of this strategy is the enlargement of the workspace by passing through singularities. At the same time this strategy requires rather complex control systems. Unlike the aforementioned reconfiguration strategies, dynamic reconfiguration type 2 (DR2) proposes to adjust geometric parameters such as strut length or modify the degree of freedom (DOF) by joint couplings or lockings e.g. in order to avoid singularities or to manage singularity loci. Theingi et al. [12] introduce singularity management by joint couplings using belt drives. The design and control of a parallel manipulator as a joint coupling testbed is employed and a first experimental study is shown. O’Brien proposes the kinematic control of singularities with passive joint brakes [6]. Another approach of DR2 was presented in [4], by means of the VARIOPOD robot. Here, pneumatic cylinders are used as additional actuators mainly to approve the modification of the workspace. Additionally, a modular frame was presented to rebuild the robot depending on the requirements. Referring to the abovementioned definitions this approach is subject of SR. A qualitative comparision of the described reconfiguration strategies is shown in Table 1 referring to main aspects, which evaluate the individual reconfiguration strategies. Additionally the necessity of constrictive modification and the scale of the effect on system properties compared to reconfiguration expense are assessed.

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Table 1 Qualitative comparison of reconfiguration approaches.

SR DR1 DR2

Duration of Constructive reconfiguration modification

Number of modifiable system properties

Costs for special Effect on sysmachine tem properties components cp. to reconfiguration expense

– o +

+ – o

o + o

– + o

– + +

+/o/–: positive/neutral/negative criteria satisfaction

Thereby dynamic reconfiguration expresses itself as a quick acting strategy with high efficiency compared to the employed costs. In this contribution a new approach for DR2 is introduced, as the modifiable system properties are even numerous. An adaptive revolute joint as a special machine component is used to reconfigure a parallel mechanism. The application spectrum especially according to an assured passing through singularities is discussed exemplarily.

3 Components for Dynamic Reconfiguration Mechanical components within a robot system have to perform tasks such as transfer of mechanical energy flow (struts) and enable specific movements (joints). In cases of purely mechanical elements, parameters are fixed by embodiment design, material selection, and manual adjustment e.g. bearing clearance. With regard to changing application requirements, parameters of components e.g. DOF or stiffness have to be adapted during process. The necessity for reconfiguration is caused by different use cases or tasks within these (see Fig. 1). For effective adaption, relevant system properties have to be identified and related to component parameters. An approach for this challenging task was introduced by Schmitt et al. [9]. Based on requirement spreads within product utilization, reconfiguration parameters are identified by means of an object-oriented system modelling. For instance, in a pick-and-place use case different tasks arise, such as picking object, moving object, and placing object. Required system parameters for these tasks may be differing e.g. high accuracy in place task and high dynamic while moving phase. This results in a set of different system configurations, which are directly related to parameters of several components (see Fig. 1). For instance, stiffness of struts and clearance in joints influence system accuracy. In order to adapt component parameters, their mechanical structure has to be augmented by actuation elements (see Fig. 1). With regard to required system characteristics, integration of actuation elements should allow compact design and low complexity without changing major component properties. Application of known

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Fig. 1 Interpendency of mechanical components and application requirements.

components e.g. pneumatic cylinders [4] or additional brakes at the joints [6] often cause higher system mass. Furthermore, several parameters depending on actual working conditions lead to uncontrollable changing. To overcome drawbacks of these conventional components, new mechanical components have to be developed. As a new type of component, an adaptive joint for dynamic reconfiguration of parallel robots is introduced. The desgin of the adaptive joint as well as its application and its influence on the kinematic behavior are described in the following. Parallel mechanisms are made up of at least two guiding chains, which include one or more passive joints. As operation forces and movements are conducted to joints, their properties are of major importance to system properties such as accuracy and workspace dimension. Hence, joint parameters are especially suitable for reconfiguration aspects. Furthermore, additional functions of adaptive joints can be used for calibration purpose as shown in [5]. Major parameters of joints are stiffness and friction. In order to enable adaption of these parameters during robot operation two effect principles were investigated as part of the research activities of the Colaborative Research Center (SFB) 562, namely high-frequency excitation and quasi-statical clearance adjustment [7, 8]. Based on these first experimental results, a new joint prototype was developed (see Figs. 2a and b). An essential part of the new adaptive joint concept is the rotary axis shown in Fig. 2). Two plain bearings (1) enable rotary movement and carry axial as well as radial loads. Outer rings are fixed by means of two bearing caps (2), which are wrapped by a tension shaft (3). To adapt joint properties, internal rings of bearings are propped by two piezoelectric actuators (4) (maximum force generation 20kN, maximum stroke 25 µm). Actuator elongation in axial direction results in simultaneous axial and radial clearance modification in the bearings. Hence, this so-called

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Fig. 2 Prototype of adaptive revolute joint based on quasistatical clearance adjustment.

quasistatical clearance adjustment causes adaption of friction and stiffness. As a result of the high power mass of piezoelectric actuators, a compact design is possible. However, actuator ceramics have to be shielded from lateral forces. Thus the actuators are glued on two guide sleeves (5), which are driven into the housing (6). Due to the fact that actuator characteristics (stroke and force) depend on the mechanical stiffness in the effective direction of the actuator, the design of the mechanical structure is quite challenging. In the present case stiffness is primary influenced by the tension shaft. In order to investigate the correlation between the stiffness and the resulting clearance modification, facile replacement of the shaft with variing diameters is possible. Achievable friction modification is essentially coupled with the preload of the bearings fixed by manually adjustment before operation. Depending on the previous friction moment caused by the preload the prototype enables continuous locking of rotary DOF by friction magnification of 120%. Compared to conventionell brakes mounted at the joint, the functionallity of the adaptive joint enables dynamic variation of the stiffness and therefore the dynamic behaviour of the structure.

4 Example of a Reconfigurable Parallel Mechanism Referring to the identified parameters for dynamic reconfiguration and the adaptive joint stated in the previous paragraphs, the reconfiguration approaches and their specific benefits are shown and validated. For this purpose a simple RRRRRmechanism is exemplarily presented (see Fig. 3). The RRRRR-mechanism is a fully closed-loop planar parallel manipulator with two DOF. The cranks at the base points A1 and A2 are actuated. The two passive rods L12 and L22 are coupled to each other by a passive revolute joint in the end-effector point C. The kinematic chain originating at base A1 and the corresponding crank is connected by the described adaptive revolute joint.

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Fig. 3 Kinematic scheme of the RRRRR-mechanism (a) and solutions of the DKP (b).

The kinematic description of the mechanism with its two closed loop chains i = 1...2 can be geometrically done with the cartesian end-effector coordinates X = [xc , yc ]T , the base coordinates A = [xAi , yAi ]T , which can be derived by LB , the actuator angles qi and the other given geometric parameters Lii with respect to the base frame {0}. In eq. (1) the kinematic model of the mechanism is described, considering the parameters and variables shown in Fig. 3.

F=


2 xAi + cos(qi ) · Li1 xc − − Li2 2 = 0. yc yAi + sin(qi ) · Li1

(1)

Expression (1) provides a system of equations which relate the end-effector coordinated X to the actuator coordinates qi . The equation for the direct kinematic problem (DKP) has two solutions, in case of the RRRRR-mechanism. They constitue to different configurations of the mechanism (see Fig. 3 b)). Changing these configurations, possibly a singularity of type 2 has to be passed. In singular positions kinematic structures are restricted by their drivability or they are not controllable. Mathematically, singularities appear, if the Jacobian Matrices JA,B JA =

∂F ∂X

∨

JB =

∂F ∂qj

(2)

become singular. Whereas singularities of type 1; type 2 are defined by det(JA ) = 0

∧

det(JB ) = 0 ;

det(JA ) = 0

∧

det(JB ) = 0.

(3)

The latter type of singularity can occur in the workspace or separate workspaces into different areas. To use these workspace areas these singularities have to be passed and hence the robot has to be reconfigured. Assembling the adaptive revolute joint in point B1 (see Fig. 3) and blocking it entirely, it is possible to treat the mechanism as a RRRR 4-bar structure, with one actuated revolute joint at base point A1 . According to the theorem of G RASHOF different kinds of 4-bar mechanisms exist, depending on the geometric aspect ratio of the links [3]:

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Fig. 4 Assured passing through singularity by dynamic reconfiguration.

lmin + lmax < l  + l 

ability to turn around

lmin + lmax = l + l lmin + lmax > l  + l 

ability to snap-through no ability to turn around.





lmin and lmax are the lengths of the shortest and the longest link in the structure and l’, l” are the lengths of the remaining links. In case of the shown 5-bar mechanism with the adaptive revolute joint, one link length can be adjusted according to the desired mechanism property. The resulting length LR can be calculated by  (4) LR = L212 + L211 − 2 · L12 · L11 · cos(Θ1 ). Beside the aforementioned possibility to reconfigure the robot, it is feasible to develop strategies to pass through a type 2 singularity in an assured manner by blocking and releasing the mechanism at point B1 by means of the adaptive revolute joint. This reconfiguration idea is shown in Fig. 4. The aspect ratios in the example are chosen by Li1 /(Li2 , LB ) = 3/4. If the mechanism is close to a singular position (Fig. 4.1), the adaptive revolute joint will be locked, here at θ1 = 40◦ . The resulting 4-bar mechanism moves on a path, which is defined by a segment of a circle of radius LR with center at A1 through the singular position (Fig. 4.2). Subsequently, the mechanism is in another configuration, which is not a singular position (Fig. 4.3) and the adaptive joint can be released.

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5 Conclusion and Further Research Reconfiguration as a general approach for flexible and efficient manufacturing systems was introduced. Influences of different component parameters on major system properties were highlighted and passive joints were identified as suitable machine components for dynamic reconfiguration. Subsequently, an adaptive revolute joint and its functionality was presented. By means of a RRRRR-mechanism the ability of reconfiguration was shown, in particular its use as a RRRR-mechanism and to pass through singularities of type 2 in an assured manner. Further research activities attend to a practical validation of the reconfiguration strategies and the detailed investigation of the adaptive joint. Achievable friction maginfication as well as the influence of tension shaft diameter have to be investigated in several tests. Furthermore the effect of other functional parameters such as stiffness will be accomplished. Further research will focus on control concepts and implementation of a testbed. Based on this results the desgin of the adaptive joint has to be optimized concering weight and compactness.

Acknowledgement The authors gratefully thank the German Research Fundation (DFG) for supporting the Collaborative Research Center SFB 562 Robotic Systems for Handling and Assembly – High Dynamic Parallel Structures with Adaptronic Components.

References 1. Budde, C., Last, P., and Hesselbach, J., Development of a triglide-robot with enlarged workspace. In Proceedings of the International Conference on Robotics and Automation, Rome, Italy, pp. 543–548, 2007. 2. Helm, M., Durchschlagende Mechanismen f¨ur Parallelroboter, Technische Universit¨at Braunschweig, Vulkan-Verlag, Essen, 2003. 3. Kerle, H., Pittschellis, R., and Corves, B., Einf¨uhrung in die Getriebelehre. Teubner Verlag, Wiesbaden, Germany, 2007. 4. Krefft, M. et al., Reconfigurable parallel robots: Combining high flexibility and short cycle times. Journal of Production Engineering, XIII(1):109–112, 2006. 5. Last, P. et al., Parallel robot calibration utilizing adaptronic joints. In Proceedings of the 2008 ASME International Design Engineering Technical Conferences (IDETC) and Computers and Information in Engineering Conference (CIE), New York, USA, 2008. 6. O’Brien, J.-F. and Wen, J.-T., Kinematic control of parallel robots in the presence of unstable singularities. In Proceedings of IEEE International Conference on Robotics and Automation, Seoul, Korea, pp. 354–359, 2001. 7. Pavlovic, N., Keimer, R., and Franke, H.-J., Improvement of overall performance of parallel robots using joints with integrated piezo-actuators. In Proceedings of the 11th International Conference on New Actuators, ACTUATOR 2008, Bremen, pp. 121–124, 2008. 8. Pavlovic, N., Keimer, R., and Franke, H.-J., Adaptronic joints based on quasi-statical clearance adjustment as a means to improve performance of parallel robots. In Proceedings of the

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19th International Conference on Adaptive Structures and Technologies (ICAST), Ascona, Switzerland, 2008. Schmitt, J. et al., Reconfigurable parallel kinematic structures for spreading requirements. In Proceedings of the IASTED – Robotics and Applications Conference, Cambridge, MA, USA, 2009. Setchi, R-M. and Lagos, N., Reconfigurability and reconfigurable manufacturing systems – State-of-the-art review. In Proceedings of 2nd IEEE International Conference on Industrial Informatics INDIN2004, Berlin, pp. 529–535, 2004. Stechert, C., Pavlovic, N., and Franke, H.-J., Parallel robots with adaptronic components – Design through different knowledge domains. In Proceedings of 12th IFToMM World Congress, Besancon, France, 2007. Theingi, Chen, I.M., Angeles, J., and Chuan, L., Management of parallel-manipulator singularities using joint-coupling. Advanced Robotics, 21(5–6):583–600, 2007.

Development of a Voice Controlled Surgical Robot C. Vaida1 , D. Pisla1 , N. Plitea1 , B. Gherman1, B. Gyurka1, F. Graur2 and L. Vlad2 1 Technical

University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected] 2 University of Medicine and Pharmacy Cluj-Napoca, 400012 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. In the paper the first developed voice-controlled parallel robot PARAMIS (PARAllel robot for Minimally Invasive Surgery) in Romania, used for laparoscope camera positioning in the minimally invasive surgery, is presented. The development of the structure focused on the achievement of a simple, lightweight, cheap solution, able to work alone or as part of a multiarm robotic system. The adopted solution for the voice commands interface is presented. Some experimental results are illustrated pointing out the behavior of the robot in a surgical procedure. Key words: parallel structure, robotic surgery, open architecture, voice control

1 Introduction The evolution of surgery, from open procedures to the chain – minimally invasive interventions – NOTES – single port surgery shows the evolution of surgical procedures towards the increase of the life quality of patients. The continuous research and achievements in the development of innovative medical equipments provide and answer the demand of surgeons for performing difficult procedures with minimal damage and discomfort for the patient. Some of the most complex tools developed for surgery are the robotic systems. Their use in surgical applications represented a challenge for many research centers which resulted in the development of several robotic systems [1, 2, 6]. Referring to endoscopic surgery, there exist several research directions which lead to different solutions aiming to increase the performances of the surgical act: the first refers to the development of laparoscope holders, the second one focuses on the development of complete robotic surgical systems while the third proposes robotic solutions acting as an assistant for the surgeon. Referring to laparoscope holders, the first achievement in this area of research was AESOP, the first robot ever to enter the surgical arena [13]. FreeHand [3] from Prosurgics is a robotic arm that positions the laparoscopic camera based on the head motions of the surgeon. Endocontrol provides ViKY, [5] a laparoscope holder with foot, hand and voice control for the camera positioning. Referring to complex robotic systems for surgery, several solutions were proposed. HERMES [2] is used to control all

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the intelligent equipments within the operating room using voice commands. The success of AESOP encouraged its developer, Computer Motion, to use the same platform for a multiarm solution, named Zeus, able to to manipulate 28 different instruments including scalpels, hooks to tie knots, scissors and dissector [2]. In the same time, Intuitive Surgical [7] launched the da Vinci robotic system, which became the first surgical robotic system which obtained clearance from the FDA to perform surgeries. The Centre for Simulation and Testing of Industrial Robots – CESTER within the Technical University of Cluj-Napoca, Romania started in 2005 a joint research with the Surgical Clinic III, within the University of Medicine and Pharmacy Cluj-Napoca, Romania, aiming to develop robotic systems based on parallel architectures for surgical applications. In 2008, the first experimental model, PARAMIS, was developed [4, 8, 10, 14, 15]. The PARAMIS robot was designed to be used either as a laparoscope holder or as part of a multiarm robotic system. The control interface allows the positioning of the laparoscopic camera using either keyboard, joystick, voice or haptic commands. Regarding the use of voice control Punt presents a study [11] demonstrating some limitations of voice control, emphasizing the need for further research. PARAMIS control interface proposes a simple solution by implementing the free to use Speech SDK 6.1 from Microsoft which proved to be a valuable and reliable tool for the task. The paper is structured into 4 sections, presenting some aspects related to the design of the structure, the control system and the voice commands interface as well as experimental results and conclusions.

2 Experimental model of PARAMIS The first task of the joint research was the development of a low cost parallel robot arm used for the positioning of the laparoscopic camera in MIS procedures that would respect several characteristics: Simple, light and stiff structure; Affordable; No electrical components close to the patient; Customizable user interface; Adaptable to a multiarm solution.These characteristics would answer to some of the limitations encountered at existing systems, namely their prohibitive cost and the rigid interface. Making an analysis of the robot task, namely the positioning of the laparoscopic camera in any desired position within the operating field, a simple structure with three degrees of freedom (thus three actuated joints or three motors) can be used.The mathematical model which involves the geometric, kinematical and dynamic study of the PARAMIS robot has already been described in [8, 9, 14]. An experimental model has been developed in close cooperation with the Institute of Machine Tools and Production Technology of the Technical University Braunschweig, as presented in Figure 1. The motion of the PARAMIS parallel robot is ensured by three actuators (q1, q2, q3) and several passive joints (4, 5), see Figure 1, including also the virtual joint of class 2 represented by the entrance point in the surgical field. This joint, which constrains the laparoscopic camera to pass through a fixed point, is called ”virtual” because it is not actually built in the robotic structure. The laparoscope

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Fig. 1 PARAMIS experimental model.

is mounted in the passive joint (5) and connected to the image processing unit and light source (6). The laparoscope is positioned inside a human torso trainer with cholecystectomy model (7) from Simulab [12] with internal organs transmitting the images to a display (8). The first two actuators (q1 and q2) are positioned on the same fixed ball screw by means of two rotary nuts while the third actuator (q3) is fixed on the base. This construction of the PARAMIS robotic system aimed to minimize the occupied volume in the neighborhood of the patient in the perspective of its integration in a multiarm surgical system which would imply the presence in the both sides of PARAMIS, of other arms manipulating surgical instruments.

3 PARAMIS Control System and User Interface The control system of PARAMIS consists in a four phase process illustrated in Figure 2. The control input allows the user to issue commands using several interfaces. The processing of these commands is achieved by the computer and sent to the PLC which generate instructions for the actuators.The control system of PARAMIS uses a CanOpen communication protocol controlled by a PLC provided by BR Automation. The use of the CanOpen protocol allows the connection of up to 127 devices to the same controller allowing the extension of PARAMIS to a multiarm robotic system using the same control system, ensuring complete compatibility, compactness and low costs. The control system commands the three intelligent actuators MCD EPOS 60W from Maxon which integrate in a single drive a motor, a gear head, an encoder, a PID controller and a driver. The selection of the gearheads ratios was achieved

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Fig. 2 PARAMIS control system. Actuation scheme (top) and user interface (bottom).

based on the displacement speeds imposed by the application, namely 1:26 for the translation joints (q1 and q2) respectively 1:156 for the rotation joint (q3). As each actuator manipulates a different mass, the possibility of individual PID controller configuration ensured an optimal behavior of each motor [14]. The use of gearheads with high ratios resolve a safety issue, namely the self-blocking of the robot during a power loss. In the same, the user can rotate with a minium effort the robot by pulling the horizontal arm, the laparoscope is than removed and the operation continued without the robot. The functions of the user interface, figure2, can be groups in several main categories: configuration, control interface selection, positioning, motion configuration and safety volume definition commands. The configuration commands are used for the initial setup of the robot: Nest, Origin, Save B. They are used at the beginning of the procedure to reset the encoders and to define the relative position between the patient and the robot. This setup time is about 1.5–2 minutes. The control interface commands allow the user to select any of the available command modes, enabling him to control the robot using different equipments. The positioning commands refer strictly to the positioning of the laparoscopic camera inside the patient. Two sets of commands (enabling large or small motions) allow the positioning on each direc-

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tion: In, Out, Up, Down, Left and Right. The displacement increment (or the camera resolution) for a motion can be set between 1 and up to 20 millimeters. Due to the fact that the surgeon is always viewing a larger area, during the experimental runs displacements below 5 millimeters were considered too small. The interface includes also three groups of commands that enable the saving of up to three points in the surgical field and a STOP command which stops the motion of the robot in any situation or configuration. The motion configuration commands allow the configuration of the motion parameters in terms of maximum displacement in any direction, and the maximum allowable speed or acceleration. The safety volume definition commands, optional, allow the definition of a volume, inside the patient, on each direction. Once a limit is set the robot will not allow any displacement beyond it. Even when all the limits are set, using the Override Limits option the surgeon can position the laparoscopic camera outside that volume until this override is cleared. The user can select from the interface one of the five control modes available. Motor actuation enables the independent control of each actuator, useful during the configuration, testing and calibration actions. Keyboard/Mouse mode enables the control of the robot with direct interactions upon the buttons and controls of the interface. Joystick control mode enables the user to position the laparoscopic camera using the motions of the stick while the other commands having other buttons assigned on the joystick. The Haptic control mode will position the laparoscopic camera following the motion of the haptic device. This mode is used as a preliminary step in the development of an active robot arm (capable of manipulating a surgical instrument). Thus experiments can be performed with the task of developing a tactile sensor (to provide feedback for the haptic device) before the actual construction of the new robotic arm.

4 PARAMIS Voice Control Interface During the experimental runs, the surgeons preferred to control the positioning of the laparoscopic camera using voice commands, Figure 3. The surgeon’s option is easily justified as it permits him to operate with both hands looking at the display and moving the camera by issuing simple commands. The learning curve spreads over a couple of minutes as the surgeon has to look only at the intraoperatory images, decide where he wants to go and express that by voice. Using voice commands to position the laparoscopic camera eliminates the need of an assistant close to the surgeon while keeping his both hands free, as presented in Figure 3. The control functions for the configuration and motion of PARAMIS have been implemented in Delphi, the same set of functions being used for all command interfaces. For the voice commands, the task is to make the computer recognize the voice of the surgeon and call the desired function. For that a simple and free to use solution has been adopted by using the Microsoft Speech SDK 6, released in 2008. The profile of the user is trained and saved on the computer (Figure 4). After a 25 minutes training session, a recognition level of 90% is achieved while after 45 minutes a recognition level of 99% is achieved. The pseudo code for the

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(a) Voice commands interface

(b) Experimental run

Fig. 3 PARAMIS voice control.

(a) Pseudo code algorithm

(b) Voice profile setup

Fig. 4 Voice control algorithm and speech setup.

voice commands introduction is illustrated in Figure 4. Using the Speech SDK implies the initialization of the Speech functions and the use of a set of variables defined under the Speech functions. For each command an instance is initialized. When the recognition function is active, the system searches for the predefined sequences and when such a line is recognized a function is called. The use of the Speech SDK brings some important advantages: any command can be modified at any time without the need of profile retraining, the training time is short, recognition accuracy is high and commands can even be customized for each surgeon. An important aspect when using voice commands is also the prevention of a faulty or involuntary motion of the robot due to an unwanted given command or a command given by another person. The speech profile of each person can be configured to work with very high accuracy which reduces to a minimum the risk of recognizing a command line given by another person. In addition the use of a quality microphone with noise-canceling and the integration within the command line of a ”spe-

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cial” trigger word (ex. PARAMIS + command) will reduce to a minimum the risk of unwanted displacements of the laparoscopic camera. Furthermore the definition of the ”safety working volume” prevents any motion that might endanger the patient during the procedure both due to a human error or an unwanted and improbable faulty interpretation of a command. The voice commands interface is closed using the ”QUIT” command which stops the voice commands interface and the control system switches to the previous interface.

5 Experimental Runs and Results The evaluation of the PARAMIS parallel robot has been achieved through a series of experimental procedures. The tests were made on a Torso Trainer first with a Cholecystectomy Model [12] and then on a pork liver. The aim of the tests was to evaluate the interaction surgeon–robot, the acceptance of the robot and the advantages and drawbacks reported during the procedure. During the first set of experiments the robot was positioned caudal with respect to the surgeon, while the second setup, the robot has been placed on the left side of the patient, while the surgeon was placed in the ”French position” for the cholecystectomy (Figure 3). Due to the very easy to use control system the surgeons adapted in a matter of minutes to the use of PARAMIS. From the surgical point of view, for PARAMIS were reported the following advantages: PARAMIS is able to manipulate the laparoscopic tool in a large area allowing the inspection and intervention in different points of the surgical field; The voice-control interface allows the surgeon to focus on the procedure without any concern regarding the robot controller; The system allows the positioning of the laparoscope with large displacement or fine positioning as the procedure imposes at a certain moment in time; The adaptability of the robot is increased as it allows the positioning, with respect to the patient, in any of the four cardinal points which allows the use of multiple intraoperative positions as well as several types of interventions (abdominal or thoracic); The possibility of recording different positions allows a fast and easy return of the camera in those specific positions. From the technical and economic point of view there are several aspects that have to be pointed out regarding PARAMIS: The cost of the robot is small compared to other existing solutions (15000 Eur); The easy to customize parameters of the robot makes it a versatile tool adaptable to the specific needs of each surgeon; The possibility to transform the system in a multiarm robot controlled from the console will allow the integration of the system in a new type of surgical robot.

6 Conclusions An experimental parallel robot for camera positioning in MIS was developed. The experimental results shown that the robot is designed to respond quickly to the surgeon voice commands and can be moved safely in the defined surgical workspace.

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Its design is focused on two aspects: to provide a minimum occupied space in the insertion point and to be useful in a multiarm robotic system. The open-architecture voice control solution is simple, safe and customizable and fulfils the specific needs of the surgeons. The control system allows continuous improvements of the robot functions without any physical modifications, making it versatile and adaptable to the specific needs of surgical arena. The team will continue to work on the structure optimization and start the construction of a second robot, which will be destined to manipulate a surgical instrument, working in cooperation with PARAMIS.

Acknowledgement The research work reported here was financed by the PNCDI-2 P4 Grant, entitled “Multidisciplinary development of surgical robots using innovative parallel structures”. It has been awarded by the Ministry of Education, Research, Youth and Sports of Romania.

References 1. Anvari, M. and Marescaux, J., Robotic surgery: Ready for prime time. Epublication: WBSurg.com, 6(10), http://www.websurg.com/ref/doi-ed01en0020.htm, October 2006. 2. Brown University, http://biomed.brown.edu/, 2004. 3. Finlay, P.A., A new miniature manipulator for laparoscopic camera control. In Proceedings World Congress on Medical Physics and Biomedical Engineering, Germany. 2009. 4. Graur, F. et al., Experimental laparoscopic cholecistectomy using PARAMIS parallel robot. In Proceedings of the 21st Conference of SMIT, Sinaia, Romania, 2009. 5. Gumbs, A., Milone, L., Sinha, P., and Bessler, M., Totally transumbilical laparoscopic cholecystectomy. Gastrointest. Surg., 13:533–534, DOI 10.1007/s1105-008-0614-8, 2009. 6. Hanly, E.J. and Talamini, M.A., Robotic abdominal surgery. The American Journal of Surgery, 188:19–26, 2004. 7. Intuitive Surgical Inc., www.intuitivesurgical.com, 2009. 8. Pisla, D., Plitea, N., and Vaida, C., Kinematic modeling and workspace generation for a new parallel robot used in minimally invasive surgery. In Advances in Robot Kinematics, Springer, 2008. 9. Pisla, D., Hesselbach, J., Plitea, N., Raatz, A., Gherman, B., and Vaida, C., Dynamic analysis and design of a surgical parallel robot used in laparoscopy, Journal of Vibroengineering, 11(2):215–225, 2009. 10. Plitea, N. et al., Innovative development of parallel robots, Acta Electrotehnica, Special Issue, 201–206, 2007. 11. Punt, M.M., Stelefs, C., Grimbergen, C., and Dankelman, J., Evaluation of voice control, touch panel control and assistant control during steering of an endoscope. Minimally Invasive Therapy and Allied Technologies, 14:181–187, 2005. 12. Simulab Inc., www.simulab.com, 2009. 13. Unger, S.W., Unger, H.M., and Bass, R.T., AESOP robotic arm, J. Surg. Endosc., 8:1131, 1994. 14. Vaida, C., Contributions to the achievement and the kinematic and dynamic modeling of a new parallel robot for minimally invasive surgery, PhD Thesis, 2009. 15. Vaida, C., Pisla, D., Plitea, N., et al., Development of a control system for a parallel robot used in minimally invasive surgery, IFMBE Proceedings Series, 2009.

Modeling and Simulation of the Tracking Mechanism Used for a Photovoltaic Platform C. Alexandru Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: [email protected]

Abstract. The paper presents the modeling and simulation of the dual-axis tracking system of azimuthal type used for a photovoltaic (PV) platform. The study is approached in mechatronic concept, by integrating the mechanical device and the control system at the virtual prototype level, during the entire design process. The mechanical device model of the tracking mechanism is developed as multi-body system by using the MBS environment ADAMS, while the DFC EASY5 is used for modeling the control system. The key-word for design is the energetic efficiency of the azimuthally tracked PV platform, which is evaluated in different intervals during the year. Through the optimal design of the tracking mechanism and of the control system (the controller and the control law), we obtained good values for the energetic efficiency throughout the year (the average value is around 34%), which justify the viability - utility of the tracking system. Key words: PV platform, tracking mechanism, multibody system, mechatronic model

1 Introduction The researches in the field of renewable energy systems represent a priority at international level because provides alternatives to a series of major problems: the limited and pollutant character of the fossil fuels, global warming, and the greenhouse effect. The photovoltaic conversion (transforming the solar radiation in electricity) is one of the most addressed topics in the field. The efficiency of the PV modules depends on the degree of use and conversion of the solar radiation. Consequently, there are two ways for maximizing the efficiency: optimizing the conversion to the absorber level and decreasing the losses by properly choosing the absorber materials; increasing the incident radiation rate by using solar trackers, the maximum degree of collecting being obtained when the solar radiation is normal on the active surface (for this paper, the second solution is considered). Basically, the tracking systems are mechanical devices – mechanisms driven by controlled rotary or linear actuators (i.e. mechatronic systems). The orientation of the PV modules, in order to intercept the maximum amount of solar radiation that reaches the ground level, may increase the efficiency up to 50% [3, 6, 7]. In practice,

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there are two solutions for developing the tracked PV arrays (groups of modules): array with individual modules, where the modules are separately mounted on individual sustaining structures; PV platforms, where the modules are mounted on a common frame (sustaining structure), the orientation being simultaneously realized by the orientation of the entire platform. The PV platforms, even they involve inconvenient concerning the construction or integration in the built environment, have the advantage of a unitary electrical management and the structure is more compact than in the case of the array with individual modules. The orientation principle of the PV platforms is based on the input data referring to the position of the Sun on the sky dome. For the design process of the tracking mechanisms, the two rotational motions of the Earth-Sun system have to be considered: the daily motion, and the yearly precession motion. The dual-axis tracking systems combine the two motions, so that they are able to follow very precisely the Sun path during the year. Depending on the relative position of the revolute axes, there are two types of dual-axis systems: polar and azimuthal. For the polar trackers, there are two independent motions, because the daily motion is made by rotating the platform around the fixed polar axis. For the azimuthal trackers, the main motion is made by rotating the platform around the vertical axis, so that it is necessary to combine this rotation with an elevation/altitudinal motion, the correlation of the motions increasing the complexity of the control process. Nevertheless, the azimuthal systems are frequently used for PV platforms because of the constructive and safety operation aspects. In the field of analysis, optimization and simulation, important publications reveal a growing interest in methods for multi-body systems (MBS) that may facilitate the self-formulating algorithms, having as goal the reduction of the processing time in order to make possible real time simulation [2, 5, 8, 11]. Based on the MBS theory, powerful virtual prototyping environments have been developed, which allow to build models of not only subsystems but also entire systems, and then to simulate their behavior and optimize the design before creating the hardware prototype. Such approach allows also to integrate the finite element models in the multibody system analysis, for the precise simulation of the tracking system with compliant components, as well as the evaluation of the stability performance (involving results of the modal analysis and of the control system design), all within the MBS environment. In these terms, the aim of the paper is to analyze and simulate the dual-axis tracking system (azimuthal-type) for a PV platform. The paper proposes the integration of the two main components of the tracking system (the mechanical device and the control system) at the virtual prototype level (i.e. modeling in mechatronic concept). The study is performed by using a virtual prototyping platform, which integrates the following software solutions: CAD – CATIA (for creating the solid model of the tracking mechanism), MBS – ADAMS (for modeling the mechanical device), DFC – EASY5 (for designing the control system) and ADAMS/Controls (for defining the input & output plants, which assure the communication between the mechanical device and the control system). The main result of the research consists in evaluating the tracking efficiency during the year.

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2 The Virtual Prototype of the Tracking Mechanism For identifying accurate mechanical configurations suitable for the tracking systems, a conceptual design method based-on the MBS theory was used. The method is based on the following stages [4]: identifying all possible graphs, by considering the space motion of the system, the type of joints, the number of bodies, and the degree of mobility; selecting the graphs that are admitting supplementary conditions imposed by the specific utilization field; transforming the selected graphs into mechanisms by mentioning the fixed body and the function of the other bodies, identifying the distinct graphs versions based on the preceding particularizations, transforming these graphs versions into mechanisms by mentioning the types of geometric constraints. The graphs of the multibody system are defined as features based on the modules, considering the number of bodies and the relationships between them. As result, a collection of possible structural schemes of tracking mechanisms was obtained, the solution in study, which is a dual-axis azimuthal tracking mechanism (Figure 1), being selected by applying the multi-criteria and the morphologic analyses. The evaluation criteria were referring to the tracking precision, the amplitude of the motion, the complexity of the system, the possibility for manufacturing and implementation. The driving source for the azimuthal motion is a linear actuator, the motion being transmitted to the sustaining pillar of the platform (containing 8 PV modules) with a stroke amplifying mechanism (for avoiding the self-blocking risk), which is a spatial four-bar mechanism (DEFG). The lower tube of the pillar is rigidly connected on ground plate, while the crank is fixed to the moving upper part of the pillar. The altitudinal motion is also generated with a linear actuator, at which the cylinder is connected to the moving part of the pillar, and the piston acts directly on the platform. In this case there is no need for a stroke amplifying mechanism, because the angular field is less than in azimuthal motion (for elevation, in Bras¸ov area, the maximum angular field is 47◦ instead of 180◦ for azimuthal motion). The 3D-solid model of the tracking mechanism was realized using the CAD environment CATIA, the geometry being transferred to the MBS environment ADAMS by using the STEP format (via ADAMS/Exchange interface). The dynamic model of the tracking mechanism takes into consideration the mass and inertial forces, the reaction in joints, and the joint frictions, which are modeled by the coefficient of dynamic friction, the friction arm, the bending reaction arm, the radius of the pin, the stiction transition velocity, the maximum stiction deformation, and the preload friction torque in the joint. In this way, in the main revolute joints of the tracking mechanism (G – for azimuthal motion, and K – for altitudinal motion), there are the following friction torques: TG = 8.4 Nm, TK = 0.5 Nm. Regarding the control process, closed loop systems (based on photo sensors, which are responsible for discrimination of the Sun position and for sending signals to the controller) and opened loop systems (based on mathematic algorithms, which provide predefined parameters for the motors, depending on the Sun positions) are traditionally used [10]. Because the orientation based on the Sun detecting

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Fig. 1 The virtual model of the azimuthally tracked PV platform (MBS ADAMS).

sensors may introduce errors for variable weather conditions, we adopted an opened loop control system, which was developed using ADAMS/Controls and EASY5. For connecting the mechanical model and the control system, the input & output plants were defined. The control forces developed by the linear actuators represent the input parameters in the mechanical model. The outputs transmitted to the controller are the azimuthal and altitudinal angles of the PV platform. With these plants, the control system model was created in EASY5 (Figure 2), the input function generators (SF/SF2) representing the databases with the imposed values of the azimuthal and altitudinal angles. For obtaining reduced transitory period and small errors, we used PID controllers for both motions. The tuning of the controllers was made by performing a MonteCarlo analysis with the EASY5 Matrix Algebra Tool (MAT). The method is based on the generation of multiple trials to determine the expected values of a random variable, by following the steps: define a domain of possible inputs, generate inputs randomly from the domain, perform a deterministic computation using the inputs, aggregate the results of the individual computations into the final result. In this case, the variables are the proportional, derivative and integral terms of the PID controller, while the objective is to improve the rise time, the overshoot, and the steady-state error, which define in fact the tracking error.

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Fig. 2 The control system of the tracking mechanism (DFC EASY5).

The control system was imported in MAT as EMX file, the tracking error being optimized by using the “minimize v” function. As MAT proceeds through the minimization, we will see the calculation converge. The values of the tuning parameters will result in a simulation that meets the design requirements, as follows: K p = 150, Ki = 10, Kd = 50. The controller design by considering the entire mechatronic system allows obtaining valuable results, which are important for the physical implementation. In the mechatronic model, ADAMS accepts the control forces from EASY5 and integrates the mechanical model in response to them. At the same time, ADAMS provides the current azimuthal and altitudinal angles to integrate the control system.

3 Results and Conclusions The PV platform can be rotated without brakes, or can be step-by-step driven, usually by rotating the platform with equal steps at every hour. The continuous orientation allows to capture the maximum incident solar radiation, but there are some important disadvantages: the operating time of the system/motors is high, with negative influence on the system reliability, including the motors wearing; there are ne-

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cessary transmissions with high ratios, which can generate constructive problems; the behavior of the system in terms of occurrence of external perturbations, such as the wind action, whose effect can be amplified if the system is moving. In our study, the key idea for optimizing the motion law is to maximize the energy gained through the step-by-step orientation, for absorbing as much as possible solar energy with minimum energy consumption. This is made according with the methodology presented in [1], considering the correlation between the maximum amplitudes of the azimuthal and altitudinal motions, the number of tracking steps, and the actuating moments. The tracking efficiency of the PV platform can be written in the following way: ε = (ET − EF ) − EC  0, in which ET is the electric energy produced by the tracked PV platform, EF is the energy produced by the same platform without tracking (fixed), and EC is the energy consumption for orienting the platform. The quantity of electric energy produced by the PV platform, with and without tracking, was obtained by integrating the incident radiation curve, and multiplying this value with the product between the conversion efficiency of the modules (14%) and the active surface of the platform (8 modules × 1.638 m2 = 13.104 m2 ). The energy consumption for realizing the imposed motion laws was determined by simulating the virtual prototype of the tracking system (described in the previous section). The incident radiation, normal to the active surface, depends on the direct radiation and the angle of incidence. The direct radiation can be estimated using different empirical methods (such as Kasten’s, Adnot’s, Hottel’s or Meliß’s model), depending on a lot of factors: the solar constant, the day number in year, the atmospheric turbidity factor (which is an important parameter for assessing the air pollution in local area, as well as being the main parameter controlling the attenuation of solar radiation reaching the Earth’s surface under cloudless sky conditions), the solar altitude angle, the solar declination, the location latitude, the solar hour angle, and the local time. For this study/paper, we used the Meliß’s model [9], which can be most easily adapted to specific meteorological conditions in the Bras¸ov area. The angle of incidence was determined from the scalar product of the sunray vector and the normal vector on platform, the mathematic model being developed according with the methodology presented in [12]. The numeric simulations were performed considering the specific values for Bras¸ov location (45.6333 North – latitude, 25.5833 East – longitude), in different days during the year. For this location, the maximum altitude angle of the sunray has an annual variation between 10◦ and 68◦ , and the homologous angle of the azimuthally tracked platform can approximate the previous variations through 14 annual steps, which involves 14 annual intervals [12]: N ∈ [27–52], [53–72], [73– 87], [88–103], [104–120] [121–140], [141–202], [203–222], [223–239], [240–254], [255–270], [271–290], [291–315], [316–326], where N is the day number in year (e.g. N = 1 for 1 January). According with the tracking strategy, for each interval we selected a representative day, in which the motion laws for the azimuthal and altitudinal motions were established (the azimuthal angle is null at the solar noon, positive in the morning and negative in the afternoon, while the altitudinal angle is null when the platform is vertically disposed).

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Table 1 The energetic efficiency of the PV tracking system throughout the year. N

EC [Wh/day]

ET [Wh/day]

EF [Wh/day]

Gain [%]

40 63 81 96 112 131 172 213 231 247 267 281 303 356

340.58 485.28 483.42 652.70 685.91 780.12 820.53 780.12 659.70 652.70 483.42 485.28 340.58 240.77

10102.05 11943.88 13863.88 14668.57 16006.77 16964.13 17932.92 16838.91 15880.73 14790.47 12800.96 11982.89 9692.30 7384.46

7520.97 8989.66 10305.65 10716.33 11282.97 11438.8 11470.24 11375.38 11190.47 10768.21 9610.99 8990.86 7223.07 5289.51

29.79 27.46 29.83 30.78 35.78 41.48 49.19 41.17 35.78 31.29 28.16 27.88 29.47 35.05

Fig. 3 The results for summer solstice day.

The results obtained in this way are systematized in Table 1, in which the total energy consumption (EC ) was obtained by summing the energy consumptions for realizing the azimuthal and altitudinal motion laws. The energetic gain is computed as percent of the tracking efficiency (ε ) from the energy produced by the PV platform without tracking (ε ·100/EC ). The average of the energetic gain throughout the year is around 34%, and this value justifies the investment involved by the tracking system. As exemplification, the results obtained for summer solstice day (N = 172), with the highest energetic gain, are shown in Figure 3. The application is a relevant example regarding the implementation of the virtual prototyping tools in the design process of the PV systems with solar tracker. One

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of the most important advantages of this kind of simulation is the possibility to perform virtual measurements in any point or area of the tracking system, and for any parameter (motion, force, energy). Using the virtual prototyping technique, we are able to optimize the mechanical structure of the tracking mechanism, choose the appropriate actuators, design the optimal controller, optimize the motion law, and perform the energy balance of the PV system. The tracking mechanism will be manufactured and implemented in the area of the Research Institute High-Tech Products for Sustainable Development from the Transilvania University, creating a real perspective for the research in the field. This will allow a relevant comparison between the virtual prototype analysis and the data achieved by measurements. At the same time, the future researches will follow to evaluate the durability and stability characteristics of the tracking mechanism with compliant components, as well as more complex control strategies based on predictive and fuzzy methods (using meteorological prognoses).

References 1. Alexandru, C., The mechatronic model of a photovoltaic tracking system. International Review on Modelling and Simulations, 0:64–74, 2008. 2. Ceccarelli, M., Challenges for mechanism design. In Proceedings of the 10th IFToMM International Symposium SYROM, Brasov, Plenary Lecture, pp. 1–14, 2009. 3. Chong, K.K. et al., Integration of an on-axis general sun-tracking formula in the algorithm of an open-loop sun-tracking system. Sensors, 9:7849–7865, 2009. 4. Comsit, M. and Visa, I., Design of the linkages-type tracking mechanisms of the solar energy conversion systems by using multi body systems method. In Proceedings of the 12th IFToMM World Congress, Besanc¸on, ID 582, 2007. 5. Eich-Soellner, E. and F¨uhrer, C., Numerical Methods in Multibody Dynamics. Teubner, 2008. 6. Guo, L. et al., Design and implementation of a sun tracking solar power system. In Proceedings of the ASEE Annual Conference and Exposition, Austin, pp. 1–11. 2009. 7. Hoffmann, A. et al., A systematic study on potentials of PV tracking modes. In Proceedings of the 23rd European Photovoltaic Conference EUPVSEC, Valencia, pp. 3378–3383, 2008. 8. H¨ohne, G. et al., Extended Virtual Prototyping. Springer, 2007. 9. Meliß, M., Regenerative Energiequellen. Springer-Verlag, 1997. 10. Mousazadeh, H. et al., A review of principle and sun-tracking methods for maximizing solar systems output. Renewable and Sustainable Energy Reviews, 13:1800–1818, 2009. 11. Shabana, A., Dynamics of Multibody Systems, 2nd edition. John Wiley & Sons, 1998. 12. Visa, I. et al., On the optimization of the PV azimuthal tracking steps. In Proceedings of the 23rd European Photovoltaic Conference EUPVSEC, Valencia, pp. 3165–3169, 2008.

The Modular Robotic System for In-pipe Inspection O. Tatar, C. Cirebea and A. Alutei Mechanisms, Precision Mechanics and Mechatronics Department, Technical University of Cluj-Napoca, 4000641 Cluj-Napoca, Romania; e-mail: {olimpiu.tatar, claudiu.cirebea, adrian.alutei}@mmfm.utcluj.ro

Abstract. The paper presents a modular system for inspection and exploration developed by the authors, which is composed of active and passive modules. The system is placed into motion using two geared DC motors and geared transmissions. The control of the modular system is achieved using a CEREBOT II circuit board trough a graphic interface developed using the DELPHI visual programming environment. Key words: in-pipe, mechanism, modular, robot

1 Introduction Recently, a great number of robot systems have been developed. These robots can access narrow spaces of the pipelines to perform inspection tasks, using electronic devices such as sensors and cameras. The utility of these type of robots proves itself to be even greater when the areas that need to be inspected are hazardous or difficult to access. Some inspection robots were designed to perform specific tasks inside pipes with a certain diameter [1, 6], but another category of robots is being developed. These types of robots can adapt their structure to the variation of the pipe diameter and they can perform several tasks (video inspection, cleaning, welding, sealing, cutting and so on) [1–5, 8]. In this paper the authors present a modular robotic system for the inspection of pipelines with diameters ranged between 130 and 200 mm. This robotic system was designed and manufactured within the Robotics Laboratory from the Technical University of Cluj-Napoca, Mechanisms, Precision Engineering and Mechatronics Department. While designing and developing this in-pipe inspection system the following objectives were considered: the development of a structure that can adapt to a large range of pipe diameters, to reduce the number of actuators, to reduce the weight of the system and to obtain a high traction force.

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Fig. 1 The photo of the modular robotic system.

2 The Modular Robotic Systems The modular system that has been developed is presented in Figure 1 [7]. The system consists of three modules: two active (drive) modules and a passive module, all connected by universal joints. The modular system has a total length of 881 mm. For inspection tasks the system uses a wireless video camera placed in the front of one of the active modules and it is surrounded by a protection system. The pipes that can be inspected using this system can have the inner diameters ranged between 130 and 200 mm. The modular system was tested within the laboratory inside PVC pipes with inner diameters of 190 and 200 mm. The velocity of the system inside an horizontal pipe section is of 20 cm/s. This system is not energetically autonomous, being powered from a voltage source trough wires.

2.1 The Modular System Structure The Drive Module The active module consists of three articulated mechanisms places at angles of 120◦ around the longitudinal axis. The elements are made of aluminium and the central shaft (Θ 10 mm) is made of steel. The articulated elements (Figure 2) of the driving module have the following lengths: h1 = 30 [mm], h2 = 60 [mm], h3 = 135 [mm], (h1 = OA, h2 = BC = DE, h3 = CF). The wheels have the radius R = 25 [mm] and the width 7 [mm] and they are fitted with rubber rings for a better grip to the travel surface. The mass of the driving module (the weight of the power feeding wires being also considered) is of 610 [g]. The force exerted by the driving module’s mechanisms on the inner surface of the pipe is generated by an extensible spring [7]. This spring is placed on the central shaft and it also ensures the module’s stability while travelling trough sections of pipe that have inner diameter variations. For the in-pipe inspection, the robot uses a wireless video mini-camera fixed inside of a protection system. The IG22 geared motor is used for placing the system into motion. The reduction gear box that is used has a reducing factor of ireducer = 19.

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Fig. 2 The structural scheme and the photo of the drive module.

Fig. 3 The motor – reducing gear box – transmission – wheel assembly.

The motion from the geared motor to the three drive wheels is achieved by using three geared transmissions. The said transmissions consist of: • 1 – single enveloping worm (z1 = 1), module m = 0.75 mm, the angle of the worm thread θw = 4◦ ; • 2, 3, 4 – gears with angled teeth (the angle of the teeth β = 4◦ ) • z2 = 39, z3 = 39, z4 = 44 – teeth. Considering nM the number of revolutions per minute of the motor and nR the number of revolutions per minute of the drive wheel, from the equation of the gear ration: n iMR = M = i12 · i23 · i34 = itransmission (1) nR we obtain the numbers of revolutions per minute of the drive wheels: nR =

z1 n z4 M

(2)

The motor-reducing gear box-transmission – driving wheel assembly is presented in Figure 3.

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Fig. 4 The 3D model, the structural scheme and the photo of the passive module.

Fig. 5 The electronic components from the passive module; 1 – Cerebot II, 2 – PmodRS232, 3 – PmodHB5.

The Passive Module The 3D model and a photo of the passive module are presented in Figure 4a. The passive module has a total length l = 244 [mm], a maximum diameter D = 110 [mm], the radius of the wheels R = 25 [mm], the width of the wheels of 7 [mm] and a mass of 700 [g]. The modular system is controlled using a Cerebot II circuit board [9]. The components of the passive module included in this system are shown in Figure 5.

Actuators Selection The motor was chosen and verified following the next procedure: the reduced resistant torque of the motor shaft was determined. The condition that the developed torque must be greater than the reduced resistant torque was placed. For determining the reduced resistant torque of the motor shaft we consider the system to be placed in the most detrimental position (vertical position), when the motor must carry the maximum load (Figure 6). The following notations were used: F¯ f F¯R M¯

fr

= the friction force between the active module’s wheel and the inner surface of the pipe (Ff = µ · FR); = the reaction force between the pipe wall and the active module’s wheel; = the active module’s roll friction torque (M f r = s · FR);

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s µ ω¯ R M fl k θ d v R kp ∆ xp G1 m1 G2 m2 F¯Rp F¯Ap M¯

f rp

M flp F¯ f p

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= the roll friction coefficient; = the slide friction coefficient; = the angular speed of the active module’s wheel; 
µ ·d·F = the friction torque inside the wheel M f l = 2 R ; = the drive module spring constant; = the angle between the elements h1 and h2 ; = the diameter of the wheel axle; = the speed of the robot (v = ωR R); = the radius of the wheel; = the passive module spring constant; = the passive module spring deformation; = the weight of the active module (G1 = m1 · g); = the mass of the active module; = the weight of the passive module (G2 = m2 · g); = the mass of the passive module; = the reaction force between the pipe wall and the passive module’s wheel, FRp = FAp = k p ∆ x; = the passive module’s spring force; = the passive module’s roll friction torque (M f rp = s · FRp); 
µ ·d·F = the friction torque in the passive module’s wheel bearing M f l = 2 Rp ; = the friction force between the passive module’s wheel and the inner surface of the pipe (Ff p = µ · FRp).

r is determined using the The reduced resistant torque from the motor shaft MraxM following relation:

1 1 · ireducer itranssmision       2kh1 µd 1 (sin θ − sin θmin ) s + × 6 3 2 tgθ    m2 g µd + m1 gR + R + 3k p ∆ x p s + 2 2

r MraxM =

(3)

Considering the relation (3) k = 0.1 N/mm2 , h1 = 30 mm, s = 1 mm, µ = 0.18, d = 4 mm, k p = 0.08 N/mm2 , ∆ x p = 6 mm, m1 = 610 g, m2 = 700 g, θmin = 15◦ r the reduced resistant torque from the motor shaft MraxM was obtained (Figure 6). The torque that the motor is developing is greater than the reduced resistant torque of the calculated motor. This is the condition that has to be fulfilled so the modular robotic system is able to travel trough the pipe. One of the main aspects in designing the drive module was to obtain a sufficient traction force so it could place the drive module into motion as well as the passive module, which carries the electronic components and other devices needed for the inspection of the pipe.

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Fig. 6 The variation of the reduced resistant torque, considering (the angle) θ .

Fig. 7 The “In Pipe Modular Robotic System 2” user interface.

In order for the drive wheel to roll without slipping on the inside surface of the pipe, the traction force must be smaller or equal with the friction coefficient or with the pressure force between the wheel and the pipe wall. The friction coefficient depends directly on the wheel material and the surface conditions of the pipe.

Control of the Robotic Modular System To control the direction of the entire robotic system, the “In Pipe Modular Robotic System 2” user interface was developed. The interface is presented in Figure 7. The general functioning scheme for the inspection and exploration robotic system is presented in Figure 8. For controlling the motor number of revolutions, a PID (proportional–integral– derivative) controller was used. One of the issues that come up when using this type of controller is the tunning. One of the developing possibilities regards using the Signal Constraint block from the MATLAB/Simulink Response Optimization environment. This block allows the optimal determination of the controller parameters. The user can settle, in a graphical way, the performances of the feedback from the step input signal (Figure 9).

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Fig. 8 The flow chart for controlling the modular robotic.

Fig. 9 The optimization of the controller parameters using the Signal Constraint block.

For modelling the behaviour of an electric DC motor into the Simulink enviroment, the relations that describe the electrical and mechanical behaviour of the motor were used. 1 dia = (ua − cm · ω − R · ia) (4) dt L 1 dω = (cm · ia − ω · b − Mext ) (5) dt J where R represents the resistance of the armature, L the electric inductance, J is the rotor’s moment of inertia, cm is the machine constant, b represent the friction r coefficient and ia is armature current, Mext – external torque (Mrax M ). Taking note of Equations (4) and (5), a flow chart was developed. This flowchart represents the model of the DC motor presented in Figure 10.

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Fig. 10 The model of the motor.

Fig. 11 Initial step response system.

Fig. 12 Step response the parameter optimization.

The characteristics of the motor are: b = 2.6 · 10−6 [Nm/ rad/s], L = 6 · 10−3 [H], R = 3[Ω ], J = I r = 1.5 · 10−6 [kg m2 ], cm = 5.7 · 10−3 [Nm/A]. Therefore, the initial response of the system with KP = 1, KI = 0, KD = 0 is presented in Figure 11. After the optimisation the resulting constants are: KP = 5.294, KI = 0.07686, KD = 0.0238. In a corresponding way, the response during an time period is also obtained (Figure 12).

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In the near future, field tests will be conducted with the system and the prototype will be adjusted according to the obtained results.

3 Conclusions This paper presents a wheeled in-pipe modular robotic system. A very impor-tant design achievement of the in-pipe modular robotic system is the adaptability to variable inner diameters of the pipes. The in-pipe modular robotic system can inspect pipes with variable diameters ranged between 130 and 200 mm. The prototype test results were successful inside PVC pipes.

Acknowledgments This work is supported by the CNCSIS (National Council of Scientific Research in Higher Education from Romania); through PNII – IDEI Project, ID 1056: Modelling, simulation and development of robotic system families used for inspection and exploration.

References 1. Horodinca, M., Doroftei, I., Mignon, E. and Preumont, A., A simple architecture for in-pipe inspection robots. In Proc. Int. Coll. on Mobile and Autonomous Systems, 10 Years of the Fraunhofer IFF, Magdeburg, Germany, pp. 61–64, 2002. 2. Jun, C., Deng, Z.Q. and Jiang, S.Y., Study of locomotion control characteristics for six wheels driven in-pipe robot. In Proceedings of the 2004 IEEE International Conference on Robotics and Biomimetics, Shenyang, China, pp. 119–124, 2004. 3. Kwon, Y.S., Jung, E.J., Lim, H. and Yi, B.J., Design of a reconfigurable indoor pipeline inspection robot. In International Conference on Control, Automation and Systems 2007, Seoul, Korea, pp. 712–716, 2007. 4. Moghaddam, M.M. and Hadi, A., Control and guidance of a pipe inspection crawler (PIC). In Proceedings International Symposium on Automation and Robotics, pp. 11–14, 2005. 5. Roh, S. and Choi, H.R., Differential-drive in-pipe robot for moving inside urban gas pipelines, IEEE Transactions on Robotics, 21(1), 2005. 6. Suzumori, K., Miyagawa, T., Kimura, M., and Hasegawa, Y., Micro inspection robot for 1-in pipes, IEEE/ASME Trans. Mechatronics, 4:286–292, 1999. 7. Tatar, M.O., Mandru, D., and Alutei, A., In pipe modular robotic systems for inspection and exploration. In Proceedings of the 5th International Conference Mechatronic Systems and Materials (MSM 2009), Vilnius, pp. 89–91, 2009. 8. Zhang, Y. and Yan, G., In-pipe inspection robot with active pipe-diameter adaptability and automatic tractive force adjusting, Mechanism and Machine Theory, 42:1618–1631, 2007.

Mechanism Design

Geometric and Manufacturing Issues of the 3-UPU Pure Translational Manipulator A.H. Chebbi and V. Parenti-Castelli Department of Mechanical Engineering – DIEM, University of Bologna, 40136 Bologna, Italy; e-mail: [email protected], [email protected]

Abstract. This paper collects the most relevant results on singularities and workspace of the Tsai manipulator, a parallel manipulator that provides its mobile platform with three degree of freedom of pure translation with respect to its fixed base. The paper investigates the influence of some geometric parameters – specifically the orientation of the base/platform revolute axes and the locations of its legs – on the structure of the manipulator and consequently on the manipulator performances, mainly in terms of singularity loci and workspace size. New promising manipulator geometries are presented and some manufacturing solutions are proposed for leg collision avoidance. Key words: parallel manipulators, geometry, singularity, leg collision avoidance

1 Introduction Great attention has been devoted to parallel manipulators (PMs) for their complementary characteristics with respect to the serial ones. The topology of parallel manipulators features a fixed base connected to a moveable platform by a number of serial chains (leg). Since many applications do not necessarily need six degrees of freedom (DOFs), manipulators with less than six DOF are also very interesting, in particular three-DOF PMs which provide the platform with a pure translation, a pure rotation or a mixed of translation and rotation have been studied in the literature [2, 3, 6]. The 3-UPU parallel manipulator [5] (Figure 1) has been extensively studied both as a three-DoF pure translational and a pure rotational manipulator (U and P are for Universal and Prismatic joint respectively). However, the influence the manipulator geometric parameters have on the manipulator performances still deserves attention. Indeed, new geometries can be devised which provide mechanisms with appealing kinematic and static features. This paper will focus on the 3-UPU pure translational manipulator, hereafter called 3-UPU TPM. The influence of both the directions of the base/platform revolute axes and the leg position is further investigated and some new geometries of the mechanism are presented, which exhibit interesting performances.

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Fig. 1 The 3-UPU TPM.

Moreover, for the cases where the three legs of the manipulator intersect, some design solutions are proposed for the leg collision avoidance. The paper is organized as follows. In Section 2, two known 3-UPU TPM geometries are recalled to show the main properties of the 3-UPU TPM. Section 3 presents four new geometries that show the influence of the directions of base/platform revolute axes and of the leg location on the singularity loci of the manipulator. Section 4 presents three design solutions that allow the leg collision avoidance. Finally some conclusions are reported.

2 Basics A schematic of a 3-UPU parallel manipulator is shown in Figure 1. It features a translational platform connected to a fixed base by three extensible legs of type UPU. The universal pair U comprises two revolute pairs with intersecting and perpendicular axes, centred at point Bi , i = 1, 2, 3 in the base and at point Ai , i = 1, 2, 3 in the platform. In order to prevent possible rotations of the platform, two conditions must be fulfilled for each leg [1, 5]: • the axes of the two intermediate revolute pairs are parallel to each other; • the axes of the two ending revolute pairs are parallel to each other. The singularity of the manipulator occurs when the determinant of the Jacobian matrix that relates the load applied in the platform and the wrench leg vanishes. This condition correspond to [1]: [s1 .(s2 × s3 )].[u1 .(u2 × u3 )] = 0

(1)

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Fig. 2 Singularity loci for the 3-UPU TPM Geometry 1.A.

where si , ui , i = 1, 2, 3, are respectively the unit vector of the i-th leg Ai Bi and the unit vector orthogonal to the axes of the universal joint in the i-th leg on the base and on the platform. Equation (1) can be satisfied when: (i) all unit vectors si , i = 1, 2, 3, become mutually parallel or coplanar [4, 7]; (ii) two out of three vectors ui , i = 1, 2, 3, are parallel. By geometric inspection, it can be seen that this condition occurs when two axes of the revolute pairs of the platform (q4i , q4 j , i = 1, 2, j = 2, 3, i
= j) projects on the two corresponding axes of the base (q1i , q1 j ), providing the projection direction is along the shortest distance of the two axes. Condition (ii) is a concise and geometric definition of singularity occurrence and it represents a powerful geometric tool for detecting this type of singularity. In the following, two known geometries of the manipulator are recalled to show the main properties of the 3-UPU TPM [1,4]. For the first geometry, defined as Geometry 1.A, the axes of the three revolute pairs in the base/platform are coplanar and intersect at three points Ci , i = 1, 2, 3 as shown in Figure 2; here, only the revolute pairs in the base/platform are represented for clarity, all the other ones are omitted. The same simplification has been adopted in the next figures. A reference system Sb fixed to the base with origin Ob (the centre of the circle with radius b defined by the points Bi , i = 1, 2, 3) is chosen. Axes x and y are on the plane π defined by the three points Ci , i = 1, 2, 3, with x axis through point B1 , z axis is normal to the plane π , while y axis is taken according to the right hand rule. According to the condition of singularity defined above, the singularity for the Geometry 1.A occurs: • •

when all vectors si , i = 1, 2, 3, are coplanar and belong to the plane π defined by the three points Bi , i = 1, 2, 3. Point O p , taken as a platform reference position, also belongs to π ; when the base and the platform have the same size; in this case all vectors si , i = 1, 2, 3, are coplanar;

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Fig. 3 Singularity loci for the Geometry 1.B of the 3-UPU TPM.

•

when point O p of the platform projects into point Op in the base, the condition (ii) is satisfied. Likewise, other conditions occur which lead to define similar points Op and O p in the base. Analytically, it can be proved that a singularity locus is a right cylinder Γ [5], with circular directrix γ and axis coincident with the z axis of Sb . Therefore, conversely, once defined the points Op , Op and O p, the circle γ is defined and the cylinder Γ is defined too. The three points can be easily found by geometrical inspections thus representing a simple and efficient method to easily find the cylinder Γ . This cylinder has radius r = 2(b − p).

In order to investigate the influence of the legs location on the singularity of the manipulator, a second geometry defined as Geometry 1.B, is shown in Figure 3. This geometry is obtained by disconnecting the platform of the Geometry 1.A from the legs and rotating it 180◦ about the z axis of Sb , which is defined as in the previous geometry, then connecting again the legs to the same corresponding platform revolute pairs. This makes the three legs intersect at one point, which is a practical drawback since the legs cannot penetrate each other. However, suitable manufacturing solutions can overcome it. Indeed, some efficient manufacturing solutions are presented in the next section. Similarly to the Geometry 1.A, the singularity loci of this geometry correspond to the plane π (z = 0) and to the cylinder Γ with axis z of Sb and radius r = 2(b + p). Also structural singularity occurs when the base and the platform have the same size [4]. It is worth noting that, for the same size of the base and of the platform of the two Geometries 1.A and 1.B, the 3-UPU TPM with Geometry 1.B has a larger cylinder of singularity than that with Geometry 1.A, therefore it allows a larger workspace free from singularity inside the cylinder.

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Fig. 4 Geometry 2.A of the 3-UPU TPM.

3 Influence of the Geometry on the Performance of the 3-UPU TPM 3.1 Influence of the Directions of Base/Platform Revolute Axes In this section, two new geometries of the 3-UPU TPM are presented. These geometries can be found when the base and the platform revolute pairs axes are not coplanar. For the first geometry defined as Geometry 2.A, the axes of two revolute pairs on the base are on the plane π and intersect in one point. The axis of the third revolute pairs is orthogonal to the plane π as shown in Figure 4. The singularity loci correspond to: • • •

the plane π (all the vectors si , i = 1, 2, 3 are coplanar). the structural singularity, i.e., the base and the platform have the same size. This singularity occurs when all the vectors si , i = 1, 2, 3 are parallel. three lines δi j , i = 1, 2, j = 2, 3, i
= j, which represent the locus of the reference point O p when two axes of the revolute pairs of the platform (q4i , q4 j ) projects on the two corresponding axes of the base (q1i , q1 j ) providing the projection direction is along the unit vector vi j of the shortest distance among the two axes. A geometrical inspection shows that the lines δ12 and δ13 are on the plane π . While the line δ22 is orthogonal to the plane π .

For the second geometry, defined as Geometry 3.A, according to the condition (ii) of Section 2, two axes of the revolute pairs on the base are mutually parallel

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Fig. 5 Geometry 3.A of the 3-UPU TPM.

and belongs to the plane π , while the third one is orthogonal to plane π as shown in Figure 5. The singularity loci correspond to the plane π and to a line δ13 (δ23 ) (locus of the platform reference point) on this plane obtained by the projection of the axes of the two revolute pairs q11 and q13 (q12 and q13 ) of the platform on the two corresponding axes of the base in the direction orthogonal to these two axes. It can be concluded that the singularity loci is the plane π .

3.2 Influence of the Leg Location Similarly to what done for the transition from Geometry 1.A to the Geometry 1.B (changing the location of the legs), a further 3-UPU TPM geometry can be devised. Indeed, by disconnecting the platform from the legs of the geometries 2.A and 3.A respectively, rotating it 180◦ about z axis, then reassembling it to the same corresponding platform revolute pairs, still keeping the same direction of the base revolute pairs, two new geometries defined as Geometry 2.B and Geome try 3.B, can be found as shown in Figures 6 and 7. These geometries lead to the intersection of the three legs at one point. Analogously to the Geometries 2.A and 3.A, the singularity loci of the Geometries 2.B is the plane π and a line orthogonal to this plane, and for the Geometry 3.B, the plane π .

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Fig. 6 Geometry 2.B of the 3-UPU TPM.

Fig. 7 Geometry 3.B of the 3-UPU TPM.

4 Manufacturing Solutions for the Leg Collision Avoidance of the 3-UPU TPM In this section, three manufacturing solutions are presented in order to avoid the leg collision in the Geometries of type B (crossed legs) of the 3-UPU TPM. Geometry 1.B is taken (for clarity) as an example of this type of 3-UPU TPM. The first manufacturing solution S1 , is to rebuilt the platform of the manipulator. This is obtained by disconnecting the platform of this geometry from the legs and rotating it by a suitable angle α about the z axis of Sb , then connecting again the legs to the platform still keeping the same base axis directions. This means to manufacture a platform with the revolute axis directions rotated of α (clockwise in the example shown in Figure 8a) with respect to the Geometry 1.B. This makes it possible to avoid the leg collision. In Figure 8a, the universal joints on the base and on the platform are represented by points for clarity, and the prismatic ones are omitted. After manufacturing the new platform, the coordinates of the center of the universal joint on the platform Ai , i = 1, 2, 3, are given by:

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Fig. 8 First (a), second (b), and third (c) manufacturing solution for the Geometry 1.B.





O p Ai = cos α O p Ai + sin α O p Ai , with O p A ⊥O p A ; and
O p A
=
O p A
(2) where Ai , i = 1, 2, 3, are the centers of the universal joints on the platform of the Geometry 1.B. The second manufacturing solution S2 , schematically shown in Figure 8b, is to rebuild both the base and the platform of the Geometry 1.B in order to have the coordinates of the centers of universal joints at the base and at the platform, respectively Bi and Ai , i = 1, 2, 3, see Figure 8b, given as follows: Ob Bi = Ob Bi + eq1i , and O p Ai = O p Ai + eq4i

(3)

where Bi and Ai , i = 1, 2, 3, are respectively the center of the universal joints in the base and in the platform of the original Geometry 1.B; q1i and q4i , i = 1, 2, 3, are respectively the unit vectors of the revolute joints on the base and on the platform, which maintain the same directions of the original Geometry 1.B; e is a given distance between the corresponding center of universal joints in the platform of the Geometry 1.B and the platform rebuilt. The third manufacturing solution S3 , schematically shown in Figure 8(c), is to rebuilt the second and the third link of each leg of the Geometry 1.B in order to change the physical position of the prismatic pairs on each leg along Ei Fi , where the coordinate of points Ei and Fi , i = 1, 2, 3 are given by: Ob Ei = Ob Bi + dq2i , and O p Fi = O p Ai + dq3i

(4)

where Bi and Ai , i = 1, 2, 3, are respectively the centers of the universal joints in the base and in the platform of the Geometry 1.B; q2i and q3i , i = 1, 2, 3, are respectively the unit vectors of the intermediate revolute joints of the i-th leg; d is a given distance between the directions of the prismatic pairs for the Geometry 1.B and the manipulator geometry after rebuilding.

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5 Conclusions After the most relevant results on singularity and workspace analysis of the 3-UPU pure translational manipulator have been recalled, the influence of some geometric parameters on the manipulator performances has been investigated. Interesting new geometries of the 3-UPU manipulator have been presented. For some cases, which would exhibit intersecting legs, efficient manufacturing solutions that prevent leg collision have been proposed.

Ackowledgment The support of AER-TECH is gratefully acknowledged.

References 1. Di Gregorio, R. and Parenti-Castelli, V., Mobility analysis of the 3-UPU parallel mechanism assembled for a pure translational motion. ASME Transactions, Journal of Mechanical Design, 124:259–264, 2002. 2. Gosselin, L. and Angeles, J., The optimum kinematic design of a spherical three-degree-of freedom parallel manipulator. ASME Journal of Mechanisms, Transmission and Automation in Design, 111:202–207, 1989. 3. Herv`e, J.M. and Sparacino, F., Structural synthesis of parallel robots generating spatial translation. In Proceedings Fifth ICAR International Conference on Advanced Robotics, pp. 808–813, 1991. 4. Parenti-Castelli, V. and Bubani, F., Singularity loci and dimensional design of a translation 3dof fully-parallel manipulator. In Proceedings of Advances in Multibody Systems and Mechatronics, pp. 319–332, 1999. 5. Tsai, L.W., Kinematics of three-degrees of freedom platform with three extensible limbs. In Advances in Robot Kinematics, Kluwer Academic Publishers, pp. 401–410, 1996. 6. Yang, P-H., Waldron, K., and Orin, D.E., Kinematic of a three-degree-of-freedom motion platform for a low-cost driving simulator. In Advances in Robot Kinematics, Kluwer Academic Publishers, pp. 89–98, 1996. 7. Zlatanov, D., Bonev, I.A., and Gosselin, C., Constraint singularities of parallel mechanisms. In Proceedings IEEE International Conference on Robotics and Automation, Vol. 1, pp. 496–502, 2002.

Workspace Determination and Representation of Planar Parallel Manipulators in a CAD Environment K.A. Arrouk1, B.C. Bouzgarrou2 and G. Gogu2 1

Laboratoire de M´ecanique et Ing´enieries, Clermont-Universit´e, Universit´e Blaise Pascal, EA 3867, BP 10448, F-63000 Clermont-Ferrand, France; e-mail: [email protected] 2 Laboratoire de M´ ecanique et Ing´enieries, Clermont Universit´e, IFMA, EA 3867, BP 10448, F-63000 Clermont-Ferrand, France; e-mail: [email protected], [email protected] Abstract. This paper presents a new method, based on a geometrical approach, for the total workspace determination, characterization and analysis of planar parallel manipulators. The geometrical algorithm proposed for total workspace computation is implemented in a CAD environment. This paper illustrates the effectiveness of such a method in robot design process. Examples of total workspace determination of several types of planar parallel manipulators are presented. Key words: CAD, planar parallel manipulator, workspace, manipulator design

1 Introduction The parallel kinematic architectures have attracted great attention from the academic and industrial communities in the last decades due to their potential performances. Parallel robotic manipulators (PRMs) could be an alternative to serial robots in several industrial applications. Indeed, PRMs could reach higher dynamic performances, due to low moving masses, and proffer superior structural rigidity and higher precision. But, from application point of view, a limited workspace and difficult geometric calibration, control and singularity determination are the major drawbacks of PRMs. Researchers have paid special attention to the determination of the workspace of PRMs to verify the accessibility to the desired poses according to the different robotic applications. The determination of the workspace for PRMs is a challenging problem due to the complexity of the kinematic equations. This paper tackles the problem of total workspace determination and characterization. It focuses on planar parallel manipulators (PPMs) for which an original method is proposed. This method pertains to a geometrical approach and emphasizes on the use of CAD environment as natural framework for developing such approach. It results in an efficient algorithm that handles geometric entities generated from a parametric design of PPMs.

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(a)

(b)

(c)

Fig. 1 Representation of the 3-RPR PRM workspace by its boundaries, (a) CAD models of 3-RPR PPM, (b) 3D perspective view of the workspace, (c) total workspace top view.

2 Overview of Method for Workspace Determination It is not possible to represent spaces of dimension higher than three. Thus, a graphical representation of the workspace will be possible only with n-3 fixed parameters, where n is the degree of freedom of the manipulator. According to the constraints that can be imposed to the parameters, various types of the workspace can be obtained as the constant orientation workspace, maximal workspace, dexterous workspace, etc. We can classify the representation methods of the workspace into two categories [2]. The first method is based on calculating the boundaries of the workspace (the workspace is represented by its envelope) as is shown in Figure 1. The second method is based on scanning, and the workspace is generally represented by its volume. There are various techniques to determine and represent the workspace of PRMs. The early literature focuses on: numerical, algebraic, and geometric methods. In the numerical methods, the workspace is covered by a regularly arranged grid in either Cartesian or polar form of nodes. Each node is then examined with respect to the kinematic constraints to see whether it belongs or not to the workspace. The accuracy of the boundaries of the workspace depends upon the sampling step used to generate the arranged grid [11]. Such a method has several drawbacks i.e. this method is computationally intensive because the computation time increases exponentially with the mesh resolution, hence it introduces uncertainty in the accuracy which is related to the sampling density [12]. The chosen step influences the precision of representation. Moreover, it requires a large data storage space. This method gives little data about the exact boundaries of the PRMs workspaces. Figure 2 clearly shows the discontinuities in the workspace of Gough– Stewart PRM which appear when very fine or very large steps are used. This method has also several advantages: discretization is usually simple to implement, all kinematic constraints can be taken into account as they can usually be simply verified for a given pose. Several applications of this method can be found in [6, 7, 13, 14,

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Fig. 2 Workspace modelled by the discretization method: (a) with fine step and (b) with large step.

17]. In these applications the numerical techniques have been used to determine the workspace of a three DOFs micromanipulator, to analyze both reachable and dexterous workspaces. The algebraic methods by discretization use the direct and inverse geometrical models to calculate the set of configurations that the manipulator can reach. These data are stored in a hierarchical structure. We cite as example the Quadtree and Octrees. These methods are based on recursive subdivision of the space [8]. They are very useful for the calculation of uniqueness domains in the workspace of PPMs. This model offers many advantages over the discretization methods [2]: less memory space is required, particularly well adapted to the representation of the complex forms of the three-dimension PPMs workspace [15], implicit adjacency graph, which facilitates the analysis of workspace connexity. But this model depends on the realization of the octree, which must be neither too fine, this causes the appearing of the discontinuities, nor too large, which on the contrary eliminates these discontinuities that may exist in reality. Octree model has been used in several robotic applications. It is used to represent the reachable joint space, the workspace and the singularities for the robots 3-RPR, 3-RRR [1, 3]. The geometrical methods are used efficiently for representing the workspace of parallel robots. The geometric approach usually allows us to establish the nature of the boundary of the workspace. The principle of these approaches is to deduce from the constraints on each kinematic chain associated with the limb separately a geometrical entity which describes all the possible locations of the end-effector that satisfy the limb constraints. Then, the robotic manipulator workspace is constituted of the intersection of all the entities for the limbs [11]. This technique is used for the calculation of the workspace of a Gough platform when its orientation is fixed (constant orientation or translational workspace) [4, 9]. Merlet [11] has pointed out that the geometrical approaches are very efficient for the determination of various types of workspace. The inverse geometric model was used to determine the boundaries workspace by this technique [4, 5]. In [10, 18], we can find several examples on the determination of the maximal workspace and the constant orientation workspace of PPMs in which geometrical methods are used.

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3 The Proposed Method As mentioned in the previous section, geometric approach has been used only for translational workspace determination. In this paper we propose a graphical representation of the total workspace of PPMs by concatenating the representation of the translation workspace and rotation capability. In other words, this method extends the geometric approach to determination and 3D representation of the total workspace of PPMs. Total workspace is considered as a volume generated from a geometric construction process. CAD software such CATIA provides powerful tools for graphical programming and geometric feature handling. Therefore, the implementation of this approach in a CAD environment becomes implicitly and naturally imposed. The approach presented in this paper demonstrates that CAD tools could be a serious alternative, relevant and effective to the complex mathematical resolution of the representation of the total workspace. We can produce the workspace as a wireframe and even solid, faster than a complex method such as the algebraic or numerical methods. In discretization method, even when a rather fine step is used, it is very difficult to obtain a good quality representation of the workspace without additional graphical manipulation. By embedding the problem of the determination of the PPMs workspaces within a CAD framework these methods will become more accessible to industry. The main contribution of this paper is to extend the geometrical approach to determine the workspace of large variety of PPMs taking into consideration the design parameters and constraints. The three-dimensional total workspace of PPMs can be generated and represented globally at a low computational cost. In addition, a very excellent visualization of the workspace is obtained.

3.1 Total Workspace Determination in CAD Environment In this section a 3-Step CAD-algorithm is used to determine the total workspace of PPMs will be presented. In step 1, the region produced by each limb for a given orientation of the mobile platform, is created. It is realized in the Part Design CATIA workbench. For 3-RPR parallel manipulator the region produced by the limb is an annular region as shown in Figure 3a. The point A1 (the extremity of one limb) is located in an annular region delimited by two circles with outer and inner radii ρmax and ρmin . The point E represents the end-effector characteristic point (EECP). It is located in an annular region deduced from a translation of the initial annular region by the vector A1 E (ϕ ) as shown in Figure 3a. All the annular regions corresponding to different values of the rotation angle ϕ of the moving platform can be generated continuously by an extrusion of the translated annular region along a helical curve (a coil or helix). The axis of the helix passes by the fixed base point A0 and is perpendicular to the plane containing the annular region. This helix has as radius of
A1 E
and a height of 2π . This step is realized in the Generative Shape Design (GSP) CATIA workbench, as shown in Figure 3a. Thus, in order to generate all the parallel plane regions produced by each limb for all orientations of the mobile

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Fig. 3 (a) 3D perspective view of generic trajectory curve used for construction the swept volume for each limb, (b) the annular regions before and after translation, (c) 3D perspective view of the swept volume for one limb of the 3-RPR PPM.

Fig. 4 (a) Three volumes reached by the limbs of 3-RPR PPM, (b) W1 ∩ W2 , (c) W1 ∩ W2 ∩ W3 (3D perspective view of the workspace), (d) top view of total workspace.

platform, a sweeping technique is applied in step 2. It consists in extruding the region generated in step 1, for a given platform orientation along the helical sweep path (see for example Figure 3b), created also in step 1. This second step is realized in the Generative Shape Design (GSP) CATIA workbench. The volume produced by one limb for all orientations of the mobile platform is obtained (see for example Figure 3c). This technique will be applied repeatedly for all the regions reached by the three limbs of the manipulator as it shown in Figure 4a. If the end-effector characteristic point EECP is chosen on the second revolute joint of the limb, the working volume associated to this limb is a cylinder in which the height represents the rotation angle of the moving platform and the base represents the translation workspace of the limb. Finally, in step 3 intersection Boolean operations are applied to obtain the common volume corresponding to the three-dimensional total workspace for the PPMs. Thus the total workspace W can be described as the intersection of three sets Wi associated with the three limbs

(W = W1 W2 W3 ). The result is shown in Figure 4 for 3-RPR parallel manipulator. This step is realized in the Part Design CATIA workbench. The intersection of all three volumes is then the maximum workspace (total workspace).

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Fig. 5 (a) Volumes reached by the limbs of 3-PPR-type PPM, (b) W1 ∩W2 , (c) W1 ∩W2 ∩W3 .

Fig. 6 (a) Volumes reached by the limbs of 3-PRR-type PPM, (b) W1 ∩W2 , (c) W1 ∩W2 ∩W3 .

The proposed method can be applied to several architectures of 3DOF PPMs as shown in Figures 5 and 6.

4 Influence of Position of the End-Effector Characteristic Point on the Manipulator Total Workspace In this section, the influence of the position of EECP on the workspace volume and total workspace shape is studied. The results obtained from the quantitative study confirm that there is no influence of the position of the EECP on the numerical value of total workspace volume. Consequently, a specific criterion to characterize and analysis the planar parallel manipulator workspace is used, this index is called workspace form complexity δ which can be defined as the ratio between the envelope surface area of the workspace A and the volume of the workspace V , i.e. δ = A/V . We can see that for the same volume of the workspace, the greater envelope surface area corresponds to the higher form complexity of the planar parallel workspace. This criterion can be used to optimize the complexity of the trajectories within the complete workspace of the manipulator, i.e. the manipulator should ideally has large workspace but low form complexity index [16]. A quantitative example will be used to illustrate the use of this index to characterize the work-

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Fig. 7 CAD models of 3-RPR PPM with the different positions of the end-effector characteristic point (EECP): 3D perspective view (Ai ) and side views of its total workspaces (Bi , Ci , Di ), i = 1, 2.

space. Figure 7 presents the workspace of 3-RPR PPMs in two cases depending on the position of EECP. For two positions of the EECP the workspace volume is the same V1 = V2 = V . The envelope surface area is calculated for the first case (1) AA = A1a + B1a + C1a , and for the second case (2) AB = A2b + B2b + C2b . With Ai , Bi and Ci we have denoted the lateral surfaces associated to the three limbs when these ones are considered as serial chains isolated from the rest of mechanism. We obtain for the both positions of the EECP on the moving platform the following relationship: δ1 /δ2 = 1.0623 which can describe the regularity of the workspace form. We can see from this value that the workspace form complexity index in the first case (1) is greater than the second case (2). The workspace has a more regular shape when the end-effector is located at the center of the mobile platform.

5 Conclusions A geometrical and graphical method for the determination and representation of the workspace of PPMs has been introduced. The associated algorithm was implemented in the CAD system CATIA using its programming interface. The proposed method gives a high quality visualization of the workspace at low computational cost. It is also independent of the discretization step and allows overcoming the calculation difficulties of an algebraic determination of the workspace. It provides a geometric description of the workspace and its boundaries with an accurate and smooth graphical representation. In order to optimize the EECP location on the mo-

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bile platform, a form complexity index was introduced, based on workspace areas and volume measurements.

References 1. Chablat, D. and Wenger, Ph., The kinematic analysis of a symmetrical three-degreeof-freedom planar parallel manipulator, CoRR abs/0705.0959, 2007, available from http://hal.archives-ouvertes.fr/hal-00144971/fr/. 2. Chedmail, P., Dombre, E. and Wenger Ph., The CAD in Robotic, Tools and Methodologies, Hermes, 1998. 3. Garcia, G., Wenger, Ph. and Chedmail, P., Computing moveability areas of a robot among obstacles using octrees. In Proc. Int. Conf. on Advanced Robotics (ICAR’89), Columbus, Ohio, USA, 1989. 4. Gosselin, C., Determination of the workspace of 6-dof parallel manipulators, ASME J. Mech. Des., 112(3):331–336, 1990. 5. Kohli, D. and Spanson, J., Workspace analysis of mechanical manipulators using polynomial discriminates, ASME J. Mechanisms, Transmissions and Automation in Design, 107:209–215, 1985. 6. Kumar, A. and Waldron, K.-J., The workspaces of a mechanical manipulator, ASME Journal of Mechanical Design, 103:665–672, July 1981. 7. Lee, K. and Shah, D.-K., Kinematic analysis of a three degree of freedom in-parallel actuated. In Proc. Manipulator, IEEE Conference on Robotics and Automation, Raleigh, NC, pp. 345– 350, April 1988. 8. Meagher, D., Geometric modelling using octree encoding, Technical Report IPL-TR-81-005, Image Processing Laboratory, Rensselaer Polytechnic Institute, Troy, New York, 1981. 9. Merlet, J.-P., Geometrical determination of the workspace of a constrained parallel Manipulator. In: Proceedings ARK, Ferrare, pp. 326–329, 1992. 10. Merlet, J.-P., Gosselin, C.M. and Mouly, N., Workspaces of planar parallel manipulators, Mechanisms and Machine Theory, 33(I/2):7–20, 1998. 11. Merlet, J.-P., Parallel Robots, Springer, Dordrecht, 2006. 12. Shawn, P.-A., Robert B.-J. and Robert L.-D., Comparison of discretization algorithms for NURBS surfaces with application to numerically controlled machining, Computer-Aided Design, 29(1):71–83, 1997. 13. Tsai, Y.-C. and Soni, A.-H., Accessible region and synthesis of robot arm, ASME Journal of Mechanical Design, 103:803–811, 1981. 14. Waldron, K.-J., Raghavan, M. and Roth, B., Kinematics of a hybrid series-parallel manipulation system, ASME Journal of Dynamic Systems, Measurement, and Control, 111:211–221, 1989. 15. Wenger, Ph. and Chablat, D., Uniqueness domains in the workspace of parallel manipulators. In Proc. IFAC-SYROCO, Nantes, 3–5 September, Vol. 2, pp. 431–436, 1997. 16. Wu-Jong, Y., Chih-Fang, H. and Wei-Hua Ch., Workspace and dexterity analyses of the delta hexaglide platform, Journal of Robotics and Mechatronics, 20(1), 2008. 17. Yang, D.-C.H. and Lee, T.-W., On the workspace of mechanical manipulators, ASME Journal of Mechanisms, Transmission and Automation in Design, 105:62–69, 1983. 18. Yu, A., Bonev, I.A. and Zsombor-Murray, P., Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots, Mechanism and Machine Theory, 43(3):364–375, 2008.

A Theoretical Improvement of a Stirling Engine PV Diagram N.M. Dehelean, L.M. Dehelean, E.-C. Lovasz and D. Perju Mechanical Engineering Faculty, “Politehnica” University of Timisoara, 300222 Timisoara, Romania; e-mail: {nicolae.dehelean, liana.dehelean, erwin.lovasz, dan.perju}@mec.upt.ro

Abstract. Power efficiency of Stirling engine is low due to different causes. One of them is the displacement function of compression piston. This is pseudo-sinusoidal and is performed by a slidercrank mechanism. The thermo-dynamic functions should require a special displacement function for compression piston. Focusing on this request the paper proposes a new dwell mechanism. Key words: Stirling engine, slider-crank mechanism, linkage drive mechanism, displacement and torque functions

1 On the Drive Mechanism for an α -Stirling Engine The PV diagram of an α -Stirling engine (Figure 1) is far away from an ideal PV diagram (Figure 2). The principle schema of the α -Stirling engine is shown in Figure 3. It could be observed that the engine has two crank rod mechanisms, one from compression cylinder and the other for expansion cylinder. The displacement functions for both pistons (cylinders) are of approximately sin shape, as it is shown in Figure 4. The majority of the mechanisms used to drive the Stirling engines are crankrod mechanisms. In [10] there is a schema of low-temperature differential solar Stirling engine powered by a flat-plate solar collector without regenerator shown in Figure 5. A modified slider-crank mechanism is analyzed in [2]; its schema is shown in Figure 6. In Figure 7 it is shown the schematics of a rhombic drive with power piston connected to the upper and displacer to the lower section. This mechanism is analyzed in [8]. In the configuration shown, the left crank will turn clockwise and the right crank counter clockwise in order to achieve proper phase lag between expansion and compression space. They turn at the same angular velocity which can be accomplished by two intermeshing counter-rotating gears. In [7] a calculator is made available online dedicated to optimize the solution for bowtie drives with unsymmetric sections. The schema of the mechanism is shown in Figure 8. It might have more choices than necessary to achieve design objectives

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Fig. 1 Real Stirling PV diagram.

Fig. 2 Ideal Stirling PV diagram.

like: a swept-volume ratio between expansion and compression space of about 1, volumetric phase lag of about 90◦ , and minimal sidewise motion of the connection points 4 and 4 . The slider-crank mechanism is part of the cause to the deficient PV diagram of a Stirling engine that uses it as a drive mechanism [1]. In [6] a calculator is made available online dedicated to a four bar mechanism for a Stirling engine as shown in Figure 9.

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Fig. 3 Schema of the α Stirling engine.

Fig. 4 The displacement functions.

2 A New Mechanism The low efficiency of α -Stirling engine is proved by the PV diagram. This statement shoves the engine developers to improve the work characteristics in order to rise-up the engine efficiency. The main way to achieve this task is to improve the engine characteristics. There are a lot of mechanisms species that could be used to obtain a dwell movement necessary to the compression cylinder. It could be use-

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Fig. 5 Schema of differential solar engine.

ful a mechanism able to perform a dwell during the whole period of expansion. The dwell of the compression piston has to enlarge the active area of the PV diagram. As a consequence it is expected an efficiency improvement. This request could be very well accomplished by a cam mechanism. But this one is too large, too complicated, expensive and unreliable. A linkage could be more reliable than cam mechanism despite the fact that the cam mechanism is more precise. The paper is focused on a dwell mechanism as a linkage one. A good mechanism was proposed in [5] and the scheme is shown in Figure 10. The reference x0y axes are placed according to the Schmidt Analysis [9]. Schmidt has made the classical Stirling Analysis in 1871. Using the same reference it can replace the sin function of compression cylinder, in the Schmidt Analysis by a new displacement function, in order to develop a new isothermal analysis of the engine. But this one is not good enough regarding the phases of the two pistons. In order to improve this situation a mechanism as shown in Figure 11 was proposed in [3]. The analysis of the dwell mechanism begun in [3] and it reveals the expected displacement functions yC (t) and yE (t) and the phase relation between them. The proper phase relation demonstrates that the mechanism is good for the application that it is focused on. Second, it must show that yE (t) has a dwell large enough level, in accordance with the expansion stroke time of the expansion cylinder. Implementing the displacement functions from [3] in a computer program and proceeds to a numerical derivation it obtains the velocity and the acceleration functions. All these functions are shown in diagram Figure 12.

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Fig. 6 Engine with two displacer pistons

The reference x0y axes are placed in accordance with the Schmidt Analysis [9]. It should be noted that the sense of the crank rotation is in accordance with the classical Schmidt Analysis calculus. Some of the initial dimensions are given as follows: θ is the crank angle, proportional to time; Vclc is the clearance volume; Vswc is the swept volume; DC is the cylinder diameter; yC (t) is the displacement of compression piston; yE (t) is the displacement of expansion piston. The parameters in equations (1) are necessary to obtain the value of yC min . Vclc = min{VC };

max{VC } = Vclc + Vswc

VC = Vclc + 0.5 ·Vswc(1 + cos θ ) YC min =

π

Vclc
2 DC 2

(1) (2)

yC min (equation 2) and h1 (equation 3) are necessary to establish the dimensional structure of entire mechanism; →

−→

→

−→

h1 = O1 A + AC − yC min

(3)

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Fig. 7 Rhombic drive mechanism.

Fig. 8 Unsymmetric sections mechanism.

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Fig. 9 A four bar mechanism.

Fig. 10 Dwell mechanism.

Figure 12 shows the two displacement functions and the velocity and acceleration only for the compression piston. It is observed that the mechanism does not perform a perfect dwell, but the angle of the dwell is more important. The dwell angle is from about 220 to 310◦. So, ∆ θ ≈ 90◦ . The new displacement function yC (t) will be inserted in Schmidt Analysis, instead of sin relation. For example the third equation in (1) from Schmidt Analysis has to be improved by replacing the term 0.5Vswe · cos(θ + δ ) by yC (t). But function yC (t) does not have an analytical expression so the replacement could be done in a computer program. Ve = Vcle + 0.5Vswe(1 + cos(θ + δ ))

(4)

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Fig. 11 New dwell mechanism.

Fig. 12 Displacement, velocity and acceleration.

3 Torque New Dwell Mechanism Calculus The torque calculus starts from the equivalent calculus in [4] and the added scheme is shown in Figure 13. This calculus is more detailed, rigorous and complete than the previous one [4]. It contains the equilibrium equations of the element (2) of the linkage, which are implemented in the computer program to obtain the diagram of the torque shown in Figure 14.

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F3 =

2F6 · tan β ; cos δ

tan δ =

621

h2 − yB − R2 · cos β xB + R2 · sin β − e

(5)

The main equation of calculus is the generic torque equation (6). The set of equations (7) represents the geometrical relations among the main geometric and auxiliary points on linkage schema (Figure 13). The equation system (8) represents the equilibrium relations of the element (2). T = RA · RCrod · sin θ − RA · RCrod cos θ 2R

BS =

h ; cos α

T S = h · tan α ;

CU = CS · cos α ;

(6)

1R

CS = b − b1 − h · tan α ;

CQ = h1 + RCrod · cos θ ;

SU = CS · sin α

s = BS + SU

(7)

where RCrod is the crank radius, R2 is the secondary crank radius and b, b1 , h, h1 , h2 , s are the geometrical dimensions of the linkage. ⎧ ⎪ ⎨ RC + RA1 − F3 · cos δ = 0 F7 + F3 · sin δ − RA2 = 0 ⎪ ⎩ F3 · cos δ (CQ − CU) + F3 · sin δ (s − RCrod · sin θ ) − RC ·CQ = 0

(8)

In the equation system (8) the first two equations represent (∑ F)x = 0 and (∑ F)y = 0. The last equation represents (∑ M)A = 0.

Fig. 13 External forces scheme.

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Fig. 14 Torque diagram.

In order to view the configuration of the torque at element (1) it implements equations (1) to system equation (8) into the computer program that was used in [4]. The torque diagram is shown in Figure 14. It could be observed that torque presents a positive maximum along the dwell angle of the compression cylinder (in the torque diagram the sign (+) is under the 0x axis).

4 Conclusions The torque diagram configuration is a reliable argument to sustain the attempt to improve the PV diagram of classical Stirling engine (with two slide-crank mechanisms, see Figure 3) using the proposed linkage (Figure 11). Movement, velocity and acceleration functions of the compression cylinder (piston) prove that the proposed linkage is able to perform a dwell in a proper phase position. Torque function of the crank presents a maximum point along the dwell angle of the compression cylinder. The two diagrams prove that the paper makes a step award to improve the Stirling engine design.

References 1. Ataera, E.O. and Karabulutb, H., Thermodynamic analysis of the V-type Stirling-cycle refrigerator, International Journal of Refrigeration, 28:183–189, 2005. ¨ Design and manufacturing of a V-type Stirling engine with 2. Batmaz, I. and S¨uleyman, U., double heaters, Applied Energy, 85:1041–1049, 2008. 3. Dehelean, N.M. and Ciupe, V., The analisis of a dwell mechanism for α -Stirling engine. In Proceedings of the 10th IFToMM International Symposium on Science of Mechanisms and Machines, SYROM’09.

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4. Dehelean, N.M., Ciupe, V. and Lovasz, E.-C., A digital model of a dwell mechanism for alpha Stirling engine. In Proceedings of the 10th IFToMM International Symposium on Science of Mechanisms and Machines, SYROM’09. 5. Dehelean, N.M., Ciupe, V. and Maniu, I., A new Mechanism to syncrhonize α -Stirling engine, Scientific Bulletin of “Politehnica” University of Timisoara, 53(67), S1:81–84, 2008. 6. Herzog, Z., Web-based program “4bar”, available from http://mac6.ma.psu.edu/stirling/drives/ 4bar/4bar.html. 7. Herzog, Z., Web-based program “bowtie”, available from http://mac6.ma.psu.edu/stirling/ drives/bowtie/index.html. 8. Herzog, Z., Analysis of the symmetric rhombic drive, available from http://mac6.ma.psu.edu/ stirling/drives/beta rhombic/index.html. 9. Herzog, Z., The Schmidt analysis of Stirling engines, available from http://mac6.ma.psu.edu/ stirling/simulations/isothermal/schmidt.html, 2008. 10. Tavakolpoura, A.R., Zomorodiana, A. and Golneshan, A., Simulation, construction and testing of a two-cylinder solar Stirling engine powered by a flat-plate solar collector without regenerator, Renewable Energy, 33:77–87, 2008.

Cylindrical Worm Gears with Improved Main Parameters T.A. Antal and A. Antal Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania; e-mail: [email protected], [email protected]

Abstract. Cylindrical worm gear sizing was done so far based on the strength conditions.The paper gives a new method for finding the geometrical dimensions of the cylindrical worm gears of ZA, ZI and ZK types based on the conditions required for that between the flanks, during the load transmission, the hydrodynamical lubrication would be kept. The module of the gear is determined based on the hydrodynamical lubrication conditions. Here, are taken into account the load to be transited, the material of the worm and that of the wheel, the surface roughness of the teeth flanks and the lubricant used. The diameter factor, the specific addendum modification and the efficiency (as high as possible) are determined based on the conditions concerning the worm shaft stiffness and the bearing strength of the worm wheel. Numerical results were obtained and given in tables based on a Matlab computing environment. Key words: worm gears, hydrodynamical lubrication, addendum modification, worm shaft stiffness, contact pressure

1 Introduction The main parameters design of the cylindrical worm gears, depending on which the geometrical dimensions are established, must be determined so that the duration of the uptime would be as long as possible. It is known that in the case of these gears, between the meshing teeth flanks, large sliding occur from which, on the worm tooth flank, significant wearing appears. Wearing reduction can be done by providing appropriate conditions in the lubrication of the gear teeth. In this paper, the cylindrical gear parameters are determined while keeping the hydrodynamical lubrication conditions between meshing flanks, a high efficiency, the appropriate conditions of operation and transmission of load. Taking account of those listed above, for a given case are presented combinations of values which can influence tooth flanks wearing and increase uptime of the gears.

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1.1 The Module Determination under Hydrodynamical Lubrication Conditions In some books and standards [3, 9] dealing with worm gears is shown that, between the meshing teeth flanks, hydrodynamical lubrication can be provided only if certain conditions are met. Based on these conditions the module can be determined, thus:
  T20.13 · λ · Ra1 + Ra2 2 1.39 . (1) mx >= 0.7 · n0.7 · E 0.03 q + z2 + 2x 21 · h∗ ·Cα0.6 · ηOM 1 red The lubrication thickness h∗ for commonly used worm gears in the transmission of type ZA, ZI and ZK is given by: u 1 q x + +  − + z2 110 36300 7.86 q + z2  √ (2q − 1) 2(0.5 + q + 1) + . + 370.4 213.9

h∗ = 0.018 +

(2)

Notations from the relations (1) and (2) are: q is the diameter factor; z1 is the number of the threads (starts) of the worm; z2 is the number of the teeth of the worm wheel; x is the specific addendum modification of the wheel; T2 is the moment of torsion on the axle of the worm wheel in Nm; λ is the safety coefficient (λ = 1 . . . 2); Ra1 and Ra2 arithmetic average roughness on the teeth of the worm and on the worm wheel measured in µm (Ra1 = 0.4 µm at corrected worm and Ra2 = 1.6 µm for the milled worm wheel); Cα = 1.7 · 10−8 m2 /N is the viscosity pressure variation exponent in the case of mineral oils; ηOM is the oil viscosity at ambient pressure and at temperature of the meshing area in Ns/m2 ; n1 is the rotative speed of the worm in min−1 ; Ered = 140144 N/mm2 is the elastic modulus of the teeth (for bronze worm whee CuSn12 and steel worm). If the load transmitted to the worm gear, the gear ratio and other factors related to operating are known, with relations (1) and (2), the mx corresponding to the hydrodynamical lubrication condition can be determined. For a proper functioning there must be considered other requirements such as a high efficiency, a reduced strain of the meshing elements, the strength of the meshing teeth and more.

1.2 Cylindrical Worm Gear Efficiency Determination Taking account of the lubrication conditions between the teeth flanks based on the recommendations of [2, 5] the efficiency can be calculated with the following relation:

Cylindrical Worm Gears with Improved Main Parameters

η=

627

1 V12 µ 1+ cos(αn ) V1 cos(β1 )

(3)

 where µ = 0.04/ 4 V12 is the friction coefficient between the flanks of the teeth in contact; V12 is the relative velocity between the teeth flanks on the rolling cylinder of the worm; V1 is the peripheral speed of the worm on the rolling cylinder of the worm; αn is the worm tooth profile angle at normal section (αn can be considered 20◦ ); β1 is the worm tooth declination angle on the rolling cylinder. Replacing in the relation (3) the corresponding velocities and angle of declination of the tooth, on the rolling cylinder, depending on the worm geometric parameters the following expression is obtained:

η=

z1 (q + 2x)cos(αn )   . 0.04 z21 + (q + 2x)2  z1 (q + 2x)cos(αn ) + π mx n 1  2 4 z + (q + 2x)2 60 × 1000 1

(4)

The parameter values from relation (4) must be determined so that the efficiency to be as high as possible but at the same time to ensure adequate strength and rigidity of the gear on the transmission capacity of the load during operation.

1.3 The Stiffness Calculation of the Cylindrical Worm Gears At cylindrical worm gears, in order to keep the proper meshing conditions between the teeth flanks, is necessary, during the transmission of the load, that the worm shaft not be deformed over the allowable limits. The bending of the worm shaft is computed considering the worm shaft laid on two supports on which are acting the Fr1 radial force and the Ft1 tangent force [2, 6–8]. l3  2 2 ≤ f . Ft1 + Fr1 (5) f= a 48E1 I1 where l is the distance between the supports (l = Ψa a, generally Ψa ≈ 1.5 . . . 2, π d2

m (q+z +2x)

is the distance between the axes in [mm]; I1 = 64w1 is in [mm]); a = x 2 2 the geometric moment of inertia in [mm4 ]; E1 = 2.1 × 105 N/mm2 is the elastic modulus of the worm; fa is the admissible bending deformation [7] ( fa = 0.004 mx at hardened worm and fa = 0.01 mx at improved worm). The Ft1 tangent force and the Fr1 radial force, at the contact point, are determined using the expressions [1]: Ft1 = where

2T1 mx (q + 2x)

and Fr1 =

2T1 tan(αn ) cos(γ ) mx (q + 2x) sin(γ ) + tan(ϕ1 )

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tan(γ ) = tan(ϕ1 ) = µ1 =

z1 , q + 2x

0.04 µ  = = cos(αn ) cos(αn ) 4 V12

and T1 =

 cos(αn ) 4

0.04  π mx n1 2 2 60×1000 z1 + (q + 2x)

z1 T2 . z2 η

Replacing the above relationships in expression (5) the bending of the worm becomes:
z21 + (q + 2x)2 Ψa3 z2 + q + 2x z1 T2 2 (α ) 1 + tan (6) f= n  2 ≤ fa . 3π E1 m2x (q + 2x)5 z2 η z + µ (q + 2x) 1

1

1.4 The Bearing Strength of the Teeth Flanks in Contact at Cylindrical Worm Gears The contact pressure between the teeth flanks can be determined using the Hertz relation.

F 1

≤ σHa . σH =

(7)

n2

Lk ρ 1 − ν12 1 − ν22  π + E1 E2 where Fn2 is the normal force on the tooth flank; Lk is the length of the contact line between the teeth flanks in contact (Lk ≈ Ψm dm1 = 0.55(q + 2)); ρ is the reduced curvature radius (Figure 1); ν1 and ν2 are the Poisson coefficients for the material of the worm and of the wheel(ν1 = 0.30 for the steel worm and ν2 = 0.35 for the bronze worm); E1 and E2 are the elastic moduli of the worm and the wheel (E1 = 2.1 × 105 N/mm2 for steel worm and E2 = 0.883 × 105 N/mm 2 for the bronze CuSn12 wheel); σHa is the admissible contact pressure of the wheel from the worm gear (σHa = 400 N/mm2 ). The normal force on the tooth of the worm wheel, at the contact point on the rolling cylinder of the worm (Figure 1) is determined by the relationship [1]: Fn2 =

2T2 1 . dm2 cos(αn ) cos(γ ) − tan(ϕ1 ) sin(γ )

(8)

where T2 is the torque on the shaft of the worm wheel  in[Nmm]; dm2 = mx z2 is z1 is the lifting angle on the diameter of the pitch circle in [mm]; γ = arctan q+2x   the rolling cylinder of the worm; ϕ1 = arctan(µ1 ) = arctan cos(µαn ) is the reduced friction angle.

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Fig. 1 The teeth flanks meshing at normal section.

as:

With the geometric element given above, the normal force from (8) can be written  z21 + (q + 2x)2 2T2 . (9) Fn2 = mx z2 cos(αn )(q + 2x − µ1z1 )

Based on Figure 1, where the normal section on the helical line of the rolling cylinder from the worm is given, the reduces radius is determined by the relationship: 1 1 2 cos2 (γ ) 1 1 1 1 = = = + = + = ρ ρ1 ρ2 ∞ CN2 rmv2 sin(αn ) mx z2 sin(αn ) =

2(q + 2x)2   . mx z2 z21 + (q + 2x)2 sin(αn )

(10)

Replacing relationships (9) and (10) in relationships (7) and taking into account the recommended relationship for the length computation of the contact line, the module can be determined as:

T2 (q + 2x) 8 1

 mx ≥  × 3 2 0.55 sin(2αn ) z2 (q + 2x − µ z ) z2 + (q + 2x)2 σHa 2



1  . ×

2 3  1 − ν1 1 − ν22 π + E E 1

2

1 1

1

(11)

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2 Computation of the Main Parameters of Cylindrical Worm Gears Based on the Described Circumstances Using Matlab The improved main parameters use to find the geometrical dimensions of the worm gears, while the hydrodynamical lubrication conditions are kept and the effciency is high, are obtained using the following Matlab code: k=1; A=[]; for q=7:1:17 for x=-1:0.1:1 vareta=eta(z1,z2,x,q,etaom,T2,n1); varf=f(z1,z2,x,q,etaom,T2,n1); varmx=mx(z1,z2,x,q,etaom,T2,n1); varfmx=fmx(z1,z2,x,q,etaom,T2,n1); if (vareta>=0.94) && (varf=varfmx) A = [A;[q x vareta varf 0.004*varmx varmx varfmx]]; k=k+1; end; end; end; try B=sortrows(A, 3); fprintf(\n’); for i=1:k-1 fprintf(’%f %f %f %g %g %f %f\n’, B(i,1), ..., B(i,7)); end; catch exception disp(exception.message); end;

The data from Tables 1, 2 and 3 were obtained for the following input data: z1 = 1, z2 = 41, etamin=0.86 for Table 1, z1 = 3, z2 = 41, etamin=0.94 for Table 2 and z1 = 2, z2 = 57, etamin=0.92 for Table 3, ηOM = 0.08 Ns/m2 , n1 = 1500 min−1 , T2 = 587.28 Nm. Two for cycles are used to control the variation of q and x main parameters. The q is variating between 7 and 17 with step 1, and x between -1 and 1 with step 0.1. The values of mx are computed using the expression (1)n. The results that have the efficiency computed using (4) higher than the given etamin value and that are satisfying (6) and (11) are stored in the A matrix. These rows are sorted ascending by the efficiency column (the third column in the A matrix), stored in B matrix and then printed. The bending from (6) is given in the f and fa columns, while the module from (11) is given in the mx and fmx columns, where fmx is the right part of inequality (11). Exceptions must be handled as the restrictions can leave the A matrix without any rows, in which case sorting would cause errors.

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Table 1 Improved main parameters for etamin=0.86, z1 = 1 and z2 = 41. q

x

η

f

fa

mx

f mx

7.0 8.0 7.0 8.0 7.0 8.0 7.0 7.0 7.0 7.0 7.0

–0.30 –0.80 –0.40 –0.90 –0.50 –1.00 –0.60 –0.70 –0.80 –0.90 –1.00

0.860263 0.861268 0.863398 0.864400 0.866555 0.867553 0.869732 0.872930 0.876149 0.879387 0.882646

0.000133342 0.000120575 0.000143302 0.000129369 0.000154629 0.000139357 0.000167578 0.000182469 0.000199704 0.000219788 0.00024337

0.0685084 0.0708454 0.0695823 0.0719952 0.0706871 0.0731795 0.0718242 0.072995 0.0742011 0.075444 0.0767257

17.127092 17.711347 17.395567 17.998789 17.671770 18.294867 17.956040 18.248743 18.550263 18.861010 19.181421

5.545293 5.545224 5.603002 5.602927 5.663142 5.663063 5.725896 5.791464 5.860068 5.931956 6.007406

Table 2 Improved main parameters for etamin=0.94, z1 = 3 and z2 = 41. q

x

η

f

fa

mx

f mx

8.0 7.0 8.0 7.0 8.0 7.0 8.0 7.0 7.0 7.0 7.0 7.0

–0.70 –0.30 –0.80 –0.40 –0.90 –0.50 –1.00 –0.60 –0.70 –0.80 –0.90 –1.00

0.940015 0.940536 0.940997 0.941501 0.941963 0.942448 0.942909 0.943373 0.944274 0.945148 0.945991 0.946799

0.000171493 0.0002058 0.000186287 0.000224881 0.000203234 0.000246888 0.000222754 0.000272418 0.000302224 0.000337261 0.000378752 0.000428281

0.0690672 0.0678777 0.0701625 0.0689317 0.0712898 0.0700157 0.0724505 0.0711309 0.0722788 0.0734609 0.0746786 0.0759338

17.266806 16.969437 17.540622 17.232929 17.822440 17.503915 18.112621 17.782723 18.069700 18.365215 18.669659 18.983449

5.347991 5.394000 5.393807 5.441178 5.440973 5.489759 5.489540 5.539794 5.591336 5.644436 5.699144 5.755505

Table 3 Improved main parameters for etamin=0.92, z1 = 2 and z2 = 57. q

x

η

f

fa

mx

f mx

8.0 7.0 8.0 7.0 7.0 7.0 7.0 7.0

–0.90 –0.50 –1.00 –0.60 –0.70 –0.80 –0.90 –1.00

0.920659 0.921442 0.922370 0.923145 0.924838 0.926520 0.928187 0.929839

0.000390703 0.000494758 0.000423755 0.000540807 0.000594099 0.000656179 0.000729011 0.00081512

0.0638548 0.0617835 0.0650492 0.062892 0.0640404 0.065231 0.0664662 0.0677488

15.963699 15.445882 16.262306 15.723000 16.010094 16.307739 16.616554 16.937209

4.447110 4.491692 4.491496 4.537768 4.585622 4.635361 4.687103 4.740972

3 Conclusions In order to obtain the hydrodynamical lubrication between the flanks of the meshing teeth, in the case of the cylindrical worm gears of ZA, ZI and ZK types, it is recommended that the module to be high and the diameter factor to be small. These

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two parameters must be determined such as the bearing capacity of the gear to be appropriate. At the same time the specific addendum modification must have a negative value. The higher the specific addendum modification in absolute value is the higher will be the efficiency of the cylindrical worm gear. The negative values of the specific addendum modification contribute to the shifting of the meshing filed to the exit of the meshing improving the work conditions based on hydrodynamical lubrication.

References 1. Antal, T.S. and Antal B., Worm gears main parameters design algorithm. Technical Review Journal, 29:3–8, 2005. 2. Drobni, J., Modern Worm Gears. Tenzor Kft/Springer, 2001. 3. Dubbel, Handbook of Machine Elements, 19th Edition, 1997. 4. Dudas, I., The Theory and Practice of Worm Gear Drives. Penton Press, London, 2000. 5. Levai, I., Interpretation of Losses in the Field of Gears with Crossed Axes. ME Anyagmozgat´asi e´ s Logistikai Tansz´ek, Miskolc, 1996. 6. Maros, D., Killmann, V. and Rohonyi V., Worm Gears. M˝uszaki K¨onyvkiad´o, Budapest, 1970. 7. Niemann, G. and Winter, H., Machine Elements, Volume III. Springer-Verlag, Berlin, 1983. 8. Shigley, J.E.., Mechanical Engineering Design. McGraw-Hill Book Company, New York, 1986. 9. ** DIN 3996, Bearing capacity calculation of cylindrical worm gears with the angle between the axes Σ = 90◦ , 1986.

Multi-Objective Optimization of Parallel Manipulators A. de-Juan1 , J.-F. Collard2 , P. Fisette2 , P. Garcia1 and R. Sancibrian1 ´ Estructural y Mec´anica, Universidad de Cantabria, Avda. de los Ingenieria Castros s/n, 39005 Santander, Spain; e-mail: [email protected] 2 Center for Research in Mechatronics, Universit´ e Catholique de Louvain, place du Levant 2, B-1348 Louvain-la-Neuve, Belgium; e-mail: [email protected] 1 Dpto.

Abstract. This paper deals with the optimal dimensional design of Delta robot. The aim is to obtain a manipulator with maximum dexterity and minimum projected area of the occupied volume on the base plane. Two different optimization strategies, which cope with assembly difficulties, have been applied to solve this problem. Comparing optimized Delta robot with and without area reduction, results show an important reduction of area maintaining an acceptable dexterity. Key words: parallel robots, dimensional synthesis, optimization, closed-loop multibody systems

1 Introduction Compared with serial robots, parallel robots present advantages in terms of speed, accuracy and stiffness [10]. Nevertheless, they have more singular configurations, smaller workspace and dexterity. Dimensional synthesis is a basic task in the design of a parallel manipulator, because its performance is highly sensitive to its dimensions. However, dimensional synthesis of closed-loop multibody systems is significantly more complex than serial ones and it is still an unfinished research area. One of the most important parameters in the design of parallel robots is the dexterity. This parameter is a performance related with the accuracy of manipulator control. It is defined by the condition number of the forward Jacobian matrix. If the value of condition number is 1 over the workspace, the robot is called fully isotropic [3]. Achieving isotropic manipulators is a common design objective [2, 4, 6, 8, 9]. However, designing only with respect to a performance index usually leads to a huge space requirements [6] or provides more singular configurations [13]. The aim of this work is to achieve a maximum dexterity and minimum projected area in Delta robot, which is a dimensional synthesis problem. First of all, the multiobjective function is defined. Then, two different strategies are described in order to optimize the manipulator. Finally, results of the optimization process are presented and discussed.

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2 Definition of the Synthesis Error Function The Synthesis Error Function (SEF) or objective function is defined as the sum of two terms: (1) SEF = η1 + ωη2 Term η1 is related with the kinetostatic performance of the Delta Robot, and term η2 is related with the projected area occupied by the robot. Term ω is a weighting factor.

2.1 Kinetostatic Performance Index: Dexterity Dexterity of a manipulator is a kinetostatic performance which is defined by the condition number κ of the forward kinematics Jacobian matrix J f . It is a measure of the efficiency in motion transmission of the manipulator. J f maps the actuated velocities q˙ into the end-effector velocities x˙ , as it is shown in Eq. (2) x˙ = J f q˙

(2)

The condition number κ is computed as the ratio between the largest σl and the smallest singular value σs of J f , as it is shown next

κ (J f ) =

σl σs

(3)

Note that the values for the condition number as it is defined in Eq. (3) may cover from 1 to ∞. For this reason, it will be used its inverse in this application. Global Dexterity Index (GDI) is a posture-independent index defined as the mean of the inverses of the condition number κ over a volume V in the Cartesian space of the end-effector [1, 9]:
1 dV GDI = V κ (4) V Volume V of Eq. (4) is the chosen workspace subset of the manipulator. It is discretized here in eight precision poses which corresponds to the vertices of a 2 cm cube. The aim here is not to compute dexterity at the most critical points, but to obtain the mean value over a given volume. For this reason, the eight vertices of the cube are chosen. Anyway, as the dimensions of the cube are small with respect to the dimensions of the robot, dexterity should not vary too much over the cube. Thus, term η1 from Eq. (1) will be

η1 = −

1 8 1 ∑    8 i=1 κ J f

i

(5)

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635

Negative sign of Eq. (5) is because the aim is to maximize the GDI.

2.2 Area Occupation Index Previous works [6, 13] show that Delta robot optimization only with respect to Eq. (5) leads to a huge design. In practice, not only a good kinematic performance is expected, but also a space occupation as small as possible [13]. A smaller design could be obtained by reducing the upper bound of a leg length. However, for practical reasons and depending on the application, the projected area is a more suitable index. The goal is not to obtain the smallest area but an acceptable area for a given application. For this reason, another term is introduced here

η2 =

1 8 (A p )i − Aw ∑ Aw 8 i=1

(6)

where (A p )i is the total projected area of the occupied volume on the base plane in each precision poses, i.e. the minimum bounding rectangle around the robot, and Aw is the projected area of the chosen robot workspace subset on the base plane, i.e. 4 cm2 in this work. This term η2 is also dimensionless, in order to maintain the dimensional balance on Eq. (1), and it is also independent on the scale of the robot.

3 Kinematic Optimization Definition of Delta robot and synthesis problem are described in this section. Then, two different strategies are presented to solve the optimization problem. The Delta robot is shown in Figure 1 invented by Reymond Clavel [5]. Only the 3 translational d.o.f. of the moving platform will be considered in this work. Design parameters of this problem, grouped in vector l, are: the lengths of the upper leg lA and the lower leg lB , the distance zc between the base and the center of the cube and the characteristic radii of the base rb and the platform r p t  l = lA lB zc rb r p

(7)

Fourteen generalized coordinates q are defined by means of relative coordinates. There are three revolute joints in each closed-loop and three linear displacement of the end-effector. As the Delta robot is a 3-d.o.f. manipulator in this work, three generalized coordinates are independent (grouped in vector u) and nine are dependent (grouped in vector v). Nine assembly constraints (h) are defined in each pose h(l, q1 (l), . . . , q8 (l)) = 0

(8)

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Fig. 1 Delta robot model.

The software tool used to obtain the kinematic equations in a symbolic form is Robotran [12], developed by the CEREM at the Universit´e Catholique de Louvain.

3.1 Definition of the Synthesis Problem The aim of this work is to reach the maximum GDI of the Delta robot when it occupies the minimum projected area. This is an optimization problem, whose solution will be the minimization of the SEF (Eq. 1) with respect to design variables l (Eq. 7). However, SEF also depends on the generalized coordinates q that are constrained by the assembly constraints h(l, q(l)) = 0. Moreover, design variables are also bound-constrained by means of a set of design inequations as b(l) ≥ 0. Thus, the SEF is evaluated on the eight precision poses and the objective function can be formulated as follows minl SEF(l, q1 (l), . . . , q8 (l)) subject to h(l, q1 (l), . . . , q8 (l)) = 0 and b(l)) ≥ 0

(9)

In order to guarantee the assembly of the manipulator, two strategies have been applied [6]. The next sections will describe them briefly.

3.2 Artificial Penalty Optimization Strategy In this approach, classical method of Newton–Raphson is proposed to solve the assembly constraints during the optimization process. When the assembly is not possible, a penalization of the objective function is performed.

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Fig. 2 Artificial penalization strategy.

A theoretical illustration with two parameters P1 and P2 is shown in Figure 2(a). Suppose that the optimizer is calling the objective function outside the closed-loop border, at point X, during the optimization process. A fixed point G is chosen inside the boundaries and the penalization is computed along the direction GX. As the value of SEF is upper-bounded, a discontinuous penalization can be envisaged. The penalized objective function is straightforwardly computed by summing this upper-bound and a linear extension starting from G along GX for example, as it is shown in Figure 2(b). In summary, the penalized objective function g(l, q(l)) to optimize could be defined as  SEF(l, q(l)) if constraints are satisfied g(l, q(l)) = (10) SEF(l, q(l)) + SEFpen(l) otherwise where SEFpen(l) is the penalized or extension term calculated along the direction GX. Design inequations b play the same role as assembly constraints h to define de feasible region. This strategy is integrated in a gradient-free direct search algorithm Nelder–Mead Simplex [11].

3.3 Best-Assembly Penalty Optimization Strategy This strategy is based on the minimization of the assembly constraints instead of their resolution with respect to dependent generalized coordinates (v) [7]. Thus, rather than solving Eq. (8), it is formulated as the following constrained optimization problem minv 21 h (l, u, v)t h (l, u, v) (11) subject to c (l, u, v) ≥ 0, where inequalities c (l, u, v) ≥ 0 are formulated in order to avoid multiple solutions and singular configurations when assembly problem is solved. Relative coordinates make its formulation easier and an unique and regular assembly solution can be achieved.

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Eq. (11) can be seen as an extension of the assembly problem. If the assembly constraints are satisfied, solution of Eq. (11) is zero, as Newton–Raphson in Artificial strategy. Otherwise, it is minimized while Newton–Raphson algorithm does not provide a consistent result to the assembly problem. With this strategy, the performance of the robot can be evaluated everywhere in the design space, even if it is not assembled. The penalized objective function g is defined as follows

such that with

g(l, q(l)) = SEF(l, q(l)) + ωb f pen (l, q(l)) t  8 1 f pen  = ∑k=1 minvk 2 h l, uk , vk h (l, u, v)  (l, q(l)) c l, uk , vk ≥ 0, k = 1, . . . , 8

(12)

where ωb is a weighted factor between the SEF and the penalty term f pen . Its main advantage is that g is always continuous and differentiable over the design space. For this reason, a gradient-based sequential quadratic programming (SQP) algorithm is used with this strategy, which is usually more efficient than direct search algorithms. Two major drawbacks appear on this strategy. The first one is the choice of the ωb . However, the final optimal solution must be an assembled configuration and then, this term vanishes. The value of ωb in this work is 104 . The second inconvenient of this strategy is the computation of the sensitivity analysis of the objective function. More information about this issue can be found in [6].

4 Application to Delta Robot: Results of Optimization An initial simulation, Sim. 1, which does not take into account the area occupation index (η2 ), i.e., ω = 0 in Eq. (1), has been performed. Then, several optimization process have been computed corresponding to different values of weighting factor, as can be seen in Figure 3, showing the sensitivity analysis of the problem with respect to weighting factor. As the goal of this optimization problem is to keep an acceptable GDI reducing as much as possible the area, Sim. 2 has been chosen as a second result with a weighting factor of ω = 0.001 in Eq. (1). Table 1 shows initial and optimal values of GDI and projected area when the end-effector is located at the center of the cube. All projected areas in this table are computed at this position. Percentages of optimal area (Aopt p ) variation are computed ini with respect to both initial area (A ) and optimal area when ω = 0 (Aopt p )ω =0 . Note that a slightly decrement of GDI, i.e., from 98.88% to 95.3%, entails a reduction of the occupied area more than 72% in both simulations. Table 2 shows values of the design variables: initial guesses and their upper and lower values and the results of the two simulations. In Sim. 1, optimal values of design variables are similar in both strategies. There is only one difference: radii of platform and base. However, it demonstrates that GDI does not depend on these values but their ratio, which tends to unit value.

Multi-Objective Optimization of Parallel Manipulators

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Fig. 3 Sensitivity analysis of weighting factor. Table 1 Optimization results. 2 GDI [%] Aopt p [cm ]

ini Aopt p −A Aini

[%]

Initial

36.11

353.81

Artificial Best-assembly

98.88 98.88

900.04 801.35

154.39 126.49

Artificial Sim.2: ω = 0.001 Best-assembly

95.34 95.31

217.93 216.47

-38.40 -38.82

Sim.1: ω = 0

opt Aopt p −(A p )ω =0 (Aopt p )ω =0

[%] Fnc. evals

957 122 -75.79 -72.99

919 199

Table 2 Values of the design variables. lA [cm]

lB [cm]

zc [cm]

rb [cm]

r p [cm]

2 10 20

2 10 20

-20 -10 -5

1 5 10

1 2 10

Initial guesses

Min. Initial Max.

Sim.1: ω = 0

Artificial Best-assembly

16.39 16.39

20 20

-11.5 -11.5

2.22 1.1

2.25 1.2

Sim.2: ω = 0.001

Artificial Best-assembly

8.16 8.13

9.93 9.89

-5.7 -5.7

1 1

1 1

Figure 4 shows schematic initial guess and optimal solutions for Delta robot graphically. It can be observed the huge reduction of occupied area. Note that results obtained with both strategies are practically similar.

5 Conclusions An optimization method for designing parallel manipulators is presented in this work. The objective function is composed by two terms, namely, dexterity and area

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Fig. 4 Delta robot: initial guess (black), results of Sim. 1 (red) and results of Sim. 2 (yellow).

occupation. The main problem which appears during the optimization of closedloop multibody systems is its assembly. Two different strategies are applied: Artificial and Best-assembly penalty optimization strategies. The first one is combined with Nelder–Mead optimization algorithm, and the second with SQP. Results show optimized Delta robot with area reduction occupies up to 75% less than optimized Delta robot without area reduction while the dexterity index is only 4% worse.

References 1. Angeles, J., Fundamentals and Robotic Mechanical Systems. Theory, Methods and Algorithms, Mechanical Engineering Series, Springer, 2006. 2. Baron, L., Wang, X., and Cloutier, G., The isotropic conditions of parallel manipulators of Delta topology. In ARK, Caldes de Malavalla, pp. 357–366, 2002. 3. Carricato, M. and Parenti-Castelli, V., Singularity-free fully-isotropic translational parallel mechanisms, Int. Journal of Robotics Research, 21(2):161–174, 2002. 4. Chablat, D. and Wenger, P., Architecture optimization of a 3-DOF parallel mechanism for machining applications, the orthoglide, IEEE Transactions on Robotics and Automation, 19(3):403–410, 2003. 5. Clavel, R., Delta, a fast robot with parallel geometry. In Proc. 18th Int. Symposium on Industrial Robots, IFS Publications, Lausanne, pp. 91–100, 1988. 6. Collard, J.-F., Geometrical and kinematic optimization of closed-loop multibody systems. PhD Thesis, Universit´e catholique de Louvain, Louvain-la-Neuve, Belgium, 2007. 7. Collard, J.F., Duysinx, P. and Fisette, P., Kinematical optimization of closed-loop multibody systems, MB Dynamics: Computational Methods and Applications, 12:159–179, 2008. 8. Gogu, G., Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations, European Journal of Mechanics A/Solids, 23:1021-1039, 2004. 9. Gosselin, C. and Angeles, J., A global performance index for the kinematic optimization of robotic manipulators, Journal of Mechanical Design, 113:220–226, 1991. 10. Merlet, J.-P., Parallel Robots, Solid Mechanics and Its Applications Series, Springer, 2006. 11. Nelder, J.A. and Mead, R., A simplex method for function minimization, Computer Journal, 7(4):308-313, 1965. 12. Samin, J. and Fisette, P., Symbolic Modeling of Multibody Systems. Kluwer Academic Publishers, Dordrecht, 2003. 13. Stock, M. and Miller, K., Optimal kinematic design of spatial parallel manipulators: Application to Linear Delta robot, ASME Journal of Mechanical Design, 125(2):292-301, 2003.

A Design Method of Crossed Axes Helical Gears with Increase Uptime and Efficiency T.A. Antal and A. Antal Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania; e-mail: [email protected], [email protected]

Abstract. The paper present an original method of designing crossed axes helical gears with addendum modification based on the sliding between the meshing teeth’s flanks. Based on the equalization of the relative sliding coefficients at the points where the meshing begins and ends the geometrical dimensions of the helical gears are determined. The efficiency is computed taking into account the friction coefficient between the tooth flanks and the declination teeth angles rolling cylinders. The numerical results are computed using Matlab by multiobjective optimization of the values for the equalized sliding coefficients and the efficiency of the helical gears. Based on this method the designer is helped to responsible choose the helical gear in terms of gauge, efficiency and uptime. Key words: helical gear, relative sliding coefficients, efficiency, multiobjective optimization

1 Introduction Helical gears allow the transmission of mechanical power between crossed axes. Their use is in the field of spatial mechanisms, robot constructions and conveyor driving systems. During operation, between the meshing teeth flanks, is a high sliding speed that produces wearing of the flanks and a decrease of the efficiency. Starting from the sliding equalization at the points where the meshing of the teeth begins and ends the paper gives a method of designing helical gears with reduced equalized sliding and good efficiency.

2 The Efficiency of Crossed Helical Gears with Addendum Modification The technical literature studies the efficiency of the crossed helical wheels without considering the addendum modification [5, 6]. If addendum modification is considered, the contact point of the teeth is computed on the rolling cylinders instead of

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the pitch cylinder. With the x1 and x2 addendum modifications, the efficiency of the gear tooth was determined based on [3] as:

η=

1 − µ1 tan(βw2 ) . 1 + µ1 tan(βw1 )

(1)

where βw1 and βw2 are the teeth declination angles on the rolling cylinders of the 1st and 2nd wheel; and µ1 = tan(ϕ1 ) = µ /cos(αn ) is the reduced coefficient of friction (µ is the friction coefficient).

3 The Relative Sliding Coefficients at Crossed Helical Gears with Addendum Modification The sliding between the teeth flanks can be evaluated with the help of ζ12 and ζ21 relative sliding coefficients. ζ12 measures the relative sliding of flank 1 in regard to flank 2, while ζ21 measures the relative sliding of flank 2 in regard to flank 1. It was proved in [1, 2] that the general expressions of the relative sliding coefficients can be expressed, in the case of helical gears, as: → → − − V 12 · V 12 ∆ S1 − ∆ S2 = → − → − . ∆ S1 →0 ∆ S1 V 1 · V 12

(2)

→ → − − V 21 · V 21 ∆ S2 − ∆ S1 = → − → − . ∆ S2 →0 ∆ S2 V 2 · V 21

(3)

ζ12 = lim and

ζ21 = lim

where ∆ S1 and ∆ S2 are the lengths covered by the contact point on the 1st surface and the 2nd surface, in ∆ t time; V12 and V21 are the relative velocities, at the contact point, of the 1st surface with respect of the 2nd one, and of the 2nd surface with respect of the 1st one; V1 and V2 are the absolute velocities of the 1st and the 2nd surface at the contact point. The sliding coefficients are determined at the A and E points which are the most far meshing points from the pitch point as here the sliding coefficients have the highest values. The sliding coefficients at the A (beginning of the meshing) and E (end of the meshing) points are [4]: A2A + B2A + CA2 . (yA + rw1 )AA + xA BA

(4)

A2A + B2A + CA2 
. − yE − rw2 u21 AE cos(Σ ) − u21 ((xE cos(Σ ) − zE sin(Σ )) BE

(5)

ζ12A = and

ζ21E =

CE2

A Design Method of Crossed Axes Helical Gears with Increase Uptime and Efficiency

where

and

643

⎫
 AA = yA
1 − u21 cos(Σ ) + rw1 + rw2 u21 cos(Σ ) ⎬ BA =
xA 1 − u21 .  cos(Σ ) + zA u21 sin(Σ ) ⎭ CA = yA − rw2 u21 sin(Σ )

(6)

⎫
 AE = yE
1 − u21 cos(Σ ) + rw1 + rw2 u21 cos(Σ ) ⎬ BE =
xE 1 − u21 .  cos(Σ ) + zE u21 sin(Σ ) ⎭ CE = yE − rw2 u21 sin(Σ )

(7)

where xA , yA , zA and the xE , yE , zE are the coordinates of the A point where the meshing begins and E point where the meshing ends; u21 = ω2 /ω1 = z1 /z2 is the gear ratio; rw1 and rw2 are the radii of the rolling circles for the 1st and the 2nd wheel; Σ is the angle between the axes of the 1st and the 2nd wheel (the sum of the βw1 and βw2 teeth declination angles on the rolling cylinders). If the x1 , x2 , z1 (number of the teeth for the 1st wheel), z2 (number of the teeth for the 2nd wheel), αn (the profile angle of the basic rack), h∗a (the height coefficient from head of the tooth), c∗ (the clearance coefficient from the head of the tooth), mn (the module), Σ and a (the distance between the axis) values are given, a set of β1 and β2 values are obtained by solving the following system of nonlinear equations:  βw1 (x1 , x2 , β1 , β2 ) + βw2(x1 , x2 , β1 , β2 ) = Σ . (8) rw1 (x1 , x2 , β1 , β2 ) + rw2 (x1 , x2 , β1 , β2 ) = a The system (8) is solved using MATLAB’s fsolve() function from MATLAB’s Optimization Toolbox. fsolve() implements an iterative method that needs starting values for the computations. The data from the following tables are obtained for: z1 = 20, z2 = 35, αn = 20◦ , h∗a = 1, c∗ = 0.25, mn = 2.5 mm and µ = 0.1 (all angles are in degrees and distances in millimeters). In the case of Table 1 the x1 and x2 values are given, a = 97 mm, Σ = 90◦ and the start values for the helix angles on the pitch cylinders are β1 = 45◦ and β2 = 45◦ . The results (unknowns) are the β1 and β2 returned by the fsolve() function. For each row from Table 1 the x1 and x2 values are given, while the start values for β1 and β2 are the same and we obtain different wheels that will keep the same a and Σ . As shown in Table 1 the system (8) gives multiple technical solutions but the values of the relative sliding coefficients at the A and E points are different as well as the efficiency. These differences, rise in time, different wearing of teeth flanks, one of the wheels having a faster deterioration then the other. The designer must choose one solution between the many that will improve some properties of the helical gear.

4 Equalization of the Relative Sliding Coefficients Equalization of the relative specific sliding is done by adding one more equation to the system (8). (4) and (5) are made equal and this new system has three unknowns, x2 , β1 and β2 , while x1 is given. The same fsolve() function is use to solve it, keeping

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T.A. Antal and A. Antal Table 1 The variation of the relative sliding coefficients at the A and E points.

x1

x2

β1

β2

ζ12A

ζ21E

βw1

βw2

Σ

a

η

–0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2

–1.600 –1.400 –1.200 –1.000 –0.800 –0.600 –0.400

42.11 42.40 42.70 43.04 43.41 43.82 44.29

51.92 51.12 50.30 49.46 48.59 47.67 46.70

2.69812 2.64975 2.59463 2.53446 2.47034 2.40285 2.33206

1.82078 1.86972 1.91917 1.96965 2.02181 2.07655 2.13518

40.431 40.898 41.395 41.925 42.498 43.124 43.816

49.569 49.102 48.605 48.075 47.502 46.876 46.184

90 90 90 90 90 90 90

97 97 97 97 97 97 97

0.802350 0.803113 0.803866 0.804606 0.805332 0.806039 0.806718

Table 2 The variation of a with equalized sliding coefficients. x1

x2

β1

β2

ζ12A

ζ21E

βw1

βw2

Σ

a

η

–1.8 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4

–1.154 –0.802 –0.358 –0.676 –0.179 –0.531 0.007 0.592

49.45 48.85 48.31 47.87 47.37 46.90 46.43 46.05

46.57 45.76 44.90 45.56 44.70 45.45 44.58 43.64

2.07495 2.14235 2.18112 2.18675 2.20763 2.21302 2.22031 2.21382

2.07495 2.14235 2.18112 2.18675 2.20763 2.21302 2.22031 2.21382

46.295 46.422 46.611 46.088 46.288 45.697 45.910 46.211

43.705 43.578 43.389 43.912 43.712 44.303 44.090 43.789

90 90 90 90 90 90 90 90

92 93 94 94 95 95 96 97

0.808292 0.808337 0.808398 0.808211 0.808289 0.808033 0.808134 0.80826

the same start values for β1 and β2 while making initially, x2 =0.15. ⎫ ζ12A (x1 , x2 , β1 , β2 ) = ζ21E (x1 , x2 , β1 , β2 ) ⎬ βw1 (x1 , x2 , β1 , β2 ) + βw2(x1 , x2 , β1 , β2 ) = Σ . ⎭ rw1 (x1 , x2 , β1 , β2 ) + rw2 (x1 , x2 , β1 , β2 ) = a

(9)

In the case of the Σ the limits are varying between 0◦ and 90◦ , however in the case of a these limits are given by the possibility of finding solutions. If working with real numbers the limits for a are hard to find, however if we put the condition of integer numbers we can find the limits of the a values that could provide us solutions. The results given in Table 2 are found by looking for an integer value of a. If we are interested in the equalization of the specific sliding with keeping a given Σ = 90◦ and a, the a values are restricted to the [92 . . . 97] limits. Based on the results from Table 2 if Σ = 90◦ and a = 95 mm it is possible to obtain helical gears with equalized sliding coefficients and the results are given in Table 3. The x1 and x2 values of the addendum modifications obtained from the equalization condition at the beginning and the end of the meshing must satisfy the interference conditions, which are the conditions concerning the cut and the undercut of the teeth. In order to maintain the thickness of the teeth on the head circles the following relations must be true: – at wheel 1:

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Table 3 The variation of the efficiency and of the equalized sliding coefficients. x1 –1.8 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4

x2

β1

β2

ζ12A

ζ21E

βw1

βw2

Σ

a

η

1.015 0.748 0.461 0.152 –0.179 –0.531 –0.904 –1.297

49.39 48.87 48.36 47.86 47.37 46.90 46.44 46.00

41.99 42.63 43.29 43.98 44.70 45.45 46.22 47.01

2.16750 2.17987 2.19079 2.20012 2.20763 2.21302 2.21592 2.21581

2.16750 2.17987 2.19079 2.20012 2.20763 2.21302 2.21592 2.21581

48.609 48.041 47.463 46.878 46.288 45.697 45.107 44.521

41.391 41.959 42.537 43.122 43.712 44.303 44.893 45.479

90 90 90 90 90 90 90 90

95 95 95 95 95 95 95 95

0.808578 0.808612 0.808577 0.80847 0.808289 0.808033 0.807703 0.807301

 Sna1 = da1

 0.5π + 2x1 tan(αn ) + inv(αt1 ) − inv(αta1 ) cos(βa1 ) ≥ 0.5mn . (10) z1

– at wheel 2:   0.5π + 2x2 tan(αn ) + inv(αt2 ) − inv(αta2 ) cos(βa2 ) ≥ 0.5mn . (11) Sna2 = da2 z2 where da1 and da1 are the diameters of the head circles for the 1st and the 2nd wheel; αt1 and αt2 are the profile angles of the basic rack at frontal planes of the 1st and the 2nd helical wheel; αta1 and αta2 are the pressure angles in the frontal planes on the addendum circles of the 1st and the 2nd helical wheel; βa1 and βa2 are the teeth declination angle on the head cylinder on the 1st and 2nd helical wheel. In order to eliminate the undercut phenomena the x1 and x2 addendum modification must be greater than the xmin1 and xmin2 values from the following formulas: – at wheel 1: xmin1 = h∗a −

z1 tan2 (αn ) . 2 cos(β1 ) tan2 (αn ) + cos2 (β1 )

(12)

xmin2 = h∗a −

z2 tan2 (αn ) . 2 2 cos(β2 ) tan (αn ) + cos2 (β2 )

(13)

– at wheel 2:

The results from Table 3 contain only the solutions that are meeting the conditions (10)–(13). Still, the designer has a large number of possibilities to choose from in order to find its desired helical gear. Ideally, the value for the ζ12A = ζ21E must be the lowest while the value for η should be the highest for the chosen row of Table 3. The lowest equalized value will increase the uptime while the highest η value will provide the maximum efficiency.

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5 Uptime and Efficiency Optimization We are interested to find a combination of the x1 , x2 , β1 , β2 values for which the lowest equalized specific sliding is minimum (first goal) while, at the same time, the efficiency is maximum (second goal). The problem is that these two goals cannot be achieve simultaneously. The first row from Table 3 has the minimum values of the equalized specific sliding, while at the second row of the same table we find the best value for the efficiency. Multiobjective optimization methods of nonlinear functions are known, however in this case these can’t be used as there are problems concerning the target of obtaining the minimal value of the equalized specific sliding: • as proved in [2] in order to solve the system (9) a transcendent equation must be solved, which might not return solutions because: – there are no solutions; – the numerical method won’t converge to the solution; – the solution is a complex value, instead of real one. • when solving the (9) system: – the system might not have solutions; – depending on the start points, the numerical method may converge to a nonzero point; – the obtained values are numerically correct however we have interferences. Genetic Algorithms (GAs) are a category of evolutionary algorithms well known to find approximate solutions to the optimization problems of difficult functions. The Optimization Tool from Matlab has the ”Multiobjective Optimization using Genetic Algorithm” solver that finds the minimum of multiple functions using a genetic algorithm. For the first goal we write the objective function (14) starting from the unpenalized function to which positive penalties are added if the solutions violate in some way the feasibility as: k

ζ12Amin = ζ12A + ∑ Ci δi .

(14)

i=1

In (14) δi is 0 if the constraint i is 0, otherwise is 1 and Ci is a positive constant imposed for the violation. For a set of x1 , x2 , β1 , β2 violations are considered if: • any of the three equations from (9) are not satisfied (no convergence to the solution); • the values of x1 or x2 are creating interferences; • computations are not possible (no convergence or complex values are obtained). For the second goal, as the solver finds only minimum values, the objective function will be written as (K is a positive constant):

ηmin = K(1 − η ).

(15)

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Table 4 Solutions for some extreme values and a middle value for the objective functions with Σ = 90◦ and a = 95mm. x1

x2

β1

β2

βw1

βw2

ζ12Amin

η

ηmin

–1.9970 –1.9943 –1.5715 –1.6427 –1.8521

1.2576 1.2545 0.7086 0.8070 1.0810

49.90253 49.89557 48.79617 48.97890 49.52201

41.39591 41.40373 42.72007 42.48946 41.83246

49.15834 49.15105 47.95884 48.16278 48.75520

40.84166 40.84895 42.04116 41.83722 41.24480

2.15397 2.15416 2.18152 2.17734 2.16404

0.808480 0.808481 0.808612 0.808611 0.808558

1.915204 1.915187 1.913885 1.913895 1.914422

Being interested to maximize the efficiency, which variate between 0 and 1, the second objective function from (15) will be minimum when the efficiency will be maximum. Pareto front graphs are used for multiobjective optimization where there is more than one solution at each run. The solutions from every run are shown in Figure 1. We can notice that when the equalized relative sliding coefficients are convenient (have low values), the efficiency is inconvenient (has also low values) and vice versa. We can use this graph to work out a compromise solution, where the equalized relative sliding coefficients are not too high and the efficiency is not very low from all the possibilities given by the obtained solutions.

Fig. 1 Pareto front of ζ12Amin and ηmin functions.

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Some of the results obtained by the Matlab application are shown in Table 4, first two rows are giving good solutions for the equalized relative sliding, the third and fourth rows are good results for the efficiency, while the last row, found with the help of the Pareto front from Figure 1, is a compromise between the two extremes.

6 Conclusions The design of helical gears based on the equalization of te relative sliding coefficients allows improvement of operating conditions. As can be seen in Table 1, these coefficients, at the points where the meshing of the teeth starts and ends are very different. The result of this condition is that the wearing of the teeth flanks will be uneven. In order to influence the uneven wearing the relative sliding coefficients are equalized as shown in Tables 2, however this process is limited to only certain domains of Σ and a. For a given Σ , a Matlab application finds the limits of the axial distances can be used to keep the equalization, while the efficiency of the gear varies very little. If the Σ and a values are given, from Table 3 we can notice that the efficiency of the gear varies. After restricting the domains for the x1 and x2 addendum modifications so that the thickness of the teeth is not too small and no undercut phenomena are appearing, using a GA, optimal values for x1 , x2 , β1 and β2 are obtained (Table 4) so that the designer can find the best gear that fill its needs in terms of efficiency and uptime.

References 1. Antal, A. and Antal, T., Program for the calculation of the sliding at helical gears. In Proceedings PRASIC’98 – National Symposium with International Participation, Volume II, Machine Parts and Mechanical Drives, Bras¸ov, Romania, pp. 235–238, 1998. 2. Antal, T.A., A new algorithm for helical gear design with addendum modification. Mechanika, 3(77):53–57, 2009. 3. Antal, T.A., Eficiency of crossed helical gears with equalized slinding coefficients. In: Annals of DAAAM for 2009 – Proceedings of 20th DAAAM International Symposium, Volume 20, No. 1, pp. 1017–1018, DAAAM International, Vienna, Austria, 2009. 4. Antal, T.A., Antal, A. and Arghir, M., Determination of the addendum modification at helical gears, at the points where the meshing starts and ends, based on the relative velocity equalization criterion. PAMM Journal, 8(1):10965–10966, 2008. 5. Niemann, G. and Winter, H., Machine Elements, Volume III. Springer-Verlag, 1983. 6. Tochtermann, W. and Bodenstein, F., Design Elements of Machine Engineering, Part 2. Springer-Verlag, 1979.

Dual Axis Tracking System with a Single Motor Gh. Moldovean, B.R. Butuc and R. Velicu Transilvania University of Bras¸ov, 500036 Bras¸ov, Romania; e-mail: [email protected]

Abstract. The main direction in the use of solar energy conversion systems is to improve their efficiency by using dual axial tracking systems. The energy required for positioning the photovoltaic platforms is significant due to the complexity of constructive solutions of dual axial tracking systems, which incorporate one driving source for each axis, and of the control systems. Thus, the interest in research, innovation, production and implementation of tracking systems increased. This paper presents a new tracking system for photovoltaic platforms actuated by a single rotary electric motor and the tracking method used in order to maximize the energy production. Key words: dual axis system, PV platform, tracking procedure, energetic efficiency

1 Introduction Continuous economic development of the world leads to an increase of energy consumption and a decrease in fossil fuels resources (natural gases, coal, oil). Exhaustion of fossil fuels, increased prices and their impact on human health and environment led to the promotion of renewable energy and creation of a suitable legal framework of their implementation [3]. From the main renewable energy resources (solar, wind, hydro and biomass), solar energy can assure twice the amount of energy that can be obtained from fossil resources (coal, natural gas, oil) [10]. Considering these, solar energy conversion systems are part of a large range of domestic and industrial applications. Photovoltaic (PV) cells are converting solar radiation into electric energy by means of the photovoltaic phenomena. The optimization of the efficiency of solar radiation conversion (into electric energy) is mainly achieved by using of special materials with high absorbant properties and by increasing the amount of solar energy on the PV cell. To improve the quantity of direct solar radiation that falls on the PV module two main directions are considered: use of systems for concentration of the radiation and use of tracking systems for better positioning of PV modules towards the sun [11].

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_74, © Springer Science+Business Media B.V. 2010

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The main factor influencing the energetic efficiency of a solar conversion system is the quantity of the direct solar radiation. In order to maximize the solar radiation received by a conversion system, tracking system are used. Energy gain by dual axis tracking systems is significant in relation with single axis ones, thus leading to an increase interest in research and inovation [1, 8], and their implementation [4, 9]. It is not recommended to use tracking systems for individual photovoltaic modules of small sizes due to the energy lost in the tracking system, which can reach 2 to 3% of the additional energy obtained [7]. This paper presents a new constructive solution (patent proposal registered under No. A/740/21.09.09) of a tracking system for medium and large size PV platforms, using a single rotative motor. It also presents the tracking methodology chosed in order to maximize the electric energy gain.

2 Tracking System Description The proposed tracking system is a dual axes azimuthal tracking system. It allows two independent rotational movements related to the following axes: a vertical axis for the azimuthal rotation and a horizontal axis for the altitudinal rotation. In the case of these kind of tracking systems, the two movements develop in sequences, at certain periods of time, during each daylight, following sun movement from sunrise to sunset. The altitudinal rotation is performed in one direction from sunrise to noon and in oposite direction from noon to sunrise. After sunset, the system is turning back to the sunrise position of the next day, through a continuous azimuthal movement. A sequence of movement during daylight positioning determines rotational angles around the two axes in the range of 5 to 15◦ , smaller values for altitudinal movement. These rotational angles are small enough in order to limitate the error of positioning and as a result to minimize the incidence angle of solar ray on the PV platform. The structural scheme of the proposed transmission of the dual axes tracking system is presented in Figure 1. The transmission is consisted of: a rotative motor with output shaft linked to the worm pinion 1, a worm gear (worm pinion 1 and worm wheel 2), coupling C1 acting as a simple brake, two spur bevel gears (bevel pinion 3 and bevel wheels 4 and 5) and two couplings C2 and C3. All three couplings are normally engaged, with friction and pressure force performed by springs during normally engaged stage and electomagnetic disengage. Bevel gears contribute with their transmission ratio to the increase of tracking precision of the altitudinal movement, relative to the azimuthal movement. An assemble view of the PV platform tracker is presented in Figure 1b with the same notations as in Figure 1a. The rotational movements performed by this tracking system are: azimuthal movement, around vertical axis I, controlled by coupling C1 and altitudinal movement, around horizontal axis II, controlled by couplings C2 and C3. In order to obtain these movements the tracking system has four distinct kinematic situations, also presented in Table 1.

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Fig. 1 Azimuthal tracking system; (a) structural diagram, (b) assemble view. Table 1 Control diagram.

A. Rotation around the vertical axis I (azimuthal movement) – with disengaged C1 coupling and engaged C2 and C3 couplings. Rotation is transmitted from motor through the worm gear 1-2 to the bevel pinion 3. The subassembly from casing 6, consisted of bevel gears 3-4-5, C2 and C3 couplings, shaft of axis II and platform 7 are moving together as one part, rotating around axis I. From the control of the motor, rotation can be performed in both directions, following the sun during daylight in one direction and repositioning for the next sunrise, in the other direction. B. Rotation around the horizontal axis II (altitudinal movement), direction B – with engaged C1 and C2 couplings and disengaged C3 coupling. The rotation of the motor shaft is transmitted through the worm gear 1–2 to the bevel pinion 3 and following to the bevel wheel 4. This wheel is engaged on the shaft of axis II through the C2 coupling and will rotate together with platform 7. C. Rotation around the horizontal axis II (altitudinal movement), direction C – with engaged C1 and C3 couplings and disengaged C2 coupling. The rotation of the motor shaft is transmitted through the worm gear 1–2 to the bevel pinion 3 and

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Fig. 2 Constructive solution of the azimuthal axis.

following to the bevel wheel 5. Bevel wheel 5 is engaged on the shaft of axis II through the C3 coupling and will rotate together with platform 7. D. Resting situation. C1, C2 and C3 couplings are engaged and the motor is off. The system is blocked without any energy consumption. A 3D model of the tracker was developed during the embodiment design phase. A partially section through the azimuthal axis I is presented in Figure 2. The main shaft 8 of the azimuthal axis is assembled on the the worm wheel 2 of the existing slew drive 1–2. The spur bevel pinion is mounted on the top of the shaft 8. Casing 6 (see Figure 1a) lays on the main shaft 8 by a ball bearing mounting. Casing 6 is assembled together with the halfcoupling 9 of C1. The halfcoupling 10 of C1 is assembled on casing 11 of the slew drive. Casing 11 is assembled on the pillar 12 (see Figure 1b) of the PV platform. Figure 3 presents the construction of the altitudinal rotational axis II. Shaft 13 assembles the gear wheels 4 and 5 through the couplings C2 and C3. At the two ends of the shaft, the platform is assembled with bushings and keys. Bevel gears 4 and 5 are free mounted on shaft 13, assembled with bolts on half couplings 14 and 15. The interior halfcouplings 16 and 17 are mounted on shaft 13 through key assemblings. The main advantages of the proposed tracking system are: reduce of the energy needed for tracking, by using a single motor and three normally engaged friction couplings (couplings control only consume energy during the disengagement stage, which is very short); increase of the accuracy of tracking by using a higher transmission ratio for the altitudinal rotation; increase of mechanical efficiency by using bevel gears instead of a second worm gear, as used in actual existing constructions; symetry of construction determine a proper load distribution and following a bet-

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Fig. 3 Constructive solution of the azimuthal axis.

ter design solution; during the resting stage, loads coming from weight, wind or snow (mainly torsion moments) are transmitted directly to the fixed part through C1 couplings without any need of supplementary brackes on motors; torque around axis II is closing on the casing through bevel pinion 3, without loading the rest of the transmission.

3 Tracking Method The energy produced by solar conversion systems depends on the amount of direct solar radiation received on their surfaces. Since the amount of available direct solar radiation is influenced by the Sun-Earth geometry, the identification and optimization of tracking methods able to follow as closely the path of the sun on the sky represents an important step in the design stage of such systems. Sun tracking can be achieved by two methods: either by using sensors to detect the brightest point on the sky, either using predefined Sun paths, derived from astronomical knowledge. The disadvantages of the systems based on the first method are related to the control system complexity and engines energy consumption, especially in sudden climate changes. The main advantages of tracking methods based on predefined Sun paths are related to the simplicity of control scheme, the possibility of tracking program optimization [2] and the substantially energy gain, these being also the reasons of their wide use in practice [12–14].

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Fig. 4 Variation of solar and photovoltaic platform angles for summer solstice.

The orientation of the photovoltaic panels can be made continuously, providing this way the maximum amount of received direct solar radiation, but in terms of high energy consumption, due to the extended running time of the driving systems. A discrete orientation (in steps) implies a change in the position of the photovoltaic panels at equal time intervals. This method principle is based on the input data related to the Sun position on the sky, described by altitude α and azimuth ψ angles [6]: (1) sin(α ) = sin(δ ) sin(φ ) + cos(δ ) cos(φ ) cos(ω ) cos(ψ ) =

sin(α ) sin(φ ) − sin(δ ) cos(α ) cos(φ )

(2)

where ϕ represents the latitude, δ declination angle and ω hour angle. The amount of solar energy received by a photovoltaic platform RPV depends on the direct solar radiation RD , determined in this case with Meliss relation [5], and on the incidence angle – the angle built by the solar ray and the surface normal: RPV = RD cos(ν )

(3)

Since the Sun’s position on the sky changes with 15◦ per hour, the position of photovoltaic platform is also modified at every one hour interval to maintain the incidence angle as close to 0. Variations of the angles described by the Sun and by the Photovoltaic platform position (α PV and ψ PV) were simulated for the equinoxes and solstices, considering the geographical parameters of Bras¸ov area (45.6◦ North latitude). The curves obtained for summer solstice day are presented in Figure 4. The amount of direct solar radiation received by photovoltaic platform, determined based on input data related to: sun position, photovoltaic platform position (α PV and ψ PV) and clear sky condition, was compared with the amount of direct solar radiation received by a tilted system. The tilt angle of the tilted system α tilt changes according to the following relation (4):

αtilt = φ − δ

(4)

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Fig. 5 Solar energy on summer solstice N = 172.

Fig. 6 Annual tracking efficiency of a tracked photovoltaic platform.

Analyzing Figure 5 we can notice a significant increase in solar energy of an azimuthally tracked photovoltaic platform compared to a tilted one. This type of tracking method provides an annual energy gain up to 24% more than tilted systems, as shown in Figure 6, thus justifying the use of predefined Sun path tracking methods. As general conclusion, by using the proposed tracking system and by selecting adequate tracking method and program, the photovoltaic platforms efficiency can be substantially increased. Further research by the authors will focus on optimization the proposed tracking system, by identifying and minimizing the factors that influence the energy efficiency.

Acknowledgments This paper is supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and by the Romanian Government under contract number POSDRU/6/1.5/S/6.

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References 1. Alava, O., Two axis solar tracker. Patent No. EP 1 998 122 A1, 2008. Available from http://v3.espacenet.com/searchResults?AP=EP+1+998+122+A1&DB=EPODOC&PD= 20081203&submitted=true&locale=en EP&TI=Two+axis+solar+tracker&IN=Alava&ST= advanced&compact=false. 2. Dinicu, V., Vis¸a, I. and Diaconescu, D., Tracking improvement of an azimuthally tracked PV panel. Bulletin of the Transilvania University of Bras¸ov, 15(50):81–86, 2008. 3. Directive 2009/28/EC. Available from http://eur-lex.europa.eu/LexUriServ/LexUriServ.do? uri=OJ:L:2009:140:0016:0062:EN:PDF. 4. Grottke, M. et al., Performance of a 230 KWp solar park in Spain with two-axis trackers from a European market leader. In Proceedings of 24th European Photovoltaic Solar Energy Conference, Hamburg, Germany, pp. 4080-4082, 2009. 5. Meliss, M., Regenerative Energiequellen, Springer-Verlag, Berlin, pp. 5–15, 1997. 6. Messenger, R. and Ventre, J., Photovoltaic System Engineering, London, pp. 21–39, 2000. 7. Mousazadeh, H. et al., A review of principle and sun-tracking methods for maximizing solar systems output. Renewable and Sustainable Energy Reviews, 13:1800–1818, 2009. 8. Neff, K., Tracking systems for solar units, Patent No. WO 2007/025618 A, 2007. Available from http://www.wipo.int/pctdb/en/wo.jsp?wo=2007025618. 9. Rindelhardt, U., Dietrich, A. and R¨osner, C., Tracked megawatt PV plants: Operation results 2008 in Germany and Spain. In Proceedings of 24th European Photovoltaic Solar Energy Conference, Hamburg, Germany, pp. 3909–3910, 2009. 10. Weston, A.H., Quantifying global exergy resources. Energy, 31(12):1685–1702, 2006. 11. Visa, I. et al., New mechanical systems for increasing the solar conversion systems in electric energy. In Bulletin of MENER Conference, Romania, Sinaia, pp. 127–134, 2008 [in Romanian]. 12. www.ades.tv. 13. www.tracker.cat. 14. www.pryusenergy.com.

The Determination of the Exact Surfaces of the Spur Wheels Flank with the Unique Rack-Bar S. Bojan, C. Birleanu, F. Sucala and I. Turcu Machine Elements and Tribology Department, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: {stefan.bojan, corina.birleanu, felicia.sucala, ioan.turcu}@omt.utcluj.ro

Abstract. Rolling tooth processing system, which is the most used procedure, is based on the principle that the surfaces in meshing are reciprocal winding surfaces in a gearing meshing, besides the two active surfaces there are also two surfaces which are reciprocally winded. The paper presents the manner in which the imaginary rack-bar is determined in the case of the spur curved gearing and the study of the relative velocities. The mathematical modelling of the processing procedure of the curved spur wheels imposed the determination of theoretical surfaces of the unique generating rack-bar teeth flanks. Key words: unique rack-bar, meshing, spur gear, head holder, curved toothing

1 Introduction This paper is included in the field of the theoretical and experimental researches based on cylindrical gear transmission with parallel axes, the aim being focused on the shape and direction of the teeth. Gear shaping is widely used to produce various gears. The traditional design scheme on ordinary gear shaper cutters is similar, and a detailed discussion is presented in [1,2]. Janninck [3,5] presented the integrated design method of shaper cutters for various applications. Rogers et al. presented an offset based method to obtain the tooth specifications required for the shaping processes and the strength of the meshing tooth pair [4]. Kim and Kim analyzed the basic theory of pinion cutter shape and developed computer software to design a pinion cutter [5]. Other researchers analyzed the tooth profile undercutting conditions of generating gears by shaper cutters, tooth-profile shifting and gear clearance [6–9]. Tsay et al. proposed a complete geometrical mathematical model of a spur shaper cutter, including the protuberance, the involutes region and semi-topping in manufacturing standard or nonstandard spur gears [9]. Bair investigated not only the profiles of the shaper cutter while considering protuberance and semi topping, but also the relationship between the shaper cutter parameters and the gear tooth profile for the manufacture of helical gears with small numbers of teeth [10]. In most of these shaper cutter related stud-

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_75, © Springer Science+Business Media B.V. 2010

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ies, tooth profile modifications are frequently applied on design and manufacture of shaper cutters for gear accuracy and quality. Researches concerning the shape of the teeth can be divided in two categories: the ones which refer to the profile curves of the teeth and the other ones which refer to the directrix curves [1]. Nowadays the cyclodale curves show interest again, but this time not only for the profile curves, but also to the directrix curves [4]. Taking into consideration the advantages of the cyclodale curves, (the capacity caring of the transmission by taking out the concentration of the load from its front parts can increase until 70– 90%, the axial forces are missing, the curve tooth can be cut with a single adjustment of the tool) the curve shape of the teeth directrix can be used in practice under the exact form or with acceptable geometrical modifications. The aim of this paper is to solve some problems regarding the achievement of some cylindrical gears with curve teeth. The processing procedure of the spur wheels curved teeth flanks is based on the unique generating rack-bar for the both wheels of a gearing. The tools head holder exerts a rotation motion, and the toothed wheel rolls on the imaginary generating rack-bar. Through the introduction of the generating rack-bar between the tools head holder and the toothed wheel, the study of the winded (proccessed) surfaces can be made in two stages: – the study of the tools head holder meshing with the rackbar, that is the rack-bar flanks generation (proccessing). The tools head holder exert a rotation motion, and the rack-bar a translation one; – the study of the rack-bar meshing with the spur toothed wheel, that is the generating (proccessing) of the wheel tooth flanks.

2 Theoretical Considerations The generating scheme is presented in Figure 1, where the tools head holder rotates around its axis zs with the angular speed ω¯ s . The toothed wheel rotates around its axis with the angular speed ω¯ 1 , needed for indexing, and its axis O1 x1 displaces with a translation velocity v¯t , achieving a continuous rolling on the plane (π ), without disturbing the indexing. According this scheme, the toothing processing is made with continuous indexing. If a plane B, perpendicular to the plane π , and solid with this, but which displaces parallel with the axis Os ys , thus that a cylinder having the radius ro , solid with the tools head holder to roll on the plane B, the points Pi respectively Pe , belonging to the cutting edge Γi , respectively Γe , situated in the plane π , will describe in this plane elongated cycloids. The same thing will be made by other points belonging to the cutting edges Γie , belonging to planes parallel to π , but exerting the same motion. Portions of the curves described in the plane π , through the plane winding on the primitive cylinder C1 , generates the directrix curves of the wheel tooth flank. The front tools head holder “S” has the cutting edges Γie – rectilinear, inclined in the same sense with respect to the axis Os Zs with the angle αn .

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Fig. 1 Generating scheme.

2.1 The Imaginary Rack-Bar Surfaces Determination It is considered the tools head holder (Figure 2) on which the presented coordinate systems are fixed and which exert a rotation motion with the angular speed ω¯ s oriented in the positive sense of the axis Os zs , and the opposite mobile body, which represents a toothed wheel with an infinite teeth number and which exert a translation motion with the velocity vt , so that no sliding speed to exist in the plane B, belonging to the opposite mobile body and the cylinder Cr of radius r0 connected to the tools head holder. Therefore: v¯t = ω¯ s × r¯o . One has to solve the problem of the surfaces SBie determination, which are processed by the tools cutting edges in the opposite mobile body. In these conditions, the plane B becomes a fixed centroide (basis), the cylinder Cr mobile centroide (rolling) and Cr generator, tangent to the plane B, instantaneous rotation axis. One has to remark that if the points Pie move along the afferent cutting edges Γsie and the instantaneous rotation centers have a corresponding motion, on the instantaneous rotation axis, it results resembling curves on the surfaces SBie . Therefore the surfaces SBie are made of the resembling curves described by the points Psie , which displaces along the afferent cutting edges, in parallel planes with Os xs ys , which contain them, the tools head holder effecting a rotation motion around the axis Os zs and a translation one parallel to Os ys . In order to understand the nature of the described trajectories by the points Psie , it is sufficient to study the trajectories in a certain plane belonging to those parallel planes fascicles. Being the selected plane, that which contains the origin of the coordinate system Os , therefore Os xs ys plane, which coincide with plane p (Figure 2). To the opposite body, which becomes fixed, a fix coordinate system OB xB yB zB is attached, so that to the moment ψie = 0 it has its axis parallel to Os xs ys zs system axis, but its origin OB displaced with the distance Os OB = ro in the positive sense of the

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Fig. 2 The directrix curve of the flank situated in the rolling-plane.

axis Os xs . The system Os xs ys zs to the moment ψie = 0 is designated by Oso xso yso zso . It can be observed that the mobile system Os xs ys zs motion, with respect to the fixed one OB xB yB zB is that from the cycloids cinematic generation. The points Psie being external to the generating cylinder will generate elongated cycloids. To the opposite body, which becomes fixed, a fix coordinate system OB xB yB zB is attached, so that to the moment ψie = 0 it has its axis parallel to Os xs ys zs system axis, but its origin OB displaced with the distance Os OB = ro in the positive sense of the axis Os xs . The system Os xs ys zs to the moment ψie = 0 is designated by Oso xso yso zso . It can be observed that the mobile system Os xs ys zs motion, with respect to the fixed one OB xB yB zB is that from the cycloids cinematic generation. The points (Figure 2) Psie being external to the generating cylinder will generate elongated cycloids. The points selection on the ordinate axis, do not affect the problem generality concerning the trajectories determination. The points Psie velocities, at a certain moment, are the same that those corresponding if the points displace with the angular speed ωs along an arc of circle with its center in IsB . Because vie are tangent to the curves Ωie , and IsB Psie are perpendicular on vie it results that the vectors IsB Pie belong to the normal planes to the surfaces SBie . Therefore the velocities of the points PSie belonging to the cutting edges Γsie are perpendicular on the normal of the surfaces SBie generated by them. In order to deduce the surfaces SBie equations, it passes from the mobile became system Os xs ys zs , in the fixed system. In order to deduce the surfaces SBie equations, it passes from the mobile became system Os xs ys zs , in the fixed system (Figure 2). The points PSie

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Fig. 3 The graphical representation of the rack-bar teeth flanks surfaces.

coordinates, in the system Os xs ys zs are given by the column matrix: ⎛ ⎞ ⎛ ⎞ xs ∓rwie cos ψie + hztgαn cos (ψie − δie ) ⎜ ys ⎟ ⎜ ∓r sin ψ + hztgαn sin (ψ − δ ) ⎟ ie ie ie ⎟ ⎜ ⎟ = ⎜ wie ⎝ zs ⎠ ⎝ ⎠ hz t 1 The coordinates of the points PBie , belonging to the surfaces SBie , superposed on the points PSie , are given by the matrix: ⎛ ⎞ ⎛ ⎞ ∓rwie cos ψie + hztgαn cos (ψie − δie ) − ro xBie ⎜ y ⎟ ⎜ ∓r sin ψ + hztgαn sin (ψ − δ ) − ro ψ ⎟ ie ie ie ie ⎟ ⎜ Bie ⎟ = ⎜ wie (1) ⎝ zBie ⎠ ⎝ ⎠ hz t 1 The relations (1) express the imaginary rack-bar flanks “generated by the tools cutting edges”. For an clear example the form of the imaginary rack-bar flanks (rwi = rwe = 9, −1 ≤ hz ≤ +1; ro = 0, 5) resulted from relations (1) is presented in Figure 3.

3 The Meshing of the Spur Wheels with Rack-bar The toothed wheels 1 and 2, composing the gearing with parallel axes, and with constant speed ratio, defined through the fixed position of the axes ∆1 and ∆2 and of the instantaneous rotation axis ∆12 , can be processed by rolling with the same generating surface SBie (Figure 4). The relative centroids R1 , R2 and π will always be tangent and will roll without sliding one over the other so that the instantaneous rotation axes ∆12 , ∆1B , ∆2B will coincide every moment. The centroids R1 and R2 are external surfaces of the pitch cylinders afferents to the wheels 1 respectively 2, and the centroid π represents the relative rolling plane, afferent to the imaginary rack-bar, becoming technological rack-bar. In order for the flanks F12ie , generated by SBie to be reciprocal winding with the linear contact, the instantaneous axes ∆1Bie and ∆2Bie need to rotate, and to overlap

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Fig. 4 The wheels meshing with the rack-bar.

over ∆12 , which is the tangent line of the rolling surfaces R1 , R2 and π . In this context the characteristics KB1 and KB2 must overlap over the characteristic K12 . The geometrical locus of the characteristic K21 (KB1 ; KB2 ) in a fixed system of coordinates it is called meshing surface. Considering a certain contact point PBie on the characteristic K12 and the overlapped points P1ie respectively P2ie belonging to the flanks F1ie , respectively F2ie , in order for these to be reciprocally winding surfaces it is necessary that the relative speed v12 of the point P2ie with respect to the point P1ie to be perpendicular to the common normal n. The relative motion of two bodies can be reproduced through the rolling of the relative axoides, which are tangent along the relative motion instantaneous axis. The angular speed ω21 is determined with the relation: ω¯ 21 = ω¯ 2 − ω¯ 1 . The orthogonal axis ∆21 intersects the common normal of the rotation axes ∆1 and ∆2 in the point O where v¯21 coincides with ω¯ 21 direction, thus can be written: v¯21(O) = h.ω¯ 21 = −ω¯ 1 × r¯1 + ω¯ 2 × r¯2

(2)

The vector v¯21(O) represents the sliding relative velocity along ∆21 axis, and h the parameter of the helicoidally relative motion. The angular speeds ω¯ B1 and ω¯ B2 are determined with the relations: ω¯ B1 =−ω¯ 1 ; ω¯ B2 =−ω¯ 2 , because the rack-bar exerts a rectilinear motion having the velocity v¯t perpendicular to ω¯ 1 and ω¯ 2 . The relative sliding velocities along the axes ∆B1 and ∆B2 are: v¯B1(O) = h1 ω¯ 1 = −ω¯ 1 × r¯1 + v¯t ;

v¯B2(O) = h2 ω¯ 2 = −ω¯ 2 × r¯2 + v¯t

(3)

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In these relations h1 and h2 are the parameters of the relative helicoidally motion. Because the gearing has parallel axes and the rack-bar exerts a rectilinear motion with the velocity v¯t perpendicular to ω¯ 1 and ω¯ 2 , the parameter h, is zero, the axes ∆12 , ∆B1 and ∆B2 are superposed and become instantaneous axes of the relative rotation motion. In this case the relations (2) and (3) become: v¯21(O) = 0; v¯B1(O) = h1 .ω¯ 1 = −ω¯ 1 × r¯1 + v¯t = 0; v¯B2(O) = −ω¯ 2 × r¯2 + v¯t = 0 (4) It is considered the superpossed points P1ie P2ie PBie , belonging to the surfaces F1ie F2ie respectively SBie , positioned through the vectors radii ρ¯ 1ie ρ¯ 2ie ρ¯ B1ie (ρ¯ B2ie , ρ¯ 12ie ). v¯1 , v¯2 and v¯t are the absolute velocities of the points P1ie P2ie . From Figure 3 it is observed that:

ρ¯ 1ie = r¯1 + ρ¯ 21ie;

ρ¯ 2ie = r¯2 + ρ¯ 21ie

(5)

Substituting and taking into consideration the relations (2) and (4) results: v¯21ie = (ω¯ 2 − ω¯ 1 ) × ρ¯ 21ie = ω¯ 21 × ρ¯ 21ie

(6)

v¯B1ie = −ω¯ 1 × ρ¯ B1ie = ω¯ B1 × ρ¯ B1ie, v¯B2ie = −ω¯ 2 × ρ¯ B2ie = ω¯ B2 × ρ¯ B2ie The relations refer to the system of coordinates Oxyz, parallel with the OB xB yB zB system. The system Oxyz is fixed having the axis Ox superposed over the fixed instantaneous rotation axes ∆B1 ∆B2 ∆12 . Because the vectors v¯B1ie respectively v¯B2ie are situated in the tangent plane of the surface SBie corresponding to the point PBie they are perpendicular to the normal of the surfaces F12ie in the point P12ie superposed over PBie . Therefore the common normal of the surfaces SBie and F12ie is situated in the front plane created by the vectors ω¯ B12 and ρ¯ B12ie . Because ω¯ B12 has the support of the relative instantaneous rotation axis ∆B12 , it results that the common normal intersects this axis. The fixed systems O10 x10 y10 z10 and O20 x20 y20 z20 with the parallel axes of the Oie xie yie zie system are attached to the wheels 1 and 2, but with the origins moved along the zie axis with −r1 for wheel 1 and +r2 for wheel 2. Are also attached to the wheels the systems O1 x1 y1 z1 and O2 x2 y2 z2 .

4 Conclusions The mathematical model of the processing procedure of the curved spur wheels is imposed by the determination of the theoretical surfaces of the unique generating rack-bar teeth flanks. In this direction, the rotation motion of the tool and the translation motion of the rack-bar have transposed from analytical relations, keeping in mind the processing procedure. From the study of the relative velocities it results that in any contact point of the conjugated flanks the common geometrical normal

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intersects the relative rotation motion instantaneous axis. The mathematical relations determined to define the unique rack bar of the active teeth surfaces with curve directrix have allowed their visualization through the PC graphical representation. The mathematical model presented in this paper has been submitted to take out a patent at the State Office for Inventions and Trademarks.

References 1. Andrei, L., Andrei, G., Epureanu, A., Oancea, N. and Walton, D., Numerical simulation and generation of curved face width gears, International Journal of Machine Tools and Manufacture, 42(1):1–6, January 2002. 2. Bojan, S., et al., The study of the relative velocities to the meshing of the spur wheels having cuved directrix with the imaginary unique rack-bar. In Proc. MET’04, Varna, Bulgaria, 23–25 September, pp. 200–203, Volume 5, 2004. 3. Chen, C-K., Chiou, S-T., Fong, S-T., et al., Mathematical model of curvature analysis for conjugate surfaces with generalized motion in three dimensions, Journal of Mechanical Engineering Science, 215(4):467–502, 2001. 4. Danieli, G.A. and Mundo, D., New developments in variable radius gears using constant pressure angle teeth, Mechanism and Machine Theory, 40(2):203–217, February 2005. 5. Kang, J., Kim, J. and Kang, B.S., Numerical analysis and design of pinion with inner helical gear by FEM G, Virtual and Physical Prototyping, 2(3):181–187, 2007. 6. Lixin, C., Hu, G. and Jian, L., Contact of surfaces and contact characteristics of offset surfaces, Frontiers of Mechanical Engineering in China, 3(3):318–324, 2008. 7. Shen-Wang, L., Cheng-Shun, H., Jiu-Bin, T. and Shen, D., Mathematical models for manufacturing a novel gear shaper cutter, Journal of Mechanical Science and Technology, 24:383–390, 2010. 8. Spitas, V., Costopoulos, T. and Spitas, C., Fast modeling of conjugate gear tooth profiles using discrete presentation by involute segments, Mechanism and Machine Theory Journal, 42(6):751–762, June 2007. 9. Tsay, C-B. and Tseng, T-J., Undercutting and contact characteristics of cylindrical gears with curvilinear shaped teeth generated by hobbing, Journal of Mechanical Design, 128(3):634– 643, May 2006. 10. Vogel, O., Griewank, A. and B¨ar, G., Direct gear tooth contact analysis for hypoid bevel gears, Computer Methods in Applied Mechanics and Engineering, 191(36):3965–3982, 2002.

Choosing the Actuators for the TRTTR1 Modular Serial Robot R.M. Gui, V. Ispas, Vrg. Ispas and O.A. Detesan Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected], [email protected], [email protected], [email protected]

Abstract. The authors present in this paper an original method of calculus and choosing the actuators of the robot’s modules. The novelty of this method resides in the fact that the whole robot dynamics and the mechanical structure of each module is considered. The paper presents the stages to be passed for the correct application of this method. First, following the kinematic diagram of the TRTTR1 modular serial robot, the dynamic equations of the robot are determined, using the Lagrange’s equations of the second kind. Then, constructive solutions conceived by the authors are adopted, for the translation, rotation and orientation modules and relations for the driving forces and moments at the output of the modules are obtained. These relations contain constructive mechanical parameters of the modules, as well as the power, rotation and moment of the actuators. By introducing these relations in the dynamic equations of the robot, the expressions of the actuators moments are obtained. Based on these expressions, the graphs of variation of these moments are drawn, imposing trapezoidal variation laws for linear and angular velocities and including the numerical values of the constructive mechanical parameters. From the moments variation graphs, the maximum values of the actuators’ moments are chosen, then, from an actuators catalogue the proper actuators are chosen. Key words: robot, dynamic equations, actuator, mechanical parameters

1 Introduction The dynamic study of industrial robots allows solving of the following problems: • choosing the actuators; • optimal modules arrangement in a robot structure such as the energy consumption to be minimal; • choosing the laws of motion of the robot’s axes in such a way that the energy consumption to be minimal. The kinematic diagram of the TRTTR1 modular serial robot is presented in Fig. 1. The robot is in the stage of design and it will be proposed to be produced and implemented at S.C. Comelf S.A., Bistrit¸a, Romania. Using Lagrange’s equations of the second kind [6]

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_76, © Springer Science+Business Media B.V. 2010

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Fig. 1 The kinematic diagram of the TRTTR1 robot.

d dt




 ∂ Ec ∂ Ec − = Qk , ∂ q˙k ∂ qk

k = 1, . . . , 5,

(1)

the steps for obtaining the differential equations of motion were presented in [3]. They are the following: 1. The determination of the kinetic energy Ec of the robot 2. The determination of the derivatives:
 d ∂ Ec ∂ Ec ∂ Ec , k = 1, . . . , 5 , , ∂ qk ∂ q˙k dt ∂ q˙k 3. The determination of the generalized forces Qk . 4. The replacing of the results 2 and 3 in the relation 1 and the obtaining of the system (2) of differential equations.   6

∑ mi

i=1

q¨1 + [m4 (l4 + q4) + m5 (l4 + l5 + q4 ) + m6(l4 + l5 + q4 )] cq2 q¨2

Choosing the Actuators for the TRTTR1 Modular Serial Robot

 +

6

667



∑ mi

sq2 q¨4 − [m4(l4 + q4 ) + m5(l4 + l5 + q4 )

i=4



+ m6 (l4 + l5 + l6 + q4) sq2 ]q˙22 + 2

5



∑ mi

cq2 q˙2 q˙4 = F1 + F4sq2 ,

i=4



[m4 (l4 + q4 ) + m5(l4 + l5 + q4 ) + m6(l4 + l5 + l6 + q4 )] cq2 q¨2 + 6

+

3

∑ J∆(i)2 +

i=2



∑ Jz(i)i + m4(l4 + q4)2 + m5(l4 + l5 + q4)2 + m6(l4 + l5 + l6 + q4 )

2

q¨2

i=4



+ 2[m4 (l4 + q4 ) + m5(l4 + l5 + q4) + m6 (l4 + l5 + l6 + q4 )] q˙2 q˙4 = M2 ,  6

6

∑ mi q¨3 − m6l6 q¨5 = F3 − ∑ Gi ,

i=3



6

∑ mi

i=3



(q¨1 + q¨4 + cq2 q˙1 q˙2 ) − [m4 (l4 + q4) + m5 (l4 + l5 + q4)

i=4



+ m6 (l4 + l5 + l6 + q4 )] q˙22 −

6



∑ mi

cq2 q˙1 q˙2 = F4 ,

i=4

−m6 l6 q¨3 + [J∆(5) + Jy(6) + m6 l62 ] q¨5 = M5 . 5

6

(2)

2 The Calculus of the Actuators In order to calculate and choose the actuators an original method is used, considering the dynamics of the whole robot. This method involves passing the following steps.

2.1 Establishing the Relations between the Driving Forces and Moments from the Output of the Robots Modules and the Moment of the Actuators of Each Module The structure of these relations depends on the construction of each module. Thus, for the translation module 1 from the base of the robot (Fig. 1), the construction described in [4] was used, where the following relations hold:

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F1 =

2 · 103 ηr1 Mm1 ,  s 1 do1 tg ϕ1 + arctg d sin ϕ b1

ϕ1 = arctg

ph

1

π · d o1

,

(3)

1

where ηr1 is the efficiency of a pair of bearings; Mm1 [N·m] represents the moment of the actuator of the module; do1 [mm] is the diameter of the ball screw cylinder where the balls centers are situated; ϕ1 [◦ ] represents the closing angle of the helix on the average cylinder; s1 [mm] is the rolling fiction coefficient; db [mm] is the 1 ball diameter; ϕ1 [◦ ] designates the contact angle between the ball and the running track; ph [mm] is the pitch of the ball screw. 1 For the rotation module 2 of the robot arm (Fig. 1), the construction presented in [2] was used, for which the following relation was established: M2 = Mm2 ηc2 ηr2 ηm2 ηs2 ic2 im2 is2 ,

(4)

where Mm2 [N·m] is the moment of the DC motor actuating the module; ηc2 , ηr2 , ηm2 , ηs2 are the efficiencies of the cylindrical gear, of a pair of bearings, of the worm gear and of the combined screw, with balls and rolls; ic2 , im2 , is2 are the transmission ratios of the cylindrical gear, the worm gear and the combined screw. For the vertical translation module 3 of the robot arm, a construction described in [1] was considered, where the following equations were deduced: F3 =

2 · 103 Mm3 ηc3 ηr3 ηm3 ic3 im3 ,
s 3 do3 tg ϕ3 + arctg d sin ϕ b3

ϕ3 = arctg

ph

3

π · d o3

.

(5)

3

In equations (5), Mm3 represents the moment of the DC motor actuating the vertical translation module, the other amounts having the significance of the amounts specified in equations (3) and (4). For the translation module 4 from the structure of the robot arm, a constructive variant presented in [1] was chosen, where relations similar to (5) were established for the driving forces F¯4 and the angle ϕ4 . The moment of the DC motor actuating the translation module is denoted by Mm4 , the other amounts having the same significance with those contained in (5). For the orientation module 5 of the gripper, a construction presented in [2] was adopted, where for the driving moment M¯ 5 , the following equations were obtained: M5 = Mm 5 ηc5 ic5 ,

(6)

where Mm5 [N·m] represents the moment of the DC motor actuating the orientation module; ηc5 represents the efficiency of the cylindrical gear; ic5 represents the transmission ratio of the cylindrical gear.

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2.2 Determination of the Moments of the Robot Modules Introducing (3), (4), (5) and (6) in the system (??) of the differential equations, the actuators’ moments yield. The expressions of these moments are: 
s1   do1 tg ϕ1 + arctg d sin 6 ϕ1

 b 1 mi q¨1 + m4 l4 + q4 Mm1 = ∑ 3 2 · 10 ηr1 i=1   6   + m5 l4 + l5 + q4 +m6 l4 + l5 + q4 cq2 q¨2 + ∑ mi sq2 q¨4 i=4

 

 − m4 l4 + q4 + m5 l4 + l5 + q4 +m6 l4 + l5 + l6 + q4 sq2 q˙22     5 6    + 2 ∑ mi cq2 q˙2 q˙4 − ∑ mi q¨1 + q¨4 + cq2q˙1 q˙2 − m4 l4 + q4 i=4

i=4



+ m5 l4 + l5 + q4



 + m6 l4 + l5 + l6 + q4 q˙22 −



6

∑ mi



  cq2 q˙1 q˙2 sq2 ,

i=4

(7)

Mm2 =

   1 m4 l4 + q4 + m5 l4 + l5 + q4 ηc2 ηr2 ηm2 ηs2 ic2 im2 is2  3 6   2 + m4 l4 + q4 + m6 l4 + l5 + l6 + q4 cq2 q¨2 + ∑ J∆(i) + ∑ Jz(i) i 2

i=2

Mm3

i=4

 2 

 q¨2 + 2 m4 l4 + q4 + m5 l4 + l5 + q4 + m6 l4 + l5 + l6 + q4    (8) + m5 l4 + l5 + q4 + m6 l4 + l5 + l6 + q4 q˙2 q˙4 ,   s3    do3 tg ϕ3 + arctg d sin 6 6 φ3 b3 = ∑ mi q¨3 − m6l6q¨5 + ∑ Gi , (9) 2 · 103ηc3 ηr3 ηm3 ic3 im3 i=3 i=3 

2

 do4 tg ϕ4 + arctg d Mm4 =

s4 b4 sin φ4





6

∑ mi



 q¨1 + q¨4 + cq2q˙1 q˙2

2 · 10 ηc4 ηr4 ηm4 ic4 im4 i=4

   − m4 l4 + q4 + m5 l4 + l5 + q4 + m6 l4 + l5 + l6 + q4 q˙22    3

−

6

∑ mi

i=4

cq2 q˙1 q˙2 ,

(10)

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670 Table 1 The constructive mechanical parameters. i

mi /Gi

li

doi

phi

dbi

ϕi

ηri

η ci

η mi

i ci

imi

1 2 3 4 5 6

50/490.5 43/421.8 41/402.2 30/294.3 8/78.5 15/147.2

0.36 0.46 0.25 0.04 0.50 0.25

0.03 – 0.0158 0.0127 – –

0.008 – 0.0035 0.0036 – –

0.0055 – 0.00198 0.00198 – –

30◦ – 45◦ 45◦

0.995 0.995 0.995 0.995 – –

– 0.97 0.97 0.97 0.97 –

– 0.78 0.78 0.78 – –

– 1 1 1 1 –

– 30 30 30 – –

Mm5 =

   1  −m6 l6 q¨3 + J∆(5) + Jy(6) + m6 l62 q¨5 . 6 ηc5 ic5 5

(11)

2.3 Numerical Data Specification The values of the generalized coordinates, velocities and accelerations are found in the graphs from Figs. 2–7, chosen according to [2, 5]. The initial values of the generalized coordinates are:

π rad. 2 (12) The values of the constructive mechanical parameters, according to [1, 4, 5] are presented in Table 1 and relations (13):

to = 0;

q10 = 0.3 m;

q20 = π rad;

q30 = 0.38 m;

J∆(2) = 0.608 kg · m2;

J∆(3) = 0.24 kg · m2;

Jz(5) = 0.007 kg · m2;

Jz(6) = 0.015 kg · m2;

2

5

Jy(6) 6

= 0.01 kg · m ; 2

2

6

ηs2 = 0.99;

Fig. 2 Generalized coordinates q1 , q3 , q4 .

is2 = 32;

q40 = 0 m;

q50 =

Jz(4) = 0.12 kg · m2; 4

J∆(5) = 0.013 kg · m2; 5

s1 = s3 = s4 = 8 · 10−6. (13)

Fig. 3 Generalized coordinates q2 , q5 .

Choosing the Actuators for the TRTTR1 Modular Serial Robot

Fig. 4 Generalized velocities q˙1 , q˙3 , q˙4 .

Fig. 6 Generalized accelerations q¨1 , q¨3 , q¨4 .

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Fig. 5 Generalized velocities q˙2 , q˙5 .

Fig. 7 Generalized accelerations q¨2 , q¨5 .

2.4 The Determination of the Characteristics of the Actuators Introducing in (7)–(11) the variation laws of the generalized coordinates, velocities [1] and accelerations from Figs. 2–7 and the values of the constructive mechanical parameters of the robot, the variation laws in time of the actuators’ moments are obtained, called the actuators’ characteristics.

2.5 Choosing the Actuators Regarding the figures of the variation laws in time, the maximum values of the actuators’ moments are chosen. Thus, Mm1 = 3.723 · 10−4 N · m;

Mm2 = 0.015 N · m;

Mm3 = 4.166 · 10−5 N · m;

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Mm4 = 8.274 · 10−6 N · m;

Mm5 = 3.089 N · m.

(14)

Then from actuators catalogue Siemens – Drehstrommotoren Kemmerich Elektromotoren, having the actuation moments just above the values (14) are chosen. The following actuators resulted: Mm1 = 0.45 N · m;

Mm2 = 0.45 N · m;

Mm4 = 0.45 N · m;

Mm5 = 3.5 N · m.

Mm3 = 0.45 N · m; (15)

3 Conclusions By combining motion differential equations of the robot with the relations established for the driving forces and moments, taking into account the mechanical structure of each module of the robot, an original method of calculus and choosing the robot’s actuators is obtained. Compared to other methods of choosing the actuators, existent in the literature, this method uses the whole robot dynamics and the real construction of each module of the robot. Working this way, the most proper actuators are chosen. Thus, the under dimensioning and the over dimensioning of these actuators are avoided.

References 1. Ispas, V. et al., Aspecte privind calculul s¸i construct¸ia modulelor de translat¸ie din structura robot¸ilor industriali modulari. In al-VIII-lea Simpozion Nat¸ional de Robot¸i Industriali, ClujNapoca, Vol. I, pp. 345–350, 1988. 2. Ispas, V., Manipulatoare s¸i robot¸i industriali, Editura Didactic˘a s¸i Pedagogic˘a, Bucures¸ti, 2004. 3. Ispas, V., Gui, R.M., and Ispas, Vrg., The dynamic equations of the industrial serial modular robot TRTTR1, Acta Technica Napocensis, I(53):7–12, 2010. 4. Petris¸or, S.M., Contribut¸ii la calculul s¸i construct¸ia modulelor de translat¸ie din structura mecanic˘a a robot¸ilor seriali modulari, PhD Thesis, Cluj-Napoca, 2009. 5. Spur, G., Auer, B.H., and Sinning, H., Industrieroboter, Carl Hauser Verlag, M¨unchen/Wien, 1979. 6. Voinea, R., Voiculescu, D., and Ceaus¸u, V., Mecanica, Editura Didactic˘a s¸i Pedagogic˘a, Bucures¸ ti, 1983.

Mechanics of Robots

Stiffness Modelling of Parallelogram-Based Parallel Manipulators A. Pashkevich, A. Klimchik, S. Caro and D. Chablat Institut de Recherche en Communications et en Cybernetique de Nantes, 44321 Nantes, France; Ecole des Mines de Nantes, France; e-mail: {anatol.pashkevich, alexandr.klimchik}@emn.fr, {stephane.caro, damien.chablat}@irccyn.ec-nantes.fr

Abstract. The paper presents a methodology to enhance the stiffness analysis of parallel manipulators with parallelogram-based linkage. It directly takes into account the influence of the external loading and allows computing both the non-linear “load-deflection” relation and relevant rankdeficient stiffness matrix. An equivalent bar-type pseudo-rigid model is also proposed to describe the parallelogram stiffness by means of five mutually coupled virtual springs. The contributions of this paper are highlighted with a parallelogram-type linkage used in a manipulator from the Orthoglide family. Key words: stiffness modeling, parallel manipulators, parallelogram-based linkage, external loading, linear approximation

1 Introduction In the last decades, parallel manipulators have received growing attention in industrial robotics due to their inherent advantages of providing better accuracy, lower mass/inertia properties, and higher structural rigidity compared to their serial counterparts [1, 2, 6]. These features are induced by the specific kinematic structure, which eliminates the cantilever-type loading and decreases deflections due to external wrehches. Accordingly, they are used in innovative robotic systems, but practical utilization of the potential benefits requires development of efficient stiffness modeling techniques, which satisfy the computational speed and accuracy requirements of relevant design procedures. Amongst numerous parallel architectures studied in robotics literature, of special interest are the parallelogram-based manipulators that employ special type of linkage constraining undesirable motions of the end-platform. However, relevant stiffness analysis is quite complicated due to the presence of internal closed kinematic chains that are usually replaced by equivalent limbs [5, 7] whose parameters are evaluated rather intuitively. Thus, the problem of adequate stiffness modelling of parallelogram-type linkages, which is in the focus of this paper, is still a challenge and requires some developments.

D. Pisla et al. (eds.), New Trends in Mechanism Science:Analysis and Design, Mechanisms and Machine Science 5, DOI 10.1007/978-90-481-9689-0_77, © Springer Science+Business Media B.V. 2010

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Fig. 1 Parallel manipulators with parallelogram-based linkages.

Another important research issue is related to the influence of the external loading that may change the stiffness properties of the manipulator. In this case, in addition to the conventional “elastic stiffness” in the joints, it is necessary to take into account the “geometrical stiffness” due to the change in the manipulator configuration under the load [4]. Moreover buckling phenomena may appear [11] and produce structural failures, which must be detected by relevant stiffness models. This paper is based on our previous work on the stiffness analysis of overconstrained parallel architectures [7, 8] and presents new results by considering the loading influence on the manipulator configuration and, consequently, on its Jacobian and Hessian. It implements the virtual joint method (VJM) introduced by Salisbary [10] and Gosselin [3] that describes the manipulator element compliance with a set of localized six-dimensional springs separated by rigid links and perfect joints. The main contribution of the paper is the introduction of a non-linear stiffness model of the parallelogram-type linkage and its linear approximation by 6x6 matrix of the rank 5, for which it is derived an analytical expression.

2 Problem Statement Let us consider a typical parallel manipulator that is composed of several kinematic chains connecting a fixed base and a moving platform (Figure 1a). It is assumed, that at least one chain includes a parallelogram-based linkage that may introduce some redundant constraints improving the mechanism stiffness. Following the VJM concept, the manipulator chains are usually presented as a serial sequence of pseudo-rigid links separated by rotational or translational joints of one of the following types: (i) perfect passive joints; (ii) perfect actuated joints, and (iii) virtual flexible joints that describe compliance of the links, joints and/or actuators. To derive a parallel architecture without internal kinematic loops, the parallelogram based linkage must be replaced by an equivalent bar-type element that has similar elastic properties. Thus, let us consider the VJM model of the kinematic parallelogram assuming its compliance is mainly due to the compliance of the longest links, which are oriented in the direction of the linkage (Figures 1b,c). In order to be

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677

precise, let us also assume that the stiffness of the original bar-elements is described by a 6 × 6 matrix Kb whose elements are identified using FEA-based modeling. In the framework of the VJM approach, the geometry of the parallelogram-based linkage can be descried by the following homogeneous transformations Ti = Tz (η d/2) Ry (qi1 ) Tx (L) Tx (θi1 ) Ty (θi2 ) Tz (θi3 ) Rx (θi4 ) Ry (θi5 ) Rz (θi6 ) Ry (qi2 ) Tz (−η d/2) Ry (−qi2 )

(1)

where η = (−1)i , Ti is a 4 × 4 homogenous transformation matrix, Tx , ..., Rz are the matrices of elementary translation and rotation, L, d are the parallelogram geometrical parameters, qi j are the passive joint coordinates, θi j are the virtual joint coordinates, i = 1, 2 defines the number of the kinematic chain, j identifies the coordinate number within the chain. It should be noted that using this notation, the closed loop equation can be expressed as T1 = T2 . For further computational convenience, the homogenous matrix equations (1) may be also rewritten in the vector form as t = gi (qi , θi ),

(2)

where vector t = (p, ϕ ) is the output frame pose, that includes its position p = (x, y, z)T and orientation ϕ = (ϕx , ϕy , ϕz ), vector qi = (qi1 , qi2 ) contains the passive joint coordinates, and vector θi = (θi1 , ..., θi6 ) collects all virtual joint coordinates of the corresponding chain. Using the above assumptions and definitions, let us derive the stiffness model of the parallelogram-based linkage.

3 Static Equilibrium Let us derive first the general static equilibrium equations of the parallelogram assuming that it is virtually divided into two symmetrical serial kinematic chains with the geometrical and elastostatic models t = gi (qi , θi ), τi = Ki θi , i = 1, 2 where the variables θi correspond to the deformations in the virtual springs, Ki is the spring stiffness matrix, and τi incorporates corresponding forces and torques. Taking into account that the considered mechanism is under-constrained, it is prudent to assume that the end-points of both chains are located at a given position t and to compute partial forces Fi corresponding to the static equilibrium. To derive equations, let us express the potential energy of the mechanism as E(θ1 , θ2 ) =

1 2

2

∑ θiT Kθ i θi

(3)

i=1

In the equilibrium configuration, this energy must be minimised subject to the geometrical constraints t = gi (qi , θi ), i = 1, 2. Hence, the Lagrange function is

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L(θ1 , θ2 , q1 , q2 ) =

1 2

2

2

i=1

i=1

∑ θiT Kθ i θi + ∑ λiT (t − gi(θi , qi ))

(4)

where the multipliers λi may be interpreted as the external forces Fi applied at the end-points of the chains. Further, after computing the partial derivatives of L(...) with respect to θi , qi , λi and setting the derivatives to zero, equations of the static equilibrium can be presented as JTθ i λi = Ki θi ;

JTqi λi = 0;

t = gi (qi , θi );

i = 1, 2

(5)

where Jθ i = ∂ gi (...)/∂ θi and Jqi = ∂ gi (...)/∂ qi are the kinematic Jacobians. Here, the configuration variables θi , qi and the multipliers λi are treated as unknowns. Since the derived system is highly nonlinear, the desired solution can be obtained only numerically. In this paper, it is proposed to use the following iterative procedure 


T Jθ i K−1 λi
t − gi + Jqi qi + Jθ i θi θ i Jθ i Jqi T
= ; θi
= K−1 θ i Jθ i λ i (6) JTqi 0 q
i 0 where the prime
corresponds to the next iteration. Using this iterative procedure, for any given location of the end-point t, one can compute both the partial forces Fi and the total external force, which allows to obtain the desired “force-deflection” relation F = f(t) for any initial unloaded configuration.

4 Stiffness Matrix To compute the stiffness matrix of the considered parallelogram-based mechanism, the obtained “force-deflection” relations must be linearized in the neighborhood of the static equilibrium. Let assume that the external forces Fi , i = 1, 2 and the endpoint location t of both kinematic chains are incremented by some small values δ Fi , δ t and consider simultaneously two equilibriums corresponding to the state variables (Fi , θi , qi , t) and (Fi + δ Fi , θi + δ θi , qi + δ qi , t + δ t). Under these assumptions, the original system (5) should be supplemented by the equations 

T Jθ i + δ Jθ i (λi + δ λi ) = Kθ i (θi + δ θi ) ;   Jqi + δ Jqi (λi + δ λi ) = 0;

(7)

t + δ t = gi (qi , θi ) + Jθ i δ θi + Jqi δ qi ; where δ Jθ i and δ Jqi are the differentials of the Jacobians due to changes in (θi , qi ). After relevant transformation, neglecting high-order small terms and expanding the 2 2 differentials via the Hessians of the function Ψi = gi (qi , θi )T λi : H(i) qq = ∂ Ψi /∂ qi , (i) 2 ∂ (i) 2 2 Hqθ = ∂ Ψi /∂ qi θi , Hθ θ = ∂ Ψi /∂ θi , the system of equations may be rewritten as

Stiffness Modelling of Parallelogram-Based Parallel Manipulators

679

Jθ i δ λi + H(i) δ qi + H(i) δ θi = Kθ i δ θi ; θθ qθ (i) Jqi δ λi + H(i) qq δ qi + Hqθ δ θi = 0; Jθ i δ θi + Jqi δ qi = δ t

(8)

After analytical elimination of the variable δ θi and defining k(i) = (Kθ −H(i) )−1 , θ θθ one can obtain a matrix equation



−1
 Jθ i · k(i) · JTθ i Jqi + Jθ i · k(i) · H(i) δ λi δt θ θ θq = T · (i) T (i) (i) (i) (i) Jqi + H(i) · k · J H + H · k · H δ qi 0 qq θi θ θ θq qθ qθ

(9)

which yields the linear relation δ λ = Kci δ t defining the Cartesian stiffness matrix Kci for each kinematic chain. Taking into account the architecture of the considered mechanism, the total stiffness matrix may be found as Kc = Kc1 + Kc2 . These expressions allow numerical computation of the Cartesian stiffness matrix for a general case, both for loaded and unloaded equilibrium configuration. However, in the case of unloaded equilibrium, the above presented equations may be essentially simplified: ⎡K ⎢ ⎢ ⎢ ⎢ ⎢ K p (q) = 2 ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

0

0

0 K22 0

0

0

K26

0

0

0

11

0 0

0 0

0

0

d 2 Cq2 K22 0 K44 + 4

0

0

0 K26 0

0 d 2 S2q K22 8

0 d 2 S2q K22

0 d

2

Cq2 4 0

K11

8 0 d 2 Sq2 K22 K66 + 4

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

(10)

where Cq = cos q, Sq = sin q, S2q = sin 2q, q = q1i (Figures 1b,c), Ki j are the elements of the 6 × 6 stiffness matrix of the parallelogram bars Kb which are assumed here to be the only elements of the linkage that posses non-negligible stiffness. In the following example, this expression is evaluated from point of view of accuracy and applicability to stiffness analysis of the parallelogram-based manipulators.

5 Illustrative Example To demonstrate validity of the proposed model and to evaluate its applicability range, let us apply it to the stiffness analysis of the Orthoglide manipulator (Figure 1a). It includes three parallelogram-type linkage where the main flexibility source is concentrated in the bar elements of length 310 mm. Using the FEA-based software tools and dedicated identification technique [9], the stiffness matrix of the bar element was evaluated as

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Fig. 2 Critical forces and buckling configurations of a bar element employed in the parallelograms of Orthoglide.

⎡ 2.20 · 104 ⎢ ⎢ ⎢ Kb = ⎢ ⎢ ⎢ ⎣

0

0

1.81 · 101

0

0

0

0

0

0

0

0

−2.84 · 10

0

0

0

0

0

7.86 · 10

0

1.25 · 10

0

0

3.48 · 104

0

0

1.25 · 104

0

2.66 · 106

0

0

0

0

5.85 · 105

1

3

0

⎤

−2.84 · 103 ⎥ 4

⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(11)

where for linear/angular displacements and for the force/torque are used the following units: [mm], [rad], [N], [N·mm]. This bar-element was also evaluated with respect to the structural stability under compression, and the FEA-modelling produced three possible buckling configurations presented in Figure 2. It is obvious that these configurations are potentially dangerous for the compressed parallelogram. However, the parallelogram may also produce some other types of buckling because of presents of the passive joints. For the parallelogram-base linkage incorporating the above bar elements, expression (11) yields the following stiffness matrix ⎡ 2.20 · 104 ⎢ ⎢ ⎢ Kp = 2 ⎢ ⎢ ⎢ ⎣

0

0

0

0

1.81 · 101

0

0

0

0

0

0

0

0

0

.0

0

0

0

0

0

⎥ ⎥ ⎥, 4 4 ⎥ 5.64 · 10 0 1.25 · 10 ⎥ ⎦ 0 2.64 · 107 0

0

−2.84 · 10

0

1.25 · 104

0

5.92 · 105

3

0

⎤

0

−2.84 · 103 ⎥

(12)

corresponding to the straight configuration of the linkage (i.e. to q = 0). Being in good agreement with physical sense, this matrix is rank deficient and incorporates exactly one zero row/column corresponding to the z-translation, where the parallelogram is completely non-resistant due to specific arrangement of passive joints. Also, as follows from comparison with doubled stiffness matrix (11) that may be used as a reference, the parallelogram demonstrates essentially higher rotational stiffness that mainly depends on the translational stiffness parameters of the bar (moreover, the rotational stiffness parameter K55 of the bar is completely eliminated by the passive joints). The most significant here is the parallelogram width d that explicitly presented in the rotational sub-block of K p . To investigate applicability range of the linear model based on the stiffness matrix (10), it was computed a non-linear “force-deflection” relation corresponding to the parallelogram compression in the x-direction. This computation was performed us-

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Fig. 3 Force-deflection relations and buckling configurations for parallelogram compression (modelling methods: VJM, FEA, and LIN – linear model with proposed stiffness matrix).

Table 1 Stiffness parameters of the Orthoglide manipulator for different assumptions concerning parallelogram linkage. Model of linkage

2x-bar linkage Parallelogram linkage (solid axis) Parallelogram linkage (flexible axis)

VJM-model

FEA-model

Ktran [mm/N]

Krot [rad/N]

Ktran [mm/N]

Krot [rad/N]

3.35 · 103

0.13 · 106

–

–

3.23 · 103

4.33 · 106

3.33 · 103

4.10 · 106

3.08 · 103

4.06 · 106

3.31 · 103

4.07 · 106

ing an iterative algorithm presented in Section 3. As follows from obtained results, the matrix (10) ensures rather accurate description of the parallelogram stiffness in small-deflection area. However, for large deflections, corresponding VJM-model detects a geometrical buckling that limits applicability of the matrix (10). Similar analysis was performed using the FEA-technique, which yielded almost the same “force-deflection” plot for the small deflections but detected several types of the buckling, with the critical forces may be both lower and higher then in the VJM-modelling. Corresponding numerical values and parallelogram configurations are presented in Figure 3. The developed VJM-model of parallelogram was verified in the frame of the stiffness modelling of the entire manipulator. Relevant results are presented in Table 1. They confirm advantages of the parallelogram-based architectures with respect to the translational stiffness and perfectly match to the values obtained from FEAmethod. However, releasing some assumptions concerning the stiffness properties of the remaining parallelogram elements (other than bars) modifies the values of the translational and rotational stiffness. The latter gives a new prospective research direction that targeted at more general stiffness model of the parallelogram.

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6 Conclusions The paper presents new results in the area of enhanced stiffness modeling of parallel manipulators with parallelogram-based linkage. In contrast to other works, it explicitly takes into account influence of the loading and allows both computing the non-linear load-deflection relation and detecting the buckling phenomena that may lead to manipulator structural failure. There is also derived an analytical expression for relevant rank-deficient stiffness matrix and proposed equivalent bartype pseudo-rigid model that describes the parallelogram stiffness by five localized mutually coupled virtual springs. These results are validated for a case study that deals with stiffness modeling of a parallel manipulator of the Orthoglide family, for which the parallelogram-type linkage was evaluated using both proposed technique and straightforward FEA-modeling.

Acknowledgement The work presented in this paper was partially funded by the Region “Pays de la Loire” (project RoboComposite).

References 1. Ceccarelli M. and Carbone G., A stiffness analysis for CaPaMan (Cassino Parallel Manipulator), Mechanism and Machine Theory, 37(5):427–439, 2002. 2. Company O., Krut S., and Pierrot F., Modelling and preliminary design issues of a 4-axis parallel machine for heavy parts handling, Journal of Multibody Dynamics, 216:1–11, 2002. 3. Gosselin C., Stiffness mapping for parallel manipulators, IEEE Transactions on Robotics and Automation, 6(3):377–382, 1990. 4. Kovecses J. and Angeles J., The stiffness matrix in elastically articulated rigid-body systems, Multibody System Dynamics, 18(2):169–184, 2007. 5. Majou F., Gosselin C., Wenger P. and Chablat D., Parametric stiffness analysis of the Orthoglide, Mechanism and Machine Theory, 42:296–311, 2007. 6. Merlet, J.-P., Parallel Robots, Kluwer Academic Publishers, Dordrecht, 2000. 7. Pashkevich A., Chablat D. and Wenger Ph., Stiffness analysis of overconstrained parallel manipulators, Mechanism and Machine Theory, 44:966–982, 2009. 8. Pashkevich A., Klimchik A., Chablat D. and Wenger Ph., Stiffness analysis of multichain parallel robotic systems with loading, Journal of Automation, Mobile Robotics & Intelligent Systems, 3(3):75–82, 2009. 9. Pashkevich A., Klimchik A., Chablat D. and Wenger Ph., Accuracy improvement for stiffness modeling of parallel manipulators. In Proceedings of 42nd CIRP Conference on Manufacturing Systems, Grenoble, France, 2009. 10. Salisbury J., Active stiffness control of a manipulator in Cartesian coordinates, in Proceedings of 19th IEEE Conference on Decision and Control, pp. 87–97, 1980. 11. Timoshenko S. and Goodier J.N., Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970.

Incorporating Flexure Hinges in the Kinematic Model of a Planar 3-PRR Parallel Robot R.J. Ellwood, D. Sch¨utz and A. Raatz Institute of Machine Tools and Production Technology, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany; e-mail: {j.ellwood, d.schuetz, a.raatz}@tu-bs.de

Abstract. In order to improve the absolute positioning abilities of the 3-PRR planar parallel robot M ICABO es, its kinematic model is adapted to take the predictable motion of flexure hinges into consideration. This is accomplished by replacing the idealized rotations within the kinematic model with a complexer rotational model. A set of kinematic equations is derived for the robot which is solved using an iterative method. A method for calibrating these equations is then explained and a simulation is used to show its success implementation. Key words: parallel robot, kinematic model, flexure hinge, compliant mechanism

1 Introduction Precision engineering and microassembly are constantly pushing the limits of available robots. Within this field it has been shown that parallel robots, at least in theory, should be able to outperform their serial counterparts. In an attempt to improve the repeatability of parallel robots, it has been shown that replacing joints which exhibit play, and have complex friction models with compliant mechanisms will improved the robots repeatability [2]. What has often been overlooked is the absolute positioning abilities of the resulting robots, which is often limited by a simplified kinematic model. These simplified models assume that the compliant mechanism will bend in a manor similar to that of an idealized rotation. Although it is possible to introduce different mechanics of materials properties into the kinematic model, the resulting equations would be too complex for real time implementation. Within this paper, the direct kinematic model (DKP) of a planar 3 degree of freedom (DOF) robot is extended to incorporate a coupled motion which is better able to model the flexure hinges. This coupled motion is realized through a modified version of that proposed by Kimball and Tsai in [5]. A discussion of these methods is, as well as its dependence on calibrated data is briefly discussed. This is followed by the derivation of the DKP for the robot, using the iterative Newton–Raphson method. A method to calibrate these kinematic equations is then presented and simulated to illustrate its ability to improve the absolute positioning of the robot.

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2 Planar 3-PRR Parallel Robot To accomplish tasks with increased precision requirements, the robot M ICABOes has been constructed at the Institute for Machine Tools and Production Technology of the Technische Universit¨at Braunschweig. Using typical revolute joints, this planar 3PRR tripod robot is able to obtain an approximate workspace of 30 × 30 mm2 and allows for 2 translational as well as an additional rotational DOF. Located underneath the tool center point (TCP) is a table which can move parts in the z direction. Due to structural limitations, the maximal rotation of the TCP is 40◦ . Driving the prismatic joints are piezo actuators which use the stick-slick effect to achieve a linear actuator. It is possible to eliminate the negative effects found within the joints by replacing these elements with compliant mechanisms. Through extensive research and design it has been found that the combination of a typical notched design and a memory shape allow composed of a copper aluminum nickel iron (CuAlNiFe) alloy are advantageous for this application [3]. The joint notches have been cut with a radius of 15mm, so that the thinnest point between them have a cross section of 0.15mm. Along with its superior deflection angle of ±30◦ , it has been shown that these hinges have approximately 1.5 · 105 cycles to failure [2]. In order to allow for a quantitative comparison of the two structures, a repeatability test according to the EN ISO 9283 has been conducted with each of the structures. It has been found that the original structure with normal joints has a repeatability of 5 µm. Conducting the same test with the structure with flexure hinges resulted in a repeatability of 3 µm.

3 Rigid Body Kinematic Model Although the introduction of flexure hinges improves the repeatability of the robot, the absolute positioning degrades proportionally to the magnitude of the deflection angle of the compliant mechanism. This lose in accuracy at increased angles is a direct result of the simplified joint model used in the direct kinematic problem. In order to quantify this, the idealized rotation model is compared with deflection data of a single flexure hinge collected using the flexure hinge test bench described in [2]. The magnitude of this difference vector between these two sets of data is thus 0.192 mm, clearly illustrating the downside of this simplified model. The parameters which will be used throughout the rest of the paper are illustrated in Fig. 1.

3.1 Flexure Hinge Model To counter this weakness, an improved rigid body kinematic model of flexure hinges is sought. The most promising of these models has been proposed by Kimball and Tsai, in which an additional joint is introduced [5]. As can be seen in Fig. 2(a), this

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Fig. 1 The M ICABO es illustrated with labeled parameters.

results in an additional length with the 1 DOF being maintained by coupling the joints with a angular correction factor k. The intermediate angle α is defined as a function of the total angle τ such that

τ = α · (1 + k).

(1)

In order to take full advantage of this new joint model, the unknown length parameters l1 , l2 , and l3 as well s the angular correction value k need to be found. Though somewhat trivial in a single joint case, when applied to the robot at hand there is a 12 parameter increase in the entire calibration process. To limit this, a modified version of the Kimball and Tsai method in which the angular correction value is set to 1 is used. Before this is applied to the entire robot model, the simple one hinge case is again explored. First the modified model is calibrated using the same method as will be discussed in Section 4, again using data from the single hinge test setup. The magnitude of the resulting difference vector is then found to be less than 1 µm, the sensor sensitivity, throughout the entire ±30◦ deflection range. Confidence gained through the simplified case rationalizes the extension of this method to the DKP for the M ICABOes . Due to the complexity of the proposed kinematic model the Newton–Raphson method is used to find the 9 unknowns, the coordinates of the TCP x, as well as the arm angles α and β respectively. This is

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y ki b i

l3

ei fi

ka l2

bi

gi

di

ai

t

bi

a l1

ai

ci

ai

x

(a) Kinematic model proposed by Kimball and Tsai.

gi

(b) Application of modified Kimball method.

Fig. 2 Kinematic models of a single joint, and applied to a joint.

accomplished by defining three vector loops which start at the origin, with each of the chains following a respective arm to the platform. It is then crucial that these 3 vector loops cross in the middle of the platform, as this enables the platform variables to be scalable during the calibration. These vectors are then combined to form vector Φ , where Φ is a function of the unknowns x, α , and β , as well as the drive coordinates q and the parameters k. A picture of the ith kinematic chain, where i is one of the three arms can be seen in figure 2(b). The TCP for loop 1 in Fig. 1 can be said to be: xTCP = q1 cos(ε1 ) + a1 cos(γ1 + ε1 ) + b1 cos(α1 + γ1 + ε1 ) + c1 cos(2α1 + γ1 + ε1 ) + d1 cos(2α1 + β1 + γ1 + ε1 ) + e1 cos(2α1 + 2β1 + γ1 + ε1 ) + f 1 cos(2α1 + 2β1 + γ1 + ε1 + π + κ1 ) π + g1 cos(2α1 + 2β1 + γ1 + ε1 + + κ1 ) 2

(2)

yTCP = q1 sin(ε1 ) + a1 sin(γ1 + ε1 ) + b1 sin(α1 + γ1 + ε1 ) + c1 sin(2α1 + γ1 + ε1 ) + d1 sin(2α1 + β1 + γ1 + ε1 ) + e1 sin(2α1 + 2β1 + γ1 + ε1 ) + f 1 sin(2α1 + 2β1 + γ1 + ε1 + π + κ1 ) π + g1 sin(2α1 + 2β1 + γ1 + ε1 + + κ1 ). 2

(3)

The six equations resulting from the 3 loops have 9 unknowns, and thus additional information is needed. This can be found when the kinematic constraint are considered, which are easily seen in the rest position of the robot. In this condition, which is illustrated in Fig. 1, both αi and βi will be zero and thus

φ = γi + 2αi + 2βi − κi βi = (φ − γi − 2αi + κi )/2.

(4) (5)

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In order to solve these equations, the partial derivative J DKP for the resulting Φ vector is taken with respect to ξ where ξ = [xT , α T ]T . With all the pieces, it is then possible to apply the Newton–Raphson method [4] such that

ξ j+1 = ξ j − J DKP (ξ j )−1 · Φ (ξ j ).

(6)

4 Parameter Calibration of the M ICABOes The presented modified Kimball and Tsai approach allows the description of the kinematic behavior of the M ICABOes with flexure hinges, with a minimum increase in additional parameters. Due to the nature of the model, a parameter calibration is required to find and use the correct geometric parameter values within the model. A parameter calibration in typically conducted in four successive steps. First, a kinematic model must be obtained which includes all necessary kinematic parameters and is extended by non-redundant parameters for adjusting possible production as well as assembly errors. The nominal geometric parameter values are defined by the construction data and are used within the IKP to calculate the drive angles for given TCP poses. Subtracting the given poses xgiven from the actual poses xactual , which must be measured, yields the residual vector r. The objective in the following identification portion is to minimize the components of r by optimizing the values of the parameter vector knominal . The final step in the parameter calibration process is the implementation of the found parameter values into the robot robot. This optimized model will then result in an improvement in the accuracy [1, 7].

4.1 Simulation of Measurement Data The validation of the calibration method which has been established for the structure of the M ICABOes can be achieved by simulating measurement data. Here the drive displacements at a selection of endeffector poses are calculated using the nominal parameters knominal within the inverse kinematic problem (IKP) qgiven = IKP(xgiven , knominal ).

(7)

The actual endeffector poses can then be obtained by computing the DKP with these given drive angles and a modification of the nominal parameter values knominal , which can be written as kmodi f ied xactual = DKP(qgiven , kmodi f ied ).

(8)

The vector xactual , which includes the coordinates of several poses, represents the result of a measurement procedure. At this point it is possible to add measurement noise, which can then test the robustness of the identification process.

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4.2 Identification by Optimization An integral portion of the calibration process is the development of a cost function F. The function F, is then the value which needs to be minimized within an optimization algorithm. Here it is a function of the residual vector r, which is the difference of the given poses xgiven to xactual . It is then made scalar by multiplying it with itself as shown in Eq. (9). (9) R = rT · r As the parameter correction between the nominal parameters knominal to the optimized or the real parameters koptimized is small, the well-known least square optimization approach of Levenberg and Marquardt (LMA) is often used for nonlinear optimization applications. The general procedure of this optimization method is presented, for instance, by Scales [6]. In Equation 10 the necessary inputs and outputs of the LMA are abstractly shown. koptimized = LMA(knominal , xactual )

(10)

Thereby, an identification Jacobian Je must be obtained, which defines the relationship between the residual vector r and the parameter adjustments δ k. This identification Jacobian Je can be computed using a simple finite difference method (FDM), which must be rebuilt in every iterative cycle of the LMA [6].

4.3 Implementation of the Optimized Parameters After the optimization algorithm, the optimized parameters koptimized can be implemented into the model of the robot. If no measurement noise is implemented within the simulation, the optimized parameters should have the same values as the modified parameters. The quality of the optimized parameters can then be checked by calculating the drive displacements for given poses and comparing these with results of an additional measurement.

4.4 Simulation Results For the simulated measurement 27 of the 35 regarded parameters of the M ICABOes structure are modified within a range of ±0.5mm and ±0.5◦ respectively. The 8 parameters which are not modified are also held constant within the LMA, as their interation would effect the scale of the whole robot structure. In order to obtain repeatable results, their orientation with respect to the base frame must not be varied. The calculated measurement data of the M ICABOes has been added with a random measurement noise of ±1 µm. This range represents the resolution of a suitable measurement system for the calibration of this robot class. The parameter calibra-

Incorporating Flexure Hinges in the Kinematic Model of a Planar 3-PRR Parallel Robot

0.35

8 7

0.25

6

0.2

−4

optimized

−x

given

5 4

i

0.15

x 10

x

0.3

|x | in mm

i

|x | in mm

xmodified − xgiven

689

3 0.1

2

0.05 0 0

1 10

20 30 40 Pose number i

50

0 0

60

(a) Before the parameter calibration.

10

20

30 40 Pose number i

50

60

(b) After the parameter calibration.

Fig. 3 Position derivations with a simulated measurement noise of ±1 µm.

5

x

modified

8

−x

given

x 10

−4

x

optimized

−x

given

7

4

3 |φ | in °

5 i

i

|φ | in °

6

2

4 3 2

1

1 0 0

10

20 30 40 Pose number i

50

(a) Before the parameter calibration.

60

0 0

10

20

30 40 Pose number i

50

60

(b) After the parameter calibration.

Fig. 4 Orientation derivations with a simulated measurement noise of ±1 µm.

tion algorithm yields a set of optimized parameters which leads to an improvement of the absolute pose accuracy. As shown in Fig. 3, the position accuracy before calibration was about 0.31mm and has been reduced to 6.4·10−4mm. In order to prevent a unit bias, the resulting position coordinates and orientation angle φ are separated. The resulting position of each pose can be seen in Fig. 3. The derivation error of the orientations is about 4.5◦ before the calibration procedure and through the optimization process, an angular accuracy of (7.5·10−4)◦ can be obtained. The results of the same 60 test poses are presented in Fig. 4.

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5 Conclusion and Outlook The introduction of flexure hinges within the parallel robot M ICABOes has improved the repeatability of the robot, with the side effect of a kinematic model degradation proportional to the magnitude of deflection. To overcome this, a new set of equations for the direct kinematic model of the robot is developed. This new model replaces the idealized rotations with a coupled angle model. The model is briefly described for a simple flexure hinge, then applied to the robot. Through three vector loops, the direct kinematic model is developed and solved using the iterative Newton–Raphson method. In order to take full advantage of this new model, a calibration process to find the unknown lengths is discussed. A simulation of the parameter calibration is then achieved by modifying the parameters and generating data using the inverse kinematic model. The successful implementation of the calibration method is then shown through the identification of the parameter changes. It is foreseen that data will be gathered from the actual system using a spacial measurement system such as a laser tracer for use in the calibration process. The resulting accuracy of the calibrated kinematic model of the robot is then dependent on the sensor resolution.

Acknowledgement The research reported here has been supported by the German Research Foundation (DFG) within the scope of both the Collaborative Research Center SFB 516 and 562.

References 1. Bernhardt, R. and Albright, S.L., Robot Calibration. Chapman & Hall, London, 1993. 2. Hesselbach, J. and Raatz, A., Pseudo-elastic flexure-hinges in robots for micro assembly. In Proceedings SPIE Microrobotics and Microassembly II, pp. 157–167, 2000. 3. Howell, L.L., Compliant Mechanisms. Wiley, New York, NY, 2001. 4. Kerle, H. and Pitschellis, R., Einfuehrung in die Getriebelehre. B.G. Teubner, Stuttgart, 2002. 5. Kimball, C. and Tsai, L.-W., Modeling of flexural beams subjected to arbitrary end loads. Journal of Mechanical Design, 124:223–235, 2002. 6. Scales, L.E., Introduction to Non-Linear Optimization. Macmillan Computer Science Series. Macmillan, London, 1985. 7. Zhuang, H. and Roth, Z., Method for kinematic calibration of stewart platforms. Journal of Robotic Systems, 10(3):391–405, 1993.

On the Dynamics of a 5 DOF Parallel Hybrid Robot Used in Minimally Invasive Surgery D. Pisla, B.G. Gherman, M. Suciu, C. Vaida, D. Lese, C. Sabou and N. Plitea Department of Mechanics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania; e-mail: [email protected]

Abstract. Considering the latest results in robotic assisted surgery it is obvious that robots became useful tools in minimally invasive surgery (MIS), providing benefits such as reduction in hand tremor, the possibility of reaching positions and places that could prove difficulty in reaching using classical instruments, navigation and workspace scaling. In this paper the inverse dynamic model for a parallel robot conceived for MIS using virtual work principle is presented. The results of the analytic dynamic model have been validated by means of a Multi-Body simulation. Key words: parallel hybrid robot, minimally invasive surgery, kinematics, dynamics, simulation

1 Introduction Parallel hybrid robots are complex multi-body systems that are difficult to model because of their closed loops added to opened ones. Dynamic modeling is essential for design specifications and advanced control of parallel hybrid robots. To solve the dynamic model, Merlet uses Lagrange formulas for the direct and the inverse dynamic model for the “left hand”, to a prototype accomplished at INRIA based on a KPS kinematic chain structure in [8]. The virtual displacement and the d’Alembert principle was used in [6] to achieve the static and dynamic analysis for a 4 DOF parallel mechanism to obtain torque references of the actuator. Zakerzadeh studies the inverse kinematics and dynamics of a new 4-DOF hybrid (serial-parallel) manipulator used for pole climbing in [13]. The hybrid manipulator consists of a one-DOF rotary mechanism in series with a 3-DOF planar 3-R PR parallel mechanism, providing 2 translational and 2 rotational degrees of freedom for the pole climbing robot. The inverse dynamic formulation is presented by the Newton–Euler approach for the robot. A generalized dynamic model for parallel robots using first order Lagrange equations on the basis of equivalent masses is proposed in [9]. A comparative study between different dynamic methods for 6-DOF parallel robots is presented in [5]. Concerning the application of robots for surgical applications, Shoham has performed extensive studies regarding: the performances of the surgeon in endoscopy,

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focused on the fulcrum effect [1]; the development of new miniaturized robots which can be mounted on the bones and a research regarding the different methods for scanning and surface recognition of bones before the surgical intervention [3]. LAPMAN [12] is a dynamic laparoscope holder with three degrees of freedom, guided by a joystick clipped onto the laparoscopic instruments. Just like other camera holders, it confers an optimal control over the visual field, the elimination of the human tremor, it helps to reduce the number of assistants in the operating room. The paper is organized as follows: Section 1 is dedicated to the description and design of the studied hybrid parallel robot for MIS. Section 2 deals with the dynamics of the PARASURG 5M robot with several explanations regarding the kinematics. Section 3 presents the dynamic model of the hybrid parallel robot using the virtual work principal. Section 4 presents the simulation dynamics results, which fit to the experimental model, using both Matlab and Adams. The conclusions are presented at the end of the paper.

2 Design Consideration of 5-DOF Parallel Robot for MIS PARASURG 5M has 5 degrees of freedom: three degrees of freedom for general positioning of the camera and a two DOF orientation module. The robot presented in [10] has just about the same structure, the only difference being the orientation module. In that case, the laparoscope will entirely rely its weight on the abdominal wall of the patient, putting a lot of pressure on the muscles and to avoid this inconvenient, an orientation module with two degrees of freedom was added. In the same time, the structure presented in this paper is a developing project, meaning that the camera will be replaced by an active instrument used, among other, for cutting, suturing, grasping, etc. Thus the orientation module becomes absolutely necessary. The robot structure is based on an already registered patent [11]. The robot has five actuated joints (two prismatic and three rotations, Figure 1a). There are also the following passive joints: two cylindrical joints (between links (1) and (3), respectively between links (2) and (3)), two prismatic joints (between links (4) and (3), respectively (5) and (3)), three rotary joints (between links (4) and (6), (5) and (7) respectively (6) and (7)). At the end of the element (7) there is the motor (10) which represents the q4 active coordinate and is mounted on element (9). On the output shaft of the motor (10) there is mounted the element (12) on which there is mounted the motor (11), denoting the q5 active coordinate. At the end of the element (12) there is mounted the element (8), representing the laparoscope (Figure 1). The geometrical parameters of the parallel robot are represented by b, h and the coordinates of point B, the incision point in the abdominal wall: XB , YB , ZB . In Figure 2 is presented the CAD model of the robot. In order to have a rotation and translation along the Z axis, a secondary shaft along the main guiding shaft was added. The overall dimensions of the robot are: 1000 mm × 1000 mm × 300 mm. Concerning the actuation, for PARASURG 5M a compact solution was achieved by using the intelligent actuator MCD EPOS 60W, which includes also a PID controller, for the

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(a) The robot structure.

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(b) The rotation angles psi and theta.

Fig. 1 The PARASURG 5M robot for MIS.

motors situated at the base of the robot (q1 , q2 and q3 ) [7]. The gear heads were selected based on the required displacements for each axis: 1:26 for the translation modules (q1 and q2 ) and 1:156 for the base rotation module. Smaller motors were chosen for the coordinates q4 and q5 : 6W DC motors, planetary gearheads with a ratio of 1:4592, encoder and epos controller. The actuator has command interfaces for RS-232 and CANOpen, allowing both the direct connection to a computer (for setup and calibration) and to a CAN network. The CAN module has been selected because using a single PLC a user can control up to 127 devices with a speed of 1 Mbit/s for cable length up to 40m. This means that additional systems (robotic arm for instrument manipulation) can be added to the system without the need of another PLC or compatibility issues. The PLC is provided by BR Automation [2]. The CPU has a 650 MHz processor, and 64 SDRAM communicating with the PC through an Ethernet link. The structure of the PARASURG 5M parallel robotic system is illustrated in Figure 3. The control input allows the user to give commands for the positioning of the laparoscope using different interfaces: joystick, microphone, keyboard, mouse and haptic device. The control processing stage is achieved within a PC that provides the user interface and processes the data from the input controls, or in other words the commands given by the user. Based on the parallel robot kinematics and its current position the motion parameters are calculated. As a safety measure, each displacement is first calculated, verified and validated and only afterwards the actual motion of the robot is permitted. The calculated motion parameters are transmitted via Ethernet to the PLC where the instructions generation takes place. Only the kinematics was implemented until now in the control algorithm.

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Fig. 2 PARASURG 5M as laparoscope holder.

Fig. 3 The structure of the PARASURG 5M robotic system.

3 Dynamics 3.1 Kinematics For the inverse geometric model, we have as inputs the position of the tip of the laparoscope (point G(XG ,YG ,ZG )) and point B(XB ,YB ,ZB ), angles ψ and θ being determined from Figure 1b and we have to find the active coordinates: q1 , q2 , q3 , q4 and q5 . Two cases have to be considered: Case 1. If XA =XB and YA =YB , when ψ = 0 and θ = 0. Case 2. If XA
=XB and YA
=YB , it yields:
ψ = arctan 2(YB − YA , XB − XA), θ = arctan 2( (XB − XA)2 + (YB − YA )2 , ZA − ZB ), (1) For both cases, the joint coordinates are obtained:

Dynamics of a 5 DOF Parallel Hybrid Robot Used in Minimally Invasive Surgery


q1 = ZA = ZG + h, q2 = q1 + (2 · b)2 − (XG2 + YG2 ), q3 = arctan 2(YG , XG ), ⎧ sin(q4 ) = (− cos(α2 ) · sin(θ ) · sin(ψ − q3 )) + sin(α2 ) · cos(θ ), ⎪ ⎪ ⎪ ⎨cos(q ) = − sin(α ) · sin(θ ) · sin(ψ − q ) − cos(α ) · cos(θ )), 4 2 3 2 ⎪ sin(q ) = − sin( θ ) · cos( ψ − q ), ⎪ 5 3 ⎪ ⎩ cos(q5 ) = sin((q4 ) + α2 ) · sin(θ ) · sin(ψ − q3) + cos(q4 + α2 ) · cos(θ ),

695

(2)

(3)

3.2 The Dynamic Model To obtain the dynamic model of the PARASURG 5M robot, the virtual work principle was used. It has the following form: n

δ W = δ qT · τ + ∑ (δ XiT · (TiTr + Tig )),

(4)

i=1

where δ XiT = [δ Xi , δ Yi , δ Zi ]T is the vector of virtual displacements for each point i of mass mi , TiTr = [−mi · X¨i , −mi · Y¨i , −mi · Z¨i ]T , where XMi = [X¨i , Y¨i , Z¨i ]T is the acceleration vector of each mass point, Tig = [0, 0, −mi · G]T , τ = [F1 , F2 , M3 , M4 , M5 ]T and δ qT = [δ q1 , δ q2 , δ q3 , δ q4 , δ q5 ]T , G being the gravitational acceleration. From Equation (4) the vector τ can be obtained. Further, we assume that the linear motors have the masses mA and mB and the rotary motor corresponding to q3 and the assemble formed by the elements (1), (2), (3) and (4) have the mass: m1 , the other rotary motors having the mass m7 and m9 . Elements (5), (6), (7), (8), (9), (12), (13), (14) have the masses m2 , m3 , m4 , m5 , m6 , m8 , m9 and the mass of the laparoscope: m10 (Figure 4). The other mechanism links are considered homogenous bars having the length: lb and l2b , l2b , l1 and h; the joints have the masses: mC , mD , mE . For the dynamic model, the theory of the lumped masses was applied. As in [4], simplifying hypothesis were used to obtain the equivalent system represented by mass points. The equivalent masses of the system being mi , i = 1, . . . , 21: m∗1 = m1 ·i2s1 c11 ·l1 ,

m∗6 =

m1 ·i2s1 ∗ c12 ·l1 , m3 i2 m5 (1 − c s2·c ), 21 22

m∗2 =

m5 ·i2s2 m5 ·i2s2 1 ∗ c21 ·l1 + mA , m5 = c22 ·l1 + mC + 6 m7 , m6 ·i2s3 m6 ·i2s3 1 ∗ ∗ c31 ·l1 + mB , m8 = c32 ·l1 + mD + 6 m8 , m9 = m6 · (1 −

= m1 (1 − c

i2s1

11 ·c12

m∗7 =

), m∗4 =

i2s3 2 1 1 2 ∗ ∗ ∗ ∗ ∗ ∗ c31 ·c32 ), m10 = 3 m7 , m11 = 6 m7 , m12 = 6 m8 , m13 = 3 m8 , m14 = m9 , m15 = m10 , m∗16 = m12 , m∗17 = m11 , m∗18 = m13 , m∗19 = 16 m14 , m∗20 = 23 m14 , m∗21 = 16 m14 . The

coordinates of equivalent mass points Xi , i = 1, . . . , 21 depend on joint coordinates q1 , q2 , q3 , q4 , q5 . X1 = 0, Y1 = 0, Z1 = s1 , X2 = l1 · cos(q3 ), Y2 = l1 · sin(q3 ), Z2 = s1 , X3 = c11 · cos(q3 ), Y3 = c11 · sin(q3 ), Z3 = s1 , X4 = 0, Y4 = 0, Z4 = q1 , X5 = l1 · cos(q3 ), Y5 = l1 · sin(q3 ), Z5 = q1 , X6 = c21 · cos(q3 ), Y6 = c21 · sin(q3 ), Z6 =

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Fig. 4 Dynamic modeling of the PARASURG 5M robot using lumped masses.

q1 , X7 = 0, Y7 = 0, Z7 = q2 , X8 = l1 · cos(q3 ), Y8 = l1 · sin(q3 ), Z8 = q2 , X9 = √ (2·b)2 −(q2 −q1 )2 + l1 ) · cos(q3 ), c31 · cos(q3 ), Y9 = c31 · sin(q3 ), Z9 = q2 , X10 = ( 4 √ √ (2·b)2 −(q2 −q1 )2 (2·b)2 −(q2 −q1 )2 3·q1 +q2 + l1 ) · sin(q3 ), Z10 = 4 , X11 = ( + l1 ) · Y10 = ( 4 √ 2 (2·b)2 −(q2 −q1 )2 q +q + l1 ) · sin(q3 ), Z11 = 1 2 2 , X12 = ((2 · b − l2 ) · cos(q3 ), Y11 = ( 2 cos(α2 ) + l1 ) · cos(q3 ), Y12 = ((2 · b − l2 ) · cos(α2 ) + l1 ) · sin(q3 ), Z12 = q2 − (2 · (2·b−l2 )·cos(α2 ) (2·b−l2 )·cos(α2 ) + l1 ) · cos(q3 ), Y13 = ( + l1 ) · 2 2 (2·b−l2 )·sin(α2 )  , X14 = ((2 · b − l2 ) · cos(α2 ) + l2 + l1 ) · cos(q3 ), sin(q3 )4, Z13 = q2 2 Y14 = ((2 · b − l2 ) · cos(α2 ) + l2 + l1 ) · sin(q3 ), Z14 = q2 − (2 · b − l2 + l2 ) · sin(α2 ),

b − l2 ) · sin(α2 ), X13 = (

X15 = XA + l3 · sin(q3 ), Y15 = YA − l3 · cos(q3 ), Z15 = q1 , X16 = XA + is4 · sin(q3 ), Y16 = YA − is4 · cos(q3 ), Z16 = q1 , X17 = XA − l4 · cos(q4 + α2 ) · cos(q3 ), Y17 = YA − l4 · cos(q4 + α2 ) · sin(q3 ), Z17 = q1 + l4 · sin(q4 + α2 ), X18 = cos(q5 ) · cos(q3 ) ·
cos(q4 + α2 ) · sin(α3 ) + sin(q4 + α2 ) · cos(α3 ) − sin(q3 ) · sin(q5 ) l52 + (l5 )2 + XA , Y18 = cos(q5 ) · sin(q3 ) · cos(q4 + α2 ) · sin() + sin(q4 + α2 ) · cos(α3 ) + cos(q3 ) ·
sin(q5 ) l52 + (l5 )2 + YA , Z18 = q1 − cos(q5 ) · (− cos(q4 + α2 ) · cos(α3 )) + sin(q4 +
α2 ) · sin(α3 ) · l52 + (l5 )2 , X19 = l6 · (cos(q3 ) · cos(q4 ) · cos(q5 ) − sin(q3 ) · sin(q5 )) +

XA , Y19 = l6 ·(sin(q3 )·cos(q4 )·cos(q5 )+ cos(q3 )·sin(q5 ))+YA , Z19 = −l6 ·sin(q4 + α2 ) · cos(q5 ) + q1, X20 = (l6 + h2 ) · (cos(q3 ) · cos(q4 ) · cos(q5 ) − sin(q3 ) · sin(q5 )) + XA , Y20 = (l6 + h2 ) · (sin(q3 ) · cos(q4 ) · cos(q5 ) + cos(q3 ) · sin(q5 )) +YA , Z20 = −(l6 + h 2 ) · sin(q4 + α2 ) · cos(q5 ) + q1 , X21 = (l6 + h) · (cos(q3 ) · cos(q4 ) · cos(q5 ) − sin(q3 ) · sin(q5 )) + XA , Y20 = (l6 + h) · (sin(q3 ) · cos(q4 ) · cos(q5 ) + cos(q3 ) · sin(q5 )) + YA , Z20 = −(l6 + h) · sin(q4 + α2 ) · cos(q5 ) + q1,

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Fig. 5 Dynamics of the PARASURG 5M robot using MATLAB.

where s1 is the Z coordinate of the  mass center of the assembly formed by the elements (1), (2), (3), (4), XA = (2 · b)2 − (q2 − q1)2 + l1 ) · cos(q3 ),YA =  I  (2 · b)2 − (q2 − q1 )2 + l1 ) · sin(q3 ), isi = ( m∆ i ), i = 1, 2, 3, 4, I∆ i being the inertia i moment, as obtained from SolidWorks software, α3 = arctan(l5 /l5 ), l2 is the distance from the end of element (8) (Figure 4) to point A, l2 the distance from the end of element (8) to the mass center of element (9), l3 the distance from the mass center of element (10) to point A, l4 the distance from the mass center of element (11) to point A, l5 and l5 coordinates of the mass center of element (13) relative to point A, h2 the distance from point A to the top point of the laparoscope (element (14)).

4 Validation of the Dynamics through a Multi-Body Simulator By using the kinematic and dynamic models for PARASURG 5M, some simulations of the end-effector motions were achieved. This trajectory is linear and similar to a real one, as the instrument is controlled by the joystick, when the surgeon wants to focus the camera in the robot workspace. The dynamic model for PARASURG 5M has been implemented in an Inverse Dynamic Modeling software (IDM), namely MATLAB and the data obtained were validated through a Multi-body Simulation software (MBS), namely Adams. The chosen data for the IDM fit the experimental model presented in Figure 4: XB = 0.65 m, YB = 0.4 m, ZB = 0.095 m, b = 0.39 m,

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h = 0.36 m, the maximum velocity being νmax = 0.01 m/s and the maximum acceleration amax = 0.005 m/s2 . The law of motion for the tip of the endoscope has a trapezoidal form for the velocities. Figure 5 presents a comparison between the results obtained from IDM (forces and torques) and the results from the MBS. Due to a very good correlation between the two curves presented in each graph, the results obtained from the MDS validate the analytical model (with an error less than 1%, mainly due to the approximations in mass distribution of the links and joints of the robot).

5 Conclusions In this paper the dynamics and some design characteristics of the PARASURG 5M hybrid parallel robot used in MIS have been presented. Starting from mass geometry considerations, which consist in replacing a given multibody system correctly by dynamically equivalent single masses, the method of virtual work equations is proposed. The algorithm was programmed in Matlab and the results were validated using Adams. As seen from above, the model is compact and easy to implement it in a computer program to use for improved control algorithms.

Acknowledgement The authors gratefully acknowledge the financial support provided by the research grants awarded by the Romanian Ministry of Education, Research and Innovation.

References 1. Ben-Porat, O., Shoham, M., and Meyer, J., Control design and task performance in endoscopic teleoperation. MIT, 9:256–267, 2000. 2. BR Automation, http://br-automation.com/, 2009. 3. Brown University, Division of Biology and Medicine, http://biomed.brown.edu/, 2009. 4. Dizioglu, D., Getriebelehre, Band 3, Dynamik. Fried Vieweg und Sohn, Braunschweig, 1966. 5. Itul, T. and Pisla, D., On the solution of inverse dynamics for 6-DOF robot with triangular platform. In Proceedings of 1st European Conference EUCOMES, Obergurgl, 2006. 6. Ma, X.L. et al., Dynamic modeling and analysis of 4-DOF parallel mechanism. J. Jian. Univ., 28:337–380, 2007. 7. Maxon Motor AG, , http://www.maxonmotor.com, 2009. 8. Merlet, J.-P., Parallel Robots. Kluwer Academic Publisher, 2000. 9. Pisla, D. and Kerle, H., Development of dynamic models for parallel robots with equivalent lumped masses. In Proceedings of 6th International Conference on Method and Models in Automation and Robotics, Miedzydroje, pp. 637–642, 2000. 10. Pisla, D., Plitea, N., Gherman, B., Pisla, A., and Vaida, C., Kinematical analysis and design of a new surgical parallel robot. In Computational Kinematics, Duisburg, pp. 273–282, 2009.

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11. Plitea, N., Pisla, D., Vaida, C., and Gherman, B., Robot chirurgical. Patent Pending No. a00525/7.07.2009, Romania 2009. 12. Pollet, R. and Donnez, J., Using a Laparoscope Manipulator (LAPMAN
)in laparoscopic gynecological surgery. In Proceedings Surgical Technology International XVII – Gynecology, San Francisco, pp. 187–191, 2008. 13. Zakerzadeh, M.R., Tavakoli, M., Vossoughi, G.R., and Bagheri, S., Inverse kinematic/dynamic analysis of a new 4-DOF hybrid (serial-parallel) pole climbing robot manipulator. In: Proceedings of the 7th International Conference CLAWAR, Madrid, pp. 919–934, 2004.

Author Index

A Alexandru, C. 575 Altuzarra, O. 165 Alutei, A. 583 Antal, A. 625, 641 Antal, T.A. 625, 641 Arrouk, K.A. 605 Atanasiu, V. 421 B Balcau, M. 277 Baldisserri, B. 295 Bayar, G. 345 Berceanu, C. 309, 335 Birleanu, C. 429, 657 Blebea, I. 467 Bojan, S. 429, 657 Bolshakova, D. 503 Bouzgarrou, B.C. 605 Brisan, C. 73 Bruckmann, T. 537 Burduhos, B. 181 Burisch, A. 117 Butnariu, S. 519 Butuc, B.R. 649 C Capustiac, A. 73 Carbone, G. 545 Caro, S. 675

Ceccarelli, M. 545 Chablat, D. 29, 675 Chebbi, A.H. 595 Ciobanu, D. 261 Cirebea, C. 583 Climescu, O. 199, 209 Collard, J.-F. 633 Comanescu, A. 235 Comanescu, D. 235 Copilusi, C. 327 Corral, J. 387 Corves, B. 147 Csibi, V.I. 125 D Danieli, G.A. 449 Dehelean, L.M. 245, 613 Dehelean, N.M. 613 de-Juan, A. 191, 633 Detesan, O. 485 Detesan, O.A. 665 Diaconescu, D. 157, 181, 199, 209 Dietz, T. 369 Diez, M. 83 Dobocan, C. 467 Drewenings, S. 117 Duma, V.-F. 475, 493 Dumitru, N. 309, 327 Dumitru, S. 335 Duta, A. 157 701

702

E Edeler, C. 109 Elisei, R. 457 Ellwood, R.J. 117, 683 F Fatikow, S. 109 Fernandez del Rincon, A. 191 Filip, D. 335 Fisette, P. 633 Flores, P. 225, 253, 397 Frunza, M. 457 Furcea, L. 457, 485 G Garcia, P. 633 Gherman, B. 567 Gherman, B.G. 691 Gogu, G. 605 Golovin, A. 269, 361 Graur, F. 457, 485, 567 Gude, M. 173 Gui, R.M. 665 Gyurka, B. 567 H Herder, J.L. 413 Herm´andez, A. 83 Hermenean, I. 157 Hern´andez, A. 165 Hesselbach, J. 557 Higuchi, M. 319 Hiller, M. 537 Huang, C. 3 Hufenbach, W. 173 Husty, M.L. 91 I Iglesias, M. 191 Inkermann, D. 557 Ispas, V. 665 Ispas, Vrg. 665 Itul, T. 39, 437 Ivan, A.I. 125 Iwaya, T. 319

Author Index

J Jaliu, C. 199, 209 Jaschinski, J. 173 K Kiper, G. 137 Klimchik, A. 675 Koku, A.B. 345 Kong, X. 3 Konukseven, E.I. 345 L Lankarani, H.M. 397 Leohchi, D. 421 Lese, D. 691 Lovasz, E.-C. 173, 245, 613 M Machado, M. 397 Macho, E. 83 Mˆandru, D. 125 Manychkin, N. 511 Marcusanu, V. 309 M˘argineanu, D. 173, 245 Marin, N. 327 Merlet, J.-P., 529 Meyer, I. 109 Mianowski, K. 303 Miermeister, P. 353 Modler, K.-H. 173, 245 Modler, N. 173 Moldovan, E.C. 245 Moldovean, Gh. 649 Morariu-Gligor, R. 277, 467 Moroz, G. 29 Mundo, D. 449 Muresan, A. 457, 485 N Nawratil, G. 47, 99 Neagos, H. 457, 485 Noveanu, S. 125 Nudo, P. 449

Author Index

O Ogata, M. 319 Ottaviano, E. 545 P Parenti Castelli, V. 295, 595 Pashkevich, A. 675 Pedrero, J.J. 217 Perez-Gracia, A. 11 Perju, D. 173, 245, 613 Perrelli, M. 449 Petuya, V. 83, 387 Pfurner, M. 91 Pinto, C. 165, 387 Pisla, A. 437 Pisla, D. 39, 117, 437, 567, 691 Pleguezuelos, M. 217 Plitea, N. 485, 567, 691 Pop, A.F. 277 Popa, D. 309 Popescu, D.I. 55 Potapova, A. 269 Pott, A. 353, 369 R Raatz, A. 117, 557, 683 Radu, C. 457 Ravina, E. 377 Reich, S. 405 Rolland, J.P. 475 Rouiller, F. 29 Rusu, L. 327 S Sabou, C. 691 Sakharov, M. 511 S´anchez, M. 217 Sancibrian, R. 633 Sandru, B. 165 S˘aulescu, R. 181, 199, 209 Schmitt, J. 557 Schramm, D. 537 Schr¨ocker, H.-P. 21 Sch¨utz, D. 683

703

Scurtu, L. 457, 485 Seabra, E. 253 Segla, S. 405 Stachel, H. 99 Stoica, A. 39 Stroe, I. 285 Sucala, F. 429, 657 Suciu, M. 691 ˘ ıgler, J. 63 Sv´ Szilaghyi, A. 457, 485 T Takeda, Y. 319 Talaba, D. 519 Tarabarin, V. 503, 511 Tarnita, D. 309, 335 Tarnita, D.N. 309 Tatar, O. 583 Tseng, R. 3 Turcu, I. 429, 657 U Umnov, N. 361 Urizar, M. 387 V Vaida, C. 485, 567, 691 van der Wijk, V. 413 Vatasescu, M. 181 Velicu, R. 649 Verl, A. 369 Viadero, F. 191 Vietor, T. 557 Visa, I. 157, 181, 261 Vlad, L. 457, 485, 567 Vukolov, A. 269, 361 W Wenger, P. 29 Z Zichner, M. 173 Zielinska, T. 303

Subject Index

3D model 511 3D printing 309 4-4-correspondence 99 A actuators 377, 665 addendum modification 625 analysis 545 arthroplasty 327 assistance robotics 529 azimuth PV tracking system 181 azimuth tracking linkage 157 B bearings 191 Bennett 3 bi-axial mono-mobile tracking system 181 biomechanics 319 brake model 369 Burmester’s focal mechanism 99 C cable 345 cable-driven parallel robot 353 CAD 605 calibration 73 cam follower mechanisms 261 cam mechanism 245, 269 cam-follower mechanisms 253 cams 225 centrifugal governor 511

chaotic regime 235 closed-loop multibody systems 633 clutch 285 compliant hinge mechanisms 173 compliant mechanism 125, 683 conceptual design 199 connection forces 327 constructors and manufacturers of models 503 contact 63 contact pressure 625 contact ratio 421 controlled friction damper 405 cost function 467 crank-slider mechanism 413 curved toothing 657 cusp 29 cyclic test 173 D deformable element 199 deployable structures 137 design 39 dexterity 165 digital photography 269 dimensional synthesis 11, 633 direct tracking efficiency 181 discrete line congruence 21 displacement and torque functions 613 driving axe 429 driving force 429 drum 345 705

706

dry contact 397 dual axis system 649 dynamic balancing 413 dynamic equations 665 dynamic friction coefficient 345 dynamic model 209, 327 dynamic simulation 309 dynamic transmission error 421 dynamics 437, 691 dynamics simulation 353 E efficiency 217, 641 elastic 285 elastic potential energy 217 elderly aid 529 energetic efficiency 649 ethics 457 experimental mechanics 377 external loading 675 F FEA 475 finite element method 309 finite screw systems 11 flexure hinge 683 fluidic muscles 377 foot design 303 four-positions theory 21 friction 369 friction coefficient 429 friction force 429 Fulbright 493 functional angles 181 G gait 361 gear dynamics 191 geometrical restriction 261 geometry 595 “goodwill” function 467 H Hamiltonian 467 hand-arm system 277

Subject Index

hardware-in-the-loop 353 head holder 657 helical gear 217, 421, 641 helicopter flight simulator 437 higher education 493 higher engineering education 503 history of science and mechanisms 511 history of science and technology 503 horse 361 human lower limb 327 hydrodynamical lubrication 625 I implant 327 incidence angle 157 incorrect position 63 ink jet printer 235 in-pipe 583 inverse dynamic model 117 inverse kinematics 91 J Jitterbug 137 K kinematic analysis 253 kinematic and force screws 63 kinematic model 295, 683 kinematics 29, 39, 83, 335, 475, 691 kinematotropic mechanism 3 knee joint 309 Kokotsakis mesh 99 L Lagrange method 209 laparoscopic liver surgery 485 laser jet printer 235 least invasive surgery 449 leg collision avoidance 595 leg control 361 leg mechanisms 545 life support 319 ligament 309 linear approximation 675 linear encoder 345

Subject Index

linkage design 545 linkage drive mechanism 613 load distribution 217 low CPV (concentration photovoltaic) system 157 low-mobility 387 lubricated spherical clearance joints 397 M machine components 557 manipulator design 605 mathematical model 73, 235 measurement 269 measurements machine-tool 277 mechanical 335 mechanical parameter 665 mechanical vibration 277 mechanism 493, 519, 583 mechanism simulation 147 mechatronic model 575 mechatronics 493 mesh stiffness 421 meshing 657 meshing stiffness 191 micro gear 117 miniaturized robot 117 minimally invasive surgery 691 minimum size of the cam mechanism 245 mini-system 125 model of mechanism 511 models of mechanisms and machines 503 modular 583 motion profiles 147 motion simulation 83 motion synthesis 319 multibody system 575 multibody system theory 261 multiobjective optimization 165, 641 multiple functions 285 N nanorobot 109 natural frequencies 387

707

Newton–Euler method 209 O open architecture 567 operation mode 3 optimal control 467 optimization 39, 225, 633 optomechanics 475 overconstrained linkage 3, 11, 99 P parallel hybrid robot 691 parallel manipulator 29, 165, 387, 595, 675 parallel mechanism 39 parallel robot 557, 633, 683 parallel structure 437, 567 parallelogram-based linkage 675 particle motion 55 partner parallel robots 73 passenger car 405 passive motion 295 passive suspension 405 peripheral devices 235 piezoactuator 109 piezoelectric actuator 125 planar parallel manipulator 605 planetary speed increaser 209 pneumatics 377 polygon scanners 475 polyhedral linkages 137 positioning system 345 probability 361 profile 269 proteins 83 PV platform 575, 649 R reconfiguration 557 relative sliding coefficients 641 remote diagnostic 269 research 493 ride comfort 405 robot 485, 583, 665 robot design 303

708

robot model 369 robot-assisted surgery 449 robotic hand 335 robotic surgery 457, 567 robotics 319, 545 roller followers 225 roller motion 253 rotating shafts 475 rotation 21 RPRP 3 S safety 285 Sch¨onflies motion group 47 Sch¨onflies-singular 47 screw surfaces 63 semi-active suspension 405 serial chain 91 servo drives 147 shaking force 413 shaking moment 413 simulation 73, 109, 191, 437, 485, 691 singularity 29, 47, 557, 595 singularity analysis 91 slider-crank mechanism 613 small hydropower plant 199 software 91 spatial linkage 181 spatial multibody dynamics 397 speed increaser 199 spherical four-bar linkage 99 spherical indicatrix 137 spur gear 657 standard tools control 449 state equations 467 steam engine 511 Stewart Gough platform 47 stick-slip 109 stiffness 387 stiffness modeling 675 Stirling engine 613 stopping behavior 369 strobelight 361 structural integrity 475

Subject Index

students 493 Study quadric 21 symbolic computation 29 synthesis 225 T teaching method 519 telemedicine 457 telesurgery 457 tensed mechanism 537 testing 377 textile-reinforced composites 173 theory of the regulation 511 tibia-fibula-ankle complex 295 torque control 117 tracking efficiency 157 tracking mechanism 575 tracking procedure 649 training platform 485 translating follower 245 transmission error 191 U unique rack-bar 657 V vibrations 55 vibratory conveying 55 vibratory hopper 55 vibratory track 55 Virtual Reality 519 virtual spherical wrist 449 voice control 567 W walker 529 walking assist 319 walking machines 303 wind tunnel 537 wire robot 73, 537 working cyclogram 429 workspace 39, 165, 605 worm gears 625 worm shaft stiffness 625