Mechanism Design for Robotics: MEDER 2021 9783030752705, 9783030752712

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Mechanisms and Machine Science

Saïd Zeghloul Med Amine Laribi Marc Arsicault   Editors

Mechanism Design for Robotics MEDER 2021

Mechanisms and Machine Science Volume 103

Series Editor Marco Ceccarelli , Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy Advisory Editors Sunil K. Agrawal, Department of Mechanical Engineering, Columbia University, New York, USA Burkhard Corves, RWTH Aachen University, Aachen, Germany Victor Glazunov, Mechanical Engineering Research Institut, Moscow, Russia Alfonso Hernández, University of the Basque Country, Bilbao, Spain Tian Huang, Tianjin University, Tianjin, China Juan Carlos Jauregui Correa, Universidad Autonoma de Queretaro, Queretaro, Mexico Yukio Takeda, Tokyo Institute of Technology, Tokyo, Japan

This book series establishes a well-defined forum for monographs, edited Books, and proceedings on mechanical engineering with particular emphasis on MMS (Mechanism and Machine Science). The final goal is the publication of research that shows the development of mechanical engineering and particularly MMS in all technical aspects, even in very recent assessments. Published works share an approach by which technical details and formulation are discussed, and discuss modern formalisms with the aim to circulate research and technical achievements for use in professional, research, academic, and teaching activities. This technical approach is an essential characteristic of the series. By discussing technical details and formulations in terms of modern formalisms, the possibility is created not only to show technical developments but also to explain achievements for technical teaching and research activity today and for the future. The book series is intended to collect technical views on developments of the broad field of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of MMS but with the additional purpose of archiving and teaching MMS achievements. Therefore, the book series will be of use not only for researchers and teachers in Mechanical Engineering but also for professionals and students for their formation and future work. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected] Indexed by SCOPUS and Google Scholar.

More information about this series at http://www.springer.com/series/8779

Saïd Zeghloul Med Amine Laribi Marc Arsicault •

Editors

Mechanism Design for Robotics MEDER 2021

123

•

Editors Saïd Zeghloul University of Poitiers SP2MI - Site du Futuroscope Poitiers Cedex 9, France

Med Amine Laribi University of Poitiers SP2MI - Site du Futuroscope Poitiers Cedex 9, France

Marc Arsicault University of Poitiers SP2MI - Site du Futuroscope Poitiers Cedex 9, France

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-75270-5 ISBN 978-3-030-75271-2 (eBook) https://doi.org/10.1007/978-3-030-75271-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

MEDER 2021, the IFToMM International Symposium on Mechanism Design for Robotics, is the fifth event of a series that started in 2010 as a specific conference activity on mechanisms for robots. The first event was held at Universidad Panamericana de Ciudad de Mexico (Mexico) in September 2010; the second was held at Beihang University in Beijing (China) in October 2012; the third event was held at Aalborg University in Aalborg (Denmark) in June 2015, and the fourth event was held at the University of Udine in Udine (Italy) in September 2018. The aim of the MEDER 2021 Symposium is to bring together researchers, industry professionals and students from broad ranges of disciplines dealing with a mechanism for robots, in an intimate, collegial and stimulating environment. Again, MEDER 2021 received increased attention, since the proceedings contains contributions by authors from all over the world. The proceedings of MEDER 2021 Symposium contains 34 papers, which were selected after reviewing for oral presentation. These papers cover several aspects of the wide field of mechanism design for robotics, from theoretical studies up to practical applications through new robot designs and prototypes. They are authored mainly from the IFToMM community coming from Bulgaria, China, Denmark, France, Germany, India, Italy, Japan, Kazakhstan, Mexico, Russia, Slovakia, Spain, Taiwan, Tunisia, UAE, UK and USA. We express our grateful thanks to the MEDER Symposium International Scientific Committee members, namely Marco Ceccarelli (University of Cassino, Italy, Chair), Ding Xilun (Beihang University, China), Grigore Gogu (SIGMA Clermont, France), Mario Acevedo (University Panamericana, Mexico), Shaoping Bai (Aalborg University, Denmark), Yukio Takeda (Tokyo Institute of Technology, Japan), Alba Perez Gracia (Idaho State University, USA), Yan Jin (Belfast University, UK), Erwin Lovasz (Politehnica University Timisoara, Romania), Said Zeghloul (Poitiers University, France) and Victor Petuya (University of Basque Country, Spain). We thank the authors who contributed with very interesting papers on several subjects, covering many fields of mechanism design for robotics and additionally for their cooperation in revising papers in due time in agreement with the reviewers’ v

vi

Preface

comments. We are grateful to the reviewers for the time and efforts they spent in evaluating the papers within a given schedule that has permitted the publication of this proceedings volume. We thank the two keynote speakers, namely Philippe Wenger from the Laboratory of Digital Sciences of Nantes (the paradigm of tensegrity and its applications in robotic design) and Xavier Bonnet from the Centre of Biological Studies of Chizé (the snake robot: unparalleled versatility of a single locomotor mode). According to the planned conference sessions, this book is organized in seven chapters, covering the major research areas of the Symposium: I. II. III. IV. V. VI. VII.

Linkage and manipulators Parallel manipulators Mechanics of robots Mechanism deign Actuators and control Innovative mechanism/robot and their applications Special session in honour of Prof. Said Zeghloul

We also acknowledge the support of the International Federation for the Promotion of Mechanism and Machine Science (IFToMM, http://iftomm.net/). We thank the TC for Robotics and Mechatronics and the TC for Linkages and Mechanical Controls of IFToMM, as well as IFToMM France, the French Section of IFToMM, for their sponsorship. We thank the University of Poitiers, in particular the Fundamental and Applied Science Faculty, for its availability to host the MEDER 2021 event. The conference received generous support from local sponsors, namely the University of Poitiers, the Grand Poitiers and the Nouvelle Aquitaine region, which were crucial to make this conference possible. Unfortunately due to the COVID-19 pandemic situation and the related travel restrictions and social distancing requirements, the in-person meeting is very limited, and both in-person and virtual attendance were combined during MEDER 2021. We thank the publisher Springer and its editorial staff for accepting and helping in the publication of this proceedings volume within the book series on Mechanism and Machine Science (MMS). May 2021

Saïd Zeghloul Med Amine Laribi Marc Arsicault

Organization

Program Chair Med Amine Laribi

Université de Poitiers, France

Program Committee Marco Ceccarelli

University of Rome Tor Vergata, Italy

vii

Introduction to the Special Session in Honour of Prof. Said Zeghloul By Marco Ceccarelli (Chair od ISC MEDER) University of Rome Tor Vergata, Rome

This MEDER 2021 special session is to honour Prof. Said Zeghloul in occasion of his near retirement after a very long prolific career. Professor Said Zeghloul is a great IFToMMist and a distinguished MMS scientist figure with a gentleman dynamic attitude. He has been a very prolific MMS scientist with a great reputation worldwide not only within the IFToMM community as referring to this conference series MEDER: Mechanism Design for Robots. He was born in 1956 in Casablanca (Morocco), but he developed his engineer scientific career in Poitiers, France. His research interests have been and still are in robot design, mechatronics, computer-aided design for robotics, dexterous grasping, humanoid locomotion and collision-free path-planning, with theory, design and practice activity whose achievements have disseminated in publications well know all around the world. He has been awarded with several honours by local, national and international institutions. He served IFToMM in different technical committees and since 2012 as Chair of France IFToMM Member Organization. Moreover, he proposed and contributed to the organization of the several conference events and in particular, being a member of the scientific committee for MEDER, for this MEDER 2021 event. I met him in 1988 at a conference when both were PhD students, and since, then we have friendly collaborations in technical and social plans with the IFToMM spirit, exchanging also students and sharing research activity. Similarly, he has been a reference person and a scientist for many other colleagues in Europe and in the world with a special care to his motherland north-Africa frames also with the aim to help and promote others in improving skills and international collaborations for personal growth and society welfare. In MEDER community, he has been soon an active actor whose figure and activity can be considered a significant reference for us and young generations to achieve good results in our life and in our career. We thank Prof. Said Zeghloul for having shared with us his brilliant experience in the scientific career, and we hope to keep his thoughts forever.

ix

Contents

Linkage and Manipulators Analytical Movement Optimization of Dual-Arm Planar Robots with Rotating Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matteo Bottin, Giulio Rosati, and Giovanni Boschetti Inverse Kinematics and Velocity Analysis of a 6-DOF Hexapod-Type Manipulator with a Circular Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexey Fomin, Anton Antonov, Daniil Petelin, Victor Glazunov, and Marco Ceccarelli Torque Minimization of Dynamically Decoupled R-R Spatial Serial Manipulators via Optimal Motion Control . . . . . . . . . . . . . . . . . . . . . . . Vigen Arakelian, Jing Geng, and Yaodong Lu Kinematic Analysis of a Coaxial 3-RRR Spherical Parallel Manipulator Based on Screw Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . Alejandro T. Cruz-Reyes, Manuel Arias-Montiel, and Ricardo Tapia-Herrera A Novel Design for an Autonomous Mobile Agricultural Fruit Harvesting Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divyansh Khare, Sandra Cherussery, and Santhakumar Mohan

3

12

20

28

38

Parallel Manipulators A Comparison of Algebraic and Iterative Procedures for the Generation of the Workspace of Parallel Robots . . . . . . . . . . . . Matteo Russo and Marco Ceccarelli

53

Symbolic Kinematics of Special 3-RSR Parallel Mechanism with Zero Coupling Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huiping Shen, Yao Tang, Tao Li, and Qingmei Meng

62

xi

xii

Contents

Inverse Kinematics and Workspace of a 3-PRRS Type Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zh. Baigunchekov, M.A. Laribi, R. Kaiyrov, and E. Zholdassov

71

Investigation of Interference-Free Workspace of a Cartesian (3-PRRR) Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isaac John, Parvathi Sunilkumar, Santhakumar Mohan, and Larisa Rybak

79

Exit Point, Initial Length and Pose Self-calibration Method for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bozhao Wang and Stéphane Caro

90

Kinematic Simulation of a Geared Planar Parallel Manipulator . . . . . . 102 Antonio-Marius-Flavius Luputi, Erwin-Christian Lovasz, Marco Ceccarelli, Sticlaru Carmen, and Ana-Maria Stoian Mechanics of Robots Design Criteria Study for Underactuated Symmetric Pinching Mechanism of Pinch Roll Machine in High-Speed Wire Rod Product Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Wang Renquan, Yao Shuangji, Marco Ceccarelli, and Yang Xiaohan Design of a Reconfigurable Novel Constant-Force Mechanism for Assistive Exoskeletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Yichen Liu, Zhongyi Li, and Shaoping Bai Aerodynamic Double Pendulum with Nonlinear Elastic Spring . . . . . . . 132 Yury Selyutskiy, Marat Dosaev, Andrei Holub, and Marco Ceccarelli Design of a Flexible Interphalangeal Joint . . . . . . . . . . . . . . . . . . . . . . . 141 Job Eli Escobar-Flores, Christopher R. Torres-San Miguel, Luis Antonio Aguilar-Pérez, and Marco Ceccarelli Mechanism Deign A Reconfigurable Underactuated Grasper with In-hand Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Carl A. Nelson Design of a Robotic Brace with Parallel Structure for Spine Deformities Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Rahul Ray, Laurence Nouaille, Briac Colobert, Laurine Calistri, and Gérard Poisson An Adaptive Drive of Spacecraft Docking Mechanism . . . . . . . . . . . . . . 168 Konstantin S. Ivanov, Dana T. Tulekenova, and Marco Ceccarelli

Contents

xiii

Driving Mechanism in Robotized Hospital Bed for Patients with COVID 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Jorge Enrique Araque Isidro and Marco Ceccarelli Synthesis of a Function Generator Six-Bar Linkage in Two Steps with Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Claudio Villegas, Mathias Hüsing, and Burkhard Corves Design and Analysis of a Device for Enlarging the Allowable Position Error for Screwing Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Hao-Tien Ku and Yu-Hsun Chen Dimensional Synthesis of a Four-Bar Linkage Assessing the Path and Reformulating the Error Function . . . . . . . . . . . . . . . . . . . . . . . . . 204 Alfonso Hernández, Aitor Muñoyerro, Mónica Urízar, and Víctor Petuya Actuators and Control Motion Control of 6-DOF Relative Manipulation Device . . . . . . . . . . . . 217 Sergey Khalapyan, Larisa Rybak, Elena Gaponenko, and Giuseppe Carbone Genetic Algorithm with Iterative Learning Control for Estimation of the Parameters of Robot Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Kaloyan Yovchev and Lyubomira Miteva Control of the Vibrations of a Cartesian Automatic Machine . . . . . . . . 236 Matteo Bottin , Giulio Cipriani , Domenico Tommasino, and Alberto Doria Innovative Mechanism/Robot and their Applications Design of the 2 D.o.F Compliant Positioning Device Based on the Straight-Line Watt’s Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 247 Jaroslav Hricko and Stefan Havlik Optimal Design of a Five-Bar Mechanism Dedicated to Assisting in the Fingers Flexion-Extension Movement . . . . . . . . . . . . . . . . . . . . . . 256 Araceli Zapatero-Gutiérrez, Med Amine Laribi, and Eduardo Castillo-Castañeda An Extendable Continuum Robot Arm to Deal with a Confined Space Having Discontinuous Contact Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Daisuke Matsuura, Ryota Shioya, H. Harry Asada, and Yukio Takeda Mobile Manipulator and EOAT for In-Situ Infected Plant Removal . . . 274 Taher Deemyad and Anish Sebastian New Variable Stiffness Joint (VSJ): Study and Simulation . . . . . . . . . . . 284 M. G. Contreras-Calderón, M. A. Laribi, and E. Castillo-Castañeda

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Contents

Special Session in Honor of Prof. Said Zeghloul Design of an 8-DoF Redundant Robotic Platform for Medical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Elie Gautreau, Aurélien Thomas, Juan Sandoval, Med Amine Laribi, and Saïd Zeghloul Serial Approach for Solving the Forward Kinematic Model of the DELTA Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Majdi Meskini, Houssem Saafi, Med Amine Laribi, Abdelfattah Mlika, Marc Arsicault, and Said Zegloul On the Optimal Design of LAWEX for a Safe Upper Arm Rehabilitation Exercising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Ines Ben Hamida, Med Amine Laribi, Abdelfattah Mlika, Lotfi Romdhane, Saïd Zeghloul, and Giuseppe Carbone A Reconfigurable 6-DoF Cable-Driven Parallel Robot with an Extended Rotational Workspace . . . . . . . . . . . . . . . . . . . . . . . . 322 Ferdaws Ennaiem, Abdelbadiâ Chaker, Med Amine Laribi, Juan Sandoval, Sami Bennour, Abdelfattah Mlika, Lotfi Romdhane, and Saïd Zeghloul Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Linkage and Manipulators

Analytical Movement Optimization of Dual-Arm Planar Robots with Rotating Platform Matteo Bottin1(B) , Giulio Rosati1 , and Giovanni Boschetti2 1

Department of Mechanical Engineering, University of Padova, Via Venezia 1, Padova, Italy [email protected] 2 Department of Management and Engineering, University of Padova, Stradella San Nicola, 3, Vicenza, Italy

Abstract. In the last years some industrial robots have been realized with two robot arms connected to a single rotating platform. Such a redundant structure allows moving the platform to adjust each robot base for optimizing each arm movement. In this paper, a planar model of two 7-axis robots connected to a rotating platform is proposed, and a novel analytical optimization procedure, retrieved from a geometrical analysis, is presented.

Keywords: Dual arm robot

1

· Kinematics · Robotics

Introduction

Redundant robots have become more and more popular in the last years thanks to new commercial solutions, especially in the field of collaborative robotics [1]. These solutions, however, mostly rely on the addition of a supplementary joint to enhance the performance of traditional robots [2], by increasing their degrees of freedom. In the field of industrial robots there is also a small branch in which it can be found a particular type of robots: dual arm robots. These are generally made up of two manipulators attached to a common platform [3]. In this way, the movements of the two robots are decoupled [4] (except for the possible mutual interference [5,6]), and the work-cell itself improves in both flexibility and capabilities [7,8]. Its flexibility can be further improved by adding a rotating platform, so that the two robot shoulders move around a common axis. In this way the two arms are capable of a wider workspace, but this solution has the downside of influencing the movement of both the arms. A proper trajectory planning of the rotating platform allows the movement of the two arms to improve the workcell throughput. This is important because commercial solutions are already available. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 3–11, 2021. https://doi.org/10.1007/978-3-030-75271-2_1

4

M. Bottin et al.

Research have already addressed the problem of the optimization of dual arm robots, but most of them are based on optimization algorithms that may not be reliable and depend on the starting conditions [9–11]. The aim of this paper is to develop an analytical method to rapidly provide a solution of the movement of the platform so that the two arms’ elbow joints perform the same displacement in moving between consecutive positions. This method can be used to choose specific arm configuration depending on the task, e.g. in case of vibrations. The proposed method is specific to 7-axis robots with spherical shoulder and wrist joints, since with such configuration it is possible to change the redundant angle value without affecting the elbow joint, adapting the configuration of the distal joints to the movement time of the elbow joint. The method, although approximated, does not rely on any optimization algorithm that may decrease performances while obtaining only local minima. Due to its fast calculation, the method can be used to optimize the work-cell throughput by using specific task sequencing algorithms [12].

2

Analytical Model

Let’s consider the kinematic scheme shown in Fig. 1 in which a manipulator is made of two arms on the edge of a rotating platform of radius R0 . The manipulator is considered to be the planar projection of two 7-axis robots mounted on a rotating platform working on a plane. An example of such robot is the Yaskawa SDA series robot, in which two 7-axis robot are attached on a common rotating base1 . Each arm is built in such a way that the first three joints and the last three joints can be considered as two different spherical joints (commonly named “shoulder” and “wrist” joints respectively), leaving the fourth joint (“elbow”) to rotate lonely. In this scheme Si is the shoulder of the arm i and Pi is the corresponding target position. The rotation of the platform is identified by the angle ϕ. For simplicity, the first arm’s shoulder (S1 ) is located on the radius defined by the angle ϕ, while the second arm’s shoulder (S2 ) is located around the same circumference of S1 (centered in platform center O) at an angular distance of α from S1 . From this scheme, the shoulder positions S1 and S2 can be easily calculated:     R0 cos(ϕ) R0 cos(ϕ + α) , S2 = (1) S1 = R0 sin(ϕ) R0 sin(ϕ + α) For simplicity the two arms are considered to be equal with link lengths a1 (first link) and a2 (second link), and the robot arms are considered to be lying on the xy-plane. In this way, joints 1-2-3 can be reduced to a single joint on the shoulder, and joints 5-6-7 are reduced on the wrist. This simplification can be justified by the fact that joints 1-2-3 act as a single spherical joint on 1

https://www.motoman.com/en-us/products/robots/industrial/assembly/sda.

Analytical Movement Optimization of Dual-Arm Planar Robot

5

Fig. 1. Scheme of the dual arm robot with rotating platform.

the shoulder, allowing the arm to increase its height. In this case, on the xyplane all the points would be projected, but the shoulder-wrist distance will only depend on the link lengths a1 and a2 . In fact, for the proposed method only shoulder-wrist distance and platform radius are needed. Since target positions P1 and P2 are known, the shoulder-wrist distance is: → − → − → − Ri = Pi − Si (ϕ)  2 2 cos(ϕ − ϕ ) + R02 − 2R0 x2P,i + yP,i Ri2 = (x2P,i + yP,i P,i )

(2)

where xP,i and yP,i are the coordinates of the target position Pi on the plane → − and ϕP,i is the angle about x-axis defined by the vector Pi . The objective of the optimization is to find an angle ϕf so that the two arms, in moving from a position Pi,0 to Pi,f , move equally the fourth joint q4,i . This is possible since in a 7-axis robot the value of q4 only depends on the shoulder-wrist distance. In fact, since a 7-axis robot is kinematically redundant, it is possible to change the redundant angle value to change the values of the other joints (except for q4 ) still maintaining functionality (e.g. with the method proposed in [13]), decoupling the problem [14]. In this scenario, it is important to impose (if possible) that the movement of the elbow joints (Δq4,i ) are equal. The equation holds:     2 2 a21 + a22 − R1,f a21 + a22 − R1,0 Δq4,i (ϕf ) = arccos − arccos (3) 2a1 a2 2a1 a2 Equation 3 is highly non linear and it is difficult to solve with analytical tools. It is possible to approximate this equation by using the appropriate Taylor series,

6

M. Bottin et al.

that, in the case of the arccosine, is arccos(x) = the first order of the series, Eq. 3 becomes: Δq4,i (ϕf ) =

π 2

− x + o(x3 ). By stopping at

2 2 − R1,f R1,0 2a1 a2

(4)

In such a way, since the two arms are equal and the objective of our optimization is to equal the two elbow joint displacements Δq4,1 and Δq4,2 : 2 2 2 2 |R1,f − R1,0 | = |R2,f − R2,0 |

(5)

where the absolute values of the movements are taken since the electric motors equipped on the robots provide equal performance in both directions. The initial system configuration is considered to be known, so Ri,0 and R2,0 are fully defined (due to the initial platform rotation ϕ0 ). To calculate ϕf , from 2 2 and R2,f : Eq. 2 and Fig. 1 it is possible to retrieve the values of R1,f 2 R1,f = (x2P 1,f + yP2 1,f ) + R02 − 2R0

= (x2P 1,f + yP2 1,f ) + R02 − 2R0 2 = (x2P 2,f + yP2 2,f ) + R02 − 2R0 R2,f

= (x2P 2,f + yP2 2,f ) + R02 − 2R0

 

 

x2P 1,f + yP2 1,f cos(ϕf − ϕP 1,f ) x2P 1,f + yP2 1,f cos(ϕf )

x2P 2,f + yP2 2,f cos(ϕf − ϕP 1,f + α − β) x2P 2,f + yP2 2,f cos(ϕf + γ)

with γ = α − β and ϕf = ϕf − ϕP 1,f . Equations 6 and 7 can be simplified by grouping the constant terms: 2 R1,f = A1 + B1 cos(ϕf ) 2 R2,f

= A2 +

(6)

B2 cos(ϕf )

(7)

(8) +

C2 sin(ϕf )

(9)

where: 2

2

2

A1 = xP 1,f + yP 1,f + R0 2

2

2

A2 = xP 2,f + yP 2,f + R0

B1 = −2R0 B2 = −2R0

 

(10)

2 x2 + yP P 1,f 1,f 2 x2 + yP cos(γ) P 2,f 2,f

C2 = 2R0



2 x2 + yP sin(γ) P 2,f 2,f

(11) By imposing the equality of Eq. 5 it is possible to calculate the value of ϕf . The modulus defines two different equations, each one with opposite signs. It results in: (12) A + B cos(ϕf ) + C sin(ϕf ) = 0 2 2 ∓ R2,0 ) A = (A1 ∓ A2 ) − (R1,0

,

B = B1 ∓ B2

,

C = ∓C2

(13)

where the sign depends on which one of the two solutions provided by the modulus is being calculated.

Analytical Movement Optimization of Dual-Arm Planar Robot

7

It is possible to apply the double-angle formulae to Eq. 12, and by solving the equation it is possible to obtain the platform angle that equals the movement of the two arms:   √ −C ± −A2 + B 2 + C 2 + ϕP 1,f (14) ϕf = 2 · arctan A−B Equation 14 produces 2 solutions, which are doubled thanks to the absolute value equality expressed in Eq. 5. It is worth to notice how the proposed method does not consider movement times nor mechanical limits. In fact, only reachability is considered, which may remove some solutions: if Eq. 5 provides no solution (i.e. the function never reaches null value), the optimal ϕf can be obtained by deriving Eq. 5 (with Eqs. 8 and 9) by ϕf . In this way it is possible to fin the base rotation angle that minimize the difference of the two elbow displacements. The solutions, in this case, are:   √ −B ∓ B 2 + C 2 + ϕP 1,f ϕf = 2 · arctan (15) C Future work will include movement times and mechanical limits for a more comprehensive method. Nonetheless, from the proposed method a first good solution can be found.

3

Simulation Results and Discussion

To test the algorithm, some simulations have been carried out. In this paragraph an example will be provided. Let’s consider an application as defined via the parameters of Table 1. The two arms are made of identical links and are placed around the platform at the same distance R0 . By applying the method described above in moving the first arm from P1,0 to P1,f and the second arm from P2,0 to P2,f , four possible ϕf values are obtained. Among these, in our application the closest to ϕ0 is chosen (#3 of Table 2). Table 1. Parameters used in the simulation

Fig. 2. Initial configuration and points to be reached

Parameter

Value

R0 α

50 [mm] 120 [◦ ]

a1 , a2

70 [mm]

ϕ0

−30 [◦ ]

P1,0

[120,0] [mm]

P2,0

[0,120] [mm]

P1,f

[80,10] [mm]

P2,f

[50,100] [mm]

8

M. Bottin et al. 1

Normalized values [ ]

0.8 2

2

2

2

2

2

0.6

(R1f-R10 )-(R2f-R20 )

0.4

(R1f-R10 )+(R2f-R20 )

2

2

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -200

-150

-100

-50

0

50

100

150

200

Fig. 3. Shape of the Eq. 5. The two lines are normalized to the respective maximum absolute values. Dashed lines provide the solutions of Eq. 14 (black) and 15 (red) with the parameters of Table 1. (Color figure online)

Table 2. Values and difference of the displacements of the arms for each solution of Eq. 14 (#1–#4) and Eq. 15 (#5–#8) with the parameters of Table 1. Some solutions are not available due to the unreachability of the final point (two red lines to the left of Fig. 4). N ϕf − ϕ0 [◦ ] Δq4,1 [◦ ] Δq4,2 [◦ ] |Δq4,1 | − |Δq4,2 | [◦ ] 1 2 3 4

−113.63 −40.43 −26.57 39.49

59.14 4.62 −6.87 −45.01

−58.17 5.24 7.61 −33.88

−0.97 0.62 0.74 −11.13

5 6 7 8

−70.10 109.90 179.53 −0.47

28.95 0.65 54.4 −27.55

−12.75 – – −0.40

−16.20 – – −27.15

The application of the method is straightforward: no parameter optimization is required. In Figs. 2 to 5 the results are shown. Figure 3 shows the two functions provided by Eq. 5, where the values have been normalized to the respective maximum absolute values. The values of ϕf are obtained from the null values of these two functions. As can be noted, the two functions are not symmetric with respect to the x-axis. This means that in some cases it could be possible that the functions do not result in null values, thus not providing any solutions (Eq. 14). As stated in previous Section, in this scenario the best solution to ϕf to be used can be obtained by finding the stationary points of the function (Eq. 15). In this case the two arms will not have the same displacement (leftmost column of Table 2, #5–#8), but it will be minimized by the approximation of the proposed method.

Analytical Movement Optimization of Dual-Arm Planar Robot

9

80

q4,1

60

q4,2

40

| q4,1|-| q4,2|

20

[°]

0 -20 -40 -60 -80 -100 -120 -200

-150

-100

-50

0

50

100

150

200

Fig. 4. Displacement of elbow joints for the right (blue) and left (red) arms. The green line shows the difference |Δq1,4 | − |Δq4,2 |. Dashed lines provide the solutions of Eq. 5 (black) and 15 (red) with the parameters of Table 1. (Color figure online) 120

120

100

100

80

80

60

60

40

40

20

20

0

0

-20

-20

-40

-40

-60

-60 -50

0

50

100

-50

0

50

100

Fig. 5. Solution #1 (to the left) and #3 (to the right) as of Table 2. Green line shows the initial position of the first shoulder; dashed lines show the starting configuration. (Color figure online)

In fact, the proposed method is approximating the displacement of the two arms. As can be noted from the #4 solution of Table 2 (leftmost black line of Fig. 4), the solutions provided by Eq. 14 do not reflect on a completely equal displacement of the two arms. This approximation, however, is very small in most of the solutions. In fact, as can be noted by Table 2, the error is usually lower than 1◦ , a very small error if it is considered that is provided by an analytical solution and is then computationally efficient (in our tests the computational time is around 4 ms for each run). Since Eq. 14 provides some solution, Eq. 15 is not used. In fact, the solution provided by this equation (#5–#8 of Table 2) are not optimal for our application. Difference between calculated and actual joint displacements may be due to the expansion of the Taylor series, which is calculated around x = 0 even if q4 can be very far from 0. However it should be noted that our objective is to find the difference of the two Taylor series, so the error should be limited (as shown by the results of Table 2).

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Conclusions

In this paper a novel analytical method for the definition of the movement of the platform of a dual arm robot is presented. This approach aims at finding the proper rotation of the platform so that the two arms, in moving between positions, perform the same elbow joint displacement. In this way none of the elbow joints will result to be the bottleneck of the movement. The other joints will be optimized exploiting redundant configurations. In fact, the elbow joint value only depends from the distance between shoulder and wrist, and hence it is decoupled from the rest of the arm. Future work will consider the overall movement time in the model.

References 1. Matheson, E., et al.: Human-robot collaboration in manufacturing applications: a review. Robotics 8(4), 100 (2019) 2. Boschetti, G.: A novel kinematic directional index for industrial serial manipulators. Appl. Sci. 10(17), 5953 (2020) 3. Smith, C., Karayiannidis, Y., Nalpantidis, L., Gratal, X., Qi, P., Dimarogonas, D.V., Kragic, D.: Dual arm manipulation-a survey. Robot. Auton. Syst. 60(10), 1340–1353 (2012) 4. Zhang, J., Xu, X., Liu, X., Zhang, M.: Relative dynamic modeling of dual-arm coordination robot. In: 2018 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 2045–2050. IEEE (2018) 5. Lewis, C.L.: Trajectory generation for two robots cooperating to perform a task. In: Proceedings of IEEE International Conference on Robotics and Automation, vol. 2, pp. 1626–1631. IEEE (1996) 6. Bottin, M., Boschetti, G., Rosati, G.: A novel collision avoidance method for serial robots. In: IFToMM Symposium on Mechanism Design for Robotics, pp. 293–301. Springer, New York (2018) 7. Huang, H.-K.: A method of machining error compensation for dual robot prototyping system using dimensional measurement data, Ph.D. dissertation, National Tsing Hua University (2000) 8. Makris, S., et al.: Intuitive dual arm robot programming for assembly operations. CIRP Ann. 63(1), 13–16 (2014) 9. Wang, M., Luo, J., Yuan, J., Walter, U.: Coordinated trajectory planning of dualarm space robot using constrained particle swarm optimization. Acta Astronautica 146, 259–272 (2018) 10. Wu, Y.-H., Yu, Z.-C., Li, C.-Y., He, M.-J., Hua, B., Chen, Z.-M.: Reinforcement learning in dual-arm trajectory planning for a free-floating space robot. Aerosp. Sci. Technol. 98, 105657 (2020) 11. Li, Y., Hao, X., She, Y., Li, S., Yu, M.: Constrained motion planning of free-float dual-arm space manipulator via deep reinforcement learning. Aerosp. Sci. Technol. 106446 (2020) 12. Bottin, M., Rosati, G., Boschetti, B.: Working cycle sequence optimization for industrial robots. In: The International Conference of IFToMM ITALY, pp. 228– 236. Springer, New York (2020)

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13. Xu, W., Yan, L., Mu, Z., Wang, Z.: Dual arm-angle parameterisation and its applications for analytical inverse kinematics of redundant manipulators. Robotica 34(12), 2669 (2016) 14. Bottin, M., Rosati, G.: Trajectory optimization of a redundant serial robot using cartesian via points and kinematic decoupling. Robotics 8 (2019)

Inverse Kinematics and Velocity Analysis of a 6-DOF Hexapod-Type Manipulator with a Circular Guide Alexey Fomin1(&), Anton Antonov1, Daniil Petelin1, Victor Glazunov1, and Marco Ceccarelli2 1

Mechanical Engineering Research Institute of the RAS, Moscow 101000, Russia [email protected] 2 LARM2: Laboratory of Robot Mechatronics, University of Rome “Tor Vergata”, Rome 00133, Italy

Abstract. The study considers inverse kinematics (position problem) and velocity analysis of a six-degree-of-freedom hexapod-type manipulator with a circular guide. The structure of the proposed manipulator provides location of all drives fixed on the base and excludes the collision of the adjacent carriages when they move along the circular guide. The inverse kinematics provides the explicit relations between the platform coordinates and the driven (controlled) ones. The velocity analysis follows next and presents a procedure to find the driven velocities based on the platform ones. Example implemented in MATLAB demonstrates the performance of the proposed algorithms. As a case study, the platform motion along a screw trajectory is considered. Keywords: Parallel manipulator
Hexapod
Circular guide kinematics
Velocity analysis
MATLAB simulation


Inverse

1 Introduction Hexapods are widely used in technology as parallel manipulators. This happens also due to the need for more precise movements of the output elements. Hexapods, due to their parallel structure, provide high rigidity and advanced load capacity. Their potential usage is directed to both ground-based and space applications, including positioning systems for satellite antennas and telescopes, movable frames for car- and airplane simulators, etc. [1]. Structurally, a hexapod is a mechanical system in which six kinematic chains connect the fixed and output links [2–6]. In this case, the composition of the kinematic chains and the number of degrees-of-freedom (DOFs) of hexapods may differ. Most of them have six DOFs, however, there are lower-mobility manipulators. For example, a hexapod presented in [7] includes an additional kinematic chain, which imposes two constraints on the end-effector movement, turning this manipulator from 6-DOF to 4DOF. Study [8] presents a hexapod, in which all six kinematic chains mounted between

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 12–19, 2021. https://doi.org/10.1007/978-3-030-75271-2_2

Inverse Kinematics and Velocity Analysis

13

the fixed and output links are coupled through an additional 1-DOF mechanism, which provides the given system with a single mobility, i.e. it operates from a single drive. Among the hexapods, a class of manipulators with a circular guide is known [9, 10]. Their key functional property is the ability to reproduce a sufficiently large rotational angle around the vertical axis. This allows increasing the workspaces and expands the possibilities of practical application of these manipulators. However, despite these advantages, the manipulators’ drives are mounted on the movable links (carriages). This causes the appearance of additional inertial loads. Moreover, the design of these manipulators does not provide the exclusion of collisions between the adjacent carriages. This study proposes a novel model of the hexapod-type manipulator with a circular guide, in which all drives stay on a fixed base and the problem of collision of adjacent carriages is constructively solved. The next section provides the design overview of this manipulator.

2 Manipulator Architecture Let’s consider the structure and functional features of the proposed manipulator [11]. Figure 1 presents its CAD (computer-aided design) model (virtual prototype). The manipulator includes: 1 is the circular guide (fixed link); 2 is the drive; 3 is the crank (driving link); 4 is the slide block; 5 is the swinging arm; 6 is the carriage; 7 is the leg; 8 is the platform (end-effector). Slide block 4 and swinging arm 5 form prismatic joint, swinging arm 5 and carriage 6 form a single link (rigid connection), leg 7 is connected on both sides with carriage 6 and platform 8 by spherical joints.

Fig. 1. (a) CAD model (virtual prototype) of the 6-DOF hexapod-type manipulator with a circular guide; (b) pair of adjacent kinematic chains coupled between fixed and output links.

In each of the six kinematic chains of the manipulator, the input motions pass from cranks 3 to slide blocks 4 and then to swinging arms 5, turning them at certain angles.

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Swinging arms 5 displace carriages 6 along circular guide 1. Then the motion transmits to legs 7 and further to platform 8. Thus, six independently actuated kinematic chains provide the change of position and orientation of platform 8. Such a design of the manipulator allows locating all drives fixed on the base. Moreover, the correctly chosen cranks lengths eliminate the possibility of collision between the adjacent carriages. Additionally, the end-effector can provide a sufficiently large rotational angle around the vertical axis for the optimal dimensions of the manipulator elements.

3 Kinematic Analysis This section will present the inverse kinematics and velocity analysis of the considered manipulator. The inverse kinematics aims to calculate the driven coordinates given the platform ones. The velocity analysis studies the relations between the platform velocities and the ones of other links, first of all, the drives. These two topics are of big importance since they form the basis for workspace analysis, study on singularities, dynamics, and control. The position of the platform relative to some stationary coordinate system OXYZ can be represented by Cartesian coordinates pP of its center, point P (Fig. 2a). Plane OXY is in the plane of carriages Ai, i = 1…6, and the center, point O, is the intersection point of the circular guide central axis with this plane. The platform orientation can be set using rotation matrix RP, which determines the orientation of local coordinate system PXPYPZP attached to the platform relative to OXYZ. Velocity tP of point P and angular velocity xP define the platform velocities.

Fig. 2. Toward kinematic analysis of the 6-DOF hexapod-type manipulator with a circular guide: (a) location of the stationary and local coordinate systems; (b) fragment of the circular guide.

Inverse Kinematics and Velocity Analysis

15

Vectors q and q_ determine the driven coordinates and velocities, respectively: q ¼ ½q1 q2 q3 q4 q5 q6 T ; q_ ¼ ½q_ 1 q_ 2 q_ 3 q_ 4 q_ 5 q_ 6 T ;

ð1Þ

where qi and q_ i are the rotational angle and drive velocity of the i-th crank (Fig. 2b). 3.1

Inverse Kinematics

Inverse kinematics (or inverse position problem) aims to find driven coordinates q for given vector pP and matrix RP, which describe the platform position and orientation. The algorithm for solving this problem is the following. First, coordinates pBi of points Bi (the centers of spherical joints 7–8), i = 1…6, relative to stationary coordinate system OXYZ can be written as: pBi ¼ pP þ RP rBi ;

ð2Þ

where rBi are the coordinates of points Bi in local coordinate system PXPYPZP. Next, we can write coordinates pAi of points Ai (the centers of spherical joints 6–7) in stationary coordinate system OXYZ: pAi ¼ ½R1 cosðai þ di Þ

R1 sinðai þ di Þ 0T ;

ð3Þ

where R1 is the radius of the circular guide; ai is the angle between axis OX and line ODi (Fig. 2b); di is the rotational angle of the i-th swinging arm. We can connect the coordinates of points Ai and Bi using length Li of the i-th leg: ðpAi  pBi Þ2 ¼ L2i :

ð4Þ

Substituting Eqs. (2) and (3) into Eq. (4), we obtain the following relation: pxBi cosðai þ di Þ þ pyBi sinðai þ di Þ ¼

p2Bi þ R21  L2i ; 2R1

ð5Þ

where pxBi and pyBi are the corresponding components of vector pBi. The equation above contains an unknown parameter di. To find it, we can apply the tangent half-angle substitution in the following form:


 ai þ di 1  ti2 2ti ti ¼ tan ; sinðai þ di Þ ¼ : ; cosðai þ di Þ ¼ 2 2 1 þ ti 1 þ ti2

ð6Þ

Using Eq. (6), Eq. (5) can be transformed to the quadratic equation below: ai ti2 þ bi ti þ ci ¼ 0:

ð7Þ

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Coefficients ai, bi, and ci are known when solving the inverse position problem. Equation (7) can have two solutions, which have the following meaning. On the one hand, each carriage (point Ai) moves along the circle. On the other hand, for the given (fixed) platform coordinates, point Ai must lie on a sphere with the center at point Bi and radius Li. In general, this sphere will have two intersection points with the mentioned circle, and Eq. (7) represents this fact. The choice of the specific solution is determined by the manipulator design. Having ti found from Eq. (7), we obtain di using the first relation in Eq. (6). Next, we calculate ∠OCiDi (Fig. 2b) from the sine theorem for triangle OCiDi and, finally, obtain the crank rotational angle, i.e., the driven coordinate qi:
qi ¼ di þ \OC i Di ¼ di þ arcsin

 di sin di ; li

ð8Þ

where di is the distance from point O to rotational axis Di of the i-th crank; li is the length of the i-th crank. Thus, we have found the solution to the inverse position problem. The algorithm above has an explicit form and is identical for all kinematic chains of the manipulator. 3.2

Velocity Analysis

The task of this section is to find the connection between the platform velocities, tP and _ The procedure uses the results of the previous section and is xP, and the driven ones q. in the following. First, we rewrite Eq. (8) in the form below: li sinðqi  di Þ ¼ di sindi :

ð9Þ

Taking time derivative of both sides of Eq. (9) and considering the geometry of triangle OCiDi, we find the angular velocity d_ i of the i-th swinging arm: li cosðqi  di Þ d_ i ¼ q_ i ; OCi

ð10Þ

where OCi is the distance from center O to the slide block (point Ci). Differentiating Eq. (3) with respect to time, we find velocity tAi of the i-th carriage: tAi ¼ d_ i ½ R1 sinðai þ di Þ

R1 cosðai þ di Þ

0 T :

ð11Þ

Combining Eqs. (10) and (11) into a single one, we express velocity tAi as: tAi ¼ ki q_ i ;

ð12Þ

where vector ki contains components that are known from the inverse kinematics.

Inverse Kinematics and Velocity Analysis

17

Next, let’s consider velocity tBi of point Bi. It depends on the platform velocities tP and xP in the following manner: tBi ¼ tP þ xP  RP rBi :

ð13Þ

The projections of velocities tAi and tBi on the direction of leg AiBi must be equal to each other since each leg is a rigid body. This condition can be written by dot product: pTAiBi tAi ¼ pTAiBi tBi ;

ð14Þ

where pAiBi = pBi – pAi. Substituting Eqs. (12) and (13) into Eq. (14) and using the properties of triple product, we obtain the following relation: pTAiBi ki q_ i

¼



pTAiBi

   tP : ðRP rBi  pAiBi Þ xP

ð15Þ

T

We can write Eqs. (15) for each kinematic chain and then group them in the wellknown form: Aq_ ¼ BV; ð16Þ where V is the six-component vector that includes platform velocities tP and xP; matrix A has elements pTAiBi ki on its main diagonal and zeros elsewhere; each row of matrix B is defined according to the right side of Eq. (15). Drive velocities q_ can be easily found from Eq. (16) by inverting matrix A. Notice that both matrices A and B are known from the inverse kinematics and depend on the manipulator configuration. In some configurations, any of these matrices or even both can degenerate, leading to manipulator’s singularities [12], calculation of which requires an additional study. 3.3

Case Study

Let’s consider an example of the kinematic analysis for the manipulator with the following parameters corresponding to the CAD model (Fig. 1): R1 = 250 mm; li = 67 mm, di = 160 mm, and Li = 222 mm for all i-th parameters; ai = 30°, 90°, 150°, 210°, 270°, 330°; rBi ¼ ½ R8 cosbi R8 sinbi 0 T , where R8 = 153 mm is the platform radius, bi is the angular placement of spherical joints 7–8, bi = 7.5°, 112.5°, 127.5°, 232.5°, 247.5°, 352.5°. Suppose that the platform has to follow the screw-like trajectory in the form: 2 6 pP ðtÞ ¼ 4

0

3

2

cosuðtÞ sinuðtÞ 1

7 6 0 5 mm; RP ðtÞ ¼ 4 sinuðtÞ 160 þ 3t 0

uðtÞ ¼ 10 þ 2t deg; t ¼ ½0; 10 s:

cosuðtÞ 0

3

7 0 5; 0

ð17Þ

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With this trajectory, platform moves along axis Z and simultaneously rotates around it. From Eq. (17), platform velocities have the following constant values: tP ¼ ½ 0

0 3 T mm=s; xP ¼ ½ 0

0 2 T deg=s:

ð18Þ

The inverse kinematics and velocity analysis have been performed according to the developed algorithms and implemented in MATLAB. Figure 3 demonstrates the results for the drive coordinates and velocities. Notice that the motions of cranks with i = 1, 3, and 5 coincide with each other, and the same is for i = 2, 4, and 6 in this case study. Driven velocities have been also calculated by the numerical differentiation (green dots in Fig. 3), and the results completely coincide with the ones obtained by the suggested analytical algorithm (red lines in Fig. 3). This verifies the correctness of the proposed methodology.

Fig. 3. Results of the kinematic analysis: blue lines are q; red lines are q_ according to the proposed algorithm; green dots are q_ by the numerical differentiation of q.

4 Conclusions The study has presented a new variation of a 6-DOF hexapod-type manipulator with a circular guide, in which all drives stay on a fixed base and the possibility of collision of adjacent carriages is eliminated. For this manipulator, an algorithm for solving the inverse position problem has been developed. It determines the analytical relations between the platform coordinates and the driven ones. Based on the obtained position

Inverse Kinematics and Velocity Analysis

19

problem solution, an algorithm for determining the drive velocities has been developed. The considered case study has demonstrated the correctness of both algorithms. In particular, the motion of the end-effector along a helical trajectory has been considered and the time dependencies for the drive coordinates and velocities have been determined. Moreover, the velocities have also been calculated by the numerical differentiation, and their values confirmed the results obtained by analytical methods. The proposed algorithms form a basis for further research on workspace construction, singularities, dynamics, and control. Acknowledgements. The reported study was carried out with the support of the Russian President Grant according to the research project MK-2781.2019.8.

References 1. Dasgupta, B., Mruthyunjaya, T.S.: The Stewart platform manipulator: a review. Mech. Mach. Theory 35, 15–40 (2000) 2. Stewart, D.: A platform with six degrees of freedom. Proc. Inst. Mech. Eng. 180(1), 371–386 (1965) 3. Deng, H., Xin, G., Zhong, G., Mistry, M.: Object carrying of hexapod robots with integrated mechanism of leg and arm. Robot. Comput. Integr. Manufact. 54, 145–155 (2018) 4. Merlet, J.-P.: Parallel Robots, 2nd edn. Springer, Dordrecht (2006) 5. Carbone, G., Shrot, A., Ceccarelli, M.: Operation strategy for a low-cost easy-operation Cassino Hexapod. Appl. Bionics Biomech. 4(4), 149–156 (2008) 6. Glazunov, V.A., Briot, S., Arakelyan, V., Gruntovich, M.M., Ngyuen, M.T.: Development of manipulators with a parallel-cross structure. J. Mach. Manufact. Reliab. 37, 176–185 (2008) 7. Chiu, Y.-J., Perng, M.-H.: Self-calibration of a general hexapod manipulator with enhanced precision in 5-DOF motions. Mech. Mach. Theor. 39(1), 1–23 (2004) 8. Fomin, A., Antonov, A., Glazunov, V.: Forward kinematic analysis of a rotary hexapod. In: Venture, G., Solis, J., Takeda, Y., Konno, A. (eds.) ROMANSY 23 - Robot Design, Dynamics and Control. ROMANSY 2020. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 601, pp. 486–494 (2021) 9. Aleshin, A.K., Glazunov, V.A., Shai, O., Rashoyan, G.V., Skvortsov, S.A., Lastochkin, A.B.: Infinitesimal displacement analysis of a parallel manipulator with circular guide via the differentiation of constraint equations. J. Mach. Manufact. Reliab. 45, 398–402 (2016) 10. Coulombe, J., Bonev, I.A.: A new rotary hexapod for micropositioning. In: Proceedings of the IEEE International Conference on Robotics and Automation, Karlsruhe, pp. 877–880 (2013) 11. Fomin, A., Antonov, A., Glazunov, V., Rodionov, Y.: Inverse and forward kinematic analysis of a 6-DOF parallel manipulator utilizing a circular guide. Robotics 10(31), 1–13 (2021) 12. Bohigas, O., Manubens, M., Ros, L.: Singularities of Robot Mechanisms. Springer, Cham (2017)

Torque Minimization of Dynamically Decoupled R-R Spatial Serial Manipulators via Optimal Motion Control Vigen Arakelian1,2(&), Jing Geng1,2, and Yaodong Lu1,2 1

LS2N-ECN UMR 6004, 1 rue de la Noë, BP 92101, 44321 Nantes, France 2 INSA Rennes/Mecaproce, 20 av. des Buttes de Coesmes, CS 70839, 35708 Rennes, France {Vigen.Arakelyan,jing.geng,yaodong.lu}@insa-rennes.fr

Abstract. This paper deals with the analytically tractable solution for input torques minimization of decoupled two degrees of freedom spatial serial manipulators based on generation of motion via “bang-bang” profile. The problem is considered in two steps: at first, the dynamic decoupling of the manipulator involves redistributing the moving mass, which leads to the decoupling of motion equations. Then, the input torques are minimized via generation of motion of links through “bang-bang” law. The efficiency of the suggested method via numerical simulations carried out with ADAMS is demonstrated. Keywords: Input torque
Dynamic decoupling
Spatial serial manipulators Common center of mass
Optimal control
“Bang-Bang” profile




1 Introduction It is known that the manipulator dynamics are highly coupled and nonlinear [1]. The complicated dynamics results from varying inertia and interactions between the different joints. The aim of dynamic decoupling of manipulators is to ensure the conditions, which allow one to obtain decoupled and linear dynamic equations. It simplifies the optimal control of manipulators. However, it does not solve the problem of reducing the loads on the actuators. The loading of manipulator actuators depends on the distribution of mass in the links as well as the efficient motion generation. In this paper, we will consider the problem of reducing the input torque of the dynamically decoupled R-R spatial manipulator via efficient motion generation. First of all, let’s consider the design approaches of dynamic decoupling of manipulators and which one is appropriate for the special R-R series manipulators. There are three ways to create the dynamically decoupled manipulators through mechanical transformations: i) via mass redistribution; ii) via actuator relocation and iii) via addition of auxiliary links. In order to eliminate the coupling and nonlinear torques via mass redistribution, the inertia matrix must be diagonalized and made invariant for all the arm configuration [1–3]. The linearization and dynamic decoupling of 3-DOF manipulators via mass © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 20–27, 2021. https://doi.org/10.1007/978-3-030-75271-2_3

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21

redistribution have been considered [3]. In this study, all of the arm constructions that yielded the decoupled inertia matrices were identified. The suggested approach is applied to serial manipulators in which the axis of joints are not parallel. In the case of parallel axes such an approach allows linearization of the dynamic equations but not their dynamic decoupling [4]. Thus, in the case of planar serial manipulators, it cannot be used. Therefore, in serial manipulator arms with an open kinematic chain structure, the inertia matrix cannot be decoupled unless the joint axes are orthogonal to each other. As regards decoupling of dynamic equations via actuator relocation, the review [1] have shown that the design concept with remote actuation is not optimal from the point of view of the precise reproduction of the end-effector tasks because it accumulates all errors due to the intermediate transmissions. Obviously, it is lot better to connect actuators directly with links than to use transmission mechanisms. The clearance, flexibility, manufacturing and assembly errors of the added transmission mechanisms have a negative impact to the robot precision. The linearization of the dynamic equations and their decoupling via redesign of the manipulator by adding auxiliary links has also been developed [5–7]. A new mechatronic design approach, which is based on the opposite motion of manipulator links and the optimal command design has been discussed in [7]. The opposite motion of links with optimal redistribution of masses allows the cancellation of the coefficients of nonlinear terms in the manipulator’s kinetic and potential energy equations. Then, by using the optimal control design, the dynamic decoupling is achieved. The proposed approach, which is a symbiosis of mechanical and control solutions, improves the known design concepts permitting the dynamic decoupling of serial manipulators whilst taking into account the changing payload. The dynamic decoupling of the manipulator by connecting to initial structure a two links group forming a Scott-Russell mechanism has been proposed in [8]. It should be noted that the dynamic decoupling via redesign of the manipulator by adding auxiliary links is a promising new approach in robotics. Thus, taking into account the structural features of the R-R spatial serial manipulator, i.e., orthogonality of the joint axes, we can confidently assert that dynamic decoupling via mass redistribution is the most convenient. With regard to torque minimization in robot manipulators, there are two different approaches. In static mode, it is easily solved by gravity compensation [9, 10]. However, in dynamic mode, solving this problem by means of mechanical modifications is quite difficult [11], therefore the efficient motion generation is applied. The solutions result in torque saturation. Several offline planning algorithms have been proposed for the minimum time trajectory, considering manipulator dynamics with torque limits of the actuators [12–17]. However, in this paper, based on simplified motion equations due to the dynamic decoupling, the torque minimization will be carried out by diminishing the maximum angular accelerations of robot links. For this purpose, the “bang-bang” law will be applied.

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2 Dynamic Decoupling of the R-R Spatial Serial Manipulator Let us consider the dynamic decoupling of the R-R spatial serial manipulator (Fig. 1).

Fig. 1. R-R spatial serial manipulator.

The manipulator consists of two orthogonal links 1 and 2 with rotating angles h1 0 0 and h2 . We will distinguish the vectors h_ 1 and h_ 2 relative angular velocities with h_ 0 ¼ dh1 =dt, h_ 0 ¼ dh2 =dt and the vectors h_ and h_ absolute angular velocities with 1

2

1

2

0 0 0 h_ 1 ¼ h_ 1 and h_ 2 ¼ h_ 1 þ h_ 2 . According to Lagrangian dynamics, the equations of motion can be written as:


 d @L @L ; i ¼ 1; 2 si ¼  dt @ h_ i @hi where, si are the torques; hi are the generalized coordinates; L ¼ K  P is the Lagrangian; K is the kinetic energy and P is the potential energy. One of the requirements for dynamic decoupling of robot manipulators [1] is the static balance of the manipulator. Therefore, the centre of mass S1 of link 1 should be on the vertical rotation axis of the manipulator (Fig. 1) and the centre of mass S2 of link 2 should be at the intersection of the vertical and horizontal rotation axes of the manipulator. In these conditions, the common centre of mass of the manipulator becomes motionless and P ¼ 0. With regard to the kinetic energy of the manipulator, taking into account the conditions of the mentioned static balancing, we can write: K ¼ K1 þ K2 ¼

 1  _2 I1 h1 þ Ix2 h_ 2x2 þ Iy2 h_ 2y2 þ Iz2 h_ 2z2 2

Torque Minimization of Dynamically Decoupled R-R Spatial Serial

23

where, I1 is the axial moment of inertia of link 1; Ix2 ; Iy2 ; Iz2 are the axial moments of inertia of link 2 relative to the corresponding coordinate axes of the system associated with link 2; h_ x2 ; h_ y2 ; h_ z2 are the components of the angular velocity h_ 2 about the same  12 axes, i.e. h_ 2 ¼ h_ 2x2 þ h_ 2y2 þ h_ 2z2 . Figure 1 shows that h_ x2 ¼ h_ 01 sin h2 h_ y2 ¼ h_ 01 cos h2 h_ z2 ¼ h_ 02 Thus, the Lagrangian of the manipulator can be rewritten as:   2  2  2  1 2 0 0 _ _ _ L ¼ K ¼ K1 þ K2 ¼ I1 h1 þ Ix2 h1 sin h2 þ Iy2 h1 cos h2 þ Iz2 h_ 02 2 As it follows from the obtained relationship, the dynamic decoupling of the manipulator can be achieved if Ix2 ¼ Iy2 ¼ I  , since in this case:   2  2  1 0  _ L ¼ K ¼ K1 þ K2 ¼ ðI1 þ I Þ h1 þ Iz2 h_ 02 2 and as a consequence s1 ¼ ðI1 þ I  Þ€h01 s2 ¼ Iz2 €h02 From the obtained equations, it follows that decreasing the maximum values € h01 and €h0 , the torques will also be reduced. To ensure it, let us consider the efficient motion 2 generation of links.

3 Motion Generation via “Bang-Bang” Profile It is obvious that the maximal values of the angular accelerations change following the   pffiffiffi motion profile [18]: for quantic polynomial profile € hmax  ¼ 10h 3t2 and for bang  bang profile €hmax  ¼ 4h t2 (Fig. 2).

24

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Fig. 2. “Bang-bang” profile used for generation of motion.

According to the proposed method, the rotation of links 1 and 2 should follow bang-bang motion profile hðtÞ, i.e. 8
2  > t tf  > > h t ð Þ þ 2 h; 0  t  i > < tf 2 " hðtÞ ¼

2 #  > tf  > > hðti Þ þ 1 þ 4 t  2 t h; 0  t  > : tf tf 2 Thus, the application of bang-bang law theoretically brings about a reduction of 31% of the maximal value of the angular acceleration. Hence, to minimize the maximum values of the angular accelerations of the rotating links 1 and 2, as a result, torques, the “bang-bang” profile should be used. Thus, by reducing the angular accelerations of the manipulator’s links, a decrease in their torques can be achieved.

Torque Minimization of Dynamically Decoupled R-R Spatial Serial

25

4 Illustrative Example and Simulation Results To create a CAD model and carry out simulations via ADAMS software, a decoupled R-R spatial serial manipulator with I1 ¼ 0:4 kgm2 ; I  ¼ 0:1 kgm2 ; Iz2 ¼ 0:14 kgm2 is considered. The initial and final values of the rotating angles h1 and h2 are the following: h1i ¼ h2i ¼ 0; h1f ¼ 90 and h2f ¼ 60 : Two types of motions for tf ¼ 0:2s are simulated: 1) with fifth order polynomial profile; 2) “bang-bang” profile.

Fig. 3. Variations of torque s1 for two studied cases.

Fig. 4. Variations of torque s2 for two studied cases.

Figure 3 and Fig. 4 show the variations of torques for two studied cases: 1) the generation of the motion with fifth order polynomial profile and 2) the generation of the motion by “bang-bang” profile. The simulation results show that the two torques have been reduced up to 30.7%.

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5 Conclusion This paper considers the problem of reducing the torques in robot manipulators with decoupled dynamics. The R-R spatial serial manipulator is considered. Firstly, the dynamic decoupling of the manipulator is carried out. To obtain decoupled and linear dynamic equations of the manipulator, the arrangement of the centers of mass and the inertia redistribution of the links are described. It is shown that in the obtained dynamically decoupled manipulator, the input torques are directly proportional to the input angular accelerations. Then, for generation of motion, it is proposed to apply the “bang-bang” profile, which allows one to reduce the maximal values of input torques. Numerical simulations carried out via ADAMS software illustrate the efficiency of the suggested solution.

References 1. Arakelian, V., (ed.) Dynamic Decoupling of Robot Manipulators. Springer, Switzerland (2018) 2. Youcef-Toumi, K., Asada, H.: The design of open-loop manipulator arms with decoupled and configuration-invariant inertia tensors. In: 1986 IEEE International Conference on Robotics and Automation, pp. 2018–2026 (1986) 3. Youcef-Toumi, K., Asada, H.: The design of open-loop manipulator arms with decoupled and configuration-invariant inertia tensors. J. Dyn. Syst. Meas. Control 109(3), 268–275 (1987) 4. Gompertz, R.S., Yang, D.C.: Performance evaluation of dynamically linearized and kinematically redundant planar manipulators. Robot. Comput. Integr. Manufact. 5(4), 321– 331 (1989) 5. Coelho, T.A.H., Yong, L., Alves, V.F.A.: Decoupling of dynamic equations by means of adaptive balancing of 2-dof open-loop mechanisms. Mech. Mach. Theor. 39(8), 871–881 (2004) 6. Arakelian, V., Sargsyan, S.: On the design of serial manipulators with decoupled dynamics. Mechatronics 22(6), 904–909 (2012) 7. Arakelian, V., Xu, J., Le Baron, J.-P.: Mechatronic design of adjustable serial manipulators with decoupled dynamics taking into account the changing payload. J. Eng. Des. 27(11), 768–784 (2016) 8. Arakelian, V., Xu, J., Le Baron, J.-P.: The design of planar serial manipulators with decoupled dynamics taking into account the changing payload. J. Robot. Mech. Eng. Res. 1 (4), 39–46 (2017) 9. Arakelian, V.: Gravity compensation in robotics. Adv. Robot. 30(2), 79–96 (2016) 10. Arakelian, V., Briot, S.: Balancing of Linkages and Robot Manipulators: Advanced Methods with Illustrative Examples. Springer, Berlin (2015) 11. Arakelian, V., Le Baron, J.-P., Mottu, P.: Torque minimisation of the 2-DOF serial manipulators based on minimum energy consideration and optimum mass redistribution. Mechatronics 21(1), 310–314 (2011) 12. Chin, F.G., McKay, N.D.: Minimum-time control of robotic manipulators with geometric path constraints. IEEE Trans. Autom. Control 130(6), 531–541 (1985) 13. Bobrow, J.E., Dubowsky, S., Gibson, J.S.: Time-optimal control of robotic manipulators along specified paths. International Journal of Robotic Research 4(3), 3–17 (1985)

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14. Arai, H., Tanie, K.: Real-time path tracking with torque limits by using a disturbance observer. IEEE Int. Conf. Robot. Autom. 3, 1859–1865 (1994) 15. Arai, H., Tanie, K., Tachi, S.: Path tracking control of a manipulator considering torque saturation: Industrial robotics. IEEE Trans. Indus. Electron. 41(1), 25–31 (1994) 16. Ohishi, K., Someno, T.: Robust robot manipulator control with autonomous consideration algorithm of torque saturation: intelligent motion control for robotics. Adv. Robot. 12(7-8), 755–769 (1999) 17. Eom, K.S., Suh, I.H., Chung, W.K.: Disturbance observer based path tracking control of robot manipulator considering torque saturation. Mechatronics 11, 325–343 (2001) 18. Khalil, W., Dombre, E.: Modeling Identification and Control of Robots. Hermès, Paris (2003)

Kinematic Analysis of a Coaxial 3-RRR Spherical Parallel Manipulator Based on Screw Theory Alejandro T. Cruz-Reyes1 , Manuel Arias-Montiel2(B) , and Ricardo Tapia-Herrera3 1

2

Division of Postgraduate Studies, Universidad Tecnol´ ogica de la Mixteca, Oaxaca, Mexico Institute of Electronics and Mechatronics, Universidad Tecnol´ ogica de la Mixteca, Oaxaca, Mexico [email protected] 3 CONACyT-Universidad Tecnol´ ogica de la Mixteca, Oaxaca, Mexico [email protected]

Abstract. This work deals with the forward and inverse kinematic analysis of a coaxial spherical parallel manipulator (SPM) by using the Euler angles representation for the mobile’s platform orientation. A unique solution for the inverse and forward kinematic problem which corresponds with the assembly mode of a physical prototype is considered. The screw theory is then used to derive the Jacobian matrix as well as to carry out the velocity analysis. Finally, the solutions from both analysis are compared with numerical results obtained by simulation using ADAMS View software for different position and velocity trajectories, showing the validity of the obtained results.

Keywords: Spherical parallel manipulator theory

1

· Kinematics · Screw

Introduction

Parallel mechanisms with spherical motion are intended to provide a threedegree-of-freedom (DOF) of pure rotation. These kinds of parallel kinematic machines (PKM) have found a wide range of applications in diverse areas such as camera systems, machine tools, telescopes, medical robots and exoskeletons [1]. Research on spherical parallel manipulators (SPM) began in 1989 with the article from Gosselin and Angeles [2], and since then extensive studies have been reported covering diverse topics about kinematic analysis, design and optimization as described in the recent review by Bai et al. [1]. In recent years, a particular configuration of SPM has captured the interest of researchers and academics in the area of robotics. The coaxial SPM of 3 DOF is used in applications where full 360◦ or unlimited rolling around an axis is required. Detailed analysis on c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 28–37, 2021. https://doi.org/10.1007/978-3-030-75271-2_4

Kinematic Analysis of a Coaxial 3-RRR SPM

29

the forward and inverse position kinematics are presented in [3]. However, to the best of the authors’ knowledge, numerical results for the velocity analysis for this kind of SPM have been rarely reported and validated. On the other hand, over the past few decades screw theory has emerged as an advantageous method to directly obtain input-output equations of velocity, acceleration, as well as carry out higher-order kinematic analysis (see [4] and references within). In this context, Wu [5,6] used the screw theory approach to characterize the stiffness and optimize the geometric synthesis of the general SPM using kinematic and dynamic indexes. Hou et al. [7] analyzed the motion feature of a coaxial SPM through screw theory. However the Jacobian matrix was obtained by means of time derivation of the constraint equations and a velocity analysis was not presented. In this paper, the forward and inverse kinematic analysis of the coaxial SPM is presented. The Euler angles representation for the mobile platform’s orientation is used. In addition, the Jacobian matrix is obtained by means of screw theory and is used to derive the velocity input-output equation. The solutions from both analyses are compared with numerical results obtained by simulation using ADAMS View software for position and velocity trajectories, thereby showing the validity of the obtained results.

2

Coaxial SPM Description

Figure 1 shows the kinematic diagram of a coaxial spherical parallel mechanism 3-RRR, which is formed using three limbs with coaxial active joints connecting a mobile platform to a fixed base. All the joint axes of the mechanism are concurrent in the origin O of the fixed XYZ reference frame, which is referred to as the center of rotation. The mobile platform has spherical movement of three DOF and its joint axes form a tetrahedron with the point O around which it rotates. The geometry of the tetrahedron is defined by the β angle. The mobile platform is connected to the fixed base through three equally spaced limbs which are numbered counter-clockwise as i = 1, 2, 3. Each one is comprised of two links, proximal and distal, where the proximal link is attached to the fixed base and the distal link is attached to the mobile platform. Furthermore, α1 and α2 are the proximal and distal links angles, respectively. The joint axes are defined by unitary vectors ui , vi and wi , directed from the rotation center to its respective joint. The coordinated or simultaneous movement of the mobile platform’s actuated joints enables the unlimited rotation propriety around an arbitrarily oriented axis [3]. The mechanism’s “home” position is established so that the proximal links are set at 120◦ from each other. The Z axis of the fixed reference frame is normal to the fixed base, while the X axis is located in the plane of the proximal 1 in the home configuration, as shown in Fig. 1. The Y axis is given by right hand rule. Furthermore, in the home configuration the mobile platform is horizontal and its normal vector coincides with the positive Z axis of the fixed reference frame.

30

A. T. Cruz-Reyes et al. Z

v1 2$31

β

o

X

α2 α1

0 1 1

$ u1

1 2 1

$

w1

θi

Fig. 1. Kinematic diagram of the coaxial 3-RRR parallel mechanism.

Input variables form the θ = [ θ1 θ2 θ3 ]T vector and are measured clockwise from the home configuration, where θ = [ 0 0 0 ]T . Unitary vectors ui , i = 1, 2, 3 of the actuated joints are defined as: ui = [ 0 0 −1 ]T . Unitary vectors wi , i = 1, 2, 3 of the intermediate joints are defined as: T  wi = cos(ηi − θi ) sin α1 sin(ηi − θi ) sin α1 − cos α1 ,

(1)

(2)

where ηi = 2(i − 1)π/3, i = 1, 2, 3. Unitary vectors vi , i = 1, 2, 3 of the mobile platform joints are used to describe its orientation. However, sometimes it is convenient to express mobile platform orientation by using a more conventional representation, such as Euler angles, for which it is necessary to calculate the rotation matrix R to obtain the vi vectors through the following relation vi = Rvi,h ,

(3)

where vi,h are the home configuration vi vectors. In the case of the roll-pitch-yaw (RPY) Euler angles convention for describing rotations around the Z, Y and X axes of the fixed reference frame, respectively, the rotation matrix R is obtained from the sequence of basic rotations R = Rx (θyaw )Ry (θpitch )Rz (θroll ),

(4)

where θyaw , θpitch and θroll are the desired RPY Euler angles.

3 3.1

Position Analysis Inverse Kinematics

The inverse kinematics problem is defined by obtaining the angular position vector θ corresponding to a given orientation of the mobile platform, defined by

Kinematic Analysis of a Coaxial 3-RRR SPM

31

unitary vectors vi , i = 1, 2, 3. Solutions for the inverse kinematics are obtained by the closure equations [2] wi · vi = cos α2 , i = 1, 2, 3,

(5)

which, for each limb, leads to a quadratic equation of the form Ai Ti2 + 2Bi Ti + Ci = 0, 

with Ti = tan

θi 2

i = 1, 2, 3,

(6)

 .

(7)

The coefficients Ai , Bi and Ci are: Ai = −vix cos ηi sin α1 − viy sin ηi sin α1 − viz cos α1 − cos α2 Bi = vix sin ηi sin α1 − viy cos ηi sin α1 Ci = vix cos ηi sin α1 + viy sin ηi sin α1 − viz cos α1 − cos α2 ,

(8)

where vix , viy and viz are the vi vector components. Quadratic equation (6) has two roots for Ti , thus for a given orientation of the mobile platform there are two solutions for each position θi , resulting in a total of up to eight total solutions, known in literature as working modes [8]. 3.2

Forward Kinematics

Forward kinematics of the coaxial SPM defines the orientation of the mobile platform described in terms of the vi unitary vectors for a given input vector θ. It is assumed that ui and wi are known a priori. Each unitary vector vi contains X, Y and Z components, resulting in nine unknown variables. An independent nineequation system is obtained from (5) with the following geometric restrictions: vi · vj = cos α3 , vi  = 1,

i, j = 1, 2, 3, i = j (9)

where α3 is the angle between the axis of the i th and j th mobile platform joints. The equation system given by (5) and (9) consisting of three linear and six quadratic equations in the vi vectors. Typically, these systems cannot be solved analytically and explicit equations cannot be obtained. For this reason it is necessary to use numeric methods to find the solution [3]. Therefore, an initial approximation x0 which contains the vi vector components is needed, so that x0 = [ v1x v1y v1z v2x v2y v2z v3x v3y v3z ].

(10)

For any proximal link there are two ways to mount the associated distal link (left-directed or right-directed) for coupling with the mobile platform. Therefore, multiple solutions are produced by solving the forward kinematics problem. This results in a total of up to eight possible solutions known in literature as assembly modes [8]. Tursynbek et al. [3] give a useful algorithm to obtain a unique solution to the forward kinematics problem that corresponds with the assembly mode of a physical prototype.

32

3.3

A. T. Cruz-Reyes et al.

Numerical Results

In order to validate the position analysis, a coaxial SPM was proposed with the following parameters: α1 = 45◦ , α2 = 90◦ and β = 90◦ . All distal links are leftdirected mounted for coupling with the mobile platform, the orientation of which is specified in RPY Euler angles. A 3-D model of the mechanism was developed in ADAMS View environment in order to compare the analytical results with a numerical approach (Fig. 2).

Fig. 2. 3D model of the coaxial 3-RRR SPM for simulation in ADAMS.

Based on the scheme in Fig. 3, a trajectory for the orientation of the mobile platform is proposed and used to solve the inverse kinematics, thereby obtaining the input angles. Next, the obtained angles are considered as inputs for the forward kinematics problem, whose outputs are the orientation of the mobile platform. Finally, output is compared with the initial proposed trajectory, verifying the accuracy of the obtained results. Forward and inverse kinematics equations were solved in MATLAB for the trajectories shown in Fig. 4. Typical trajectory

Inverse kinematics

Input angles

Forward kinematics

Verification trajectory

Fig. 3. Schematic representation of the kinematic problem.

The desired mobile platform trajectory shown in Fig. 4 corresponds to a third order polynomial angular displacement in the yaw angle (X axis), that varies from 0◦ to 30◦ in one second. In Fig. 5 the comparison between the inverse kinematics solution obtained in MATLAB and ADAMS View for the desired trajectory is presented. Furthermore, the same comparison was made for the forward kinematics solution, as shown in Fig. 6. By analyzing both results, it can be observed that the analytical and numerical solutions are practically identical.

Kinematic Analysis of a Coaxial 3-RRR SPM

33

Proposed trajectory examined for the coaxial 3-RRR SPM

30

Orientation (deg)

25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 4. Third-order polynomial trajectory for the mobile platform orientation. Inverse kinematic solution of the coaxial 3-RRR SPM for the given trajectory

20

Position (deg)

10 0 -10 -20 -30 -40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 5. Comparative of the inverse kinematic solution. Forward kinematic solution of the coaxial 3-RRR SPM for the given trajectory

30

Orientation (deg)

25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Fig. 6. Comparative of the forward kinematic solution.

0.9

1

34

4

A. T. Cruz-Reyes et al.

Velocity Analysis

The equation that represents the kinematic restrictions imposed by the limbs of a parallel mechanism is f (θ, X ) = 0, where θ and X are the vectors of the active joints and the variables of the mobile platform, respectively [9]. The derivative of this expression with respect to time is ˙ Aω = B θ,

(11)

where θ˙ is the velocity vector of the actuated joints and ω is the vector indicating the mobile platform angular velocity, furthermore A=

∂f ∂f and B = − . ∂X ∂θ

(12)

The general Jacobian matrix of a parallel mechanism is defined as the matrix that represents the transformation that maps the velocity vector of the actuated joints from the angular velocity of the mobile platform, so that from Eq. (11) θ˙ = Jω,

where

J = B −1 A,

(13)

for a non-singular B matrix. In this work, the velocity analysis will be performed by means of screw theory. The velocity state of the mobile platform on the fixed reference frame, Vo , can be expressed as a screw through any kinematic chain of the robot [4] as   ω (14) = 0 ω1i 0 $1i + 1 ω2i 1 $2i + 2 ω3i 2 $3i , VO = vo where, for the i th chain, the screw a $bi models the link b movement with respect to the link a; a ωbi represents the magnitude of the linear and angular velocity of the link b referenced to the link a and vo is the linear velocity of the mobile platform seen from the fixed base. The revolute joints screws are:       ui wi v 0 1 $i = , 1 $2i = , 2 $3i = i . (15) 0 0 0 It can be proven that screws 1 $2i and 2 $3i are simultaneously reciprocal to a screw of the form   0 r $i = . (16) (wi × vi )T Equation (16) can be seen as the screw that represents a virtual prismatic joint, whose dual part is perpendicular to the primal part of 1 $2i and 2 $3i . Applying the Klein form [4] on both sides of the Eq. (14), and taking into account that the velocity magnitude 0 ω1i is related to the i th actuated joint, i.e., 0 ω1i = θ˙i , it follows   Vo $ri = θ˙i 0 $1i $ri (17) T ˙ (wi × vi ) ω = (ui × wi ) · vi θi ,

Kinematic Analysis of a Coaxial 3-RRR SPM

35

 where ∗ ∗ denotes the Klein form. Equation (11) is obtained from writing (17) in matrix form, where ⎤ ⎡ (w1 × v1 )T A = ⎣ (w2 × v2 )T ⎦ (w3 × v3 )T   B = diag (u1 × w1 ) · v1 (u2 × w2 ) · v2 (u3 × w3 ) · v3 . (18) Furthermore, from Eq. (13) T  J = B −1 A = j1 j2 j3 ,

ji =

(wi × vi )T . (ui × wi ) · vi

(19)

ji is the i th row of the Jacobian matrix being an orthogonal vector to the plane defined by the vi and wi vectors, i.e., is an orthogonal vector to the plane defined by distal link. 4.1

Numerical Results

In order to validate the SPM velocity analysis, the same mechanism parameters and desired trajectory as the position analysis are used. In this case, the velocity profile corresponding to the trajectory is shown in Fig. 7. In Fig. 8 the comparative between the inverse kinematics solution for the velocity obtained in MATLAB and ADAMS View for the desired trajectory is presented. The comparative was made for the forward velocity solution which can be seen in Fig. 9. By analyzing both results, it can be observed that the solution obtained by the two different methods is practically identical. The aforementioned ensures that the velocity analysis using the screw theory was carried out correctly, and the power of this tool in the resolution of the first order SPM kinematics can be observed. Proposed velocity trajectory examined for the coaxial 3-RRR SPM

0.8 0.7

Velocity (rad/s)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 7. Second-order polynomial trajectory for the mobile platform velocity.

36

A. T. Cruz-Reyes et al. Inverse kinematic solution of the coaxial 3-RRR SPM for the given trajectory

Velocity (rad/s)

0.5

0

-0.5

-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 8. Comparative of the velocity inverse analysis solution. Forward kinematic solution of the coaxial 3-RRR SPM for the given trajectory

0.8

Velocity (rad/s)

0.6

0.4

0.2

0

-0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 9. Comparative of the velocity forward analysis solution.

5

Conclusions

In this work, the position analysis of the coaxial 3-RRR SPM was addressed by using geometrical procedures and numerical methods proposed in the literature in order to obtain an interpretation of the multiple solutions generated and select the one that corresponds with the desired specific case. The velocity problem was solved by means of infinitesimal screw theory. The use of screws to obtain the velocity state of the mobile platform simplifies the analysis, using the reciprocal screw theory and the Klein form operation. The aforementioned avoids the need to compute the passive joints rates and finds the mechanism’s Jacobian matrix in a simpler manner. This procedure provides as result an identical expression for the coaxial SPM mechanism’s Jacobian matrix as reported in earlier works, where the mechanism’s closure equations are derived with respect to time.

Kinematic Analysis of a Coaxial 3-RRR SPM

37

The data validation was carried out with ADAMS View software by comparing the results of the mechanism 3D simulation with those obtained through the analytical equations, thereby obtaining identical results for both position and velocity analysis. This comparative allows one to observe and confirm the effectiveness of the methods proposed for the solution of the mechanism kinematics, and the potential that these tools have in the analysis of spherical parallel mechanisms. As a future work, the obtained results can be extended for the dynamical analysis and compare the performance of the manipulator with other configurations as reported in [10].

References 1. Bai, S., Li, X., Angeles, J.: A review of spherical motion generation using either spherical parallel manipulators or spherical motors. Mech. Mach. Theory 140, 377–388 (2019) 2. Gosselin, C.M., Angeles, J.: The optimum kinematic design of a spherical threedegree-of-freedom parallel manipulator. J. Mech. Transm. Autom. Design 111(2), 202–207 (1989) 3. Tursynbek, I., Niyetkaliyev, A., Shintemirov, A.: Computation of unique kinematic solutions of a spherical parallel manipulator with coaxial input shafts. In: Proceedings of IEEE 15th International Conference on Automation Science and Engineering (CASE), New York, USA, pp. 1524–1531. IEEE (2019) 4. Gallardo-Alvarado, J.: Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory, 1st edn. Springer, Switzerland (2016) 5. Wu, G.: Multiobjective optimum design of a 3-RRR spherical parallel manipulator with kinematic and dynamic dexterities. Model. Ident. Control 33(3), 111–122 (2012) 6. Wu, G.: Stiffness analysis and optimization of a co-axial spherical parallel manipulator. Model. Ident. Control 35(1), 21–30 (2014) 7. Hou, Y., Hu, X., Zeng, D.: Kinematics analysis of a 3-RRR spherical parallel mechanism with coaxial input shafts. Appl. Mech. Mater. 404, 237–243 (2013) 8. Bonev, I., Chablat, D., Wenger, P.: Working and assembly modes of the Agile Eye. In: Proceedings of the 2006 IEEE International Conference on Robotics and Automation ICRA 2006, New York, USA, pp. 2317–2322. IEEE (2006) 9. Taghirad, H.D.: Parallel Robots: Mechanics and Control, 1st edn. CRC Press, Boca Raton (2013) 10. Wu, G., Caro, S., Bai, S., Kepler, J.A.: Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator. Robot. Auton. Syst. 62(10), 1377–1386 (2014)

A Novel Design for an Autonomous Mobile Agricultural Fruit Harvesting Robot Divyansh Khare(B) , Sandra Cherussery, and Santhakumar Mohan Indian Institute of Technology, Palakkad, Kerala 678623, India [email protected] https://iitpkd.ac.in/people/santhakumar

Abstract. This paper focuses its attention on the mechanical design of autonomous agricultural robots used for the purpose of harvesting fruits in plantations. The paper analysis the present mechanical designs, identifies the research gaps and proposes a novel design of a fruit plucking robot. A design consisting of a four wheeled mobile base, a manipulator and an end-effector has been proposed. An open-chain rotary-rotaryprismatic (RRP) manipulator attached to the mobile base and having two additional active rotary joints at the end-effector at the free-end has been designed. A hose has been provided to transfer fruits from the tree to the basket. The chassis of the mobile base is equipped with four solid rubber power (motorized) wheels controlled by skid steering method and a detachable basket for collecting fruits. A counterweight mechanism has been designed to provide static stability. The paper addresses the issues present in the current mechanical designs with a brief discussion on the system architecture. Keywords: Agricultural robot · Autonomous fruit harvesting Manipulator design · End-effector · Powered cutter

1

·

Introduction

The past few years have shown a stark increase in the demand for intelligent robots capable of performing complex tasks without any human intervention in almost every industry. The agriculture industry is not untouched by it. A lot of effort has gone into developing autonomous robots for harvesting crops, identifying and removing weed, spraying chemicals and more. Here we have focused our attention on the design of robots used for harvesting fruits [1]. Several such robots have been developed to date but most of them harvest the fruits in a manner that causes damage to either the fruit or the tree or both. We have attempted

Supported by Ministry of Electronics and Information Technology, Government of India. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 38–49, 2021. https://doi.org/10.1007/978-3-030-75271-2_5

A Novel Design for an Autonomous Mobile Agricultural Fruit

39

to develop an autonomous robotic solution for fruit harvesting by causing minimum damage to the tree and the fruit while making the fruit collection as easy as possible for the plantation workers. The paper presents a generalised design of an autonomous fruit harvesting robot that addresses the mentioned issues and can be extended in one or more functionalities as per the requirement. The design consists of a robust mobile base capable of traversing through regular and irregular ground equipped with a basket for collecting the fruits and a manipulator on the top in an RRP serial configuration having an end-effector with two active rotary joints providing two more degrees of freedom at the end of the prismatic joint. A hose-like structure has been provided between the end-effector and the chassis to facilitate the transfer of fruits from the tree to the basket. The paper starts with a review of the present designs of autonomous robots used for fruit plucking with a discussion on their pros and cons and identifies the major issues existing in these designs in the second section. The third section identifies the design requirements and proposes a possible design for the robot. A detailed discussion on the design implementation can be found in the fourth section. The architecture of the system has been discussed briefly in the fifth section with concluding remarks and acknowledgements at the end.

2

Motivation

To understand the problem better, many such robots were studied. Depending on the type of fruit and environment, a large variety of agricultural robots are employed. Few of these robots based literature are briefed below and different commercialized industrial robots are compared through a table. A fruit picking stick is discussed in [2] that employs solar-based renewable energy for powering the motor and gear mechanism. The 14 ft long and 2.6 kg weight stick can be easily used in an outdoor environment for manual picking. The dropped fruit is collected in a small basket near the cutter which is later emptied manually. A similar mechanism is briefed in [3], where a box type mechanism is employed for plucking fruits especially the thorny ones like that of the cactus pear fruit. Being specific to the type of fruit a detailed study of the engineering properties of the fruit was carried out before building the box dimensions and capacity. The efficiency and the harvesting capacity were found to be 93.86% and 10.08 kg/h respectively i.e., 32.76% more than the traditional method. To reduce the time in fruit plucking and collecting a tree trunk shaker based mechanism is detailed in [4]. An inverted - umbrella type sheet covers the tree’s main trunk and a motorized shaker at the middle shakes off the fruits from tree branches. The harvesting time per tree was 30 s and the overall detaching percentage was found to be 80%. A stereo-vision based automatic robotic arm is discussed in [5]. A 3 DOF robotic arm is fitted onto a mobile platform that can easily traverse longitudinally. The camera is placed on the mobile platform to detect fruit through the image processing technique. The overall workspace of the environment is limited to the length of the arm and is tested in a laboratory environment. The regulation of force applied to the gripper is discussed in

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[6]. A three-finger single actuated gripper is developed for the same. The prototype gripper operates with a stepper motor and pressure profile sensors are attached to each finger. The tactile based sensing reduces the dependency on the vision system for grasping the fruit. A fuzzy controller is employed to regulate the equivalent force of grasping. Furthermore, study on improving the grip time. An autonomous system that harvests most types of crops with peduncles is discussed in [7]. The proposed gripper is attached with the Fetch robot that is based on Open source ROS. The setup is tested in the laboratory environment for different artificial plants. The failures in the attachment of the fruit are overcome using adjustments in direction of crops. Other than these, some of the commercialized techniques are enlisted in the Table 1. Table 1. Industrial fruit harvesting robots Company

Product

Type of fruit

Harvesting method Pros

Cons

Abundant Robotics

Apple harvester

Apple

Vacuum suction plucking

Fast and efficient

Suction can be damaging, Heavier in weight

Dogtooth Soft fruit technologies harvester

Straw berry

Holding and cutting

Efficient detection Fruit specific of fruits

FF Robotics FFRobot

Apple, mango, Twisting and avocado plucking

Detect damaged, unripe fruits through image processing

Time consuming

AGvision

Straw berry

Multiple robotic arms

High maintenance, Heavy vehicle

SW6010

Holding and cutting

After studying these robots and a few others, the following observations were made: 1. Most of these robots are employed for harvesting a particular fruit and hence their designs are restrictive and not fit for customization. 2. The mechanical design usually consists of a mobile base and a manipulator over the mobile base performing the tasks of picking and plucking. The manipulator design is inherently complex. 3. Most of these robots are implemented by simply integrating commercial robotic arms (designed for industries and shop floors) with a mobile base providing an expensive solution for the problem of harvesting fruits autonomously. 4. The harvesting action is performed by one of the following methods: – Plucking by gripping: Many robots use a gripper (traditional/soft) as the end-effector. The gripper grips the fruit and it either twists it or pulls it towards itself or does both to separate it from the branch. This is the action that causes damage to the fruit and the tree. – Plucking by vacuum suction: This method utilises the presence of a vacuum to suck the fruit into a tube separating it from its branch. The effects of this method are also similar to that of plucking by gripping.

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– Shaking and collecting: Some robots collect the fruits by shaking the whole tree and collecting the fallen fruit in an inverted umbrella-like structure. This method causes great damage to the fruits and the tree as the tree and its roots get weakened by the shaking action and the fruits falling also suffer some damage. 5. In almost all such robots, vision is primarily used for sensing. Also, most of the literature gives an efficient way of collecting fruits into a basket near to the cutter but all these require timely unloading of the fruits, which requires time and labour. Taking all these aspects into consideration, an efficient design has been proposed consisting of a mobile base, a manipulator and an end-effector through this paper.

3

Design Considerations

After studying the present designs, the following considerations were arrived upon for the robot’s design: 1. The robot should be able to harvest multiple fruits. 2. The manipulator should be simple in construction and control. 3. The manipulator should be able to reach tall trees like mango and orange trees. 4. The weight of the manipulator should be easy to balance. 5. From an overall perspective, the design should be, simple, easy to maintain, robust, modular and fit for multiple use purposes. As per the above considerations, it was decided that the robot will mainly consist of three components: the mobile base, manipulator and the end-effector. The mobile base has been made by adding four wheels powered by electric motors to an aluminium chassis and follows the skid steering mechanism for manoeuvring. To harvest the fruits, the manipulator has been kept in an RRP configuration rather than in a 3/4/5-R configuration. This has been done to decrease the complexity of the manipulator which is present when three or more revolute joints are present in the manipulator. The RRP configuration follows the spherical coordinate system and avoids the possibility of having multiple solutions. The initial two revolute joints help in macro orienting the manipulator by forming an elbow joint and the prismatic joint extends to reach the required position. The total extendable height of the manipulator has been decided to be 6 m. A 2 degree of freedom end-effector consisting of two active rotary joints has been designed to incorporate the cutter and harvest the fruits. Harvesting different fruits might require a different kind of cutting action. For example, fruits like mangoes need to be cut along with their stem because if the stem is separated from the fruit, in the separation region, the latex sap coming out of performs decaying action on the fruit. Other fruits like oranges and apples [8]

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simply are plucked with or without the stem. Hence, due to the requirement of different cutting actions, the end-effector has been designed to accommodate a variety of cutters that can be changed as per need, thereby providing the required customization.

Fig. 1. (From left to right) Physical representation and frame diagram of the robot

A hose (flexible pipe-like structure) has been provided between the endeffector and the chassis to allow for easy transfer of fruits from the tree to the basket. A similar mechanism is given in [9]. A representation of the robot is shown in Fig. 1.

4

Mechanical Design

The mechanical design has been done upholding the above-mentioned considerations. A complete design can be seen in Fig. 2. The mechanical design of the robot consists of three sections: chassis design, manipulator design and endeffector design.

Fig. 2. Robot views

A Novel Design for an Autonomous Mobile Agricultural Fruit

4.1

43

Chassis Design

A robust chassis has been created using Aluminium 6060 T-slotted cross-sections. It can be assembled using T-nuts and bolts and has been designed in a manner that leaves space for internal mechanisms and electronic components. A provision has been made in the chassis for incorporating a detachable basket for fruit collection. Transmission and Steering System. The robot is equipped with four motorized solid rubber wheels connected to the chassis to facilitate smooth motion. The power transmission and steering is done by using four electric motors which provide controlled motion to the four solid rubber wheels. A skid steering mechanism is used for manoeuvring the robot. 4.2

Manipulator Design

A representation of the manipulator has been shown in Fig. 3. The first two revolute joints have been implemented using DC servo motors and a belt drive system and the prismatic joint has been implemented using two, two-stage telescopic arms actuated by a lead screw. Two telescopic arms have been used to provide stability and strength to the overall fruit plucking mechanism and are mounted over a rotatable plate the rotation of which forms the first revolute joint. A hole has been provided in the plate to allow the hose to reach the basket. The telescopic arms are connected to the plate through another revolute joint. Towards the open end of the telescopic arms, a provision has been made to hold the end-effector equipped with a cutter.

Fig. 3. Manipulator design

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Additional Details: The manipulator has been mounted over a plate which is being rotated by a motor creating the first revolute joint. The robot can be moved along the periphery of the tree in steps to harvest the fruits and in each step the rotary joint of the plate can be used to orient the manipulator. The second rotary joint can be used to adjust the height of the manipulator and the prismatic joint to extend and retract as per requirement. As per analysis, the robot should be able to harvest the fruit with a 60-degree rotation in either direction from the mean line and a 45-degree rotation of the second revolute joint as shown in Fig. 4.

Fig. 4. Rotation angles of first and second revolute joints respectively

Telescopic Mechanism: The telescopic mechanism has been implemented using three sections that contract and expand simultaneously to achieve the required length (Fig. 5). The middle section contains a timing belt pulley system. This belt is connected with the top and bottom sections. The middle section is actuated by a lead screw driven by a motor. which in turn causes the whole arm to expand or contract.

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Fig. 5. Telescopic mechanism design

Static Balancing: The telescopic arms can be extended to a height of 6 m which may cause the system dynamics to change and be difficult to control. For the same purpose, a counter-weight mechanism has been added to balance the dynamic effects of the two telescopic arms. This has been implemented by using two links connected to sliders which are in turn connected together through a revolute joint forming an ‘X’ like structure. Two more links have been added to one of the links of the ‘X’ structure forming a parallelogram mechanism to restrict the degree of freedom of the mechanism to one. The other link of the ‘X’ structure can be considered to be the first section of the telescopic mechanism. Figure 6 explains the same by using slotted links, a pin and an extended link representing the telescopic mechanism for the sake of simplicity. This allows the system to move along with the telescopic arm’s motion and shift the counterweight’s location accordingly, thus balancing the weight.

Fig. 6. Static balancing mechanism

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End-Effector Design

Fig. 7. End-effector design

The end-effector, with 2 degrees of freedom, is to be attached at the end of the telescopic arms (Fig. 7). Here, we have portrayed a simple scissors-like cutter operated by a motor for performing the cutting action, but various kinds of cutters can be designed for specific fruits and can be attached to the end-effector [10]. The telescopic arms help in the macro positioning of the robot as per the fruit location and the end-effector, having two additional degrees of freedom helps in the micro-positioning of the cutter and cuts the fruit. A camera can also be attached to the end-effector to have better results with the chosen vision technique. The fruit then slides into a hose and gets transferred to the basket below. The end-effector consists of two active rotary joints, with the axis of one joint being perpendicular to the axis of the other, and a top plate with a motor attached to accommodate the cutter.

5

Architecture

The general architecture comprises of the sensing and perception, motion planning, robot controller, and the actuation part shown in Fig. 8. The mobile base camera with the LiDAR sensor gives the pose of the mobile manipulator concerning its environment.

Fig. 8. General architecture of the system

A Novel Design for an Autonomous Mobile Agricultural Fruit

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The end-effector camera inputs images to the image processing algorithm in the system which then, sorts and produces the position and orientation of the target fruit, jointly forming the sensing and perception part of the system. The motion planning part decides the trajectory of the mobile base and the joint velocity of the manipulator. The robot controller optimizes the path and the joint velocity to be followed, thereby, actuating the motor drivers to reach the desired target. The relative change in the pose of the robot and target fruit can be updated from the encoder feedback and camera feedback.

Fig. 9. Flowchart of the process

The general workflow of the system is as shown in Fig. 9. The initial pose of the robot remains till the target is updated by the sensing and perception part. The robot then advances to the target tree and locates the target fruit produced by the system algorithms. The target fruit approached with the desired velocity optimized by the robot controller. The cutting mechanism actuates to pluck the fruit. Later, the robot progresses to the initial pose till the next target is fed.

6

Concluding Remarks

This paper proposes a novel autonomous agriculture robot to be employed in the harvesting of fruits. The mechanical design with the system architecture has been discussed. The robot’s chassis has been designed using Aluminium 6060 T-slotted cross-sections. The robot is equipped with four powered solid

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rubber wheels. The manipulator has an RRP configuration and has been implemented using two arms each having a two-stage telescopic mechanism actuated by a lead screw with an end-effector at the free end. The end-effector has 2◦ of freedom with the capability of having various kinds of cutters/grippers at the end. A static balancing mechanism has been added to provide stability to the robot’s structure during manipulation. With these features, the robot is simpler to control, possesses more stability and offers a degree of customisation for different kinds of fruits by having different cutters/grippers at the end-effector. The present design of the robot gives the user an advantage over the other robots in a number of ways. First, it allows the user to approach tall trees and pluck the fruits either selectively or in a bunch. Second, the manipulator has the RRP configuration which follows the spherical coordinate system thereby removing the presence of multiple solutions during inverse kinematics. Third, the robot can be used for harvesting various kinds of fruits simply by changing the gripper. This allows for customization as per need. The paper does not account for the dynamic effects, control and motion planning along with the vision methods to be deployed, which remain reserved for future work in this project. The expected results of this project are the prototype development and field testing. Acknowledgement. This research is part of the development of a mobile manipulator for harvesting funded by the Ministry of Electronics and Information Technology, Government of India.

References 1. De-An, Z.J., Jidong, L., Wei, J., Ying, Z.: Design and control of an apple harvesting robot. Biosyst. Eng. 110(2), 112–122 (2011) 2. Chandru, M., Gowthamvishal, R., Karlmarx, M., Hariharansudarson, S.: Design and fabrication of multiple type fruit picking stick using renewable energy source. In: International Conference of Recent Innovations in Science and Technology. ISBN 978-93-87793-15-6 (2018) 3. Makwana, A.D., Vaibhav, R., Shukla, S., Tiwari, V.K.: Design and development of a device for harvesting of thorny fruits. Pharma Innov. J. 8(10), 173–177 (2019) 4. Torregrosa, A., Molina, J.M., Perez, M., Orti, E., Xamani, P., Ortiz, C.: Mechanical harvesting of ornamental citrus trees in Valencia, Spain. Agronomy 9(12), 827 (2019) 5. Lokhande, P., Gawand, S., Mandavkar, S., Kadam, V., Kharade, P.A.: Agriculture based fruit plucking robot. Int. Res. J. Eng. Technol. (IRJET) 6(4) (2019). e-ISSN: 2395-0056 6. Dimeas, F., Sako, D.V., Moulianitis, V.C., Aspragathos, N.A.: Design and fuzzy control of a robotic gripper for efficient strawberry harvesting. Robotica 33(5), 1085–1098 (2015) 7. Zhang, T., Huang, Z., You, W., Lin, J., Tang, X., Huang, H.: An autonomous fruit and vegetable harvester with a low-cost gripper using a 3D sensor. Sensors 20(1), 93 (2020) 8. Chiu, Y.C., Chen, S., Lin, J.F.: Study of an autonomous fruit picking robot system in greenhouses. Eng. Agric. Environ. Food 6(3), 92–98 (2013)

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9. Williams, H.A., Jones, M.H., Nejati, M., Seabright, M.J., Bell, J., Penhall, N.D., Barnett, J.J., Duke, M.D., Scarfe, A.J., Ahn, H.S., Lim, J., MacDonald, B.A.: Robotic kiwifruit harvesting using machine vision, convolutional neural networks, and robotic arms. Biosyst. Eng. 181, 140–156 (2019) 10. Davidson, J.R., Mo, C.: Mechanical design and initial performance testing of an apple-picking end-effector. In: ASME International Mechanical Engineering Congress and Exposition American Society of Mechanical Engineers, vol. 57397, p. V04AT04A011 (2015)

Parallel Manipulators

A Comparison of Algebraic and Iterative Procedures for the Generation of the Workspace of Parallel Robots Matteo Russo1(&) 1

and Marco Ceccarelli2

Faculty of Engineering, University of Nottingham, Nottingham, UK [email protected] 2 LARM2: Laboratory of Robot Mechatronics, University of Rome “Tor Vergata”, Rome, Italy [email protected]

Abstract. Computing the workspace of parallel manipulators is critical to characterize their behavior. Usually, workspace is obtained by iterating the robot’s forward kinematics for a discrete range of poses, resulting in a cloud of reachable points. Here we propose an alternative algebraic formulation that is based on a recursive generation of volumes for solids of revolution to evaluate each limb’s workspace. The robot’s workspace is then obtained as the intersection of the limbs’ workspaces. With a practical example on the CaPaMan design and on a 3-UPR architecture, both the numerical and algebraic methods are discussed with their features and limits. Keywords: Kinematics manipulators


Mechanism analysis
Workspace
Parallel

1 Introduction The architecture of parallel robots consists of two platforms—one fixed, one mobile— that are connected by multiple kinematic chains, called limbs, that determine their relative motion [1, 2]. When compared to serial robotic architectures, which are made of a single kinematic chain from base to end-effector, parallel robots usually perform better in speed and repeatability. Furthermore, as each limb is usually actuated by a single motor, the moving masses can be limited to move with a lower inertia and higher efficiency than in the case of serial robots. However, the complex kinematics and the multiple physical constraints on the mobile platform result in a smaller workspace that can be further split into distinct working volumes by singular poses and assembly configurations [1]. These issues make the workspace of a parallel robot difficult to be represented with its geometrical shape, so that, rather than using the full reachable workspace, many researchers opt to operate a parallel robot in a subset of it that corresponds to suitable geometrical shapes, such as cylinders or spheres [3]. In general, the workspace is calculated in a recursive way, by iterating the forward kinematic problem of the robot with discretized pose parameters ranging from their minimum to maximum values. This workspace generation procedure results in a point © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 53–61, 2021. https://doi.org/10.1007/978-3-030-75271-2_6

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cloud that includes the reachable poses of the robot, and as such it may be difficult for an easy geometrical interpretation unless a subset with a “familiar”, easy-to-use shape (e.g. cylinder, cube, sphere) is used. Furthermore, while a good approximation of the workspace can only be obtained with a fine discretization interval, large intervals are needed for efficient computation with shorter times to solution [1, 2]. An algebraic approach for the generation of the workspace of robots has been often proposed as an alternative to the popular iterative solution. This approach provides an exact solution for workspace volume and topology by generating the workspace geometrically as a solid of revolution when considering revolute motion generation [4, 5]. A rich literature on the topic can be found for serial manipulators, but the application of this method to parallel robots has been limited because of the complexity in defining and modelling their boundary conditions and constraints [1, 2]. Most of the applications of algebraic approaches for the determination of the workspace of parallel robots are related to planar parallel manipulators, as a generation procedure can be simplified from volumes to areas [6–8]. A solution has been also provided for a few parallel structures, such as Delta Robots [9, 10], spherical manipulators [11, 12], and others [13, 14]. However, a general approach has not been proposed yet, as only structure-specific formulations have given solutions that are available in literature. In this paper, the characteristics of iterative and algebraic approaches are discussed and compared. First, a general algorithm is presented to generate the workspace of a parallel robot for each of those methods. Then, two examples are reported as referring to the CaPaMan design [15, 16] and a 3-UPR design [17–20] to compare the solutions and to discuss advantages and disadvantages of each of them.

2 Workspace Determination In this section, two main approaches to generate the workspace of a parallel manipulator are described: an iterative method and an algebraic one. The iterative method evaluates workspace boundaries as a point cloud of reachable configurations by discretizing the motion parameters and computing the position of the end-effector for each parameter combination. Conversely, the algebraic approach evaluates the workspace volume by generating a volume of reachable points for each limb of the robot and then obtaining the overall workspace of the robot as the intersection of such volumes. 2.1

Workspace Discretization

A discrete approach for workspace evaluation can be obtained
from the forward kinematic problem of a robot. The continuous range of motion qi;min ; qi;max of each active joint parameter qi of actuation vector q can be discretized into a set of n points with an interval of Dqi from each other, as


   qi;min ; qi;max ! qi;1 ; qi;2 ; . . .; qi;n ; ;

ð1Þ

A Comparison of Algebraic and Iterative Procedures

qi;j ¼ qi;min þ

n X

ðj  1ÞDqi ; Dqi ¼

j¼1

qi;max  qi;min n

55

ð2Þ

The forward kinematic problem (FKP) x ¼ fFKP ðqÞ can be used to compute the endeffector pose x corresponding to a given combination of the m active joint parameters. Thus, by considering all the possible point combinations given by the discretization (2) of each joint parameter, the workspace can be generated as a point cloud of nm poses. The exponential nature of the number of resulting poses means that the determination of a workspace with the discretization method can be extremely taxing from a computational point of view. Out of the two parameters that influence the number of loops, m is an intrinsic parameter that is defined by the robot architecture and it cannot be modified, whereas n is a function of the chosen discretization interval, as formulated in (2), and it can only be reduced at the expense of point density, which reduces the resolution of the workspace and risks missing critical points where the manipulator behaves singularly. However, this approach is widely used because of its simplicity as in [2], as it only requires knowledge of the forward kinematic problem of the robot. 2.2

Algebraic Geometrical Approach

In the algebraic geometrical approach, the workspace of a parallel manipulator can be computed by intersecting the workspace volumes that is generated by each of its limbs. This method does not require any a priori modelling of the manipulator but only its topological layout and constraints, in contrast with the previously described approach, which requires the FKP. The algebraic approach starts by determining the workspace Ki of each limb, which is obtained as a volume of revolution by considering the rotation of each link of the limb’s kinematic chain around the axes of its revolute joints. For clarity, a practical example is here reported in Fig. 1 for a UPS kinematic chain (universal-prismatic-spherical joints, with the prismatic joint qi actuated by a linear motor as per Gough-Stewart platform [1, 2]). The universal and spherical joints are decomposed into a series of revolute joints (i.e., two consecutive revolute joints #i1 and #i2 along the x- and y-axes for the universal joint, and three consecutive revolute joints #i3 , #i4 , and #i5 along the x-, y-, and z-axes respectively for the spherical joint). Thus, the limb workspace can be obtained as due to the sequence of the joint motions as Ki ¼ Rotx ð#i1 Þ
Roty ð#i2 Þ
Trz ðqi Þ
Rotx ð#i3 Þ
Roty ð#i4 Þ
Rotz ð#i5 Þ

ð3Þ

where Rotk is a rotation matrix around the k-axis and Trk is a translation matrix along the k-axis. The resulting workspace is a family of spheres S that can be expressed as Si : ðx  x0i Þ2 þ ðy  y0i Þ2 þ ðz  z0i Þ2 ¼ ri2

ð4Þ

These spheres are all centered on the extremity of the limb fixed onto the base in ðx0i ; y0i ; z0i Þ and they are characterized by variable radius ri as family parameter, which corresponds to the length of the linear actuator in the limb and is thus related to its

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motion parameter and its lower and upper boundaries. By considering the actuation constraints, the limb workspace can be determined as the union of these spheres as S Ki ¼ [ rrmax i ¼rmin i

ð5Þ

Once the limb workspaces Ki have been defined, the overall workspace of the parallel manipulator K can be computed as their intersection [21] as K ¼ \m i¼1 Ki

ð6Þ

When compared to the previous method, the algebraic geometrical approach is computationally efficient and provides an exact solution rather than an approximated one. However, the highly coupled behavior of different limbs of a parallel manipulator makes the implementation of self-collision constraints difficult and can result in unexpected operational failure. A first solution can represent a theoretical reachable workspace of the robot, without considering limb collisions.

Fig. 1. Algebraic geometrical generation of workspace volume: a. A kinematic scheme with axes of rotation of the joints of each limb; b. An example workspace K of a parallel manipulator obtained as a solid of a parallel of revolution by following the procedure in Eqs. (3–5).

3 Numerical Examples In this section, the above-mentioned methods are compared by referring to two examples of parallel manipulators: a lower-mobility mechanism that was designed for earthquake simulations, CaPaMan [15, 16], and a robotic leg mechanism [17–20]. 3.1

CaPaMan

The CaPaMan parallel robot has been conceived to simulate the 3D motion modes of earthquakes [16]. As shown in Fig. 2, CaPaMan is a lower-mobility manipulator with three limbs with a PaPS kinematic chain based on a four-bar parallelogram (Pa) [15].

A Comparison of Algebraic and Iterative Procedures

57

With reference to the diagram in Fig. 2a and the formulation in [15], CaPaMan’s kinematics can be written as x¼

y3  y2 rp z1 þ z2 þ z3 pffiffiffi  ð1  sin uÞ cosðw  #Þ; z ¼ ; 2 3 3 y ¼ y1  rp ðsin w cos # þ cos w sin u sin #Þ

ð7Þ

where u ¼ arccos

2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3ð z 3  z 2 Þ z21 þ z22 þ z23  z1 z2  z2 z3  z3 z1 ; w ¼ arctan ; 2z1  z2  z3 3rp # ¼ arcsin

2ð y 1 þ y 2 þ y 3 Þ  w; 3rp ð1 þ sin uÞ

ð8Þ

where yk ¼ bk cos qk and zk ¼ bk sin qk + hk are function of motion variable qk and geometrical parameters bk and hk . By using (7), the workspace of CaPaMan with b1 ¼ b2 ¼ b3 ¼ 100 mm; h1 ¼ h2 ¼ h3 ¼ 100 mm; rp ¼ 50 mm has been computed by discretizing actuation variables fq1 ; q2 ; q3 g with a step Dq ¼ p=36 rad, with the results shown in Fig. 3; and by using the proposed algebraic geometrical method, with results shown in Fig. 4.

Fig. 2. CaPaMan, Cassino Parallel Manipulator [15]: a. A kinematic scheme with main parameters; b. A prototype at LARM laboratory of University of Cassino.

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Fig. 3. Computed discretized workspace K of CaPaMan in Fig. 2: a. side view; b. top view.

Fig. 4. Computed algebraic workspace K of CaPaMan in Fig. 2: a. side view; b. top view.

3.2

3-UPR Mechanism

The 3-UPR parallel manipulator in this example has been designed for the locomotion system of service robots [17–20]. The mechanism design is shown in Fig. 5, and its performance is analyzed in [2, 20]. With reference to the diagram in Fig. 5a, the FKP of the system can be formulated as   1  1 2 2 l1 þ l2  2l23 ; z ¼ x ¼ pffiffiffi l21  l22 ; y ¼ 6a 2 3a

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l23  x2  ðy  aÞ2

ð9Þ

By using (9), the workspace of a 3-UPR mechanism with a ¼ 100 mm, li;min ¼ 200 mm; li;max ¼ 300 mm has been computed by discretizing actuation variables fl1 ; l2 ; l3 g with a step Dl ¼ 10 mm, with the results shown in Fig. 6; and by using the proposed algebraic geometrical method, with results shown in Fig. 7.

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Fig. 5. A 3-UPR parallel mechanism for robotic legs [20]: a. A kinematic scheme with main parameters; b. A prototype at LARM2 laboratory of University of Rome Tor Vergata.

Fig. 6. Computed discretized workspace K of the mechanism in Fig. 5: a. side view; b. top view.

Fig. 7. Computed algebraic workspace K of the mechanism in Fig. 5: a. side view; b. top view.

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Results and Discussion

Both the proposed algebraic geometrical method and the discretized approach in the examples above have been computed in MATLAB 2021a on a Windows 10 computer with a 3.70 GHz 6-core CPU (AMD Ryzen 5 5600X). In both cases, the algebraic approach computes a general geometrical solution in less than 0.01 s, whereas the discretized workspace is obtained after 3.20 s for the CaPaMan and 0.40 s for the UPR mechanism. Thus, the algebraic geometrical method is computationally more efficient than the discretized one and provides a solution rather than an approximated one as from the other procedure. However, the discretized workspace can include self-collision constraints that have not be implemented in the algebraic formulation up to now. Furthermore, the geometrical approach only results in the position workspace of the manipulator, without giving information on the orientation. In conclusion, the algebraic method is more precise and efficient for an evaluation and representation of the workspace, whereas the discretized workspace enables the inclusion of self-collision and can provide further information on the manipulator behavior that cannot be seen with the algebraic geometrical approach.

4 Conclusions In this paper, two different methods for the determination of the workspace of parallel manipulators are compared: an iterative method based on the discretization of pose parameters, and a geometrical approach with an algebraic formulation. After briefly introducing the methods, advantages and disadvantages of both approaches are outlined through two examples on lower-mobility parallel manipulators with three degrees of freedom. Even if the discretized approach provides more information on the workspace, the algebraic formulation provides a faster exact solution that can be used for a fast representation of the workspace without the need of a kinematic model, as well as for an exact determination of the workspace boundaries.

References 1. Tsai, L.W.: Robot Analysis. Wiley, New York (1999) 2. Ceccarelli, M.: Fundamentals of Mechanics of Robotic Manipulation. Springer, Heidelberg (2004) 3. Russo, M., Herrero, S., Altuzarra, O., Ceccarelli, M.: Kinematic analysis and multi-objective optimization of a 3-UPR parallel mechanism for a robotic leg. Mech. Mach. Theor. 120, 192–202 (2018) 4. Ceccarelli, M.: A formulation for the workspace boundary of general N-revolute manipulators. Mech. Mach. Theor. 31(5), 637–646 (1996) 5. Gu, H., Ceccarelli, M.: Ring workspace topology of three-revolute manipulators. Int. J. Model. Simul. 31(2), 143–153 (2011)

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6. Li, T., Ceccarelli, M.: A geometric approach for workspace determination of planar n-DOF parallel manipulators. In: 4th International Congress Design and Modeling of Mechanical Systems (2011) 7. Yahya, S., Moghavvemi, M., Mohamed, H.A.: Geometrical approach of planar hyperredundant manipulators: inverse kinematics, path planning and workspace. Simul. Model. Pract. Theor. 19(1), 406–422 (2011) 8. Kim, D.J., Chung, W.K., Youm, Y.: Geometrical approach for the workspace of 6-dof parallel manipulators. Proc. Int Conf. Robot. Autom. 4, 2986–2991 (1997) 9. Liu, X.J., Wang, J., Oh, K.K., Kim, J.: A new approach to the design of a DELTA robot with a desired workspace. J. Intell. Robot. Syst. 39(2), 209–225 (2004) 10. Di Gregorio, R., Zanforlin, R.: Workspace analytic determination of two similar translational parallel manipulators. Robotica 21(5), 555 (2003) 11. Bulca, F., Angeles, J., Zsombor-Murray, P.J.: On the workspace determination of spherical serial and platform mechanisms. Mech. Mach. Theor. 34(3), 497–512 (1999) 12. Alamdar, A., Farahmand, F., Behzadipour, S., Mirbagheri, A.: A geometrical approach for configuration and singularity analysis of a new non-symmetric 2DOF 5R spherical parallel manipulator. Mech. Mach. Theor. 147, (2020) 13. Hu, B., Cui, H., Shi, D., Zhang, D., Wang, A., Wang, Y., Zhang, Q.: Reachable workspace determination for a spatial hyper-redundant manipulator formed by several parallel manipulators. J. Mech. Sci. Technol. 33(2), 869–877 (2019) 14. Aboulissane, B., El Haiek, D., El Bakkali, L., El Bahaoui, J.: On the workspace optimization of parallel robots based on CAD approach. Procedia Manufact. 32, 1085–1092 (2019) 15. Ceccarelli, M.: Historical development of CaPaMan, Cassino parallel manipulator. In: New Trends in Mechanism and Machine Science, pp. 749–757. Springer, Dordrecht (2013) 16. Selvi, Ö., Ceccarelli, M.: Interpretation of earthquake effects on mechanism operation: an experimental approach. J. Naval. Sci. Eng. 8(2), 31–45 (2012) 17. Ceccarelli, M., Cafolla, D., Russo, M., Carbone, G.: HeritageBot platform for service in cultural heritage frames. Int. J. Adv. Robot. Syst. 15(4), 1729881418790692 (2018) 18. Ceccarelli, M. et al.: HeritageBot service robot assisting in cultural heritage. In: 2017 First IEEE International Conference on Robotic Computing (IRC), pp. 440–445. IEEE (2017) 19. Russo, M., Cafolla, D., Ceccarelli, M.: Design and experiments of a novel humanoid robot with parallel architectures. Robotics 7(4), 79 (2018) 20. Russo, M., Ceccarelli, M., Takeda, Y.: Force transmission and constraint analysis of a 3-SPR parallel manipulator. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 232(23), 4399–4409 (2017) 21. Chkhartishvili, L.S.: Volume of the intersection of three spheres. Math. Not. 69(3–4), 421– 428 (2001)

Symbolic Kinematics of Special 3-RSR Parallel Mechanism with Zero Coupling Degree Huiping Shen(&), Yao Tang, Tao Li, and Qingmei Meng School of Mechanical Engineering, Changzhou University, Changzhou, China

Abstract. A special 3-RSR antenna parallel mechanism (PM) with zero coupling degree is designed. The symbolic forward position solutions of the 3-RSR PM are obtained by using the kinematic modeling method based on topological characteristics. Meanwhile, the inverse position solutions of the PM are solved. The workspace of the PM is efficiently calculated based on the symbolic forward position solutions not on inverse position solutions that is traditionally used by most of researchers. The results lay a foundation for dimension synthesis and dynamic analysis of the 3-RSR PM. Keywords: Parallel mechanism Coupling degree


Symbolic solution
Kinematic modelling


1 Introduction 3-RSR antenna mechanism is an interesting parallel mechanism (PM). Kinematics of the mechanism was analyzed in reference [1, 2]. G. R. Dunlop studied the closed-form forward position solution of a 3-RSR Parallel Robot with a common structure (both the moving and base platforms are equilateral triangles, and the lengths of six links are not equal to each other), and the closed-form inverse position solutions of its simplified structure (the moving and base platforms are equilateral triangles, the driving links and passive links of each chain have the same length) [3]. Chen carried out the kinematics and dynamics modeling analysis of the 3-RSR Parallel Robot System [4]. Besides, Duan analyzed the kinematic and mechanical properties of the 3-RSR antenna PM [5]. In addition, Shan studied the continuous rotation axis problem of a novel 3-RSR PM and analyzed its kinematics [6]. The authors of the paper have calculated that the coupling degree of the 3-RSR antenna PM is 1, which is not conducive to the subsequent type synthesis and dynamic analysis. In order to solve this problem, the topological coupling degree-reducing design of the 3-RSR antenna PM was carried out, which leaded to a novel 3-RSR antenna PM with zero coupling degree but the same DOF as well as output motion. Zero coupling degree of the PM ensures that symbolic forward position solutions of the PM are obtained. In this paper, a novel 3-RSR antenna PM with zero coupling degree is obtained by using the topological coupling-reducing principle [7] and design method of parallel mechanism based on POC equations [8]. Then, the symbolic forward and inverse position solutions of the PM is solved based on its topological characteristics. Finally, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 62–70, 2021. https://doi.org/10.1007/978-3-030-75271-2_7

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the workspace is analyzed based on the symbolic forward position solution not on inverse position solutions, which needs much less calculations.

2 The Antenna PM with Zero Coupling Degree The antenna PM with zero coupling degree is composed of a Hybrid-Single-OpenChain I (HSOC I) and a simple chain II, as shown in Fig. 1. The HSOC I is composed of a 3R−2S spatial mechanism (R11S12R13S22R21) and a revolute joint R23, which is connected in series with R13. The end output of this chain is three translations and three rotations. The chain II is R31−S32−R33. The topological structure of the two chains can be represented as: HSOC1 {−(R11−S12−R13−S22−R21)−R23−}, with R11||R21, and SOC2 {−R33−S32−R31−}. The design procedures and the calculation process of this PM with zero coupling degree are omitted due to the limitation of the space of the paper.

Fig. 1. An antenna PM with zero coupling degree

3 Position Analysis 3.1

Descriptions of the Kinematics Modeling

The kinematic modeling of the antenna PM is shown in Fig. 2. In Fig. 2(a), let the base platform be a rectangle with a length of 2a and a width of 2b, and set a base coordinate system O-XYZ with the midpoint O of A1A2 on the base platform 0 as the origin of coordinates. The Y-axis is along the A1A2 direction, and the Z-axis is vertical upward. Then the X-axis is determined by the right-hand rule. Let the moving platform 1 be a rod with length of l5, and the midpoint O1 on the moving platform 1 as the origin of the moving coordinate system O1−X1Y1Z1. The positive direction of X1 is along the direction of C3C1. So that the direction of Y1 is perpendicular to line C3C1 and the Z1axis meets the right-hand rule.

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(a) Parameter marking

(b) Annotation of YOZ projection surface

(c) Annotation of XOZ projection surface

Fig. 2. Kinematics modeling of the PM

Let three actuated inputs in this way: h1 is the angle between A1B1 and the positive direction of Y axis. h2 is the angle between A2B2 and the positive direction of Y axis, and h3 is the angle between A3B3 and the positive direction of X axis. Let the midpoint of line B1B2 be D1 that is changing point, and A3 is the midpoint of the opposite side of line A1A2 of the base platform 0. Let A1B1 = A2B2 = l1, B1C1 = B2C1 = l2, A3B3 = l3, B3C3 = l4, C1C3 = l5. As in Fig. 2(b), for the projection of the loop {A1−B1−C1−B2−A2}, Let C1D1⊥B1B2. The angle between B1B2 and the positive direction of Y-axis is a2. And the angle between the moving platform and line n which is parallel to Y-axis is a4. In Fig. 2(c), the angle between C1D1 in the projection plane of XOZ plane and the positive direction of Z axis is a3, and having C3C1⊥C1D1. The position of the moving platform 1 can be described by using Euler angle. The specific transformation matrix can be obtained as follows: first rotate a2 around X axis, then rotate a3 around current B1−B2, and finally rotate a4 around X1 of the moving platform 1. 3.2

Analysis of Forward Position Solution

Solving of the forward position solution of the PM can be described as: given the input angles h1, h2 and h3 of three actuated joints, and to derive the coordinates (x, y, z) of O1 on the moving platform in the base coordinate system and the Euler angles a2, a3 and a4. In this condition, it is easy to know that A1 = (0,−b,0), A2 = (0,b,0), A3 = (−2a,0,0), B1 = (0,l1cosh1 − b,l1sinh1), B2 = (0,l1cosh2 + b,l1sinh2), and B3 = (l3cosh3 − 2a,0,l3sinh3).

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(1) The solving of position of SKC1 In the first loop {A1 − B1 − C1 − B2 − A2}, a2,, lB1 B2 (= S) and lC1 D1 (= T) can be obtained as a2 ¼ arctan S¼

l1 ðsin h2  sin h1 Þ 2b þ l1 ðcos h2  cos h1 Þ

ð1Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½l1 ðcos h2  cos h1 Þ þ 2b2 þ l21 ðsin h2  sin h1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T¼ 4l22  S2 =2

ð2Þ

Therefore, the coordinates of C1 are defined by 8 > < xC1 ¼ Tsina3 yC1 ¼ l1 ðcosh1 þ cosh2 Þ=2  Tsina2 cosa3 > : zC1 ¼ l1 ðsinh1 þ sinh2 Þ=2 þ Tcosa2 cosa3

ð3Þ

(2) The solving of position of SKC2 The coordinates of points C3 and O1 are obtained from the length of the link l5 and its rotation angle. C3 ¼ ðxC1  l5 cosa3 ; yC1  l5 sin a2 sin a3 ; zC1 þ l5 cos a2 sin a3 Þ

ð4Þ

l5 cosa3 l5 sin a2 sin a3 l5 cos a2 sin a3 ; yC 1  ; z C1 þ Þ 2 2 2

ð5Þ

O1 ¼ ðxC1 

According to the constraint of link length C3B3 = l4, it can be deduced that a11 sin a3 þ a12 cos a3 þ a13 ¼ 0 a3 ¼ 2 arctan½ða11 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a211 þ a212  a213 Þ=ða13  a12 Þ

where, a11 ¼ 2d11 T2D1 yD1 l5 sina2 þ 2d13 l5 cosa2 a12 ¼ 2ðd11 l5 yD1 T sin a2 þ d13 Tcosa2 Þ 2 2 a13 ¼ d11 þ d13 þ l2C1D1 þ l25 þ y2D1  l24 ;

ð6Þ

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d11 ¼ 2a  l3 cosh3 ; d13 ¼ l1 ðsin h1 þ sin h2 Þ=2  l3 sinh3 ; yD1 ¼ l1 ðcos h1 þ cosh2 Þ=2: So far, the coordinates of O1 (x, y, z) can be obtained. Here the main variables are taken from the moving platform. Firstly, the pose parameter a3 of the moving platform is selected as main variable (intermediate variable). Secondly, the main variable a3 of the moving platform is obtained by substituting a3 into the SKC2. Thirdly, T, S and a2 are obtained in the SKC1. Finally, the calculated T, S, a2 and a3 are substituted into Eqs. (3)–(5), and the coordinates O1 are obtained. From x ¼ f1 ðh1 ; h2 ; h3 Þ; y ¼ f2 ðh1 ; h2 ; h3 Þ and z ¼ f3 ðh1 ; h2 ; h3 Þ; it can be seen that the PM has no input-output motion decoupling property. (3) Calculate the rotation angle of the moving platform Giving, where ①, ② and ③ are the order of rotation and transformation around the current axis. Expand it to 2

ca3 4 sa2 sa3 ca2 sa3

sa3 sa4 ca2 ca4  sa2 ca3 sa4 sa2 ca4  ca2 sa4 ca3

3 sa3 ca4 ca2 sa4  sa2 ca3 ca4 5 sa2 sa4 þ ca2 ca3 ca4

where sa2 = sina2, ca2 = cosa2, sa3 = sina3, ca3 = cosa3,, sa4 = sina4, ca4 = cosa4. Find Y1 direction vector through C1 B3  C1 C3 ; C1 B3  C1 C3 ¼ ða21 ; a22 ; a23 ÞT Where, a21 ¼ d21 þ d22 ; a23 ¼ d25 þ d26 ; d21 ¼ 2l5 Tca2 sa2 sa3 ca3  yD1 l5 ca2 ca3 ; d22 ¼ zD1 l5 sa2 sa3  l5 l3 ca2 sa3 sin h3 ; d25 ¼ 2al5 sa2 sa3  l5 l3 sa3 sa2 cos h3 ; d26 ¼ yD1 l5 ca3 þ l5 Tsa2 ; yD1 ¼ l1 ðcos h1 þ cos h2 Þ=2; zD1 ¼ l1 ðsin h1 þ sin h2 Þ=2; Therefore, a4 can be calculated as  a4 ¼ arccot

 a23 sin a3 þ cos a2 cot a3 a21 sin a2

ð7Þ

So far, a2, a3 and a4 are all obtained. 3.3

Analysis of Inverse Position Solution

Solving of the forward position solution of the PM can be described as: given the input angles h1,h2 and h3 of three actuated joints, and to derive the coordinates (x, y, z) of O1 in base coordinate system and Euler angles a2, a3 and a4 are given, and the input angles h1, h2 and h3 are to be calculated.

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The coordinates of the points C1 and C3 can be obtained by using Eqs. (3) and (4). On the basis of the three link-length constraints B1C1 = B2C2 = l2 and C3B3 = l4, h1, h2 and h3 can be calculated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a231 þ a232  a233 h1 ¼ 2arctan a33  a32 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a41  a241 þ a242  a243 h2 ¼ 2arctan a43  a42 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a51  a251 þ a252  a253 h3 ¼ 2arctan a53  a52 a31 

ð8Þ ð9Þ

ð10Þ

where a31 ¼ a41 ¼ 2zC1 l1 ; a32 ¼ 2l1 ðyC1 þ bÞ; a33 ¼ l22  l21  x2C1  ðyC1 þ bÞ2  z2C1 ; a42 ¼ 2l1 ðyC1  bÞ; a43 ¼ l22  l21  x2C1  ðyC1  bÞ2  z2C1 ; a51 ¼ 2zC3 l3 ; a52 ¼ 2l3 ðxC3 þ 2aÞ; a53 ¼ l24  l23  ðxC3 þ 2aÞ2  y2C3  z2C3 :

4 Examples Given the parameters of the PM as (unit: mm): l1 = 150, l2 = 200, l3 = 150, l4 = 200, l5 = 400, a = 200, b = 150. Three input angles are (unit: rad) given as: h1 = 1.3707, h2 = 1.6707, h3 = 1.1707. The results of the forward position solution calculated from Eqs. (1)–(7) by using MATLAB programming is shown in Table 1. Table 1. The forward position solutions of the PM x(mm) y(mm) z(mm) a2(rad) a3(rad) a4(rad) 1 −187.35 5.93 317.27 0.008 0.078 0.606

The data of the forward solutions in Table 1 are substituted into the inverse position solution Eqs. (8)–(10), which leads to the results of the inverse solutions by using MATLAB programming, shown in Table 2. In Table 2, the solutions of No.5 is h1 = 1.3708, h2 = 1.6708 and h3 = 1.1707 that is consistent with the three input angles, which verifies the correctness of the forward and inverse solutions.

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h1(rad) 1.3708 0.8159 1.3708 0.8159 1.3708 0.8159 1.3708 0.8159

h2(rad) 2.3612 2.3612 2.3612 2.3612 1.6708 1.6708 1.6708 1.6708

h3(rad) 1.1707 1.1707 1.8913 1.8913 1.1707 1.1707 1.8913 1.8913

5 Workspace Analysis of Decoupling Optimized Model The reachable workspace of PM refers to the working area of the end effector considering the joints, angle range and non-interference of the links. The traditional workspace calculation is based on the inverse position solution formula of a PM and on adopting the three-dimensional search method of discrete space. That is to say, it is necessary to estimate and set a search range in advance, and then search out all the points that meet the constraint conditions of inverse position solution formula through MATLAB programming. The three-dimensional diagram composed of these points is the workspace of the PM. However, since that the search range presupposed is much larger than the actual workspace it is difficult to estimate its size, so it need a large amount of calculation. Since the PM has symbolic forward position solution, it can be directly used to calculate the workspace with less calculation, and the calculation of the workspace boundary is more accurate. For this purpose, the range of three input angles are determined as 0.1p  h1 0.9p, 0.1p  h2  0.9p and −p/3  h3  p/3. According to the forward solution Eqs. (1)–(7), the three-dimensional workspace of the PM can be directly obtained by using MATLAB programming, as shown in Fig. 3.

(a) 3D workspace

(b) CAD model drawing of mechanism

Fig. 3. 3D sketch of the workspace

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Some section views of the workspace are shown in Fig. 4.

(a) YOZ section(x=-200mm)

(b) XOY section(z=290mm)

(c) XOZ section(y=0mm)

Fig. 4. Workspace of the PM

It can be seen from Figs. 3 and 4 that ① the PM can realize a large range of three directions of movement, and ② the workspace of the PM is continuous and regular.

6 Conclusions The kinematics model and symbolic forward position solutions of 3-RSR PM with zero coupling degree is established based on its topological characteristics. In addition, the inverse position solutions are solved and verified by numerical example. Based on the symbolic forward position solution, the workspace of the PM is obtained, which shows that its workspace is large and regular. At the same time, it is shown that the workspace calculation method based on symbolic forward position solutions has the advantages of less calculation and accurate workspace boundary. The PM not only has the same motion mode as the traditional 3-RSR PM, but also its coupling degree is 0. Therefore it is easy to get its symbolic forward position solution, which provides a easy foundation for the dimension synthesis and dynamic analysis of the 3-RSR PM with zero coupling degree. The stability and stiffness of this configuration need to be analyzed latter, so the PM is a possible antenna mechanism. Acknowledgments. This work thanks to the support from NSFC (No. 51975062).

References 1. Angeles, J.E., Hommel, G.E.: Computational kinematics. Artif. Intell. 51(1–3), 381–416 (1993) 2. Peruzzini, W., Ouellet, A., Hui, R.: Iftomm. Proceedings of Ninth World Congress on the Theory of Machines and Mechanisms. Politecnico di Milanso, Italy, pp. 214I–2145 (1995) 3. Qingzeng, F., Dunlop, G.R., Jones, T.P.: Position analysis of a 3-DOF parallel manipulator. Mech. Mach. Theory 32(8), 903–920 (1997) 4. Chen, W.K.: Kinematics and dynamics modeling and simulation of 3-RSR parallel manipulators. East China Jiaotong University (2006)

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5. Duan, Y.B., Hou, R.W., Deng, Y.J., et al.: Kinematic and mechanical analysis of 3-DOF antenna parallel mechanism. Mach. Manuf. 57(06), 63–66 (2019) 6. Shan, Y.X., Li, S.H., Li, F.J., et al.: 3-RSR parallel mechanism with continuous rotating shaft and its kinematics analysis. Mach. Design 36(05), 40–44 (2019) 7. Shen, H.P., Zhu, X.R., Yin, H.B., et al.: Structure decoupling principle and design method of parallel mechanism. J. Mech. Eng. 52(23), 102–113 (2016) 8. Yang, T.L., Liu, A.X., Luo, Y.F., et al.: Theory and Application of Robot Mechanism Topology. Science Press, Beijing (2012)

Inverse Kinematics and Workspace of a 3-PRRS Type Parallel Manipulator Zh. Baigunchekov1,2(&), M.A. Laribi3, R. Kaiyrov1, and E. Zholdassov1 1

3

Al-Farabi Kazakh National University, Almaty, Kazakhstan 2 Satbayev University, Almaty, Kazakhstan Department of GMSC, Prime Institute, CNRS - University of Poitiers, ENSMA - UPR 3346, Poitiers, France [email protected]

Abstract. In this paper, methods of workspace analysis of a 3-PRRS type parallel manipulator are described. The equations of spheres and circles on these spheres, along which the center of the moving platform can move, are derived, and it is shown that the total reachable area of these spheres is the workspace of the considered parallel manipulator. Numerical examples of defining the workspace of the 3-PRRS type parallel manipulator are presented. Keywords: Parallel manipulator


Moving and fixed platform
Workspace

1 Introduction Most parallel manipulators with six degrees of freedom (DOF) have six legs [1–3]. Parallel manipulators with six DOF and three legs or tripods, in comparison with parallel manipulators with six legs or hexapods, have a larger workspace and less singular configurations. The following types of tripods are known: 3-URS [4], 3-ESR [5], 3-PRPS [6], 3-RES [7–9], 3-PPSR [10], 3-PRPS [11], 3-CRS [12], 3-CCC [13]. We have developed a novel tripod of a 3-PRRS type with six DOF (Fig. 1) which in comparison with the existing tripods has a large workspace.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 71–78, 2021. https://doi.org/10.1007/978-3-030-75271-2_8

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Fig. 1. 3-PRRS type parallel manipulator.

In [14, 15], the geometry of this parallel manipulator was studied and the kinematics were solved. This paper is devoted to a workspace analysis of a 3-PRRS type parallel manipulator.

2 Workspace Analysis Figure 2 shows a diagram of a 3-PRRS tripod with vectors characterizing its geometry and kinematics, where the O0 U0 V0 W0 and PXP YP ZP coordinate systems are connected to a fixed and moving platform, respectively.

Fig. 2. Coordinate systems and vectors.

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The coordinates of the centers of spherical joints O4;i are determined by the following equations 9 bi cci þ si sci  fi sci sh2;i  gi sci sh23;i ¼ UO4;i > = bi sci  si cci þ fi cci sh2;i þ gi cci sh23;i ¼ VO4;i ; i ¼ 1; 2; 3; ð1Þ > ; ci þ ai þ fi ch2;i þ gi ch23;i ¼ WO4;i where h23;i ¼ h2;i þ h3;i : Multiplying the first and second equations of system (1) by cci and sci , respectively, and adding them, we obtain the following three equations of the planes along which three dyads of the RRS type move cci
UO4;i þ sci
VO4;i þ bi ¼ 0; ði ¼ 1; 2; 3Þ:

ð2Þ

Let us consider an algorithm for determining the workspace of the considered tripod by solving an inverse kinematic problem. In [14, 15], the transformation matrix TOP between the O0 U0 V0 W0 and PXP YP ZP coordinate systems was obtained.

TOP

where, R OP

⎡ 1 0 0 0⎤ ⎢ X t t12 t13 ⎥ = ⎢ P 11 ⎥, ⎢ YP t21 t22 t23 ⎥ ⎣⎢ Z P t31 t32 t33 ⎥⎦

ð3Þ

⎡ t11 t12 t13 ⎤ = R y (θ ) Rx (ψ ) Rz (ϕ ) = ⎢t21 t22 t23 ⎥ , t11 ¼ chcu þ swshsu; ⎢ ⎥ ⎣t31 t32 t33 ⎦

t12 ¼ chsu þ swshcu; t13 ¼ cwsh; t21 ¼ cwsu; t22 ¼ cwcu; t23 ¼ sw; t31 ¼ shcu þ swchsu; t32 ¼ shsu þ swchcu; t33 ¼ cwch: Determine the coordinates of the spherical joints in the absolute coordinate system O0 U0 V0 W0

⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ 1⎤ ⎡ ⎢ h⎥ ⎢U ⎥ ⎢U ⎥ ⎢ O4,1 ⎥ = Т ⎢h ⎥ , ⎢ O4,2 ⎥ = Т ⎢ 2 ⎥ , ОР ⎢ 0 ⎥ ⎢ V ОР ⎢ h 3 ⎥ ⎢ VO4,1 ⎥ O4,2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎥ 0 ⎦⎥ ⎢W ⎢ WO ⎢ ⎥ ⎣ O ⎢⎣ 4,1 ⎥⎦ ⎢⎣ 4,2 ⎥⎦ ⎣ 0 ⎦

⎡ 1 ⎤ ⎡ 1 ⎤ ⎢ h ⎥ ⎢U ⎥ ⎢ O4,3 ⎥ = Т ⎢ 2 ⎥ ОР ⎢ h 3 ⎥ ⎢ VO4,3 ⎥ ⎢− 2 ⎥ ⎢ ⎥ WO ⎢ ⎥ ⎢⎣ 4,3 ⎥⎦ ⎣ 0 ⎦

ð4Þ

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From Eqs. (2), taking into account (3), (4), we obtain three constrained equations of the moving platform 9 ðXP þ h
t11 Þcc1 þ ðYP þ h
t21 Þsc1 þ b ¼ 0 > = p ffiffi ffi p ffiffi ffi ½XP þ h2 ð 3t12  t11 Þcc2 þ ½YP þ h2 ð 3t22  t21 Þsc2 þ b ¼ 0 > pffiffiffi pffiffiffi ; ½XP  h2 ð 3t12 þ t11 Þcc3 þ ½YP  h2 ð 3t22 þ t21 Þsc3 þ b ¼ 0

ð5Þ

where n is an angle between U0 and the direction of the vector s1 , c1 ¼ p2 þ n, p 1 c2 ¼ 3p 2 þ n, c3 ¼ n  6, n ¼ s ðb=hÞ. After subtracting the third equation from the second equation of system (5), previously divided the second and third equations into cc2 and cc3 , respectively, we obtain XP ¼

 h t11 þ ð3  4c2 nÞt22 þ hsð2nÞt12  2bsn: 2

ð6Þ

In the same way, after subtracting the third equation from the second equation of system (5), previously divided the second and third equations into sc2 and sc3 , respectively, we obtain YP ¼

 h t21 þ ð1  4c2 nÞt12  hsð2nÞt22  2bcn: 2

ð7Þ

From the first equation of system (4), taking into account (6) and (7), we obtain ðt11 þ t22 Þsn þ ðt12  t21 Þcn ¼

2b : h

ð8Þ

From Eq. (8), taking into account the values of t11 ; t12 ; t21 ; t22 , we obtain Acu þ Bsu 

2b ¼ 0; h

ð9Þ

where A ¼ ðcw þ chÞsn þ swshcn; B ¼ swshsn  ðcw þ chÞcn: From Eq. (9), we obtain two values of the angle u for two assemblies of each legs of the tripod u1;2 ¼ c1

  C þ a; r

ð10Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   where a ¼ tan1 BA ; r ¼ A2 þ B2 ; C ¼ 2b h. We use the right assembly solution for each RRS dyad. Further, we arbitrarily set the values of w and h from (10) and obtain u, then from Eqs. (6) and (7) we determine the coordinates XP and YP .

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For known values of XP and YP , from Eq. (4) it is need to determine UO4;i ; VO4;i . To determine WO4;i , it is need to specify ZP in any form, i.e., ZP has no restrictions, unlike XP and YP . From Eq. (4) we obtain pffiffiffi 3 3 2 3 2 XP þ h2 3t12  t11 UO4;2 XP þ ht11 pffiffiffi 7 6V 7 6 7 6 7 6 h 7 4 O4;1 5 ¼ 4 YP þ ht21 5; 4 VO4;2 5 ¼ 6 4 YP þ 2 3t22  t21 5; p ffiffi ffi   WO4;1 WO4;2 ZP þ ht31 ZP þ h2 3t32  t31 2 3 p ffiffi ffi   2 3 XP  h2 3t12 þ t11 UO4;3 pffiffiffi 7 6V 7 6 h 7 4 O4;3 5 ¼ 6 4 YP  2 3t22 þ t21 5: p ffiffi ffi   WO4;3 ZP  h2 3t32 þ t31 2

UO4;1

3

2

ð11Þ

From the orthogonally condition for the matrix ROP , we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2  t2 > = 1  t11 21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2  t2 > ; ¼  1  t12 22

t31 ¼  t32

ð12Þ

Because of the sign before t31 ; t32 we can get four different ( þ t31 ; þ t32 ; t31 ; t32 ; þ t31 ; t32 ; t31 ; þ t32 ) combinations of WO4;i , for the same values of UO4;i ; VO4;i . Only two combinations ( þ t31 ; þ t32 ; t31 ; t32 ) out of four coincide with the required constraint conditions (4), i.e., there are two different orientations of the moving platform for one point PXP YP ZP . Some points can be reached by the center of the moving platform with only one orientation of the moving platform.

3 Numerical Results With the following values of constant parameters: ai ¼ 15, bi ¼ 8, ci ¼ 5, fi ¼ 60, gi ¼ 70, h ¼ 43, c1 ¼ p=2 þ n, c2 ¼ 7p=6 þ n, c3 ¼ n  p=6, n ¼ s1 ðb=hÞ we draw the workspace of the tripod. To do this, we set the angles w and h, and determine the angle u. Using formulas (6) and (7), we determine XP and YP . Thus, in the given three independent variables w; h; ZP , free constrained variables XP ; YP ; u can be defined. Figure 3 shows the workspace of the tripod, with fixed translational drives motions s1 ¼ s2 ¼ s3 ¼ 50 and given ZP from 0 to 170 with a step of 5, and w and h change in two cycles from p=2 to p=2, with a step p=50. The two orientations of the moving platform are shown in red “o” and blue “*”, i.e., red-blue dots show the workspace that center of the moving platform can to reach in two orientations.

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Fig. 3. Workspace of the tripod.

Figure 4 shows the workspaces of the PM in six planes in different positions of the prismatic kinematic pairs, when ZP changes from 60 to 135 with a step of 15, and w and h change in two cycles from p=2 to p=2 with a step of p=50. This figure also shows the effect of prismatic kinematic pairs on the change in the workspace in the vertical direction, i.e., at smaller values of the distance between the prismatic kinematic pairs, the workspace increases in the upper planes, and at large values of the distances between the prismatic kinematic pairs, the workspace increases in the lower planes.

s1 = s2 = s3 = 40

s1 = s2 = s3 = 60

Fig. 4. Influence of prismatic kinematic pairs motion on the tripod workspace.

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4 Conclusions Workspace of a 3-PRRS type parallel manipulator is defined, in this paper. It is shown, that three legs of the parallel manipulator move in circles, and the center of the moving platform moves in three spheres relative to the centers of the spherical kinematic pairs. The total reachable area of the three spheres is the workspace of the considered parallel manipulator. The numerical results of the 3-PRRS type parallel manipulator’s workspace analysis are obtained. Acknowledgment. This work is supported by the grant « 2020/AP08857522 – “Development of RoboMech class parallel manipulators” (2020–2022) of the Ministry of Education and Science Kazakhstan.

References 1. Merlet, J.-P.: Parallel Robots. Kluwer Academic Publishers (2000) 2. Hunt, K.H.: Structural kinematic of in-parallel-actuated robot-arms. ASME J. Mech. Trans. Autom. Des. 105, 705–711 (1983) 3. Liu, X.-J., Wang, J.: Parallel Kinematics: Type, Kinematics, and Optimal Design. Springer, Heidelberg (2014) 4. Clearly, C., Uebel, M.: Jacobian formulation for a novel 6-DOF parallel manipulator. IEEE Int. Conf. Robot. Autom. 3, 2377–2382 (1994) 5. Tsai, L.-W., Tahmasebi, F.: Synthesis and analysis of a new class of six-degree-of-freedom parallel manipulators. J. Robot. Syst. 10(5), 561–580 (1993) 6. Alizade, R.I., Tagiyev, N.R., Duffy, J.A.: Forward and reverse displacement analysis of a 6DOF in-parallel manipulator. Mech. Mach. Theory 29(1), 115–124 (1994) 7. Collins, C.L., Long, G.L.: The singularity analysis of an in-parallel hand controller for forcereflected teleoperation. IEEE Trans. Robot. Autom. 11, 661–669 (1995) 8. Mimuza, N., Funabashi, Y.: A new analytical system applying 6 DOF parallel link manipulator for evaluating motion sensation. In: IEEE International Conference on Robotics and Automation, pp. 227–333 (1995) 9. Ebert, U., Gosselin, C.M.: Kinematic study of a new type of spatial parallel platform mechanism. In: ASME Design Engineering Technical Conference, Atlanta, 13–16 September 1998 10. Byun, Y.K., Cho, H.S.: Analysis of a novel 6-DOF 3-PPSP parallel manipulator. Int. J. Robot. Res. 16(6), 859–872 (1997) 11. Behi, F.: Kinematic analysis for a six-degree-of-freedom 3-PRPS parallel mechanism. IEEE J. Robot. Autom. 4(5), 561–565 (1988) 12. Khali, D., Lee, S.-H., Tsai, K.-Y., Sandor, G.N.: Manipulator configurations based on Rotary-Liner (R-L) actuators and their direct and inverse kinematics. J. Mech. Trans. Autom. Design 110, 397–404 (1988) 13. Baigunchekov, Zh., Izmambetov, M.: Inverse kinematics of six - DOF three – limbed parallel manipulator. In: Advances in Robot Design and Intelligent Control, Proceedings of the 25th Conference on Robotics in Aple-Adria-Danube Region (RAAD 2016), pp. 171–178. Springer (2016)

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14. Baigunchekov, Zh., Laribi, M.A., Mustafa, A., Kaiyrov, R., Amanov, B., Kassinov, A.: Geometry and inverse kinematics of 3- PRRS type parallel manipulator. In: Berns, K., Görges, D., (eds.) RAAD 2019, AISC 980, pp. 12–18. Springer Nature, Switzerland, AG (2020) 15. Baigunchekov, ZZh, Kaiyrov, R.A.: Direct Kinematics of a 3-PRRS type parallel manipulator. Int. J. Mech. Eng. Robot. Res. 9(7), 967–972 (2020) 16. Laribi, M.A., Romdhane, L., Zeghloul, S.: Analysis and dimensional synthesis of the DELTA robot for a prescribed workspace. Mech. Mach. Theory 42 (2007) 859–870

Investigation of Interference-Free Workspace of a Cartesian (3-PRRR) Parallel Manipulator Isaac John1(B) , Parvathi Sunilkumar1 , Santhakumar Mohan1 , and Larisa Rybak2 1 IIT Palakkad, Palakkad, India {prj56,santhakumar}@iitpkd.ac.in, [email protected] 2 Belgorod State Technological University, Belgorod, Russia

Abstract. This work focuses on determining the link interference-free workspace of a Cartesian (3-PRRR) parallel manipulator. An algorithm for obtaining the largest volume cuboid that inscribes the interferencefree workspace is proposed, which can be used as a performance index for the mechanism under investigation. The effect of link dimensions and the size of the moving platform on the interference-free workspace is also analysed in this work. Keywords: Link interference volume inscribed cuboid

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· Interference-free workspace · Largest

Introduction

Parallel manipulators have numerous advantages such as high speed manipulation, high stiffness, high payload capacity over their serial counterparts. Since the actuators of most of the parallel manipulators are mounted close to the base, errors due to deflection of links are minimal. Hence, the manipulation is achieved not only with high speed, but also with high accuracy. These inherent advantages have led to the tremendous research of parallel manipulators of various degrees of freedoms and motion patterns. Despite this, only a few examples are commercialized and used in practice owing to their limited and singularity riddled workspace. There are, of course, exceptions such as the Cartesian parallel manipulator called Tripteron [1], whose reachable workspace does not have interior singularities. The motion of the Tripteron is fully decoupled, where each degree of freedom can be independently controlled by a single actuator. These added benefits of the Tripteron have led to its implementation in robotic rehabilitation systems, where the manipulator works in close proximity with human beings [2]. Link interference is a common issue found in parallel manipulators [3]. Estimating the interference-free workspace is important as it can serve as a performance index for the manipulator workspace. It is also a matter of safety when the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 79–89, 2021. https://doi.org/10.1007/978-3-030-75271-2_9

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mechanism is working in close proximity with human beings. Previous researches have formulated various algorithms for obtaining the interference-free workspace of parallel manipulators. A geometrical approach based on the intersection of line segments was used on the self-interference detection on Tripteron parallel manipulator [4]. A case-independent methodology based on interval analysis was proposed in [5]. Although generalized, such methods are computationally expensive. A methodology using intersection of finite cylinders was employed for detecting self-collision in 6D workspace [6]. A self-collision avoidance algorithm based on cumulative danger field was proposed in [7]. A case-independent algorithm for self-interference detection of parallel manipulator via triangle-totriangle intersection tests was proposed in [8]. By enclosing the links of a manipulator in swept sphere volumes (SSV) and enclosing the objects in the workspace in oriented bounding boxes (OBB), a collision avoidance algorithm based on the intersection of SSVs with OBBs was elaborated in [9]. For Cartesian manipulators, determining the interference-free workspace alone is not sufficient for evaluating its workspace. Cartesian manipulators are evaluated based on the dimensions of their workspace which is in the form of a rectangular parallelepiped/cuboid. This paper focusses on developing algorithms for obtaining the interferencefree workspace as well as to obtain the maximum volume cuboid inscribing the interference-free workspace of a Cartesian parallel manipulator (a variant of a specific class of Cartesian parallel manipulators called Tripteron). The dimensions of the largest inscribed cuboid can be used for selecting the appropriately sized links and the moving platform edge lengths. Two types of interference are identified in this work, namely, inter-limb interference where the links of different limbs interfere and intra-limb interference, where the links of the same limb interfere. Inter-limb interference can be kept to a minimum if the edge length of the moving platform is kept above a minimum value. Intra-limb interference can be eliminated by limiting the range of rotation of the passive revolute joints. As the application focusses on rehabilitation, the inter-limb interference are to be kept to a minimum and hence the algorithm for obtaining the largest inscribed cuboid is developed for this condition. 1.1

Description of the Mechanism

The schematic diagram of the 3-DOF Tripteron parallel manipulator is shown in Fig. 1. Here, a right-angled triangle shaped moving platform is connected to the fixed frame using three PRRR kinematic chains. The prismatic joints in each of the limbs are actuated for controlling the three Cartesian degrees of freedom while the revolute joints remain passive. Including the moving platform, the mechanism consists of ten moving links. The motion of the moving platform is fully decoupled, where each degree of freedom can be controlled independently by the corresponding actuator. The inverse kinematic modelling of the Tripteron parallel manipulator is carried out in the previous work [10].

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Fig. 1. 3-PRRR parallel manipulator

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Link Interference

In this section, algorithms for obtaining the interference-free workspace and the largest volume cuboid that inscribes the interference-free workspace are discussed. 2.1

Algorithm 1- Interference Free Workspace

Algorithm 1 describes the general outline to identify regions of workspace with and without interference of the Tripteron parallel manipulator. Step 1 defines a cuboidal domain XYZ, whose limits are determined by the link lengths of the manipulator. Step 2 filters out the set of all points that satisifies the inverse kinematics conditions of the mechanism as XYZ fil. The coordinates satisfying the inverse kinematics (XYZ fil) are then fed into a collision detection function in step 3. In the function, each link is enclosed by a sphere swept line (SSL) [11] defined by pair of points within the link, and a radius. As the links are considered to be of constant cross sections, the sphere radius can be assumed to be equivalent to the radius of the link’s cross section for cylindrical links or equivalent to half the width of the links with square cross sections. The pair of points are chosen such that they are at an offset from the extreme ends of the links. As a rule of thumb, the offset distance should be higher than the collision threshold value. The function scans through all the points in (XYZ fil) and checks whether the minimum distance between all pairs of vectors are higher than the collision threshold value or not. The collision threshold value can be described as the sum of radii or two cylindrical links

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under consideration or sum of half-widths of two links with rectangular cross section under consideration. If the minimum distance between any two pairs of vectors are less than the collision threshold value, link interference is said to have occurred and subsequently, the set with col XYZ are updated with the current X, Y and Z coordinates. If the minimum distance between all pairs of vectors are greater than the collision threshold value, then the current coordinates are interference-free and subsequently, the set without col XYZ are updated with current X, Y and Z coordinates. Figure 2 shows the workspace regions with and without interference of the Cartesian parallel manipulator for a set of link lengths. Step 4 carries out further manipulations on with col XYZ and the inverse kinematics of each of the limbs considered separately, which in turn gives the revoulte joint limits θ2lim , φ2lim and ψ2lim above which the intra-limb interference occurs. Step 5 considers the joint limits from step 4 to obtain the vertex space of the three limbs (RR chains) as L1 vertex, L2 vertex and L3 vertex for limbs 1, 2 and 3 respectively. The intersection of the three vertex spaces, where each of them has a shape of a quarter cylinder, yields the intra-link interference-free workspace of the manipulator. The vertex spaces serve as a basis for algorithm 2 for obtaining the largest volume cuboid that inscribes the interference-free workspace.

1.6

Interference free workspace Region with link interference

1.4 1.2

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1 0.8 0.6 0.4 0.2

1 1

0.5

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0.5 0

0

X axis

Fig. 2. Link interference in the manipulator workspace

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Algorithm 1. Interference detection for the manipulator workspace Input: L = vector of link lengths, t = Interference threshold Output: Interference free workspace, interference workspace, vertex space 1: [XYZ] = Domain(L)  cuboidal domain  Inverse Kinematics 2: [XYZ fil] = Inv Kin(XYZ) 3: [with col XYZ, without col XYZ] = col det(XYZ fil, L, t) 4: Run the inverse kinematics of each limbs on ‘with col XYZ’ to obtain the revolute joint limits θ2lim , φ2lim and ψ2lim 5: Considering joint limits θ2lim , φ2lim and ψ2lim obtain the vertex space of the three limbs. 6: Function [XYZ] = Domain(L) 7: x = 0 → (k3 + l23 + l33 ) 8: y = 0 → (k3 + l23 + l33 ) 9: z = 0 → min({k1 + l11 + l21 }, {k2 + l12 + l22 }) 10: XYZ = [x y z] 11: end 12: Function [with col XYZ, without col XYZ] = col det(XYZ fil,L,t) 13: Define a pair of points on each links, at an offset from the nearest joint. 14: Define line segments as vectors between the point pairs for each link. 15: for (i=1:size(XYZ fil)) 16: XC = XYZ fil(i,1); YC = XYZ fil(i,2); ZC = XYZ fil(i,3); 17: if min distance between any two pair of vectors < ‘t’ 18: Update ‘with col XYZ’ with ‘XC, YC, ZC’ 19: end 20: if minimum distance between all pairs of vectors > ‘t’ 21: Update ‘without col XYZ’ with ‘XC, YC, ZC’ 22: end 23: end 24: end

2.2

Algorithm 2- Largest Inscribed Cuboid

Here, the algorithm for determining the largest volume cuboid that inscribes the interference-free workspace of the Tripteron parallel manipulator is discussed. The algorithm is suitable for Tripteron configurations where the inter-limb interference is minimal. It has been already established from the previous algorithm that the intra-limb interference-free workspace of the mechanism can be obtained by the intersection of the vertex spaces of the three limbs. This algorithm successively intersects the vertex spaces of the three limbs and based on the general behaviour, transformations are applied until the largest volume inscribed cuboid is obtained. Steps 1–3 deal with the intersection of vertex spaces of limbs 1 and 2. Steps 1 and 2 derive L1C and L2C, the largest volume cuboid inscribed within the vertex spaces L1 vertex and L2 vertex respectively. The largest cuboid that fits in the vertex space of the first limb is shown in Fig. 3(a). Step 3 derives the largest cuboid that fits in the intersection of L1 vertex and L2 vertex as inters1.

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(a)

(b) without_col_XYZ cuboid

1.4

L_3vertex_T without_col_XYZ inters3

1.4 1.2

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Z axis

1 0.8 0.6

0.8

0.6

0.4 0.4

0.2 0.2

1 1 1

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0

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(c)

X axis

0.5

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00

0.2

0.4

0.6

0.8

1

1.2

1.4

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(d)

Fig. 3. Various stages of Algorithm 2. 3(a), the largest square cross section that fits in the intersection of limbs 1 and 2 vertex spaces. 3(b), the largest cross section that fits in the intersection of all the three limb vertex spaces. 3(c), the largest height for the maximal cuboid that inscribes the interference free workspace. 3(d), the largest cuboid that inscribes the interference free workspace

In step 4, the coordinates of the set defined as inters2 are obtained by intersecting the coordinates of inters1 obtained in step 3 and L3 vertex, the vertex space of the third limb. As some portion of the cross section of inters1 lies outside the vertex space of the third limb, step 5 determines the coordinates of the largest cross section of inters1, CS, that fully fits the vertex space of the third limb as shown in Fig. 3(b).

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Algorithm 2. Largest volume cuboid inscribed in the collision free workspace Input: Vertex spaces of the three limbs as L1 vertex, L2 vertex, L3 vertex, Link lengths as ‘L’, Interference free workspace as ‘without col XYZ’, Revolute joint limits θ2lim , φ2lim and ψ2lim Output: Largest volume cuboid that inscribes the interference free workspace as ‘cuboid’ 1: L1C = largest cuboid (L1 vertex, L, θ2lim ) 2: L2C = largest cuboid (L2 vertex, L, φ2lim ) 3: inters1 = intersect(L1C, L2C, ‘rows’)  Standard MATLAB command 4: inters2 = intersect(inters1, L3 vertex, ‘rows’) 5: CS = cross section(inters2) 6: L3 vertex T = Points in L3 vertex that lies within the cross section defined by ‘CS’ 7: inters3 = intersect(without col XYZ, L3 vertex T, ‘rows’) 8: cuboid = largest vol (L3 vertex T, inters3, CS) 9: Function [L1C] = largest cuboid (L1 vertex, L,θ2lim ) 10: obtain r min and r max from L and θ2lim ) 11: Calculate ‘side’ from the following equation of the circle x2 + (y − k1 )2 = r max2 where x = r min + side, y = side 12: L1C = points of L1 vertex that lies inside the cross section defined by the square of size ‘side 13: end 14: Function [CS] = cross section(inters2) 15: Select any slice in the cross section and determine all the corner points 1- 5. (refer Fig.3(b)) 16: Incrementally decrease the y coordinate of side 1-5 from point 5 and x coordinate of side 1-2 from point 2 until length of side 1-5 becomes equal to length of side 2-3 and length of side 5-4 equal to that of side 1-2 17: if (side 1-5=side 2-3 and side 5-4=side 1-2) 18: xmin = x 1; xmax = x 2; ymin = y 1; ymax = y 5; 19: CS = [xmin, xmax, ymin , ymax]; 20: end 21: end 22: Function [cuboid] = largest vol (L3 vertex T, inters3, CS) 23: obtain xmax, ymax from CS 24: zmax = maximum height of all points of inters 3 at (xmax, ymax) 25: cuboid = set of all points in L3 vertex T that lies below zmax 26: end

In step 6, all the points of L3 vertex that lie outside the cross section defined by CS (step 5) are deleted and the resulting set of coordinates is saved as L3 vertex T. With this, the maximum square cross section of the largest cuboid is determined. Steps 7 and 8 return the height of the largest cuboid inscribed in the interference-free workspace.

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In step 7, the interference-free workspace obtained in algorithm 1, labeled as without col XYZ is intersected with L3 vertex T and the resulting set of coordinates as shown in Fig. 3(c) is saved under the label inters3. In step 8, the maximum of the Z coordinates of all points with X,Y coordinates at xmax and ymax of the set inters3 is obtained, labelled as zmax . The set of all points in L3 vertex T having Z coordinates less than zmax is identified and is saved under the label cuboid. The set of all coordinates under the label cuboid defines the largest volume cuboid that inscribes the interference-free workspace of the Tripteron parallel manipulator as shown in Fig. 3(d).

3

Results and Discussions

In this work, the link lengths corresponding to the scale factor r = 1 are as follows: k1 = k2 = k3 = 0.1 m, l11 = l21 = l12 = l22 = l13 = l23 = 0.75 m, p = 0.75 m. The effect of link lengths corresponding to different scale factors (r = 0.5:1.5) on the dimensions of the largest cuboid inscribed is shown in Fig. 4. Based on the requirements of the workspace dimensions, an appropriate scaling factor can be chosen. 1.8 1.6

Length of the cuboid cross section height of the cuboid

1.4

length (m)

1.2 1 0.8 0.6 0.4 0.2 0.5

1

1.5

Scaling factor

Fig. 4. Influence of link length on the cuboid dimensions

The size of moving platform has an effect on inter-link interference workspace. For sufficiently large size of the moving platform, the inscribed cuboid and the interference space remain as separate regions. With the reduction of the moving platform size, the interference space grows and at a particular threshold, these two regions intersect. It is observed that the growth of the collision space with the

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Fig. 5. Effect of moving platform size on the volume of maximal inscribed cuboid/rectangular parallelepiped

reduction of moving platform size is contributed by the inter-limb interference. The reduction in moving platform size has no effect on the intra-limb interference. Two different cases of moving platform dimensions are being discussed in this section. Case 1: p1 = p2 . Here, the edges of right angled triangle shaped moving platform are considered to be of equal lengths. For this particular case, the link dimensions corresponding to the scaling factor r = 1 as in Fig. 4 are considered. The size of the moving platform edges, initially considered to be of same lengths as of the RR linkages, is reduced by multiplying with a different scaling factor ‘k’. The variation of volume of the largest cuboid inscribed with respect to k is represented by the curve ABC of Fig. 5. It is observed that the volume of the cuboid remains constant until the scaling factor k reaches a threshold value kt , below which the volume decreases (curve ABC). This is due to the intersection of interference space and the largest cuboid inscribed in the interference-free workspace. The shape of the cuboid also changes to a rectangular prism below the threshold kt . Case 2: p1 = p2 . Refer Fig. 5 again for the variation of the cuboidal volume with respect to end effector size. As long as the scaling factor k of p1 is greater than kt , the volume of the cuboid is given by the ordinate of the red line segment ‘DC’. If the scaling factor of p1 goes below the threshold value kt , the volume of the rectangular prism is given by the ordinate of the line segment AB, corresponding to the abscissa of p1 ’s scaling factor. The size of p2 has no effect on the volume.

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Conclusions and Further Works

The evaluation of Cartesian parallel manipulators is carried out based on the dimensions of a rectangular parallelepiped that inscribes its total reachable workspace. However, link interference might lead to several points of the reachable workspace unattainable. This paper investigates the influence of link interference on the Tripteron Cartesian parallel manipulator and proposes an algorithm for finding its interference and interference-free workspace. A second algorithm is also proposed to determine the largest volume cuboid that inscribes the intra-link interference-free workspace. The algorithm can be applied so long as the inter-link interference space does not intersect with the largest cuboid inscribed. The effect of the link lengths and the size of the moving platforms on the volume of the maximal cuboid inscribed is also investigated in this work. As a future work, the algorithms proposed here can be used for the design of the active mechanism for the lower limb rehabilitation systems [2]. The workspace estimation of the passive orthosis is carried out in [12]. For the integrated system, the active parallel mechanism must ensure that the movement of the passive orthosis must be within its workspace. As the link lengths of the orthosis are determined using clinical data and patient’s lower limb dimensions, the link lengths of the active mechanism can be selected such that the maximal cuboid inscribed fully encloses the workspace of the orthosis. Acknowledgements. This research is partly assisted by the Russian Science Foundation (RSF), Russia, the agreement number 19-19-00692 and partly assisted by the Council of Scientific and Industrial Research (CSIR), India, the project number 22(0829)/19/EMR-II. The sponsorship of the first author is provided by CSIR.

References 1. Kong, X., Gosselin, C.M.: Type Synthesis of Parallel Manipulators. Springer, Heidelberg (2007) 2. Mohan, S., Sunilkumar, P., Rybak, L., Malyshev, D., Khalapyan, S., Nozdracheva, A.: Conceptual design and control of a sitting-type lower-limb rehabilitation system established on a spatial 3-PRRR parallel manipulator. In: International Conference on Robotics in Alpe-Adria Danube Region, pp. 345–355. Springer (2020) 3. Merlet, J.P.: Parallel Robots, vol. 128. Springer (2005) 4. Anvari, Z., Ataei, P., Masouleh, M.T.: The collision-free workspace of the tripteron parallel robot based on a geometrical approach. In: Computational Kinematics, pp. 357–354. Springer (2018) 5. Kaloorazi, M.H.F., Masouleh, M.T., Caro, S.: Collision-free workspace of parallel mechanisms based on an interval analysis approach. Robotica 35(8), 1747–1760 (2017) 6. Merlet, J.P., Daney, D.: Legs interference checking of parallel robots over a given workspace or trajectory. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006, pp. 757–762. IEEE (2006)

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7. Liu, Y., Yu, C., Sheng, J., Zhang, T.: Self-collision avoidance trajectory planning and robust control of a dual-arm space robot. Int. J. Control Autom. Syst. 16(6), 2896–2905 (2018) 8. Danaei, B., Karbasizadeh, N., Masouleh, M.T.: A general approach on collision-free workspace determination via triangle-to-triangle intersection test. Robot. Comput. Integr. Manuf. 44, 230–241 (2017) 9. Bottin, M., Boschetti, G., Rosati, G.: A novel collision avoidance method for serial robots. In: IFToMM Symposium on Mechanism Design for Robotics, pp. 293–301. Springer (2018) 10. Sunilkumar, P., Choudhury, R., Mohan, S., Rybak, L.: Dynamics and motion control of a three degree of freedom 3-PRRR parallel manipulator. In: European Conference on Mechanism Science, pp. 103–111. Springer (2020) 11. Ericson, C.: Real-Time Collision Detection. CRC Press, Boca Raton (2004) 12. Malyshev, D., Nozdracheva, A., Dubrovin, G., Rybak, L., Mohan, S.: A numerical method for determining the workspace of a passive orthosis based on the RRRR mechanism in the lower limb rehabilitation system. In: European Conference on Mechanism Science, pp. 138–145. Springer (2010)

Exit Point, Initial Length and Pose Self-calibration Method for Cable-Driven Parallel Robots Bozhao Wang and St´ephane Caro(B) CNRS, Laboratoire des Sciences du Numerique de Nantes, UMR CNRS 6004, 1, rue de la No¨e, 44321 Nantes, France {Bozhao.Wang,Stephane.Caro}@ls2n.fr

Abstract. A calibration process simulation for the interest variable values, including Cartesian coordinates of exit points, moving platform poses and initial cable lengths is performed. The simulation considers the modelling of the pulleys at exit points, and is carried out using nonlinear least square method. The effects of calibration tuning parameters and of measurement pose number on calibration quality are analyzed. As a result, the calibration quality increases with the decrease of tuning parameters and the increase of measurement pose number. The use of Jacobian matrix of the interest function fi,j also leads to a better calibration quality.

Keywords: Cable-Driven Parallel Robot

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· Calibration · Optimization

Introduction

Cable-Driven Parallel Robots (CDPRs) are a type of parallel robot that are actuated by cables. [1] studied reconfigurable Cable-Driven Parallel Robots (RCDPRs) whose cable connection points on the base frame can be positioned at a discrete set of possible locations, and the ways to optimize the sequence of discrete reconfigurations allowing the moving-platform to follow a prescribed path. A novel concept of RCDPRs that consists of a classical CDPR mounted on multiple Mobile Bases is known as Mobile CDPR. In [2], the authors developed a methodology to trace the wrench-feasible-workspace of mobile CDPRs. Robot calibration is performed in purpose of increasing the accuracy of robot positioning by sensor data without changing the robot design. Among different robot configurations, the positioning accuracy can be affected by a large number of sources. Most previous works on the model-based calibration are dedicated to the kinematic process [3,4]. Other than the importance on robot positioning accuracy, calibration will also contribute on several parts, for example motion control, robot production evaluation, etc. [5]. Well-designed experiments can help calibrate quality and efficiency. [6] developed a method to enhance the measurements in geometric parameter identification, the method implemented c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 90–101, 2021. https://doi.org/10.1007/978-3-030-75271-2_10

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irreducible geometric model and took into account of different sources of errors. [7] proposed a method to select manipulator configurations that allow the user to essentially improve calibration quality. [8] studied the geometric and elastostatic calibration of robotic manipulator using partial pose measurements, which do not provide the end-effector orientation, the proposed method improved the calibration efficiency. [9] proposed a concept of the user-defined test-pose to evaluate the calibration experiments quality. Sensitivity analysis is critical to model-based control robustness, in [10], the author performed sensitivity analysis of the elasto-geometrical model to their geometric and mechanical uncertainties, and carried out a case study with a reconfigurable CDPR ‘CAROCA’. Among different calibration approaches for model parameters, the nonlinear least square (NLLS) method used in this work is a basic parameter identification method. In [4], this method is used for kinematic calibration for a new developed CDPR with large workspace. Several different methods derived from NLLS are tested in [11]. Another similiar method is orthogonal distance regression (ODR) [11]. In addition to parameter identification, a sensitivity analysis of CDPR is performed in [12] to decrease the computing time of the calibration process. In order to perform CDPR calibration, measurements are required. Two types of sensors exist. The first one is the internal sensors, including motor encoders and cable force sensors. Self-calibrations, as the case in this work, are realised by the help of internal sensors [11]. The moving-platform of a CDPR is finally connected to the motors through the cables, and encoders can provide the angular position and velocity of the motors. The force sensors provide directly the cable tensions. The second category is the external sensors, for example cameras [13], which can directly measure the position and orientation of the MP. In [14], a look-and-move calibration procedure is developed based on a wireless camera. A low-cost, and efficient calibration method using camera is described in [15]. The sensors could be used alone or in various combinations [16]. In [3], the calibration is carried out with only motor encoders, on a 3-degree-of-freedom CDPR with a tetrahedral platform, which is used as a haptic interface mechanism. In [17], the robot calibration is carried out while the Cartesian coordinates of the exit points expressed in the moving-platform frame are supposed to be known. Some other contributions using multiple sensors have been made [14,15,18,19]. Self-calibrations, as the case in this work, are realised by the help of internal sensors [11]. In this paper, a calibration process simulation for the interest variable values, including Cartesian coordinates of exit points, moving platform poses and initial cable lengths is performed, the robot studied has a 3-DoF, point-mass moving-platform and 4 cables. The calibration simulation takes into account the modelling of pulleys at the exit points, and is performed with encoders alone. The effects of both calibration tuning parameters and measurement pose number on the calibration quality are analyzed. The calibrations with and without Jacobian matrix are compared. The cable sagging, elasticity and mass are neglected in the calibration process.

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In Sect. 2, the modelling of the CDPR used in this paper is introduced, the identification problem performed is presented. In Sect. 3, the robot calibration methodology is introduced and new formulas to represent the calibration results are defined. Some simulation results are analyzed and discussed in Sect. 4. Finally, conclusions and possible future works are presented in Sect. 5.

2

CDPR Modelling

The CDPR considered in this work is illustrated in Fig. 1. The mechanism consists of a 3 DoF point-mass end-effector M , controlled by the m = 4 cables; each cable will pass around a pulley fixed at an arbitrary exit point Ai , i = 1, 2, 3, 4. The cable lengths are controlled by motors. Each motor is equipped with an encoder that measures its angular position and velocity. A number of MP poses are to be defined within the base frame, including one home pose Ph . The cable exit points positions are also not known exactly. Both the poses and exit points are to be identified by the identification method presented in this paper.

Fig. 1. Architecture of the cable-driven parallel robot under study

The ith loop of the CDPR mechanism under study is represented in Fig. 2, i = 1, 2, 3, 4. The vectors from the origin O to the pulley and the platform are denoted as ai and p respectively. The distance from the pulley exit point to the platform is denoted as lci , or the ith estimated cable length. The parameters assocatiated to the ith pulley are described in Fig. 3. The cable starts from the encoder to the exit point Ai , then wrapped on part of the pulley sheave with the length lsi to Ai , then ends on the moving platform Pi . The radius of the pulley used is rp = 0.075 m. The total cable length from exit point to the moving platform calculated as [20]: lti = lsi + lci

(1)

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Fig. 2. The ith loop of the cable-driven parallel robot

where lsi is the length of the cable wrapped on pulley sheave, from Ai to Ai , lci is the cable length from Ai to Pi , and lti is the total cable length from exit point to the moving-platform. Equation (1) can be calculated directly from the Cartesian coordinates of cable exit points and poses with the following steps. First, lsi is calculated by: lsi = rp (π − αi )

(2)

Then, the pulley center oi can be described by ai and the pulley radius rp : oi = ai + rp b Ri xp with

⎤ cosθi −sinθi 0 b Ri = ⎣sinθi cosθi 0⎦ 0 0 1

(3)

⎡

(4)

where oi is the vector pointing from the base frame origin to the pulley center, Ri is the rotation matrix from base frame to the pulley frame {x, y, z} shown in Fig. 3, and xp is the unit vector along the x-axis of the mentioned pulley frame. Li is the vector pointing from the exit point directly to the cable anchor point, θi is the angle between x-axis of base frame and the pulley, and it can be calculated by:

b

θi = atan2(Lyi , Lxi )

(5)

where Lxi and Lyi are the x and y components of Li respectively, expressed in the base frame. The vector pointing from the pulley center to the anchor point is denoted as mi : mi = pi − oi

(6)

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The length Li is expressed as: lci =



mi mi T − rp 2

(7)

The angle relations around the pulley are: αi = −βi + γi

(8)

where βi and γi angles take the form:

and

βi = −atan2(lci , rp )

(9)

a − p  iz iz γi = arcsin ||mi ||2

(10)

Fig. 3. Parameterization of the ith pulley

As a result, Eq. (1) can be expressed as:  lti = rp π − atan2( (pi − ai + rp b Ri xp )(pi − ai + rp b Ri xp )T − rp2 , rp )

aiz − piz − arcsin( ) (11) ||pi − ai + rp b Ri xp ||2  + (pi − ai + rp b Ri xp )(pi − ai + rp b Ri xp )T − rp2

3

Identification Methodology

As introduced before, the end-effector is connected to the motors by cables. The information provided by the encoders (the motor angular positions) is used to

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perform the CDPR calibration. Here, m = 4 cables control the point-mass endeffector. At first, the initial cable lengths from exit points to the end-effector at home pose are denoted as li,0 . It should be noted that li,0 are not known exactly, and are to be identified. In order to implement calibration process, the pointmass end-effector is moved to n different positions within the workspace, meanwhile the variations in the cable lengths are obtained from the motor encoder measurements. Thus, the real cable lengths li,j are calculated by adding the cable length variations Δli,j to the initial cable lengths li,0 , namely, li,j = Δli,j + li,0

(12)

Similarly to [20], in real experiments the variations in the cable lengths obtained from the motor encoder measurements Δli,j are inputs of the identification problem. In this simulation, Δli,j is calculated from the assumed actual Cartesian coordinates of home pose and measurement poses. For the Cartesian coordinates of cable exit points, MP poses and the initial cable lengths, only approximate values are known in the beginning. Therefore, they are considered as the outputs of the identification problem. The known values of the identification problem are: mn cable length variations Δli,j . The unknowns of the problem: m initial cable lengths li,0 , 3m Cartesian coordinates of cable exit points ai,x , ai,y , ai,z , and 3n Cartesian coordinates for the point-mass end-effector position pj,x , pj,y , pj,z . The unknowns form a vector x for the variables of the calibration problem: x = [a1,x , a1,y , a1,z , . . . , am,x , am,y , am,z , l1,0 , . . . , lm,0 , p1,x , p1,y , p1,z , . . . , pn,x , pn,y , pn,z ] = [a1 , . . . , am , l0 , p1 , . . . , pn ]

(13)

The size of x is dependent of the number of measurement poses. If n poses are used, x will be a (4m + 3n)-dimensional vector. Compared to the real cable length li,j , Eq. (1) gives the estimated cable lengths derived from the geometric model of the robot mechanism. As a consequence, a system composed of mn equations is obtained [3,20]: fi,j (x) = (lsi + lci )2 − (Δli,j + li,0 )2    = rp π − atan2( (pi − ai + rp b Ri xp )(pi − ai + rp b Ri xp )T − rp2 , rp ) − arcsin(

 aiz − piz ) b ||pi − ai + rp Ri xp ||2

 + (pi − ai + rp b Ri xp )(pi − ai + rp b Ri xp )T − rp2

2 − (Δli,j + li,0 )2 ,

i = 1, . . . , m, j = 1, . . . , n

(14)

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In order to solve the nonlinear system of equations expressed in Eq. (14), the number of equations should be larger than or equal to the number of unknowns, namely, mn ≥ 3m + m + 3n

(15)

From Eq. (15), for a CDPR with m = 4 cables and a point-mass end-effector, the number of measurement poses should be at least equal to n = 16. Equation (14) provides the squared difference of estimated cable length and real cable length. The nonlinear least-square solver in Matlab is used to solve the curve fitting problems of the following form, which contains the defined function Eq. (14):  m n

2 fi,j (16) min x

i=1 j=1

In order to simulate the calibration process, arbitrary errors err are added on each real variable value, to obtain xepsi : xepsi = x + err = [a1,x + err1 , a1,y + err2 , . . . , pn,z + err16+3n ]

(17)

The lsqnonlin function takes the starting value xepsi as input, and outputs the calibrated values x∗ that satisfies Eq. (16): x∗ = [a∗1 , . . . , a∗m , l∗0 , p∗1 , . . . , p∗n ]

(18)

a∗1 , . . . , p∗n

where are all the simulated identified values. The other input arguments include the objective function fi,j , the lower bound lb and upper bound ub for xepsi , and optimization options that could be tuned to have a better control of the calculation process. Apart from xc , the other output arguments used are the final solution values of fi,j , saved in a vector fresi : fresi = F(x∗ ) = [F1 (x∗ ), . . . , Fj (x∗ ), . . . , Fn (x∗ )]

(19)

with Fj (x∗ ) = [f1,j (x∗ ), . . . , fi,j (x∗ ), . . . , fm,j (x∗ )]

(20)

The size of fresi is the same as the number of interest function fi,j , therefore fresi is mn-dimensional. The following criterion, named μ, is defined to assess the identification quality: mn |fresi,h | μ = h=1 (21) mn For each calibration, the process is performed 500 times with different err applied. Then 500 calibrated results x∗l , l = 1, 2, . . . , 500, are obtained. x∗l = [a∗l1 , . . . , a∗lm , l∗l0 , p∗l1 , . . . , p∗ln ], l = 1, 2, . . . , 500

(22)

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To prevent any single deviated value to produce a large effect on the final result, the mean value of them is taken, noted as x∗M : 500

x∗l,k , k = 1, 2, . . . , 4m + 3n (23) 500 To examine the calibration quality, x∗M is compared with the real variable values x, noted as . Smaller  values indicate better calibration quality. x∗M,k

=

l=1

 = [1 , . . . , k , . . . , 4m+3n ] k = |x∗M,k − xk |,

k = 1, 2, . . . , 4m + 3n

(24) (25)

In order to speed up the convergence of the algorithm, the following Jacobian matrix can be used in lsqnonlin [3]: ⎡ ⎤ ∂f1,1  ∂f1,1 ∂f1,1  . . . . . . ∂l1,0 ∂pn ⎢ ∂a1 ⎥ ∂f ⎢ .. . .. ⎥ . . . . .. .. (26) =J=⎢ . . ⎥ ∂x ⎣ ⎦ ∂fn,m  ∂fn,m ∂fn,m  . . . ∂l1,0 . . . ∂pn ∂a1

4 4.1

Simulation Results Effect of Tuning Parameters

There are a list of tuning parameters inside the option argument of lsqnonlin function. Among those parameters, ‘MaxFunctionEvaluations’, ‘MaxIterations’, ‘FunctionTolerance’ and ‘StepTolerance’ have obvious and important inflluences on the calculation. ‘MaxFunctionEvaluations’ and ‘MaxIterations’ are the maximum number of function evaluations and iterations allowed respectively, and with both the default value of 200 times of starting vector length. If the number is not high enough, the optimization will end prematurely without reaching the desired termination tolerance. Here the two parameters are set to be 120000 and 8000 respectively. ‘FunctionTolerance’ and ‘StepTolerance’ are the termination tolerances for the function value and interest variables respectively, with both with the default values of 1e−4. The two parameters are set to be 1e−20 and 1e−12 according to their effect on the identification, measured by identification quality μ. The result is shown in Fig. 4. Identification error μ decreases with the decrease of ‘FunctionTolerance’ and ‘StepTolerance’. ‘FunctionTolerance’ causes the result to decrease rapidly while ‘StepTolerance’ have a less stronger effect on μ. At the area where the square root of ‘F unctionT olerance’ is less than 1e−8 (m) and ‘StepTolerance’ is less than 1e−6 (m), μ remains at the same level of magnitude, around 1e−5 (mm). When ‘FunctionTolerance’ and ‘StepTolerance equal to 1e−20 and 1e−12 respectively, the value of μ is favourable.

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Fig. 4. TolFun, TolX influence on identification error

4.2

Effect of Number of Poses

For the identification problem studied in this paper, the minimum number of measurement poses is 16. In this paper, up to 40 poses were performed to obtain more sets of cable length variations from the encoders, in order to achieve higher calibration quality. The result is shown in Fig. 5. The identification quality increases with the increase of number of poses.

Fig. 5. Influence of number of measurement moving-platform poses on the identification quality

4.3

Results Analysis

The difference between real and calibrated variable values  is calculated. For the Cartesian coordinates of exit points and moving platform poses, almost all the result values locate around or under 0.1 (mm), and have a decreasing trend.

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Examples of the y coordinate of exit point A1 , for the case of using Jacobian matrix are shown in Fig. 6. Noted by Matlab, the convergence times are 4327 s and 17278 s for calibration simulation process with and without Jacobian Matrix, respectively. The results show that Jacobian matrix contributes to a faster convergence and higher calibration quality.

Fig. 6. The error in y-coordinate of the first exit point as a function of the number of measurement poses, with Jacobian Matrix

5

Conclusions and Future Work

A simulation of the calibration process of interest variables using lsqnonlin function was carried out in this paper. The robot, which includes a 3-DoF, point-mass moving-platform and 4 cables, was studied considering the pulley modelling. The calibration process was assumed to be performed with encoders, the only one type of sensor alone. The effect of calibration tuning parameters on the calibration quality is analyzed. Identification error μ decreases with the decrease of ‘FunctionTolerance’ and ‘StepTolerance’. The effects of measurement pose number on calibration quality and calibrated interest variable values are calculated. With more poses used, the calibration quality would improve, and the errors on the calibrated interest variable values would decrease. The use of Jacobian matrix would result in a better interest variable error. There are uncertainties in the geometric parameters, for example geometrical errors on pulley radius, the mechanical errors concerning cable elasticity etc. Sensitivity analysis of these uncertainties will be performed to further improve the calibration quality. This work considered the pulley modelling at the exit points, however the cable elasticity and cable sagging effects were not. The CDPR studied in this work contained only 4 cables and a simple point-mass moving-platform. In future works, real experiments will be carried out, and more complex CDPRs will be considered.

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Acknowledgement. This work was supported by the ANR CRAFT project, grant ANR- 18-CE10-0004, https://anr.fr/Project-ANR-18-CE10-0004

References 1. Gagliardini, L., Caro, S., Gouttefarde, M., Girin, A.: Discrete reconfiguration planning for cable-driven parallel robots. Mech. Mach. Theory 100, 313–337 (2016) 2. Rasheed, T., Long, P., Caro, S.: Wrench-feasible workspace of mobile cable-driven parallel robots. ASME J. Mech. Robot. 12(3), 031009 (2020) 3. Fortin-Cˆ ot´e, A., Cardou, P., Gosselin, C.: An admittance control scheme for haptic interfaces based on cable-driven parallel mechanisms. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 819–825 (2014) 4. Qian, S., Bao, K., Zi, B., Wang, N.: Kinematic calibration of a cable-driven parallel robot for 3D printing. Sensors 18(9), 2898 (2018) 5. Elatta, A., Gen, L., Zhi, F., Daoyuan, Y., Fei, L.: An overview of robot calibration. Inf. Technol. J. 3, 74–78 (2004) 6. Wu, Y., Klimchik, A., Caro, S., Furet, B., Pashkevich, A.: Geometric calibration of industrial robots using enhanced partial pose measurements and design of experiments. Robot. Comput. Integr. Manuf. 35, 151–168 (2015) 7. Klimchik, A., Caro, S., Pashkevich, A.: Optimal pose selection for calibration of planar anthropomorphic manipulators. Precis. Eng. 40, 214–229 (2015) 8. Klimchik, A., Wu, Y., Caro, S., Furet, B., Pashkevich, A.: Geometric and elastostatic calibration of robotic manipulator using partial pose measurements. Adv. Robot. 28, 1419–1429 (2014) 9. Klimchik, A., Pashkevich, A., Wu, Y., Caro, S., Furet, B.: Design of calibration experiments for identification of manipulator elastostatic parameters. J. Mech. Eng. Autom. 2, 531–542 (2012) 10. Baklouti, S., Caro, S., Courteille, E.: Sensitivity analysis of the elasto-geometrical model of cable-driven parallel robots. In: 3rd International Conference on CableDriven Parallel Robots, CableCon 2017, Quebec City, Canada, August 2017 11. dit Sandretto, J.A., Daney, D., Gouttefarde, M.: Calibration of a fully-constrained parallel cable-driven robot. In: CISM International Centre for Mechanical Sciences, Courses and Lectures, vol. 544, no. 2007, pp. 77–84 (2013) 12. Reicherts, S., Blume, S., Reichert, C., Schramm, D.I.D.D.: Sensitivity analysis of the design parameters for the calibration of cable-driven parallel robots. PAMM 16(1), 859–860 (2016) 13. Lessanibahri, S., Cardou, P., Caro, S.: A cable-driven parallel robot with an embedded tilt-roll wrist. ASME J. Mech. Robot. 12(2), 021107 (2020) 14. Zavatta, M., Chianura, M., Pott, A., Carricato, M.: A vision-based referencing procedure for cable-driven parallel manipulators. J. Mech. Robot. 12(4), 1–7 (2020) 15. Tremblay, N., Kamali, K., Cardou, P., Desrosiers, C., Gouttefarde, M., Otis, M.J.: Eye-on-hand calibration method for cable-driven parallel robots. Mech. Mach. Sci. 74, 345–356 (2019) 16. Garrido-Jurado, S., Mu˜ noz-Salinas, R., Madrid-Cuevas, F.J., Medina-Carnicer, R.: Generation of fiducial marker dictionaries using mixed integer linear programming. Pattern Recogn. 51, 481–491 (2016) 17. Lau, D.: Initial length and pose calibration for cable-driven parallel robots with relative length feedback. Mech. Mach. Sci. 53, 140–151 (2018) 18. Tang, G.R., Mooring, B.W.: Plane-motion approach to manipulator calibration. Int. J. Adv. Manuf. Technol. 7(1), 21–28 (1992)

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19. Zhuang, H., Liu, L., Masory, O.: Autonomous calibration of hexapod machine tools. J. Manuf. Sci. Eng. Trans. ASME 122(1), 140–148 (2000) 20. Picard, E., Caro, S., Claveau, F., Plestan, F.: Pulleys and force sensors influence on payload estimation of cable-driven parallel robots. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2018), Madrid, Spain, October 2018

Kinematic Simulation of a Geared Planar Parallel Manipulator Antonio-Marius-Flavius Luputi1, Erwin-Christian Lovasz1(&), Marco Ceccarelli2, Sticlaru Carmen1, and Ana-Maria Stoian1 1

Politehnica University of Timişoara, Timişoara, Romania [email protected], {erwin.lovasz,carmen.sticlaru,ana.scut}@upt.ro 2 University of Rome Tor Vergata, Rome, Italy [email protected]

Abstract. The paper presents a kinematic simulation of a planar parallel manipulator with geared linkages with a structure 3-R(PRRGR)RR. The kinematic positional analysis of the planar parallel manipulator is performed with an analytical formulation for calculation in MATLAB and with designed manipulator in Creo Parametric software. The inverse kinematic positional analysis is used for generating by Tool Center Point a combined motion on an oblique trajectory with rotation of the mobile platform. The rotation angles of leg elements and the stroke of the linear actuators obtained with the Creo Parametric software are compared in order to validate the analytical formulation for calculation of inverse kinematic in MATLAB. Keywords: Parallel planar manipulator Kinematics


Geared linkages
Linear actuators


1 Introduction A parallel manipulator is mechanism with closed loop kinematic chains, where the endeffector (mobile platform) is bound to the base platform (frame) through legs (kinematic chains), each of them with the degree of freedom zero [1]. To avoid redundant mechanism [2] the number of actuated joints used is equal to the number of degrees of freedom the parallel manipulator. Usual a mobile platform needs to be hold by at least two kinematic chains. Each of the kinematic chains has to have at least one driving actuator which should incorporate sensors. These sensors measure the angle of rotation or in case of a linear actuator, the stroke. The branch of parallel manipulator is relatively new, at middle of the twenty century. One of the first of its kind was a planar parallel manipulator with a four-bar mechanism design for grasping object of different shapes [3]. The forward kinematic problems were solved for a wide range of planar parallel manipulators by several authors [4–6], etc. The applications for the parallel manipulators are significantly increasing in industry, medicine and new branches that require accuracy and high speed. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 102–110, 2021. https://doi.org/10.1007/978-3-030-75271-2_11

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Parallel manipulators exhibit salient advantages in accuracy and dynamic response compared to the serial counterparts as pointed for example [7]. Other advantages of the parallel architecture are low inertia, lighter weight, stabile capacity, high speed, considerable stiffness and large ratio between the platforms load and the weight of the manipulator. However, these parallel structures come also with drawbacks in reduced workspace of the end effector and singularity characteristics [8]. The present paper focuses on the inverse kinematics analysis of a planar parallel manipulator geared linkages design. Based on the mathematical model of the manipulator, the inverse kinematics is investigated to determine the strokes of the actuators, angles of rotation of the joints, position and orientation of the mobile platform for an operation and characterization.

2 A 3-R(PRRGR)RR Planar Parallel Manipulator with Geared Linkages The planar parallel manipulator using geared slider-crank with linear actuators has a proper transmission, an approximately constant transmission function for wide range and a large rotation angle as pointed out in [9]. The planar parallel manipulator of present interest can be named as 3-R(PRRGR)RR, where P, R and G stand for prismatic, revolute and gear kinematic pairs, respectively.

Fig. 1. Kinematic schema of the planar parallel manipulator 3-R(PRRGR)RR using geared slider-crank with linear actuators [10]

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The legs used a geared linkages as kinematic chain, which contain as basic structure a slider-crank connected in parallel with a spur gear train. The drive element is a linear actuator 2 and the driven element is the gear 5, which is fixed with the connection element to mobile platform. Figure 1 shows a kinematic scheme of the 3-R(PRRGR)RR design [11, 12]. Good transmission ratio is obtained when self-locking condition is satisfied for the linear actuators. Previous works show that the motions generated from geared linkages using linear actuation are influenced by the large rotation angle and a transmission ratio approximately constant in a very large range [10–12]. In this paper the problem of the inverse kinematics analysis of the parallel mechanism is attached with a formulation for calculation in MATLAB, which have been verified with a specific simulation of the CAD model in Fig. 2.

Fig. 2. A CAD model of the planar parallel manipulator 3-R(PRRGR)RR using geared slidercrank with linear actuators

2.1

Inverse Positional Kinematic Approach

The mobile platform of planar parallel manipulator 3-R(PRRGR)RR has 3 degrees of freedom: one rotation, around the z-axis, and two translation, along the x-axis and

Fig. 3. The kinematic scheme for inverse kinematic of the planar parallel manipulator

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y-axis. Figure 3 shows a model for one leg of the 3-R(PRRGR)RR manipulator from which the inverse positional kinematic is formulated with a basic/first solution in [9]. Referring to Fig. 3 the inverse positional kinematic analysis allows the computing of the linear actuators stoke si by given the TCP parameters (xM(t), yM(t), a(t)) with t 2 [0,1]. The lengths of mobile and fixed platform l0 and l6, the frame joints coordinates (xAi, yAi), the gear ratio q and the link lengths of the chains l1, l5 are the characteristics of the planar parallel platform design. As proposed in [9] the generalized coordinates h1i and h5i , i ¼ 1; 3 can be expressed as h1i ðtÞ ¼ 2
arctan

B1i ðtÞ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt Þ A21i ðtÞ þ B21i ðtÞ  C1i ; A1i ðtÞ  C1i ðtÞ

ð1Þ

where   ð6Þ ð6Þ A1i ðtÞ ¼ 2
l1
xM ðtÞ þ xCi
cosaðtÞ  yCi
sinaðtÞ  xAi ;   ð6Þ ð6Þ B1i ðtÞ ¼ 2
l1
yM ðtÞ þ xCi
sinaðtÞ þ yCi
cosaðtÞ  yAi ; ð6Þ

ð6Þ

:

ð2Þ

C1i ðtÞ ¼ l25  l21  ðxM ðtÞ þ xCi
cosaðtÞ  yCi
sinaðtÞ  xAi Þ2 ð6Þ

ð6Þ

ðyM ðtÞ þ xCi
sinaðtÞ þ yCi
cosaðtÞ  yAi Þ2 and h5i ðtÞ ¼ 2 arctan

B2i ðtÞ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt Þ A22i ðtÞ þ B22i ðtÞ  C2i ; A2i ðtÞ  C2i ðtÞ

ð3Þ

where   ð6Þ ð6Þ A2i ðtÞ ¼ 2
l5
xM ðtÞ þ xCi
cosaðtÞ  yCi
sinaðtÞ  xAi ;   ð6Þ ð6Þ B2i ðtÞ ¼ 2
l5
yM ðtÞ þ xCi
sinaðtÞ þ yCi
cosaðtÞ  yAi ; ð6Þ

ð6Þ

:

ð4Þ

C2i ðtÞ ¼ l21  l25  ðxM ðt þ xCi
cosaðtÞ  yCi
sinaðtÞ  xAi Þ2 ð6Þ

ð6Þ

ðyM ðtÞ þ xCi
sinaðtÞ þ yCi
cosaðtÞ  yAi Þ2 The relative rotation angle of the gear 5, v5i ðtÞ can be obtained from v5i ðtÞ ¼ h5i ðtÞ  h1i ðtÞ þ Dv5 i ¼ 1; 3 where Dv5 is a constructive angle.

ð5Þ

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The motion equation of the geared slider-crank can be given as: w5i ðtÞ þ q
ðu5i ðtÞ  w5i ðtÞÞ  v5i ðtÞ ¼ 0

ð6Þ

q ¼ r4 =r6

ð7Þ

where

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi ½2
l6
ðs0 þ si ðtÞÞ2  ðs0 þ si ðtÞÞ2 þ l26  l24 h i 2
l6
ðs0 þ si ðtÞÞ  ðs0 þ si ðtÞÞ2 þ l26  l24

u5i ðtÞ ¼ 2
arctan

2

ð8Þ

and w5i ðtÞ ¼ arcsin

l6
sinðu5i ðtÞÞ l4

ð9Þ

The transmission function v5i ðsi ðtÞÞ can be computed using the Eqs. (6)–(9). In order to compute an analytical equation for the inverse transmission function of the geared linkage si ðv5i ðtÞÞ an interpolation Lagrange function with a polynomial of 3rd order was performed 0

1

C 4 B 4 X Y B ðv5i ðtÞÞ  ðv5i ðtÞÞj C C: B si ðv5i ðtÞÞ ¼ Bðsi ðtÞÞk
ðv5i ðtÞÞk  ðv5i ðtÞÞj C A k¼1 @ j¼1 j6¼k

ð10Þ

where the parameter pairs ðsi ðtÞÞk ; ðv5i ðtÞÞj;k were recorded from the calculated transmission function.

3 Numerical Example A numerical example is discussed for the parallel manipulator according to the constructive dimensions of a linear actuator Firgelli L12-30-50-6-R for the reported example. The values of the design parameters are listed in Table 1.

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Table 1. The parameters for simulating the model in Fig. 2 Frame platform length (0) Mobile platform length (6) Chain link length (1, 1′, 1″) Chain link length (5, 5′, 5″) Coupler length (3, 3′, 3″) Carrier length (4, 4, 4″) Gear ratio Initial stroke Maximum stroke

l0 = 240 mm l6 = 40 mm l1 = 150 mm l5 = 100 mm l3 = 100 mm l4 = 45 mm q=2 s0 = 82 mm hr = 30 mm

Figure 4a shows the transmission function of the geared slider-crank with linear with linear actuators and Fig. 4b the interpolated transmission function used for stroke computation of the linear actuator according the relationship (10).

Fig. 4. The transmission function v5i ðsi Þ (a) and the interpolated inverse function si ðv5i Þ (b)

In order to obtain the inverse transmission function of the geared linkage with linear actuation the Lagrange function was used, detailed presented in reference [10]. For the Lagrange interpolation function the following nodes are used: 8 si1 ¼ 0 mm; vi1 ¼ 208:60 > > > < s ¼ 10 mm; v ¼ 169:51 i2 i2 > si3 ¼ 20 mm; vi3 ¼ 124:90 > > : si4 ¼ 30 mm; vi4 ¼ 59:35

ð11Þ

For the combined motion in rotation and translation of the mobile platform with respect to the reference system OX0Y0 the equations are 8   > < xM ðtÞ ¼ xi þ xf  xi
t y M ð t Þ ¼ yi þ yf  yi
t >   : aðtÞ ¼ ai þ af  ai
t

ð12Þ

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The coordinates (xM, yM, a) of the mobile platform for a combined motion are varying over time and being independent of each other. All the initial and final parameters are listed as from the simulation in Table 2. Table 2. Parameters for combined motion Initial Final xi = 94.29 mm xf = 142.61 mm yi = 50.71 mm yf = 103.53 mm ai = 0.00° af = 30.00°

The CAD design used for simulation in Fig. 5 illustrate the corresponding trajectory of the mobile platform from the initial coordinates (xi, yi, ai) to the final coordinates (xf, yf, af) of the reference point M.

Fig. 5. The initial position (a), the final position (b) and the trajectory (c) of the mobile platform

The trajectory of the mobile platform does not pass through the centroid of the frame platform as result the strokes of the three linear actuators do not cross all in a single point, but because the centroid is close to the trajectory of the mobile platform the strokes are close. In Fig. 6 are shown the strokes of the three linear actuators obtained from MATLAB and Creo Parametric. The graphs results of both simulations plotted in MATLAB are identical, that validate the mathematical formulation of the positional kinematic problem of the considered planar parallel manipulator.

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Fig. 6. The simulations corresponding the strokes of the geared linkage with linear actuation of the mobile platform for combined motion.

Figure 7 shows the variation of the angles of the chain links during the considered combined motion of the mobile platform. In the combined motion of the mobile platform are no sudden jumps of the linear actuators stroke and the angular stroke of the chains links. That means the linear actuators are able to follow the motion.

Fig. 7. Computed results of the simulation of combined motion in terms of angles

4 Conclusions The mathematical model of the planar parallel manipulator 3-R(PRRGR)RR is formulated and programed in a MATLAB program to generate simulations. Based on the dimensions of the planar parallel manipulator a CAD design was created in Creo Software. The simulation results of the CAD design were compared with the ones in the MATLAB simulations, reaching to the conclusion that both data are identical validating the mathematical formulation of the positional kinematic problem of the considered planar parallel manipulator. The planar parallel manipulator 3-R(PRRGR) RR avoid the singularities of 1st order. Further will be studied the workspace and dexterity of them comparing with 3-RRR parallel manipulator and a prototype will be manufactured for several experimental tests.

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References 1. Tsai, L.W.: Robot Analysis: The Mechanics of Serial and Parallel Manipulators. John Wiley & Sons, New York 2. Chiaverini, S., Oriolo, G., Walker, I.D.: Kinematically redundant manipulators. In: Siciliano, B., Khatib, O. (eds.) Springer Handbook of Robotics. Springer, Berlin, Heidelberg. https:// doi.org/10.1007/978-3-540-30301-5_12 (2008) 3. Yi, B.-J., Ra, H.Y., Lee, J.H., Hong, Y., Park, J.S., Oh, S.-R., Suh, I., Kim, W.: Design of a parallel-type gripper powered by pneumatic actuators. In: International Conference on Intelligent Robots and Systems, Takamatsu (2000) 4. Chandra, R., Zhang, M., Rolland, L.: Solving the forward kinematics of the 3RPR planar parallel manipulator using a hybrid meta-heuristic paradigm. In: IEEE International Symposium Computational Intelligence Robotics and Automation (CIRA), pp. 177–182, Daejeon, Korea (South) (2009) 5. Hayes, M.J.D., Husty, M.L., Zsombor-Murray, P.J.: Solving the forward kinematics of a planar three-legged platform with holonomic higher Pairs. ASME J. Mech. Des. 121(2), 212–219 (1999) 6. Arellano, H.V., Rivera, M.M.: Forward kinematics for 2 DOF planar robot using linear genetic programming. Res. Comput. Sci. 148, 123–133 (2019) 7. Pandilov, V., Dukovski, Z.: Comparison of the characteristics between serial and parallel robots. Acta Tehn. Corviniensis – Bull. Eng. Tome VII [2014] 8. Ceccarelli, M.: Fundamentals of the mechanics of robots. In: Fundamentals of Mechanics of Robotic Manipulation. International Series on Microprocessor-Based and Intelligent Systems Engineering, vol. 27. Springer, Dordrecht 9. Lovasz, E.-C., Modler, K.-H., Pop, C., Pop, F., Mărgineanu, D.T., Maniu, I.: Type synthesis and analysis of geared linkages with linear actuation. Mechanika 24(1), 108–114 (2018) 10. Grigorescu, S., Luputi, A.-M.-F., Maniu, I., Lovasz, E.-C.: Novel planar parallel manipulator using geared slider-crank with linear actuation as connection kinematic chain. New Trends Mech. Mach. Sci. Springer Mech. Mach. Sci. 89, 496–505 (2020) 11. Lovasz, E.-C., Pop, C., Grigorescu, S., Gruescu, C., Mǎrgineanu, D., Maniu, I.: Kinematics of the planar parallel manipulator using geared linkages with linear actuation as kinematic chains 3-R(RPRGR)RR, In: 14th IFToMM World Congress Proceedings, pp. 493–498 (2015) 12. Lovasz, E.-C., Grigorescu, S., Mărgineanu, D., Gruescu, C., Pop, C., Ciupe V., Maniu, I.: Geared linkages with linear actuation used as kinematic chains of a planar parallel manipulator. In: Proceedings of the Third MeTrApp Conference 2015, Series: Mechanism and Machine Science, vol. 31, pp. 21–31. Springer (2015)

Mechanics of Robots

Design Criteria Study for Underactuated Symmetric Pinching Mechanism of Pinch Roll Machine in High-Speed Wire Rod Product Line Wang Renquan1, Yao Shuangji2(&), Marco Ceccarelli3, and Yang Xiaohan2 1

CERI Long Product Co. Ltd., Beijing 100176, China School of Vehicles and Energy, Yanshan University, Qinhuangdao 066004, China Department of Industrial Engineering, University of Rome Tor Vergata, 00133 Rome, Italy 2

3

Abstract. Inspired by the underactuated robotic hand mechanism and its feature of shape adaptive and compliant envelop grasping, the paper proposed underactuated symmetric pinching mechanism to solve the problem of pinch rollers idle and slipped or pinched too tightly. Pinch roll machine is a key device in High-Speed wire rod product line. It is designed with actuated symmetric pinch mechanism. We discuss the possibility of the underactuated symmetric pinch mechanism applied in the pinch roll machine. Three kinds of underactuated symmetric pinch mechanism solutions are presented. When the mechanism implement the synchronous symmetric pinch action, a displacement can occur at the underactuated joint, and the instantaneous clamping force can be buffed by the spring system, so as to reduce the dynamic impact of clamping force. A suitable design criteria is deduced from statics and kinematics research. The design method can keep the contact force do not change when the rod dimension is variable in the passing line. Thus, it can protect the rod not to be damaged while it is pinching and conveying, and avoids the fatigue damaging caused by excessive pinching impact force. The research method is novel and feasible. The design result will be beneficial to improve service life of the pinch roll machine. Keywords: Underactuated mechanism
Symmetric pinch
Pinch roll
Highspeed rod line
Design criteria

1 Introduction The pinch roll machine located between finishing mill and cooling bed, or between the water-cooling sets in the high speed rod wire production line. Its main function is to pinch steel rod smoothly through the water-cooling section, or smoothly reduce the speed into the cold bed input roller table. The ideal steel rod transmission line is a straight line, the ideal steel bar transmission line is straight, but because of the reasons of interference, the steel rod is conveying along the transmission line and shaking at the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 113–121, 2021. https://doi.org/10.1007/978-3-030-75271-2_12

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same time. The pinch roll machine can also prevent the rod shaking around the by a suitable pinch force and pinch speed. Thus, pinch roll machine will ensure the steel rod keeping micro tension with upper and lower mills while it is conveying [1–3]. Figures 1 and 2 are two main linkage symmetric pinch mechanisms in pinch roll machine. In Figs. 1 and 2, 1 and 1′ are a pair of gears with the same gear parameters. The gear 1 is fixed on one input shaft o, gear 1′ is fixed on the other shaft o′, a motor is connected with the shaft; 2 and 2′ are gears with the same gear parameters, which connected on two working shafts and coupling with a pair of rollers 5 and 5′; 3 is a linear actuator (It may be air cylinder or linear motor.); 6 is the conveyed steel rod, its diameter is A; 4a is a pair of sector gears synchronization mechanism, which have the same parameters; 4b is a multi-linkage synchronous mechanism, which can carry out synchronous action.

Fig. 1. Gear symmetrical pinch and conveying mechanism

Fig. 2. Linkage synchronous pinch and conveying mechanism

In the mechanism of Fig. 1, gear 1 has the same speed ratio as gear 1′, so it is constant speed transmission. In addition, input shaft o drives gear 1 also engages with working shaft and gear 2 at the same time; On shaft o′, gear 1′ engages with working shaft gear 2′. Thus the mechanism can realize constant rotation on working shaft 2 and 2′. The linear actuator 3 drives the sector gear 4a to output synchronously equal speed ratio through connecting links. The working shaft 2 and 2′ are rotated around the center points of gears 1 and 1′ symmetrically to open or close the rollers gap A by 5 and 5′. Thus, the pair of rollers 5 and 5′ can pinch and convey steel rod 6. In the mechanism of Fig. 2, gear 1 has the same speed ratio as gear 1′, so it is constant speed transmission. In addition, input shaft gear 1 also engages with working shaft gear 2 at the same time; input shaft gear 1′ engages with working shaft gear 2′. Thus the mechanism can realize constant rotation on working shaft 2 and 2′. The linear actuator 3 drives the multi-linkage synchronous mechanism 4b to output synchronously equal speed ratio through connecting links. The working shaft 2 and 2′ are rotated

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around the center points of 1 and 1′ symmetrically to open or close the rollers gap A by 5 and 5′. Thus, the pair of rollers 5 and 5′ can pinch and convey steel rod 6. The main structural characteristics of the two kinds of pinch rolls are as follows: a motor transmits rotating motion coupling with the input shaft on gear 1. The two working shafts coupling with working shaft gear 2 and 2′ are rotated through one-stage gear increase transmission, and the pair of pinch rollers 5 and 5′ are connected with the two working shafts. Thus, the pinch rollers 5 and 5′ can rotate with the equal speed ratio because of the equal ratio gears in the transmission system. When the hot metal detector in upstream production line detects the head of the incoming rolled rod passes through the pinch roll machine, the PLC system gives signal order to close the gap of the pinch rollers, and the two rollers clamp the rod and transfer into downstream. The pinch rollers gap between 5 and 5′ are fixed by mechanical position limitation in the machine. It can be computed that the number of degrees of freedom of the mechanism in Figs. 1 and 2 are two. The mentioned pinch roll machines of Figs. 1 and 2 have two motions: pinch rollers rotation and pinch rollers gap open & close. There are also have two input actuators: motor rotation on shaft o and linear actuator 3. Thus, it can be concluded that the mentioned two kinds of pinch roll machines are actuated mechanisms.

2 Existing Problems Discussion During long-term production and operation of the pinch roll machines, it is found that the two kinds of equipment have 2 main disadvantages, which can be summarized as: Firstly, the pinch rollers gap is determined by mechanical position limitation. As shown in Fig. 3, 7 is the mechanical position limitation in the pinch roll machine. If a gap size is fixed the opening and closing mechanical position limitation is determined and it can’t be changed when it is working. Thus, it can only pinch and convey a certain steel rod with diameter as A. But the diameter of actual steel rod are varied within a certain range as (A + dA) for the whole length rolled steel rod. That will lead to the pinch rollers idle and slipped or pinched too tightly, then finally fail to pinch [4].

a) Ideal pinching and conveying

b) Actual pinching and conveying

Fig. 3. Pinching and conveying situation comparison

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Secondly, the opening and closing process of pinch rollers is driven by the linear actuator rapidly. After reaching the gap position, the pinch rollers are stopped by hitting against the mechanical position limitation. The gap open and close frequently, the hitting force makes great impact to cause some parts damage. Based on the above two reasons, we believe that the actuated pinch roll linkage mechanisms are not satisfied for actual steel rod pinch and conveying. Underactuated symmetric pinching mechanism design for pinch roll machine should be considered as a solution to keep all-time pinching and reducing the dynamic impact force[5–7]. As we researched before [8, 9], underactuated robotic finger or gripper mechanisms have a notable feature which can be summarized as “shape-adaptive envelop grasp”. Underactuated robotic hand can envelop grasp objects by compliant contacting objects’ unregular outside surface. The underactuated mechanism solution even can be achieved low-cost and easy-operation. With the idea of “shape-adaptive envelop grasp”, the underactuated mechanism with the feature of “shape-adaptive pinching and conveying” is proposed for pinch roll machine and applied in High-Speed wire rod product line.

3 Underactuated Pinching and Conveying Mechanisms The solution for changing actuated pinching and conveying mechanism to underactuated pinching and conveying mechanism is carried out. The main method is adding underactuated elements, such as spring and damper, into the linkage actuated pinching and conveying mechanism. In this paper, three kinds of underactuated pinching and conveying mechanisms are proposed, as shown in Fig. 4. In which Fig. 4 a) and b) are evolved from Figs. 1 and 2, 4 c) is a new designed underactuated pinching and conveying mechanism. The mechanism in Fig. 4 a) includes spring elements 8 and 8’ and the fixed rotation shaft joints o and o’ are changed from the mechanism in Fig. 1. It can move in the linear direction as the spring 8 and 8′. The mechanism in Fig. 4 b) includes a spring elements 8, and the fixed rotation joint C is changed from the mechanism in Fig. 2. The joint C is located on a slider, and it can move in horizontal direction. The mechanism in Fig. 4 c) is a symmetrical structure, the steel rod passing line is the symmetric line. The linear actuator 3 is fixed in the rod passing line. This mechanism includes component 7, it is connected by two springs elements and linear sliders. The spring box 7 is connected with the linear actuator 3, 7′ is the mechanical position limitation in the system. Link 4 and 4′ are jointed with the linear sliders. All the three kinds of mechanisms are 2 actuators, but the mechanisms in Fig. 4 a) and b) have 3 degrees of freedom, the mechanisms in Fig. 4 c) have 4 degrees of freedom. Thus, they are underactuated pinch and conveying mechanisms.

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a) underactuated gear symmetrical pinch and conveying mechanism

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b) underactuated linkage synchronous and conveying mechanism

c) new designed underactuated linkage symmetrical pinch and conveying mechanism

Fig. 4. Three kinds of underactuated linkage pinch and conveying mechanism

In order to design the underactuated pinch roll machine and applied in the wire rod rolling production line, the proposed three kinds of underactuated linkage pinch and conveying mechanisms are considered the possibility mainly from the following 3 aspects: 1) structure design; 2) components and parts design and manufacture; 3) components and parts assembly. Finally, the mechanism in Fig. 4 3) are selected as the pinch roll machine design basement.

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4 Design Criteria Study Based on Statics and Kinematics A new design scheme of underactuated linkage symmetrical pinch and conveying mechanism in Fig. 4 c) are provided as shown in Fig. 5. Some of the components in it have been introduced in Sect. 3. Other new parameters related with statics and kinematics as: D 1: The standard pitch diameter of gear 1 and 1′; D 2: The standard pitch diameter of gear 2 and 2′; D: The diameter of the pinch rollers 5 and 5′; 7′: The mechanical position limitation in the mechanism; O and O′: The rotation joints of gear 1 and 1’; 0 O2 and O2 : The rotation joints of gear 2 and 2′; The length of link from joint O to joint O2. L12: 0 L4 and L4 : The length of link from joint O2 and O2 to its slider; 0

a: b: N: F: H 1:

0

Because of symmetry, the link L12 ¼ L12 ; L4 ¼ L4 . The above half of the symmetrical pinch and conveying mechanism is divided as the statics and kinematics study aim. Symmetrical pinch and conveying mechanism The angle of L12; Symmetrical pinch and conveying mechanism The angle of L4; The contact force between pinch roller and steel rod; The spring force on the slider; The distance between slider and steel rod passing line

A is the steel rod diameter, and it’s also the rollers’ gap distance between 5 and 5′. Its infinitesimal diameter change is defined as dA; S is the spring compression connected with the slider, and the spring compression infinitesimal change is defined as dS;

Fig. 5. A new design scheme of underactuated pinch and conveying mechanism

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When the pinch roll machine is working in an ideal pinching and conveying condition, the pinching system is in static equilibrium state. The torque equilibrium equation can be deduced from principle of virtual work and formulated as F
dS ¼ N
dA

ð1Þ

In the coordinates with the origin joint O, S and A can be expressed as S ¼ L12 cos a þ L4 cos b

ð2Þ

1 A ¼ L12 sin a þ D 2

ð3Þ

According to the geometrical relationship of the mechanism, L12 sin a þ L4 sin b þ H1 ¼ D1

ð4Þ

The relationship between angle a and b can be deduced and obtained as D1  H1  L12 sin a L4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD1  H1  L12 sin aÞ2 cos b ¼ 1  L24 sin b ¼

ð5Þ

ð6Þ

Let’s define Q ¼ ðD1  H1  L12 sin aÞ, and substitute Eq. (2) with Eq. (6), S can be formulated as 

Q2 S ¼ L12 cos a þ L4 1  2 L4

12

ð7Þ

Through differential operation of Eq. (7), an equation of dS and da can be formulated as  1 L12 Q Q2 2 dS ¼ L12 sin a
da þ 1  2 cos a
da L4 L4

ð8Þ

Through differential operation of Eq. (3), an equation of dA and da can be formulated as dA ¼ L12 cos a
da

ð9Þ

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Thus, da ¼

dA L12 cos a

ð10Þ

Substitute Eq. (8) with Eq. (10), a function with dS and dA can be obtained and formulated as dS ¼ FðaÞ
dA ¼

 1   1 L4  Q2 L3 4 Q  tan a
dA 2

ð11Þ

It is the relationship between the pinched steel rod diameter infinitesimal change dA and the spring compression infinitesimal change dS, in which   2 3 FðaÞ ¼ L1 4  Q L4 Q  tan a

ð12Þ

Substitute Eq. (1) with Eq. (11), the equation can be deduced as kS
F(a) þ N ¼ 0

ð13Þ

in which, k is the elastic coefficient of the spring. Thus, when the pinched steel rod diameter has an infinitesimal change dA, and correspondingly, the rollers’ gap distance between 5 and 5′ adjusted dA. In order to maintain a constant value of the contact force N between pinch roller and steel rod, the function F(a) in Eq. (12) should satisfy: F(a) ¼

N kS

ð14Þ

Because function F(a) includes all the design parameters in Fig. 5, Eq. (14) can be a design criteria for the new design scheme of underactuated pinch and conveying mechanism.

5 Conclusion and Prospection A novel underactuated symmetric pinching mechanism of pinch roll machine are proposed in this paper. This underactuated steel rod pinching and conveying mechanism is inspired by underactuated robotic hand. It has the feature of “shape-adaptive pinching and conveying”. The mechanism design scheme is considered with the aspects of structure design, parts manufacture and components assembly. The key problem of design criteria to maintain a constant contact force for variable diameter steel rod pinching and conveying are study. The design criteria deduced from the mechanism’s statics and kinematics equations. Further work will be carried out to obtain a suitable solution for the mechanism design parameters with optimum design method. The proposed mechanism and design criteria is also looking forward to be applied in practice in high-speed wire rod product line.

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References 1. Frauenhuber, K., Moser, F.: Product and maintenance efficiency by applying a new concept for pinch-roll change focus on the rolling and processing industries. Metallurgia Italiana (2014) 2. Primetals Technologies, Limited. Intelligent Pinch Rolls at Laying Head Functional Description. Siemens VAI Metals Technologies (2015) 3. Wang, R., Yao, S.: Clamping force design and calculation for intelligent pinch roll on highspeed bar rolling line. Steel Roll. 37(04), 56–61 (2020). (in Chinese) 4. Kishore, K., Mukhopadhyay, G.: Root cause failure analysis of pinch roll bearing at hot strip mill. J. Fail. Anal. Prev. 19(1), 219–229 (2019) 5. Adeoye, O.J., et al.: Development of a motorized sheet metal rolling machine. Adv. Res. 17 (1) (2018) 6. Gong, Y.: The improvement plan of control system of pinch roll in finishing area of high speed wire rod. Dissertation of Dalian University of Technology, China (2016) 7. Yao, S., Ceccarelli, M., Carbone, G., Ma, B.: Force analysis and curve design for laying pipe in loop laying head of wire rod mills. Chin. J. Mech. Eng. 32(1), 32 (2019) 8. Yao, S., Ceccarelli, M., Carbone, G., Dong, Z.: Grasp configuration planning for a low-cost and easy-operation underactuated three-fingered robot hand. Mech. Mach. Theory 129, 51–69 (2018) 9. Yao, S., Ceccarelli, M., Lu, Z.: Underactuated elements design criterion for envelop gripper mechanism. Mech. Mach. Sci. 66, 432–442 (2019)

Design of a Reconfigurable Novel ConstantForce Mechanism for Assistive Exoskeletons Yichen Liu1, Zhongyi Li2, and Shaoping Bai2(&) 1

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China 2 Department of Materials and Production, Aalborg University, 9220 Aalborg, Denmark [email protected]

Abstract. Constant-force mechanisms can be used for gravity counterbalance and force regulation. While many constant-force mechanisms have been designed, most of them have a limited range of force adjustment, which brings very limited capability to maintain an equilibrium state when load changes. In this paper, a novel design of reconfigurable constant-force mechanism is proposed. The mechanism is designed from the concept of hinged-lever constantforce mechanism. Through adjusting the configuration parameter, a large adjustment range of the equilibrium state is achieved. Mathematical models are developed to simulate the equilibrium performance changed with configuration parameters. A prototype is constructed and validates the working principle of the novel design, demonstrated with a concept of assistive arm exoskeleton. Keywords: Constant-force
Compliant joint configuration
Assistive exoskeleton


Equilibrium state
Multi-

1 Introduction Constant-force mechanisms (CFM) are able to provide nearly-constant output forces over a prescribed deflection range by appropriate structure design and can reduce the need for sophisticated control systems with passive elastic elements [1]. The CFMs (or quasi-zero stiffness mechanisms) are essential for gravity counterbalance [2] and force regulation [3], which have found their applications in overload protection mechanisms [4], biomedical manipulations [5], exoskeletons and robot end-effectors [6]. Some CFM designs can be found in literature. A constant-force compression mechanism was proposed in [7] by introducing leaf springs. A constant-force doubleslider mechanism was designed to adjust the output force magnitude by changing its spring length [8]. A compliant CFM for force regulation of robot end-effectors operating in an unknown environment was developed in [9] and a constant-force bistable mechanism can snap to the stable equilibrium state when an unknown contact force exceeding its critical load [4]. A method to generate constant output forces from a constant stiffness is introduced in [10]. A shape optimization of a constant-force spring was reported in [11].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 122–131, 2021. https://doi.org/10.1007/978-3-030-75271-2_13

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Of most CFMs designed [12], the hinged-lever constant-force mechanism (HLCFM) was commonly used owing to the concise mathematical model. Examples include four-bar mechanism designed with three-stage stand, which can counterbalance the gravitational force [13]. A HLCFM with seven rigid links and two spring links to balance a mass statically was presented in [14]. By applying the counterweight matching method, HLCFM has been widely used in spatial parallel mechanism and a statically balanced direct drive arm was further developed in [15]. Moreover, a static balancing parallel robot by applying a counterweight for gravity compensations was proposed in [16]. However, it is noticed that HLCFM has a limitation of adjustment range to maintain the equilibrium state [17]. In this paper, a new design is proposed to improve the equilibrium capability by incorporating a compliant joint with multi-configuration. Through changing the stiffness mode parameter (configuration) and the pretension of spring, the new mechanism can increase the range of equilibrium state significantly. Mathematical models of the mechanism are established, with which stiffness analysis is conducted to evaluate the influence of pretension and configuration. The mathematical models can be used to identify a specific pretension to achieve the equilibrium state. Experiments with a prototype of the mechanism made by 3D printing validate the models.

2 Design Modeling and Stiffness Simulation 2.1

Basic Concept of HLCFM

In this section, we first outline briefly the statics of HLCFM, upon which a new design is introduced. Figure 1(a) shows the schematic diagram of HLCFM. The mechanism consists of three main components: a fixed link locked in the preselected position, a rotatable lever with a hinge joint, and a spring. An external load Fe is applied at point C.

Fig. 1. (a) Schematic diagram of HLCFM and (b) the single module of novel constant-force mechanism

Let k be the spring stiffness and l0 represents the initial length of spring. At any equilibrium state, the external load Fe must satisfy:

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Fe ¼

k
l3
l1 l0
ð1  Þ l2 ls

ð1Þ

By assuming a zero initial length, Eq. (1) becomes: Fe ¼

k
l3
l1 l2

ð2Þ

which means that the mechanism achieves a constant-force behavior. On the other hand, torque Te under equilibrium condition is a function of deflection angle a, Te ¼ k
l3
l1
sin a

ð3Þ

Equation (2) implies that Fe is related only to its structural parameters and independent to the position parameters. However, due to the constraints of the mechanism construction and size, adjustment of link length and spring stiffness is rather limited, which makes conventional HLCFM impractical to maintain equilibrium capability in a large adjustment range. Moreover, a zero-initial-length spring physical does not exist, which implies that only an approximate equilibrium can be expected. 2.2

Design of a Reconfigurable Constant-Force Mechanism

In this work, a novel multi-configuration constant force mechanism was designed by transforming HLCFM into a reconfigurable pulley-cable mechanism. As shown in Fig. 1(b), a single module of the mechanism is designed with two fixed pulleys as well as one movable pulley mounted on the rotatable lever. All pulleys are coupled through an elastic wire of stiffness K. Let the torque and the corresponding tension along the wire denoted by T and Ft respectively. Referring to Fig. 1(a), Ft is expressed as: Ft ¼ K
dls þ Fp

ð4Þ

where dls and Fp are the elongation and the pretension of the spring. By introducing J = ∂ls/∂a, we have ð5Þ

T ¼ J
Ft The mechanism stiffness Ke= dT/da is found as Ke ¼ J




 dFt dJ d^J ^ d^J
Ft ¼ Kl23 ^J 2 þ
dls þ Fp l3 þ da da da da |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} Kes

ð6Þ

Kep

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ^ls ¼ k2  2k cos a þ 1 and ^J ¼ k sin a=^ls with k = l1/l3, k 2 (0,1). As can be seen from Eq. (6), the spring stiffness coefficient K influences Kes while the spring pretension Fp dominates Kep.

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Fig. 2. Simplified structure of the multi-configuration constant force mechanism

A simplified model of multi-configuration constant force mechanism is shown in Fig. 2, where the fixed pulleys are mounted on the fixed pedestal while the movable pulleys are fixed on the rotatable output turntable. Each two fixed pulleys and one movable pulley coupled through the wire compose a basic module. The linear spring is connected along the wire. The number of movable pulleys on the rotatable output turntable, noted by N, represents also the number of configurations. In deriving the stiffness Kec of multi-configuration constant force mechanism based on the basic module, we introduce a torque Tb which is generated by a single module. The stiffness can be obtained as: Kec ¼

dTb dJ1 dJ1 ¼ J1
K
J1 þ
K
dl þ
Fp da da daffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflffl K1

ð7Þ

K2

where l represents the length of equivalent spring and the Jacobian J1 = ∂l/∂a. Given that multi-configuration design is utilized in the constant-force mechanism, the corresponding stiffness will be modified due to the geometry changes. Based on the stiffness model reported in [18], Eq. (7) is finally obtained as

 d^J ^ d^ J 2 ^
dls þ 2NFp l3 J þ Kec ¼ 4N da da |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 2

Kl23

K1

ð8Þ

K2

The torque generated from the mechanism for any deflection is determined as: Tc ¼ N 2 KdlJ1 þ NFp J1

ð9Þ

Comparing Eqs. (6) and (8), we obtain the nominal stiffness k = 4 N2K. Substituted the stiffness into Eq. (2), the equilibrium condition becomes: Fe ¼

4
N 2
K
l3
l1 l2

ð10Þ

Moreover, under the equilibrium state, the relationship between torque Te and deflection angle a is:

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Te ¼ 4
N 2
K
l3
l1
sin a

ð11Þ

Comparing Eq. (2) and Eq. (10), we can find that a major difference is the configuration coefficient N2. By adjusting the parameter N, the equilibrium state can be achieved in a broad adoption range. Figure 3 shows three different configurations N = 1, 2, 3 respectively.

(a) N=1

(b) N=2

(c) N=3

Fig. 3. Three configurations of the constant force mechanism

2.3

Simulations

The mathematical model in Eq. (9) describes the torque-deflection relationship changed with configurations and pretensions. Simulations were conducted to reveal the characteristics of the new mechanism. In the simulation, parameters are set as K = 0.4 N/mm, L1 = 15 mm, L2 = 120 mm, L3 = 30 mm. Substituted the parameters into the equilibrium condition Eq. (10), the novel mechanism can provide constant forces Fe = 6, 24, 54 N under each configuration N = 1, 2, 3 respectively. Figure 4 displays the torque-deflection behaviors with the variation of pretensions. In the figure, the torque curves of Te are denoted by yellow dash-dot lines and the theoretical pretensions Fp = 12, 25, 36 N can be identified by matching with the contours under configurations N = 1, 3 respectively. From the simulations, it can be seen that novel constant-force mechanism is able to generate the equilibrium force Fe under a pretension Fp. By increasing the number of modules engaged, the range of equilibrium force is increased proportionally.

Fig. 4. Torque-deflection configurations

relationship

and

equilibrium

condition

curves

under

two

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3 Prototype and Testing In this work, a prototype of a multi-configuration constant force mechanism was constructed, with which performance testing was conducted for the new mechanism. By varying pretensions, we obtained a series of torque-deflection relationships under three configurations. 3.1

Prototype Design and Test Rig

Fig. 5. Test rig with (1) fixed pulley, (2) fixed pedestal, (3) linear spring, (4) load cell, (5) fixed endpoint, (6) pretension-winch, (7) wire, (8) movable pulley, (9) rotatable output turntable, (10) support-base, (11) rotatable pedestal, (12) torque sensor, (13) encoder, (14) payload.

The prototype, developed at Aalborg University, Denmark, is shown in Fig. 5. The linear spring is connected along the wire, which is regarded as infinite stiffness wrapped through the pulleys. One end of the wire is fixed on the pretension-winch, then the controllable pretension can be obtained by rotating pretension-winch. A load cell (model: Forsentek FS01-10 kg; accuracy: 0.1 N) is used to measure the pretension. A torque sensor (model: Forsentek FTE-20NM) with an accuracy of 0.04 N
m is fixed on the rotatable pedestal to measure the torque generated by the applied force. Meanwhile, an absolute encoder (model: RLS) with an accuracy of 0.5° is used to acquire the angle deflection a. During the whole execution of experiments, the sensor data is collected by an Arduino DUE board and transmitted to the computer for processing. 3.2

Experiments

The torque-deflection relationships of the novel constant-force mechanism under configurations N = 1, 2, 3 were obtained in the experiments. A standard linear spring with the stiffness of 0.4 N/mm was applied during the tests and the other main system parameters were set as l1 = 15 mm, l2 = 120 mm, l3 = 30 mm. The tests were conducted for a range of 1.2 rad. The tests under configuration are depicted in Fig. 6. A series of applied pretensions (Fp = 10, 25 N) were conducted in tests. In the figure, results with a series of applied pretensions are shown by blue dots respectively, showing together with the simulated torque-deflection curves indicated by the green lines and the equilibrium condition

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indicated by the red lines. As we can see, when the applied pretension equals the theoretical value of 12 N, the mechanism can achieve a quasi-equilibrium state. Results with the configuration of N = 2, 3 are shown in Figs. 7 and 8 respectively. For N = 2, the pretension to achieve equilibrium condition equals 25 N which is same as the theoretical value. For configuration N = 3, the pretension that achieves equilibrium condition is equal to 35 N, which is close to theoretical result of 36 N. It is seen from Figs. 6, 7 and 8 that the test results generally agree with the simulated torque-defection curve and change with the applied pretensions. When the applied pretension approaches the simulated equilibrium pretension, the fitting curve is coincident with the equilibrium condition curve approximately, which is confirmed with theoretical values obtained by the simulation. Under this equilibrium condition, the mechanism can provide the equilibrium torque to balance an applied external force Fe over any deflection. The equilibrium pretension obtained by the test fits with the one obtained by the mathematical simulation in general.

(a) Fp=10 N

(b) Fp=25 N

Fig. 6. Test results for varying pretension with N = 1

(a) Fp=10 N

(b) Fp=25 N

Fig. 7. Test results for varying pretension with N = 2

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(a) Fp=10 N

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(b) Fp=35 N

Fig. 8. Test results for varying pretension with N = 3

4 A Compact Design of a Passive Upper-Body Exoskeleton The proposed mechanism enables innovative compact designs of passive exoskeletons to overcome the gravitational and inertial effects during moving. Herein, a case illustrating an application of the mechanism in a passive upper-body exoskeleton is included, as shown in Fig. 9. In the figure, the exoskeleton mounted the arm of a human model through two cuffs, and is able to overcome the gravitational and inertial loads from forearm. This is helpful to reduce muscle activities of the human arm. The exoskeleton can be used to assist the elderly whom may suffer from degradation of the musculature, and also to manual works in load carrying and lifting.

Fig. 9. CAD model of a passive upper-body exoskeleton

5 Conclusion This paper proposes a new constant-force mechanism designed with a multiconfiguration constant force mechanism. Mathematical models were developed, which reveal the capability of generating constant forces in a large range. Varying torque-deflection relationships can be obtained by adjusting initial pretension and configurations. In addition, the constant-force behavior was achieved corresponding to a theoretical equilibrium pretension. A prototype of multi-configuration constant force mechanism was constructed to validate the models and equilibrium condition.

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The novel design of reconfigurable constant force mechanism allows maintaining equilibrium state in a broad payload range. By introducing the reconfigurable design, a high performance constant-force mechanism is obtained to contour balance a variety of payloads. For future work, the effect of friction on stiffness behavior and the application in exoskeletons will be studied. Moreover, automatic implementation of reconfiguration will be considered. Acknowledgments. The project is in part supported by Innovation Fund Denmark through Grand Solutions project EXO-AIDER. The first author acknowledges the financial support by Chongqing University for his visiting study at AAU.

References 1. Bossert, D., Ly, U.-L., Vagners, J.: Experimental evaluation of a hybrid position and force surface following algorithm for unknown surfaces. In: Proceeding of IEEE International Conference on Robotics and Automation, pp. 2252–2257 (1996) 2. Tolman, K.A., Merriam, E.G., Howell, L.L.: Compliant constant-force linear-motion mechanism. Mech. Mach. Theory 106, 68–79 (2016) 3. Tian, Y.L., Zhou, C.K., Wang, F.J., et al.: Design of a flexure-based mechanism possessing low stiffness and constant force. Rev. Sci. Instrum. 90(10), (2019) 4. Pham, H.T., Wang, D.A.: A constant-force bistable mechanism for force regulation and overload protection. Mech. Mach. Theory 46(7), 899–909 (2011) 5. Wang, P.Y., Xu, Q.S.: Design of a flexure-based constant-force XY precision positioning stage. Mech. Mach. Theory 108, 1–13 (2017) 6. Berselli, G., Vertechy, R., Vassura, G., et al.: Design of a single-acting constant-force actuator based on dielectric elastomers. J. Mech. Rob. 1(3), (2009) 7. Boyle, C., Howell, L.L., et al.: Dynamic modeling of compliant constant-force compression mechanisms. Mech. Mach. Theory 38(12), 1469–1487 (2003) 8. Nahar, D.R., Sugar, T.: Compliant constant-force mechanism with a variable output for micro/macro applications. In: 2003 IEEE International Conference on Robotics and Automation (Cat No. 03CH37422), Taipei, Taiwan, pp. 318–323, 14–19 September 2003 9. Lan, C.C., Wang, J.H., Chen, Y.H.: A compliant constant-force mechanism for adaptive robot end-effector operations. In: 2010 IEEE International Conference on Robotics and Automation, pp. 2131–2136 (2010) 10. Pedersen, C.B., Fleck, N.A., Ananthasuresh, G.K.: Design of a compliant mechanism to modify an actuator characteristic to deliver a constant output force. J. Mech. Des. 128(5), 1101–1112 (2006) 11. Meaders, J.C., Mattson, C.A.: Optimization of near-constant force springs subject to mating uncertainty. Struct. Multidiscipl. Optim. 41(1), 1–15 (2010) 12. Wang, P.Y., Xu, Q.S.: Design and modeling of constant-force mechanisms: a survey. Mech. Mach. Theory 119, 1–21 (2018) 13. Nathan, R.H.: A constant force generation mechanism. J. Mech. Transm. Autom. Des. 107 (4), 508–512 (1985) 14. Jenuwine, J.G., Midha, A.: Design of an exact constant force generating mechanism. In: Proceedings of the 1st National Applied Mechanisms & Robotics Conference, p. 10B-4 (1989) 15. Kazerooni, H., Kim, S.: A new architecture for direct drive robots. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 442–445 (1988)

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16. Russo, A., Sinatra, R., Xi, F.F.: Static balancing of parallel robots. Mech. Mach. Theory 40 (2), 191–202 (2005) 17. Lambert, P., Herder, J.L.: An adjustable constant force mechanism using pin joints and springs. In: New Trends in Mechanism and Machine Science, pp. 453–461. Springer, Cham (2017) 18. Li, Z., Bai, S.: A novel revolute joint of variable stiffness with reconfigurability. Mech. Mach. Theory 133, 720–736 (2019)

Aerodynamic Double Pendulum with Nonlinear Elastic Spring Yury Selyutskiy1(&) , Marat Dosaev1 , Andrei Holub1 and Marco Ceccarelli2 1

,

Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia {seliutski,dosayev}@imec.msu.ru 2 Department of Industrial Engineering, University of Rome Tor Vergata, Rome, Italy [email protected]

Abstract. Dynamics of an aerodynamic double pendulum is considered under the assumption that the spring in the first joint is nonlinear elastic. This object is an aeroelastic system with two degrees of freedom, which can be considered as a source of energy. Numerical simulation of the behavior of the pendulum is performed. It is shown that it is possible to choose parameters in such a way as to ensure the instability of the equilibrium and the existence of limit cycle oscillations. The influence of characteristics of the joint spring upon the amplitude and frequency of the cycles is analyzed. In particular, it is shown that the amplitude non-monotonically depends on the parameter characterizing the cubic nonlinearity of the stiffness. In the same time, frequency monotonically increases as this parameter increases. The presence of limit cycle oscillations means that the system in question is capable of harvesting energy from the flow. However, thorough parametric analysis is still required in order to determine the optimal design and parameters of such systems. Keywords: Wind power
Aeroelasticity
Oscillations
Limit cycle
Stability

1 Introduction The desire to develop renewable energy sources invariably leads to the use of various mechanisms for these purposes. The areas of wind and hydropower are actively developing. The most common wind and hydroelectric power plants convert the movement of air or water into the rotational movement of the working body (turbines with a vertical or horizontal axis). Different types of wind-receiving elements are used for different tasks. Mechanical systems that experience the combined action of elastic forces and aerodynamic (or hydrodynamic) loads are called aeroelastic systems. An intensive study of these systems began in various countries in the 1930s (e.g., [1, 2]), which was stimulated by intensive progress in the aviation industry. Later it was suggested that oscillations appearing in aeroelastic systems can be used for flow energy harvesting. Reviews of various approaches and constructions intended © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 132–140, 2021. https://doi.org/10.1007/978-3-030-75271-2_14

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to generate electricity flow-induced oscillations are provided in [3, 4]. In [5], aeroelastic systems coupled with piezoelectric elements are discussed, and estimations of the output power are given. In a whole range of works [6–10], flutter-type oscillations are considered, that is, flow-induced oscillations of wing-like bodies elastically mounted in such a way that they has one rotational and one translational degree of freedom. In particular, [8] have shown experimentally that flutter-based generators have high potential and provided preliminary recommendations for improving the performance of such devices. Apart from flutter, other types of flow-induced oscillations are also considered as a source of power. For instance, energy harvesting using galloping (i.e., oscillations of a bluff body in the direction transversal to the flow) is studied in [11], and body shapes are proposed that allow optimizing the output power. The possibility to extract energy from translational oscillations of bluff bodies induced by vortices shed from the body surface is discussed in [12, 13]. Parameters of the system are identified, which are the most important in terms of influence upon the efficiency of power production. In [14], an aeroelastic system with two translational degrees of freedom is investigated where the electric power is generated using piezo-elements. Dynamic response and average power output of the device are numerically studied. An aeroelastic system with two rotational degrees of freedom, or aerodynamic double pendulum, is discussed in [15]. It is supposed that springs in joints of the pendulum are linearly elastic. It is shown that it is possible to select the values of the parameters so that the zero equilibrium (when both pendulum links are stretched along the flow) is unstable. A numerical and experimental study of limit cycle oscillations arising in this system is performed, and the potential of using this device for energy production is outlined. It should be noted that the aerodynamic double pendulum can be considered as a promising additional source of energy for autonomous robotic systems, in particular, for robots moving on the surface of planets with atmosphere. When the robot is at rest, the pendulum can be connected with an electric generator in order to recharge the battery. When the robot is moving, the pendulum can be switched to idle mode (disconnected from the generator), thereby so that practically no additional aerodynamic drag is produced. In this work, a spiral nonlinear (with a linear and cubic term) spring with linear damping is fixed in the first joint of an aerodynamic double pendulum. A dynamic system is obtained that describes the behavior of the corresponding mechanical system. Numerical simulation is performed in order to analyze the effect of parameters describing stiffness and damping upon the limit cycles arising in this dynamic system. The non-monotonic dependence of the amplitude of the limit cycle on the stiffness coefficients has been demonstrated. Ranges of values of parameters are proposed that are more efficient from the perspective of energy harvesting.

2 Aerodynamic Double Pendulum A double aerodynamic pendulum (see Fig. 1) consists of two links, O1 O2 0 0 00 0 000 O2 O1 and O2 O3 O3 O2 , that represent rectangular frames made of thin rigid rods.

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The first link is elastically mounted using spiral spring. On the second link there is fixed a thin wing with symmetrical airfoil. We assume that axes of both joints of the pendulum are vertical.

Fig. 1. A scheme of the double aerodynamic pendulum with an elastic joint frame: a) top view; b) side view.

An aerodynamic double pendulum (see Fig. 1) consists of two links, O1 O2 0 0 00 0 000 O2 O1 and O2 O3 O3 O2 , that represent rectangular frames made of thin rigid rods. The first link is elastically mounted using spiral spring with a non-linear stiffness characteristic. The pendulum is placed in the steady horizontal airflow. The joint spring is not deformed when the first link is directed along the wind. We assume that the moment exerted by the spring in joint O1 is given by the following expression: Mel ¼ ju  j3 u3 :

ð1Þ

Here j and j3 are coefficients characterizing the spring stiffness. The energy of oscillations of the pendulum can be converted into electric power, e.g., by means of a piezoelement connected to the first link, or another electric generator. The load applied to the pendulum by the electricity generating device can be simulated by damping (like in [8]) in the first joint. This means, that the moment generated by the load is as follows: _ Mel ¼ du:

ð2Þ

We assume that the flow interacts only with the wing. In order to describe the corresponding aerodynamic load, we use the quasi-steady model [e.g., 16–18]. In the context of this approach, the aerodynamic forces can be reduced to the drag force D and lift force L applied in the mid-chord C of the wing, as well as to the moment Mz about this point. These values are given by the following expressions:

Aerodynamic Double Pendulum with Nonlinear Elastic Spring

1 D ¼ qrVC Cd ðaÞVC ; 2

1 L ¼ qrVC Cl ðaÞez  VC ; 2

1 Mz ¼ qrbVC2 Cm ðaÞ; 2

135

ð3Þ

where q is the air density; r is the wing area; b is the wing chord; ez is the unit vector normal of the ascending vertical; VC is the airspeed of the point C (that is, the velocity of point C with respect to the incoming flow); a is the instantaneous angle of attack, i.e., the angle between VC and the wing chord (see Fig. 1); Cd, Cl, Cm are nondimensional drag, lift, and moment coefficients, correspondingly.

3 Equations of Motion We introduce a fixed coordinate system O1 XY with the abscissa axis directed along the wind velocity. As generalized coordinates, we use the following values: u, the angle 00 between O1 O2 and O1 X, and h, the angle between O2 O3 and O1 X. Referring to Fig. 1, VC and a satisfy the following kinematical relations: VC cos a ¼ V cos h þ l1 u_ sinðu  hÞ; _ VC sin a ¼ V sin h þ l1 u_ cosðu  hÞ þ r h;

ð4Þ

where l1 is the length O1 O2 of the first link; r is the distance between point C and the inter-link joint; and V is the flow speed. Consequently, taking into account relations (1)–(3), the equations of motion of the pendulum can be expressed in the following form:
€ þ ml1 l2 cosðh  uÞ€h  ml1 l2 sinðh  uÞh_ 2 J1 þ ml21 u   qrVC l1 Cd ðaÞ V sin u þ l1 u_ þ r cosðh  uÞh_ ¼  2   qrVC l1 _ Cl ðaÞ V cos u þ r sinðh  uÞh_  ju  j3 u3  du;  2
J2 þ ml22 €h þ ml1 l2 cosðh  uÞ€ u þ ml1 l2 sinðh  uÞu_ 2   qrVC r Cd ðaÞ V sin h þ r h_ þ l1 cosðh  uÞu_ ¼  2 qrVC r qrVC2 b þ Cl ðaÞðV cos h þ l1 sinðh  uÞu_ Þ þ Cm ðaÞ; 2 2

ð5Þ

where J1 is the moment of inertia of the first link, J2 is the moment of inertia of the second link about its center of mass G, m is the mass of the second link, and l2 is the distance from the axis of the inter-link joint to point G. In order to reduce the number of parameters, we make these equations dimensionless using the following values as characteristic mass, time, and length, correspondingly: M = qrb/2, T = b/V, L = b. Denoting the derivative with respect to nondimensional time with dot and retaining the previous designations for dimensionless variables and parameters, we get the following non-dimensional equations of motion

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€ þ ml1 l2 cosðh  uÞ€h  ml1 l2 sinðh  uÞh_ 2 J1 þ ml21 u   ¼ VC l1 Cd ðaÞ sin u þ l1 u_ þ r cosðh  uÞh_   _  VC l1 Cl ðaÞ cos u þ r sinðh  uÞh_  ju  j3 u3  du;
J2 þ ml22 €h þ ml1 l2 cosðh  uÞ€ u þ ml1 l2 sinðh  uÞu_ 2   ¼ rVC Cd ðaÞ sin h þ r h_ þ l1 cosðh  uÞu_

ð6Þ

þ VC rCl ðaÞð cos h þ l1 sinðh  uÞu_ Þ þ VC2 Cm ðaÞ: For symmetrical airfoils Cl ð0Þ ¼ Cm ð0Þ ¼ 0, hence, it follows from Eqs. (5) that u ¼ h ¼ 0 is equilibrium position of the pendulum. In order for the considered pendulum to be able to harvest flow energy, it should perform a periodic motion. This means, that stable equilibriums are undesired. In [15] it was shown that, in case of linear spring without damping, this equilibrium becomes unstable if certain conditions upon other parameters are satisfied. In such situation, there exist attracting limit cycles. As was mentioned before, damping in the first joint simulates not so much losses due to friction as useful load applied to the pendulum. Therefore, it is important to study the influence of damping coefficient d upon characteristics of oscillations (amplitudes and frequencies). Besides, it is interesting to analyze the effect of nonlinear stiffness upon these characteristics.

4 Limit Cycles As we consider the motion of the pendulum in the air, and the air density is much smaller than the density of pendulum elements, it seems reasonable to assume that the pendulum is heavy. This means that dimensionless masses and moments of inertia of its links are much greater than 1. For numerical simulation, we may choose the values of non-dimensional parameters provided in Table 1 below. For the aerodynamic load, we use static characteristics of the standard NACA 0012 airfoil from [19]. Table 1. Values of parameters used for numerical simulation. l1 l2 r m J1 J2 2 2 2 50 50 10

Methods based on Lyapunov functions (e.g., [20]) encounter certain difficulties in such problems, while the use of well-known perturbation methods seems to be efficient. We obtain sufficient stability conditions of the discussed equilibrium from the Hurwitz criterion. These conditions determine the boundaries of the domain of instability of the

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equilibrium in the space of parameters. In Fig. 2, this domain is shown (with grey color) in the plane of parameters ðd; jÞ.

Fig. 2. Domain of instability of the equilibrium (grey).

It should be noted that, for relatively small damping (d < d1  14), the equilibrium is unstable when the spring stiffness j is low enough and becomes asymptotically stable as j increases. For large damping (d > d2  20), the equilibrium is asymptotically stable for all j. For d1 < d < d2, it is asymptotically stable for small j (like in point A1 in Fig. 2) but if the stiffness becomes larger than a certain value, the stability is lost (an example is point A2). Thus, increase in stiffness can result in destabilization. The equilibrium is unstable in a certain range of values of stiffness, and becomes stable again, when the stiffness becomes larger than the upper boundary of this interval (like in point A3 in Fig. 2). This effect is similar to the one mentioned in [15] for a double aerodynamic pendulum with linear spring without damping. When parameters belong to the instability area, limit cycles appear in the system. These oscillations can be used for power harvesting. Let us discuss now the influence of parameters of the spring upon characteristics of limit cycles. For a rather wide range of values of parameters, oscillations are practically uni-periodic, so that time dependences of angles are well enough described with the following formulae: u ¼ Au sinðxt þ Uu Þ;

h ¼ Ah sinðxt þ Uh Þ:

ð7Þ

This means that both links of the pendulum perform harmonic oscillations with the same frequency x, with a phase shift, and different amplitudes Au and Ah. From the point of view of energy harvesting, it is desirable to have large amplitude Au of the first link and, in the same time, high frequency x of oscillations. Let us discuss the influence of parameters of the system upon these values.

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It seems natural to suppose that the amplitudes of cycles will decrease with the increase in j3. However, numerical simulation shows that the situation is much more intricate. In order to obtain amplitudes and phases of limit cycles for different values of parameters, numerical integration of Eqs. (5) was performed using the standard RungeKutta method of the 4th order. In Fig. 3, the amplitudes and frequencies of oscillation of pendulum links are shown against j3 for different values of the spring stiffness and damping. Boxes correspond to j = 1, diamonds to j = 5, circles to j = 10, and crosses to j = 15. Let us analyze these dependences bearing in mind that, in order to obtain more power, it is necessary to maximize the amplitudes and the frequency of the cycle.

Fig. 3. Amplitudes and frequencies of limit cycles vs. j3 for different j and d.

One can readily see that, for small j, amplitudes of both links depend on j3 in a non-monotonous way: they increase for low j3 and decrease for larger ones. However, for large enough j, these dependences become monotonically decreasing. In the area of small j3, dependences of Au, Ah on j are also non-monotonic. In the same time, for larger j3, these dependences are monotonically decreasing. The cycle

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frequency x monotonically increases with j, which is quite expectable. Moreover, in all considered cases (except for d = 1, j = 10), it increases monotonically with j3, too. The effect of damping is much more straightforward: its increase results in decrease in cycle amplitudes and frequency. It is interesting to note, however, that, for larger d, the dependence of Au, Ah, and x upon j3 becomes weaker. Thus, for d = 10, these values remain almost constant in the considered range of j3. When d becomes large enough, the equilibrium becomes asymptotically stable, and cycles disappear. It should be emphasized that limit cycles are preserved even for large enough values of damping. This means that it is possible to harvest power using this pendulum even when the load applied to it is relatively high.

5 Conclusions Due to the presence of limit cycles, the aerodynamic double pendulum has a potential to be used as a flow energy harvester. In order to increase the output power, it is necessary to choose parameters of the system in such a way as to increase the product of the amplitude and frequency of limit cycle oscillations. Hence, based on the above analysis, it is reasonable to use springs with moderate stiffness characteristics: j about 5 and j3 from 20 to 40 (dimensionless values). However, selection of optimal (from the perspective of the output power) parameters requires further analysis. This research is partially supported by the Interdisciplinary Scientific and Educational School of Moscow University “Fundamental and Applied Space Research”.

References 1. Theodorsen, T.: General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496 (1935) 2. Grossman, E.P., Flutter, J.M.: Central Aero-Hydrodynamic Institute. Report 186 (1935). (in Russian) 3. Abdelkefi, A.: Aeroelastic energy harvesting: a review. Int. J. Eng. Sci. 100, 112–135 (2016) 4. McCarthy, J., Watkins, S., Deivasigamani, A., John, S.: Fluttering energy harvesters in the wind: a review. J. Sound Vibr. 361, 355–377 (2016) 5. Erturk, A., Vieira, W.G.R., De Marqui, C., Inman, D.: On the energy harvesting potential of piezoaeroelastic systems. Appl. Phys. Lett. 96, 184103 (2010) 6. Bryant, M., Garcia, E.: Modeling and testing of a novel aeroelastic flutter energy harvester. ASME J. Vib. Acoust. 133(1), 011010 (2011) 7. Dimitriadis, G., Li, J.: Bifurcation behavior of airfoil undergoing stall flutter oscillations in low-speed wind tunnel. AIAA J. 47(11), 2577–2596 (2009) 8. Pigolotti, L., Mannini, C., Bartoli, G., Thiele, K.: Critical and post-critical behaviour of twodegree-of-freedom flutter-based generators. J. Sound Vib. 404, 116–140 (2017) 9. Bao, C., Dai, Y., Wang, P., Tang, G.: A piezoelectric energy harvesting scheme based on stall flutter of airfoil section. Eur. J. Mech. B. Fluids 75, 119–132 (2019) 10. Eugeni, M., Elahi, H., Fune, F., Lampani, L., Mastroddi, F., Romano, G.P., Gaudenzi, P.: Numerical and experimental investigation of piezoelectric energy harvester based on flagflutter. Aerosp. Sci. Technol. 97, 105634 (2020)

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11. Yang, Y., Zhao, L., Tang, L.: Comparative study of tip cross-sections for efficient galloping energy harvesting. Appl. Phys. Lett. 102, 064105 (2013) 12. Mannini, C., Marra, A.M., Massai, T., Bartoli, G.: Interference of vortex-induced vibration and transverse galloping for a rectangular cylinder. J. Fluids Struct. 66, 403–423 (2016) 13. Barrero-Gil, A., Pindado, S., Avila, S.: Extracting energy from vortex-induced vibrations: a parametric study. Appl. Math. Model. 36(7), 3153–3160 (2012) 14. Wu, Y., Li, D., Xiang, J., Da Ronch, A.: A modified airfoil-based piezoaeroelastic energy harvester with double plunge degrees of freedom. Theor. Appl. Mech. Lett. 6(5), 244–247 (2016) 15. Selyutskiy, Y., Holub, A., Dosaev, M.: Elastically mounted double aerodynamic pendulum. Int. J. Struct. Stab. Dyn. 19(5), 1941007 (2019) 16. Klimina, L., Lokshin, B., Samsonov, V.: Parametrical analysis of the behavior of an aerodynamic pendulum with vertical axis of rotation. In: Modelling, Simulation and Control of Nonlinear Engineering Dynamical Systems, State-of-the-Art, Perspectives and Applications, pp. 211–220. Springer, Heidelberg (2009) 17. Samsonov, V.A., Dosaev, M.Z., Selyutskiy, Y.D.: Methods of qualitative analysis in the problem of rigid body motion in medium. Int. J. Bif. Chaos. 21(10), 2955–2961 (2011) 18. Okunev, Y.M., Privalova, O.G., Samsonov, V.A.: On the motion of a finned body in a resisting medium. Autom. Remote Control 74, 1326–1333 (2013) 19. Sheldahl, R.E., Klimas, P.C.: Aerodynamic characteristics of seven symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamic analysis of vertical axis wind turbines. Technical report SAND-80–2114, Sandia National Labs., Albuquerque, NM (USA) (1981) 20. Aleksandrov, A.Y., Tikhonov, A.A.: Asymptotic stability of a satellite with electrodynamic attitude control in the orbital frame. Acta Astronaut. 139, 122–129 (2017)

Design of a Flexible Interphalangeal Joint Job Eli Escobar-Flores1, Christopher R. Torres-San Miguel1(&), Luis Antonio Aguilar-Pérez1, and Marco Ceccarelli2 1

2

Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, SEPI, Zacatenco, 07738 Mexico City, Mexico [email protected] Department of Industrial Engineering, Laboratory of Robot Mechatronics, University of Rome Tor Vergata, 00133 Rome, Italy

Abstract. In human prostheses, rigid parts commonly use hinges, shafts, ball joints, and bearings to connect. Nevertheless, human anatomy body parts are flexible, and the movement comes from bending flexible parts. In particular, the hand is the member in charge of interacting with the environment. For this reason, when an individual suffers an amputation, his abilities are greatly diminished. Therefore, a prosthesis that restores much of his basic physical abilities is required. This problem can be solved using flexible joints. The human hand has 27 degrees of freedom (DF) or more, of which 9 DF correspond to the interphalangeal joints which can be replaced with the design made in this research since it can be adapted to the different dimensions of each finger. Keywords: Biomechanics
Finger prosthesis
Flexible joints
Spring design

1 Introduction The human hand’s skill is of great value to human beings, to explore and interact with the world [1]. For this reason, when an individual suffers an upper limb amputation, he or she is greatly diminished in their physical capabilities. One of the main challenges of robotics is the replication of the human hand [2], for applications in Biomechanics [3], healthcare [4], and industry [5]. These challenges cannot be met using traditional (rigid) parts such as hinges, shafts, ball joints, or bearings [6]. Looking at nature, most things are very flexible, and the movement comes from bending the flexible parts [7]. Incorporating different functions in fewer parts offers advantages and requires simultaneous design for movement and force behaviour [8, 9]. This difficulty is further increased by the fact that simplified linear equations are not suitable to define their movement [10, 11]. The advantages include that a flexible element has a spring and a hinge [12]. The human hand has 27 or more DOF [13]. Only 16 DOF are needed to perform everyday tasks, of which 9 DOF correspond to the interphalangeal joints. For this reason, a parameterised flexible joint was designed to replace the interphalangeal joints. Below is a history of robotic hands that have used flexible mechanisms. Table 1 show robot hands summarised.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 141–148, 2021. https://doi.org/10.1007/978-3-030-75271-2_15

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Author Raphael [14] Ishikawa komuro [15] Dyneema [16] I-limb [17] Microrrobótica-sssa [18] Underactuated robotic and prosthetic hands [19] Luke [20]

Year 2009 2009 2009 2010 2014 2017

DOF 14 6 15 10 10 10

2019

2

Actuator Pneumatic extrinsic Electrical extrinsic Electrical extrinsic Electrical extrinsic Electrical extrinsic Manual tendons

Materials Acrylic and polyester Acrylic Composite materials Composite materials Composite materials ABS plastic

Extrinsic electric

Rubber

The main problem with hands with flexible joints is that there are few models, so there is no reference in robotic hands’ design using flexible joints.

2 Design Requirements The interphalangeal joints have a function similar to a hinge, which is established between the phalanges. There are three phalanges and two interphalangeal joints in the last four fingers of the human hand. For this prototype, the proximal and distal interphalangeal joint, which will be replaced by a flexible joint designed in this work. The bars that simulate the phalanges will be 12 mm  15 mm and will have the values cited in Table 2 in length, the ends of the phalanges that join with a joint will have a circular cut of 7.5 mm radius, the material will be 7075-T6 aluminium due to its excellent mechanical properties and easy machining. The joints will be round bars with a diameter of 15 mm  15 mm long. The material will be SAE 841 self-lubricating bronze because the bars that simulate the phalanges will slide on the surface. Next, in Fig. 1, the prototype is observed in the initial or rest position (A), in the intermediate position (B) and in the final or maximum flexion position (C). Table 2. Prototype length [21] Soft tissue posterior to Distal phalanx the distal phalanx (mm) (mm) 3.84 15.82

Middle phalanx (mm) 22.38

Proximal phalanx (mm) 39.78

Metacarpal (mm) 68.12

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Fig. 1. A - Rest position, B - intermediate position, C - maximum deflection

The analysis was carried out using the Solidworks® Motion Add-in to simulate and analyse the assembly’s motion accurately. The analysis provides displacement, velocity, acceleration, force, and moment values for the selected entities to be measured as a function of time. The proposed spring works under tension and is seen in Fig. 1B and C, it has a coefficient of restitution of 0.625 N/mm [22], and its dimensions are 3 mm diameter wire, 100 mm minimum length, 0.5 mm diameter, made of AISI 302 stainless steel. The tendon lever arm is 90.7 mm, and the angle of incidence is 2.52° from vertical. The movement was simulated by a rotary motor applied to each moving part, represented with red arrows in Fig. 1A. The amount of movement is shown in Table 3. Table 3. Degree of bending per piece Part Metacarpophalangeal joint Proximal phalanx Proximal interphalangeal joint Middle phalanx Distal interphalangeal joint Distal phalanx

Flexion 43° 43° 51° 51° 36° 36°

3 Experimental Test Model The prototype was mounted on a base to facilitate its handling, allowing flexionextension movement and hyperextension. The prototype’s performance was by means of a linear actuator, whose stroke is 25 mm and has a force of 30 N. As shown in Fig. 2C, the movement of the bars that simulate the phalanges is homogeneous because

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internally, the spring can be traversed so that the same resistance value is obtained in each of the joints. This is very important at the moment of apprehension of objects.

Fig. 2. A - Rest position, B - intermediate position, C - maximum deflection

4 Results The force transmitted by the tendon is performed by the bending movement correctly and is calculated as follows: • Obtain the twisting force as a function of time (Fig. 4A), using movement analysis using the Solidworks® Motion add-on. • Obtain the tendon force as a function of time (Fig. 4B), transforming the graph of torsional force as a function of time (Fig. 4A) with the diagram in Fig. 3.

Fig. 3. Boundary conditions.

• Obtain the angular displacement as a function of time (Fig. 4C), by analysing movement using the Solidworks® Motion plug-in. • Obtain the angular displacement vs force transmitted by the tendon (Fig. 4D) by joining the graphs represented in Fig. 4B and C.

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Fig. 4. A - Twisting force versus time, B - force transmitted by the tendon against time, C - angular displacement vs time, D - angular displacement vs. force transmitted by the tendon.

Figure 4A shows the torque generated by the rotary motor by simulating the joint’s flexion, from the rest position (Fig. 1A), to the maximum flexion (Fig. 1C) in a 5-s interval. Figure 4B shows the force transmitted by the tendon from the resting position (Fig. 1A) to maximum bending (Fig. 1C) within 5 s. The angular displacement generated by the rotary motor from the rest position to maximum deflection within 5 s is shown in Fig. 4C. The angular displacement versus the force transmitted by the tendon is shown in Fig. 4D, with an interval of 0° to 224° in a maximum flexion position. The calculation of the distance the actuator must travel to control the amount of flex of the prototype is calculated as follows: • Obtain the linear displacement of the tie against time (Fig. 5A) through the movement analysis using the SOLIDWORKS Motion plug-in. • With the linear displacement of the tie against time (Fig. 5A) and angular displacement against time (Fig. 4C), is obtained the linear displacement of the tie against angular displacement (Fig. 5B).

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Fig. 5. A - Linear tendon displacement vs time, B - linear tendon displacement vs angular displacement.

As it is mentioned, the prototype is parameterised and with this methodology it can be reached the last three fingers of the hand. Figure 4 is suitable to choose the correct linear actuator and the graph in Fig. 5 is used to control the degree of bending of the prototype. The resting position shown in Fig. 1A is achieved when the tendon has no displacement but a preload of 0.05 N. The intermediate position represented in Fig. 1B is achieved when the tendon has a displacement of 10 mm. The maximum flexion observed in Fig. 1C is obtained when the tendon has a displacement of 23 mm. With these results, the linear actuator must generate a maximum force of 1.7 N and a minimum displacement of 23 mm.

5 Conclusion For interphalangeal joints, the relatively wide range of flexion and extension makes it challenging to use flexible materials that differ from springs due to the relatively large stress generated on the material. The design presented in this work facilitates flexible joints’ manufacture since it is composed of simple geometries and commercial parts. It is important to mention that the parts of the prototype are held together by means of the springs. For this reason, it is expected that the prototype adopts the geometry of the pressed object, avoiding the use of systems that control each one of the phalanxes in particular. The design of this prototype is the starting point for the development of a robotic hand that uses flexible joints to be as anthropomorphic as possible and can be taken as a reference in the manufacture of prostheses and orthoses. Acknowledgements. The authors are grateful for the financial support for this work provided to the Government of Mexico by the Consejo Nacional de Ciencia y Tecnología (CONACYT), the National Polytechnic Institute. The authors also acknowledge the support of project 20210282 and an EDI grant, all from SIP/IPN.

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References 1. Kivell, T.L.: Evidence in hand: recent discoveries and the early evolution. Philos. Trans. Roy. Soc. B, Biol. Sci. 370, 1–11 (2015). 20150105 2. Leal Naranjo, J.A., Ceccarelli, M., Torres-San Miguel, C.R.: Mechanical design of a prosthetic human arm and its dynamic simulation. In: Rodić, A., Borangiu, T. (eds.) International Conference on Robotics in Alpe-Adria Danube Region RAAD 2016: Advances in Robot Design and Intelligent Control, vol. 540, Issue no. 108, pp. 482–490. Springer, Cham (2016) 3. Xu, Z., Todorov, E.: Design of a highly biomimetic anthropomorphic robotic hand towards artificial limb regeneration. In: de 2016 IEEE International Conference on Robotics and Automation (ICRA), Stockholm (2016) 4. Chen, W., Xiong, C., Yue, S.: Mechanical implementation of kinematic synergy for continual grasping generation of anthropomorphic. IEEE/ASME Trans. Mechatron. 20(3), 1249–1263 (2015) 5. Odhner, L.U., Jentoft, L.P., Claffee, M.R., Corson, N., Tenzer, Y., Ma, R.R., Buehler, M., Kohout, R., Howe, R.D., Dollar, A.M.: A compliant, underactuated hand for robust manipulation. Int. J. Robot. Res. 33(5), 736–752 (2014) 6. Cao, L., Dolovich, A.T., Schwab, A.L., Herder, S.L., Zhang, W.: Toward a unified design approach for both compliant mechanisms and rigid-body mechanisms: module optimisation. ASME J. Mech. Des. 137(12), (2015) 7. Liu, Y., He, K., Chen, G., Chen, X., Leow, W.R.: Nature-inspired structural materials for flexible electronic devices. Am. Chem. Soc. 117(20), 12893–12941 (2017) 8. Sivapuram, R., Dunning, P.D., Kim, H.A.: Simultaneous material and structural optimisation by multiscale topology optimisation. Struct. Multidiscip. Optim. 54, 1267–1281 (2016) 9. Chen, X., Jia, Y., Deng, Y., Wang, Q.: Dynamics behavior analysis of parallel mechanism with joint clearance and flexible links. Hindawi Shock Vib. 2018, 17 (2018). Article ID 9430267 10. Kuo, Y.-L.: Mathematical modeling and analysis of the Delta robot with flexible links. Comput. Math Appl. 71(10), 1973–1989 (2016) 11. Zhang, X., Sørensen, R., Rahbek Iversen, M., Li, H.: Computationally efficient dynamic modeling of robot manipulators with multiple flexible-links using acceleration-based discrete time transfer matrix method. Robot. Comput.-Integr. Manuf. 49, 181–193 (2018) 12. Helal, M., Alshennawy, A.A., Alogla, A.: Optimal design of a positioning flexible hinge compliant micro-gripper mechanism with parallel movement arms. Int. J. Eng. Res. Technol. 10(2), 2105–2128 (2017) 13. Savescu, A., Cheze, L., Wang, X., Beurier, G.: A 25 degrees of freedom hand geometrical model for better hand attitude simulation. SAE Tech. Pap. 113(1), 8 (2004) 14. Desarrollan una mano robótica impulsada por aire comprimido, [En línea]. http://www. tendencias21.net/Desarrollan-una-mano-robotica-impulsada-por-airecomprimido_. (Accessed 20 Apr 2021) 15. Una mano robótica de alta velocidad con una destreza increíble Hizook, [En línea]. http:// www.tuexperto.com/2009/09/14/una-mano-robotica-de-alta-velocidad-con-unadestreza. (Accessed 20 Apr 2021) 16. Mano robótica alemana super resistente, [En línea]. http://www.fayerwayer.com/tag/ dyneema/. (Accessed 20 Apr 2021) 17. i-LIMB Pulse, la nueva mano biónica de Touch Bionics, [En línea]. https://www.engadget. com/es/2010/05/06/i-limb-pulse-la-nueva-mano-bionica-de-touch-bionics/. (Accessed 20 Apr 2021)

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18. la primera mano biónica que permite sentir, [En línea]. https://gestion.pe/tecnologia/ lifehand-2-primera-mano-bionica-permite-sentir-3327-noticia/Lifehand2. (Accessed 20 Apr 2021) 19. Annick, M., Thierry, L., Cl´ement, G.: Underactuated tendon-driven robotic/prosthetic. Robot. Sci. Syst. (2021) 20. LUKE, el brazo protésico de DARPA, llega a los primeros usuarios, [En línea]. https:// computerhoy.com/noticias/life/luke-brazo-protesico-darpa-llega-primeros-usuarios-65001 21. Buryanov, A., Kotiuk, V.I.: Proportions of hand segments. J. Morphol. 28(3), 755–758 (2010) 22. Greene, W., Heckman, J.: The Clinical Measurement of Joint Motion. AAOS, Greene, Illinois (1994)

Mechanism Deign

A Reconfigurable Underactuated Grasper with In-Hand Manipulation Carl A. Nelson(&) Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA [email protected]

Abstract. One of the hallmarks of a dexterous robotic hand is the ability to perform in-hand manipulation (i.e., without re-grasping). In the field of robotic hand design, however, there are competing interests between dexterity, simplicity, and reconfigurability. It can be difficult to achieve all of these objectives simultaneously. This paper presents the design of a simple underactuated grasper which uses a gimbal and parallelogram mechanism to achieve in-hand manipulation while maintaining grasp. The new grasper also integrates elements which make it reconfigurable or metamorphic, and is readily adapted for different types of robotic fingers. Keywords: Underactuated grasper


Robotic grasping
In-hand manipulation

1 Introduction In robotic grasping, there is a fundamental conflict of objectives: simple designs are desirable for cost reduction and robustness, but generally provide limited dexterity, and greater dexterity generally comes at the cost of increased complexity and cost. When fully dexterous hands, such as those described in [1, 2], are over-matched to the task, underactuated graspers (e.g., [3–5]) are often favored for their relative mechanical simplicity and the avoidance of computationally heavy control strategies. However, inhand manipulation (i.e., manipulation of a grasped object without relinquishing the grasp, see Fig. 1) has been identified as a key capability that tends to be difficult with underactuated graspers, with only a few examples of success in the literature; only recently [6] was an underactuated hand and manipulation strategy presented to accomplish this with a planar grasper without a priori knowledge of the object shape. Reconfigurable or metamorphic hands [7] can provide more extensive capabilities to deal with objects of varying shape and size. A grasper combining the benefits of simplicity (associated with underactuation), the dexterity of in-hand manipulation, and the ability to easily reconfigure the grasper could be seen as a desirable mix of characteristics.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 151–158, 2021. https://doi.org/10.1007/978-3-030-75271-2_16

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Fig. 1. In-hand manipulation: the grasped object is reoriented by relative motion of the fingers and palm without loss of grasp.

In this paper, a new grasper design is presented which aims to capture the various positive characteristics just mentioned: simplicity, reconfigurability, and the capability of performing in-hand manipulation. The design is illustrated, kinematics are discussed, and testing with a prototype is shown.

2 Grasper Design Inspired by how the human hand achieves basic in-hand manipulation (see Fig. 1), the design goal is to implement a robotic hand in which the underactuated fingers can perform a gross translational motion through interaction with the palm, such that the grasped object is reoriented. Rather than focus on the finger design or aspects of underactuation, attention will be placed on the interaction between the palm and fingers. With a fixed palm, underactuated fingers can be caused to achieve a similar inhand manipulation effect [6], but this involves a coupled translation and rotation of the grasped object. For improved usability, our design goal is to achieve something closer to a pure rotation as illustrated in Fig. 1.

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Fig. 2. Parallelogram coupling between the palm and finger: planar version with revolute joints (left); spatial version (including revolute, spherical, and universal joints as labeled) with 2-DOF palm motion as input (right).

To achieve the desired motion, a parallelogram mechanism is adopted to couple the palm and finger motions. As illustrated in Fig. 2 (left), a planar parallelogram mechanism would allow the base of the finger (first phalanx) to be advanced/retracted without significant rotation, leaving the underactuation characteristic of the finger (in the more distal phalanges) to conform to the grasped object as the object is reoriented. Considering the case of a non-planar grasp, a modified parallelogram as shown in Fig. 2 (right) allows the revolute-jointed underactuated finger to primarily respond to angular displacement of the palm which occurs in the plane normal to the revolute joint axes. The net result is that multiple fingers can be configured as part of the hand mechanism to interact on arbitrarily shaped (non-prism) objects and achieve in-hand manipulation by acting in their respective planes to maintain contact. Ideally the finger actuation would be decoupled from the palm actuation (i.e., the grasp closure can be maintained independent of the input for in-hand manipulation). In this context, the palm can be modeled as a parallel manipulator consisting of a gimbal platform driven by three tendons (effectively a cable-driven U-3SPS architecture, see Fig. 3). Thus, for a hand with n 1-DOF underactuated fingers, the total number of tendon actuators is 3 + n. To achieve reconfigurability, the palm is equipped with circular-arc tracks in which fingers can be slidably mounted (see Fig. 4). This allows for both planar and spatial functionalities to be accomplished (e.g., simple pinch grasp vs. tripod grasp). Although automatic reconfiguration of the finger layout on the palm (i.e., dynamic reconfiguration without direct human intervention) is beyond the scope of this paper, one could envision various ways of accomplishing this, for example, using a ring gear with actuators mounted on the palm, or using cables and remote actuation (i.e., on the robot forearm) to slide the finger mounts in the tracks.

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Fig. 3. Palm modeled as a cable-driven U-3SPS manipulator (U joint at the origin, with cables L1,2,3 acting as SPS chains).

Fig. 4. Finger configuration for a planar pinch grasp (left) and for a spatial tripod grasp (right); fingers labeled as F1,2,3 with red squares always move in planes passing through the center of the palm.

To decouple the finger actuation from the palm actuation, the finger tendons need to pass through a path which has fixed length with respect to the palm motion parameters. The only point on the palm which is fixed with respect to the forearm is the center of motion at the universal joint or gimbal. However, routing the finger tendons through the gimbal to the palm center is relatively complicated and involves passing the tendons along or through the gimbal pivots themselves. A simpler alternative is to allow an incomplete decoupling of the finger motion from the palm motion, routing the finger tendons along the same paths as the three palm control tendons. In this case, any displacement of a palm tendon would need to be matched by the same displacement of the finger tendon to avoid unintended finger displacement; fortunately, this is a very simple (linear) kinematic scheme to achieve. Figure 5 shows both options on a schematic of the palm.

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Fig. 5. Finger tendon routing options: along same path as palm angle tendons (left side); through gimbal axes to center of palm (right side). Gimbal axes are shown in green, finger mount locations in red, and tendons in orange.

3 Analysis The lower routing point of cables L1,2,3 is aligned on the fixed y axis (using the notation illustrated in Fig. 3) at a distance d below the palm. The palm can be considered as an equilateral triangle with radius r (distance from the origin to each vertex). With the universal joint at the center of the palm embodied as a gimbal, the motion sequence is a rotation h1 about x followed by a rotation h2 about z. The position of the cable attachment point in the fixed reference frame is therefore 2

1

6 P ¼ 40 0

0 cos(h1 ) sin(h1 )

0

32

cos(h2 ) sin(h2 )

76 sin(h1 ) 54 sin(h2 ) 0 cos(h1 ) 2

0

32

rcosðaÞ

3

76 7 0 54 0 5 1 rsinðaÞ 3 cos(a) cos(h2 ) 6 7 ¼ r 4 cos(a) cos(h1 ) sin(h2 ) + sin(a) sin(h1 ) 5 cos(h2 ) 0

ð1Þ

cos(a) sin(h1 ) sin(h2 )  sin(a) cos(h1 ) where a is the angular position of the cable attachment point in the local x-z plane of the palm, and a1,2,3 = a0 + {0, 2p/3, 4p/3}. The respective cable lengths then simplify to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 P2x þ Py þ d þ P2z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ r 2 þ d 2 þ 2dr ðcos(a) cos(h1 ) sin(h2 ) + sin(a) sin(h1 )Þ

L¼

ð2Þ

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This provides an inverse kinematic solution for the three cable lengths given two desired palm angles (as with cable-driven robots in general, to ensure cables remain in tension, the number of cables has to exceed the number of degrees of freedom by one, and the third tendon input is not independent of the other two).

4 Prototype and Testing A concept prototype was fabricated mostly with acrylic using a laser cutter. Thread was used for tendons, and screws for pin joints. Wire hooks were used to complete the parallelogram mechanism while allowing finger reconfiguration. Two different grasps were tested, a planar pinch grasp and a tripod grasp. In each case, the palm was actuated to induce object reorientation. The three finger inputs (one tendon per finger) were kept constant during the palm actuations. These tests are illustrated in Figs. 6 and 7.

Fig. 6. Prototype grasper in tripod grasp; composite rotation at palm induces object reorientation (as shown by movement of logo relative to fingers).

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Fig. 7. Prototype grasper implementing planar pinch grasp: initial grasp (top); reoriented grasp (bottom). Relatively small palm inclination results in object reorientation (as shown by movement of logo relative to fingers).

It should be noted that the input-output relationships of the palm, fingers, and manipulated object and complicated, and depend on the object geometry, the finger pad geometry, the finger mechanism, and the nature of the underactuation and the related equilibrium conditions for grasping (see, for example, [6]). The details of these relationships are outside the scope of this paper, as no specific finger design is proposed; however, these experiments show qualitatively that the proposed palm design can be used to achieve in-hand manipulation and that the grasper can be easily reconfigured for different types of grasps.

5 Conclusions A novel grasper design using a gimbal-based palm mechanism has been proposed to achieve reconfigurability and in-hand manipulation without undue complexity in design or operation. Feasibility of the concept has been shown through preliminary

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testing with a concept prototype. The results illustrate the basic functionality, and the concept is anticipated to work independent of finger design. Future work includes further testing and refinement to demonstrate robust utility. Acknowledgments. Thse author acknowledges support from the UNL College of Engineering.

References 1. Jacobsen, S.C., Wood, J.E., Knutti, D.F., Biggers, K.B.: The UTAH/M.I.T. Dextrous hand: work in progress. Int. J. Robot. Res. 3(4), 21–50 (1984) 2. Butterfaẞ, J., Grebenstein, M., Liu, H., Hirzinger, G.: DLR-Hand II: next generation of a dextrous robot hand. In: Proceedings of the 2001 IEEE International Conference on Robotics & Automation, Seoul, Korea, 21–26 May, pp. 109–114 (2001) 3. Birglen, L., Laliberté, T., Gosselin, C.: Underactuated Robotic Hands. Springer, Berlin (2008) 4. Dechev, N., Cleghorn, W.L., Naumann, S.: Multiple finger, passive adaptive grasp prosthetic hand. Mech. Mach. Theory 36, 1157–1173 (2001) 5. Odhner, L.U., Jentoft, L.P., Claffee, M.R., Corson, N., Tenzer, Y., Ma, R.R., Buehler, M., Kohout, M., Howe, R.D., Dollar, A.M.: A compliant, underactuated hand for robust manipulation. Int. J. Robot. Res. 33(5), 736–752 (2014) 6. Ospina, D., Ramirez-Serrano, A.: Sensorless in-hand manipulation by an underactuated robot hand. J. Mech. Robot. 12, 051009-1–051009-13 (2020) 7. Dai, J.S., Wang, D., Cui, L.: Orientation and workspace analysis of the multifingered metamorphic hand—metahand. IEEE Trans. Robot. 25(4), 942–947 (2009)

Design of a Robotic Brace with Parallel Structure for Spine Deformities Correction Rahul Ray1,2(&), Laurence Nouaille1, Briac Colobert2, Laurine Calistri2, and Gérard Poisson1 1

PRISME Laboratory, University of Orléans, INSA CVL, Orléans, France [email protected] 2 Proteor, Dijon, France

Abstract. This paper describes the mechatronics design and tests of a robotic brace prototype specially dedicated for the treatment of idiopathic scoliosis. These developments have been motivated by the limitation of the current rigid braces used for this purpose which do not adapt to changes in the skeletal system in response to treatment. This new robotic brace has been designed taking into account the human biomechanics properties and more specifically to be able to exert “three-point pressure” in order to be effective for treatment of scoliosis. Based on a double Stewart-platform, it was produced by rapid prototyping and equipped with 12 linear actuators, 12 linear position sensors and multi-axis force/torque sensors. Position and force controls approaches are implemented on a dedicated control device. This robotics brace has been used on healthy persons to test its motion range and its acceptability. The prototype version allows the force applied on the human body to be adjusted to correct the scoliosis, then the mechanical concept was validated. Keywords: Scoliosis


Spine deformity
Robotic brace
Human torso

1 Introduction Scoliosis is a disease characterized by an abnormal spinal curvature, see Fig. 1a. Braces have been used to treat scoliosis since the 1500’s with the use of padded iron corsets [1]. While it is a globally accepted practice in the treatment of scoliosis, systematic, randomized, group studies on the effectiveness of spine braces have been performed only recently. The hard design and complex structure of these braces makes difficult to wear for a long period of time as they interfere with daily life activities [2]. To allow for greater mobility, the SpineCor brace was proposed which uses a series of electric actuators and elastic straps to correct the curvature with varying levels by applying forces [3]. However, this brace is relatively new and has not been widely adopted by orthotists. There are some aspects of the current design that can be improved. First, the required power to continuously apply force on the vertebrae when using system mounted batteries. The recommended duration of wearing a conventional rigid brace is between 12–16 hours per day. Second, due to the rigid transmission of the mechanism which can impact daily activities. One of active brace is from Japet, which does not deal with correcting abnormal posture of the spine it is rather focused on back pain © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 159–167, 2021. https://doi.org/10.1007/978-3-030-75271-2_17

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reliefs during works or other activities based on daily life performances [4]. Similarly, an active brace, called Rose, is designed for the correction of the spinal curvature and for the treatment of scoliosis [5]. This brace can potentially contribute to high adaption ability and provide a better human-machine interface for the user. In the robotic brace, the required range of motion spans a 6-D space of position and orientation. This is an important consideration in the use of a robotic brace for human with scoliosis deformities, and this is what motivates the development here presented. Section 2 of this paper is dedicated to the description of medical specifications. Section 3 presents the mechanical design of the robotic brace. Section 4 describes mechatronics integration approach in the brace. Section 5 shows simulation and results.

2 From Medical Specifications to Brace Concept The ideal approach for the treatment of scoliosis is to apply pressure at the certain part of the spine based on the type of spinal deformity, Fig. 1b [6]. Many experimental results have been reported to evaluate the effect of soft braces compared with hard braces, there are insufficient evidences to draw definite conclusions about effectiveness and intervention safety although the treatment of scoliosis using soft braces is supported by some papers which provide a small body of evidence [7]. In robotic brace application, the required range of motion spans workspace of position and orientation plays a major role for effective treatment of scoliosis. This robotic brace to be designed had to present some key features: pressure on precise area on human spine continuously applied force/torque, high mobility in position and orientations, light weight and wearable exoskeleton, real time motion control. We focus on the configuration where effects on the workspace were analyzed based on the applications of the robotic spine brace. The goal of the robotized brace is to provide controlled motion and force at a selected ring of the human spine. To accomplish this goal, the brace is composed of three printed elliptical corrective rings (upper, middle and lower) designed from a scan from subject torso, Fig. 1c. Two adjacent rings are connected by Stewart parallel mechanism driven by six electric actuators. Each Stewart mechanism offers 6 DOF corrective forces. The two Stewart platforms connected in series, can be controlled in position or force mode independently. The proposed robotic brace can be attached to the human body at different cross sections as rings. This designed robotic brace has tight dimensional constraints due to small distance between rings. Therefore, the brace system can adjust the rehabilitation scheme along with status of the patients by controlling the proposed robotic spinal brace flexibly.

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Upper thoracic ring Mid thoracic ring Pelvic ring

(a)

(b)

(c)

Fig. 1. (a) Scoliosis spinal deformity in dorsal and sagittal views. (b) Posture correction via the 3-point pressure applied on upper thoracic, mid thoracic and pelvis of human spine. (c) 3D printed robotic brace. Pictures (a) and (b): scoliosis patients from Smartist software© Proteor.

3 Mechanical Design of Robotic Brace 3.1

The Mechanism Architecture

The kinematic diagram below represents the pelvis ring (with a fixed base role), the mid thoracic ring (middle) and the upper thoracic ring of the robotic brace.

Fig. 2. Geometry and parameterization of Stewart platform: *parallel robots, mechanics and control by Hamid D. Taghirad (2013), [8, 9] and Matlab, The Mathworks, Inc.

The three rings lie in three different planes. Each ring center is located at the center of the human spine. In the kinematic chain, each leg consists of one universal joints at point Ai, a linear series actuator acting in the direction of vector qi and another universal joint at point Bi. 3.2

Kinematic Constraints

In this part we present the models that have been integrated into the control loops. The Stewart platform consists of two parts connected by six variable length legs which can be applicable according to stroke length of actuators [9, 10]. To obtain kinematic

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models the base frame (on pelvis) is considered to be the reference with orthogonal axes x, y, z. The architecture of each platform follows the kinematic structure of 6-6 Stewart-Gough platform where all the legs share identical kinematic chains (see Fig. 2). The platform presents 6-DOF with respect to the base. The origin of the platform frame can be defined by a 3-translation displacement with respect to the base, one for each axis. Three rotations then define the platform orientation, also with respect to the base [11]. A set of three angles is used in the following sequence to define the coordinate frame with respect to the Pitch-Roll-Yaw formalism [12] where rotation about fixed x axis by w, followed by rotation about fixed y axis by h and followed rotation about fixed z axis by / [13]. As per Stewart platform and its geometry and parameters the rotation matrix of the platform relative to the base is defined by: P

RB ¼ Rx ðwÞ: Ry ðhÞ: Rz ð/Þ

For the li leg, coordinate qi of the anchor point pi with respect to the base reference framework are given by equation: qi ¼ T þ P RB : pi

ð1Þ

The translation vector T gives the displacement of the Stewart platform origin frame with respect to the base reference frame, and pi is the vector defining the coordinates of the anchor point pi . Similarly the length of the li leg is given by: li ¼ T þ P RB :pi  bi

ð2Þ

Where bi is the vector defining the coordinates of the lower anchor point Bi these equations give the lengths of the six legs to achieve the desired position and attitude of the platform [13]. The architecture of each parallel platform follows the kinematic structure of 6-6 Stewart-Gough platform where all the legs share identical kinematic chains. Each leg connecting the fixed base denoted A to the moving platform denoted B forms a kinematic loop which can be expressed as: qi ¼ pB  ai þ RB :bi

for i ¼ 1; . . .:; 6

ð3Þ

Each leg length has uniquely determined for position and orientation of the moving Stewart platform. For kinematic analysis of the joint and for XB of the moving platform where XB is redefined with screw coordinates as: XB ¼ ½PxB ; PyB ; PzB ; sx ; sy ; sz ; HT

ð4Þ

In which sx, sy, sz and H are obtained from rotation matrix where, r32  r23 2:sinðHÞ r11 þ r22 þ r33  1 H ¼ cos1 2

Sx ¼

r32  r23 2:sinðHÞ

Sy ¼

Sz ¼

r32  r23 2:sinðHÞ

ð5Þ

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The equations to solve for forward kinematics are: Ei ¼ l2i þ qi T qi ¼ l2i þ ½pB  ai þ RB :bi T ½pB  ai þ RB :bi  ¼ 0

ð6Þ

E7 ¼ ^s: ^s ¼ s2x þ s2y þ s2z ¼ 1

ð7Þ

The principle of duality of kinematic, the forces and moments applied by the rings are related to the joint forces required at the actuators to maintain the equilibrium by the transpose of the Jacobian matrix, under the assumption that leg applies a force only along the leg axis.

4 Robotic Brace Integration 4.1

Actuator

Linear actuators are integrated in this robotic brace, which are capable of high flexibility, are configurable and compact platform with an optional on-board microcontroller. The actuators have no built-in controller, but do provide an analog position feedback signal that can be input to an external controller. Position of the actuators stroke can be monitored by providing any stable low and high reference voltage for reading position signal. The linear actuators we choose have for reference 210:1– 12 V. They are capable of a peak force of 45 N and a peak speed of 5 mm/s with 5 cm stroke length, gearing ratio of 210 allowing a peak power point of 62 N with 3.2 mm/s (see Fig. 3b). Maximum input voltage of this linear actuator is 12 V with standby current of 3.3 mA which can be operated up to –10 °C to +50 °C based with maximum force lifting capability which lead to choose this actuator model compared with other existing linear actuators for designing our spine brace. Each actuator presents a universal joint at the base and at the top (see Fig. 3a). Explicit expressions for movement of the actuators are provided in mechanical design parts and similar formulation can be found in [8–10].

Fig. 3. (a) Actuator with universal joint and stroke length out. (b) Actuators load and current.

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Control of the Stewart Platform

Two control modes have been implemented to pilot the robot: a motion control and a force one. Each leg has built in sensors to record its displacement and the force exerted by the actuators. These measurements are used in both modes to control the 12-DOF robot. The models presented above in Sect. 3.2 have been used in this part. The upper and lower parallel platforms have their own motion and force controller which can be operated independently. Standard closed loop PID controllers has been used for these controls. The detailed approach has been unveiled in [10]. 4.3

Sensors

The multi-axis force/torque sensors are used in this robotic brace which is capable to measure six components of force and moment (Fx, Fy, Fz, Mx, My, Mz) using a monolithically instrumented sensor. It measures the precise force and moment which brace applies on the vertebrae to solve spinal deformity and vice-versa measures the force and moment exerted by the body on the brace.

5 Simulation and Results 5.1

Evaluation of Position Controller

The controllers were tested to determine positioning accuracy of the structure. In addition, a range of motion study was performed by a human subject with the position controller. The brace-frame origin is located at the center of the bottom segment with the x-axis to the right of the patient, the y-axis towards the anterior, and the z-axis vertically upwards [14]. Displacements and rotations reported are relative to the corresponding axes of the respective segments in the neutral position, in the bottom base-frame. Rotation angles /, h, w are about z, y, and x respectively following the same rotation sequence. The position controller was tested by moving the brace through a range of translations and rotations. Rotation about z and translation about y in transverse plane was tested, denoted as three-point mode, as the traditional three-point correction in spine braces [15, 16]. Flexion and lateral bending were tested to compare bending in brace. The last series of test were for isolated translations and rotations. For both, the upper Stewart platform either performed the same task follow or opposite task mirror as the lower Stewart platform. The motion was recorded thanks the Qualisys® motion capture system. The translation and rotation of the middle and top segments relative to the neutral position were recorded and analyzed in the bottom base-frame.

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(a)

(b)

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(c)

Fig. 4. (a) Flexion and extension in sagittal plane. (b) Left and right lateralization in coronal plane. (c) Rotation around the vertical axis z.

5.2

Determination of the Range of Motion

The range of motion (ROM) was tested to determine the motion allowed by the brace in transparent mode (Zero desired force). A healthy individual, without scoliosis, had markers placed on their anterior superior iliac spines, sacrum, sternoclavicular joints, and the inferior border of the manubrium. The subject bent to his maximum in the frontal plane (lateral bending), Fig. 4b; sagittal plane (flexion/extension), Fig. 4a; and around the vertical axis, Fig. 4c. The maximum change in the angle between the pelvis and the sternum were then compared for all three motions without brace (Fig. 5) and with brace (Fig. 6).

Fig. 5. Motion captured position without brace. From left to right: flexion/extension, lateral bending and axial rotation

Fig. 6. Motion captured position with brace. From left to right: flexion, extension, lateral bending, and axial rotation

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Validation of the Force Control

The force control validation utilized a test bed where the middle and bottom rings were attached to each other and mounted with six linear electric actuators in the brace’s neutral position (see Fig. 7). The wrench capability is calculated on the human spine based on the three-point-pressure, where a minimum amount of force is applied on the spine to put pressure for the correction of abnormal curvature of the human spine. The brace was driven to follow a force whose direction is one of the main ones of the frame (–40 N  Fx, Fy, Fz  40 N) and (–1.5 Nm  Mx, My, Mz  1.5 Nm) driven in 3D force. The calculated force opposed by the actuator is compared with the desired force command on the actuator to evaluate tracking error.

Fig. 7. (a) Rear view of the robotic brace. (b) Front view of robotic brace on body. (c) Force applied on upper thoracic, mid thoracic and pelvis to correct scoliosis.

6 Conclusion and Future Works This paper presents a new robotic brace for patients with scoliosis particularly designed to improve the patient’s quality of daily life. As scoliosis is a complex threedimensional deformity, it is necessary for an adequate correction of this disease and therefore for the robot to be efficient, to be able to apply three-dimensional controllable corrective forces on the human spine. The robotic brace designed and prototyped is based on human biomechanics and medical needs and, in the present version, weighs up to 3.5 kg. It is composed of two 6-DOF Stewart-platforms associated in a serial structure, so 12-DOF are controlled. It permits to continuously exert three-point pressure on the human body and, at the same time, allowing patient’s usual movements (flexion, bending and rotation). The prototype was designed and dimensionally optimized from a scanner to fit on the human torso; mechanical parts have been produced via rapid prototyping technique. Several experiments (motion and forces tests) were carried out with healthy human to validate the prototype kinematic architecture choice, the actuators and sensors performance and the control strategies: position and force ones. The prototype will be improved by choosing other materials and to enhance its compliance and its lightweight property. As the future device to be used will be adapted to each patient, and the range of motions will depend on severity of the disease, it will soon be necessary to perform other experiments with different persons (adults and

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adolescent) and different pathology levels in order to size a series of devices. Further, over a longer period, medical tests will be carried out to evaluate the effectiveness of the robotized approach for treatment of scoliosis, compared with traditional bracing techniques. Acknowledgements. The work described in this paper was supported by the Association Nationale de la Recherche Technologique (ANRT), France and Proteor Company, Dijon, France.

References 1. Fayssoux, R., Cho, R., Herman, M.: A history of bracing for idiopathic scoliosis in North America. Clin. Orthop. Relat. Res. 468(3), 654–664 (2010) 2. Price, C.T., Scott, D.S., Reed Jr., F.E., Riddick, M.: Night-time bracing for adolescent idiopathic scoliosis with the Charleston bending brace: preliminary report. Spine 15(12), 1294–1299 (1990) 3. Coillard, C., Vachon, V., Circo, A.B., Beausejour, M., Rivard, C.H.: Effectiveness of the spinecor brace based on the new standardized criteria proposed by the scoliosis research society for adolescent idiopathic scoliosis. J. Pediatr. Orthop. 27(4), 375–379 (2007) 4. Japet Medical Device Company. Japet brace. https://www.japet.eu (2016). Accessed 16 Mar 2019 5. Park, J.H., Stegall, P., Agrawal, S.K.: Dynamic brace for correction of abnormal posture of human spine. https://doi.org/10.1109/icra.2015.7140029 (2015) 6. Weiss, H.R., Werkmann, M.: Retraction note: soft braces in the treatment of adolescent idiopathic scoliosis - review of the literature and description of a new approach. Scoliosis 8, 7 (2013) 7. Niu, X., Yang, C., Tian, B., Li, X., Zheng, S., Cong, D., Han, J., Agrawal, S.K.: Investigation of robotic brace of patients with idiopathic scoliosis - review of the literature and description of novel brace. J. Mech. Med. Biol. 18(8), 1840038 (2018) 8. Taghirad, H.: Parallel Robots. CRC Press, Boca Raton (2012) 9. Tasi, L.W.: Robot Analysis. Wiley, Hoboken (1999) 10. Ray, R., Nouaille, L., Poisson, G., Colobert, B.: Design of robotic braces for patients with scoliosis. In: Zeghloul, S., Laribi, M., Sandoval Arevalo, J. (eds.) RAAD 2020, Mechanism and Machine Science, vol. 84, pp. 1–8. Springer, Cham (2020) 11. Ophaswonge, C., Murray, R.C., Agrawal, S.K.: Design of a parallel architecture robotic spine exoskeleton. In: DETC2017-67842 (2017) 12. Park, J.H., Stegall, P., Agrawal, S.K.: Wrench capability of Stewart platform with series elastic actuators. J. Mech. Robot. 10, 021002–1 (2018) 13. Merlet, J.: Direct kinematics of parallel manipulators. IEEE Trans. Robot. Autom. 9(6), 842– 845 (1993) 14. Gao, X.S., Lei, D.L., Zhang, G.F.: Generalized Stewart-Gough platforms and their direct kinematics. IEEE Trans. Robot. 21(2), 141–151 (2005) 15. Innocenti, C.: A novel numerical approach to the closure of the 6-6 Stewart platform mechanism. In: Proceedings of the Fifth International Conference on Advanced Robotics, pp. 852–855 (1991) 16. Souissi, M., Wang, Z., Amokrane, W., Poisson, G.: Dynamic simulation of vertebral column and control of the posture using a parallel mechanism. In: 14th International Conference on Informatics in Control, Automation and Robotics ICINCO, Madrid, 26–28 July (2017)

An Adaptive Drive of Spacecraft Docking Mechanism Konstantin S. Ivanov1(&), Dana T. Tulekenova2, and Marco Ceccarelli3 1

Almaty University of Power Engineering and Telecommunications, Almaty, Kazakhstan 2 Al-Farabi Kazakh National University, Almaty, Kazakhstan 3 University of Rome Tor Vergata, Rome, Italy [email protected]

Abstract. Adaptive drive is a self-regulating gear mechanism with constant engagement proposed of gear wheels and with variable gear ratio. The paper summarizes the fundamentals of the theory of power adaptation of the gear drive and presents an adaptive drive for a docking mechanism of the spacecraft. Keywords: Adaptive gear drive mechanism
Drive
Simulation


Power adaptation
Spacecraft
Docking

1 Introduction A docking device is a controlled electromechanical system for connecting a moving spacecraft with an orbital station (OS) [1, 2]. Docking device comprises docking mechanism and controlled electromechanical drive. The docking mechanism is a mechanical system of interconnection between the elements of the spacecraft and the orbital station, adapting to the continuously and spontaneously changing conditions of their interaction. The controlled electromechanical drive comprises an electric motor, a toothed differential transmission mechanism, control electromechanical devices and a control system. The control system shall ensure adaptability of interacting elements of spacecraft and OS to continuously and unpredictably changing dynamic coupling conditions. Solving the problem of functioning a complex electromechanical system requires the use of a multifunctional bulky control system, which leads to the use of a large number of control devices and reduced reliability of the docking apparatus. Several complex transmissions can be considered with design issues as reported for example in [5–10]. In this paper is proposed to significantly simplify the controlled electromechanical drive of the docking mechanism by replacing the gear differential mechanism with a new planetary transmission with two degrees of freedom. The main purpose of this new planetary transmission with two degrees of freedom is related to the capability at adapting the operation to variable loading conditions by preserving efficiency and input-output load ratio.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 168–178, 2021. https://doi.org/10.1007/978-3-030-75271-2_18

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2 Description of the Docking Mechanism The proposed docking of spacecraft is implemented by active and passive docking units and starts from the moment of their first contact [1, 2]. On one of the units, the main guiding element is the head of the movable pin, and on the other - a fixed cone with a receiving seat. When spacecraft approach, the head, after several contacts with the cone, enters the receiving seat, at the same time its latches go beyond the stops of the seat and a coupling of the units is formed. After that, energy of spacecraft approach is absorbed, units are leveled and tightened, their rigid connection is formed. During docking, the rocking body can be turned relative to the receiving cone of the passive unit. But their aligned position at all three angles must be ensured before the docking planes approach for a time sufficient to absorb the remaining energy of the relative angular motion of spacecraft. In compliance with this condition, units swing relative to each other in universal joint of coupling mechanism and turn relative to their longitudinal axes. Energy of such rotation is absorbed by dampers in parallel kinematic circuits of rocking body of coupling mechanism. As a rule, special devices are not used to dampen the turns of spacecraft about their longitudinal axes due to small moments of inertia relative to these axes, complication and increase in the dimensions of the design of docking mechanisms [1, 2]. The movement of the pin relative to the rocking body is driven by a drive which can only be contained within the same body. Since the trajectory of the ends of the equalizing arms is uniquely determined by the length of the free part of the pin, rotation from the drive can be transmitted to the cam defining the appearance of this trajectory. These features of the kinematics of the central type docking mechanisms are characteristic of the structures that were developed in the 70s of the last century at the Central Design Bureau of Mechanical Engineering (CCBM) [1, 2] and NPO (currently Rocket and Space Corporation - RSC) Energia [1]. Connecting units of RSC Energia active with connecting mechanism (Fig. 1a) and passive with receiving cone (Fig. 1b) with minor modifications are used to date. Simplified kinematic diagram of the coupling mechanism is given in Fig. 1c. Figure 1 shows the pin 1, which is a hollow screw, and through the guide bearings 3 enters the rocking housing 4 connected by a universal hinge 5 to the fixed housing 6, the base of the docking mechanism. The ball-screw converter 7 transforms the translational movement of the frame into a rotary one, which is transmitted to the coil spring mechanism 8 with a limited one-way stroke, the friction brake 9 and the magnetic brake 10 of the axial damper 11. Extension of pin to initial position before connection and its retraction into swinging housing is performed by drive 12. Transmission of rotation from axial damper 11 to shafts of electric motors 12 is blocked by locking clutches 9 connected to these shafts. The initial position sensor 13 generates a signal to stop the drive when the pin is extended before docking. At the final stage of coupling units tightening, three leveling levers 19 are used, the trajectory of which is determined by the shape of the cam 20, the rotation of which is uniquely associated with the rotation of the ball-screw converter nut and the length of the free part of the pin.

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a)

b)

c) Fig. 1. a) The active docking unit with the docking mechanism [1]; b) passive unit with receiving cone and socket [1], c) simplified kinematic diagram of coupling mechanism [2].

DC drives are used in electromechanical docking mechanism. In the drive of the docking mechanism (Fig. 2a), two backup motors 1 are connected to the output shaft 5 through locking clutches 2, a planetary gear (differential) 3 and two planetary gears 4. Locking clutches prevent transmission of motion from differential main wheels to electric motors. AC motors are used in the drives of the docking mechanism. These motors for the applied power range have a large starting torque to nominal ðMP =MH ¼ 5  10Þ ratio and therefore they have good overload capacity, which favorably affects the reliability of the drives, but requires the use of safety clutches. To power these motors, 2 DC converters with voltages of 27 V and 115 V and a frequency of 400 Hz are used. The transducer has a weight of 15 kg. The drive has two motors connected through a differential electric motor power 100 W, weight 2.5 kg. The drive, excluding the voltage converter, weighs 10 kg. Motor output shaft rotation speed 415 rpm, reduced weight of all rotating elements 170 kg. The drive of the coupling mechanism is switched on for tightening when there is a signal about coupling and actuation of the touch sensor.

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Fig. 2. a) Kinematic diagram of the drive; b) planetary mechanism; c) planetary reduction gear box

The signal to switch off the drive comes from the joint alignment sensors. For the drive to pull the frames with maximum force, it is switched off a few seconds after the sensor is activated. The rod is held in the tightened position by a locking clutch that breaks the reduction gear after the motor is turned off. 1. 2. 3. 4.

Disadvantages of the existing drive: Complexity of design including two motors and complex control system. Large dimensions and weight. Large energy losses associated with matching drive parts with controls. Relatively low reliability associated with many interacting drive elements and control system.

In this regard, the task is to significantly improve the drive of the docking apparatus by using an adaptive transmission mechanism (gear variator), which has the property of independently (without a control system) changing the speed of the drive shaft depending on the load.

3 Adaptive Gear Variator Description The adaptive gear variator (Fig. 3a) comprises a post 0, an input carrier H1 , a closed loop of gears 1-2-3-6-5-4 and an output carrier H2 . Closed circuit comprises inlet satellite 2, unit of sun wheels 1–4, unit of ring wheels 3–6 and outlet satellite 5 [3]. Number of teeth: External forces applied to the initial links: FH1 and RH2 . Moments: MH1 and MH1 on the initial links H1 and H2 . Figure 3 shows the linear velocity plan of the kinematic circuit Vi i ¼ 1; 3; 4; 6 on the right, x5 ¼ x2 – angular velocities of links 5 and 2, P – is instantaneous center of velocities of intermediate link 5.

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Fig. 3. a) - b) The proposed adaptive gear variator with linear speed plan, c) Link 5 with applied forces, d) Joint of satellite 5 in Fig. 2a with balancing friction moment

The line of the angular velocity of the input carrier intersects the line of the angular velocity of the output satellite 5 at point s, which defines the end of the velocity vector Ss, which is the same for the input carrier and the output satellite. The point S occupies constant position at any speed values of other points of the mechanism [6]. This point determines the position of the excessive coupling in the gear mechanism as a contact point of the gears 8 and 7. Implementation of additional redundant communication is performed by pair of gears 8, 7 transmitting motion from input carrier to satellite 5 in parallel with transmission through input satellite 2 and units of wheels 3–6 and 1–4. Perform power analysis of link 5, to which all external forces are transmitted (Fig. 3c). Force FH1 ¼ MH1 =rH1 , attached at the point (Fig. 3), through reactions at points B, C, D is transferred to the point E as a reaction R65 ¼ 0:5MH1 r3 =rH1 r6 and to the point G as a reaction R45 ¼ 0:5MH1 r1 =rH1 r4 (Fig. 4). These reactions are converted to force 0 FH1 ¼ R65 þ R45 , attached at point S at a distance of GS ¼ 2r5 u=ðu þ 1Þ, where u ¼ R45 =R65 . Force R6 through a reaction at a point K is moved to a point K link 5. We compose the equilibrium conditions of link 5. P 1) Moment Equation Relative to Point P: MP ¼ 0: Or 0

FH1 ðSK þ KPÞ  RH2 KP ¼ 0:

ð1Þ

From this equation by the specified external forces and size SK ¼ e you can specify the position of the point P   0 KP ¼ F1 e= RH2  FH1

ð2Þ

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2) Equation of forces of link 5: X F ¼ 0:

0

FH1  RH2 þ R05 ¼ 0

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ð3Þ

Here R05 - some additional force providing balance of link 5 at arbitrarily specified 0 forces FH1 and RH2 . For simultaneous fulfillment of conditions (1) and (3) force R05 must be applied at a fixed-point P. This force serves as a reaction from the post to the link 5. From Eq. (3) we determine this reaction R05 at a fixed-point P 0

R05 ¼ RH2  FH1 :

ð4Þ

Reaction R05 can only be created by a real connection to the rack [6]. In the dual-moveable kinematic chain in question, this would lead to a change in the state of the kinematic chain, which is unacceptable for adaptive transmission. No real reaction R05 ¼ 0 leads to imbalance. P For example, when a link is stopped H2 equilibrium condition MK ¼ 0 will take a form 0

FH1 e ¼ R05 KP:

ð5Þ

When R05 ¼ 0 condition (5) is not fulfilled and it is impossible to transfer movement with a change in the force and speed of the output link without additional support on the housing. However, unbalanced force on intermediate link can be replaced by friction force [6]. This replacement provides motion detection in the absence of additional support. Accept in calculations that force FS ¼ Mf =r7 replaces the balancing friction moment. This will allow the additional transmission to be replaced by the calculated friction moment in the K hinge of the appropriate design (Fig. 5). The balancing friction moment created in the hinge K is determined by the formula Mf ¼ RH25 f q:

ð6Þ

Where f - the specified coefficient of friction in the connecting joint, q - radius of connecting joint journal, RH2 - output resistance force acting on output satellite in connecting joint. To equilibrate the gear kinematic chain or link 5 (Fig. 3c), replace the moment of unbalanced force R05 with the friction moment in friction joint K (Fig. 3d). Moment Mf on link 5 replaces the moment of unbalanced reaction R05 according to the condition of equality of these moments with respect to a fixed-point K: Mf ¼ R05 KP 0

ð7Þ

From Eq. (5), (6) we get Mf ¼ FH1 e. This moment depends only on FH1 and at FH1 ¼ const has a constant value, ensuring equilibrium at any value of force RH2 .

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The friction moment converts the excess bond at the point into a real bond and provides motion detectability when there is only one input. It can be said that the friction moment performs the power activation of the excess coupling. When the position of the point P is found by Eq. (2) and the initial speed of one initial link is set, for example, VH1 , the linear velocity plan (Fig. 1) will have a unique configuration and will allow to determine all kinematic parameters of the circuit. The condition of equilibrium of external forces on the principle of possible works (or capacities) for the defined kinematic chain has the form MH1 xH1 þ MH2 xH2 ¼ 0:

ð8Þ

This condition shows that the forces on the initial links must have opposite signs. This means that one of the initial links (example, link - H1 ) must have positive power and be input, other (link -H2 ) must have negative power and be output. Thus, the moment of friction in the joint of the satellite converts the excess connection at the point into a real connection and allows for the determination of the movement of the two-movable toothed kinematic chain in the absence of additional support on the post. Equilibrium of double-movable circuit with one input without additional support on post is possible and considered double-movable kinematic circuit is mechanism with one input. From Eq. (8) considering the signs follow xH2 ¼ MH1 xH1 =MH2

ð9Þ

Equation (9) defines a power self-control when constant parameters of input power, the output angular velocity is adapted to the variable output load. The mechanical system in question is a self-regulating gear mechanism.

4 Description of Adaptive Drive of Docking Mechanism In the proposed design, in comparison with the known design (Fig. 2), an adaptive gear variator is introduced into the design scheme of the docking mechanism drive, which significantly simplifies the drive and control means (Fig. 4). The adaptive drive contains an electric motor 1, a stop clutch 2, an adaptive gear variator 3 and an output shaft 4. Adaptive gear variator 3 allows you to eliminate the following nodes, shown in Fig. 2: additional brake motor 1 with stop clutch 2, differential 3, connecting drives from two motors and two planetary gears 4. The structure of the docking mechanism is unchanged.

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Fig. 4. Kinematic diagram of adaptive drive

This allows using the developed algorithms for modeling the dynamics of solid systems described by the given system of equations and transformation matrices [1] in the mathematical description of the docking mechanism.

5 Mathematical Model of Adaptive Drive The dynamics equations given in [1] remain at their core. The parameters of the control elements to be eliminated are excluded from these equations: an electric motor performing the function of an electromagnetic brake, friction couplings and corresponding sensors. The computed parameters of the points in Fig. 5 are listed in Table 1. Table 1. Variator output shaft parameters Point A B C C’ C’’ D E Moment of resistance, Nm 7.5 2.5 15 25 30 1 0 Anular velocity, rad/s 0 36 5.55 3.33 0 83.3 40

Main analysis results:

Fig. 5. The diagram of dependence of the angular velocity on the moment of resistance

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The main difference between the existing and proposed dynamic models is the presence of an additional degree of freedom in the adaptive gear mechanism of the gear variator. At the same time, power activation of excess coupling by use of friction moment of friction introduces real coupling into two-movable kinematic chain, turning it into mechanism adaptive to power load. Thus, in the mathematical model of dynamics [1] it is necessary to add only the mathematical condition of power adaptation in the form of Eq. (9), which binds angular speeds of the drive. Consider the adaptive drive dynamic modes created by the adaptive gear variator according to Eq. (9).

6 Analysis of Drive Modes Main results of analysis of motion modes and comparison of adaptive and existing drive: 1) At the start (section AB) in the adaptive drive, as well as in the existing drive, the characteristic of the engine at the start coincides with the characteristic of the output shaft of the drive, considering the gear ratio equal to 1. Acceleration occurs with an increase in the output angular velocity x and the output starting moment of the resistance M. 2). In the operating mode of variable resistance movement (section BC) in the existing drive, the parameters of the engine x and M are changed using its control system. Signal for change of engine parameters, depending on variable moment of resistance M, is transmitted by control system of coupling mechanism, which contains clutch, electromagnetic and friction brake. Two complex control systems of the docking mechanism and the drive mechanism interact. In the adaptive drive, the engine operates without a control system, the variable resistance moment in section BC is overcome by the adaptive variator autonomously due to self-regulation. At the same time, the adaptive drive can independently switch to the multiplier mode (section BD) with a decrease in the resistance moment, which significantly extends the control range. As a result, an adaptive self-regulating drive operating without a control system will have complete and instantaneous adequacy to a variable moment of resistance and trouble-free operation. In case of emergency overload (point C’), the output shaft of the drive will be stopped when the engine continues to operate, and the mechanism will go into a state with one degree of freedom. In addition, an engine operating without a control system has a higher efficiency, which allows to reduce the weight of the drive, both due to greater efficiency and due to the lack of braking and acceleration means. In an adaptive drive compared to an existing drive, there are the following advantages: 1) One motor is used. The motor performing the function of electromagnetic braking is not required; 2) Complete adaptation, immediate and adequate response to load change without control and adjustment system. Lack of emergency situations; 3) Elimination of engine control system; 4) Increased reliability and quality of drive operation; 5) Simplified design; 6) Reduced dimensions and weight of the drive. The developed adaptive drive of the docking device is a fundamentally new selfregulating transmission mechanism. This mechanism connects the spacecraft to the

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space station under conditions of continuous and spontaneous changes in the forces and positions of the interacting working bodies. The adaptive action of the mechanism is based on the use of a kinematic chain with two degrees of freedom. According to the laws of mechanics, such a circuit in the presence of only one input (motor) is an indeterminate mechanical system. However, the performed force analysis allowed us to find a way to achieve the definiteness of motion by using the calculated moment of friction (formula 6), which creates an additional mobile connection while maintaining two degrees of freedom. In space conditions, the friction coupling is quite reliable (compared, for example, with hydraulic coupling). The resulting adaptive transfer mechanism has a connection between the input and output power parameters (8). Formula (9) defines the possibility of instantaneous reaction of the mechanism to any change in the moment of resistance MH2 at constant input power of the motor MH1 xH1 – change in the moment of resistance MH2 leads to a change in the angular velocity of the output shaft xH2 . Thus, the transmission mechanism provides complete adaptation, instantaneous and adequate response to changes in the load without a control system and adjustments. In addition, the presence of two degrees of freedom makes it possible to completely stop the output shaft under the action of the maximum moment of resistance (Fig. 5). This means that there are no accidents in case of jamming of the interacting elements. In this case, the elimination of jamming is possible by reversing. The adaptive transmission mechanism operates without a complex gear ratio control system, which provides the implementation of discrete signals that create the required gear ratio. Each signal corresponds to a parameter sensor (force or position) and is converted into an executive program action. The result of the action should be checked by feedback and corrected. A large number of sensors and controls reduces the accuracy and reliability of the control. Duplicating the controls to ensure reliability further complicates the controlled transfer mechanism. In contrast to the controlled mechanism, the adaptive transmission mechanism, which operates without a control system, has a simple design and simple operation, which increases the reliability and quality of work (accuracy and adequacy).

7 Conclusion An adaptive drive with two degrees of freedom has been studied from aspects of mechanical design and kinematic modeling. Design of the adaptive drive is shown in kinematic scheme. The improved mathematical model with an adaptive drive reduces loads in the entire range of initial docking conditions and makes it possible to realize the given external design limitations and the possibility of docking to units with different shape of the receiving cone due to insignificant narrowing of the set of permissible combinations of parameters from this range [3, 4]. Design Changes: 1) A scheme for driving a coupling mechanism with an adaptive drive based on design requirements for the drive has been developed. The drive is significantly simplified. Brake engine, extra degree of freedom, differential mechanism,

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planetary mechanisms (2) and control system changing gear ratio to output shaft are removed. 2) The main design parameters of the adaptive drive (gearing modulus, number of teeth, range of gear ratio adjustment, etc.) are calculated. 3) Dynamic analysis of driving modes is performed. 4) The analysis confirmed the advantages of the designed drive design.

References 1. Golubev, Yu.F., Yaskevich, A.V.: Computer simulation of the dynamics of central-type docking mechanisms for spacecraft. Preprints named after Keldysh M.V., vol. 89, pp. 40. Moscow (2019) 2. Syromyatnikov, V.S.: Docking devices of spacecraft. Engineering, 216 p. Moscow (1984) 3. Ivanov, K., Ceccarelli, M., Yestemessova, G., Nurgizat, Y., Balbayev, G.: Design and characterization of a gearbox joint for manipulators. In: IFToMM World Congress on Mechanism and Machine Science. Advances in Mechanism and Machine Science, pp. 2261– 2270. Springer, Nature Switzerland (2019) 4. Ivanov, K., Ceccarelli, M., Ozhiken, A., Cafolla, D., Gonzalez-Cruz, C.: Design and experiences of a planetary gear box for adaptive drives. In: Transactions of EUCOMES 2018, pp. 284–291. Springer (2018) 5. Balbayev, G., Ceccarelli, M., Carbone, G.: Design and numerical characterization of a new planetary transmission. Int. J. Innov. Technol. Res. 2(1), 735–739 (2014) 6. Carbone, G., Mangialardi, L., Bonsen, B., Tursi, C., Veenhuizen, P.A.: CVT dynamics, theory and experiments. Mech. Mach. Theor. 2(1), 409–428 (2007) 7. Kaharman, A., Ding, H.: A methodology to predict surface wear of planetary gears under dynamic conditions. Mech. Based Des. Struct. Mach. 493–515 (2010) 8. Thirumurugan, R., Muthuveerappan, G.: Critical loading points for maximum fillet and contact stresses in normal and high contact ratio spur gears based on load sharing ratio. Mech. Based Des. Struct. Mach. 39(1), 118–141 (2011) 9. Muni, D.V., Muthuveerappan, G.A.: Comprehensive study on the symmetric internal spur gear drives through direct and conventional gear design. Mech. Based Des. Struct. Mach. 37 (4), 431–461 (2009) 10. Fetvaci, C.: Definition of involute spur gear profiles generated by gear-type sharper cutters. Mech. Based Des. Struct. Mach. 38(4), 481–492 (2010)

Driving Mechanism in Robotized Hospital Bed for Patients with COVID 19 Jorge Enrique Araque Isidro and Marco Ceccarelli(&) Laboratory of Robot Mechatronics, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy [email protected], [email protected]

Abstract. This paper presents a mechanism solution for a robotized bed with movable segments also for lateral movements of patients, the feasibility of the proposed mechanism solution is discussed by using simulation results for a typical operation. The main focus is addressed on mechanical aspects that nevertheless are affected by control actions and sensor feedbacks. Keywords: Robotics


Hospital beds
Mechanisms
Design

1 Introduction Due to the COVID 19 pandemic and all the new challenges that have emerged to combat this virus and thus to prevent the number of patient deaths from increasing the way it has, they have been presented different solutions to help sick patients, taking into account necessary biosecurity measures and to protect health professionals. Among these solutions proposals there is the automation of the most used elements for treatments, as well as robotic devices. In this paper the attention is addressed to hospital beds allowing a patient in it to perform movements that will help to limit consequences of the virus effects. A design of mechanisms is proposed that allows a patient in movements such as turning from left to right. COVID 19 is a virus that has affected the whole world, causing people to have serious respiratory problems that can lead to death. These patients are taken to hospitals to carry out necessary treatments even in Intensive Care Unit (ICU) [1]. Solutions for patient treatments have different objectives such as they try to find a way that a hospital bed can generate movements for certain postures. The bed must help a patient for adapting better position to treatments and helping the doctors or nurses to make less physical effort. This can be the case of even adopting in a bed a sensor that allows knowing other types of patient parameters such as temperature and the position in which they are [2]. Hospital bed designs present solutions with mechanical structures that very limitedly are obtained using a synthesis of mechanisms. Some solutions try to solve certain movements without the need for a large array of sensors, [3, 4]. This design approach

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 179–186, 2021. https://doi.org/10.1007/978-3-030-75271-2_19

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can be directed to low-cost solutions with a proper design with actuators with specifications that are easy to find within market product. Recently other investigations have been carried out with the aim of finding solutions for the transport of patients within the hospital, without having the need for them to get out of bed. Different types of systems with wheeled designs and navigation systems have been proposed to get such a bed mobility and handling that help the safe transport of patients. As reported for example in [5, 6]. But, according to the authors’ knowledge, no particular attention has been addresses to redesign the hospital bed to enhance mobility and comfort of imbedded patients as for the COVID19 treatments. In a previous work [1] the design of a bed with segments was presented that allows patients in hospitals to have better mobility when in bed. The aim is to make easy treatments to which they are subjected and more comfortable the hospitalization stays in the COVID19 circumstances. It has been experienced that lateral movements of patients during clinical treatments can be very helpful and a segmentation of hospital beds can be a feasible solution within the current frames and procedures during COVID19 treatments. A discussion is left about the type of mechanisms that can be used to fulfill this purpose, taking into account that they must work under some restrictions within the design of the bed, among other sanitarian aspects. Research that helps to automated hospital beds can be focused on improving the quality of life of patients during their stay. They also help health personnel avoiding direct contact with patients who have a contagious disease. This paper shows a design of a driving mechanism in a segmented bed that allows the patient’s trunk to move from right to left in the same way as the lower limbs. A series of considerations are also presented that serve to carry out the analysis of data, which will help to consider the control that must be applied for a good performance of the robotic hospital bed. The operation characterization is analyzed through a simulation to evaluate the performance and effect of a driving mechanism in moving properly the bed segments. The simulation of the mechanism gives numerical values of parameters that can be used for an improved design of motion-controlled bed segments. The results obtained from the simulation of the mechanism are discussed also to better characterize the original conceptual design as presented in [1].

2 Problem and Requirements Patient management in UCI poses a series of challenges for health personnel. Those are patients, who may suffer consequences for being in bed a long time without movement. Health personnel have additional risk in case the patient suffers from a disease caused by viruses as COVID 19 with difficulties in the treatments when having beds that do not help or help little to facilitate these tasks. The principal aim of the ongoing work related to this paper is on searching robotized hospital bed solutions for better patient life quality according the main aspects that are summarized in Fig. 1.

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Fig. 1. A block diagram for problems and requirements in COVID19 patient treatments

A robotized hospital bed can be proposed with the bed segments to the movements that can help the imbedded patient status, [1]. With such a robotized bed patient can move with lateral movements with a certain degree of inclination as shown in Fig. 2. In addition, a robotized hospital bed can be provided of components to help the motion development, like motion sensors, pressure sensor and force sensor, even for a motion control within hospital bed structure in suitable control box.

Fig. 2. A conceptual design of a robotized bed with movable segments [1].

Pressure sensors can be used for knowing the patient situation with the body and the segmented bed in contact. This can be important to take care the physiological disease occurring by long time in same position, such sores and muscle degradation.

3 The Proposed Driving Mechanism A kinematic diagram of the proposed mechanism is shown in Fig. 3 with the parameters to understand the necessary requirements for the specific application in a hospital bed. The main design parameters in the mechanism kinematic can be recognized as in

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Fig. 3 as S: Linear actuator stroke, FC: Linear actuator force, FL: joint reaction force with X and Y components, TR: reaction joint torque with X and Y components, M: Patient mass.

Fig. 3. A kinematic scheme of the bed driving mechanism with parameters

The proposed bed driving mechanism is actuated by a linear actuator that do not require much effort to move. The mechanism is designed in a CAD model with colors for a better understanding of the explanation in Fig. 4. The parts of the frame are in red and blue, and they are designed as metal sheets. The blue piece has holes that allow the mechanism to be adjusted to a bed by means of screws, and the base allows to hold a mobile link is in green. The bed segment frame is represented as the yellow link, The linear actuator can be electrical, as functional for control facilities and convenient power supply M.

Fig. 4. A CAD design of the proposed driving mechanism in Fig. 3

The design purpose of this mechanism is to achieve that with a short displacement and an acceptable force a linear actuator can move the segment of the robotic hospital bed taking into account the weight of the patient. In this case, the calculation of speed

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and acceleration is not so relevant, because they must be small to avoid traumatic actions for the patient that may affect their health.

4 Performance Analysis The CAD design Fig. 4 is used to size and check the design performance of the proposed driving mechanism. The movement of the mechanism with the segment in Fig. 5 shows the expected performance for hospital beds, in terms of smooth motion and proper inclination.

Fig. 5. A snapshot of the simulated operation

In the graphical position Fig. 6 of the linear actuator start point doesn’t begin in 0 mm, it has the start point of 46.06 mm, this is because of the reference point in the construction of the mechanism. This can be seen in the mechanism kinematic scheme exactly in point B and the distance with the base support in Fig. 3. In 3 s of the actuator displacement the driving mechanism shows a maximum linear actuator position of 83.56 mm. This difference of 37.5 mm, with the beginning gives the actuator stroke to choose a correct actuator. The plots in Fig. 7 show the velocity and acceleration that the linear actuator as referring to point B in Fig. 3. The velocity increases but not linearly all the time. It increases linearly when the acceleration is constant, in this case when the acceleration is 10 mm/s2, increasing 5 mm/s each 0.5s. When the acceleration isn’t constant increasing 2.5 mm/s. The acceleration behavior is shown linear in a short time from 0 mm/s2 until 10 mm/s2, remains constant in the largest time of driving mechanism action at 10 mm/s2 and later decreases until 0 mm/s2 again.

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Fig. 6. Computed stroke displacement of the driving linear actuator for the simulation in Fig. 5.

Fig. 7. Computed velocity and acceleration of the driving actuator for the simulation in Fig. 5

The actuation force acting at point B is displayed in Fig. 8. At the beginning, the force required is major but later this force is less for the reaction of the segment and mechanism. The maximum force value is almost 19 N. The plots in Fig. 9 show the components of reaction force at point E of Fig. 3 as union between the mechanism and the segment. The plots in Fig. 10 show the torque in point A as to be considered for the hospital bed structure analysis.

Driving Mechanism in Robotized Hospital Bed

Fig. 8. Computed simulation result for actuation force of the driving linear actuator

Fig. 9. Computed simulation result for joint reaction force components at point E

Fig. 10. Computed simulation result for joint reaction torque components at point A

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5 Conclusions A design of a robotized hospital bed is presented with a solution of driving mechanism to ensure fairly simple robust design that can be used for COVID 19 patient. The feasibility of the proposed design is evaluated by a simulation whose results give parameter values for an efficient operation for segment motions with a maximum acceleration of 10 mm/s2 for each patient body segments. The linear actuator displacement is computed for a stroke of 37.5 mm and an actuation force of 19 N, when considering a load of 100 N on a simulated bed segment. The simulation results give indication for design parameters of a prototype of the hospital bed to be built, for future testing and practice. Future work is planned for optimizing the mechanism as function of a closed-loop control of the driving actuator.

References 1. Ceccarelli, M., Sarmati, L., Ambrogi, V.: A robotized hospital bed for COVID-19 patients in intensive care treatments. In: 2020 IEEE IFAC International Conference on Automation, Teleconference, Paper 200 (in print) 2. Vazquez-Santacruz, E., Gamboa-Zuniga, M.: An intelligent device for assistance in caring for the health of patients with motor disabilities. In: Proceedings of the International Conference on Natural Computing, Mexico, vol. 2016, pp. 868–874 (2016) 3. Pavlovi, N.D.: Development of mechanisms for adjusting positions of a multifunctional bed. In: 13th World Congress in Mechanism and Machine Science, Guanajuato pp. 1–6 (2011) 4. Ching-Hua, W., et al.: Hospital bed with auxiliary functions of lateral positioning and transferring for immobilized patients. In: IECON Proceedings of the Industrial Electronic Conference, Taipei, pp. 2991–2995 (2007) 5. Wang, C., Savkin, A.V., Clout, R., Nguyen, H.T.: An intelligent robotic hospital bed for safe transportation of critical neurosurgery patients along crowded hospital corridors. IEEE Trans. Neural Syst. Rehabil. Eng. 23(5), 744–754 (2015) 6. Guo, Z., Xiao, X., Yu, H.: Design and evaluation of a motorized robotic bed mover with omnidirectional mobility for patient transportation. IEEE. J. Biomed. Heal. Inform. 22(6), 1775–1785 (2018)

Synthesis of a Function Generator Six-Bar Linkage in Two Steps with Genetic Algorithm Claudio Villegas1,2(&), Mathias Hüsing1, and Burkhard Corves1 1

RWTH Aachen University, Aachen, Germany [email protected] 2 Universidad del Bio-Bio, Bio-Bio, Chile

Abstract. Much research have been carried out to synthesize linkages, yet a few is concentrated on the dynamic behavior of six bar linkages in the early stages of the design process. Therefore, we present a two-step synthesis method for function generator six-bar linkages that includes constraints related to the dynamic behavior of the mechanism. The first stage focuses on matching eight given precision positions and the second stage on minimizing the reaction forces. Both minimization problems are solved with a genetic algorithm in MATLAB™. The results are compared with a multi-body simulation in MSC ADAMS™. Keywords: Error minimization
Force minimization
Structural constraints Optimization of mechanisms




1 Introduction Industrial processes require frequently non-linear movement, which can be provided by mechanisms [1] as six-bar linkages, among others. This kind of movement generates nonlinear forces and vibration with high harmonic components that could excite the natural vibration modes of the structure [2], having great impact on the dynamic behavior of the mechanism. Therefore, taking in account the dynamic behavior of mechanisms is essential in the design process of linkages. When not included in the first stage of the process, it could lead to undesirable recalculation while designing linkages [3]. Traditionally, the design of mechanisms is developed in separate stages that tackle the kinematics, the dynamics and other aspects. The kinematic synthesis of six-bar linkages given the positions of the input link to match a certain amount of angular positions or/and angular velocities of the output link has been studied in [4–7] as a kinematic problem. Other studies have been carried out on the topics of kinetostatic analysis of mechanisms [8], energy efficiency [1] and dynamic analysis of six-bar linkages [9]; however, none of them include an indicator of dynamic behavior in the synthesis stage. The design of a mechanism requires to fulfill not only prescribed positions but other characteristics as low reaction forces and vibration levels. To synthesize linkages, many optimization approaches have been applied and several of them use single-objective methods as in [10]. When more than one objective need to be considered in the design © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 187–194, 2021. https://doi.org/10.1007/978-3-030-75271-2_20

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process, multi-objective optimization methods are employed [11] as well as multi-level decomposition [12]. In [13], a six bar linkage was synthesized using a non-dominated sorting GA with four kinematic objective functions. In [12], a general multidisciplinary optimization procedure to synthesize mechanisms is presented. Using commercial multibody simulation (MBS) and finite elements analysis (FEA) software, they synthesize mechanisms separating the problems in two levels, the mechanism level and the structural level. This work is the only one we found that constraints structural-related variables as stress or eigenfrequency for six-bar linkages. However, due to the use of commercial MBS an FEA software, only 10 configurations per cycle are analyzed. This kind of time-domain discretization could hide big peaks of force or stress in the analysis. Using a two-step decomposition, we synthesize a Watt-II six-bar linkage with MATLAB™. Its crank has full-rotation capability. Its output link must pass over eight precision positions with minimized force. To achieve this condition, two objective functions are used; for the first step, the matching error and for the second, the maximum kinetostatic force on the joints. As constraints of the first step problem, a minimum transmission angle is set for both loops of the six-bar linkage, the Grashof condition is imposed. Moreover, 180 configurations are analyzed in each cycle to map the position of the rocker. In the second step, the stresses and the deflections are calculated with kinetostatics and used as constraints. Moreover, and also as a constraint in the second step, the first minimum eigenfrequency is calculated. Furthermore, the same 180 configurations are evaluated in the second step.

2 Methods Figure 1 shows the Watt-II six-bar linkage synthesized in this study for eight precision points. Each precision point is defined by an ordered pair ðpi ; qi Þ. Here, pi is the angular position of the input link for the i-th precision point starting from an arbitrary angular position us and qi is the angular position of the output link for the same position starting from an arbitrary angular position ws . The angular positions us and ws are to be calculated. u. represents the input link angular position measured clockwise, starting horizontally and pointing to the left of the Fig. 1. wðuÞ is the angular position of the output link when the input link is on the position u. The output angle is measured as in the same way as u. ðui ; wi Þ is the ordered pair corresponding to ðpi ; qi Þ measured as u and w, respectively. Each link length of this mechanism is denoted by lj ðj ¼ 1; 2; 3; 4; 5; 6; 7; 8Þ. Links 1 and 5 correspond to the fixed links. Link 2 is the input link and rotates with constant angular speed X. Links 4 and 6 are fixed to each other on point B0 , pinned to the ground and their longitudinal axes have an angular difference of b0 . Link 8 is the output link and swings an external body. This body has constant inertia and is attached to the link on point D0 .

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Fig. 1. Synthesized six-bar linkage

minXL Err ð X Þ subject to

ð1Þ

l1  lmin;1 \0

ð1:aÞ

l2  lmin;2 \0

ð1:bÞ

Gðl1 ; l2 ; l3 ; l4 Þ\0

ð1:cÞ

imagðwÞ ¼ 0

ð1:dÞ

minXS FM ð X Þ subject to

ð2Þ

SM  Sf \0

ð2:aÞ

fm  fl \0

ð2:bÞ

Rd;M  1\0

ð2:cÞ

The mechanism described in the previous paragraph is synthesized by means of a constrained two-step optimization (Eqs. 1 and 2). Equations in the group 1 correspond to the linkage level that optimizes the precision of the mechanism. Equations in the group 2 correspond to the structural level that optimizes the maximum reaction force on the joints and constrains the stresses, first eigenfrequency and deflection of the structure. This problem has two objective functions. One of them is the error in the position of the rocker with respect to the precision points Err (Eq. 3) and the other one is the maximum reaction force on the joints of the linkage FM , which is calculated with kinetostatics. Err ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðwi  wðui ÞÞ2

ð3Þ

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For the first step, the independent variables of the optimization are grouped in XL (Eq. 4), with rlj defined as the ratio between the links in one loop of the six-bar mechanism and the frame of that loop (Eqs. 5 and 6). XL ¼ ½rl2 ; rl3 ; rl4 ; rl6 ; rl7 ; rl8 ; us ; b0 ; ws T

ð4Þ

rlj ¼

lj ; j ¼ 2; 3; 4 l1

ð5Þ

rlj ¼

lj ; j ¼ 6; 7; 8 l5

ð6Þ

For the second step, the independent variables are included in XF , where cs;j is the cross-sectional area of the j-th link, ksh is the stiffness of the shaft that connects the input link with a motor and k46 is the stiffness added to the union of the links 4 and 6. The shape of the cross-section is square to reduce the number of variables in the problem, thus cs;j is replaced by the edge length of the cross section by the edge length of the cross section es;j . Moreover, the stiffness ksh and k46 are only used to calculate the constraint of inequality 2.b. Furthermore, rlf is defined as the aspect of the first loop and varies between in the open interval [0,1]. Thus, the links length of the first loop are calculated as in Eq. 9 and that of the second loop, as in Eq. 10, where lf is the total length of the frame.  T XF ¼ cs;3 ; cs;4 ; cs;6 ; cs;7 ; cs;8 ; ksh ; k46 ; rlf rlf ¼

l1 lf

ð7Þ ð8Þ

lj ¼ rlj rlf lf ; j ¼ 2; 3; 4

ð9Þ

  lj ¼ rlj 1  rlf lf ; j ¼ 6; 7; 8

ð10Þ

In the constraints of the first step, the maximum deviation of the transmission angle between links 3–4 (l1 Þ and between 7–8 (l2 ) are less than the limits lmin;1 and lmin;2 , respectively (in Eqs. 1.a and 1.b). These two deviation limits are chosen by the designer. On the same step, the Grashof condition for the first loop Gðl1 ; l2 ; l3 ; l4 Þ (in Eq. 1.c), and the rotatability of the crank (Eq. 1.d) are calculated. In the constraints of the second step, the maximum kinetostatic stress on the links SM , the maximum allowable fatigue strength of the material Sf , the minimum eigenfrequency of the mechanism fm , the minimum allowable first eigenfrequency fl , an indicator of the maximum deflection on the links Rd;M , and the length of the frame lframe are shown (in Eqs. 2.a–2.c). To fulfill in Eq. 2.b, fl is calculated in every iteration as in [3]. Rd;M is obtained with the Eq. 11.

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Rd;M

 dj DL;j ¼ max ; j ¼ 2; 4; 6; 8 lj

ð11Þ

Ftj l3j 3EIj

ð12Þ

dj ¼

where dj is calculated as in Eq. 12, Ftj is the transverse force on the moving end of the j-th link, Ij is the area moment of the link, E the elasticity modulus of the material and DL;j is the maximum allowable ratio between dj and lj , which is selected by the designer. The minimization problem consist of inner and outer problems. The former ones are the minimization and maximization of the variables with subindex m and M, respectively. The latter ones are the functions Err and FM . On the one hand, the inner problems are solved with the evaluation of them in one rotation of the crank. For this evaluation, 180 points are used as a mapping. Then, a local SQP optimization [3] starts at the best value found in the mapping process. On the other hand, the outer problems are solved with a genetic algorithm (GA), which is contained into the GA function in MATLAB™. First, the problem in equations group 1 is solved eight precision points, and then, this results are used as constant values to solve the optimization in equations group 2. The resulting mechanism is analyzed with a flexible multibody simulation in MSC ADAMS™ and its forces on the joints, stresses on the links, output angle deviation and vibration levels of the links are shown.

3 Results and Discussion The output link of the mechanism describes a sinusoid function in 60° as the input link rotates 160° at constant 120 rpm. For this task, eight precision points are chosen equally spaced. The total length of the frame is 400 mm. All links have the same material properties, that is E = 200 GPa, Sf = 170 MPa, and density of 7800 kg/m3. The values of Dj are chosen as in [3] and shown in Table 1. The mechanism moves an external body with an inertia of 1e4 kg–mm2. The deviation of the transmission angle from 90° for the first loop of the mechanism is constrained arbitrarily to be less than 80°. Similarly, for the second loop, 85°.

Table 1. Deflection ratio Dj used in this work Link 2 4 6 8 Value 120 160 120 160

Table 2 shows the link lengths and the edge length of the links’ cross sections lj . On the table, the low values of the edge length of links 3 and 4 stand out. This occurs due to the lack of a constraint that limits the buckling effect in the optimization. The

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Fig. 2. Function described by the output link of the mechanism

resulting values of the constructive angles are us ¼ 359:66 , b0 ¼ 285:24 and ws ¼ 273:49 . Moreover, the stiffness of the shaft obtained is 238200N  m and the torsional stiffness added to joint B0 is k46 ¼ 2:8  106 N  m. This mechanism obtained matches the given task with an error of 0.63°. Figure 2 shows the required task positions and the function generated by the output link. The maximum force on the joints and the maximum stress on the links are shown in Table 3. The first row contains the results of the optimization and the second row, that of the MBS. The maximum vibration level is found on link 6 and its value is 0.17 mm/s. The maximum deviation of the output angle in the flexible model with respect to a rigid model is 0.09°. All the calculations in the optimization process were made with a Ryzen 7 4800H processor and the time required for them was around 18 h.

Table 3. Comparison of maximum forces and stresses between our method and a MBS Method FM N SM MPa Optimization 56.49 34.28 MBS 50.09 47.43

4 Conclusions Our results show that a function generator six-bar linkage can be synthesized by a twostep optimization. This method minimizes the error of the output link position and the kinetostatic force of the mechanism. The constraints of the method ensure that the

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mechanism operates within a safe range of stress and with low levels of vibration. It is notorious, that the cross sections of links 3 and 7 are small and should be increased because of the buckling risk. These low values can be explained by the way the model was made. In the model, every linkage transmits the force on the plane of the mechanism. Therefore, no eccentric forces are modeled. This fact should be included in further improvements of this method, in order to quantify eccentric forces and buckling. Another interesting point to be taken in account is de difference between the stress calculated in the optimization and that of the MBS. This could be explained by the calculation method of the internal forces in our model. In the optimization it is assumed that the inertia forces act at the same time as the reaction forces and this does not occur in a MBS. The MBS considers the phase of each force in the simulation. As future work is proposed to solve this problem with a non-dominated sorting genetic algorithm with inclusion of buckling effect in the model.

References 1. Schwarzfischer, F., Hüsing, M., Corves, B.: The Dynamic Synthesis of an energy-efficient Watt-II-mechanism. In: Carvalho, J.C.M., Martins, D., Simoni, R., et al. (eds.) Multibody Mechatronic Systems, vol. 54, pp. 213–222. Springer, Cham (2018) 2. Yuan, L., Rastegar, J.S.: Kinematics synthesis of linkage mechanisms with cam integrated joints for controlled harmonic content of the output motion. J. Mech. Des. 126, 135–142 (2004). https://doi.org/10.1115/1.1637646 3. Villegas, C., Hüsing, M., Corves, B.: Synthesis of function generator four-bar linkages: minimization of the joint-forces constraining structural-related quantities. In: Pisla, D., Corves, B., Vaida, C. (eds.) New Trends in Mechanism and Machine Science, vol. 89, pp. 314–321. Springer, Cham (2020) 4. Plecnik, M.M., McCarthy, J.M.: Synthesis of a Stephenson II function generator for eight precision positions. In: Volume 6A: 37th Mechanisms and Robotics Conference. American Society of Mechanical Engineers (2013) 5. Gezgin, E., Chang, P.-H., Akhan, A.F.: Synthesis of a Watt II six-bar linkage in the design of a hand rehabilitation robot. Mech. Mach. Theor. 104, 177–189 (2016). https://doi.org/10. 1016/j.mechmachtheory.2016.05.023 6. Chung, W.Y., Chiang, C.: Fourth-order synthesis of Watt-II six-bar function generators. Mech. Machine Theor. 25, 417–426 (1990). https://doi.org/10.1016/0094-114X(90)90078-X 7. Plecnik, M.M., McCarthy, J.M.: Kinematic synthesis of Stephenson III six-bar function generators. Mech. Mach. Theor. 97, 112–126 (2016). https://doi.org/10.1016/j. mechmachtheory.2015.10.004 8. Jomartov, A., Tuleshov, A.: Vector method for kinetostatic analysis of planar linkages. J. Braz. Soc. Mech. Sci. Eng. 40. https://doi.org/10.1007/s40430-018-1022-y 9. Mukasheva, A., Japayev, S., Abdraimova, G., et al.: A dynamic analysis of six-bar mechanical press. Vibroengineering Procedia 13, 249–254 (2017). https://doi.org/10.21595/ vp.2017.18853 10. Khorshidi, M., Soheilypour, M., Peyro, M., et al.: Optimal design of four-bar mechanisms using a hybrid multi-objective GA with adaptive local search. Mech. Mach. Theor. 46, 1453–1465 (2011). https://doi.org/10.1016/j.mechmachtheory.2011.05.006

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11. McDougall, R., Nokleby, S.: Grashof mechanism synthesis using multi-objective parallel asynchronous particle swarm optimization. In: Proceedings of the Canadian Society for Mechanical Engineering Forum 2010, Victoria, British Columbia, Canada (2010) 12. Chen, T.-Y., Yang, C.-M.: Multidisciplinary design optimization of mechanisms. Adv. Eng. Softw. 36, 301–311 (2005). https://doi.org/10.1016/j.advengsoft.2004.10.013 13. Zhai, P., Wang, P., Li, R., et al.: Multi-objective optimization of six-bar mechanisms using NSGA-II. In: 2012 IEEE International Conference on Computer Science and Automation Engineering (CSAE). IEEE, pp. 494–497 (2012)

Design and Analysis of a Device for Enlarging the Allowable Position Error for Screwing Task Hao-Tien Ku and Yu-Hsun Chen(&) Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan {m10603410,yhchen}@mail.ntust.edu.tw

Abstract. To ensure repeatability in a screwing task, the robot manipulators have to be calibrated regularly. If the allowable position error of the manipulator is increased, the frequency of stopping a production line and calibrating the manipulators can be lower. This paper presents an innovative device with the functions of hole detecting, screw positioning, and screw locking. The mechanism design concepts with a leading-screw, a six-bar linkage, and a clutch with a screwdriver are proposed. By scanning an area near the screw hole, the bolt can be precisely positioned at the location of the screw hole detected by a plunger. The motion analysis is carried out through the vector loop method and a computer-aid simulation. Keywords: Screw positioning design


Manipulator screwing task
Mechanism

1 Introduction Robot manipulators are widely used for screw mounting tasks to improve productivity and quality stability in the industry in recent years. However, errors inevitably occur during robot arm movement creating a distance between the actual arrival point of the manipulator and the target point. Transmission error accumulates as the robot arm executes a task repeatedly. In order to solve this problem and improve the motion repeatability of the robot arm, many calibration methods for robot manipulators were developed [1–10]. Although there are existing methods to calibrate the repeatability of a manipulator, the production line has to be suspended for correction, which inevitably reduces the production line capacity and causes additional costs. Another point of view of solving this problem is by adding adaptive mechanisms/devices on the end effector to adjust the error, Lee [11] proposed a new type of variable remote center compliance (VRCC) that can easily change the position of the compliance center, and lock itself according to the working environment. Lai and Zhu [12] proposed an analytical model for a class of RCC mechanisms in order to evaluate the compliance properties and rotational precision of the mechanism. Kim et al. [13] proposed a robotic assembly integrated with a

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 195–203, 2021. https://doi.org/10.1007/978-3-030-75271-2_21

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displacement sensor on an RCC device to detect the three translation and three rotation components generated by external forces and torques applied to the device. In this study, an innovative mechanical device for enlarging the allowable position error is proposed. Even if there is a distance between the arm’s actual arrival point and the target point, it can still complete the task of screwing. In the following sections, the applied mechanisms are introduced, and the performance analysis is made to verify the feasibility.

2 Design Concepts Different from most viewpoints that improving arm precision and calibrating, the solution proposed in this study is a device that enlarges the allowed tolerance amount of error of the robot arm. In order to perform the functions of hole detecting, screw positioning, and screw locking, the design requirements are listed as following: 1. A detector is driven to go around an area near the screw hole. 2. Once the detector reaches the hole, it is moved away, and a bolt is moved to that position where the hole is located. 3. A screwdriver is moved to touch the bolt to screw it inside the workpieces. 4. All of the link in the device return to the initial position. The input motion of the device is provided by the robot arm when it presses downward. As shown in Fig. 1(a) and (b), the device is attached to the flange of robot arm through a lead screw. For the 1st design requirement, a detector mechanism is composed of the leading screw, the casing of the device, and a plunger at the bottom. When the arm goes downward, the device with the plunger rotates on the axis of the lead screw. There is a distance s between the central lines of the leading screw and the plunger, so the plunger moves along a circle with radius s to searching the screw hole. The original area of allowed position error e is shown in Fig. 2(a), and it can be 1 mm with a chamfer for an M6 screw, for example. By applying the proposed device, as shown in Fig. 2(b), a circle area with radius s is generated as the allowed position error. When the offset s is bigger than the error tolerance e, the allowed position error is enlarged. For instance, if s is set as 3 mm, Emax is 4 mm and Emin is 2 mm, where Emax means the vector from the central lines of the leading screw to that of the plunger is parallel to the original error. Emin means the case when the two vectors are in opposite direction. For the 2nd design requirement, a bolt-positioning linkage is presented, including a trigger mechanism nearby the plunger, a socket that holds the bolt, and several transmitting links, as shown in Fig. 1(b). When the plunger reaches the hole, it extends by a small amount, and this small extension is used as a signal that acts the movement of the screw-positing linkage, as shown in Fig. 3. The trigger link is connected to the frame with a revolute joint (point OT ). At the initial position, Fig. 3(a), the three points are almost collinear, including the joint between the spring and the trigger link (T), the pivot of the link (OT ), and the pivot of the spring (OS ). Therefore, the tensioning spring has its nearly largest elastic potential pulling the trigger link clockwise at the initial position. When the plunger moves downward slightly, the trigger link is pulled by the spring to reach its final position quickly.

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Fig. 1. Design concept

Fig. 2. Position error of the screwing task

Once the trigger mechanism is acted, the bolt-positioning linkage makes the plunger move upward and the socket with a bolt move downward to the hole. Figure 4 shows the positioning mechanism with six links and seven revolute joints. The input link is the trigger link driven by the spring, and the output is the link with socket and bolt. If the final position of the bolt is the initial position of the plunger, the bolt can be positioned at the location of the screw hole detected by the plunger.

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Fig. 3. The plunger and trigger mechanism

Fig. 4. The screw-positioning mechanism

For the 3rd design requirement, a screwdriver is equipped under the leading screw with a clutch nearby, as shown in Fig. 1(b). The clutch is originally engaged the leading screw and the casing of the device. When the trigger acts, the clutch disengages and switches to transmit the rotation of leading screw to the screwdriver. With the robot arm move downward, the screwdriver contacts the bolt and screw it inside the hole.

3 Motion Analysis In order to check the motion of the bolt-positioning mechanism can achieve the expected function, the vector loop method [14] is used, as shown in Fig. 5. The mechanism is divided into two independent loops to determine the movement of the plunger and the socket with a bolt. The extreme points of the plunger and socket are labeled with point P and Q, respectively. Therefore, the equations representing each loop are expressed as follow. The parameters ri ði ¼ 1  8Þ is the magnitude of the vector ri , and hi ði ¼ 1  8Þ is defined as the angle between vector ri and x-axis, as shown in Fig. 5(b).

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Fig. 5. Schematic diagram of the mechanism

r1 þ r2 ¼ r3 þ r4

ð1Þ

r5 þ r6 ¼ r7 þ r8

ð2Þ

For the first loop, Eq. (1) can be extended as: r1 cosh1 þ r3 cosh3 þ r4 cosh4 ¼ r2 cosh2

ð3Þ

r1 sinh1 þ r3 sinh3 þ r4 sinh4 ¼ r2 sinh2

ð4Þ

The input and output parameters are h1 and h3 , respectively. Since the length of the links ri ðiÞ = 1−4 and the fixed angle h4 are given, the angular position of the output link h3 can be determined. h3 ¼ 2tan

1

B 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! B2  C 2 þ A2 CA

ð5Þ

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where A ¼ 2r1 r3 cosh1 þ 2r3 r4 cosh4 B ¼ 2r1 r3 sinh1 þ 2r3 r4 sinh4 C ¼ r12  r22 þ r32 þ r42  2r1 r4 cosðh4  h1 Þ Therefore, the initial position of the plunger P can be expressed as: P ¼ ðPx ; Py Þ ¼ ðMx þ r9 cos h9 ; My þ r9 sin h9 Þ

ð6Þ

where M (Mx ; My ) is a fixed joint at the casing of the device, and the angular position h9 can be expressed by h3 : h9 ¼ h3 þ a For the second loop, the input and output parameters are h5 and h7 , respectively. The length of the links ri ðiÞ = 5−8 and the fixed angle h8 are given. Similar, the angular position of the output link h7 can be determined through Eq. (2): h7 ¼ 2 tan

1

E 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! E 2  F 2 þ D2 FD

ð7Þ

where D ¼ 2r7 r8 cos h8  2r5 r7 cos h5 E ¼ 2r7 r8 sin h8  2r5 r7 sin h5 F ¼ r52 þ r72 þ r82  r62  2r5 r8 cosðh8  h5 Þ h5 ¼ h1 þ b Moreover, the output of the whole mechanism is the position of the socket with bolt, represented by the point Q and the vector r10 . The final position of the socket Q can be expressed as Q ¼ ðQx ; Qy Þ ¼ ðNx þ r10 cos h10 ; Ny þ r10 sin h10 Þ

ð8Þ

where N (Nx ; Ny ) is a fixed joint at the casing of the device, and the angular position h10 can be expressed by h7 : h10 ¼ h7 þ c

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Table 1. Dimension synthesis Vector no. Magnitude (mm) Vector no. Magnitude (mm) Fixed angle (Degree)

r1 8.75 r6 88.8 a 134

r2 14.64 r7 9 b 79

r3 5.3 r8 109.2 c 144

r4 22.15 r9 64.3

r5 28.8 r10 63.45

With the consideration of design specifications, the feasible dimensional parameters of the mechanism are defined, as listed in Table 1. The input motion of this mechanism can be measured as h1 from 69 to 99°. In order to verify the function of the bolt-positioning mechanism, the software Inventor is used for building a 3D model for motion simulation. Figure 6 shows the motion trajectory of the points P and Q, and the numerical results of both theory and simulation are shown in Fig. 7. The initial position of point P is (47.8850, −12.4449) and the final position of the point Q is (47.7825, −12.0094). As the result, the mechanism positions the bolt lands at the same coordinate where the plunger detected a

Fig. 6. Motion simulation

Fig. 7. Numerical results

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hole with the error of about 0.1 and 0.44 mm in the x- and y-axis, respectively. Although the error in y-direction is not small enough, it will be eliminated after the screw-locking task.

4 Conclusions To lower the frequency of suspending and calibrating the robot manipulator, a device for enlarging the allowed position error is presented in this paper, including the functions of hole detecting, screw positioning, and screw locking. Due to the mechanism structure, this innovative device is cheaper than a vision system, and it is easier to be assembled to the robot arm than most RCC devices. Three relevant mechanisms are proposed to achieve the functions, including the leading screw engaged to the casing of device, a 6-bar linkage with a plunger and a socket, and a clutch with a screwdriver. The critical 6-bar linkage (bolt-positioning mechanism) is analyzed to verify its function by both theory and simulation. To sum up, this mechanism is able to put the socket with a bolt at the position where the plunger reached a screw hole with an error smaller than 0.1 mm in the horizontal direction. Although the design can be improved through dimensional synthesis, the feasibility of the mechanism is checked to execute the screwing task. The investigation of sensitivity with respect to dimensional errors and stress-strain analysis will be conducted as the next steps of this study.

References 1. Roth, Z., Mooring, B., Ravani, B.: An overview of robot calibration. IEEE J. Robot. Autom. 3(5), 377–385 (1987) 2. Wu, C.Y.: Rotation angle of the robot arm calibration system and calibration method. Master Dissertation, Department of Mechanical Engineering, National Chung Hsing University, Taichung (2004) 3. Veitschegger, W.K., Wu, C.H.: Robot calibration and compensation. IEEE J. Robot. Autom. 4(6), 643–656 (1988) 4. Nubiola, A., Bonev, L.A.: Absolute robot calibration with a single telescoping ballbar. Precis. Eng. 38(4), 472–480 (2014) 5. Cai, Y., Gu, H., Li, C., Liu, H.S.: Easy industrial robot cell coordinates calibration with touch panel. Robot. Comput. Integr. Manuf. 50, 276–285 (2018) 6. Liu, Y., Xi, N., Zhao, J., Nieves-Rivera, E., Jia, Y., Gao, B., Lu, J.: Development and sensitivity analysis of a portable calibration system for joint offset of industrial robot. In: 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, St. Louis, MO, pp. 3838–3843 (2009) 7. Ginani, L.S., Motta, J.M.S.T.: Theoretical and practical aspects of robot calibration with experimental verification. J. Braz. Soc. Mech. Sci. Eng. 33(1), 15–21 (2011) 8. Li, A., Wang, W., Wu, D.: Calibration of a robot-based measuring system. In: IEEE International Conference on Robotics and Biomimetics (ROBIO), Guilin, pp. 1361–1364 (2009) 9. Zhong, X.L., Lewis, J.M.: A new method for autonomous robot calibration. In: Proceedings of 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan, vol. 2, pp. 1790–1795 (1995)

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10. Omodei, A., Legnani, G., Adamini, R.: Calibration of a measuring robot: experimental results on a 5 DOF structure. J. Robot. Syst. 18(5), 237–250 (2001) 11. Lee, S.C.: Development of a new variable remote center compliance (VRCC) with modified elastomer shear pad (ESP) for robot assembly. IEEE Trans. Autom. Sci. Eng. 2(2), 193–197 (2005) 12. Lai, L.J., Zhu, Z.N.: Modeling and analysis of a compliance model and rotational precision for a class of remote center compliance mechanisms. Appl. Sci. 6(12), 388 (2016) 13. Kim, U., et al.: Displacement sensor integrated into a remote center compliance device for a robotic assembly. IEEE Access 9, 43192–43201 (2021) 14. Yan, H.S., Wu, L.Y.: Mechanism, 4th edn. Tung Hua Book Co. Ltd, Taipei (2014)

Dimensional Synthesis of a Four-Bar Linkage Assessing the Path and Reformulating the Error Function Alfonso Hernández, Aitor Muñoyerro, Mónica Urízar, and Víctor Petuya(&) Department of Mechanical Engineering, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain [email protected]

Abstract. The optimal dimensional synthesis approach presented in this paper aims to enhance one of the earliest and efficient methods—gradient methods. The process to achieve the optimal mechanism bases on characterizing the circuits and branches of the path traced by the coupler point, establishing the corresponding actuation mode according to some indices that avoid using penalty functions. The proposed optimal synthesis approach redefines the error function by including an additional characteristic: the slope of the tangent to each point of the trajectory. It will be demonstrated that incorporating the tangent characteristic permits better fitting to the desired curve than by defining additional points. Keywords: Dimensional synthesis
Four-bar linkage
Optimization methods
Path and motion generation

1 Introduction According to the latest reviews [1] dedicated to dimensional synthesis, the main procedures optimize an error function constituted by the quadratic differences between the points of the discretized prescribed path, and the corresponding points of the generated path. The objective function measures the quality of the mechanism in relation to the fulfillment of the synthesis requirements. Hence, the interest in locating the minimum of this function arises. The bibliography describes a series of methods that can be classified into two large groups: local and global methods. Local methods are the most popular and effective ones. They apply the null gradient condition to the objective function, resulting in a non-linear system of equations that includes the passive variables. This often leads to a linearization of the error function, which ends up being solved with an iterative Newton-Raphson method. For this, it is necessary to start from an initial approximation that leads to a single solution (a relative minimum of the function) of all the possible ones of the non-linear system. A disadvantage of this type of method is that the final solution reached depends largely on the initial approximation. The convergence of these methods is in itself good, but it even improves when the analytical expressions of the gradient are obtained, totally or partially [2, 3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 204–214, 2021. https://doi.org/10.1007/978-3-030-75271-2_22

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Other methods [4] focus on the creation of libraries of mechanisms, or on strategies for the recognition of curves [5], to find the best starting mechanisms for optimization. Related to global methods, some works such as [6], combine classic genetic algorithms with other methods, to enhance the efficiency in finding the optimal mechanism. In the last years, more and more researchers are proposing alternatives to the classical error function based on the sum of squared distances in an effort to resolve some of its problems, such as its sensitivity to translation, rotation, and scaling. An example of a novel objective function, is the circular proximity function [7], which measures the similarity of a group of points to a circular curve and has been successfully used in differential evolution. In relation to other new objective functions, the turning function [8] characterizes the path in terms of a series of tangent angles at points along the curve. The main novelties of the work presented in this paper are: • Formulation of the possible cases using indices, which by its very definition avoids the occurrence of circuit defects without requiring the use of penalty functions. • Reformulation of the error function by including as characteristics, in addition to the coordinates of the prescribed points, the slope of the tangent to the curve at each of these points. This approach will make it possible to define the curve adequately with a smaller number of specified characteristics.

2 Synthesis Equations Figure 1 represents the dimensional parameters (a1 ; a2 ; . . .; a9 Þ and the input variable, u, of the four-bar linkage that will be used in optimization. It should be noted that ða1 ; . . .; a4 Þ must be always positive because they stand for bar lengths, while ða5 ; . . .; a9 Þ can be also negative since they have vector meaning. Figure 1 also shows the local ðOA ; X0 ; Y0 Þ and global ðO; X; Y Þ reference systems, as well as the secondary variables (h; w). Starting from the Freudenstein equation, Eq. (1), the expression relating the passive variable, h, with the dimensional parameters and the input variable can be obtained. The complete derivation can be found in [9]. L1 þ L3 cosh þ L2 cosu ¼ cosðh  uÞ

ð1Þ

where L1 ¼

a24  a21  a22 þ a23 2
a1
a2 a4 L2 ¼ a2 a4 L3 ¼ a1

ð2Þ

By eliminating the variable h from Eq. (1), the following expression is obtained:

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1   B þ K
 B2  A
C @ A h ¼ 2
arctan A 0

ð3Þ

Where: A ¼ L1  L3 þ ðL2 þ 1Þ cosu B ¼ sinu

ð4Þ

C ¼ L1 þ L3 þ ðL2  1Þ cosu The index K in Eq. (3) can take on values of þ 1 or 1 according to the branch of the path of point C on the coupler. The use of this index as well as another index, M, which will serve to identify the circuit by which point C is moved, will be explained in the next section.

Fig. 1. Parameters of the four-bar linkage.

Next, Eqs. (5) and (6) establish the coordinates of the coupler point with reference to the local system of reference ðOA ; X0 ; Y0 Þ:  p x0 ¼ a1 cosu þ a5 cosh þ a6 cos h þ 2

ð5Þ

Dimensional Synthesis of a Four-Bar Linkage

 p y0 ¼ a1 sinu þ a5 sinh þ a6 sin h þ 2

207

ð6Þ

By completing the transformation of the local coordinate system ðOA ; X0 ; Y0 Þ to the global system ðO; X; Y Þ, the coordinates ðx; yÞ of the coupler point C with respect to the global reference system are obtained (Eqs. (7) and (8)). In this way, the translation and rotation variables ða7 ; a8 ; a9 Þ are already included. x ¼ x0 cos a7  y0 sin a7 þ a9

ð7Þ

y ¼ x0 sin a7 þ y0 cos a7 þ a8

ð8Þ

It must be taken into account that the 9 dimensional parameters ða1 ; . . .; a9 Þ are not enough to define its configuration in each synthesis position, since there is a DoF associated with an input variable, in this case the angle u, whose value must be also defined. By minimizing the error function, the input variable ui for each i point will take the value that minimizes the distance di between the path of the coupler point Ci , and the desired point Cid of the synthesis. This is known as unprescribed timing dimensional synthesis.

3 Path Characterization of the Coupler Point of a Four-Bar 3.1

Analytical Formulation of Branches and Circuits

First, it should be remembered that if the linkage satisfies the Grash of criterion, it will generate a path made up of two circuits, whereas if it does not meet this criterion, it will have a path made up of only one circuit. Although a number of articles have clarified the difference between a circuit and a branch [10], there still seems to be some confusion. A circuit is a closed curve that the coupler point can traverse completely without having to disassemble the mechanism. A circuit defect occurs when the synthesis obtains approximation points in different circuits of the mechanism’s path. On the other hand, a branch is the part of a circuit traversed by the coupler point while the input, in this case necessarily a rocker input, moves between its two limits (blocking positions). A circuit defect as described above is inadmissible under any circumstances. However, some authors [10] also consider as defective those solutions that propose points belonging to different branches. This situation is called a branch defect. Nevertheless, in our work, this type of solution is allowed because the necessary considerations to ensure the correct functioning are implemented. That is, guaranteeing a correct order at the time of assigning the sequence of input parameters, and ensuring that the input parameter remains within the motion range when performing unprescribed timing. In the four-bar linkage, three situations can occur: 1. One circuit formed by two connected branches (rocker-rocker) See Fig. 2a. 2. Two circuits, each corresponding to one single branch (crank-rocker or double crank). See Fig. 2b.

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3. Two circuits, each formed by 2 connected branches (rocker-rocker or rocker-crank). See Fig. 3. In each situation, the type of input is underlined. The difference between situations 2 and 3 is rooted precisely in the input. Next, a procedure is presented to detect the three possible situations. The analysis is performed using the zero points of the discriminant function ðB2  A
CÞ presented in Eq. (3). By expanding and manipulating this expression as described in [9], the secondorder equation, Eq. (9), is obtained with roots ua and ub and ua  ub .

Fig. 2. Examples of situation 1 (a) and situation 2 (b)

Fig. 3. Different branches and circuits for situation 3.

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E
u2  F
u þ G ¼ 0

209

ð9Þ

where u ¼ cosu E ¼ L22 F ¼ L1
L2 þ L3

ð10Þ

G ¼ L21 þ L23 þ 1 Letting I be the range of motion of the input, the following cases arise: Table 1. Range of inputs as a function of the roots of the discriminant of Eq. (3). Case (a)

Condition ½ua ; ub   ½1; 1

Range of inputs I ¼ ½0; 2p

(b)

½ua ; ub   ½1; 1

I ¼ M
½arccosðub Þ; arccosðua Þ where M ¼ 1

(c) (d)

ua \  1; ub 2 ð1; 1Þ ua 2 ð1; 1Þ; ub [ 1

I ¼ ½arccosðub Þ; 2p  arccosðub Þ I ¼ ½arccosðua Þ; arccosðua Þ

Classification Situation 2: two circuits, each with one branch Situation 3: two circuits, each with two branches Situation 1: one circuit with two branches

Motion is impossible (e) ua [ 1 or ub \  1 Note: cases (c) and (d) are grouped together as one situation because they are two analogous cases, but they do not occur simultaneously

A new index (see Fig. 3), M, has been introduced, which appears in case (b) of Table 1. This table completes the classification presented by Ma and Ángeles in [9], in which one of the two possible circuits in situation 3 was omitted, specifically the option: M ¼ 1, which establishes the range I ¼ ½arccosðub Þ; arccosðua Þ of case (b). 3.2

Treatment of Synthesis Considering Branches and Circuits

This section explains how to approach the synthesis of the three situations. A more detailed analysis can be found in the work [11] of the authors. Situation 1. In this case, the branches K ¼ þ 1 and K ¼ 1 remain connected, forming a single circuit (see Fig. 2a). For each prescribed point, there are two possible solutions, and each one can be located on a different branch. Situation 2. Equation (3) gives the value of the passive variable h, which combined with Eqs. (7) and (8) yields the values of the synthesis variables ðx; yÞ. Because K can take on two values, there always exist two possible solutions (see Fig. 2b). To avoid circuit defects, solutions in the same circuit must always be chosen, K ¼ þ 1

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or K ¼ 1: Evidently, the solution of interest is the one that gives the smallest total error. Situation 3. As can be seen in Fig. 3, there are two possible motion ranges for the input parameter. Within each circuit, there are two branches, with the extremities of each defined by the blocking positions. To perform the synthesis, two steps are needed: – First, the index M is defined, which will be determined by the values of the input parameter. One of the circuits will be created with the values of the input parameters included in the I range associated with the index M ¼ þ 1, while the I range for the other circuit is given by M ¼ 1 (see Table 1). All the synthesis points must be in the same circuit. – Second, inside the circuit in question, for each input parameter, there will be two possible solutions, K ¼ þ 1 or K ¼ 1. The most suitable solution for each point can be chosen (the one giving the smallest error). 3.3

Avoiding Order Defect

An order defect occurs when the coupler point, while tracing the path, does not traverse the precision points in the same order as established by the sequence of prescribed points. In situation 2, order defect is avoided by simply assigning an ordered sequence of input parameters in the start mechanism. However, in situations 1 and 3, the value of the index K must be considered.

Fig. 4. a) Curve of circuit 1; b) Movement of the input bar to verify the correct order.

To explain how the point order is controlled, let us see Fig. 4b. Three sections appear: Section 1: All the precision points from the starting point until the first blocking position is reached (ua ), will have associated with them ordered input parameters. Section 2: Once the first blocking position has been reached, the input movement is reversed, and the bar continues to move until it reaches the other blocking position, ub . Section 3: Once the second blocking position has been reached, the movement of the input bar is reversed once again, and the circuit is completed.

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4 Error Function Reformulation by Adding the Slope of the Curve This study proposes to modify the error function to include another characteristic, the slope of the tangent to the desired curve at each point, which will be characterized by the sine and cosine of the angle ai that the tangent forms with the horizontal axis. There are several reasons why it is interesting to add this new characteristic. First, it adds additional information about the shape of the path at a point, helping to better define the shape of the target curve. In addition, adding a point as a prescribed characteristic along with its slope has a lower computational cost than two points (a half). On the other hand, it is possible to obtain the analytical partial derivatives of this characteristic, so it can be easily included within the gradient method. In this way, the error function can be reformulated as: E ¼ Ep þ Es

ð11Þ

where the first term, Ep , includes the distances between the coordinates of the points, while the second term, Es , includes the error in the slope of the path. As for the optimization algorithm, minimizing the error function E with respect to the design variables expressed as the null gradient of this function gives rise to the following system of equations: ! rE ¼ 0 !



@E @sj

 ¼ f0g

ð12Þ

 where the vector sj contains the parameters to be optimized. Note that j now ranges up to n þ N, meaning that the vector can include not only the dimensional parameters, but also the input parameters:  T sj ¼ ½a1 a2 . . .an u1 u2 . . .uN 

ð13Þ

8 9 E1 >

> @E @E1 @E2 @EN < . = .. ¼0 ¼0! ... @sj @sj @sj @sj > ; : > EN

ð14Þ

By setting,

Where each term Ei stands for: ~ Ei ¼









xi  xdi ; yi  ydi ; dh cos adi  cosðai Þ ; dh sin adi  sinðai Þ

ð15Þ

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And thus:

2

2

2 Ei2 ¼ xi  xdi þ yi  ydi þ ðdh Þ2
cos adi  cosðai Þ

2 þ ðdh Þ2
sin adi  sinðai Þ

ð16Þ

By expanding Ei into a first-order Taylor series and manipulating it, the following linear system is finally obtained: ½Dm fðDsÞm g ¼ fGgm

ð17Þ

The procedure solves iteratively Eq. (17). It corresponds to a linear system in which m the vector  fðDsÞ g is the vector of unknowns, which includes the increments of the vector sj . Each iteration corrects the vector of variables to be optimized so that it converges on an optimal solution. The typical stopping criterion is too end the iterative n process when the norm of the difference vector between

ðDsÞm þ 1

and fðDsÞm g is

less than a given tolerance (let us say, 1e−6).  At this point, it is worth wondering how to define vector sj for the initial iteration. The initial mechanisms can be obtained from a previously generated curve atlas. From here, a candidate mechanism is manually identified, or can be adjusted by “trial and error”. In addition, it may be useful to carry out a previous stage where only specific parameters (translation, rotation, and scaling) are optimized, for example by the Procrustes function in MATLAB. An alternative procedure is to prescribe a limited series of points for which an exact solution exists, for example using GIM software, or by exact classical, graphical or analytical methods. Finally, a third option is using heuristic methods, such as genetic algorithms, to find regions that are promising at a global level, to subsequently obtain the corresponding local minimums by gradient methods.

5 Demonstrative Example The objective is to obtain the red right triangle whose legs measure 3 units of length, starting from an initial mechanism that traces the “D”-shaped path shown in Fig. 5a. As illustrated in Figs. 5b and 5c, one possibility is to increase the number of characteristics in the prescribed path. In the case shown in Fig. 5b, which includes the slopes of the tangents as prescribed characteristics, a very good result was obtained with a total of 27 characteristics, that is, using only 9 prescribed points and their associated tangent slopes. However, to achieve a similar result using only the ðx; yÞ coordinates of the points (Fig. 5c), 36 characteristics would have to be prescribed.

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Fig. 5. a) Desired path; including (b) and without including (c) the tangent slope

6 Conclusions This study has proposed an optimal dimensional synthesis procedure for the general case of the four-bar linkage, which perfects the classical formulation of the gradient method. On the one hand, the error function is reformulated to include as additional characteristic the slope of the tangent to the curve at each prescribed point. On the other hand, indicators have been established to adequately identify the branches and circuits in each case. Moreover, the use of these indicators makes it possible to define the actuation mode in each case, without recourse to penalty functions. The great majority of published studies limit the mechanism design to four-bar linkages with crank input, but in this study, thanks to the complete characterization of circuits and branches, designs with rocker input that produce no defect can be obtained. The reformulation of the error function, by adding the slope of the tangent to the curve at each prescribed point, enables a closer adjustment to the desired curve, and achieves this with a smaller number of prescribed characteristics. According to the usual classical approach, each additional point implies adding two more characteristics (coordinates), but if the tangent is included for an existing point, this amounts to adding only one characteristic. Acknowledgments. This work was funded by Project IT949-16 (Departamento de Educación, Política Lingüística y Cultura, Regional Government of the Basque Country) and Project DPI2015-67626-P (MINECO/FEDER, UE) (Ministerio de Economía y Competitividad, Spanish Government).

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References 1. Lee, W.T., Russell, K.: Developments in quantitative dimensional synthesis (1970-present): four-bar path and function generation. Inverse Probl. Sci. Eng. 26(9), 1280–1304 (2018) 2. Mariappan, J., Krishnamurty, S.: A generalized exact gradient method for mechanism synthesis. Mech. Mach. Theory 31(4), 413–421 (1996) 3. Sancibrián, R., Juan, A., Sedano, A., Iglesias, M., García, P., Viadero, F., Fernández, A.: Optimal dimensional synthesis of linkages using exact jacobian determination in the SQP algorithm. Mech. Based Design Struct. Mach. 40, 469–486 (2012) 4. McGarva, J.R.: Rapid search and selection of path generating mechanisms from a library. Mech. Mach. Theory 29(2), 223–235 (1994) 5. Sánchez Marín, F.T., Pérez González, A.: Global optimization in path synthesis based on design space reduction. Mech. Mach. Theory 38(6), 579–594 (2003) 6. Laribi, M.A., Mlika, A., Rhomdane, L., Zeghloul, S.: A combined genetic algorithm–fuzzy logic method (GA–FL) in mechanisms synthesis. Mech. Mach. Theory 39, 717–735 (2004) 7. Sun, J., Liu, W., Chu, J.: Dimensional synthesis of open path generator of four-bar mechanisms using the haar wavelet. J. Mech. Des. 137, 8 (2015) 8. Nadal, F., Cabrera, J.A., Bataller, A., Castillo, J.J., Ortiz, A.: Turning functions in optimal synthesis of mechanisms. J. Mech. Des. 137, 10 (2015) 9. Ma, O., Angeles, J.: Performance evaluation of path-generating planar, spherical and spatial four-bar linkages. Mech. Mach. Theory 23, 257–268 (1988) 10. Balli, S.S., Chand, S.: Effects in link mechanisms and solution rectification. Mech. Mach. Theory 37, 851–876 (2002) 11. Hernández, A., Muñoyerro, A., Urizar, M., Amezua, E.: Comprehensive approach for the dimensional synthesis of a four-bar linkage based on path assessment and reformulating the error function. Mech. Mach. Theory 156, 104126 (2021)

Actuators and Control

Motion Control of 6-DOF Relative Manipulation Device Sergey Khalapyan1 1

, Larisa Rybak2(&) , Elena Gaponenko2 and Giuseppe Carbone3

,

Stary Oskol Technological Institute n.a. A.A. Ugarov NUST MISiS, 42, Makarenko mkr., 309516 Stary Oskol, Russia 2 Belgorod State Technological University n.a. V.G. Shukhov, Kostukova 46, 308012 Belgorod, Russia 3 University of Calabria, 87036 Rende, Calabria, Italy

Abstract. The article discusses a robotic system based on a 6-DOF relative manipulation device consisting of two modules. The lower module is based on a parallel planar 3-RPR mechanism, and the upper module is based on a parallel 3-RPS tripod module. Such robotic system can be used in mechanical engineering as a robot-machine for the mechanical processing of parts. The mathematical model was built in the MATLAB package to analyze the prospects for using a 3-RPS robot. An alternative description of the rotation of the robot’s mobile platform is proposed to organize a simpler control of it within the framework of a system of interacting robots. A technique has been developed to compensate for the movable tripod platform’s horizontal displacements and take into account the geometric dimensions of the replaceable tool and the relative positions of the 6-DOF system modules during the control process. The results of the mathematical simulation are presented. Keywords: 3-RPS mechanism
6-DOF robotic system
Parallel robot model
Parallel robot control

1 Introduction Robotic systems, in which mechanisms with parallel kinematics are used as interacting modules, are currently a promising research area [1–4]. The urgency of the problem of organizing effective control of such systems is beyond doubt. The disadvantages of the parallel mechanisms include the limited volume of the workspace and the kinematic and dynamic coupling of the drives. One of the options for reducing the significance of these shortcomings is the development of manipulation mechanisms, consisting of two modules and performing joint relative manipulation [5, 6]. The number of degrees of freedom of such mechanisms is equal to the sum of the degrees of freedom of the modules. Such an organization will make it possible to jointly work out each point of the workpiece surface with the necessary orientation of the output link of the joint relative manipulation device in the moving coordinate system and the movement of the coordinate system itself. One of the mechanisms can correct some movements of the other. In the research [7], it became necessary to check © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 217–225, 2021. https://doi.org/10.1007/978-3-030-75271-2_23

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the control system’s operability and develop the corresponding algorithms based on the robotic system (RS) model. The authors developed such a model [8] for a 3-RPR planar robot. The second mechanism could be the 3-RPS mechanism (tripod) [9].

2 RS Model Development Let us consider a RS scheme in the form of a 6-DOF robot, which can be used as a robot machine for processing of parts (Fig. 1). The upper module is designed based on a tripod with prismatic drive joints, rotary joints D1 ; D2 ; D3 and spherical joints E1 ; E2 ; E3 . The center of the movable platform of the tripod module can perform translational movement along the z-axis and rotational movement around the x and y axes and additional displacements of the end effector when it rotates about the horizontal axes imposed by the kinematic chains of the tripod module. The lower module is based on a planar 3-RPR mechanism with prismatic joints and rotary joints A1 ; A2 ; A3 performs translational movement along the x and y0 axes, rotational around the z-axis. The end effector is fixed on the tripod module’s movable platform, and the workpiece to be processed on the movable platform of the 3-RPR mechanism. The z and z0 axes lie on one straight line perpendicular to the fixed base of the tripod module D1 D2 D3 and the platform of the planar 3-RPR mechanism and passing through the center of the circumscribed circle of the triangle A1 A2 A3 . A 3D model of a 6-DOF robot is shown in Fig. 2 (1 - module for installing the tool; 2 - module for installing the part; 3 - frame).

Fig. 1. Schematic 6-DOF of the robot

Fig. 2. 6-DOF robot 3D model

We construct a mathematical model of the tripod to test the algorithms’ efficiency proposed in [7], for the control of the RS. The mechanisms’ input coordinates are the rods’ lengths, which are changed under the electric motors’ action. Output is moving the center of the movable platform relative to the center of the fixed platform and its rotation.

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The tripod model (Fig. 3) consists of three subsystems LEG-1, LEG-2, LEG-3, corresponding to three rods connected with one end to the fixed base at the points defined by the blocks Base1, Base2, Base3, and the other to the movable Platform. The tasks for each rod L1, L2 and L3 are transferred for implementation to the corresponding subsystems. Each of the LEG-1, LEG-2, LEG-3 subsystems (Fig. 4) contains a kinematic diagram including a Revolute Joint, a Prismatic Joint, and a Spherical Joint. The prismatic joint position information is used as a feedback signal in the rod length control loop. The control action generated by the regulator is decomposed into two components: the absolute value, which is converted into a pulse width modulated signal in the Controlled PWM Voltage block, and the action sign used in the H-Bridge block to change the polarity of the output voltage and reverse the DC Motor electric motor. The linear velocity and applied force alignment is carried out using the Ideal Translational Velocity Source and Ideal Force Sensor blocks.

Fig. 3. Mathematical model of a tripod in the MATLAB package

Fig. 4. Contents of the subsystems of the tripod LEG-1, LEG-2, LEG-3

According to [7], to control the robot, it is necessary to represent the task for the position of the movable platform in the form of a set of tasks along the length of each rod L1, L2, L3. For this, the formulas for solving the inverse kinematics of the mechanism are used. Traditionally, to describe the rotation of a movable platform in such formulas, Tait – Bryan angles are used, which may be quite acceptable for setting a single robot’s position, but inconvenient for organizing control of interacting RS modules. Due to the specific orientation of the tripod’s pivot joints, it cannot rotate around the z-axis perpendicular to the fixed base [10]. It is much more convenient to use another method, which provides for the description of the rotational motion of the movable platform as its only rotation from the initial horizontal position. We need to define the axis and angle of rotation. Since rotation about a vertical axis is not possible,

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the w-axis of rotation always lies in the horizontal plane. Then it can be uniquely determined by the angle between it and the positive direction, for example, the x-axis. Based on the solution of the inverse kinematics of the tripod, changes in the lengths of the rods have the form L1 ¼ ðRrðð2 cos2 f0; 5Þl þ cos nÞÞ2 þ ðz0 r sin fsin nÞ2 Þ0;5  L2 ¼

 Rr

sin2 f 

ð1Þ

! pffiffiffi  2    2 0;5 pffiffiffi 3 sin 2f l þ cos n þ z0 þ 0;5r sin f þ 3 cos f sin n 2

ð2Þ L3 ¼

pffiffiffi  0;5 pffiffiffi 3 sin 2fÞl þ cos nÞÞ2 þ ðz0 þ 0;5rðsinf 3 cosfÞsin nÞ2 ðRrððsin2 f þ 2 ð3Þ

where L1 ; L2 ; L3 are the lengths of the rods, R and r are the radii of the circles passing through the vertices of the fixed base and the movable platform, respectively, z0 is the distance between the center of the platform and the plane of the base (Fig. 5), n is the angle between the w – axis of the platform rotation and the positive direction of the x; f is the platform rotation angle, l ¼ 1cos n. There is also a “parasitic” displacement of the platform along the x and y-axis (Fig. 6). As part of the RS, such “parasitic” displacements must be compensated by additional displacements of the second module.

Fig. 5. The tripod geometric parameters

Fig. 6. Rotation and “parasitic” displacement of the mobile platform: E01 E02 E03 – original state; orange triangle – rotation; E1 E2 E3 – new platform position

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x0 ¼ 0:5 r l cos 2f

ð4Þ

y0 ¼ 0:5 r l sin 2f

ð5Þ

3 Development of a RS Model Taking into Account the Geometry of the End Effector and the Relative Position of the Modules Above ratios are valid for the geometric center of the movable platform of the tripod, and for a system of interacting robots, it is more important to track the movement of the end effector, which is used to process the part moved by the planar mechanism. The tool projections on the x and y axes will be determined only by its length and the angle of inclination of the platform to the corresponding axis. Thus, the correction of the task for the planar mechanism will have two components: to level the tripod’s parasitic movements and take into account the projections of the tool. If we set its vertical position z and the angle of inclination to the vertical axis p  n, then the position of the platform center z0 is determined in the form z0 ¼ z  d cosðp  nÞ ¼ z þ d cos n

ð6Þ

where d is the length of the tool. The absence in (3) of the angle f, which determines the axis of rotation of the movable platform, is explained by the independence of the tool’s vertical projection from the direction of its inclination. In this case, the direction of inclination makes it easy to determine the required of rotation angle of the planar mechanism to orient the workpiece. It is necessary to add the projection of the tool to the corresponding coordinate axes to determine x and y coordinates of the end effector and the corresponding correction of the tasks of the planar robot to the parasitic movements of the tripod x0 and y0 : Dx ¼ x0 þ d cos ðf þ p=2Þ sin n ¼ x0  d sin f sinn

ð7Þ

Dy ¼ y0 þ d sinðf þ p=2Þ sin n ¼ y0 þ d cos f sinn

ð8Þ

The received task for the 3-RPR robot after correction must be translated into the coordinate system ðx0 ; y0 ; z0 Þ, associated with the planar mechanism, which determines the modules’ relative position. The planes of the modules’ bases must be parallel to ensure the operation of the 6-DOF robot. Let the z and z0 axes coincide for simplicity. The distance between the zero marks on each of them and the workpiece’s dimensions and position should be taken into account when forming a given trajectory of motion. We only need to know the angle d between the x0 and x axes to convert the corrected values of x and y to x0 and y0 :

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x0 ¼ ðx þ DxÞ cos d þ ðy þ DyÞ sin d

ð9Þ

y0 ¼ ðx þ DxÞ sin d þ ðy þ DyÞ cos d

ð10Þ

Then, taking into account (4), (5), (7) and (8): x0 ¼ ðx  d sin f sin n þ 0:5r l cos 2fÞ cos d þ ðy þ d cos f sin n  0:5r l sin 2fÞ sin d ð11Þ 0

y ¼ ðx þ d sin f sin n þ 0:5r l cos 2fÞ sin d þ ðy þ d cos f sin n  0:5r l sin 2fÞ cos d ð12Þ The angle u of rotation around the z-axis in the coordinate system associated with the planar mechanism decreases by d : u0 ¼ u  d. The obtained values x0 ; y0 ; and u0 are used to calculate the lengths of the planar mechanism’s rods using the corresponding formulas for solving the inverse kinematics [8].

4 Numerical Experiments The mathematical model of the tripod was supplemented with a planar robot model (Fig. 7) to check the possibility of RS control based on the above relations [8].

Fig. 8. Displacement of the center of the tripod movable platform (orange line) and the end effector (blue line), m

Fig. 7. The mathematical model of the robotic system: red frame – the planar robot with a stand and a workpiece, blue frame – the tripod with a tool

Fig. 9. The trajectory of movement of the center of the workpiece, m

Motion Control of 6-DOF Relative Manipulation Device

Fig. 10. Trajectory of the relative movement of modules, m

Fig. 11. Relative movement of the tool along a given path

Fig. 12. Distance from the end effector to the surface of the processed sphere, m

Fig. 13. Error of relative movement of modules along the x-axis, m

Fig. 14. Error of relative movement of modules along the y-axis, m

Fig. 15. Error of relative movement of modules along the z-axis, m

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Geometric characteristics of a planar robot: R0 ¼ 1 m, r 0 ¼ 25 cm, L0max ¼ 1 m, L0min ¼ 50 cm; tripod: R = r = 30 cm, Lmax ¼ 1 m, Lmin ¼ 50 cm. On the movable platform of the planar mechanism on a special stand Solid2 with an effective height of 5 cm, a workpiece Solid1 in the form of a ball with a radius of 10 cm was installed. The Solid9 and Solid3 tools for processing it 10 cm long were fixed in the center of the movable platform of the tripod perpendicular to the surface of the latter. The Base8 block determines the modules’ relative position (the distance between the fixed bases and the angle d). Transform Sensors allow determining the position in coordinates (x, y) of the center of the movable platform of the tripod (X0, Y0), the end effector (X1, Y1), the center of the workpiece (X2, Y2), movement (X, Y, Z) of the end effector relative to the center of the workpiece and the distance d between them. In the course of the experiment, the task was set to draw a spiral line with the end effector on the surface of the workpiece. The experiment showed that the center of the movable platform of the tripod and, moreover, the end effector during movement significantly deviate from the line of centers of the fixed bases of the RS (Fig. 8). However, as a result of the task correction, the planar mechanism, moving along a complex trajectory (Fig. 9), compensates for these deviations. As a result, the modules’ relative movement occurs following the specified trajectory (Fig. 10, 11). The estimation of the RS movement’s accuracy was carried out by controlling the distance (Fig. 12) from the end effector to the surface of the processed sphere (processing depth). The largest error was 0.007 mm or 0.007% of the sphere radius. Figure 13, 14 and 15 shows the errors of the relative movement of the modules along the x, y, z coordinates. The maximum error is only 0.026 mm. But the range of z change is much smaller, it is equal to 30 mm. The maximum relative error is 0.09%. The smaller value of the error is explained by the fact that one robot participates in movement along this coordinate, and two along the rest. The error in the movement of the robotic system along each of the coordinates does not exceed 0.2% of the corresponding range of movements. Thus, in the course of the experiment, the high accuracy of RS control was demonstrated.

5 Conclusions The work describes the RS models developed in the MATLAB environment. An alternative description of the rotation of the movable platform of the 3-RPS robot to simplify control within the 6-DOF system is proposed. A technique has been developed to compensate for the horizontal displacements of the movable tripod platform and take into account the geometric dimensions of the tool and the relative position of the 6DOF system modules during the control process. RS simulation confirmed the possibility of using a 3-RPS robot in its composition and the effectiveness of the proposed technique. Acknowledgements. This work was supported by the state assignment of Ministry of Science and Higher Education of the Russian Federation under Grant FZWN-2020-0017.

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References 1. Golin, J., Frantz, J., Simas, H.: Kinestatic analysis of serial-parallel cooperative robotic systems. In: 23rd ABCM International Congress of Mechanical Engineering, pp. 1–7. SWGE Sistemas, Uberlandia (2015) 2. Yeshmukhametov, A., Kalimoldayev, M., Mamyrbayev, O., Amirgaliev, Y.: Design and kinematics of serial/parallel hybrid robot. In: 3rd International Conference on Control, Automation and Robotics (ICCAR), pp. 162–165. IEEE, New York (2017) 3. Hu, B.: Formulation of unified Jacobian for serial-parallel manipulators. Robot. Comput.Integr. Manuf. 30(5), 460–467 (2014) 4. Guo, W., Li, R., Cao, C., Gao, Y.: Kinematics analysis of a novel 5-DOF hybrid manipulator. Int. J. Autom. Technol. 9(6), 765–774 (2015) 5. Glazunov, V.A., Lastochkin, A.B., Shalyukhin, K.A., Danilin, P.O.: Analysis and classification of relative manipulation devices. J. Mach. Manuf. Reliab. 38(4), 379–382 (2009) 6. Pashchenkova, V.N., Sharapov, I.V., Rashoyan, G.V., Bykov, A.I.: Construction of a working area for the manipulation mechanism of simultaneous relative manipulation. J. Mach. Manuf. Reliab. 46(3), 225–231 (2017) 7. Khalapyan, S., Rybak, L., Malyshev, D.: Two-stage method for controlling the movement of a parallel robot based on a planar three-revolute-prismatic-revolute mechanism. Adv. Intell. Syst. Comput. 1126, 368–379 (2020) 8. Khalapyan, S.Y., Rybak, L.A., Kuzmina, V.S., Kholoshevskaya, L.R., Ignatov, A.D., Popov, M.V.: The study of the accuracy of the robot movement along a given path considering the workspace boundaries, velocity and inertial properties of the drive. J. Phys: Conf. Ser. 1582(1), 1–7 (2020) 9. Khalapyan, S.Y., Glushchenko, A.I., Rybak, L.A., Gaponenko, E.V., Malyshev, D.I.: Intelligent computing based on neural network model in problems of kinematics and control of parallel robot. In: 3rd Russian-Pacific Conference on Computer Technology and Applications (RPC), pp. 1–5. IEEE, New York (2018) 10. Lee, K.-M., Shah, D.K.: Kinematic analysis of a three-degrees-of-freedom in-parallel actuated manipulator. IEEE J. Robot. Autom. 4(3), 354–360 (1988)

Genetic Algorithm with Iterative Learning Control for Estimation of the Parameters of Robot Dynamics Kaloyan Yovchev(&) and Lyubomira Miteva Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria [email protected], [email protected]

Abstract. The industrial robotic manipulators have to execute trajectory movements repetitively and with high trajectory tracking precision. In most cases the model parameters of the dynamical model of the robotic system cannot be calculated precisely and there is always an uncertainty in the model of its dynamics. This requires that the robotic control system combines multiple control strategies in order to achieve high trajectory tracking precision. This research investigates how genetic algorithm (GA) can be applied for estimation of the parameters of the mathematical model of a robotic manipulator. The GA requires definition of a solution domain and a fitness function for evaluation of the solution domain. The Constrained Output Iterative Learning Control (COILC) is a biologically inspired method which uses consecutive iterative executions of the same trajectory and uses a learning operator in order to reduce the overall trajectory tracking error. It is proven to be robust and to generate a convergent process when the correct estimation of the dynamics model parameters of the robot are incorporated within the ILC learning operator. The COILC method can be applied safely to robotic manipulators with state space constraints even if the learning operator is randomly chosen. This allows the set of possible dynamics model parameters to be selected as the solution domain and the COILC method to be used as a fitness function for the GA. This leads to an algorithm for estimation of the model parameters of the robot which are best suitable for using an ILC method. This approach is validated through a computer simulation. Keywords: Genetic algorithm
Robot dynamics Iterative learning control
Robotic manipulators


Constrained systems


1 Introduction The industrial robotic manipulators have to execute trajectory movements repetitively and with high trajectory tracking precision. This can be achieved with a precise dynamical model of the robotic manipulator. In most cases the model parameters cannot be calculated precisely and there is always an uncertainty in the model of the dynamics. This requires that the robotic control system combines multiple control strategies in order to achieve high trajectory tracking precision. In research [1] the computed torque control is combined with fuzzy control. Another approach is the use of an implicit Lyapunov function control method as a type of continuous feedback © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 226–235, 2021. https://doi.org/10.1007/978-3-030-75271-2_24

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control [2]. More computationaly efficient method is the Iterative Learning Control (ILC) which was introduced by Uchiyama and Arimoto [3–5]. It is a biologically inspired method which uses consecutive iterative executions of the same trajectory and uses a learning operator in order to reduce the overall trajectory tracking error. For nonlinear systems the robustness and convergence are proven in 1989 by Heizinger et al. [6]. The Constrained Output ILC (COILC) is proposed as a method which can easily be incorporated within the control system of hard constrained robotic systems [7–9]. The convergence of the COILC process depends on the selection of the learning operator. This selection depends on the correct estimation of the parameters of the dynamical model of the robot. The ILC is a biologically inspired method of iterative self-learning. The goal of this research is to propose a combination of COILC and genetic algorithm (GA) in order to find the best estimation of the model parameters. The GA introduced in 1975 by Holland [10] are one of the first population-based stochastic algorithms proposed in history [11]. They are probabilistic search procedures designed to work on large spaces involving states which can be represented by numbers or strings [12]. They implement optimization strategies by simulating evolution of species through natural selections [13]. They are a combination of two processes: selection of the best individuals for the next generation and formation of the next generation through crossover and mutation techniques [14]. The dynamical model parameters can be encoded into an unsigned integer number in order to create an individual for the GA. The idea of this research is to use the COILC method in the selection process of the GA. There is a relation between the COILC convergence and the estimated dynamical parameters. If the GA selects the individuals with the best COILC convergence it can improve the estimation of the parameters and after several generations it will find the optimal parameters for applying COILC for precise trajectory tracking. This paper is structured as follows. Section 2 describes a SCARA-type robotic manipulator and formulates the problem. Section 3 combines COILC and GA and proposes an algorithm for estimation of the parameters of the robot. Section 4 validates the proposed algorithm through a computer simulation.

2 Problem Formulation For the purposes of this research, we will consider the first two links of a SCARA-type robot. In order, to compare the proposed algorithm for COILC performance enhancement to our previous results we will be using the model of SEIKO Instruments TT-3000. The kinematics of this robot is as shown in Fig. 1, where l1 ¼ 25, l2 ¼ 22 cm.

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Fig. 1. Kinematics of SEIKO Instruments TT-3000 robot.

The Lagrange’s formulation of dynamical equations of motion of this robot is as follows [15]: a11 ðq2 Þ€q1 þ a12 ðq2 Þ€q2  2bsinq2 q_ 1 q_ 2  bsinq2 q_ 22 þ d1 q_ 1 þ fc1 ¼ u1 ; a12 ðq2 Þ€q1 þ a11 ðq2 Þ€q2  2bsinq2 q_ 21 þ d2 q_ 2 þ fc2 ¼ u2 ;

ð1Þ

a11 ðq2 Þ ¼ a þ 2bcosq2 a12 ðq2 Þ ¼ a21 ðq2 Þ ¼ v þ bcosq2 a22 ¼ v
 a ¼ I1 þ m1 s21 þ I2 þ m2 I12 þ s22 b ¼ m2 l1 s2 v ¼ I2 þ m2 s22 ;

ð2Þ

where:

and di ; i ¼ 1; 2 is viscous damping, fci is Coulomb’s friction, li is the specified length of the link of the robot, si is the position of the center of the mass, mi is the total mass of the link, and Ii is the inertia of the link about its center of mass. If t 2 ½0; T  denotes time, where ½0; T  is the robot tracking time interval, ul is the feed-forward term of the control law, l ¼ 1; . . .; N is the current iteration number, N is the total number of iterations, ql is the actual output trajectory and qd is the desired output trajectory, then the COILC control algorithm is defined in [8] as follows: For the robotic system defined in (1) with the hard constraints defined as Qmin  qi ðtÞ  Qmax i i ; i ¼ 1; 2; we can apply the following ILC procedure for any attainable desired trajectory qd : a. Set the initial iteration number l ¼ 0 and begin the iterative procedure. b. Starting from the initial position ql ð0Þ the system is tracking the desired trajectory under the control ul ðtÞ until the first moment Sl for which there exists j : 1  j  m or qlj ðSl Þ ¼ Qmax or the end position ql ðT Þ is reached. and either qlj ðSl Þ ¼ Qmin j j When t ¼ Sl ; Sl 2 ð0; T , the tracking process stops. c. After the current tracking performance has finished, the learning controller updates the feed-forward control term according to the following learning update law:

Genetic Algorithm with Iterative Learning Control for Estimation

ul þ 1 ðt; Sl Þ ¼ ul ðt; Sl1 Þ þ ul ðt; Sl Þ; u0 ðt; S1 Þ u0 ðtÞ; Lðql ðtÞ; tÞð€qd ðtÞ  €ql ðtÞÞ; t 2 ½0; Sl ; Sl 2 ð0; T ; ul ðt; Sl Þ ¼ 0; 8t 2 ðSl ; T 

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ð3Þ

  d. If the output error (the maximum iteration tracking error) dql ðtÞ1  is less than or equal to an acceptable tracking accuracy, then exit from the learning procedure, else set l ¼ l þ 1 and go to step (b). The offline computed feed-forward term ul þ 1 decreases the tracking error of the robot’s motion on the next iteration. The feed-forward controller is based on the update control law (3) that iteratively improves the feed-forward control term. In (3) Lðql ðtÞÞ; l ¼ 0; 1; . . .; N is the learning operator and u0 ðtÞ  0 is the initial feed-forward control input. The robustness and convergence of the ILC update law (3) is proven in [16] if b ðqÞ; LðqÞ  A

ð4Þ

b is the corresponding estimate of the inertia matrix A. For the considered TTwhere A   3000 robot with Eqs. (1) and (2): A ¼ aij ; 1  i; j  2. Furthermore, from (2) it follows that the convergence of the ILC process depends on the correct estimation of the parameters a; b and v of the dynamics model. On the other hand, if the ILC process is optimal and convergent when the learning operator as defined in (4) is used it follows b is correctly estimated. This will lead to a faster convergence that the inertia matrix A when another desired trajectory is used. The goal of this research is to try to find parameters a; b and v for which the COILC method have the minimum total convergence time by using genetic algorithm. For the genetic algorithm a genetic representation of the solution domain and a fitness function for evaluation of the solution domain is required. In the following section we present how a genetic algorithm can be applied together with the COILC method in order to find optimal estimation of the parameters of the dynamics model of the robot.

3 Genetic Algorithm with ILC for Estimation of the Parameters of the Dynamics For the described use case we will consider the three dynamics model parameters a; b and v from (2) and will encode them as three 16-bit unsigned integer numbers. This will allow all of them to take values between 0 and 65536. The group of a total of 48 bits will represent a single individual of the population (Fig. 2). As stated in Sect. 1 for application of genetic algorithm the following operations must be defined: initial initialization, selection, crossover, mutation and terminal condition.

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Fig. 2. Example individual of the genetic algorithm.

3.1

Initialization of the First Generation

The first generation will be randomly initialized 48-bit unsigned integer numbers. The individuals for the initial population should be chosen accordingly to expected estimations of the parameters of the dynamics of the robot in order to reduce the required number of generations. For the described robot the parameter a will be selected within the range 2000 to 8000 kg.cm2 and parameters b and v will be between 1000 and 2000 kg.cm2. 3.2

Selection

For each consecutive generation only the best individuals of the current generation are selected. The selection is done by defining a fitness function which calculates the total time needed for convergence when COILC is applied with the learning operator given by the corresponding individual of the generation. The maximum iterations of the ILC procedure must be limited because the learning operator is randomly chosen, and this can result in a non-convergent ILC process. If the ILC process is non convergent or solution cannot be found for the chosen maximum number of iterations than the fitness function returns a predefined maximum value. The robotic manipulator has state space constraints, but this is handled by the COILC method. The fitness function value of all individuals is evaluated, and the individuals are sorted in ascending order of their fitness value. For the next generation the fitness the top 50% of the sorted individuals are preserved. The other 50% percent are generated by the cross over operation. 3.3

Crossover Operation

For the crossover operation two parent individuals from the selected 50% with best fitness value are used to generate two new child individuals. During the crossover the parent individuals are split to two parts of 16 and 32 bits each which are later combined. The 16-bit part corresponds to the parameter a and the 32 bits to the other two parameters b and v. 3.4

Mutation Operation

After the crossover operation the individuals of the new generation are sorted in ascending order of the unsigned integer numbers represented by them. Afterwards, while there are two equal numbers in the generation, one of them is mutated by random flip of a single bit.

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Terminal Condition

The terminal condition of the generational process will be a combination of manual inspection and reaching a fixed number of generations. The proposed genetic algorithm relies on the correspondence between the ILC convergence and the correct estimation of the inertia matrix of the controlled robotic system. Through the process inspired by the natural selection the algorithm will find the optimal parameters which correspond to the estimation of the inertia matrix of the robot by selecting those for which the ILC process have better convergence. The actual performance of the proposed algorithm is evaluated in the next section through a computer simulation.

4 Simulation Results The described in [9] computer simulation of TT-3000 robot is used for assessment of the performance of the genetic algorithm. The simulation uses two different sets of the parameters from Eqs. (1) and (2) as presented in Table 1. These sets are estimated in [15] through identification methods. The standard set is used for the simulated robotic arm and the other set of experimental estimations are used for calculating the learning b ðqÞ. operator of the COILC procedure LðqÞ  A Table 1. TT-3000 parameters for the computer simulation. a 6596.2 Experimental estimations [kg.cm2] ^a Standard parameters [kg.cm2]

b 1505.6 ^ b

v 1353.4 ^ v

d1 17.29 d^1

d2 12.49 d^2

fc1 6.46 ^fc1

fc2 3.44 ^fc2

6555.3 1471.8 1847.8 17.29 12.49 8.13 5.51

The robot has to execute the trajectories shown in Fig. 3. If the COILC is applied with the experimental estimation’s parameters set from Table 1, then the execution time for achieving maximum tracking error of 0.02 is 83 s for trajectory A and 103 s for trajectory B. For the genetic algorithm a generation consisting of 120 individuals is considered. They are randomly initialized and afterwards 15 generations are created. The fitness function is the minimum required time for the COILC process to achieve maximum tracking error of 0.02 when trajectory A is considered. The sorted evaluation of the fitness function of each individual through different generations is shown in Fig. 4.

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Fig. 3. Desired trajectories (red and black lines) planned in robot workspace (blue dots).

Fig. 4. Fitness value comparison of selected generations.

After 15 generations the best fitness value was 29.1 s. Table 2 presents the best two individuals from selected generations. The genetic algorithm found set of parameters for which the total COILC execution for trajectory A was reduced from 83 to 29.1 s (about 2.9 times faster execution time).

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Table 2. Individual comparison from selected generations. Generation 1 1 2 2 15 15

Individual representation 32049171138104 30163684557856 25155706356780 26783510365447 24193633813516 24502871327752

a [kg.cm2] 7462 7023 5857 6236 5633 5705

b [kg.cm2] 1910 1972 1265 1439 1267 1265

v [kg.cm2] 1592 1056 1068 1287 1036 1032

Fitness value [s] 53.24 57.71 38.80 48.50 29.10 29.10

For the next experiment the trajectory B was used. Comparison of the convergence of the COILC process is shown in Fig. 5 when using the estimated set and the optimal set of parameters found by the genetic algorithm (the individual 24193633813516 from generation 15, which has the best fitness value and the lowest integer representation). The parameters of the genetic algorithm reduced the COILC execution time from 103 to 58.2 s (about 1.8 times faster execution time). Comparison of the tracked trajectories in workspace by selected COILC iterations is shown in Fig. 6. With the estimated set of parameters, the first few iterations were aborted due to reaching of the workspace limits. When the set of parameters found by the genetic algorithm are used, the tracked trajectories were close to the desired and COILC converges faster. It is worth noting that the parameters found by the genetic algorithm differs from those used for the simulation of the virtual robotic manipulator. This is due to the fact that the COILC learning operator does not take into account the friction but the COILC process compensates also the error caused by the friction.

Fig. 5. COILC convergence with a. model parameters from experimental estimation; b. model parameters found by the genetic algorithm.

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Fig. 6. Comparison of COILC iterations with a. model parameters from experimental estimation; b. model parameters found by the genetic algorithm.

5 Conclusion This research investigates how a genetic algorithm can be applied for estimation of the parameters of the mathematical model of robotic manipulator. The genetic algorithm requires definition of a solution domain and a fitness function for evaluation of the solution domain. The Constrained Output Iterative Learning Control (COILC) is proven to be robust and to generate a convergent process when the correct estimation of the inertia matrix of the robot is used as a learning operator. The COILC method can be applied safely to robotic manipulators with state space constraints even if the learning operator is randomly chosen. This allows the set of possible inertia matrices to be selected as the solution domain and the COILC method to be used as a fitness function for the genetic algorithm. This leads to an algorithm for estimation of the dynamical model parameters of the robot. This approach is validated through a computer simulation and implicates that the COILC can be used both for trajectory tracking and for model parameter estimation for robotic manipulators. Acknowledgement. The research presented in this paper was supported by the Bulgarian National Science Fund under the contract # KP-06-M47/5 – 27.11.2020.

References 1. Song, Z., Yi, J., Zhao, D., Li, X.: A computed torque controller for uncertain robotic manipulator systems: fuzzy approach. Fuzzy Sets Syst. 154(2), 208–226 (2005) 2. Wang, X., Hou, B.: Trajectory tracking control of a 2-DOF manipulator using computed torque control combined with an implicit lyapunov function method. J. Mech. Sci. Technol. 32, 2803–2816 (2018) 3. Uchiyama, M.: Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. SICE (Soc. Instrum. Contr. Eng.) 14(6), 706–712 (1978) 4. Arimoto, S., Kawamura, S., Miyazaki, F.: Iterative learning control for robot systems. In: Proceedings of IECON, Tokyo, Japan (1984)

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5. Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robot. Syst. 1(2), 123–140 (1984) 6. Heizinger, D., Fenwick, B., Paden, B., Miyazaki, F.: Robust learning control. In: Proceedings of 28th Conference on Decision and Control, Tampa, FL (1989) 7. Yovchev, K., Delchev, K., Krastev, E.: State space constrained iterative learning control for robotic manipulators. Asian J. Control 20(3), 1145–1150 (2018) 8. Yovchev, K., Delchev, K., Krastev, E.: Constrained output iterative learning control. Arch. Control Sci. 30(1), 157–176 (2020) 9. Yovchev, K., Miteva, L., Delchev, K., Krastev, E.: Iterative learning control of hard constrained robotic manipulators. In: Zeghloul, S., Laribi M., Sandoval Arevalo J. (eds.) Advances in Service and Industrial Robotics, RAAD 2020. Mechanisms and Machine Science, vol 84. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48989-2_52 10. Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) 11. Mirjalili, S.: Genetic algorithm. In: Evolutionary Algorithms and Neural Networks. Studies in Computational Intelligence, vol. 780. Springer, Cham (2019). https://doi.org/10.1007/ 978-3-319-93025-1_4 12. Goldberg, D., Holland, J.: Genetic algorithms and machine learning. Mach. Learn. 3(2), 95– 99 (1988) 13. Haldurai, L., Madhubala, T., Rajalakshmi, R.: A study on genetic algorithm and its applications. Int. J. Comput. Sci. Eng. 4(10), 139 (2016) 14. Razali, N., Geraghty, J.: A genetic algorithm performance with different selection strategies. In: Proceedings of the World Congress on Engineering, vol II (2011) 15. Shinji, W., Mita, T.: A parameter identification method of horizontal robot arms. Adv. Robot. 4(4), 337–352 (1990) 16. Delchev, K.: Simulation-based design of monotonically convergent iterative learning control for nonlinear systems. Arch. Control Sci. 22(4), 371–384 (2012)

Control of the Vibrations of a Cartesian Automatic Machine Matteo Bottin , Giulio Cipriani , Domenico Tommasino(B) , and Alberto Doria Department of Industrial Engineering, University of Padova, 35131 Padova, Italy [email protected], [email protected] https://research.dii.unipd.it/mmsa/

Abstract. This paper deals with the control of the vibrations of an automatic cutting machine with a cartesian kinematic chain. A particular application of the method of structural modifications is considered, since a small number of frequency response function is available. The Sherman-Morrison formula is used to predict the effect of a tuned vibration absorber (TVA) on the vibrations of the machine generated by the cutting tool; the advantages and the limits of this method are discussed. In order to improve the cutting speed, the cutting head moves along the linear guides with relevant accelerations that can excite the TVA, motion induced vibrations are analyzed with a lumped element model. Keywords: Cutting · Tuned vibration absorber machine · Dynamic modeling

1

· Automatic

Introduction

The increase in the working speed of robots and automatic machines is a typical trend of manufacturing industry. Sometimes high levels of noise and vibrations are detected in machines modified in order to increase the working speed, this happens because the increased speed leads to the excitation of modes of vibration that were not considered in the first design of the machine. Vibrations of the modified machine can be controlled carrying out structural modifications, e.g. increasing stiffness or adding lumped masses or tuned vibration absorbers (TVA). In recent years some effective methods for predicting the effects of structural modifications starting from the measured frequency response functions (FRFs) have been developed [4,10,12,14]. These methods are based on the modal analysis approach and typically require a large set of experimentally measured FRFs. In actual industrial applications, usually small slots of time are available for carrying out experimental tests on existing (and working) machines and robots. For this reason, the engineer in charge of vibration control has to predict the effect of structural modifications using a very limited set of measured FRFs. This paper focuses on this particular aspect of the method of structural modifications. Only 2 measured FRFs are used for predicting the effect of a c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 236–243, 2021. https://doi.org/10.1007/978-3-030-75271-2_25

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TVA on an automatic cutting machine. The next section briefly describes the cutting machine and the experimental tests. In Sect. 3 the fundamentals of the method of structural modifications are summarized. In Sect. 4 the ShermanMorrison formula is used to predict the effect of a TVA on the vibrations of the machine generated by the cutting tool; the advantages and the limits of this method are discussed. The cutting head moves along the linear guides with relevant accelerations that can excite the TVA; hence motion induced vibrations are analyzed in Sect. 5 adopting a lumped element approach. Finally, conclusions are drawn.

2

Experimental Tests on the Cutting Machine

The cutting machine considered in the framework of this research has a cartesian kinematic chain. The cutting head moves at high speed in the Y and Z directions, whereas the cutting tool operates in the X-direction, as shown in Fig. 1. The cutting tool is operated as a pneumatic actuator by inflating and deflating the air inside the cylinder. The high-speed switching needed for cutting is the cause of noisy vibrations. Moreover, the tool speed was increased from 8000 rpm to 13000 rpm to make the machine able to cut hard materials.

Fig. 1. Scheme of the cutting machine.

Unfortunately, the decision came up after the design and construction of the machine. It turned out that at 13000 rpm, the system is excited near a natural frequency and vibrations increase reaching annoying levels. The experimental tests involved a modal analysis on the existing system and the solution is proposed trying to minimize modifications. The experimental analysis was subjected to two major restrictions: 1. To reduce day-offs, it had to be carried out in just one working day. 2. The actual system’s geometry prevents excitation parallel to surfaces and the location of sensors in many points, hence the measures were limited to two points, one on the tool and one on its seat. The analysis is carried out using an instrumental hammer and a triaxial accelerometer, shown in Fig. 2. The excitation is performed near the accelerometer and along its Z-axis.

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(a)

(b)

Fig. 2. (a) Testing equipment and (b) scheme of the proposed solution (in orange the TVA added to the tool).

The FRFs, shown in Figs. 3 and 4, are calculated from the measured accelerations and hammer force, processing the measured signals by means of LabView.

Fig. 3. FRFs measured on the tool.

Figures 3 and 4 highlight the presence of a resonance peak at 187.7 Hz, which can be excited by the tool operating at 13000 rpm (216.7 Hz). Vibrations are critical radially (along accelerometer Z-direction) and, to a lesser degree, axially (along accelerometer X-direction). The FRFs measured on the seat highlight that the mode at 187.7 Hz involves only the tool.

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Fig. 4. FRFs measured on the tool seat.

3

Structural Modifications

The measurement of a set of FRFs between some response coordinates (pn ) and some excitation coordinates (qn ) is the basis of modal analysis methods. The measured FRFs not only allow the identification of the modal parameters of the system [2,5,8], but also allow the prediction of the effect of modifications of the structure without building an actual prototype. The great advantages in terms of reduction in development times and costs have fostered the research in this field and the results of many relevant researches have been published in recent years [3,9,12,13]. The Sherman-Morrison method is a linear algebra technique [6] that have been successfully used to solve many engineering problems including structural modifications [12]. In particular, the effects of structural modifications in milling machines [7] and trusses [11] have been predicted with this method. In [10] the effect of a TVA on a multi-DOF mechanical system has been predicted by means of the Sherman-Morrison method, and in [1] some experimental results have been shown. The Sherman-Morrison approach can be used to predict the effect of lumped masses, stiffnesses and TVAs in dynamic systems described by many coordinates, but can it be used even if there are only a response, an excitation and a modification coordinate. For this reason, the Sherman-Morrison method is suited to industrial applications in which few FRFs are available. The measured FRFs of a vibrating system can be collected in a receptance matrix [A(ω)]. It is the inverse of the dynamic stiffness matrix [Z(ω)], which is related to the mass [M ], stiffness [K] and damping matrices [C] of the dynamic system: (1) [A(ω)] = [[K] + iω[C] − ω 2 [M ]]−1 = [Z(ω)]−1

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Modifications in [Z(ω)] lead to modifications in [A(ω)] that can be calculated with the Sherman-Morrison formula, if [A(ω)] is a square non-singular matrix. In particular, if the modification in [Z(ω)] can be expressed as the product of two vectors {u} and {v}: [ΔZ(ω)] = {u}{v}T

(2)

The modified receptance matrix [Am (ω)] is given by: [Am (ω)] = [A(ω)] −

([A(ω)]{u})({v}T [A(ω)]) 1 + {v}T [A(ω)]{u}

(3)

In particular, if only one modification at coordinate r is carried out adding a lumped element, all the elements of vectors {u} and {v} are zero except the rth elements. The rth element of {v} is the dynamic stiffness of the lumped element (zl (ω)), whereas the rth element of {u} is 1. When experimental results relative to only a response, an excitation and a modification coordinate are available the modified transfer function (apqm (ω)) between the response and the excitation coordinate can be calculated according to this equation [4]: apqm (ω) =

apq (ω) + zl (ω)(arr (ω)apq (ω) − apr (ω)arq (ω)) 1 + zl (ω)arr (ω)

(4)

In the specific case considered in this research, the structural modification is a TVA located at coordinate r, hence this dynamic stiffness [12] has to be considered in Eq. 4: zl (ω) =

(−ω 2 ma ka − iω 3 ma ca ) (ka − ω 2 ma + iωca )

(5)

In which ka , ca and ma are the stiffness, damping and mass of the TVA.

4

Prediction of the Effect of a TVA by Means of the Sherman-Morrison Formula

The TVA consists of a steel cantilever beam with a tip mass ma . The stiffness, damping and mass properties of the TVA determine the natural frequency and the damping ratio of the device:  ka ωn = (6) ma ζa =

2

√

ca ka ma

(7)

Mass ma is related to the mass (M) of the original machine, and typically ma = (0.05 ÷ 0.1)M [14]. The natural frequency of the TVA is tuned to the frequency of excitation or to the natural frequency of the machine, hence Eq. 6 makes it

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possible to calculate the stiffness of the cantilever beam. Damping essentially depends on the material of the beam. Typical values of ζa for a steel beam are in the range 0.01 ÷ 0.05. Stiffness ka refers to bending deformation of the cantilever in the plane perpendicular to the cantilever’s surface, hence the TVA is mounted on the tool as shown in Fig. 2b. The modification due to the TVA is applied to the Z-coordinate. Figures 5a and 5b compare the measured FRF of the tool with the FRF predicted by means of the Sherman-Morrison formula, considering the effect of the vibration absorber tuned to the natural frequency of the tool (fn = 187.7 Hz). Figure 5a compares aXZ (ω) with aXZm (ω), whereas Fig. 5b compares aZZ (ω) with aZZm (ω). The TVA parameters are ma = 0.05 kg, ka = 69544 Nm−1 and ca = 5.90 Nsm−1 . Figures 5a and 5b show that the TVA reduces by about 66% the magnitude of the FRFs at the natural frequency and splits the first resonance peak in two peaks. Nevertheless, the second peak is still high (0.22 g/N at 211.7 Hz in X; 0.38 g/N at 210.1 Hz in Z). Typically, when the dynamic system is dominated by only one resonance peak, the TVA generates two new peaks having similar heights [8]. The different heights shown in Figs. 5a and 5b are due to the presence of higher modes of vibration.

(a) aXZ (ω), aXZm (ω), fn = 187.7 Hz

(b) aZZ (ω), aZZm (ω), fn = 187.7 Hz

(c) aXZ (ω), aXZm (ω), fn = 216.7 Hz

(d) aZZ (ω), aZZm (ω), fn = 216.7 Hz

Fig. 5. Comparison between the measured FRF and the predicted FRF, considering a vibration absorber tuned to fn .

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Parametric simulations were carried out in order to evaluate the tuning frequency of the absorber, which has the largest effect on the first resonance peak. The largest reduction in the first resonance peak was obtained with a tuning frequency of 216.7 Hz, which corresponds to the cutting speed of the tool at 13000 rpm. Figures 5c and 5d consider the vibration absorber tuned to the frequency f = 216.7 Hz. The parameters of the TVA are ma = 0.05 kg, ka = 92693 Nm−1 and ca = 6.81 Nsm−1 . Figures 5c and 5d shows that the TVA splits the first resonance peak in two smaller peaks and reduces the magnitude of the FRFs by about 42% at 216.7 Hz. This reduction is assumed enough to embedd a prototype of the TVA in the real machine to validate the models.

5

Effect of Machine Acceleration

Whilst the structure of the cartesian robot holding the tool is considered to be infinitely rigid for our dissertation (due to its bulky structure), it is true that the cartesian robot must move within the workspace to reach the working positions. Therefore, fast movement are performed by the robot, and high accelerations are exerted to the tool both in Y and Z axis, which can excite tool vibrations. To analyze this phenomenon, a lumped element two-DOFs model of the cutting head equipped with the TVA has been developed. To test if the absorber does influence the excitation of the tool during the movement, a velocity profile along Z-axis is applied. This velocity profile is symmetric and trapezoidal with a maximum speed of 2 m/s, an acceleration time of 0.2 s and a total movement time of 4 s. Results of this analysis are shown in Figs. 6a and 6b. The absorber slightly influences the response of the tool (Fig. 6a), but the difference is so limited to be considered negligible. In fact, the acceleration of the tool is much lower than the one of the absorber (Fig. 6b). Despite the high fluctuations of both tool and absorber, the damping allows to reduce the vibrations in less than 80 ms. This behaviour could be further improved if smoother trajectory profiles would be used, in which no discontinuities in accelerations would be exerted, e.g. third/five degree polynomials.

(a)

(b)

Fig. 6. (a) Comparison of the amplitude of acceleration of the tool with and without absorber; (b) Acceleration amplitude of the response of the absorber.

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Conclusion

The results presented in this paper show that the Sherman-Morrison method is suited to develop a TVA for an actual machine, when few measurements are available. A series of parametric simulations made it possible to improve the tuning of the TVA. Finally, since the TVA is installed on a cartesian robot, a dynamic analysis of its behavior during robot movement was carried out. Future work will implement the TVA on the real machine for validation.

References 1. Betters, E.D., et al.: Dynamic stiffness modification by internal features in additive manufacturing. Precis. Eng. 66, 125–134 (2020) 2. Bottin, M., et al.: Modeling and identification of an industrial robot with a selective modal approach. Appl. Sci. 10(13), 4619 (2020) 3. Braun, S., Ram, Y.: Modal modification of vibrating systems: some problems and their solutions. Mech. Syst. Sig. Process. 15(1), 101–119 (2001) 4. C ¸ akar, O.: Calculation of receptance of a structure modified by mass and grounded spring. In: 2nd International Conference on Advances in MAE (2015) 5. Cossalter, V., et al.: Experimental analysis of out-of-plane structural vibrations of two-wheeled vehicles. Shock Vibr. 11(3, 4), 433–443 (2004) 6. Egidi, N., Maponi, P.: A Sherman-Morrison approach to the solution of linear systems. J. Comput. Appl. Math. 189(1–2), 703–718 (2006) 7. Gibbons, T.J., et al.: Chatter avoidance via structural modification of tool-holder geometry. Int. J. Mach. Tools Manuf 150, 103514 (2020) 8. Inman, D.J.: Engineering Vibrations, 2nd edn. Prentice Hall, Upper Saddle River (2001) 9. Mottershead, J.: Structural modification for the assignment of zeros using measured receptances. J. Appl. Mech. 68(5), 791–798 (2001) 10. Ozer, M.B., Royston, T.J.: Application of Sherman-Morrison matrix inversion formula to damped vibration absorbers attached to multi-degree of freedom systems. J. Sound Vibr. 283(3–5), 1235–1249 (2005) 11. Rezaiee-Pajand, M., et al.: Reanalysis of 2D and 3D truss structures considering simultaneous variations in topology, geometry and size. Eng. Comput. 1–19 (2020). https://doi.org/10.1007/s00366-020-01209-2 12. Sanliturk, K.: An efficient method for linear and nonlinear structural modifications. In: Proceedings of ESDA 2002 (2002) 13. Adamson, L.J., Fichera, S., Mottershead, J.E.: Aeroelastic stability analysis using stochastic structural modifications. J. Sound Vibr. 477, 115333 (2020) 14. Raze, G., Kerschen, G.: H∞ optimization of multiple tuned mass dampers for multimodal vibration control. Comput. Struct. 248, 106485 (2021)

Innovative Mechanism/Robot and their Applications

Design of the 2 D.o.F Compliant Positioning Device Based on the Straight-Line Watt’s Mechanisms Jaroslav Hricko(&)

and Stefan Havlik

Institute of Informatics, Slovak Academy of Sciences, Banská Bystrica, Slovakia {hricko,havlik}@savbb.sk Abstract. The first inspiration in designing any compliant mechanisms usually provides classic kinematic structures with revolute joints. This paper deals with the design of the X-Y micro-positioning device inspired by specific Watt’s configuration. Such arrangement enables straight-line compliant motions of the endplate in a given range. Considering that, the proposed mechanism is symmetric enables simplified calculation of the kinematic and stiffness analyses that are indicated in this work. The optimized device enables motion in actuated axis with minimal parasitic displacements that are less than 0.65% for the maximum displacement. A short review of compliant structures inspired by straight-line linkages is given. Keywords: Compliant mechanisms stage
Watt’s linkage


Straight-line kinematics
Positioning

1 Introduction Constructions of robotic mechanisms, in general, belong to two principal groups. The first and most extended group are mechanisms composed of a set of discrete mechanical components. Each component/part performs a specific function in the mechanical structure and corresponds to given requirements. Typical components are basic building blocks, as links, joints (revolute or prismatic), and actuators [1]. This approach enables build various kinematic structures with several D.o.F. The other approach to construction consists of building compliant mechanisms usually made from one piece of elastic material [2]. In these mechanisms, the desired motions are reached by elastic deflection of their flexural parts/segments, or, the flexural body, as a whole. It should be said that such kind of mechanisms is usually designed for specific applications, as micro/small motion positioning devices in limited D.o.F. The desired performance can be achieved by constructions of compact compliant mechanisms designed as task-oriented structures with a limited range of motions in particular degrees of freedom. Besides micro-engineering, a wide range of applications is in the food industry [3], medicine [4], space, etc. Anyways, there is a similarity between both design approaches. The design of compliant positioning mechanisms frequently goes out from the similarity with classic mechanisms where revolute joints are frequently replaced by elastic hinges of various forms. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 247–255, 2021. https://doi.org/10.1007/978-3-030-75271-2_26

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common features enable the application of known kinematic structures, for instance, the prismatic joints could be replaced by parallelograms (others examples of compliant replacements of the kinematic chains are shown in Fig. 1). It is clear that the design procedure should include careful calculations with force and stresses/strains analysis. It is shown a brief overview of some proposals of straight-line compliant mechanisms inspired by known kinematic structures. In building the straight-line mechanisms the prismatic joints are replaced by parallelograms working in a very limited range angle of revolute motion. This paper analyzes straight-line kinematic structures built as the compact compliant mechanisms. Then the design of the X-Y positioning stage based on the Watt’s linkage is presented. The procedure includes and minimizes the impact of cross/ parasitic errors due to rotation of the input arm.

a)

b)

c)

d)

Fig. 1. Equivalence of joints a) revolute, b) prismatic, c) equivalent universal, d) spherical [5].

2 Compliant Straight-Line Mechanisms A straight-line mechanism is a mechanism that provides motion at least a point on its component in a straight line. Various straight-line linkages and mechanisms have been developed for since very long time [6]. But there are few flexural structures to be suitable for real applications. The straight-line output movement of the end part corresponds often only to the small range input motion. Especially, in the case of the compliant mechanisms, one should choose such a configuration of the linkage that result in minimal rotations of flexural joints. Some examples of straight-line mechanisms built on compact compliant structures are linkages by Hoeken, Scott-Russell, Peaucellier-Lipkin, Roberts [7–16]. The specific kinematics that can produce straight-line movement is the pantograph. This is not exactly the straight-line mechanism, but its construction enables the desired movement in some limits. There are many research works using such linkage (see Fig. 2a) for the design of the mount systems [7, 8]. This mechanism provides decoupled translational motion in 3 D.o.F with the fixed orientation of the endplate and exhibits a significantly high workspace to size ratio [8]. The work [9] describes the design of the robotic leg with the self-adaptive capabilities that combine the Hoeken linkage with a pantograph. Hoeken linkage is frequently used in the construction of compliant grippers. For this purpose, it is required parallel motions of fingers [10].

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The disadvantage of the straight-line linkages is that the straight-line output motion is related to the output point. Unfortunately, the rotation of this point is present too. For reduction of such rotation, it is added parallelogram, as in [10] (see Fig. 2b). In the design of micro-grippers with Scott-Russell’s mechanism, a nearly straight-line parallel jaw motion in a relatively large stroke of the fingers is achieved [11, 12]. An example of nano-positioning mechanisms based on the Scott-Russell straightline mechanism is presented in [13]. Such a mechanism has only one moving axis for manipulation tasks. In order to change the range of straight-line motions, or force, there are specific mechanical reducers/amplifiers. Some examples of such mechanisms bring works [14, 15]. The 2 D.o.F positioning stage based on Robert’s mechanism with a working range larger than 12 mm in each axis is proposed in [16]. The stage shows parasitic motions of the positioning platform less than 1.7% with respect to the input motion.

L1

L2 M

approx. straight-line motion

L1

OUTPUT

INPUT

a)

b)

c)

Fig. 2. Examples of compliant straight-line devices, a) pantograph [7], b) compliant microgripper by Hoeken [10], c) Watt’s linkage

The simplest straight-line linkage that consists of a chain of three rods has been invented by James Watt (see Fig. 2c). All rods are connected by revolute joints, where the two outer bars have equal lengths and are fixed. The free ends of longer bars are connected by a middle rod. The point in the middle of such rod moves approximately in a straight line. Considering that this midpoint rotates during movement and it is a problem to attach a parallelogram to reduce this rotation, this mechanism is not often used in the design of compact compliant devices. An example of utilization of the Watt’s linkage in the form of the compliant device is elaborated in the form of the 2 D. o.F positioning stage.

3 Design of the 2 D.o.F Positioning Platform Compliant X-Y mechanisms are base for many positioning and sensing devices. One of the important design conditions is to minimize the parasitic deflections, i.e. the movement of the end platform should exactly reflect motions of the actuated axis without displacements/rotations in other directions. This practically means that in the ideal case, the relation between input motions and motions of the endplate represents the diagonal transformation matrix. The design study of the positioning device is presented as an illustrative example.

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The task is to design the planar mechanism for the X-Y micro-positioning table working within the range of displacement ±0.5 mm with frequency up to hundred motion cycles per second. External dimensions of this mechanism should be approx. 20  20 mm. For the purpose of this design study building the model/prototype made by the 3D printing technology was chosen. Considering technological possibilities of 3D printing the minimum thickness of flexure of the hinge is 0.5 mm for the nozzle with a diameter of 0.25 mm is proposed. As a material the polyamide PA2200 with Young’s modulus E = 1650 MPa and tensile strength at yield ry = 48 MPa is used. The device should be actuating by PZT actuators. According to the previous discussion, the mechanisms should satisfy the following features: – Motions in particular axes should be maximally decoupled; i.e. the input/actuated displacement in one direction has no, or, minimal influence on displacement in the other direction. – Parameters and characteristics in both directions are identical. – The mechanisms should work within a given frequency range with minimal positional/trajectory errors. The proposed kinematics is a combination of the lever mechanism that amplifies the input displacement of piezoelectric actuators and the Watt mechanism. Considering studies [17, 18] it needs to respect the symmetry and minimal cross deflection effects within the whole range of motions. The kinematic chain of one arm and the structure of the proposed mechanisms are shown in Fig. 3.

θ4

θ3

uxOUT yH

H

θ2

xH

θ1 uxIN

y x

a)

b)

c)

Fig. 3. a) Kinematic chain of one arm of the proposed 2 D.o.F mechanism (solid line – initial configuration, dotted line – moved configuration), b) structure of the x-y flexure, c) coordinate systems for calculation of compliance/stiffness

As the flexural joints, the two-side circular notches are considered for calculations. In the first step, it was necessary to calculate the angle of deflections in particular joints. Because the whole mechanism is symmetric, the calculation of these angles on one leg is sufficient as is shown in Fig. 3. Rem.: In further relations, the number of the joint corresponds to the index of the angle h. The distance between joints 1 and 2 in the

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y-axis is labeled as L1, the length between joints 2 and 3 is equal to 4 * L1 and the distance in the x-axis between joints 3 and 4 is labeled as L2. The displacements of each joint are denoted by uxi and uyi, where index i means the number of the corresponding joint. The simplified calculations of angles hi are then as follow sin h1 ¼

uxIN L1

ð1Þ

sin h1 ¼

ux3 4L1

ð2Þ

ux3 ¼ 4uxIN

ð3Þ

h3 ¼ h4  h1

ð4Þ

uy3 L2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uy3 ¼ 5L1  25L21  u2x3

ð5Þ ð6Þ

uxOUT ¼ L2  L2 cos h4

ð7Þ

sin h4 ¼

By substitution of Eqs. (1) to (6) into (7), the dependence between input uxIN and output uxOUT displacement can be calculated. However, as the angle h4 is very small, therefore the dependence between input and output displacement can be simply expressed as ux3, which is the output motion of the mechanism. It means that the ratio of distances between joints 1 to 2 and 2 to 3 approximately determine the dependence between input and output displacement. According to a brief kinematic analysis, maximal rotation is in joint 1. It means that the angle h1 should be smaller than hMAX according to [2] jhmax j ¼

  eamd bl kM2 6kM1 bh

ð8Þ

where eadm is allowable elastic strain expressed as ry/E, variables bl is the elastic hinge length ratio, bh link height ratio and the variables kM1 and kM2 are design coefficients depending on the joint form/contour. For the chosen form of joints are 0.1079 and 0.52, respectively. The Eq. (7) can not be directly used for dimensioning of the hinge in joint 2. As can be seen in Fig. 3, this joint has to compensate rotation h2 and parasitic displacements in x and y-axes. In calculation of the hinge, the maximum stress should be taken into account [19] rmax ¼

    kta 6ktb  KxFx ux2 þ 2 LKyFy þ KyMz uy2 þ LKyMz þ KhzMz h2 wt wt

ð9Þ

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where kta and ktb are the stress concentration factors, w is the width of the flexure hinge (thickness of the plate), t is the thickness of the hinge and Kij are elements of the stiffness matrix where index i symbolize displacement or rotation in particular axes, and index j specify the load which effects deflections. Performing flexural analysis of the compliance matrix related to H reference system should be calculated for each kinematic configuration. Because the mechanism constitute serial (eq. (10)) and parallel (eq. (11)) kinematic connections the transformation of compliance/stiffness matrices are calculated by [20, 21] C1 ¼

X

 T Ti1 Ci Ti1

ð10Þ

 T Ti1 Ki Ti1

ð11Þ

n

K1 ¼

X n

where Ti1 is the transformation matrix from i-th coordinate system to coordinate system in O1, C1 and K1 are compliance/stiffness matrices related to O1 respectively. For the calculation of the compliance CmechH and/or stiffness KmechH of the proposed mechanism is utilized sketch in Fig. 3c. Considering the mechanism symmetry, only one leg is calculated, and consequently, according to (11) can be calculated desired compliance/stiffness. Because compliance of each flexure hinge is expressed to its end, is in this place located local coordinate system, marked as Oi. The compliance of the one leg related to the point H is  1 CH ¼ T2H T12 K11 TT12 þ K22 TT2H þ T3H C33 TT3H þ T4H C44 TT4H

ð12Þ

where Tij are transformation matrices from i-th coordinate system to j-th coordinate system, Cii/Kii are compliance/stiffness matrices of corresponding flexure hinge located near i-th local coordinate system. As is shown in Eq. (12), it combines both Eqs. (10) and (11), because the connection of the hinges in points O1 and O2 is parallel, but then are flexure hinges connected in the serial configuration. Because the configurations of the other legs to the calculated leg are parallel then the compliance of the whole device is expressed as  1 CmechH ¼ 4C1 H

ð13Þ

Using the above procedure it was possible to evaluate how the final flexural performance of the mechanisms depends on its dimensions and material parameters. The FEM analysis has been used to optimize the proposed mechanism. As has been mentioned, the classical kinematic structure based on the Watt linkage was utilized in the conversion to one elastic body. However, replacement of the revolute joints by its compliant replacement brings some problems like a restricted rotation of the hinge or parasitic displacements of the flexure joint. Even though direct replacement of joints gives some results, the aim of the optimization has found out the position and rotation of hinges that enable to minimize parasitic displacements of the output plate. Hence that was developed up to 10 models of 2 D.o.F positioning stage with various lengths

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of rigid arms and orientation/rotation of the flexure hinges. Comparing results of calculations for selected three modified variants of principal mechanisms brings Table 1. Here, the case c) exhibits minimal cross displacement error of the central square. Subsequently, performing simulations on this model, the geometry of the mechanisms, form, and dimensions of joints were optimized and system dynamics (frequency characteristics) were verified. Table 1. Comparing of calculations results for three modified variants of the proposed 2 D.o.F. positioning mechanism. a)

b)

c)

Kinematic configuration

Motion x [mm] Cross motion y [mm] Cross error x y [%] Max. stress [MPa]

0,4814 0,0808 16,8064 13,707

0,7416 0,0881 11,8836 15,617

0,5300 0,0034 0,6507 15,599

As shown results, the configuration c) from Table 1 provides relative independent displacement in the actuated axis. The parasitic displacement, in this case, is only 0.65% of the displacement in the actuated x-axis. When the final form of the proposed mechanism is known, the stress distribution in the flexure hinges should be solved. As is indicated in Fig. 4a, the maximum stress is located only in hinges 1 and 2. Such design expects an equal dimension of all hinges. In the case, that dimensions of the lever mechanism could differ, then all mechanism’s hinges can be stressed similarly. This case is shown in Fig. 4b, where the dimensions of compliant joints of the lever mechanism are up to 33% bigger than the dimensions of the flexure hinges of the Watt’s mechanism.

4 Conclusion The compact compliant mechanism offers very precise positioning in very small dimensions and this all usually in a monolithic body. The implementations of straightline linkages to structures of such devices provide output movement, although such devices are composite from the elastic replacement of a revolute joint. The paper shows a short description of the design study of the X-Y positioning stage based on the Watt straight-line and lever mechanism that enables the adaptation of a small input stroke of the PZT actuator. The significant advantage of the proposed mechanism lies in minimal parasitic cross deflections; less than 0.65% for maximal displacement. It is indicated an example of the FEM model, where the increasing of the dimensions of the flexure hinges of the build-in lever mechanism can distribute stresses to all flexure hinges.

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The future work should be focused to verify the results from analytical and numerical models with 3D printed prototypes of the proposed mechanism. An example of the 3D printed physical model from polypropylene is shown in Fig. 4c.

a)

b)

c)

Fig. 4. Stress distribution of proposed X-Y positioning stage a) all flexure hinges have equal dimensions; b) the dimensions of the flexure hinges of the build-in lever mechanism are bigger; c) 3D printed physical model of the proposed mechanism. Acknowledgment. This work was supported by the national scientific grant agency VEGA under project No.: 2/0155/19 – “Processing sensory data via Artificial Intelligence methods” and by project APVV-14-0076 – “MEMS structures based on load cell”.

References 1. Vagaš, M., Varga, J.: Design of modular gripper for industrial robot. Appl. Mech. Mater. 436, 351–357 (2013). https://doi.org/10.4028/www.scientific.net/amm.436.351 2. Zentner, L., Linß, S.: Compliant systems: mechanics of elastically deformable mechanisms, actuators and sensors. De Gruyter Oldenbourg (2019). ISBN: 978-3-11-047974-4. https:// doi.org/10.1515/9783110479744 3. Hussain, I., Malvezzi, M., Gan, D., Iqbal, Z., Seneviratne, L., Prattichizzo, D., Renda, F.: Compliant gripper design, prototyping, and modeling using screw theory formulation. Int. J. Robot. Res. (2020). https://doi.org/10.1177/0278364920947818 4. Libu, G.B., Bharanidaran, R.: Design of compliant gripper for surgical applications. Aust. J. Mech. Eng. (2019). https://doi.org/10.1080/14484846.2019.1701396 5. Hricko, J.: Conversion of Peaucellier-Lipkin straight-line mechanism to compact compliant device. Int. J. Mech. Control 16(1), 31–38 (2015). ISSN: 1590-8844 6. Kempe, A.B.: How to Draw a Straight Line; A Lecture on Linkages. Macmillan and co, London (1877) 7. Ishii, Y., Thümmel, T., Horie, M.: Dynamic characteristic of miniature molding pantograph mechanisms for surface mount systems. Microsyst. Technol. 11, 991–996 (2005). https://doi. org/10.1007/s00542-005-0525-5 8. Elgammal, A., et. al.: Parametric design and analysis of a new 3D compliant manipulator for micromanipulation. In: 2017 IEEE International Conference on Advanced Intelligent Mechatronics (AIM) (2017)

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9. Fedorov, D., Birglen, L.: Geometric optimization of a self-adaptive robotic leg. Trans. Can. Soc. Mech. Eng. 42(1), 49–60 (2018). https://doi.org/10.1139/tcsme-2017-0010 10. Beroz, J., Awtar, S., Bedewy, M., Sameh, T., Hart, A.J.: Compliant microgripper with parallel straight-line jaw trajectory for nanostructure manipulation. In: Proceedings of 26th American Society of Precision Engineering Annual Meeting, Denver, CO (2011) 11. Liao, Y.G.: Design and analysis of a robot end-effector for straight-line pick-and-place motion. J. Appl. Sci. Eng. Technol. 4 (2011). ISSN: 1933-0421 12. Zhu, J., Hao, G.: Design and test of a compact compliant gripper using the Scott-Russell mechanism. Archiv. Civ. Mech. Eng 20, 81 (2020). https://doi.org/10.1007/s43452-02000085-3 13. Tian, Y., Shirinzadeh, B., Zhang, D., Alici, G.: Development and dynamic modelling of a flexure-based Scott-Russell mechanism for nano-manipulation. Mech. Syst. Signal Process. 23(3), 957–978 (2009). https://doi.org/10.1016/j.ymssp.2008.06.007 14. Hricko, J., Havlik, S.: Compliant mechanisms for motion/force amplifiers for robotics. In: Advances in Intelligent Systems and Computing: Advances in Service and Industrial Robotics, RAAD 2019, vol. 980, pp. 26–33 (2020). ISSN: 2194-5357. https://doi.org/10. 1007/978-3-030-19648-6_4 15. Tian, Y., et al.: A novel XY nano positioning stage with a three stage motion amplification mechanism. In: 2019 IEEE International Conference on Manipulation, Manufacturing and Measurement on the Nanoscale (3M-NANO), Zhenjiang, China, pp. 206–210 (2019). https://doi.org/10.1109/3m-nano46308.2019.8947353 16. Wan, S., Xu, Q.: Design and analysis of a new compliant XY micropositioning stage based on Roberts mechanism. Mech. Mach. Theory 95, 125–139 (2016). https://doi.org/10.1016/j. mechmachtheory.2015.09.003 17. Havlik, S., Hricko, J.: Some quality measures in designing compliant mechanisms for robotic devices. Mech. Mach. Sci. 84, 438–447 (2020). ISSN: 2211-0984. https://doi.org/10. 1007/978-3-030-48989-2_47 18. Havlik, S., Hricko, J.: About the accuracy of fast moving robotic devices based on compliant mechanisms. In: Advances in Intelligent Systems and Computing: Advances in Robot Design and Intelligent Control, vol. 540, pp. 162–170. Springer, Wroclaw (2016/2017). ISBN 978-3-319-49057-1. ISSN: 2194-5357 19. Lobontiu, N.: Compliant Mechanisms: Design of Flexure Hinges, CRC Press (2003). ISBN: 0849-313-678 20. Sciavicco, L., Siciliano, B.: Modeling and Control of Robot Manipulators. Electrical Engineering Series. McGraw-Hill International Editions (1996). ISBN: 0-07-114726-8 21. Li, Y., Xu, Q.: Design and analysis of a totally decoupled flexure-based XY parallel micromanipulator. IEEE Trans. Robot. 25(3), 645–657 (2009)

Optimal Design of a Five-Bar Mechanism Dedicated to Assisting in the Fingers FlexionExtension Movement Araceli Zapatero-Gutiérrez1(&), Med Amine Laribi2, and Eduardo Castillo-Castañeda1 1

Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada Unidad Querétaro, Instituto Politécnico Nacional, Santiago de Querétaro, Mexico 2 Institut PPRIME, Département Génie Mécanique et Systèmes Complexes, Université de Poitiers, Poitiers, France

Abstract. The recovery after a stroke requires appropriate rehabilitation processes. Recovering hand functionality is essential for the reintegration of the patient into his habitual life. This article presents an innovative finger rehabilitation device capable of producing the real flexion-extension trajectory. The proposed device architecture is based on a five-bar mechanism. An optimal design problem solved using the gradient optimization method provides the geometric parameters for the five-bar mechanism. Keywords: Finger rehabilitation Gradient optimization method


Five-bar mechanism
Optimal design


1 Introduction Strokes represent a global public health problem. Only in the US is the leading cause of long-term disability. People with stroke usually present paralysis or weakness in the upper extremities [1]. Between 2001 and 2005, the average cost during the first year for rehabilitation services due to strokes was 7318 dollars [2–5]. In traditional therapy, three exercises are performed for the complete rehabilitation of the hand and wrist. 1. Open and close the hand: the physiotherapist manipulates the patient’s fingers to perform flexion-extension movements. 2. Grip movement: to increases the strength and precision of the hand and 3. Wrist flexion-extension movement [5]. The fingers’ repetitive flexion-extension movements have shown significant hand functionality recovery [6, 7]. Robotic assistive technology has a high potential to contribute safely and reliably in limb rehabilitation processes [8–10]. Repetitive movements with precision and minimal error in trajectory tracking can be performed. Since the 1990s, developments in assistance robots for after stroke have been increasing, representing a significant complement to traditional physical therapy [4]. The hand’s structure consists of twenty-seven bones, eight on the wrist, five on the metacarpus and fourteen on the fingers, called phalanges. Each finger, except for the thumb, has three phalanges (distal, middle and proximal) and two joints: distal © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 256–264, 2021. https://doi.org/10.1007/978-3-030-75271-2_27

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257

(DIP) and interphalangeal (PIP). The thumb has two phalanges and one joint: the interphalangeal joint (IP). Each proximal phalanx articulates with the metacarpal through the metacarpophalangeal joint (MCP) [11, 12], as shown in Fig. 1a. In this paper, a finger rehabilitation device’s dimensions, with a five-bar mechanism configuration, are computed through an optimization process. The device focuses on following the flexo-extension movement of fingers, except the thumb. The flexion-extension movement of the four fingers is similar in shape. However, it varies in amplitude according to the length of the finger. The trajectories shown in Fig. 1b are real trajectories of a subject that were obtained through the motion capture system Qualisys Track Manager and processed using Matlab.

Fig. 1. a) Hand structure. b) Real trajectories of the fingers.

Section 2 contains a brief review of the state of art focused on rehabilitating the fingers, commercial devices, and developments based on planar mechanisms. The kinematic analysis of a five-bar mechanism proposal for rehabilitation device is also presented. Section 3 presents the gradient optimization method and the results. Section 4 presents the conclusions of this work.

2 Fingers Rehabilitation Devices Some of the commercially available finger rehabilitation devices are, for example, CyberGrasp, Hand of Hope, Gloreha and Amadeo [13]. This last one dominates the fingers rehabilitation robot assistive market. It uses a series of mechanisms that produce a linear trajectory in the plane to provide flexion-extension movement assistance to the fingers [14]. Other developments are based on planar mechanisms (not commercial), [15–19] presents a twelve-bar, eight-bar, six-bar and four-bar, respectively, exoskeleton mechanism of one degree of freedom (DOF). The work presented in [20] proposes a system of three five-bar mechanisms to mobilize the five fingers. It is divided as follows: one mechanism for the thumb, one for the index and middle fingers, and another mechanism for the ring and pinky fingers. All the five-bar mechanisms have the same configuration.

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The five-bar-mechanism is well-known for its capability to reproduce complex trajectories with greater accuracy. Due to this characteristic, a single five-bar mechanism of two DOF is proposed to produce the four long fingers’ flexion-extension movement with rehabilitation purposes. One DOF mechanism cannot follow point by point a real finger flexion-extension movement; two DOF solve this problem. A real finger trajectory is used as a reference trajectory, obtained from the recorded flexion-extension movements from several healthy subjects through a motion capture system. The method to get the real trajectory can be consulted in [21]. The technical requirements for the single five-bar mechanism are listed below: 1) The mechanism should follow the reference trajectory with high accuracy. 2) Only one configuration of the five-bar-mechanism will be considered. 3) The mechanism will reproduce the flexion-extension movement of the four long fingers simultaneously. 2.1

Kinematical Analysis for the Five-Bar Mechanism Proposal

Figure 2a shows a five-bar symmetric mechanism, where L1 ; L2 ; L3 ; L4 represent the lengths of the links with L1 ¼ L4 L1 ¼ L3 and L2 ¼ L3 L2 ¼ L4 L2 ¼ L4 L2 ¼ L4 . The length L0 is the distance between points A and E, which are fixed. The points B and F describe circular trajectories of radius L1 and L2 , respectively. The actuated joints correspond to h1 and h2 , and the point C corresponds to the mechanism end-effector. The forward kinematics equations for the point C are expressed by Eq. (1) [22]. Figure 2b shows the change in mechanisms position according to the finger position.

Fig. 2. a) Five-bar mechanism. b) Five-bar mechanism position related with finger position.


C¼

Cx Cz

 ð1Þ

where, 1 Cx ¼ L1 cos h1 þ ½L0 þ L1 ðcos h2  cos h1 Þ  ½L1 ðsin h2  sin h1 Þ 2

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffi# L22 1  H2 4

Optimal Design of a Five-Bar Mechanism Dedicated to Assisting

1 Cz ¼ L1 sin h1 þ ½L1 ðsin h2  sin h1 Þ þ ½L0 þ L1 ðcos h2  cos h1 Þ 2

259

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffi# L22 1  H2 4

and, H 2 ¼ L20 þ 2L0 L1 ðcos h2  cos h1 Þ þ 2L21 ½1  cosðh1  h2 Þ

ð2Þ

Figure 3a shows the desired real reference trajectory D that the mechanism’s endeffector must follow. Through inverse kinematics analysis, the values of h1 and h2 are computed. From Fig. 3b, h1 is composed by the sum of the b and c angles, as given in Eq. (3). h2 is made up of the subtraction between the x and r angles, as given in Eq. (4). The right triangles formed by the ACE points are useful to compute the c and x angles, as indicated in the Eqs. (5) and (6) [23]. To compute the c and x angles, it is necessary to consider the coordinates of the real reference trajectory D point by point to assure that point C matches with it, D comes from the real trajectory and C corresponds to the mechanism end-effector.

Fig. 3. a) The real reference trajectory. b) five-bar mechanism inverse kinematics.

h1 ¼ b þ c

ð3Þ

h2 ¼ x  r

ð4Þ

c ¼ atan2ðDz ; Dx Þ

ð5Þ

x ¼ p  atan2ðDz ; L0  Dx Þ

ð6Þ

Solving the inverse kinematics of the five-bar mechanism is necessary to define the configurations of points B and F, which are called working modes. This configuration can be either up or down, with a sign (+) or (−) for B and F. Figure 4 shows the possible working modes of the five-bar mechanism. For this five-bar mechanism proposal, only working mode (+ +) is considered. Using the law of cosines and the properties of the sine and cosine, the u and a angles are defined through Eqs. (7) and (8). While b and r angles are defined by Eqs. (9) and (10) [22].

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Fig. 4. Working modes: a) (+ +), b) (+ −), c) (− +), d) (− −).

u ¼ atan2ðsin u; cos uÞ

ð7Þ

a ¼ atan2ðsin a; cos aÞ

ð8Þ

where, L21 þ L22  ðD2x þ D2z Þ 2L1 L2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin u ¼ B 1  cos u2

cos u ¼

cos a ¼

L23 þ L24  ½ðL0  Dxi:N Þ2 þ D2zi:N  2L3 L4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin a ¼ F 1  cos a2

b ¼ atan2ðL2 sinðp  uÞ; L1 þ L2 cosðp  uÞÞ

ð9Þ

r ¼ atan2ðL3 sinðp  aÞ; L4 þ L3 cosðp  aÞÞ

ð10Þ

3 Synthesis Problem The geometric parameters of a five-bar mechanism were obtained through an optimization process based on the fingertips trajectory generation. The middle finger’s data set is used as the reference trajectory to formulate the objective function and obtain the optimal parameters of the mechanism. The end-effector, located at the point C, must follow the specified trajectory D with a minimum error.

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3.1

261

Formulation of the Problem

The minimization of the error function EðI Þ expressed in Eq. (11) is defined as the sum of the square distance between the ith position of the desired path and the coordinates of the C point of the mechanism, where N is the total number of points. The design vector I, defined by Eq. (12), contains the set of variables of the five-bar mechanism computed during the optimization process. In the iteration process, each design variable is constrained to the minimum and maximum values defined in Table 1 by the bounding interval BI expressed in Eq. (13). The first two parameters of the design vector I were defined in the kinematic analysis section; Xt and Zt correspond to an offset applied on the x and z-axis, respectively, to place the real trajectory in the working space of the five-bar mechanism. The optimization problem is subject to four constraints inequations R1 , R2 , R3 and R4 which lead to feasible solutions H. N 1X E ðI Þ ¼ N i¼1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðCxi  Dxi Þ2 þ ðCzi  Dzi Þ2 ;

ð11Þ

Minimize EðI Þ; Subject to: I ¼ ½L1 ; L2 ; L0 ; Xt ; Yt 

ð12Þ

  I 2 BI ¼ I min ; I max

ð13Þ

fR1 ; R2 ; R3 ; R4 g 2 R where,  2  L1 þ L22  ðD2xi:N þ D2zi:N Þ  \1 R1 ¼   2L L 1 2

  L2 þ L2  ½ðL  D Þ2 þ D2  0 xi:N  3 4 zi:N  R2 ¼  \1   2L3 L4 R3 ¼ jHi j\1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi  L 2 1   2 R4 ¼   \1  Hi2 4 If the restrictions R3 and R4 are not respected, they will produce an indeterminacy within the direct kinematics during point C calculation.

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3.2

Gradient Implementation

The gradient method, implemented under MATLAB by the fmincon function, is applied to find the optimal values of the design vector I leading to the minimum of the objective function E ðI Þ. This function is used to solve optimization problems of restricted nonlinear parameters. It finds the minimum of a scalar function of several variables that begin with an initial estimate. This method is also known as nonlinear programming [24]. The upper, lower bounds and the initial guess I 0 for the fmincon function are expressed in Table 1. A maximum of 150 iterations (M axI ) with 2000 allowed evaluations (M axFE ) of the function was chosen. Table 1. Lower, Upper boundaries and the initial guess. BI min

I I max

I L1 100 350 I0 101

L2 L0 Xt Yt 100 100 80 80 350 350 300 300 101 101 81

81

Table 2 shown the optimal parameters that were obtained according to the optimization algorithm. Figure 5a shows the vector diagram of the five-bar mechanism in the xz plane. Figure 5b shows the mechanism’s CAD. The mechanism’s first design consists of placing a rigid bar attached to the mechanism’s point C, located in the yz plane. This rigid bar will have slides on the xy plane that will be adjusted according to the fingers’ lengths. The adjustable slide will provide a similar movement in the other fingers but with a different length. The hand’s position will be with the palm down, as is shown in Fig. 5b. The five-bar input angles and the quadratic error are shown in Fig. 5c.

Table 2. Optimal parameters for the five-bar mechanism. I [mm]

Initial input angles [degrees] L2 L0 Xt Zt h1 h2 L1 101.09 108.67 101.20 82.69 92.29 153.55 83.07

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Fig. 5. a) The optimal solution of the five-bar mechanism. b) Kinematic spatial configuration of the five-bar mechanism. c) Input angles (degrees) and quadratic error (mm).

4 Conclusion This paper presents the optimal dimensions of a five-bar mechanism using gradient method optimization. The five-bar mechanism is designed to assist in the fingers’ flexion-extension movements (except the thumb) during the hand rehabilitation process. The mechanism follows a real flexion-extension trajectory with minimal error; the real flexion-extension trajectory was obtained through a motion capture system. This paper investigates only one working mode of the five-bar mechanism. Future works will focus on the study of compactness and safety aspects as well as the development of the first experimental prototype. Acknowledgments. The authors are grateful to the Pprime Institute of the University of Poitier, France and the Instituto Politécnico Nacional (IPN), México. As well as the Consejo Nacional de Ciencia y Tecnología (CONACYT), México, for the facilities provided for this research.

References 1. Jhons Hopkins Medicine. Health Home. Conditions and Diseases. Arm care after stroke. https://www.hopkinsmedicine.org/health/conditions-and-diseases/stroke/arm-care-after-astroke. Accessed 8 Jan 2020 2. Benjamin, E.J., et al.: Heart Disease and Stroke Statistics—2019 Update A Report From the American Heart Association (2019). https://doi.org/10.1161/CIR.0000000000000659 3. Laakso, L.: The role of physiotherapy in palliative care. Aust. Fam. Physician-RACGP 35 (10), 781 (2006) 4. Yue, Z., et al.: Review article hand rehabilitation robotics on poststroke motor recovery. Behav. Neurol. 1–20 (2017). https://doi.org/10.1155/2017/3908135 5. McConnell, A.C., et al.: Robotic devices and brain-machine interfaces for hand rehabilitation post-stroke. J. Rehabil. Med. 49(7), 449–460 (2017). https://doi.org/10.2340/165019772229 6. Carey, J.R., et al.: Analysis of fMRI and finger tracking training in subjects with chronic stroke. Brain 125, 773–788 (2002) 7. Carey, J.R., et al.: Comparison of finger tracking versus simple movement training via telerehabilitation to alter hand function and cortical reorganization after stroke. Neurorehabil. Neural Repair. 21, 216–232 (2007). https://doi.org/10.1177/1545968306292381

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8. Kwakkel, G., et al.: Effects of robot-assisted therapy on upper limb recovery after stroke: a systematic review. Neurorehabil. Neural Repair. 22(2), 111–121 (2008). https://doi.org/10. 1177/1545968307305457 9. Krebs, H.I., et al.: Forging Mens et Manus : The MIT Robotic Therapy. Springer International Publication (2016). https://doi.org/10.1007/978-3-319-28603-7 10. Fasoli, S., et al.: Effects of robotic therapy on motor impairment and recovery in chronic stroke. Arch. Phys. Med. Rehabil. 84, 477–482 (2003). https://doi.org/10.1053/apmr.2003.50110 11. Jhons Hopkins Medicine. Health Home. Treatments, Tests and Therapies. Anatomy of the hand. https://www.hopkinsmedicine.org/health/treatment-tests-and-therapies/anatomy-ofthe-hand. Accessed 8 Jan 2020 12. Borobia, C.: Valoración del daño corporal. Miembro superior, Masson Elsevier, Barcelona, España (2006) 13. Maciejasz, P., et al.: A survey on robotic devices for upper limb rehabilitation. J. Neuroeng. Rehabil. 11(3), 1–29 (2014) 14. AMADEO®: The hand therapy world champion. https://tyromotion.com/en/products/ amadeo/. Accessed 06 Jan 2020 15. Díaz Montes, J.C., et al.: Design of a twelve-bar planar mechanism for finger orthosis. Revista Mexicana de Ingeniería Biomédica 39(1), 52–64 (2018) 16. Wolbrecht, E.T., Reinkensmeyer, D.J.: Single degree-of-freedom exoskeleton mechanism design for finger rehabilitation. In: 2011 IEEE International Conference on Rehabilitation Robotics, pp. 1–6. IEEE, Zurich (2011). https://doi.org/10.1109/ICORR.2011.5975427 17. Bataller, A., et al.: Evolutionary synthesis of mechanisms applied to the design of an exoskeleton for finger rehabilitation. Mech. Mach. Theory 105, 31–43 (2016). https://doi. org/10.1016/j.mechmachtheory.2016.06.022 18. Sands, D., Mccormack, J.: Design method for a reconfigurable mechanism for finger rehabilitation. In: Proceedings of the IASTED International Conference on Robotics and Applications (2010). https://doi.org/10.2316/P.2010.706-017 19. Yu, J.J., et al.: Mechanical design and simulation of a finger rehabilitation robot. Appl. Mech. Mater. 366, 805–811 (2013). https://doi.org/10.4028/www.scientific.net/amm.365366.805 20. Huang, Y.Y., Low, KH: Initial analysis and design of an assistive rehabilitation hand device with free loading and fingers motion visible to subjects. In: 2008 IEEE International Conference on Systems, Man and Cybernetics (SMC 2008), pp. 2584–2590 (2008) 21. Zapatero-Gutiérrez, A., Castillo-Castañeda, E., Laribi, M.A.: On the optimal synthesis of a finger rehabilitation slider-crank-based device with a prescribed real trajectory: motion specifications and design process. Appl. Sci. 11(2:708), 1–24 (2021). https://doi.org/10. 3390/app11020708 22. He, G., Lu, Z.: Optimization of planar five-bar parallel mechanism via self-reconfiguration method. Chin. J. Aeronaut. 18(2), 185–192 (2005) 23. Cortés Torres, E., de J. et al.: Desarrollo de un robot paralelo 5R para la impresión en 3D. In: Memorias IV Congreso Internacional de Ingeniería Mecatrónica y Automatización - CIIMA 2015. pp. 12–22 (2015) 24. The MathWorks, Inc.: Optimization Toolbox for Use with MATLAB, User’s Guide. Version 2, pp. 4-32–4-44 (1999/2000)

An Extendable Continuum Robot Arm to Deal with a Confined Space Having Discontinuous Contact Area Daisuke Matsuura1(&), Ryota Shioya1, H. Harry Asada2, and Yukio Takeda1

2

1 Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan [email protected] Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract. This paper proposes a concept for constructing a flexible extendable arm by feeding a number of intermediate moving plates along a flexible feed screw and deploying each of them at a certain contactable area. The mechanism is aiming to bear a practical working load to support human’s labor by utilizing a bracing on surrounding walls. Generally, a narrow and confined workspace which human worker’s arms can hardly reach does not provide a continuum and smooth wall surface and is composed of many discontinuous areas instead. Therefore, the mechanism deploys each intermediate plate at a target contactable area, and pushes rest of the plates forward. In order to bend the arm according to the workspace, tendon-driving technique is employed. In this paper, a basic composition of the arm is explained and a simple geometric calculation scheme to obtain a configuration of the arm to avoid insufficient condition is proposed. Finally, several experiments with a prototype setup are performed to demonstrate the feasibility of the proposed concept. Keywords: Flexible robot arm
Extendable mechanism Flexible feed screw
Utilization of contacting to objects


Tendon drive


1 Introduction When we look around our living environment and factory environment, there are so many various narrow and confined workspaces. It does not cause a problem when constructing a facility from scratch because an equipment can be installed before building surrounding walls, but maintenance of those equipment becomes typically harder than installation because human workers can hardly reach a narrow and confined workspace surrounded by many walls and equipment. This problem, called Last OneFoot Problem [1, 2], can generally be seen in various maintenance and repair work sites of human labors, and should be solved to elongate the lifetime of old facilities. In order to solve this problem, thin and lightweight robotic arms will become an effective solution. Research and development of such kind of robotic arms can be classified into several typical classes, and from the point of view of mechanism design, hyper redundant arm [3–5] and continuum arm [6–10] can be said as the most popular © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 265–273, 2021. https://doi.org/10.1007/978-3-030-75271-2_28

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approaches. A biggest issue of these type of robotic arms is that achieving high lengthto-width ratio is difficult because their load bearing capacity is rapidly reduced as the increase of the length due to the cantilever structure. Considering this issue, making a robot bracing on the surrounding wall becomes an effective solution to achieve high aspect ratio and adapting to the workspace shape [11] just like endoscopes. However, just lying a wall does not guarantee an enhancement of stiffness to achieve an effective load bearing capacity to perform a heavy-duty task. Moreover, a typical working space of a human labor does not have smooth and entirely continuum surface that the arm can slide on it without being stuck. Thus, an arm should be actively bent to jump over a fragile or completely hollow section, and should be able to choose a suitable contact area and brace on that area so that the robot itself and a reaction force due to an endeffector’s task can be supported. In summary, a robot arm capable of achieving a large length-to-width ratio for passing through a confined and narrow workspace while bearing a meaningful working load to support a human’s labor should satisfy the following requirements; (1) Flexible enough to be bent along the workspace shape. (2) Stiff enough to achieve a practical load bearing capacity by utilizing bracing on contactable areas. (3) Capable of transporting number of moving plates and deploying them on certain discontinuous contactable areas. Considering above three requirements, the authors came up with a concept of a novel extendable flexible arm mechanism shown in Fig. 1. The mechanism is composed of (a) a flexible feed screw which works as a backbone and a driving element, (b) a base plate and several moving plates that can move along the flexible feed screw by using a feed nut and (c) tendon cables connected to a base plate and moving plates. Since each nut on a base plate and intermediate plates is driven by an electric motor, the whole mechanism establishes a differential drive system. Namely, when only a nut on a base plate is driven while another nut on an intermediate plate is blocked, the feed screw and intermediate plate are fed together and achieve an extension motion. In opposite, when a nut on an intermediate plate is synchronously rotated in a same direction to that on the base plate while the intermediate plates’ rotation is constrained due to the bracing, the plate will be fed in the opposite direction against the feed screw’s motion. As the result, the intermediate plate stays at a constant location while the feed screw and rest of the intermediate plates are extended. This principle is experimentally confirmed by using a prototype flexible arm as shown in Fig. 2. The detail of the prototype will be explained in the following Sect. 3. Thanks to the differential driving and tendon cables, the arm can be extended and bent, and intermediate plates can be deployed one by one from the lowest side. Since the feed screw is much thinner than the diameter of the intermediate plate, the arm can be bent largely, but can be stiff by minimizing the length of the air-hovering section close to the end effector by utilizing the bracing. As the result, the arm achieves high extension ratio, can easily been inserted into a narrow space, and effective length of the arm which deforms due to the external load can be minimized. However, the mechanism also has some drawbacks due to the flexible composition and tendon driving. In the following sections, a geometric calculation scheme of the arm’s configuration to

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avoid insufficient configurations will be discussed, and some experimental results will be shown to demonstrate the feasibility and usefulness of the concept.

2 A Geometric Analysis and Evaluation Indices to Determine a Workspace Capable of Avoiding Insufficient Configurations In this section, some principles of the flexible arm’s behavior will be discussed under considering geometry and statics so that the length of the flexible screw and tendon cables to achieve a given target position and orientation of an end effector can be determined. In order to simplify the discussion, a minimum unit of the arm composed of a base- and moving plates, a feed screw and couple of tendon cables connecting them will be called a segment, and considered as a basic component of the arm. In case of a flexible mechanism driven by tendon cables, the arrangement of the cables against the flexible screw dominates the behavior of the system. Two different cable arrangements will therefore be considered to determine an achievable workspace. When a flexible screw receives only a pure moment at its end, it will form a perfect arc. When the base- and moving plates are assumed to be orthogonal to the feed screw, the DoF of a segment in a planar case can be said as two, since a specified position (x, y) or one of the positions (x or y) and orientation of the moving plate, h, can be achieved. Because of this basis, at least two segments should be connected in series to achieve a given target position and orientation of an end effector. When antagonistic drive using the feed screw’s spring back force and the tendon cables’ tension is considered, essential number of cables and feed screw for each segment is (DoF + 1), so 1screw and 2- (in case of planar arm) or 3- (in case of spatial arm) cables can be said sufficient. In case of a planar two segment arm, a position and orientation of end effector and one of the x- or y- (typically, x-) position of the second moving plate can be specified. As the result, the system becomes 4-DoF including 1-redundancy. It can therefore be said that there is a necessity to determine a unique configuration of the arm among infinite candidates. Based on the above-mentioned principle, a geometric analysis scheme to achieve a target position and orientation by a two-segment arm can be discussed. When the orientation of the end effector is assumed to possess perpendicular to the ground, arbitrary couple of arcs #1 and #2 can be drawn against a couple of given end-effector’s horizontal and vertical positions against the origin, X2 and Y2, as shown in Fig. 3. Since the two arcs are sharing a common-tangential line at a connecting point, P1, the two arcs always satisfy a constraint condition,
 X22 þ Y22 2X2 Y2 R¼ and h ¼ asin 2 ; 2Y2 X2 þ Y22

ð1Þ

where R and h are the sum of the radii of the two arcs and the orientation of the intermediate moving plate, respectively. With respect to an arbitrary-given horizontal position of P1 represented by X2, corresponding radii, r1 and r2, and center of the arcs, C1 and C2, can be determined. Three examples are illustrated in Fig. 3. Among these

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Fig. 1. The composition and important features of the proposed extendable flexible robot arm.

(a)

(b)

(c)

(d)

Fig. 2. Snapshots of a prototype arm while performing an extending and bending motion.

cases, all of the three (red/pink/blue) arcs satisfy the above constraint condition, and thus are potential candidates of a backbone curve of the arm. The remaining problem is how to choose a unique solution among them. In order to define a reasonable evaluation index to determine a unique solution, the avoidance of insufficient conditions should be considered. Regarding this, there are two representative conditions in which the arm hardly bear neither the external load nor its own weight. These conditions are illustrated in Fig. 4. Firstly, when a tendon cable crosses the flexible screw, the segment’s load bearing functionality will be partially lost so that the direction of an acceptable external torque will becomes unipolar (case A). Secondly, when the angle of a tendon cable and the moving plate becomes less than 90°, the cable no longer can be kept in tension (case B). To evaluate how an arm’s condition is close to each of these two insufficient conditions, evaluation indices are respectively derived. For the case (A),      ~ arc O1;j O2;j ; line Ai;j Bi;j 2 ; E1;j ¼ min D

ð2Þ

~ is the minimum distance between the arc connecting O1 and O2 and line where D connecting A1(2, …) and B1(2, …), normalized by one of the smallest cable connection

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point radius on the base or moving plate, and i and j are the number of cable and segment, respectively. For case (B), E2;j ¼

N Y i¼1

"

# 1 1 : 1 þ eðhi;j hthr Þ

ð3Þ

By using these two indices, a final evaluation index is determined as following, Etotal ¼

M Y j¼1

! E1;j

M Y

! E2;j :

ð4Þ

j¼1

This index varies between 0 and 1, and should be maximized to avoid the insufficient configuration at most. In order to reveal the workspace of the arm, distribution of Etotal among the arm’s reachable region is examined with respect to the two different cable arrangements as shown in Fig. 5. Intuitively, the arrangement #1 in the figure seems to be suitable to construct an arm having many intermediate plates since it forms a naturally tapered shape. Another arrangement #2 is, on the contrary, more concentrated to construct an arm having only two segments since the first segment is suitable to generate force rather than moment thanks to the short distance of the cable connection point and screw. The second segment is, on the other hand, suitable for generating the moment rather than the force since the distance between the cable connection point and screw is large. Distances between each cable connection point from the center of each plate with respect to each arrangement is written in Table 1. The distribution of Etotal with respect to the two cable arrangements is plotted in Fig. 6. It can be seen that the border of the colored (reachable) region in case of the arrangement #1 is bigger than that of #2, but the region in which Etotal marks highest value close to 1 is almost the same. In addition, both of them covers 500 mm distant range from origin which almost covers a typical adult labor’s arm length. On the other hand, when the percentage of the first segment length against the entire arm length is compared, that in arrangement #2 marked significantly smaller value than arrangement #1 as shown in Fig. 7. It has been seen in a preliminary experiment that when the first segment becomes too long, deflection of it due to the moment generated by the second segment’s cables’ tensions acting on the end effector becomes too large, and that makes the positioning of the end effector impossible. To avoid this situation, the cable arrangement #2 seems to be more suitable than the arrangement #1.

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r1

’ r2 r1+ R=

(Arc#1)

P2 (X2,Y2)

case (A) Tendon cable is acrossing the screw.

P1 (X1,Y1)

O

X

r2 (Arc#2)

o2 o2’ o2”

Case (B) Too shallow tendon cable angle.

Fig. 3. Geometric analysis of the two-segment arm.

Fig. 4. Insufficient configurations

(a) Arrangement #1

(b) Arrangement #2

Fig. 5. Two different cable arrangements used for the workspace evaluation.

Table 1. Pitch circle radii of the cable connection point Arrange. #1 Base plate Middle plate End plate

Cable #1 90 mm 20 mm –

Cable #2 65 mm 40 mm 20 mm

Arrange. #2 Base plate Middle plate End plate

Cable #1 100 mm 10 mm –

Cable #2 75 mm 30 mm 70 mm

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(a) Arrangement #1

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(b) Arrangement #2

Fig. 6. Distribution of Etotal among the reachable region of the end effector.

(a) Arrangement #1

(b) Arrangement #2

Fig. 7. Ratio of the first segment length against the entire arm length.

3 Experimental Validation Using a Prototype Setup A spatial prototype setup including two segments has been fabricated as shown in Fig. 2 for demonstrating the feasibility of the proposed concept. This setup includes a flexible screw having a M7 metric thread (1=400 dia.) and of one-meter long, six-tendon cables driven by servo motors (Dynamixel MX-106, ROBOTIS inc.), two 3D-printed moving plates having cable connection points according to the cable arrangement #2. For measuring the arm’s configuration, strain gauges (for cable tension), a linear encoder (for the feed screw’s length) and two USB cameras (for the position and orientation of the moving plates) are used. In the following experiments, lengths of the feed screw and cables in tension, and tensions of cables that were passively released accordingly were manually controlled while observing the positions of the two moving plates. Firstly, positioning capability of the arm has been demonstrated by achieving a target position of the end effector, (xp, yp) = (450, 100) millimeters and a target orientation, hP = 0. Although this position seems to be achievable since it is included in the highest Etotal region in Fig. 6, there is an error factor because of the compressive stress caused by the cable tension as mentioned in the previous section. It was therefore seen that only when the intermediate plate was braced on the wall behind the arm, the target position could be achieved as shown in Fig. 2(d). This result points out the difference between the result of the simple geometric analysis and actual physical system’s behavior, but also demonstrates the effectiveness of the concept that positioning capability of the arm can be enhanced by utilizing the bracing. Secondly,

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effectiveness of the bracing for enhancing the load bearing capacity of the arm has been demonstrated. In this experiment, a deflection of the end effector’s position due to gravitational load has been examined with and without bracing condition after locating the end effector to a specific position. As shown in Fig. 8, the deflection of the end effector with respect to several different amount of weight on the end-effector and with or without bracing the arm on the side walls was measured. In addition, the bracing condition was differed between a single-sided support for the intermediate plate and a double-sided support for the intermediate plate and the end effector. It can be seen that in Fig. 8 that under the single-side support condition, the deflection of the end effector, DY, was reduced by 14% against a non-bracing condition. In case of double-side support condition, the reduction ratio became 96% which is significantly large against the single-side case. From this result, it can be said that bracing the moving plates on the surrounding object can enhance the load bearing capacity of the arm after it has extended.

Fig. 8. Effect of the bracing under different condition; Single-side bracing at the middle plate (L), and Double-side bracing at the middle and end plate (R)

4 Conclusion Aiming to solve the Last-One-Foot Problem in our daily and industrial environments, this research has been carried out to propose a concept of a novel extendable continuum robot arm, and to grab a behavior of the arm on the basis of geometry and statics. As the result, following conclusions have been obtained. (1) A composition of extendable robot arm including a flexible feed screw, many moving plates having a motorized nut and tendon cables has been proposed. (2) Based on geometry and statics, DoF and essential number of cables for an arm with two segments have been determined. A calculation scheme of the arm’s

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configuration to achieve a given end-effector’s position and orientation has been figured out based on the maximization of total product of two evaluation indices to avoid insufficient conditions. (3) Based on the proposed concept and calculation scheme, a prototype robot arm has been fabricated. By using the prototype, several experiments have been conducted so that positioning capability and effectiveness of the bracing on surrounding objects to enhance the arm’s load bearing capability are demonstrated. Acknowledgement. This work was supported by a MIT-TokyoTech-NSK joint research program, “The Last One Foot Problem of Robot Manipulation: Novel End-Effector Units for Assembly and Inspection in Narrow Space”, funded by NSK Ltd.

References 1. Shikari, A., Asada, H.: Triple scissor extender robot arm: a solution to the last one foot problem of manipulation. IEEE Robot. Autom. Lett. 3(4), 3975–3982 (2018) 2. Yan, T., Teshigawara, S., Asada, H.H.: Design of a growing robot inspired by plant growth. In: 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 8006– 8011 (2020) 3. Shungen, M., Kobayashi, I., Hirose, H., Yokoshima, K.: Control of a multijoint manipulator “Moray Arm”. IEEE/ASME Trans. Mechatron. 7(3), 304–317 (2002) 4. Wang, Y., Chirikjian, G.S.: Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N). IEEE Trans. Robot. Autom. 20(3), 399–408 (2004) 5. Lafmejani, A.S., Doroudchi, A., et al.: Kinematic modeling and trajectory tracking control of an octopus-inspired hyper-redundant robot. IEEE Robot. Autom. Lett. 5(2), 3460–3467 (2020) 6. Jones, B.A., Walker, I.D.: Kinematics for multisection continuum robots. IEEE Trans. Robot. 22(1), 43–55 (2006) 7. Renda, F., Giorelli, M., Calisti, M., Cianchetti, M., Laschi, C.: Dynamic model of a multibending soft robot arm driven by cables. IEEE Trans. Robot. 30(5), 1109–1122 (2014) 8. Hawkes, E.W., Blumenschein, L.H., Greer, J.D., Okamura, A.M.: A soft robot that navigates its environment through growth. Sci. Robot. 2(8), 1–8 (2017) 9. Frazelle, C.G., Kapadia, A., Walker, I.: Developing a Kinematically similar master device for extensible continuum robot manipulators. ASME J. Mech. Robot. 10(2), 1–8 (2018). Article id 025005 10. Oliver-Butler, K., Till, J., Rucker, C.: Continuum robot stiffness under external loads and prescribed tendon displacements. IEEE Trans. Robot. 35(2), 403–419 (2019) 11. Bakker, D.L., Matsuura, D., Takeda, Y., Herder, J.L.: Design of an environmentally adaptive continuum manipulator. In: Proceedings of 14th IFToMM World Congress, pp. 1–10 (2015)

Mobile Manipulator and EOAT for In-Situ Infected Plant Removal Taher Deemyad(&)

and Anish Sebastian

Idaho State University, Pocatello, ID 83209, USA {deemtahe,sebaanis}@isu.edu

Abstract. This paper discusses in detail the design of a roguing mechanism as an attachment to an Autonomous Ground Vehicle (AGV) for agricultural purposes. This mechanism is a part of a larger project for detection and removal of potato plants infected by virus Y (PVY). An unmanned aerial vehicle (UAV), equipped by hyperspectral camera and high precision GPS system, will fly over a potato field to locate a sick plant within a few centimeters and communicate the GPS coordinates to an AGV. This AGV will navigate to the target location and rogue (remove) the infected plant automatically. However, currently there is no roguing mechanism available off the shelf, which can be quickly attached to existing farm equipment to automate the process. Because of significant crop losses every year due to PVY, roguing in a timely manner is extremely important to the farming community. After multiple design iterations and simulations for the end-effector, a novel roguing mechanism was designed. Significant effort was also expended in assuring the designed mechanism can be easily connected to existing farm equipment. A 1/3rd scale prototype model was built for motion analysis. This prototype has been tested in a laboratory setting with promising results. Operational and design details of this mechanism have been successfully tested. The actual, field-size model construction is currently underway. Keywords: Mobile manipulator
Roguing mechanism
Agricultural robot Autonomous system
Agricultural virus removal
End of arm tooling




1 Introduction The population of the world has gone from 3 billion in the 1960’s to 7.7 billion in 2019. This doubling in the population size has put significant stress on agricultural capabilities. More and more farm equipment and farming methods are turning to automation and smart technologies. To feed this fast-growing population, we will need between 100–110% increases in agricultural products by 2050. To achieve this challenging goal, significant improvements will have to be made to the present agroecosystem while minimizing the negative impact on the global ecosystem [1]. Recent improvement in the field of robotics and autonomous systems have paved the path forward for researchers to apply these advances to the field of agriculture. The end goal being significant improvements in efficiency, reduction of waste and development of smart autonomous systems aimed at increasing throughput [4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 274–283, 2021. https://doi.org/10.1007/978-3-030-75271-2_29

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Using mechanical systems and humans in agriculture is a traditional approach. Nowadays, with advances in the field of robotics, farmers are embracing fully or partially automated systems in various farming tasks; such as pesticide/herbicide spraying, irrigation and harvesting. Application of automation in other areas of farming, for example; identifying the size, shape of the produce, monitoring the growing conditions and its effects on the health of a crop is still in its preliminary stages. The complexities associated with different types of crops make it a big challenge for engineers to find an appropriate solution for each individual product. In this research, we are focusing on roguing sick plants from a field. This narrows down our tasks to, grasping, and lifting mechanisms. Research in these areas mostly focus on two major components: Control and Kinematic design. In this paper, we focus on kinematic design and specifically discuss the design of an end effector to be used for roguing. In the case of kinematic design, based on the target object and task, there are various options which one can consider as a designer. For roguing mechanisms, grasping the plant by a gripper, and pulling it out of the ground could be a method that can be applied. In [5], a 7 degree of freedom (DOF) robotic system, a serial link manipulator with 6 DOF is mounted on a prismatic base and the entire assembly connected to an end-effector, was used for grasping apples. For adaptively grasping objects of various shapes, and in light of recent advances in multi-purpose universal grippers (MPUG) [6], end-effector designs could be based on a fully-actuated system. One such example is the universal gripper, a combination of linkage-driven underactuated gripper with suction gripping system, to grasp objects of irregular shape in different environments [7]. Soft robots are becoming more and more popular in robotic systems nowadays and can be considered as a possible option for grasping fruits and vegetables. Soft end-effectors and grippers are playing an everimportant role in grasping and handling soft fruits and vegetables. The amount of applied force for each task is an important factor too. To mimic human grasping capabilities, some elastic structures with variable stiffness are able to apply varying degrees of force, for a successful grasp, without causing any damage to the perishables [8, 9]. Pin array gripper is another example of universal manipulators that are made from an array of slidable pins in the gripper. These can conform to the shape of many objects in both vertical and horizontal directions using a single motor [10]. Vacuum based grippers are also used as a main technique or in combination with other methods in some universal manipulators to improve grasping capabilities [11]. Optimization methods can be applied in designing grippers as well for minimizing the weight and size of mechanisms [12–14]. For the problem we face, we elected to use a digging mechanism over grippers to get rid of infected plants from the field. In [15], a novel digging mechanism was analyzed and optimized based on finite element simulations. In addition to grasping mechanisms, extraction/digging mechanisms can be an alternative option. As of writing this document we have come across only one such digging mechanism for harvesting that is presented in [15]. In order to satisfy the objective for this project of extracting the sick potato plants, which have most of the plant still buried in the soil, from the field autonomously while minimizing cross-infection. We were compelled to design a universal extraction

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mechanism. This novel roguing mechanism could be used for roguing other plants too, by modifying the hydraulic forces needed to penetrate the soil and by minor modifications to the collection bucket. Widespread acceptance of the designed mechanism would only be possible if it can be easily attached to existing farm equipment. This was addressed by designing a universal attachment for the rouging unit.

2 Project Overview and Simulated Model of Potato Field in the Lab The general idea of this project is shown in Fig. 1. First, the UAV equipped with a hyperspectral cameras and high precision GPS flies over a field and takes images. These images, after analysis, indicate the location of sick plants in the field. The associated GPS location is sent to an autonomous ground vehicle (AGV) with precision RTK GPS [2, 3]. This AGV then navigates to the target location and has been proven in field tests to be precise to a few centimeters. This AGV will have the custom roguing mechanism, which can extract a sick plant from the ground, without infecting nearby plants. This will significantly reduce widespread pesticide spraying and discarding of nearby uninfected plants.

Fig. 1. Communication between Unmanned Air Vehicles (UAV) and Autonomous Ground Vehicles (AGV)

One of the first steps in designing the prototype was understating the details of the environment the AGV will be operating in. This knowledge will dictate the type of materials which can be used for the AGV, the drive mechanism, its onboard navigational system and finally the end-effector itself. We visited a local potato field and, based on our observations, created a simulated model of a field in SolidWorks®. This model was used in making calculations for our design. This simulated model is shown in Fig. 2. As can be seen in Fig. 2, a typical potato field is fraught with rough terrain and deep irrigation ruts. The distance between the top of two crests/bumps is around 60 cm. The plants are grown on top of the crests/bumps and the average distance between any two plants is about 30 cm. The hatched area in Fig. 2 reflects the amount of material (soil, plant roots etc.) that needs to be cleared around an infected plant.

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Fig. 2. Simulated shape and size of the potato field

3 Analysis the Possible Options for Roguing Mechanism 3.1

Grasping Mechanism Design Methodology

The first iteration of the prototype design involved analyzing the forces that could be applied to the connections between the stem and the tubers and shortlisting any grippers which may satisfy the requirements mentioned previously. There are different motions that can be applied by a gripper mechanism (for example: a human hand) for pulling plants out of the soil. We analyzed motions of different mechanism to see if the entire plant and tubers could be pulled out of the soil without breaking the plant. For this we conducted experiments to identify the maximum gripping force which might snap the stem thereby leaving the tubers in the soil. One of the challenges faced here, was that the forces varied significantly between plants. If we were to proceed with a design based on the gripping force itself, we would have to find actuators with a wide range to manipulate the force on the stem. (a) Grasping the plant by the stem and moving it several times up and down (vertical shaking). With this motion, we tried to loosen the soil around the tubers and the base of the plant before pulling the plant completely out (Fig. 3-a). As shown in Fig. 3-b, the connection between stem, potato tubers and roots broke due to this motion. The potato tubers and its roots remained in the soil, which does not satisfy one the design requirements - complete removal of the plant.

Fig. 3. (a) Vertical motion (b) Connection between roots, tubers and stem of plant were broke

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(b) Grasping the plant by the stem and moving it several times front and back (horizontal shaking). This motion was used to facilitate loosening of the connection between the soil and tubers (Fig. 4-a) before pulling the plant out of the soil. This test was repeated 3 times. As it is shown in Fig. 4-b, in the first attempt, the whole plant including the stem, potato tubers and roots came out of the soil and test was successful. But in the next two attempts the test failed and connection between stem and the tubers snapped.

Fig. 4. (a) Horizontal motion (b) Connections between roots, tubers and stem of plant remained intact in the first attempt but failed in the next two.

(c) Grasping the plant by the stem and twisting it around the stem axis, and at the same time pulling the plant out of the soil (Fig. 5-a). As shown in Fig. 5-b, we observed the same results as the other two methods, the connection between stem, potato tubers and roots were broken. Therefore, the potato tubers and roots remained in the soil again failing to facilitate the desired outcome.

Fig. 5. (a) Twist & vertical motion (b) Connections between roots, tubers and stem of plant broke

3.2

The Best Options

According to the results in the last two sections, between all possible alternatives, we shortlisted three mechanisms as the best choices for roguing potatoes. The morphology chart for these three mechanisms is presented in Table 1. The first design is made of a cage and blade for cutting soil. The cage goes down while surrounding the target plant. Then the cutting blades cut the soil (which includes

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potato tubers and roots of the plant) from one side and goes up and dumps all of the material in a disposal container (Table 1 – design A). The second design is made of two rake-shaped arms. When these arms reach the top of a plant, it goes down and closes from both sides to cut the soil and grab the infected potato tubers, roots, and body of the plant. Then, it will pull up and dump all of the collected material in a disposal container (Table 1 – design B). The third design is a combination of a pinching gripper and a cutting blade. In this design, first, the entire mechanism goes down into the soil and gripper closes to engulf the top part of plant, while the blade cuts the soil grabbing the potato tubers and plant roots. Finally, the mechanism lifts up and discards all the collected material into a disposal container (Table 1 – design C). Table 1. Morphology chart

However, each of above designs had some issues. In design “C”, because of variety in the shape of the bushes it would be hard to grasp only the top portion of the plant which includes all branches, leaves, and stem by the gripper completely. Also, cutting that much soil with just a single blade will require applying significant force on the blade. In design “A”, we would have a better performance for collecting the entire plant, but still have the same issue, significant forces on one blade. While in design “B” the soil would be cut easier by splitting the forces over two arms. However, because of the shape of the rakes and the sides of the arms being open, there is a significantly high risk of losing a part of the infected plant in the field. Therefore, the best design would be a combination of these three mechanisms which includes the advantages of all of them while eliminating their weaknesses.

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4 Final Design 4.1

Parts of the Mechanism

As discussed in the previous section, the final design was generated based on combining the strengths of the three designs previously described. The mechanism contains a cage to making sure no part of the infected plant (including tubers, stem, leaves, roots, etc.) will remain in the field after roguing by this mechanism. Also, we will utilize two cutting blades to apply less force over each blade to reduce overall wear and chances of failure. In addition, two attached hydraulic actuators will be used to provide the necessary force to pierce the soil to a specified depth. Figure 6 shows all the components for this mechanism. Detailed information about the dimensions and the size of the parts along with detailed drawings can be found in [16]. These components in general can be considered to fall into one of the following five categories: i. Frame: This is the main structure of the mechanism which, holds the sick roots, stems, leaves, etc. ii. Movable Components: This includes all components required for actuation and to bring about the desired motion of the blades. iii. Cutting Sections: This includes all parts performing the actual digging action in the field and facilitate cutting the soil with ease. iv. Mechanism Strength: To improve the overall rigidity of the system we included a T-Shape Angle Bar on the sides of the main body and Inside Frame Blades which connect two sides of main body to each other (back-front). v. Attachments: This mechanism is designed so as to easily attach to various type of tractors or other farm equipment. Three possible adapters were designed to attach it to tractors: from the back, the Top and from the sides where they can be screwed to the holes which are placed on the sides of Main Body Top Frame.

Fig. 6. Final design components

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Motion Study

The complete assembly of this roguing machine is presented in Fig. 7. Figure 7-a shows the mechanism when the moving blades are open. Yellow arrows show the direction of motion of the hydraulic actuators which are connected to the arms. These hydraulic arms will actuate the opening and closing of the jaws while also providing sufficient force to pierce the soil. For closing the blades, the hydraulic system moves down and pushes the blades along the curved rails. Figure 7-b shows the machine with the blades in the closed position.

Fig. 7. Actuation of blades by hydraulic system in final design

5 Prototype Model Figure 8 shows a prototype of the final design of the roguing machine which was built to 1/3rd scale of the actual model. The prototype was used for testing the motion and functionality of the system. There will be some differences between the final design and this prototype to accommodate in filed operation. First, the final design will be about three times larger than this prototype. In the final design, all major components are divided into sub-systems for ease of replacement and cleaning. Other small changes, are related to improving the functionality of system and attachments needed to integrate the prototype to commonly used farm equipment.

Fig. 8. Prototype of the mechanism (1/3 of the real size)

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Figure 8-a shows the prototype connected to an ABB robotic platform, Fig. 8-b shows the tracks along which the jaws will move, Fig. 8-c shows the jaws in a completely closed position, Fig. 8-d shows the teeth at the end of the jaws to help pierce the soil, Fig. 8-e shows a bottom view of the rouging mechanism with the jaws in a completely closed position with the teeth interlocking, Fig. 8-f shows the bucket in which the sample to be removed will be collected and disposed, off the field. The catch bucket grills can be modified based on the type of sample being collected. Figure 9 shows the roguing process of a sample plant by the prototype mechanism. In Fig. 9-a, the robotic arm pushes the mechanism down and the soil is cut by the fixed blades. In Fig. 9-b and c, the moving blades will be closed and infected plant along with tubers in the soil will be collected in the catch bucket. In Fig. 9-d, the robotic arm moves the mechanism up with the plant and soil inside to be discarded off the field.

Fig. 9. Process of roguing the sample plant by prototype mechanism

6 Conclusion In this paper, a novel mechanism for a roguing was discussed. Multiple designs and grasping types were considered, all relevant mechanisms were analyzed and three mechanisms based on primary analysis were considered for study. On comparing advantages and disadvantage of each of these three options, the final mechanism was designed utilizing the best combination of each design’s strength and capabilities. Although this mechanism is designed to work in a potato field, because of it is versatility and ease of coupling to farm equipment, it could potentially be used for roguing other crops and work in different environments.

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Finally, a prototype was built to 1/3rd scale of the actual model and successfully tested for motion and it performed exceptionally well to achieve the desired results. In the near future, we are going to build a real sized model to test it in the potato field.

References 1. Tilman, D., Balzer, C., Hill, J., Befort, B.L.: Global food demand and the sustainable intensification of agriculture. Proc. Natl. Acad. Sci. 108(50), 20260–20264 (2011) 2. Deemyad, T., Moeller, R., Sebastian, A.: Chassis design and analysis of an autonomous ground vehicle (AGV) using genetic algorithm. In: Intermountain Engineering, Technology, and Computing Conference (i-ETC). IEEE (2020, in press) 3. Moeller, R., Deemyad, T., Sebastian, A.: Autonomous navigation of an agricultural robot using RTK GPS and Pixhawk. In: Intermountain Engineering, Technology, and Computing Conference (i-ETC). IEEE (2020, in press) 4. Han, X., Thomasson, J.A., Wang, T., Swaminathan, V.: Autonomous mobile ground control point improves accuracy of agricultural remote sensing through collaboration with UAV. Inventions 5(1), 12 (2020) 5. Silwal, A., Davidson, J.R., Karkee, M., Mo, C., Zhang, Q., Lewis, K.: Design, integration, and field evaluation of a robotic apple harvester. J. Field Robot. 34(6), 1140–1159 (2017) 6. Choi, M.S., Lee, D.H., Park, H., Kim, Y.J., Jang, G.R., Shin, Y.D., Park, J.H., Baeg, M.H., Bae, J.H.: Development of multi-purpose universal gripper. In: 2017 56th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), pp. 1421–1424. IEEE (2017) 7. Kang, L., Seo, J.T., Kim, S.H., Kim, W.J., Yi, B.J.: Design and implementation of a multifunction gripper for grasping general objects. Appl. Sci. 9(24), 5266 (2019) 8. Duckett, T., Pearson, S., Blackmore, S., Grieve, B., Chen, W.H., Cielniak, G., Cleaversmith, J., Dai, J., Davis, S., Fox, C., From, P.: Agricultural robotics: the future of robotic agriculture. arXiv preprint arXiv:1806.06762 (2018) 9. Hughes, J., Culha, U., Giardina, F., Guenther, F., Rosendo, A., Iida, F.: Soft manipulators and grippers: a review. Front. Robot. AI 3, 69 (2016) 10. Mo, A., Zhang, W.: A novel universal gripper based on meshed pin array. Int. J. Adv. Rob. Syst. 16(2), 1729881419834781 (2019) 11. Fujita, M., Ikeda, S., Fujimoto, T., Shimizu, T., Ikemoto, S., Miyamoto, T.: Development of universal vacuum gripper for wall-climbing robot. Adv. Robot. 32(6), 283–296 (2018) 12. Deemyad, T., Hassanzadeh, N., Perez-Gracia, A.: Coupling mechanisms for multi-fingered robotic hands with skew axes. In: Mechanism Design for Robotics (MEDER). Springer (2018) 13. Deemyad, T., Heidari, O., Perez-Gracia, A.: Singularity design for RRSS mechanisms. In: USCToMM Symposium on Mechanical Systems and Robotics (MSR) Conferences. Springer (2020) 14. Deemyad, T.: Design of five-fingered underactuated hand for two-position tasks. Master’s thesis, Idaho State University (2016) 15. Yang, Y., Kou, T., Peng, S., Wang, C., Hu, G.: Optimization design of digging mechanism for onion harvester based on finite element simulation technology. In: IOP Conference Series: Materials Science and Engineering, vol. 688, no. 4, p. 044071. IOP Publishing (2019) 16. Deemyad, T., Sebastian, A., Perez-Gracia, A.: Provisional patent application for “Agricultural Roguing Machine”. Application Serial No.: 63/104,937. Filing date: 23 October 2020

New Variable Stiffness Joint (VSJ): Study and Simulation M. G. Contreras-Calderón1(&), M. A. Laribi2, and E. Castillo-Castañeda1 1

Instituto Politécnico Nacional, CICATA-Unidad Querétaro, Querétaro, Mexico 2 Department of GMSC, Pprime Institute, CNRS – University of Poitiers, ENSMA – UPR 3346, Poitiers, France

Abstract. Variable stiffness joints are devices that allow changing the joint stiffness of a mechanism. The development of this joint has increased, and various applications have emerged, most to improve human-robot interaction. This paper presents a proposal design of a new variable stiffness joint (VSJ). The working principle consists of antagonist extension springs with a rack and pinion mechanism. The springs are attached to a set of racks that allow varying the spring length, the main factor in varying the stiffness. The racks engage with a set of gears; one has a motor attached, which is responsible for varying the joint stiffness. Its mathematical model describes the joint behavior to maintain a constant force at the end of a link. Its simulation is performed with a constant force placed at the end of the link to evaluate the joint’s performance in a rotational situation. Keywords: Variable stiffness joint interaction
Compliant joint


Antagonist springs
Human-robot

1 Introduction The development of variable stiffness joints has increased in recent years, and numerous applications have emerged; among the most studied are those where robots share the workspace with a human. Traditional mechanisms are rigid, while devices with flexible elements improve human-robot interaction and include high-precision, low-step, and low-friction movements [1]. Mechanical compliance can be included mainly by two mechanisms: the use of flexible transmission elements or the incorporation of links of light or elastic materials [2]. The use of variable stiffness joints absorbs vibrations; it reduces the impact and external disturbances of a mechanism. The characteristic of each joint depends on the mechanism that is implemented to modify the stiffness. Have been developed various designs, which it is grouped into joints with a series configuration [3–5] and an antagonistic configuration [4, 6, 7]. Joints with series configuration incorporate a flexible element in the transmission mechanism and can be independently compressed by a second actuator. Joints with an antagonistic configuration, two actuators of a flexible nature, or flexible transmission elements with a non-linear force-elongation characteristic, are coupled to the same joint but © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 284–293, 2021. https://doi.org/10.1007/978-3-030-75271-2_30

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oppositely [2]. The main characteristic property of a VSJ is the exit stiffness, which can be varied independently of the exit position. By replacing the conventional joints with the variable stiffness joints, the device can perform various tasks while varying its stiffness at the output or keeping the output constant. One of the difficulties of using such actuators is the considerable increase in complexity in trajectory planning and control, particularly in such highly dynamic movements [8]. Variable stiffness joints can also be used in tasks where it is required not only to damp movements but also in those where a specific force is desired at the device output, and thus generate a force without the need for motors. For a device to offer variable resistance to movement, it is required motors with high torque but can be replaced by VSJ to reduce energy consumption and apply all the advantages mentioned above. Some VSJ designs, such as the AwAS [9], in a traditional design of two springs placed in an antagonistic configuration, the stiffness is changed by controlling the spring’s tension. Extending both springs makes the joint stiffer; relaxing both springs makes the joint more flexible; this joint has been tested in ASIBOT service robot [8]. The case of a VSJ used for humanrobot interaction has a timing transmission belt, which is tensioned by springs and nonlinearly connects the main shaft to the pair of opposing pulleys are connected to motors, reverse-actuated DC motors with position control. Concordant angular variations generate displacements and variations in stiffness [10]. Other joints are based on the cantilever beam principle as the variable stiffness joint in [11] the support point of the beam is movable. This distance is related to the variation of the stiffness. The joint presented in [12] uses a lever, in which the deflection depends on the position of the pivot. The design of a variable stiffness joint proposed in [13] can provide joint stiffness independent of the rotation angle; It consists of a position frame, an output shaft attached to a guide link, and two spring blocks. It has a linear compression spring installed in each spring block and restricts the output rotation shaft relative to position, pushing the guide link on both sides. This paper presents the design of a variable stiffness joint, whose mechanism is based on a pair of antagonist springs attached to a set of gears and racks and presented the joint simulation in a case of rotation to analyze its performance.

2 Variable Stiffness Joint Proposal The design requirements for the Variable Stiffness Joint (VSJ) is that it maintains a constant stiffness at the end of the link independently of the rotation angle. A VSJ proposal whose design incorporates antagonist springs R1 and R2 mounted in a link (Fig. 1) with lengths l1 and l2 respectively. Starting from the equilibrium position (Fig. 1a), the joint rotates anti-clockwise (Fig. 1b), where spring R1 is in extension and R2 is compressed. When the joint rotates clockwise (Fig. 1c), the spring R1 is compressed, and the spring R2 is in extension. It uses the characteristics of the individual springs to determine the relationship between the joint stiffness and the force at the end of the link.

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a)

b)

c)

Fig. 1. VSJ diagram a) Equilibrium position b) anti-clockwise rotation c) Clockwise rotation.

The lc ¼ li  xm is the compression length and le ¼ li þ xm is the extension length. The joints stiffness depends on the spring’s length, by the extension or compression. The spring is extended a length xm , which depends on the rotation angle and the link radius. It is deduced that the greater the angle of h, the stiffness in the joint increases. The objective is to maintain a constant force at the link end independently; the spring length must be varied if the spring’s length in extension is controlled, it is controlled the stiffness at the end of the link. The way to control the spring’s length is by moving the ground where the springs are attached; that is, the distance from the ground is adjusted (xli ). Then the total length (XTi ) of the spring is the sum of xm and xli , as shown in Fig. 2. Adjusting the distance from the ground allows modifying the spring length and can be achieved in different ways: with an endless screw, linear actuators, or rotational motors. For this VSJ proposal, a motor is chosen to transmit the movement through a set of gears (Fig. 3).

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Fig. 2. Adjusting the distance from the ground

Fig. 3. Adjusting the spring length with a motor

The joint design is considered the vary of the spring’s length; one of its ends has to be attached to the link, while the other end is attached to the mechanical ground. The link has a gear (Pe ) in the rotational joint, and it engages with two racks (C3 and C4 ), from it is held the springs (R1 and R2 ), and they are positioned parallel to the link, which allows the linear transmission of the joint rotation to the springs. According to Fig. 3, when the link rotates anti-clockwise, the rack C4 extends the spring R1 , while the rack C3 compresses the spring R2 because of the same rotation. When the rotation is clockwise, then spring R2 is extended, and R1 is compressed. The other end of the springs R1 and R2 are attached to two racks C1 and C2 , respectively; are mesh these racks with a set of gears (P1 , P2 and P3 ). Gears P1 and P2 mesh to racks C1 and C2 , respectively. The gear P3 has a coupled motor, which allows its rotation; in this way, the pinion P3 transmits the movement to the gears P1 and P2 , which allows the racks C1 and C2 to extend the springs, modifying their length. The racks C1 , C2 , C3 , and C4 supports are fixed to the ground. Then motor causes the extension/compression of the springs, while the set of gears coupled to the motor varies the spring’s length, which ensures the constant force at the end of the link (Fig. 4).

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a)

b) Fig. 4. a) Variable stiffness joint CAD proposal b) Explosive view.

It uses the extension springs in the design that absorb and store energy and create resistance to a tensile force. Extension springs are attached at both ends to other components. They are frequently used to provide return force to components that extend on actuation [14]. The main innovation in the proposed joint is the set of gears and racks that ensure a linear movement of the springs during the link rotation. Also, the relationship between the spring’s length and the motor rotation makes it possible to maintain the desired force at the end of the link.

3 VSJ Mathematical Model The new variable stiffness joint considers the extension and compression of the springs to generate stiffness. The spring that gives the joint stiffness is that extends due to rotation. In the case of anti-clockwise rotation, as shown in Fig. 5, is the spring R1 . The total length of the springs is XTi , which is the sum of the length xm (see Fig. 2) that depends on the rotation angle and the motor’s extension (xli ). xli must vary through the gear P3 rotation to obtain a constant force at the end of the link. Hooke’s law for spring analysis states that the force F is proportional to the extension: F ¼ kx Where k is the spring constant and x corresponds to the spring length that is extended

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xli ¼ length by motor xm ¼ rh

ð1Þ

XTi ¼ xli þ ðsignhxm Þ i ¼ 1; 2

ð2Þ

Where XTi is the total extension/compression of the spring. Then for the case of anticlockwise rotation as in Fig. 5:

Fig. 5. VSJ diagram.

XT1 ¼ xl1 þ xmove XT2 ¼ xl2  xm

Extension Compression

ð3Þ

It is obtained at the end of the link a force Fi ¼ kðxli þ signðhÞxm Þ

ð4Þ

The control of the length of the spring through the motor coupled to the gear P3 transmits the movement to the gears P1 and P2 , connected to the racks C1 and C2 (Fig. 6).

Fig. 6. Gears set

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xli is known with desired force at the end of the link, with this value is characterized the motor position, and in this way, it can control the position of the gear. So: XTi ¼

F k

xli þ ðsignðhÞxm Þ ¼ xli ¼

ð5Þ F k

  F þ ðsignðhÞxm Þ k

ð6Þ ð7Þ

The applied torque due to the springs is: s ¼ r ðF1  F2 Þ

ð8Þ

s ¼ rk ðxl1  xl2 Þ þ 2kr 2 h Finally, who controls the length of the springs is the rotation of the gear P3 . The relation between the length xli and the rack pitch PC1;2 is determined; it can be the rack C1 or C2 . Two auxiliary variables, a and c, are used to facilitate the relationship of the equations. a¼

xli PC1;2

c ¼ aMP1;2

ð9Þ

The rotation angle ;P3 of gear P3 is determined by: ;P3 ¼

cZP1;2 ZP3

ð10Þ

Where, MP1;2 : is the gear pitch module P1 and P2 . ZP1;2 is the number of gears teeth P1 and P2 . ZP3 is the number of gear teeth P3 .

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4 Simulation To evaluate the variable stiffness joint behavior. It simulates an oscillatory rotation in Matlab; angular displacement h is shown in Fig. 7a. It considers two opposing springs with a length of 0.05 m, with a constant k = 3771 Nm. The constant force applied is F = 30 N. The behavior of xli is shown in Fig. 7b, which is an antagonistic behavior for the pair of springs. The maximum length is 0:031m to 0:008 m.

a)

b)

Fig. 7. a) Angular displacement h b) xli length of springs.

The gears P1 and P2 used in the VSJ design are considered identical, it uses elements from the SolidWorks toolbox and it´s adjust to the joint design. The simulation was performed with the following specifications, P1 and P2 have a modulus of MP1;2 ¼ 1:375, number of teeth ZP1;2 ¼ 14. It is considered identical to the racks with a pitch of PC1;2 ¼ 0:00432 m. Gear P3 , has ZP3 ¼ 10. The position of gear P3 is controlled to obtain a constant force at the end of the link; this gear’s behavior to maintain a constant force of 30 N is shown in Fig. 8a, The gear values p3 generate a sinusoidal behavior, oscillating between 14 and −4°. In Fig. 8b, the behavior of the torque is expected at the trajectory described by h. The maximum torque value is 6 Nm:

a)

b)

Fig. 8. a) P3 angular position b) Torque behavior.

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5 Conclusions In this paper, was presented a new variable stiffness joint. The design was made with two springs acting in an antagonistic way, attached at one end to a rotational joint and the other end to a set of gears and racks. One of the springs called P3 is the one that finally controls the force at the end of the link. The joint allows maintaining a constant force at the exit of a link regardless of the rotation direction, varying its extension length. By controlling this length, the output force can be kept constant. The springs’ extension/compression can be done in different ways in this work, and it was presented with a set of gears with a motor. The simulation carried out for a particular situation of rotation reflects the behavior of the springs’ lengths, and the motor position is calculated for the rotation of P3 . It was shown that the mathematical model describes the behavior of the joint. The VSJ allows to maintain a constant effort applied on the end of the link despite the angle position change.

References 1. Howell, L., Magleby, S., Olsen, B.: Compliant Mechanisms. Wiley, New York (2013) 2. Medina, J., Jardón, A., Balaguer, C.: Control desacoplado de un actuador de rigidez variable para robots asistenciales. Revista Panamericana de Automática e Informática Industrial 13, 80–91 (2016) 3. Pratt, G., Williamson, M.: Series elastic actuators. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, p. 399. IEEE, Pittsburgh (1995) 4. Tsagarakis, N.G., Laffranchi, M., Cannella, F., Caldwell, D.G.: Antagonistic and series elastic actuators: a comparative analysis on the energy consumption. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, St. Louis, USA, pp. 5678– 5684. IEEE (2009) 5. López-Martínez, J., Giménez-Fernández, A., García-Vallejo, D., Jardón-Huete A., BalaguerBernaldo C.: Diseño y simulación de un actuador de rigidez variable. Asociación Española de Ingeniería Mecánica. XIX Congreso Nacional de Ingeniería Mecánica (2012). ISSN 0212-5072 6. Yuwang, L., Xiagang, L., Zhongqiu, Y., Jinguo, L.: Design and analysis of spring parallel variable stiffness actuator based on antagonistic principle. Mech. Mach. Theory 140, 44–58 (2019) 7. English, C., Russell, D.: Mechanics and stiffness limitations of a variable stiffness actuator for use in prosthetic limbs. Mech. Mach. Theory 32(1), 7–25 (1999) 8. Braun, D.J., Howard, M., Vijayakumar, S.: Exploiting variable stiffness in explosive movement tasks. In: Robotics: Science and Systems VII, vol. 7, pp. 25–32 (2012) 9. Jafari, A., Tsagarakis, G., Vanderborght, B., Caldwell, G.: A novel actuator with adjustable stiffness (AwAS). In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, pp. 4201–4206 (2010) 10. Schiavi, R., Bicchi, A., Tonietti, G.: Design and control of a variable stiffness actuator for safe and fast physical human/robot interaction. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, pp. 526–531 (2005)

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11. Contreras-Calderon, M.G., Castillo-Castañeda, E.: Design of a variable stiffness joint for a five-bar-mechanism. In: Advances in Service and Industrial Robotics. Results of RAAD, Switzerland AG, pp. 54–63 (2020) 12. Robinson, M.: A compliant mechanism-based variable-stiffness joint. Master thesis, Utha (2015) 13. Kim, B., Song, J.: Hybrid dual actuator unit: a design of a variable stiffness actuator based on an adjustable moment arm mechanism. In: IEEE International Conference on Robotics and Automation Anchorage Convention District, Anchorage, Alaska, pp. 1655–1660 (2010) 14. Lee springs. https://www.leespring.mx/es/conoce-mas-resortes-de-extension. Acceded 08 Feb 2021

Special Session in Honor of Prof. Said Zeghloul

Design of an 8-DoF Redundant Robotic Platform for Medical Applications Elie Gautreau, Aurélien Thomas, Juan Sandoval(&), Med Amine Laribi, and Saïd Zeghloul Department of GMSC, Pprime Institute CNRS, ENSMA, University of Poitiers, UPR 3346, Poitiers, France {elie.gautreau,aurelien.thomas}@etu.univ-poitiers.fr, {juan.sandoval,med.amine.laribi, said.zeghloul}@univ-poitiers.fr

Abstract. Redundant robots can provide improved kinematic performances compared to non-redundant robots, becoming suitable to perform complex tasks in several areas of application. Improved manipulability or dexterity and a larger workspace are some of the advantages of using redundant robots. This paper aims to present the design and development of an 8-DoF redundant robotic platform for medical applications. The platform is composed of a 7-DoF anthropomorphic cobot, i.e. Franka Emika, whose base is coupled to a 1-DoF linear axis. The aim of the linear axis is to enlarge the workspace of the robot. Robot-assisted Doppler sonography is presented as an example of medical application, where the robot, employed as probe-holder, is able to traverse the whole patient’s body during the exam. Keywords: Redundant robots


Human-robot interaction
Medical robots

1 Introduction A robot is considered as redundant when it has more degrees of freedom (DoF) than required to perform a given task. This feature has a direct impact on the dexterity of the robot: it allows the robot to be more flexible in the realization of the imposed task, enabling multiple joint configurations for a given position and orientation of the endeffector. This is an essential feature when performing tasks in unstructured or dynamically varying environments often encountered in complex industrial applications and service robotics scenarios [1]. Redundancy is also commonly exploited to consider kinematic constraints during the task execution, such as remote center of motion (RCM) constraints [2], obstacle avoidance [3, 4] or joint-limit avoidance [5] or to optimize a performance index, e.g. manipulability index [6]. Given the aforementioned benefits, redundant robots are gradually being introduced in several medical applications. We can cite the Versius surgical system [7], a robot used in hernia repair surgeries, prostatectomies and hysterectomies, or the MicroSurge system [8], a robot intended for minimally invasive surgery on the abdominal and thoracic regions. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Zeghloul et al. (Eds.): MEDER 2021, MMS 103, pp. 297–304, 2021. https://doi.org/10.1007/978-3-030-75271-2_31

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In the case of Doppler sonography, classical exams often require the practitioner to adopt uncomfortable postures that might induce musculoskeletal disorders. Adding a remote haptic-control to manage a robotic probe-holder in contact with the patient gives the practitioner the possibility to conduct the exam while reducing the incidence of musculoskeletal disorders. Previous researches carried out at the Pprime Institute led to propose a robot-assisted Doppler sonography platform using a 7-DoF redundant robot [9, 10]. Although the proposed platform is suitable to perform the overall classical sonography exams, the robot’s workspace is not large enough to go through the whole patient’s body during the exam. In order to increase the robot’s workspace, a new kinematic solution is proposed in this paper, by adding a 1-DoF linear axis to translate the base of the robot. Besides the increased workspace, the proposed solution has a larger number of degrees of redundancy, enhancing to implement more complex control strategies. Preliminary validations of the new platform are presented here, through a dynamic simulator. Moreover, the assembled prototype is detailed. The paper is organized as follows. We describe in Sect. 2 the kinematic and dynamic modelling of the platform as well as the proposed control design. Section 3 reports preliminary simulation results. The mechanical design of the prototype is discussed in Sect. 4. In the last section, conclusions summarize the paper content with future work considerations.

2 Robot Modelling In this section, we depict the kinematic and dynamic model of the proposed 8-DoF robotic platform. These mathematical models are integrated into the simulator presented in Sect. 3. 2.1

Denavit-Hartenberg Parameters

Although the Franka Emika cobot (7-DoF) is originally supplied with its own DenavitHartenberg (D-H) parameter table [11], the new proposed platform (8-DoF) is composed of the cobot and a linear axis. Therefore, we have extended the D-H parameters to fit with the new platform, as explained below. Figure 1 presents the new D-H parameter table, where joints i ¼ 1; . . .; 7 and joint i ¼ 0 correspond to the Franka robot joints and the linear axis, respectively. The platform with the reference frames positioning is also shown, according to the D-H convention [11]. Through this description of the D-H parameters, we proceed in the following to define the kinematic and dynamic models of the platform.

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Fig. 1. D-H parameter table and reference frames positioning in the redundant robotic platform

2.2

Kinematic and Dynamic Modeling

Considering a task-space dimension of m ¼ 6 for the proposed n ¼ 8 DoF robot, the non-square Jacobian matrix JðqÞ 2