Multibody Mechatronic Systems: MuSMe 2021 9783030887513, 3030887510

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Mechanisms and Machine Science

Martín Pucheta Alberto Cardona Sergio Preidikman Rogelio Hecker   Editors

Multibody Mechatronic Systems MuSMe 2021

Mechanisms and Machine Science Volume 110

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 Institute, 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

Martín Pucheta Alberto Cardona Sergio Preidikman Rogelio Hecker •

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Editors

Multibody Mechatronic Systems MuSMe 2021

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Editors Martín Pucheta CIII Facultad Regional Córdoba - Universidad Tecnológica Nacional & CONICET Córdoba, Argentina

Alberto Cardona CIMEC Universidad Nacional del Litoral CONICET Santa Fe, Argentina

Sergio Preidikman IDIT Universidad Nacional de Córdoba CONICET Córdoba, Argentina

Rogelio Hecker Facultad de Ingeniería Universidad Nacional de La Pampa & CONICET General Pico, Argentina

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-88750-6 ISBN 978-3-030-88751-3 (eBook) https://doi.org/10.1007/978-3-030-88751-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

Dedication To the family, friends, and colleagues of Prof. Ricardo Emmanuel Campa Cocom who passed away after COVID-19 disease on 19th November 2020 at the age of 49 years. He was an author of the 7th MuSMe and a generous and recognized researcher from the Instituto Tecnológico de La Laguna, Torreon Coahuila, Mexico.

Preface

The International Symposium on Multibody Systems and Mechatronics (MuSMe 2021) will be the seventh event of a series that started in 2002 as a joint activity of the FeIbIM Technical Commission for Mechatronics (currently, Robotics and Mechanisms) and the IFToMM Technical Committees for Multibody Dynamics, and Robotics and Mechatronics. The MuSMe 2020 Conference was planned as an in-person meeting to be held in Córdoba, Argentina, in October 2020, but due to the COVID-19 pandemic the Organizing Committee postponed its realization, in virtual modality, to October 2021. The International Symposium is a conference initiative to bring together researchers from the broad array of disciplines referring to multibody systems and mechatronics. Modern systems can be considered as integrated systems that can be properly studied, designed, and operated using mechatronics viewpoints, but considering the multibody architecture from the outset results in systems that are physically more realistic and accurate. In particular, the aim of the MuSMe Symposium is to exchange views, opinions, and experiences, and stimulate integration between mechatronics and multibody systems disciplines, a forum to facilitate communication, understanding, and the sharing of ideas and experience between researchers and students. The proceedings of this conference contains two volumes with papers written by the authors from all around the world; 39 papers were published in 2020, and 23 papers were submitted and evaluated in 2021. The contributions address mainly to synthesis of mechanisms and robots, kinematics, static and dynamic analysis, control of mechatronic systems, mechatronic systems for assistive technology and rehabilitation, multibody modeling and simulation, prototypes and experimental validations, railway and vehicle dynamics, and energy harvesting. This proceedings can be of interest to researchers, graduate students, and engineers specializing in or working with mechatronics. We believe that the reader will take advantage of these papers and that they will increase his or her motivation for teaching and researching mechatronic systems.

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Preface

We would like to express our grateful thanks to the members of the International Scientific Committee of the Symposium for cooperating enthusiastically for the success of the MuSMe initiative, in particular to Prof. Marco Ceccarelli (University of Rome Tor Vergata, Italy), President of the IFToMM from 2016 to 2019, and to the authors who have contributed with interesting papers in several subjects, covering many fields of mechatronics and multibody systems. We are grateful to the reviewers for the effort and time spent evaluating the papers. Finally, we greatly acknowledge the participation of Bahram Ravani (University of California, Davis, USA) as jury in the Springer Best Paper Awards Committee. The Organizing Committee would like to thank the Córdoba Regional Faculty of the National Technological University, Argentina, for supporting MuSMe 2021 and hosting the event. We would also like to thank the sponsorship of IFToMM— the International Federation for the Promotion of Mechanisms and Machine Science, FeIbIM—the Iberoamerican Federation of Mechanical Engineering, CONICET—National Scientific and Technical Research Council from Argentina, and Springer, for their financial support, acknowledging that without their partnership it would not be possible to organize this meeting. The Editors

October 2021

7th MuSMe Organization

Local Organizing Committee (from Argentina) Chair Martín Pucheta

CIII/UTN-FRC and CONICET, Argentina

Co-chair Alberto Cardona

CIMEC/UNL-CONICET, Argentina

Members Gastón Araguás Gonzalo Perez-Paina Claudio Paz Alejandro Gallardo Javier Salomone Sebastián Giusti Augusto Romero Federico Cavalieri Alejandro Albanesi Sergio Preidikman Bruno Roccia Rogelio Hecker Ricardo Carelli

CIII–UTN-FRC, Argentina CIII–UTN-FRC, Argentina CIII–UTN-FRC, Argentina CIII–UTN-FRC and CONICET, Argentina GIDMA–UTN-FRC, Argentina GIDMA–UTN-FRC and CONICET, Argentina GIDMA–UTN-FRC, Argentina CIMEC and UTN-FRSF, Argentina CIMEC and UTN-FRSF, Argentina IDIT/UNC-CONICET, Argentina IDIT/UNC-CONICET, Argentina UN La Pampa and CONICET, Argentina INAUT/UNSJ-CONICET, Argentina

International Scientific Committee Chair Mario Acevedo, Mexico

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Members Marco Ceccarelli, Italy Jorge Ambrosio, Portugal Paulo Flores, Portugal Martín Pucheta, Argentina Silvia Rodrigo, Argentina Josep María Font Llagunes, Spain Pietro Fanghella, Italy Mario Fernández Fernández, Chile Glauco Caurin, Brazil Daniel Martins, Brazil Manfred Husty, Austria Vicente Mata Amela, Spain Carlos Munares Tapia, Peru Eusebio Hernández, Mexico Honorary Members Osvaldo Penisi, Argentina Joao Carlos Mendes Carvalho, Brazil Tatu Leinonen, Finland

Springer Awards Committee Marco Ceccarelli, Italy Rogelio Hecker, Argentina Barham Ravani, USA

Sponsor

7th MuSMe Organization

Contents

Synthesis of Mechanisms Synthesis of a Tip-Tilt-Piston Flexure System with Decoupled Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alejandro G. Gallardo and Martín A. Pucheta Synthesis and Sensitivity Analysis of a Prosthetic Finger . . . . . . . . . . . . Nícolas Arroyo, Del Piero Flores, Diego Palma, Renzo Solórzano, and Elvis J. Alegria Synthesis of an Exoskeleton and Walking Pattern Identification for a Rehabilitation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanpierrs Figueroa, Joseph Díaz, Gustavo Quino, and Elvis J. Alegria

3 13

24

Mechanisms Analysis Analysis of a Proposal for a Self-aligning Mechanism for Cartesian Robot in Greenhouses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lucas Bernardon Machado, Antonio Carlos Valdiero, Henrique Simas, and Daniel Martins

37

Historical and Technical Analysis of Harmonic Drive Gear Design . . . . Vivens Irakoze, Marco Ceccarelli, and Matteo Russo

46

Design of Variable Moment of Inertia Flywheel . . . . . . . . . . . . . . . . . . . Vigen Arakelian

56

Kinematics of a Robotic System for Rehabilitation of Lower Members in Hypotonic Infants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marco García, Esther Lugo-González, Manuel Arias-Montiel, and Ricardo Tapia-Herrera

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Contents

Modelling and Simulation of Multibody Systems Modeling and Simulation of Frictional Contacts in Multi-rigidBody Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paulo Flores Modelling Spherical Joints in Multibody Systems . . . . . . . . . . . . . . . . . Mariana Rodrigues da Silva, Filipe Marques, Miguel Tavares da Silva, and Paulo Flores Multibody Dynamics Modeling of Delta Robot with Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohamed Elshami, Mohamed Shehata, Qingshun Bai, and Xuezeng Zhao

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Railway and Vehicle Dynamics Railway Dynamics with Curved Contact Patch . . . . . . . . . . . . . . . . . . . 105 Filipe Marques, Hugo Magalhães, João Pombo, Jorge Ambrósio, and Paulo Flores On the Utilization of Simplified Methodologies for the Wheel-Rail Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 José Ferreira, Paulo Flores, and Filipe Marques Optimization Tools Applied in the Design of a Hydraulic Hybrid Powertrain for Minimal Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . 122 Társis Prado Barbosa, Aline de Faria Lemos, Luiz Otávio Ferreira Gonçalves, Ricardo Poley Martins Ferreira, Leonardo Adolpho Rodrigues da Silva, and Juan Carlos Horta Gutiérrez Determination of the Effect of Sloshing on the Railcar-Track Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Juan Carlos Jauregui-Correa, Frank Otremba, Jose A. Romero-Navarrete, and Gerardo Hurtado-Hurtado Mechatronic Systems for Energy Harvesting A Three-Dimensional Piezoelectric Timoshenko Beam Model Including Torsion Warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Emmanuel Beltramo, Bruno A. Roccia, Martín E. Pérez Segura, and Sergio Preidikman On the Effect of Hardening/Softening Structural Non-linearities on an Array of Aerodynamically Coupled Piezoelectric Harvesters . . . . 151 Bruno A. Roccia, Marcos L. Verstraete, Luis R. Ceballos, Grigorios Dimitriadis, and Sergio Preidikman

Contents

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Hybrid System Design for Energy Harvesting from Low-Amplitude Ocean Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Gabriel Gutiérrez-Diaz, Arturo Solis-Santome, and Christopher René Torres-SanMiguel Robot Design and Optimization Modelling of a Hydrodynamically Actuated Manipulator Based on Strip Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 R. Santiesteban Cos, J. A. Carretero, and J. Sensinger Shaking Force Balancing of the 2RRR PPM Specifying Tool’s Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Mario Acevedo and Ramiro Velázquez Energy Optimization of a Parallel Robot in Pick and Place Tasks . . . . . 191 Juan Pablo Mora, Juan Pablo Barreto, and Carlos F. Rodriguez Gravity Compensation of Articulated Robots Using Spring Four-Bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Vu Linh Nguyen Mechatronic Design Structural Design and Sliding Mode Control Approach of a 4-DoF Upper-Limb Exoskeleton for Post-stroke Rehabilitation . . . . 213 Johan Nuñez-Quispe, Alvaro Figueroa, Daryl Campusano, Johrdan Huamanchumo, Axel Soto, Ebert Chate, Jesus Acuña, Juan Lleren, Jose Albites-Sanabria, Leonardo Paul Milián-Ccopa, Kevin Taipe, and Briggitte Suyo Mechatronic Design of a Planar Robot Using Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Alejandra Rios Suarez, S. Ivvan Valdez, and Eusebio E. Hernandez Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Synthesis of Mechanisms

Synthesis of a Tip-Tilt-Piston Flexure System with Decoupled Actuators Alejandro G. Gallardo and Mart´ın A. Pucheta(B) Centro de Investigaci´ on en Inform´ atica para la Ingenier´ıa (CIII) and Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Facultad Regional C´ ordoba, Universidad Tecnol´ ogica Nacional, Maestro M. L´ opez esq. Cruz Roja Argentina, X5016ZAA C´ ordoba, Argentina {agallardo,mpucheta}@frc.utn.edu.ar

Abstract. Parallel flexure mechanisms are used in high-precision positioning systems for a wide range of scientific, medical, and industrial applications. In particular, tip-tilt-piston mechanisms have great relevance in the field of optics for steering light and laser beams. The controllability of the flexure system is optimized when actuators are decoupled such that the output of any actuator does not affect the output of the other actuators. This work presents a systematic procedure to design flexure mechanisms with decoupled actuators. Screw Theory and Linear Algebra are used to manipulate the motion and constraint screw systems in combination with a graph representation of the mechanism. The methodology is illustrated with the redesign of two parallel tip-tilt piston flexure stages where the actuators have directions: (i) parallel to the rotations plane; (ii) normal to the rotations plane. The solutions are validated using finite element analyses. The designed flexure systems have very simple structures, high accuracy, and controllability. Keywords: Precision mechanisms · Tip-tilt piston flexible mechanism · Screw theory · Graph theory · Finite element analysis

1

Introduction

Flexible tip-tilt-piston mechanisms have great relevance due to their application in the field of optics and precise positioning of measurement and scientific instruments, in biomedical and electronic devices, and in a wide range of industrial applications. These mechanisms have three degrees of freedom, two coplanar rotations, and a translation perpendicular to the axes of rotation. Among the several techniques for designing flexible mechanisms [6], the methodology based on exact constraint using Screw Theory [1,3,8,9] is the more relevant in last years because its great potential to be automated by computational techniques. Recently, this methodology has been applied in conjunction with Graph Theory [10] for the systematic design flexible mechanisms with high complexity in terms of their topology. Flexible mechanisms generally have complex geometric design constraints such as spatial restrictions, maximum volume c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 3–12, 2022. https://doi.org/10.1007/978-3-030-88751-3_1

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A. G. Gallardo and M. A. Pucheta

of the mechanism, specific clamping surfaces, and specified actuation volume. Hopkins et al. [4,5] addresses the synthesis of mechanism subject to a decoupled actuation space. He considers that the actuation forces must be decoupled since it allows a better controllability of the mechanism. On the other hand, Pucheta and Gallardo [7] implements the synthesis of mechanisms considering spatial design restrictions. This work presents a synthesis method to design flexible tip-tilt-piston mechanisms with decoupled actuators and spatial design requirements. This method is a simplification of the method proposed by Sun and Hopkins [10]. This simplification is the consequence of imposing that the actuators of the mechanism must be decoupled. As the movement of an actuator must not interfere with the movement of another actuator, the topology of the mechanism can be split into parallel and/or series sub-mechanisms. The synthesis of two mechanisms with different actuation spaces is proposed, the spatial constraints are taken from Hao and He [2], and the designs obtained are verified through finite element simulations. This article is organized as follows: Sect. 2 introduces the screw theory background and also describes the steps of the methodology for designing a flexible mechanism system. Section 3 details the steps for designing a tip-tilt-piston example and Sect. 4 summarizes the conclusions.

2

Theoretical Background and Methodology

This work uses the screw theory definitions and nomenclature previously presented by Pucheta and Gallardo [1]. The infinitesimal displacement screw or Twist is represented by the 6 × 1 vector     θ θ t= = (1) δ c × θ + pθ where, θ is a vector that coincides with the screw direction and collects rotational displacements, δ is the linear displacement, c is a location vector going from the origin to any point of the screw axis, and p is the pitch of the screw. The Wrench screw is defined as     f f w= = (2) m r × f + qf where, f is a vector in the direction of the screw, m represents the moment around the screw axis, r is a location vector going from the origin to any point of the screw axis, and q is the pitch of the screw. A freedom space T = [t1 , t2 , . . . , tn ] is generated by n = rank(T) unit basis screws and its complementary space is the constraint or wrench space W   O3 I3 W = [w1 , w2 , . . . , w6−n ] = QT⊥ , Q = , (3) I3 O3 where, O3 is a 3 × 3 block of zeros, I3 is a 3 × 3 identity matrix, and T⊥ is null space of T.

Synthesis of a Tip-Tilt-Piston Flexure System

5

In this work, the methodology proposed by Sun and Hopkins [10] is adapted to develop a synthesis approach based on these steps: • Step 1: Identify the desired motion and design constraints. • Step 2: Represent the topology of the mechanism by its graph GT where the vertices are the bodies and the edges represent the freedom space allowed between a pair of bodies. Each freedom space has an associated complementary constraint space as shown in Eq. (3). • Step 3: Analyze the freedom space for every path from the ground body G to the stage body S. • Step 4: Compute the physical implementation of the constraint beams that allow the freedom space for each edge of the graph and also meet spatial design constraints.

3

Case Study: Tip-Tilt Piston Flexure Stage

The methodology is illustrated with the redesign of two tip-tilt piston flexure stages where the actuators have directions: (i) parallel to the rotations plane; (ii) normal to the rotations plane. The first case is validated with the mechanism presented by Hao and He [2] and is described in Subsect. 3.1. The second case is validated with the mechanism presented by Hopkins et al. [5] and is described in Subsect. 3.2. 3.1

Case 1: Actuators on a Plane Parallel to the Rotations Plane

A Tip-tilt piston flexure stage has three desired motions: two orthogonal rotations and one translation perpendicular to the rotations space as shown in Fig. 1(a).

Fig. 1. Problem definition: (a) Sketch of the stage with the set of desired motions (rotations Rx and Ry , and translations Tz ); (b) Graph representation of the parallel stage; (c) Parallel mechanisms build with beam flexure elements; (d) Actuation pattern of three rotations (R1 , R2 , and R3 ) located on a hoop that also produce the desired motion.

Step 1: A tip-tilt-piston mechanism has three degrees of freedom equal to the rank of the Twist System composed by three unit screws multiplied by their motion magnitudes

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⎡ ⎤T ⎡ ⎤ 100000 θx [T] = ⎣0 1 0 0 0 0⎦ · ⎣θy ⎦ δz 000001

(4)

The constraint space is the Wrench System Eq. (5) which is reciprocal to [T] ⎡ ⎤T ⎡ ⎤ 100000 fx [W] = ⎣0 1 0 0 0 0⎦ · ⎣ fy ⎦ 000001 mz

(5)

In its simplest form, the constraint space can be implemented as a parallel flexure represented by the graph shown in Fig. 1(b), where G represents the ground body and S represents the stage where the guided body (a mirror, a robotic end effector, or another device) has to be mounted. A parallel flexure system with such degrees of freedom can be build by beam flexure elements as shown in Fig. 1(c). The goal is to actuate the mechanism in a decoupled way. Therefore, each actuation will contain an intermediate body which links the stage and the ground. As the freedom space has rank 3, three intermediate bodies (B1 , B2 , and B3 ) are added as shown in Fig. 2. The design constraints and geometrical data of the problem are taken from Hao and He [2]. In this case, the proposed actuation forces are all translational (linear); they are located on the same plane and their directions are symmetrically arranged at 120◦ with respect to each other. This requirement in the actuation of the mechanism can be expressed by the input motions ⎡ ⎤T 0 0 0 −1 0 0 (6) [T]A = δ · ⎣0 0 0 cos(30) − sin(30) 0⎦ . 0 0 0 cos(30) sin(30) 0

Fig. 2. Spatial design constraints: (a) spatial arrangement of the rigid bodies of the flexible mechanism; (b) dimensions of the rigid bodies. (c) Graph representation of the mechanism.

The constraints of the actuation system determines the geometrical arrangement of the intermediate bodies as shown in Fig. 2(a). Additionally, a set

Synthesis of a Tip-Tilt-Piston Flexure System

7

of geometric design constraints are defined as depicted in Figs. 2(a) and (b): r1 = 17, 5 mm; r2 = 37, 5 mm; r3 = 42, 9 mm; h1 = 10 mm; h2 = 14 mm; l1 = 16 mm y l2 = 10 mm. Once the input motions are defined, it is necessary to establish which individual output motion is obtained for each input. The freedom space were previously described by three basis screws in Eq. 4. Nevertherless, another three basis screws can be chosen as generators. In this example, the freedom space can be build as three rotations around three coplanar axes arranged as shown in Fig. 1(d). In terms of screws it can be expressed as ⎤T ⎡ ⎤ 0 −1 0 0 0 r θ1 [T]F = ⎣ cos(30◦ ) sin(30◦ ) 0 0 0 r⎦ · ⎣θ2 ⎦ θ3 − cos(30◦ ) sin(30◦ ) 0 0 0 r ⎡

(7)

Note that these new generator twists are conveniently chosen to be symmetrically arranged at 120◦ with respect to each other, as well as the input motions. In this way, the submechanisms that link the platform and the intermediate bodies are also symmetrically arranged at 120◦ . As the input motion TAi (i = {1, 2, 3}) is applied, the output motion TF i is obtained. Step 2: The selected graph must have decoupled actuation. This implies that there must be as many paths between vertices G and S (that represent the ground and the stage bodies, respectively) as the number of desired degrees of freedom. For each path between G and S, there must exist at least one intermediate body where the actuation is to be applied. The simplest graph meeting these topological requirements is shown in Fig. 2(c). Step 3: The input motion TA1 must produce the motion TF1 on the platform. The path that connects G and S and passes through B1 is (e1 −e4 ); see Fig. 2(c). Each edge of the graph represents a submechanism that joints two rigid bodies. Then, the freedom space for each edge is denoted [j]ei . The screws that represents this freedom space are normalized with respect to its direction vector such that [j]ei is multiplied by a vector Xi that collects the motion magnitudes. The addition of the freedom spaces of edges e1 and e4 must result in TF1 . [j]e1 X1 + [j]e4 X4 = TF1

(8)

The body B1 must also move in agreement with its associated input motion, such that [j]e1 X1 = TA1 . Therefore, the unknown is the freedom space of the edge e4  [j]e4 X4 = TF1 − TA1 = 0 −1 0 δ/θ1 0 r θ1 (9) When the input motion TA2 is applied on body B2 , the platform motion is TF2 . By considering the same path as before it results in [j]e1 X1 + [j]e4 X4 = TF2

(10)

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A. G. Gallardo and M. A. Pucheta

Because no motion is applied on B1 , then TA1 = 0, X1 = 0, and [j]e4 X4 = TF2 .

(11)

Also, there is no motion applied on B3 such that [j]e4 X4 = TF3 .

(12)

As a result, [j]e4 must have a minimum of 3 degrees of freedom: ⎤T ⎡ ⎤ 0 −1 0 δ/θ1 0 r θ1 [j]e4 X4 = ⎣ cos(30◦ ) sin(30◦ ) 0 0 0 r⎦ · ⎣θ2 ⎦ θ3 − cos(30◦ ) sin(30◦ ) 0 0 0 r ⎡

(13)

By following a similar analysis for the paths (e2 − e5 ) and (e3 − e6 ), the freedom spaces for the edge e5 and e6 are [j]e5 X5 = [TF1

(TF2 − TA2 )

TF3 ]

(14)

[j]e6 X6 = [TF1

TF2

(TF3 − TA3 )]

(15)

As a result, the freedom spaces [j]e4 , [j]e5 and [j]e6 must have a minimum of 3 degrees of freedom each one. Step 4: The methodology used for the construction of the constraint spaces by using beam elements was previously reported by the authors [1,7] and is based on Blanding’s rules. Firstly, the constraint space between the intermediate bodies and the ground is analyzed. The freedom space of the edge e1 is a translation along the x axis. Therefore, the beams axes must be located in planes perpendicular to the translation direction, see Fig. 3(a). In this case, the constraint space can also be implemented as flexure elements of blade type. As proposed by Hao and He [2], a number of redundant blades increase the accuracy and reduces the of parasitic motions; four blades per intermediate body are shown in Fig. 3(c). In this figure, the mechanisms for edges e2 and e3 are the same as for e1 but rotated 120◦ around the z axis to further provide symmetry. Secondly, the constraint space between the intermediate bodies and the platform is analyzed. The freedom space for edge e4 was computed in Eq. 13. Such space was represented in Fig. 3(b) as three rotations: R2 and R3 are respectively TF2 and TF3 , and J4 is TF1 whose axis was displaced and amount of h along the z axis. To build the mechanism associated to this edge, the chosen beam elements must connect the rigid body B1 to the platform S and must be parallel or must intersect the rotation axes of Eq. 13. Two beams that satisfy those requirements are shown in Fig. 3(b). This mechanism composed by two flexures permits the freedom space required in Eq. 13 plus a translation perpendicular to the plane containing both beams. The same design for e4 is used for the mechanisms that represent edges e5 and e6 . These mechanisms are rotated 120◦ around the z axis, clockwise and counterclockwise, respectively. The freedom spaces for e4 ,

Synthesis of a Tip-Tilt-Piston Flexure System

9

Fig. 3. Physical implementation of the constraint spaces by using beam elements: (a) Minimal number of beams necessary for the constraint space of edge e1 that will be replaced for blades; (b) Beams satisfying the constraint space for edge e4 ; (c) Model of the tip-tilt-piston mechanism with beam and blade flexure elements; (d) Rotation TF1 generated by a force with direction TA1 and magnitude of 15 N; (e) Rotation of the platform around the x axis generated by actuation TA2 and −TA3 with magnitude of 15 N each one; (f) Translation of the platform along the z axis generated by actuation TA1 , TA2 , and TA3 .

e5 , and e6 surpass in one the number of degree of freedom required. Therefore, so it should be verified the platform S does not increase the rank of its freedom space. Lastly, the freedom space of the platform S is computed as the intersection of the freedom space of each path existing between G and S. The freedom space for the path (e1 −e4 ) is {[T]F ∪TA1 ∪t1 }, where t1 is the translation perpendicular to the flexures of e4 . For the path (e2 −e5 ) the freedom space is {[T]F ∪TA2 ∪t2 } and for the path (e3 − e6 ) the freedom space is {[T]F ∪ TA3 ∪ t3 }. The intersection of the freedom spaces for the paths is exactly [T]F . Therefore, the platform freedom space of S meets the requirements of Step 1.

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3.2

A. G. Gallardo and M. A. Pucheta

Case 2: Actuators Directions are Normal to the Rotations Plane

The same problem solved in the previous subsection is here developed with an alternative layout of the direction of the input motion provided by the linear actuators, they are normal to the rotations plane. Step 1: The freedom space was defined in Eq. 7. The design constraints are also those established in the Fig. 2(a). The input motion for each of the intermediate bodies, Bi (i = {1, 2, 3}), is a translation along an axis parallel to the z axis  T Ta = δ 0 0 0 0 0 1

(16)

Step 2: The graph of the topology is also that represented in Fig. 2(c). Step 3: For the path (e1 − e4 ), the motion of the platform S must be TF1 when the input motion Ta is applied on the body B1 . The freedom space of the edge e1 must be equal to the input motion, [j]e1 X1 = Ta . Therefore, T  [j]e4 X4 = TF1 − Ta = 0 −1 0 0 0 (r − δ/θ1 ) θ1

(17)

By following a similar analysis, the freedom space for the edge e4 is [j]e4 X4 = [(TF1 − Ta ) TF2

TF3 ]

(18)

Step 4: The edge e1 must permit a translation, as in the previous case study, but along an axis parallel to the z axis, see Fig. 4(a). The constraint space is again physically implemented as a set of four flexures of blade type connecting each intermediate body with the ground; see Fig. 4(c). The freedom space that must posses the edge e4 is drawn in Fig. 4(b). The three degrees of freedom are coplanar rotations. The constraint space can be implemented either as set of three flexures of beam type or as one flexure of blade type. The submechanisms that represent the paths (e2 − e5 ) and (e3 − e6 ) are equal to that of the path (e1 − e4 ) but rotated 120◦ around the z axis, clockwise and counterclockwise, respectively. In this case, the freedom space of each edge is exactly the one determined in Step 3. Therefore, the platform S will have exactly the desired movements. It is not necessary to verify it as in the previous case.

Synthesis of a Tip-Tilt-Piston Flexure System

11

Fig. 4. Physical implementation of the constraint spaces by using beam elements: (a) Minimal number of beams necessary for the constraint space of edge e1 that will be replaced for blades; (b) Beams satisfying the constraint space for edge e4 ; (c) Model of the tip-tilt-piston mechanism with all flexure elements of blade type; (d) Rotation TF1 generated by a force with direction TA1 and magnitude of 15 N; (e) Rotation of the platform around the x axis generated by actuation TA2 and −TA3 with a magnitude of 15 N each one; (f) Translation of the platform along the z axis generated by actuation TA1 , TA2 , and TA3 .

4

Conclusions

A systematic synthesis of flexure systems with decoupled actuators satisfying spatial design constraints was presented. Screw Theory and Linear Algebra were used to manipulate the motion and constraint screw systems in combination with a graph representation of the mechanism. The methodology was illustrated with the redesign of two parallel tip-tilt piston flexure stages where the actuators have directions: (i) parallel to the rotations plane; (ii) normal to the rotations

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plane. The analytical solutions were validated using finite element analyses. The solutions can be combined with redundant constraints applied to the platform to further reduce parasitic motions. Acknowledgements. The authors acknowledge the financial support from Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas for the Ph.D. fellowship granted to the first author. We also acknowledge to Universidad Tecnol´ ogica Nacional (UTN) through projects PID-UTN 7819 and 4967. We gratefully acknowledge the friendly assistance of Rodrigo Tom´ as Gonz´ alez and Juan Augusto Bernad from UTN-FRC.

References 1. Gallardo, A.G., Pucheta, M.A.: Synthesis of precision flexible mechanisms using screw theory and beam constraints. Int. J. Mech. Robot. Syst. 4(4), 277–304 (2018) 2. Hao, G., He, X.: Designing a monolithic tip-tilt-piston flexure manipulator. Arch. Civil Mech. Eng. 17(4), 871–879 (2017). https://doi.org/10.1016/j.acme.2017.04. 003 3. Hopkins, J., Culpepper, M.: Synthesis of multi-degree of freedom, parallel flexure system concepts via freedom and constraint topology (FACT) - Part I: Principles. Precis. Eng. 34(2), 259–270 (2010) 4. Hopkins, J., McCalib Jr., D.: Synthesizing multi-axis flexure systems with decoupled actuators. Precis. Eng. 46, 206–220 (2016). https://doi.org/10.1016/ j.precisioneng.2016.04.015 5. Hopkins, J.B., Panas, R.M., Song, Y., White, C.D.: A high-speed large-range tiptilt-piston micromirror array. J. Microelectromech. Syst. 26(1), 196–205 (2017). https://doi.org/10.1109/JMEMS.2016.2628723 6. Howell, L.L., Magleby, S.P., Olsen, B.M.: Handbook of Compliant Mechanisms. Wiley, New York (2013) 7. Pucheta, M.A., Gallardo, A.G.: Synthesis of hybrid flexible mechanisms using beams and spatial design constraints. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 47–56. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-60372-4 6 8. Qiu, C., Dai, J.S.: Analysis and Synthesis of Compliant Parallel Mechanisms— Screw Theory Approach. STAR, vol. 139. Springer, Cham (2021). https://doi.org/ 10.1007/978-3-030-48313-5 9. Su, H., Dorozhkin, D.V., Vance, J.M.: A screw theory approach for the conceptual design of flexible joints for compliant mechanisms. ASME J. Mech. Rob. 1(4), 041009, 8 p. (2009) 10. Sun, F., Hopkins, J.: Mobility and constraint analysis of interconnected hybrid flexure systems via screw algebra and graph theory. J. Mech. Robot. 9(3) (2017). https://doi.org/10.1115/1.4035993

Synthesis and Sensitivity Analysis of a Prosthetic Finger N´ıcolas Arroyo, Del Piero Flores, Diego Palma(B) , Renzo Sol´orzano, and Elvis J. Alegria Universidad de Ingenieria y Tecnologia - UTEC, Jr. Medrano Silva 165, Barranco, Lima, Peru {nicolas.arroyo,delpiero.flores,diego.palma, renzo.solorzano,ejara}@utec.edu.pe

Abstract. A prosthetic finger based on a one degree of freedom mechanism is synthesized in this paper using a genetic algorithm optimization method. The features of this method are defined so that the mechanism performs an anatomical movement of a hand finger. Then, a sensitivity analysis is proposed to identify which links most affect the end-effector position. Some simulation results are presented to analyze this proposal. Keywords: Prosthetic finger · Sensitivity analysis mechanisms · Genetic algorithms

1

· Synthesis of

Introduction

Researchers are currently interested in bionic prosthesis because it covers a part of the body aesthetically, and it imitates biomechanical processes. They are particularly interested in replicating functionalities of biological extremities; therefore, it has improved the people’s lives quality [1]. Prosthetic mechanisms can be based on robotic systems with multiple degrees of freedom (DOF), each controlled by an actuator [2]. However, the control of each actuator and a correct determination of the inertial properties of the links make the implementation of these systems difficult, especially when the mechanisms are small enough to be manufactured and actuated by standard methods. In this paper, we are interested in a finger prosthesis, which in addition to being a small mechanism, must be light and anatomical. Underactuated linkages for finger prosthesis are preferred for small mechanisms in the literature, see for example [4]. One-DOF mechanisms are mainly convenient for the actuator control since it becomes a simple motor fixed speed control [8]. For instance, an experimental prosthetic mechanism of an aesthetically human finger with flexion movements is proposed in [7]. Another useful reference, where a three-finger prosthetic hand is focused on the opening and closing movements but not in the mechanical analysis, was proposed in [6]. In both references, the proposed mechanism is referred to as the Toronto mechanism [16]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 13–23, 2022. https://doi.org/10.1007/978-3-030-88751-3_2

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In this paper, the Toronto mechanism is analyzed and synthesized using genetic algorithms to allow the finger to grab a specific object naturally. Furthermore, we propose to drive this mechanism using a slider-crank due to a linear movement requirement [7]. Additionally, a sensitivity analysis is done to discover which variables have a significant causality in the end-effector trajectory [15]. These variables could be manipulated to reach the desired trajectory as a re-configurable version of this proposal. This mechanism was chosen among the other one-degree of freedom ones mentioned by [8], such as the Yeong-Jin Choi or the Rodriguez finger mechanisms, since it simulates a desired natural motion of a real finger and has a simple design of four bar linkages, instead of other complex mechanisms that add springs in some joints to keep the finger in a specific position. The latest is not necessary for the Toronto mechanism. The simplicity of its design also helps the synthesis process and the selection of optimization parameters. This paper is organized as follows. The kinematic and kinetic analysis of the Toronto mechanism is shown in Sect. 2. Section 3 focuses on determining the mechanism parameters using genetic algorithms and the mechanism’s sensitivity analysis. Finally, some conclusions are shown.

2

Kinematic and Kinetic Analysis

The phalanges and linkers compose the Toronto mechanism. The first ones are represented by three triangular links, which correspond to the proximal, middle, and distal phalanges, see Fig. 1. The second ones are the three remaining links that allow the finger movement. Three parameters: length, height, and the perpendicular distance from the axis are also shown in Fig. 1 for each triangular link. The length and angle parameters of the remaining links intersection are shown, as well. An additional slider-crank (blue links) is used as an actuated driver responsible for generating the linear movement necessary to drive the finger. We propose this slider-crank based on a kinetic analysis shown in Sect. 2.2. The grounded points of this mechanism are J1 = [0, 0] mm (reference frame) and J9 = [30, 11] mm. The parameters values are presented in Table 1, which are based on the suggestions shown in [16]. Table 1. Parameters values of the mechanism Angles [◦ ]

Distances [mm] Proximal Middle

Distal

Lower lengths

PL

PA

ML

MA

DL

DA

L

45

11

25

8

21

6

10 10

a1

b1

b2

a3

20 10 10

b3

a4

41 7

b4

γ1

γ2

γP

γM

γD

21 110 110 47.73 38.66 30.96

Synthesis and Sensitivity Analysis of a Prosthetic Finger

15

Fig. 1. Toronto mechanism’s parameters driven by an slider-crank {J1 , J2 , J3 }.

2.1

Non-linear Position Analysis of the End-Effector

First, the equations representing the mechanism kinematic must be established. For this purpose, a vector position analysis is done [3,5,11,14]. Specifically, vectors following each of the link’s orientations. Then, the position analysis of the Toronto mechanism is determined by the following equations: a1 e2 + b1 e3 − d1 e0 = 0,

(1)

a1 e2 + b1 e3 − b2 e4 − a2 e7 − p19 e19 = 0, a1 e2 + b1 e3 − b2 e4 + a3 e45 + b3 e5 − c3 e8 − PL e7γP − p19 e19 = 0,

(2) (3)

a1 e2 + b1 e3 − b2 e4 + a3 e45 + b3 e5 + a4 e5γ + b4 e6 − c4 e9 −ML e8γ − PL e7γ − p19 e19 = 0,

(4)

  where 0 = 0 0 , ei is the ith unit vector of its associated link, e.g. a1 e2 in (1) is to J1 , see Fig. 1, and the vector of unknowns is the position vector of J2 respect   q = θ3 θ4 θ5 θ6 θ7 θ8 θ9 d1 . Considering a vector Φ(q) ∈ R8×1 that contains the nonlinear equations (1)–(4), then the Jacobian of Φq can be calculated by deriving this vector with respect to q: ⎡ ⎤ J11 0 0 0 0 0 0 −e0 ⎢J21 J22 0 0 J25 0 0 0 ⎥ ∂Φq ⎥ =⎢ , (5) Jq = ⎣J31 J32 J33 0 J35 J36 0 0 ⎦ ∂q J41 J42 J43 J44 J45 J46 J47 0 8×8 where J11 = J21 = J31 = J41 = b1 n3 , J22 = −b2 n4 , J25 = −a2 n7 , J32 = J42 = −b2 n4 + a3 nθ4 −γ1 , J33 = b3 n5 , J35 = −PL n(θ7 −γP ) , J36 = −c3 n8 , J43 = b3 n5 + a4 n(θ5 −π−γ2 ) , J44 = b4 n6 , J45 = −PL n(θ7 +γP ) , J46 = −ML n(θ8 +γM ) , and J47 = −c4 n9 . The vector ni is the ith unit normal vector referred to ei . Since this is a system of non-linear equations, the unknowns can be calculated iteratively using the Newton-Raphson method, i.e. considering that in the i−th iteration Δqi = −Jq,i −1 Φ(qi ), where Δqi+1 = qi + Δq. Once the orientation angles of each link have been calculated, it is possible to calculate the position of each pin at any instant of time. However, in this case the vector equation of interest to determine the end-effector position J12 is a1 + b1 − b2 + a3 + b3 +

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a4 + b4 − c4 + DL = 0, where a1 is the associated vector of the link a1 , i.e. a1 = a1 e2 , and it is similar for the other ones. 2.2

Kinetic Analysis

For the kinetic analysis, the inertial properties of each link are determined. A CAD software allow us to determine their inertial properties with respect to the ˆ direction, considering a plastic ABS material center of mass of each link in the k for the prosthesis. These values are shown in Table 2. Table 2. Inertial properties of each link Link

Crank J2 , J3 J3 , J4 , J5 J5 , J6 , J7 J7 , J8 J4 , J9 , J10 J6 , J10 , J11 J8 , J11 , J12

Mass [g]

0.123 0.229 0.219

0.516

I [g · mm2 ] 1.782 9.638 7.386

99.181

1.176

0.564

0.415

10.965 177.783

0.239

30.204

16.704

Subsequently, the free body diagram of each link is done. Applying the Newton’s second law to all the links of the mechanism links, the required torque (Fig. 2a) by the crank to drive the mechanism during its whole trajectory is obtained and it is shown in Fig. 2b. For this analysis, it is considered an external 1N force applied in the point J12 , see Fig. 2a. This force is perpendicular to the J8 , J12 projection which simulates the reaction caused by holding an object with the prosthesis fingers. According to Fig. 2b, the maximum torque is about 0.11 N.m. Then, a slidercrank is proposed to generate the linear movement of the mechanism since it does not need a high torque to move and occupies less space than other alternatives because of its simplicity.

Fig. 2. Torque required by the crank to drive the mechanism. External force (a), and driving torque vs crank angle (b).

Synthesis and Sensitivity Analysis of a Prosthetic Finger

3

17

Synthesis and Sensitivity Analysis of the Mechanism

The proposed prosthesis is fitted to a specific predefined trajectory for opening and closing hand movements since it has one degree of freedom [16], which is convenient to simplify the control problem to a motor fixed speed control. However, it may be a drawback when a human finger needs to follow different trajectories, for instance, when it grabs things of different sizes, see Fig. 3b. We solve this problem adjusting some link dimensions to reconfigure the finger’s trajectory. These links, which mainly influence the mechanism trajectory, are identified in this paper using a genetic algorithm optimization method. This process corresponds to a sensitivity analysis of the mechanism, as we call in this paper [10,15]. Although changing the dimension of links can be understood as adding degrees of freedom to the system, this process does not necessarily imply adding actuators. As can be seen, many possibilities open up as a result of this proposal, from simple mechanisms with interchangeable parts to more sophisticated ones with multiple linear actuators. 3.1

Desired Trajectories

Reference points are determined from different paths that a real human finger follows. Therefore, five desired trajectories with broader and narrower amplitudes are defined and shown in Fig. 3a. These points are obtained by following the trajectory of real fingers as they are opened and closed while holding objects of different sizes, see Fig. 3b. The number of points was limited to five due to computational limitations, since the amount of points increases the computational time exponentially does as well. Also, the points are established every 45◦ of the crank so that it would occupy the 180◦ that it requires to close or open. They are compared with the actual trajectory based on the kinematic analysis shown in Sect. 2. 3.2

Optimization Using Genetic Algorithms

Genetic algorithms are computational techniques for heuristic search inspired by natural evolution [12]. They pretend to make optimization by reducing a cost function, which maximizes the fitness of the individuals in a population. In order to achieve a minimum cost function, the set of variables which forms an individual finger and defines its movement is determined. The unknown variables are shown in (6) and their initial values are established in (7). The values of the lengths are in mm and the angles in degrees. x = [a1 , b1 , b2 , a3 , b3 , γ1 , a4 , b4 , γ2 , J1x ], ◦

◦

xinit = [10, 20, 10, 10, 41, 110 , 7, 21, 110 , −30].

(6) (7)

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Fig. 3. Initial end-effector trajectory, dashed line, and five alternative end-effector paths references (a). Closing movements for different object’s sizes (b).

The remaining variables are intentionally fixed because they give a typical proportion of the finger phalanges to the linkage. Then, an analytical function calculates the extreme position of the finger using a kinematic analysis. Once the position

is obtained, the cost function is calculated using the Euclidean distance: n f (x) = i=0 [(Pxi − Pxdi )2 + (Pyi − Pydi )2 ], where Pxi and Pxdi are the actual and desired positions of P = J12 in the x-axis, respectively; similarly for Pyi and Pydi . The next stage consisted of establishing restrictions to the variables that define the mechanism. These values are shown in (8) and (9) which are the parameters with smaller and larger amplitude, respectively. These bounds are obtained through trial and error until the finger trajectory was not completely deformed during optimization. lbound = [5, 15, 8, 5, 36, 90◦ , 5, 18, 70◦ , −40],

(8)

ubound = [15, 25, 12, 11, 66, 130◦ , 13, 28, 130◦ , −30].

(9)

The relationships between variables are also considered; these refer to the restrictions necessary for a correct movement of this mechanism, such as Grashof’s condition, shown in (10), in four-bar linkages to allow the complete revolution of the crank. S + L ≤ R + Q.

(10)

where S is the distance of the shorter link, L the longest and variables R and Q correspond to the remainder links [5,14]. These are defined as the vector inequality restrictions Arest xinit ≤ Brest , whose values are shown in (11). The

Synthesis and Sensitivity Analysis of a Prosthetic Finger

19

inequality constraints were obtained so that the added slider crank can give a complete revolution. The other sub-mechanisms do not require these constraints because they are non-Grashof, i.e., they do not need to satisfy (10). ⎡ ⎡ ⎤ ⎤ 1 −1 1 0 01×5 0 0 ⎢1 −1 0 0 01×5 −1⎥ ⎢ 25 ⎥ ⎢ ⎥ ⎥ Arest = ⎢ (11) ⎣1 1 0 0 01×5 1 ⎦ , Brest = ⎣ 0 ⎦ . 12.4 0 0 0 1 01×5 0 Finally, the MATLAB function ga(func, nvars, A, b, Aeq, beq, lb, ub) is used in the genetic algorithm optimization [13,17]. This function requires eight parameters as reference, which are shown in Table 3. Based on that, the ga() function finds the local minimum x to f (x), subject to the linear inequalities Arest x ≤ Brest . Since there are not linear equalities, A and b were set as empty vectors. In addition, the optimal solution found must be in the range lbound ≤ x ≤ ubound . In order to find an optimal x, the genetic algorithm uses a random initial population, called parents. Then, it creates a sequences of new populations, called children, based on the current population either by making random changes on the parent (mutation) or by combining a pair of parents (crossover ), and evaluating its fitness scores. After that, the current population is replaced by the children one to form the next generation. This process is repeated until the conditions established are met by the current population and, therefore, the local minimum x is found [9]. Table 3. Parameters of the MATLAB function ga() MATLAB function inputs Description

3.3

Values f (x)

func

Cost function

nvars

Number of variables

10

A

Matrix A of liner inequalities

Arest

b

Matrix b of linear inequalities

Brest

Aeq

Matrix A of linear equalities

[]

beq

Matrix b of linear equalities

[]

lb

Lower bounds on the design variable x (8)

ub

Upper bounds on the design variable x (9)

Sensitivity Analysis of the Optimized Mechanism

A sensitivity analysis consists of investigating the effects of the parameter changes on a mathematical model’s solution [10]. Hence, this can be used now to determine which parameters have a greater variation to the initial trajectory. Therefore, multiple paths are performed using the genetic algorithm method

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and the optimized variables for each path are shown in Table 4. Nevertheless, the most noticeable consequence of modifying all the mechanism variables was that the initial finger configuration became non-anatomical, despite the cost function being relatively small. Figure 4 shows the finger configuration for the fourth path, and it is possible to notice that the finger is stretched and curved out. These characteristics could affect the primary function of the prosthesis since the finger must look as anatomical as possible. Table 4. Parameters that changed for each desired trajectory a1

b1

b2

a3

xinit 10

20

10

10 41

Variable x Initial

23.7 8.9

b3

γ1

a4

b4

γ2

110◦

7

21

1100◦

Path 1

x1

9.6

Path 2

x2

10.4 21.2 11.1 9.9 43.5 103.130◦ 8.8

Path 3

x3

8.5

21

Path 4

x4

10.1 20

Path 5

x5

9.5

9.8 9.1

23.7 9.1

9.2 44.8 104.850

◦

11

9.7 43.2 114.020◦ 6.8 8.2 45.5 104.280

◦

21.4 73.910 20

J1,x −30 ◦

22.9 83.8410◦ −31.6

11.3 22.7 84.620◦

8.5 41.9 120.890◦ 6.5

−33.1

105.420◦ −32.1 −30.9

22.6 103.130◦ −34.6

Fig. 4. Optimized mechanism but with a non-anatomical shape.

Based on Table 4, the variables which were modified the most in the trajectories are verified. A graphic of the mean variation of each variable of the five paths compared to their initial value established in (7) can be seen in Fig. 5. The variables that varied the most were b1 , a4 and γ2 , considering a threshold of λ = 10% of variation. This percentage represents the average total variation. In other words, the averages shown in Fig. 5 are used to detect which variables influence the most. For this reason, b1 and a4 should be adjusted to the mechanism adaptation. For instance, a reconfigurable version of the mechanism could be designed using tiny linear actuators for links b1 and a4 , and a rotary actuator,

Synthesis and Sensitivity Analysis of a Prosthetic Finger

21

such as a servo motor, to change the angle γ2 ; in order to have the capacity of picking things of different shapes and sizes comfortably but adding some DOF. It should be emphasized that small variations of links imply a greater demand for manipulation precision. For example, the link a1 requires a resolution of approximately 0.1mm, which is not easy to reach for a linear actuator. It is worth noticing that the threshold λ = 10% shown in Fig. 5, is proposed only for small links such as Toronto’s mechanism. Notice that for bigger mechanisms, it may be convenient that reduced variation in the actuators yields significant variations in the end effector, then their reconfigurable links must be those with a degree of affectation lower than the threshold λ. In such manner, it was achieved to configure the genetic algorithm restrictions for varying the selected variables. Consequently, it was possible to find configurations for the finger trajectory to pass through most of the desired points, as shown in Fig. 6.

Fig. 5. Porcentual variation of each one of the modified parameters

Fig. 6. Optimized path of the finger with anatomical shape

22

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Conclusions

In this paper, the synthesis and sensitivity analysis of the Toronto mechanism used in prosthetic hands have been done, which simulate the flexion movement of a finger. A slider-crank was conveniently proposed as a driver, i.e. to generate the necessary linear movement and to drive the mechanism. The position of each link was determined using the Newton-Raphson method. With these results, the trajectory of the point of interest J12 was obtained. Furthermore, the synthesis and sensitivity of the mechanism have been studied by optimizing it with genetic algorithms, so the end-effector tracks a group of desired points. It was accomplished to differentiate the variables that influence the most in the trajectory and define the necessary restrictions to some variables to keep them constant and maintain the mechanism finger shape. The finger trajectory cannot be changed significantly due to the mechanism restrictions; however, it would be possible to modify the trajectory with reconfigurable links, or adding actuators, to grab things of different sizes.

References 1. Basumatary, H., Hazarika, S.M.: State of the art in bionic hands. IEEE Trans. Hum. Mach. Syst. 50(2), 116–130 (2020) 2. Bottin, M., Rosati, G.: Trajectory optimization of a redundant serial robot using cartesian via points and kinematic decoupling. Robotics 8(4), (2019). https://doi. org/10.3390/robotics8040101 3. Castro, O., C´espedes, A., Alegria, E.J.: An adaptable four-bar mechanism for databased model validation. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 119–127. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-60372-4 14 4. Ceccarelli, M., Yao, S., Carbone, G., Zhan, Q., Lu, Z.: Analysis and optimal design of an underactuated finger mechanism for robotic fingers, pp. 242–256 (2012). https://doi.org/10.1177/0954406211412457 5. Constans, E., Dyer, K.B.: Introduction to Mechanism Design: With Computer Applications. CRC Press, Boca Raton (2018) 6. Dechev, N., Cleghorn, W., Naumann, S.: Multi-segmented finger design of an experimental prosthetic hand. In: Mechanism and Machine Theory (2000) 7. Dechev, N., Cleghorn, W., Naumann, S.: Multiple finger, passive adaptive grasp prosthetic hand. Mech. Mach. Theory 36(10), 1157–1173 (2001) 8. Difonzo, E., Zappatore, G., Mantriota, G., Reina, G.: Advances in finger and partial hand prosthetic mechanisms. Robotics 9(4), 80 (2020) 9. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning, 1st edn. Addison-Wesley Professional, Boston (1989) 10. Grenga, T., Paolucci, S., Valorani, M.: Sensitivity analysis and mechanism simplification using the G-scheme framework. Combust. Flame 189, 275–287 (2018) 11. Gutarra, A., Palomino, S., Alegria, E.J.: Hexapod walking mechanism based on the Klann linkage for a 2DoF amphibious robot. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 302–310. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-60372-4 34

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12. McCall, J.: Genetic algorithms for modelling and optimisation. J. Comput. Appl. Math. 184, 205–222 (2005) R For Engineering Students 13. Messac, A.: Optimization in Practice with MATLAB: and Professionals, 1st edn. Cambridge University Press, Cambridge (2015) 14. Norton, R.L.: Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 6th edn. McGraw-Hill Series in Mechanical Engineering (2020) 15. Saltelli, A.: Sensitivity analysis for importance assessment. In: Proceedings of the 3rd International Symposium on Sensitivity Analysis of Model Output, pp. 3–18 (2001) 16. Sospedra, B.: Dise˜ no mec´ anico de pr´ otesis de mano multidedo antropom´ orfica infractuada. Ph.D. thesis, Universitat Jaume I (2015) 17. Stephen Boyd, L.V.: Convex Optimization. Cambridge University Press, Cambridge (2004)

Synthesis of an Exoskeleton and Walking Pattern Identification for a Rehabilitation System Yanpierrs Figueroa1(B) , Joseph D´ıaz1 , Gustavo Quino2 , and Elvis J. Alegria1 1

University of Engineering and Technology UTEC, Jr. Medrano Silva 165, Barranco, Lima, Peru {yanpierrs.figueroa,joseph.diaz,ejara}@utec.edu.pe 2 University of Oxford, Parks Road, Oxford OX1 3PJ, UK [email protected]

Abstract. This paper proposes a modified Theo-Jansen mechanism to develop an exoskeleton for people with muscle weakness in the lower extremities. The synthesis of this modified mechanism is performed using genetic algorithms to adjust its movement to a conventional human walking or jogging. The movement patterns of real leg’s trajectories are experimentally obtained through image processing. According to their relevance, these patterns are analyzed and weighted, to optimize a weighted cost function. Furthermore, the kinematic and kinetic analysis of the mechanism are briefly shown. Keywords: Synthesis of mechanisms · Theo-Jansen linkage Exoskeleton · Genetic algorithm · Movement patters.

1

·

Introduction

Lower limb physical rehabilitation is a process to regain the leg’s motor skills due to, for example, an injury or illness. Technological systems based on mechanisms and robots have been shown to improve the efficiency of physical rehabilitation based on adequate movement control. In addition, these systems manage to follow the desired trajectory, which must be adjusted to a typical leg movement. A physical rehabilitation system is usually based on a serial robot or an underactuated mechanism [1]. In the first one, each joint is usually associated with a one-degree-of-freedom actuator. The position control is determined by its inverse kinematics, whose performance depends mainly on three factors: the robustness of each motor control [2], the stiffness of the links, and correct distribution of masses to reduce the robot’s vibration [5]. These factors are often complex to address and also make this type of equipment more expensive. On the other hand, the second one proposes a one-degree-of-freedom mechanism driven by a single motor at a constant speed [3,4,15,16], which greatly simplifies the required control to a motor constant speed control. Based on interconnected links, this mechanism produces repetitive movements at the same c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 24–33, 2022. https://doi.org/10.1007/978-3-030-88751-3_3

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25

frequency as the rotation of the crank. This repeatability is a desired characteristic in a lower limb rehabilitation process [14]. In addition to the ease of motor control, the rigidity and mass distribution requirements in underactuated mechanisms are often simpler than those of a serial robot, which reduces costs [12]. Some mechanisms for walking movements based on linkages are presented in the literature. For example, a Stephenson III six-bar linkage that mimics the ankle movement of one leg based on a combined direct solution of the synthesis equations with an optimization strategy is shown in [4]. This mechanism does not consider the movement of the knee, which is also an important point to analyze in order for the mechanism to better mimic a walk. Thus, a mechanism that adjusts simultaneously to the movement of the knee and the ankle, relative to a fixed point located on the hip, allows a more adequate rehabilitation process. This also turns the mechanism into an exoskeleton that can be attached to a leg at three points: the hip, knee and ankle. In this paper we are interested on a low-cost portable rehabilitation system, based on a one-degree-of-freedom linkage, so that the patient can use it at home. Our rehabilitation system is based on a modified Theo-Jansen mechanisms (TJM), which is portable and attachable to a conventional treadmill. The TJM crank is attached to a transmission shaft to drive the mechanism. The Fig. 1 shows a 3D model of the proposed system. The top part of the structure, is detachable to allow the patient to be comfortably installed into the treadmill. Furthermore, an adjustable seat is included for the patient’s rest, when entering or leaving the treadmill. This system considers the active and passive modes, depending on the patient’s autonomy. Two main contact points between the patient and the rehabilitation system: the ankle and the knee, which should adjust the natural movement of a human leg, are being considered also. An adjustable fasteners is included in both contact points to ensure safety and better performance of the mechanism. This paper focuses on the analysis and synthesis of a modified TJM, here referred to as Exo-Jansen, so that a more anatomical movement is achieved. The TJM is performed as an exoskeleton for the lower body using a modified Theo-Jansen mechanism for two rehabilitation exercises: walking and jogging. The modified TJM shown in Fig. 1, after an optimization process, based on real walking patterns of both the knee and ankle, achieves a human-like walking. The walking patterns are obtained using a motion tracking algorithm and then calibrated. These patterns define trajectories used as reference in the genetic algorithms optimization of the Exo-Jansen. Experimentally, we note that the cost function requires two reference points: the ankle (point P) and the knee (point F). Likewise, the initial lengths of each link are obtained from the experimental measurement of four people with average heights among 1.65 to 1.75 m for the anatomical purpose described. Section 2 introduces the kinematic and kinetic analysis of the Exo-Jansen, respectively. The optimization analysis is shown in Sect. 3. Finally, some conclusions and discussions are presented.

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Fig. 1. CAD model of the system (left) and diagram of the leg mechanism (right)

2

Kinematic and Kinetic Analysis of the Exo-Jansen

A kinematic analysis is conducted to determine the position, velocity and acceleration of the entire mechanism in function of the crank kinematic state [7]. To analyze the mechanism kinematic, we divide the Exo-Jansen into three simple fourbar linkages, as [8,16] lays out, defined by the points: ABCD, ABGD, and DHF E, each group of points forms a loop, so there is three loops in the entire linkage, see Fig. 1. Each loop is characterized by a vector equation, which is used in position, velocity, and acceleration analysis. The angles θ1 , θ2 , θ3 , θ4 , θ5 , θ6 , θ8 , and θ9 are the orientation of the links AD, AB, BC, DC, BG, DG, EF , and HF , respectively. To begin the analysis, it is necessary to establish the unknowns in each loop that solve vector equations. Taking into account that: link AB is the crank, link AD is grounded, all the link’s lengths are a priori known, links DG and GH are collinear, the geometrical relationships {θ7 = θ4 + δce , θ10 = θ9 + δlp } exist, where δce and δlp are constants; then the unknowns of the Exo-Jansen are: θ3 , θ4 , θ5 , θ6 , θ8 , and θ9 . Furthermore, the angular velocity and angular acceleration related with each one are also unknown and must be calculated. Once the unknowns are established, the kinematic analysis is straightforward. For the analysis of position, which is nonlinear in the unknown variables, the Newton-Raphson method is used. For the velocity and acceleration analysis, which is a system of linear equations, the position vector equations are derived once and twice respectively, then the unknowns are matrix cleared to solve a simple linear equations system for each fourbar linkage loop. The vector equation of each loop for each kinematic analysis are in Table 1. For a more detailed analysis of this, see [6,15].

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Table 1. Kinematic equations of the Theo Jansen mechanism shown in Fig. 1 Variables Kinematic equations (ABCD) {θi }i=3,4

ae2 + be3 − de1 − ce4 = 0

{ωi }i=3,4 aω2 n2 + bω3 n3 − cω4 n4 = 0

{αi }i=3,4 aα2 n2 − aω22 e2 + bα3 n3 − bω32 e3 − cα4 n4 + cω42 e4 = 0 (ABGD) {θi }i=5,6

ae2 + ge5 − me6 − de1 = 0

{ωi }i=5,6 aω2 n2 + gω3 n5 − mω6 n6 = 0

{αi }i=5,6 aα2 n2 − aω22 e2 + gα3 n5 − gω32 e5 − mα6 n6 + mω62 e6 = 0 (DHFE) {θi }i=8,9

(m + h)e6 + le9 − ee7 − f e8 = 0

{ωi }i=8,9 (m + h)ω6 n6 + lω9 n9 − eω4 n7 − f ω8 n8 = 0

{αi }i=8,9 (m + h)α6 n6 − (m + h)ω62 e6 + lα29 e9 − eα4 n7 + eω42 e7 − f α8 n8 = 0

The kinetic analysis is about determining the forces and torques acting on the internal parts of the mechanism. The analysis process is based on the free body diagram corresponding to each element of the Exo-Jansen, see Fig. 2. A dynamic analysis is performed using Newton’s laws applied to rigid bodies that allows obtaining the equations shown in Table 2, where Fi and Wj are the internal force and weight of the links which are the only known forces, i = {A, B1 , B2 , C, D, E, F, G, H} and j = {crank, BC, DCE, DGH, BG, EF, P }, respectively; and the ith unitary vector si is a π/2 rotated vector of ri , shown in Fig. 2. An external force acting on link FHP is nor considered here, because the Exo-Jansen is not expected to touch the ground, as well as the forces generated from the interaction with the patient are not considered in this work either. Furthermore, to solve the kinetic equations, the matrix system is used together with the kinematic analysis found in Sect. 2, to calculate the unknowns of the linear equation system, see [13]. Table 2. Links Kinetic Equations Crank AB

Link BC

FA − FB1 − FB2 + Wcrank = mcrank acrank −sTgB1 (FB1 + FB2 ) + T = Icrank αcrank

−FC + FB1 + WBC = mBC aBC −sTgC1 FC + sgB2 FB1 = IBC αBC

Link DCE

Link BG

FC − FE + FD1 − FD + WDCE = mDCE aDCE T T −sT gC FC − sgE FE + sgD (FD1 − FD ) = IDCE αDCE

FB1 − FG + WBG = mBG aBG sTgB3 FB1 − sTgG1 FG = IBG αBG

Link EF

Link FHP

FE − FF + WEF = mEF aEF sTgE2 FE − sTgF1 FF = IEF αEF

FF + FH + WP = mP aP sTgF2 FF + sTgH2 FH = IP αP

2

1

1

Link DGH FD − FD1 + FG − FH + WDGH = mDGH aDGH T sgD2 (FD − FD1 ) + sTgG2 FG − sTgH1 FH = IDGH αDGH

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Fig. 2. Link’s free body diagram.

In this analysis the effect of gravity is taken into account. For this, it is necessary to know the mechanical properties of the element to be analyzed, such as mass m, mass moment of inertia I, and distance to the center of mass in the x-axis CMx , as well as in the y-axis CMy . These properties appear in Table 3, it should be noted that the material chosen for the parts is 20 mm thick aluminum. Table 3. Link’s Mechanical properties Link

m[kg] I[kg/cm2 ] CMx [cm] CMz [cm]

AB

0.02

3.68

0

0

BC

0.01

3.06

12.9

0

DCE 0.02

6.13

4.14

7.585

BG

0.02

3.68

10.4

0

DGH 0.01

2.77

22

0

EF

0.08

FPH 0.04

2.1

24.1

0

77.28

3.398

24.323

Figure 3 (left) shows the trajectory of point P, based on Table 1, which is similar to an ankle trajectory, but requires to be optimized for human-like walking and jogging movements. Moreover, Fig. 3 (right) shows the torque required due to the movement of the crank for a complete cycle of motion, which presents a sinusoidal-motion that evidences the smooth behavior of the system. Likewise, it is possible to identify the minimum and maximum torque necessary to start the

29

5 -75 y-axis [cm]

Crank Torque [Nm]

Synthesis Walking Pattern Identification of the Exo-Jansen

0

-5

0

100 Crank Angle

200

300 2

-80 -85 -90

-30

[deg°]

-20

-10

0

10

x-axis [cm]

Fig. 3. Trajectory of point P, left, and torque required for the crank, right.

mechanism, with an approximate peak magnitude of 4.2 Nm, this would facilitate the identification of the ideal actuator that will be in charge of performing the movement.

3

Computational Synthesis of the Mechanism

In this section, the synthesis of the Exo-Jansen mechanism is done using a gradient-based optimization method so that trajectories for both walking and jogging are fitted to two reference paths, which are obtained experimentally. Representative points of the desired paths are selected and used to calculate a cost function. The variables to optimize are some link’s lengths and angles of the mechanism. It is worth noting that the crank drives the mechanism at a constant speed. Therefore, the mechanism adjusts the knee and ankle to the desired position and the desired speed of a typical leg movement. A typical angular velocity of the crank for walking and jogging, obtained experimentally, are 53.5RPM and 222RPM, respectively. 3.1

Real Movement Patterns Sampling

The real movement trajectories are sampled using a motion tracking algorithm. To do this, real walking and jogging sequences are recorded on video using a treadmill, see Fig. 4. Two point markers with a phosphorescent color are located on the knee and ankle. These points are filtered using a thresholding process so that only the trajectories of the ankle and knee are sampled, which are properly calibrated considering that point A, see Fig. 1, is the system reference. In this paper, the following points are conveniently selected for the optimization process: Walking

Pankle

Walking

Pknee

Jogging

Pankle

Jogging

Pknee

  = (7, −83.3); (15.4, −79.8); (6.2, −70.6); (−35.3, −79.9); (−6.4, −83.5) ,   = (−9.3, −43.2); (−38, −43.5) ,   = (7.5, −79.5); (14.6, −73.8); (2.5, −67.1); (−33.2, −79.2); (−6.4, −83.5) ,   = (−11.9, −42.7); (−41.4, −40) .

(1) (2) (3) (4)

This point is placed in the origin with the whole set. All the process is shown in Fig. 4. For more information see [10].

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Fig. 4. Tracking of walking and jogging trajectory, left and right, respectively (a), and calibrated movement patterns (b).

3.2

Cost Function with Weighting Factor

The cost function f is determined by the Euclidean norm between each desired point and the actual point, according to points P and F , ankle and knee respectively. The cost function is expressed in (5), which depends on the weigh factor λi , i = 1, . . . , n assigned to each desired point. So we can adjust the relevance of the ith reference point, in the optimization process, by adjusting λi . In this project, we propose, λ = [1.25, 0.9, 1.25, 0.9], for the desired points (1)–(4). f (X) =

n 

λi



(Pxi − Pxi d )2 + (Pyi − Pyi d )2 ,

(5)

i=0

This constrained nonlinear multivariable function is optimized using the Matlab function, fmincon; which finds the minimun values. For details see [9]. 3.3

Links Measurement Using Optimization

In Fig. 5(a) the desired points are shown. For the links optimization, boundaries are required, otherwise this method would search a solution into a big range of possibilities, making the process slower and even having singularity problems. Furthermore, boundaries are important to impose an anatomic behavior to the mechanism. For these reasons we fixed the lengths of links a and d, and the

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remaining should be optimized. In this work, the adjustable links where bounded with ±10% of their initial lengths, see Table 3. As a result of the optimization process, it is shown Fig. 5(b) and Fig. 5(c). We can see that the costs, in both, are low compared with their initial costs.

Fig. 5. Exo-Jansen before and after the optimizations. Walking desired points (yellow marks), jogging desired points (red marks), and actual trajectories (blue dots).

The restrictions for the optimization were collected experimentally, i.e., measurements of the lower body were taken from 4 people between 1.68–1.76 m tall and with an average physical build, to keep the proposed model as close to reality as possible, these measurements were taken from the joint points of the proposed TJM (Table 4). Table 4. Link’s boundaries for optimization Link Lower boundary Initial value Upper boundary a

4.5

4.5

4.5

b

20

22.5

3.68

c

19

21

23

d

15.8

15.8

15.8

e

15

17.5

20

f

57

58.5

60

g

22

25

28

h

19

23

27

l

8

8.5

9

m

16

19

22

p

40

41

42

32

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Discussions on the Optimization Method

As shown in the optimization method, two points of the knee were considered as constraints of our model, to achieve both an anatomical trajectory of the ankle and the knee, as well as to restrict the sizing of the TJM towards a more anatomical figure. However, this does not guarantee to avoid problems of hyperextension or hyper compression of the knee, to solve these problems the knee joint can be replaced by a six-bar mechanism, such as the one proposed in [4], which ensures a smoother and more natural motion for that point.

4

Conclusions

The synthesis of an exoskeleton and walking pattern identification for a rehabilitation system was shown in this paper. The kinematic and kinetic analysis was firstly shown. For the synthesis process, movement patterns are required, then we collect data experimentally for the reference points. A weighted cost function was proposed also to impose that some reference points are more important than others, so that an anatomical movement is reached. Furthermore, the mechanism was optimized for both rehabilitation exercises, walking and jogging, decreasing the initial costs by 89.8% for walking and 93.6% for jogging. Also, the optimization of point F, which simulates the patient’s knee, was arranged, all accompanied by restrictions in the sizing of the links that make up the mechanism, to maintain anatomical proportions to ensure greater patient comfort and safety when performing the movement. Our main future work is the adaptability of the Exo-Jansen for other users of different sizes by adding a proportionality constant.

References 1. Dollar, A.M., Herr, H.: Lower exrewmity exoskeletons and active orthoses: challenges and state-of-the-art. IEEE Trans. Rob. 24, N1 (2008) 2. Lee, J., Kim, H., Jang, J., Park, S.: Virtual model control of lower extremity exoskeleton for load carriage inspired by human behavior. Auton. Robot 38, 211– 223 (2015). https://doi.org/10.1007/s10514-014-9404-1 3. Nansai, S., Rojas, N., Elara, M.R., Sosa, R., Iwase, M.: On a Jansen leg with multiple gait patterns for reconfigurable walking platforms. Adv. Mech. Eng. 7, 1–4 (2015) 4. Tsuge, B., Plecnik, M., McCarthy, J.: Homotopy directed optimization to design a six-bar linkage for a lower limb with a natural ankle trajectory. J. Mech. Robot. 8(6), 1–7 (2016) 5. Acevedo, M., Gherrero, M.: An exoskeleton for enhancing strength and endurance during walking. Industrial and Robotics Systems, LASIRS (2019) 6. Constans, E., Dyer, K.: Introduction to Mechanism Design: With Computer Applications. CRC Press, Boca Raton (2018) 7. Bovi, G., Rabuffetti, M., Mazzoleni, P., Ferrarin, M.: A multiple-task gait analysis approach: kinematic, kinetic and EMG reference data for healthy young and adult subjects. Gait Posture 33(1), 6–13 (2011)

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8. Cuadrado, K.: An´ alisis Cinem´ atico y cin´etico del mecanismo de Theo Jansen. Dise˜ no y construcci´ on de juguete prototipo, Escuela Polit´ecnica Nacional (2018) 9. Li, J.: The optimization design of the four-bar linkage based on MATLAB. In: 2015 International Conference on Intelligent Systems Research and Mechatronics Engineering (ISRME), Zhengzhou, China (2015) 10. MathWorks Student Competitions Team. Computer Vision for Student Competitions: Object Detection using Blob Analysis, GitHub (2021). https://github.com/ mathworks/auvsi-cv-blob 11. Landmine and Cluster Munition Monitor: Monitor, The-monitor.org (2021). http://www.the-monitor.org/en-gb/home.aspx. Accessed 10 Mar 2021 12. Sabaapour, M.R., Lee, H., Afzal, M.R., Eizad A., Yoon, J.: Development of a novel gait rehabilitation device with hip interaction and a single DOF mechanism. In: 2019 International Conference on Robotics and Automation (ICRA), pp. 1492– 1498 (2019) 13. Norton, R., Rios, M., Osornio, C., Acevedo, M.: Dise˜ no de maquinaria. McGrawHill, Mexico, D.F. (2013) 14. Shao, Y., Xiang, Z., Liu, H., Li, L.: Conceptual design and dimensional synthesis of cam-linkage mechanisms for gait rehabilitation. Mech. Mach. Theory 104, 31–42 (2016) 15. Castro, O., C´espedes, A., Alegria, E.J.: An adaptable four-bar mechanism for databased model validation. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 119–127. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-60372-4 14 16. Gutarra, A., Palomino, S., Alegria, E.J.: Hexapod walking mechanism based on the Klann linkage for a 2DoF amphibious robot. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 302–310. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-60372-4 34

Mechanisms Analysis

Analysis of a Proposal for a Self-aligning Mechanism for Cartesian Robot in Greenhouses Lucas Bernardon Machado(B) , Antonio Carlos Valdiero, Henrique Simas, and Daniel Martins Federal University of Santa Catarina, Florianópolis, Brazil [email protected], {antonio.valdiero, henrique.simas,daniel.martins}@ufsc.br

Abstract. The advances in technologies that include agriculture have reached relevant proportions, when dealing with the process of automation of agricultural greenhouses, made in “smart greenhouses”. In general, the 3DoF Cartesian robots PPP-type Gantry perform development in a movement composed of three translations, with their axis of movement coinciding with a Cartesian reference coordinate system. This work proposes and develop a self-aligning analysis in a Gantry robot, verifying its redundant mobility and constraints highlighting the respective advantages and properties, both in the mechanism assembly and in full operation, avoiding misalignments, since the robot will move along rails. The development base uses the Reshetov method to analyze the parameters of the mechanism, to propose a self-aligned conception and its possible representations. Keywords: Cartesian robot · Greenhouses · Reshetov · Self-alignment

1 Introduction The use of manipulators in agriculture has grown significantly, a fact that is directly associated with the improvement of operating conditions, thus guaranteeing new standards of efficiency and agility in activities, both planting and harvesting, in agricultural greenhouses. Agriculture has past several moments of evolution and improvement in the 20th century, it started with Agriculture 1.0 characterized by animal traction, right after Agriculture 2.0, it replaced animal traction with combustion engines, following by Agriculture 3.0 the Global Positioning System (GPS) was developed and finally Agriculture 4.0 incorporated connectivity and automation with the use of vehicles, machines and robots, agreeing on precision agriculture with automation and robotics [12]. According to the International Federation for the Promotion of Mechanism and Machine Science (IFToMM), a mechanism is defined as a “system of bodies designed to transform movements into forces for one or more bodies to be moved” [3]. Kinematics can be defined as the branch of dynamics that deals with movement on its own, isolated from forces associated with movement [2, 14]. Gantry Cartesian robots have three prismatic joints (PPP) resulting in a movement composed of three translations, whose axes of motion coincide with a Cartesian reference coordinate system, that is, we have 3DoF of work and operation. Such a mechanism © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 37–45, 2022. https://doi.org/10.1007/978-3-030-88751-3_4

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operates in a suspended manner in relation to the ground, ideal for optimizing the working space and operation inside the greenhouses. Its use encompasses several practices, namely: planting, harvesting, spraying, weed control, localized application of fertilizers and pesticides [9]. This work studies the main movements (mobility and self-alignment) of a Cartesian robot during its operation in protected cultivation, since the robot moves through linear guides (rails). The main objective is to develop a self-alignment analysis of the moving mechanism, where mainly are analyzed components of the rails, the pulleys and the robot’s chassis. Due to manufacturing and assembly errors, the rails often do not provide a linear (irregular) displacement to the mechanism, thus causing misalignment problems that causes lock in the rollers, affecting the entire functioning of the robot. Thus, an analysis based on the method proposed by Reshetov [11] whose objective is to eliminate the redundant constraints by increasing the DoF (Degree of Freedom) of the Cartesian robot, thus improving its functioning, design and excessive movement restrictions. This article is organized as follows. Section 2 presents an analysis of the geometry of the Cartesian robot, analyzing its mobility and restrictions, then Sect. 3 presents the development of the Reshetov method, proposing a possible self-alignment solution. Section 4 presents the analyzed proposal and finally, Sect. 5 presents the final conclusion.

2 Cartesian Robot Geometry for Agricultural Greenhouses For the analysis and development of the work in question, it was necessary to determine some restrictions and design requirements, directly linked with the constructive (dimensions) and operational (operational requirements of the activities to be carried out) aspects of the greenhouses, in accordance with the Table 1: Table 1. Design constraints. Width (m)

Ceiling height (m)

Total height (m)

6.50

3.00

4.50

7.00

3.50

5.00

8.00

4.00

5.50

10.00

4.50

6.00

Length (m)

Lubrification

Operating temperature (ºC)

Operating speed (m/s)

Weight transported (kg)

30.00

Low

0 - 40

1

3

The structure in which the analyzed mechanisms move is shown in Fig. 1, where the rails (irregular, that is, a non-linear trajectory) are located at the top of the greenhouse, thus composing an irregular path to be traversed. The blue rail was considered the reference rail for analyzing the problem, which is less irregular than the black rail. The analyzed robot has 6 links and 8 joints (Fig. 2a). The kinematic pairs formed from the contact between the pulleys and the rail are two pairs of linear contact, that is,

Analysis of a Proposal for a Self-aligning Mechanism

39

Fig. 1. Simplified greenhouse structure with rails and benches.

it forms a line of contact, thus there are four kinematic pairs of type IV [11] that allow a translation along the x-axis and a rotation around the z-axis [11]. The other kinematic pairs, of the pulley assembly with the chassis of the mechanism, are all of revolution, the only movement of the pulley in relation to the assembly. Then, the representation of the mechanism was defined by means of a graph (Fig. 2b), where the links are represented by points and the joints by the edges. Thus, it is possible to define some analysis requirements for mobility, such as the number of independent circuits, that is, loops that have exclusive joints [4, 6, 7, 13].

a-)

b-)

Fig. 2. Schematic drawing of links and joints (a) and graph of the mechanism (b).

The number of independent circuits (V) is given through the graph (Fig. 2a) or according to the Euler Equation (Eq. 1) [6, 7], where j is the number of joints and n is the number of determined links in Fig. 1 [1, 7]: v−1=j−n v−1=8−6→v =3

(1)

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Performing the analysis in space, that is, λ = 6, it is possible to calculate the mobility (M) of the system using the following equation that considers the performance redundancies (q) [1, 7]: j fi M − q = (n − j − 1) × λ + i=1   M − q = (n − j − 1) × λ + fa + fb + fc + fd + fe + ff + fg + fh

(2)

One must remember the freedoms and restrictions of joint according to Reshetov [11]. Thus, the joints “a, b, c, d” are type IV [11] joints (linear contact) where they release only two movements (a translation along the x-axis and a rotation around the z-axis). The joints “e, f , g, h” are all rotary joints, that is, they release only one rotation movement around the y-axis [11]. Therefore, according to Eq. 2 the mobility of the system was defined [1, 7]: M − q = (n − j − 1) × λ +

j i=1

fi

M − q = (6 − 8 − 1) × 6 + (2 + 2 + 2 + 2 + 1 + 1 + 1 + 1) M − q = (−3) × 6 + (12) → M − q = −6 When manipulating the Euler equation (Eq. 1), we can obtain the same relation for mobility: j M = −v × λ + fi i=1   M = −v × λ + fa + fb + fc + fd + fe + ff + fg + fh M = −3 × 6 + (4 + 4 + 4 + 4 + 1 + 1 + 1 + 1) = −18 + 12 = −6

(3)

To determine the restrictions and extra mobility of the mechanism in question, the method proposed by Reshetov [11] can be applied. The main objective of Reshetov’s self-aligning analysis is to eliminate the constraining constraints of the system, thus increasing the degrees of freedom (DoF) of the mechanism.

3 Analysis of the Mechanism Using the Reshetov’s Method The Reshetov method [11] aims to help the problem in Fig. 3, where there are restrictions on the displacement of the pulleys in relation to the irregular track, since they were designed to satisfy the ideal conditions of the linear track. Thus, the pulleys are subject to undesirable trajectories, precisely because they have freedom of rotation only in relation to such, and it would be necessary a freedom of translation to follow the irregular path without compromising the functioning of the mechanism. Such a method is a technique of analysis where the freedoms and restrictions found in the mechanism are all organized in a table (table method). The table consists of two columns, one representing the freedom of rotation and the other the freedom of translation, according to each joint in the system. So, after completing this, it is possible to determine all the freedoms and restrictions present, since the mobilities can be rearranged

Analysis of a Proposal for a Self-aligning Mechanism

41

Fig. 3. Schematic figure of the mechanism on the rails.

according to what needs to be filled in the table [8, 11]. It is worth mentioning that in this distribution of extra mobilities (if contains) cannot be rearranged for the same axis, for example, if we have one rotation left on the x-axis, such rotation can only be reallocated for a translation on the y or z- axis [5–7]. Finally, the number of absent (missing) freedoms indicates the number of restrictions of the mechanism and, consequently, the number of extra freedoms, after satisfying the previous condition, indicates the mobility of the mechanism [8]. 3.1 Analysis for the Robotic Mechanism The analysis for the mechanism that couples the Cartesian robot is separated into three parts, one for each loop (independent circuit). Thus the circuits are: abef , cg, dh, as shown in Fig. 2. Therefore, the freedoms were distributed according to the joints defined in Sect. 2 (mechanism geometry), taking into account the referential coordinate axis of Fig. 1, therefore, we have the following analysis [8, 11] (Fig. 4): For circuit 1 (a, b, e, f ), a rotation around the y-axis (f - in green) was replaced by a translation in the z-axis (in pink), and a rotation around the z-axis (b - in green) was replaced by a translation along the y-axis (in yellow), so the extra mobilities were rearranged and what “came out” was a translation around the x-axis (b - in blue), which represents the mechanism’s mobility. Therefore, the rotations and translations that have not received any mobility are considered as system constraints (in red), thus, there is mobility equal to 1 and a number of restrictions equal to 7, which proves the value found for the relation of Eq. 2, where [7]: M − q = (n − j − 1) × λ +

j i=1

fi = −6 → 1 − 7 = −6

(4)

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Fig. 4. Reshetov’s method.

4 Self-aligned Mechanism Proposal Using a step-by-step guide to obtain kinematic chains [10] and a guide to obtain all possible structures [5] leads to a more in-depth approach. Therefore, there are some viable possibilities that serve as a new conception for a self-alignment of the mechanism, eliminating redundant constraints and providing adequate mobility for the robot to move on the irregular track without compromising the efficiency of operation and safety of the operation, just as the mechanism is supposed to in relation to the soil and

Fig. 5. Proposed self-aligned mechanism.

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Fig. 6. Prototype design of cartesian robot in greenhouse for future validating experimentally.

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the greenhouse benches. The proposal for a self-aligning mechanism for the problem in question, according to the step-by-step mentioned above [7] suggests the following solution: replace the revolution joints that connect the pulley with the chassis with a type IV joint [11], which allows both a rotation around the z-axis and a translation along the x-axis, thus eliminating the difficulty of locomotion on the irregular track. This new joint suggested can be prismatic (allows translation) combined with a revolute joint, thus obtaining a new kinematics structure, as shown in the following Fig. 5: The next activity of the project is the experimental validation of a prototype, whose design is shown in Fig. 6 (real scale). After modeling, the next step to be performed is the construction of a reduced scale model, assisting in the most diverse analyzes related to greenhouses, in addition to serving as a learning tool.

5 Conclusions This paper presented an analysis of mobility and restrictions of a new manipulative mechanism for agricultural greenhouses used for the most diversified activities in protected cultivation. From an analysis using the mobility and Euler equations, it was found the presence of undesirable constraints for this case, since the mechanism found it difficult to move on non-linear guides (irregular rails). The use of a self-aligning mechanism for greenhouse allows the construction of prototypes with low costs, not-complex assembly and adequate precision. Thus, the Reshetov method was used, which in turn proved the existence of extra mobility in the mechanism and its respective redundant restrictions. Based on the problem it was proposed some changes in joints of the mechanism, in order to eliminate redundant constraints and unwanted mobility. It is important to note that there are several options of a proposal for self-alignment in relation to the analyzed mechanism. Therefore, the choices described in the article did not provide significant changes in the structure as a whole (it would need a more in-depth analysis of the number, synthesis and type), but small changes in some joints that have already partially met the need for the theme displayed. As future work, analysis will be carried out for the complete system, which involves its activation, trajectory and control of greenhouse parameters (humidity, brightness, ventilation, irrigation), thus, new components will be subject to a more in-depth self-aligning analysis. Acknowledgments. The authors would like to express their gratitude to CNPQ (National Council for Scientific and Technological Development) and UFSC (Federal University of Santa Catarina) for the support this project (SIGPEX Number: 202002173, 202023266 and 202002437) with scientific initiation, master and doctoral research.

References 1. Carreto, V.: Study of self-aligning mechanisms using static and kinematic dependence analysis. Ph.D. thesis, Federal University of Santa Catarina, Brazil (2010) 2. Hunt, K.H.: Kinematic Geometry of Mechanisms. Oxford University Press (1978)

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3. Ionescu, T.G.: IFTOMM definitions. Mech. Mach. Theory 38, 767–776 (2003) 4. Li, Z., Lou, Y., Zhang, Y., Liao, B., Li, Z.: Type synthesis, kinematic analysis, and optimal design of a novel class of Schonflies-motion parallel manipulators. IEEE Trans. Autom. Sci. Eng. 10(3), 674–686 (2012) 5. Martins, D., Murai Hideki, E.: Mechanisms: synthesis and analysis with applications in robotics. EdUFSC, Florianópolis (2019) 6. Martins, D., Guenther, R., Simas, H.: Hierarchical singularity analysis of an articulated robot. In: Proceedings of the Brazilian Congress of Mechanical EngineeringIX COBEM, Uberlandia (2001) 7. Martins, D.: Hierarchical kinematic analysis of manipulator robots. Ph.D. thesis, Federal University of Santa Catarina, Brazil (2002) 8. Meneghini, M., Artmann, V.N., Simoni, R., Simas, H.: Mobility analysis and self-alignment of a novel asymmetric 3T parallel manipulator. In: Pucheta, M. Cardona, A., Preidikman, S., Hecker, R. (eds.) Multibody Mechatronic Systems 2020, MUSME. Cordoba, Argentina (2020) 9. Porsch, M.R.M.H., Rasia, L.A., Thesing, N.J., Pedrali, P.C., Valdiero, A.C.: Low-cost robotic manipulator for family agriculture. J. Agric. Stud. 7(4), 225–239 (2019) 10. Pucheta, M., Ulrich, N., Cardona, A.: Combined graph layout algorithms for automated sketching of kinematic chains (2012). https://doi.org/10.1115/DETC2012-70665 11. Reshetov, L.: Self-aligning Mechanism. MIR, Moscow (1979) 12. Santos, T., Esperidião, T., Amarante, M.: Agriculture 4.0. Res. Action Mag. 5(4), 122–131 (2019) 13. Simoni, R., Martins, D., Carboni, A.: Enumeration of kinematic chains and mechanisms. J. Mech. Eng. Sci. 223(4), 1017–1024 (2009) 14. Tsai, L.W.: Mechanism Design: Enumeration of Kinematic Structures According to Function. CRC Press, Boca Raton (2000)

Historical and Technical Analysis of Harmonic Drive Gear Design Vivens Irakoze1 , Marco Ceccarelli1(B)

, and Matteo Russo2

1 Department of Industrial Engineering, University of Rome Tor Vergata, Rome, Italy

[email protected] 2 Faculty of Engineering, University of Nottingham, Nottingham, UK

[email protected]

Abstract. This paper presents a historical and technical analysis of the Harmonic Drive Gear Design by taking into consideration the original patent by Clarence Walton Musser, the inventor of “Strain Wave Gear” and by studying a performance evaluation of it. The main original design peculiarities are highlighted, and the main characteristics are outlined using a numerical analysis of the operation principle and its efficiency. Keywords: History of gears · Mechanical transmissions · Harmonic drive gear · Modelling · FEA analysis · Student paper

1 Introduction The gear technology addresses interest also in terms of historical investigations to give merits to achievements and to track evolutions towards new solutions, as reported in a wide literature with papers, books and encyclopedia chapters. A specific interest is addressed to Harmonic Drive gear design as an emblematic example of modern achievements that have been requested and stimulated improvements and new solutions in mechatronic machines. In this paper the peculiar history of Harmonic Drive design is presented referring to its inventor Clarence Walton Musser and his original patent designs. In addition, a technical analysis is worked out to explain the original principles of structure and operation of a Hormonic Drive by also giving an example of an evaluation of today efficient solutions.

2 Original Concept by Musser In January 1955 Clarence Walton Musser announced the new design and then in 1957 received a patent for a mechanism called “Strain Wave Gearing” which was later referred to as “Harmonic Drives”, since a use of flexible geared wheel with harmonic response. Official information about the patent was issued in 1959 as it is illustrated in Fig. 1, [1]. At that time even the combination of the words “Strain Wave Gears” was a puzzle © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 46–55, 2022. https://doi.org/10.1007/978-3-030-88751-3_5

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and raised a lot of questioning since mechanical engineers were used only to the law of “rigid-body mechanics” for traditional gears systems. This meant that flexibility was considered driven out from the gearing system and design.

Fig. 1. Main drawings in the patent of 1959 by Clarence Walton Musser [1].

In 1960, almost 5 years after the basic patent was released, Walton Musser published a paper “Breakthrough in Mechanical Design: The Harmonic Drive” [2] where for the first time, he explained the operation principles and mechanism as “A continuous deflection wave generated in a flexible spline element achieves high mechanical leverage between concentric parts”. Clarence Walton Musser was born on 1909 in Lancaster, Pennsylvania (USA). He grew up on a farm where he spent many hours in his father’s workshop which led him to studying topics of his specific interest. At the age of 10, Walton Musser sketched and crafted his first invention that was on a rubber band gun. He studied in several universities including Chicago Technical College, MIT, and University of Pennsylvania. In the early 1930’s he worked in a tool and die shop under a cruel foreman who gave him the worst jobs during a period he later recalled as “some of the best training” of his career. This experience taught Musser to work under pressure and to solve difficult problems on his own as a self-education that paid off when he began work on his many other inventions [5].

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.a)

b)

Fig. 2. (a) A portrait of Clarence Walton Musser; (b) The title page of 1959 patent, [1].

In 1935 he started his own company designing and manufacturing special machinery as a prolific inventor. The greatest of those inventions, which came in 1957 can be considered leverages with elastic deflection. In 1955 Walton Musser invented the strain wage gear and received a patent as shown in Fig. 2(b). As a research adviser at United Shoe Machinery Corp., he explored non-rigid-body mechanics, using controlled deflection as an operating medium for manufacturing.

3 Structure and Operation A Harmonic Drive (HD) is made of three main components shown in Fig. 3. These components include: a Circular Spline, which is a rigid crowned ring; a Flexspline, which is a flexible thin-walled cylinder whose teeth mesh with the circular spline teeth; and the Wave Generator which is a thin raced ball bearing fitted on the elliptical plug generating a high input torque to the whole system. The Harmonic Drive is basically dependent on the elastic behavior that is due to the flexibility of metal. This property is a result of significant flexibility of the flexspline walls at the open end and its teeth positioning radially around its outside. The flexspline fits tightly over the wave generator so that it deforms with the shape of the rotating ellipse but does not rotate with the wave generator. On the other side the circular spline is a rigid circular ring that is expected to be stationary regardless of the flexspline movements. The flexspline and wave generator are placed inside the circular spline, meshing the teeth of the flexspline and those of the circular spline [3]. The joint connecting flexspline and wave generator can be considered as a pseudo-revolute kinematic pair. In fact, the kinematic element of the flexspline is flexible and adapts to the quasi-elliptic shape of the wave generator. A kinematic structure of the single stage harmonic drive gearing can be represented by means of the graph shown in Fig. 4(a), bearing similarities with the one of a basic epicyclic gear (EG) drive in Fig. 4(b). The meshing gears are those cut on the spline and flexspline whereas the wave generator plays a role almost like a gear carrier (coaxial). However, within the basic

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Fig. 3. The basic structure of a Harmonic drive, [7]

Fig. 4. Kinematic design of: a) single stage harmonic drive; (b) a conventional epicyclic gear drive kinematic structure.

EG the axes of the revolute joints are on different levels (i.e., not coaxial). With the simplification discussed above, all the HD revolute joints axes are placed on the same level [6]. It is assumed that the class of combined harmonic-epicyclic gear trains herein considered obey the Grubler’s degrees-of-freedom as m = 3(l − 1) − 2JR − JG

(1)

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where m represents the degree-of-freedom; l, the number of links; JR and JG are the number of revolute and gear pairs, respectively. A simple analysis reveals that, in its simplest form, the HD device has l = 3 links, two coaxial revolute joints (JR = 2) and one gear pair (JG = 1). Therefore, referring to Fig. 4a, the basic model of HD, once the support frame is considered, features a single degree-of-freedom as per m = 1. The dimensions and weight of the Harmonic Drive parts are smaller than those of other transmission devices with similar kinematic and power characteristics. For this reason, the distinctiveness of the Harmonic Drive spans a wide range of application in early and modern devices. Regardly, there has been several developments in the field of industrial engineering that offer improved options in terms of performance and efficiency. The zero-backlash property is a feature that makes it popular in robotic system yet. Among these, there is a new class of joint torque sensors embedded in Harmonic Drive using order tracking method as for example in robot arm of Fig. 5, [8].

Fig. 5. An example of integrated Harmonic Drive and torque sensor in robot joint, [8].

4 Harmonic Drive Model A model of the harmonic drive has been elaborated with Autodesk Fusion360engineering software 2021 [4]. After modelling, all the parts are assembled in Autodesk Inventor as shown in Fig. 6. Generally, all the parts are made of steel as material except for the flexspline which is made of a flexible stainless steel. Color appearances in Fig. 6 were added to easily differentiate the components and the parts from each other.

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a)

51

b)

Fig. 6. The modelled Harmonic gear drive assembly: (a) mechanical design; (b) a cross-section view

5 Finite Element Analysis The here-in presented 3D simulation focuses on the displacements, reaction forces and stress analysis for a case with a given input torque of 100 Nm. Data are reported in terms of maximum and minimum values of these parameters expected to be below the yield strengths of the assumed materials for a safe operation. In particular, the circular spline and wave generator are assumed made of steel with yield strength of 207 MPa and shear modulus of 80.76 GPa, and the Flexpline is assumed made of stainless steel with yield strength of 250 MPa and shear modulus of 74.23 GPa. A simulation was carried out for a typical operation using Autodesk Nastran-2021 engineering simulation software. The circular spline was locked using constraints and the flexible spline was loaded with a torque of 100 Nm, first in the direction of rolling of the gear, then in the opposite direction for the wave generator. The Fig. 7 shows the computed displacements and deformations after the simulation has been completed with numerical results and Fig. 8 illustrates the variation of the displacements recorded on the wave generator’s bearings. The geometry of the FEA mesh is automatically generated in the used simulation package with the peculiarity to design smaller nodal distance as function of the expected deformation instressed areas. Figures 9 and Fig. 10 show results on the applied load and contact forces analysis, respectively. Firstly, the contact forces are considered between the teeth of the flexspline and the circular spline. However, the interference between the gear teeth is very large due to the software’s limitations. Thus, the reaction force values that were extracted from one flexspline tooth (it is expected similar action to the all the teeth) are not accurate and the relative stress error was of 0.1974 MPa, which is indeed a small value.

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a)

b)

Fig. 7. Computed displacement and deformation in a simulated typical operation: (a) Front view; (b) Rear view

Fig. 8. Graph showing recorded displacements of bearing vs nodal distance

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Fig. 9. Trend of computed applied loads on the flexspline’s tooth

In the stress analysis, the maximum stress value by Von Mises, was computed as 167.5 MPa in the region of the bearings and outer flexible bearing ring. This is a well satisfactory result considering the ultimate tensile strength of about 345 MPa for the assumed steel in the simulated mechanical design (Fig. 11).

Fig. 10. Computed reaction forces on the flexspline’s tooth

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Fig. 11. Computed Von Mises stress across the harmonic drive’s main components for a simulated typical operation

6 Conclusion This paper summarizes a historical-technical evaluation of Harmonic Drive gearing design by outlining the sources of the conception by its inventor Clarence W. Musser and the main operation characteristics. It took quite a while to have the Harmonic Drive transmission to be well understood and then it has been and it is yet widely used in many mechatronic systems. A typical operation has been simulated to characterize the performance in terms of teeth stress well below the strength limits, motion high transmission ratios, and reaction forces within the mechanical design.

References 1. Musser, C.W.: US Patent No. 2,906,143 (1959) 2. Musser, C.W.: Breakthrough in mechanical drive design: the harmonic drive. Mach. Des. 14, 160–172 (1960) 3. Timofeyev, G.A., Kostikov, Y.V., Yaminsky, A.V, Podchasov, Y.O.: Theory and practice of harmonic drive mechanisms. Sci. Eng. 468, 012010 (2018) 4. Autodesk Nastran: Autodesk (2021). https://www.autodesk.com/products/nastran/overview. Accessed 16 Mar 2021 5. Musser, C.W. (2020). http://www.waltmusser.org/. Accessed 08 Oct 2020 6. Tao, T., Jiang, G.: A harmonic drive model considering geometry and internal interaction. In: ARCHIVE Proceedings of the Institution of Mechanical Engineers. Part C Journal of Mechanical Engineering Science, pp. 203–210, 16 December 2015

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7. Toulas, B.: How does a Harmonic Drive work? Why are they used?, September 2019. https:// www.engineeringclicks.com/harmonic-drive/. Accessed 28 June 2020 8. Jung, B.-J., Kim, B., Koo, J.C., Choi, H.R., Moon, H.: Joint torque sensor embedded in harmonic drive using order tracking method for robotic application. IEEE/ASME Trans. Mechatron. 22(4), 1594–1599 (2017)

Design of Variable Moment of Inertia Flywheel Vigen Arakelian(B) LS2N/ECN, UMR 6004 and MECAPROCE, INSA de Rennes, 20 av. des Buttes de Coesmes, CS 70839, 35708 Rennes, France [email protected], [email protected]

Abstract. This paper considers the problem of a mechanical system design with a variable moment of inertia. The suggested system consists of a rotating disc with the ability to change the angle of inclination by means of a rod and a slider mounted on the rotation axis. Design equations and techniques are described, making possible the dynamic substitution of the mass of the connecting rod by two concentrated masses. By application of the suggested dynamic model of the coupling rod, a weightless link with two concentrated masses has been obtained. This allows a more precise estimation of the influence of the connecting rod position on the moment of inertia about the rotation axis. A numerical example illustrates the proposed design of variable moment of inertia flywheel. Keywords: Flywheel · Mechanism design · Energy storage · Variable moment of inertia · Dynamic substitution of masses

1 Introduction The energy storage by flywheel consists in storing kinetic energy thanks to the rotation of a heavy object (a wheel or a cylinder), generally moved by a drive mechanism, and then to restore this energy as a generator. Due to its very simple configuration, the flywheel has had many applications in the past and poses new possibilities for innovative use in the future. New approaches to design of flywheels have continued to attract the attention of researchers and different solutions are constantly being reported. One of the most challenging areas in this field is the development of variable-inertia flywheels. One of the earliest variable inertia flywheels was the flyball governor by James Watt. Ullman [1] proposes four original variable-inertia flywheel solutions, which are variations of Watt’s concept. One attempt to apply the use of a variable-inertia flywheel to a moving vehicle, albeit a toy, has been proposed in [2]. Gulia [3] proposed one of the most interesting design concept for a variable-inertia flywheel called band flywheel. This concept was further developed in [4, 5]. Ullman also analyzed the concept of a band variable-inertia flywheel applied it to powering an inertia load [6–8]. In [9], by means of computer simulation, the potential of a Band Variable-Inertia Flywheel as an energy storage device for a diesel engine city bus was also evaluated. It has been shown that the regenerative braking system reduces brake wear by a factor of five in comparison with the conventional vehicle. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 56–63, 2022. https://doi.org/10.1007/978-3-030-88751-3_6

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A flywheel of regulated moment of inertia comprising a heavy rim, which is a link for half-clutches, set on the shafts of electric motor and initial mechanism segment was also developed [10]. In the proposed system, the heavy sliders can travel in radial grooves of the rim and form rectilinear sliding pairs with the latter. Spokes form rotating kinematic pairs with half clutches and sliders. Fluidic variable inertia flywheel was also developed [11–14], which can acquire variable inertia through the valve’s control over the fluid flow. A kind of variable inertia flywheels was proposed by Elliott, Mintah and Lapen [15], the centrifugal force can change the displacement of the mass block and the control system can lock the position of mass block with hydraulic fluid and control valve. The variable inertia flywheel starts with small inertia and rotate at high speed with big inertia improving the stability of machinery. The inertia of flywheel in the invention of Jayakar and Das [16] can be changed by filling and draining. Based on the flywheel concept proposed by Jauch [17] applied to wind turbine and the flywheel design developed by Hamzaoui et al. [18] and Yang et al. [19] applied to hydraulic system, in [20], a variable inertia flywheel for diesel generator, was considered. This paper proposes a new solution, which consists in adjusting the moment of inertia with respect to its axis of rotation by setting the inclination angle of the flywheel disc.

2 Design of the Variable Inertia Flywheel When solving various technical problems, it becomes necessary to use flywheels with a variable moment of inertia. Thereby, it is often required to adjust the moment of inertia based on dynamic behavior of the mechanical system. Consider an adjustable flywheel based on the variation of the inclination of the disc with respect to its axis of rotation (Fig. 1). It consists of a disc 1 connected to its rotating axis 2 by means of revolute joint 3. The slider 4 and the rod 5 are connected by spherical joints. The special form of the connecting rod 5 is discussed below. Thus, the slider 4 and the connecting rod 5 carry out the variation of the inclination angle α of the disc 1. Let us see how this system works. Suppose that it is necessary to ensure the given moment of inertia Iz1 . The relation between Iz1 and Ix , Iz , i.e. the moment of inertia about the rotation 1 1 axis z of the disc and its axes of symmetry is as follows [21]: Iz1 = Ix sin2 α + Iz cos2 α

(1)

  Ix =Iy = m1 R2D /4 + h2D /12

(2)

Iz = 0.5m1 R2D

(3)

1

with

1

1

1

1

where, m1 is the mass of the disc 1; RD and hD are the radius and the width of the disc 1 (Fig. 2). Please note that Ix y = Ix z = Iy z = 0, since the axes of the coordinate 





1 1

1 1

1 1

system x1 y1 z1 directed along the axes of symmetry of the disc 1.

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Fig. 1. Variable inertia flywheel.

From (1), we determine the inclination angle α of the disc 1, which ensure the given inertia moment Iz1 :    1/2 (4) α = cos−1 Iz1 − Ix1 / Iz − Ix1 1

Thus, knowing the angles α and β (see Fig. 1), we determine the displacement d to adjust the position of the disc 1: 1/2  1/2  −2 − lOC cosϕ − lCD 1 − (lOC sinϕ − e)2 lCD (5) d = (lOC + lCD )2 − e2 where, lOC is the distance between the centers of joints O and C; lCD is the length of the connecting rod 5; e is the eccentricity of slider guide 5 (see Fig. 1). Thus, by controlling the inclination angle α of the disc 1 through the slider’s displacements, a given moment of inertia Iz1 can be provided.

Fig. 2. Disc’s parameters about its axes of symmetry.

However, the position of the connecting rod 5 will affect the moment of inertia of the system about axis z should be taken into account. To simplify this task, consider a special

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shape of the connecting rod 5 making possible the dynamic substitution of its mass by two concentrated masses located in the joint centers C and D. Standard connecting rod represents two spheres connected with a cylinder. Such a shape of the coupler is not optimal for the realization of the dynamic modeling by masses substitution [22]. Figure 3 shows a special coupling rod shape composed by two elements: the segments of a sphere and a cylinder.

a)

b)

Fig. 3. A special coupler shape: (a) CAD model in 3D; (b) drawing in 2D.

Fig. 4. Segment of a sphere.

Fig. 5. Circular cylinder.

Segments of sphere (Fig. 4). – Masse: 

m5 = π h2 (3R − h)ρ/3

(6)

where, ρ is the coefficient of the material density, which is same for all elements of the coupler; h and R are geometric parameters shown in Fig. 4. 

– The location of the center of masses S of the segment of a sphere can by found by the expression: 

xS = 0.25h(8R − 3h)/(3R − H )

(7)

   Ix x = 0.1m5 h 5R(4R − 3h) + 3h2 /(3R − h)

(8)

– Mass moments of inertia:

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     Iy y = Iz z = 0.5Ix x + m5 h2 8R(5R − 3h) + 3h2 / 80(3R − h)2

(9)

Circular cylinder (Fig. 5). – Mass: 

m5 = π r 2 Lρ

(10)

where, r is the radius of the circular cylinder and L its length (see Fig. 5). 

– The location of the center of masses S of the segment of a sphere can by found by the expression: 

xS = 0.5L

(11)

– Mass moments of inertia: 

Ix x = 0.5m5 r 2

(12)

     Iy y = Iz z = 0.25m5 r 2 + m5 L2 /12 = m5 3r 2 + L2 /12

(13)

Taking into account that the suggested coupler shape is symmetric and the rotation around the longitudinal axis is locked by a pin, the condition of the dynamic substitution of the coupler mass by the concentrated masses m5C and m5D located at the centers C and D of the spherical joints can be reduced to the expression:  2   2Iz z + 2m5 R − xS + 0.5l + Iz z = (m5B + m5C )(0.5l)2 

(14)



where, m5C = m5D = m5 + 0.5m5 . Introducing the relationsips (6)–(13) into Eqs. (14), we obtain the folloxing equation:    2π h2 h 5R(4R − 3h) + 3h2 h2 8R(5R − 3h) + 3h2 · + · (3R − h) 3 20 3R − h 80 (3R − h)2       2  + 2π h2 (3R − h)/3 × R − xS + 0.5l + π r 2 L 3r 2 + L2 /12     = (0.5l)2 2 π h2 (3R − h)/3 + π r 2 L (15) From which we determine the radius of the connecting cylinder:  −2 −b ± b2 − 4ac r1,2 = 0.25a

(16)

where, a = 3π L/12

(17)

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b = π L3 /12 − π Ll 2 /4

(18)

  

 c = 2π h2 (3R − h)/3 0.05h 5R(4R − 3h) + 3h2 /(3R − h) +        h2 /80 8R(5R − 3h) + 3h2 / (3R − h)2 + 2π h2 (3R − h)/3 ×  2    R − xS + 0.5l − 0.5l 2 π h2 (3R − h)/3

(19)

Thus, after applying such a special rod shape, we obtain a system of two point masses m5C and m5D dynamically equivalent to the coupling rod 5. This makes it easy to express the moment of inertia of all elements of the system in relation to the z-axis: 2 sin2 ϕ + m5D e2 Iz = Iz1 + Iz2 + Iz4 + m5C lOC

(20)

Let us consider a numerical example.

3 Illustrative Example Let us consider the design of the suggested system ensuring Iz = 4.5 kgm with following parameters of links: RD = 0.3 m; hD = 0.05 m; m1 = 110 kg; β = 20◦ ; lOC = 0.18 m; lCD = 0.3 m; e = 0.08 m; Iz2 = 0.1 kgm2 ; Iz4 = 0.08 kgm2 . By selecting the geometrical parameters R = 0.04 m and h = 0.06 m, we find from (16) the radius r = 0.0083 m of the connecting cylinder of the rod 5. Then, we determine the concentrated masses: m5C = m5D = 2 kg.

Taking into account that Ix =Iy = m1 R2D /4 + h2D /12 = 2.27 kgm and Iz = 1

1

1

0.5 m1 R2D = 4.95 kgm2 , from (20), we determine the inclination angle = 30◦ of the disc 1, which ensure the given value of the moment inertia Iz . Thus, by setting a new value of the moment of inertia Iz , the new angle of inclination α will be determined that will adjust the moment of inertia. It is obvious that the change in the moment of inertia Iz will be limited by the capabilities of the system.

4 Conclusions During the last century, flywheel mechanical energy storage has advanced from simple inertial rotating machines operating at low speeds to fully integrated electro-mechanical systems. These systems are now becoming enabling technologies for many applications. One of the ways to develop the new concepts is the creation of flywheels with a variable moment of inertia. Adjustment of the flywheel moment of inertia depending on the operating mode of the machine expands their technological capabilities. The paper proposes a new solution, which consists in adjusting the flywheel moment of inertia by setting the inclination angle of the disc. A heavy disc that is the flywheel mounted on a rotating axis through a revolute joint. This allows one to set the required angle of inclination of the cylinder in relation to its axis of rotation. Thus, the disc

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inclination angle determine the moment of inertia of the flywheel. The paper describes in detail how to determine this inclination angle of the disc to establish the required inertia of flywheel. A numerical example illustrates the variable moment of inertia flywheel. It should be noted that in the paper a cylindrical disc is considered. However, it is obvious that it is possible to apply a similar approach to other shapes, such as wheels, coiled band mechanisms, systems with moving sliders, etc. The paper also discusses a connecting rod with spatial shape that allows the dynamic substitution of the rod’s mass by two concentrated masses located on its joint axes. This allows a more precise estimation of the influence of the connecting rod position on the moment of inertia about the rotation axis. However, observation shows that the influence of additional links of the suggested flywheel system, such as connecting rods and slider, on the change in the moment of inertia is not significant. Therefore, the use of a connecting rod with a special shape can be considered as an additional design step.

References 1. Ullman, D.G.: A Variable Inertia Flywheel as an Energy Storage System. Ph.D. Dissertation, Ohio State University (March 1978) 2. Lin, S.T.: Variable Inertia Flywheel. US Patent 3,968,593, 13 July 1976 3. Gulia, N.V.V: Variable Moment-of-Inertia Flywheel. SU 1182724/24-27 (July 1969) 4. Dvali, R.R., Gulia, N.V.: A coiled band mechanism for the recovery of a vehicle’s mechanical energy. J. Mech. 3(3), 113–118 (1969) 5. Gulia, N.V.: Centrifugal Accumulator, SU 1131894/25-28 (November 1969) 6. Ullman, D.G., Velkoff, H.R.: An introduction to the variable inertia flywheel, VIF. J. Appl. Mech. 46(1), 186–190 (1979). https://doi.org/10.1115/1.3424494 7. Ullman, D.G.: The Band Type Variable Inertia Flywheel and Fixed Ratio Power Recirculation Applied to it. Presented at the Mechanical and Magnetic Energy Storage Technology Meeting (October 1978) 8. Ullman, D.G., Corey, J.: Continued Development of the Band Variable Inertia Flywheel. Report of Union College, Schenectady, New York (1980) 9. Moosavi-Rad, H., Ullman, D.G.: A band variable-inertia flywheel integrated-urban transit bus performance. SAE Trans. J. Commer. Veh. 99(Section 2), 933–942 (1990) 10. Eliseev, S.V., et al.: Flywheel of regulated moment of inertia. RU Patent 2498127C1, 10 November 2013 11. Van de Ven, J.: Fluidic variable inertia flywheel. In: 7th International Energy Conversion Engineering Conference, 2–5 August 2009, Denver, Colorado (2009) 12. Dugas, P.J.: Variable Inertia Flywheel. US 2011/0277587, 17 November 2011 13. Van de Ven, J.: Fluidic Variable Inertia Flywheel and Flywheel Accumulator System. US Patent 8,590,420, 26 November 2013 14. Dugas, P. J.: Variable inertia flywheel. US 20150204418, 23 July 2015 15. Elliott, Ch., Mintah, B., Lapen D.: Variable Inertia Flywheel. US20090320640, 31 December 2009 16. Jayakar, V., Das, S.K., Variable Inertia Flywheel. US 20120291589, 22 November 2012 17. Jauch, C.: A flywheel in a wind turbine rotor for inertia control. Wind Energy 18(9), 1645– 1656 (2015) 18. Hamzaoui, I., Bouchafaa, F., Talha, A.: Advanced control for wind energy conversion systems with flywheel storage dedicated to improving the quality of energy. Int. J. Hydrog. Energy 41(45), 20832–20846 (2016)

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19. Yang, S., Xu, T., Li, C., Liang, M., Baddour, N.: Design, modeling and testing of a twoterminal mass device with a variable inertia flywheel. J. Mech. Des. 138(9), 095001 (2016). https://doi.org/10.1115/1.4034174 20. Zhang, Y., et al.: Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Rep. 6, 58–68 (2020) 21. Uicker J.J., Jr., Pennock, G.R. and Shigley J.E.: Theory of Machines and Mechanisms, p. 752. OUP, USA (2003) 22. Arakelian, V., Briot, S.: Balancing of Linkages and Robot Manipulators. Advanced Methods with Illustrative Examples. Springer, Heidelberg (2015)

Kinematics of a Robotic System for Rehabilitation of Lower Members in Hypotonic Infants Marco García1(B) , Esther Lugo-González2 , Manuel Arias-Montiel2 , and Ricardo Tapia-Herrera3 1

Universidad Tecnológica de la Mixteca, Carretera a Acatlima Km. 2.5, Acatlima, 69000 Huajuapan de León, Oax, México 2 Institute of Electronics and Mechatronics, Universidad Tecnológica de la Mixteca, Carretera a Acatlima Km. 2.5, Acatlima, 69000 Huajuapan de León, Oax, México {elugog,mam}@mixteco.utm.mx 3 CONACYT-Universidad Tecnológica de la Mixteca, Carretera a Acatlima Km. 2.5, Acatlima, 69000 Huajuapan de León, Oax, México [email protected]

Abstract. This paper develops the kinematics of a parallel mechanism with topology 5R. This mechanism focuses on the lower limb rehabilitation of hypotonic infants. At first, the inverse kinematics is solved to generate circular and ellipsoidal trajectories with the objective of demonstrating that specific trajectories, such as flexion/extension, can be generated during the rehabilitation for an infant with hypotonia. The performed analysis to obtain the dimensions of the links is shown, as well as the equations of the proposed mechanism motion. The velocity analysis is also obtained, which will be later used to design and implement the control system necessary to generate the movement in the rehabilitation therapy. As a numerical result, the inverse kinematics validation is presented by performing simulations in ADAMS View software.

Keywords: Inverse kinematics Hypotonia

1

· Parallel robot · Trajectory ·

Introduction

Hypotonia is a condition characterized by a deficiency of muscle tone which mainly occurs in infants. Palliative treatment is a passive therapy, and to the best of the authors’ knowledge, there are no robotic devices for infants with this pathology as the dimension, weight, materials, or mobility values must be very specific [1]. The basic passive therapy applied to the lower limbs of hypotonic infants focuses on the hip, knee and ankle, movements generated on the sagittal plane. For this condition, parallel robots have been used to generate movements and specific trajectories in this type of rehabilitation. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 64–73, 2022. https://doi.org/10.1007/978-3-030-88751-3_7

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The 5R parallel manipulator is a plane mechanism with two degrees of freedom that has been widely studied using analytical and numerical methods, as reported in [8,11]. An application of a 5R parallel manipulator is found in [4], where a lower limb rehabilitation device was developed with the proposed mechanism providing exercises for knee rehabilitation. For lower limb mobilization, there are devices such as the plane Lambda [3] mechanism which is composed of 3 links and 3 active joints: 2 translational and one rotational. Furthermore RECOVER [6] is a parallel robotic system designed for lower limb rehabilitation. This kinematic model was formulated to achieve a direct correlation between the robot’s active joints and anatomical joint angles. It is composed of two modules: the hip-knee module that has 2 DOF and facilitates flexion/extension movements of the hip and knee, while the ankle module with 2 DOF performs flexion/extension and inversion/eversion of the ankle. In another study, Calin Vaida et al. in [12] proposed a parallel robot that has a modular construction allowing hip, knee and ankle mobility. The kinematics of the robotic system was presented and singularities were found. A detailed kinematic analysis was performed to demonstrate that the robot performs the required movements safely through numerical simulations. Moreover, the HyPO [9] is a robot designed with parallel links instead of gear systems or timing belt systems to transmit the torques generated by the actuators to the knee and hip joints. The links are four-bar mechanisms in which one link is connected to the actuator to transfer the generated torque. All the aforementioned parallel robots and mechanisms are designed for adults and are generally used for rehabilitation after a stroke or surgery. For children (between 3 and 12 years old), exoskeleton robots have been mainly developed to support walking. This is achieved by generating movement in the sagittal and/or frontal plane, and some of these designs are those presented by [2,5,7,10]. These works develop hip, knee and ankle movements, mainly generating flexion/extension in each limb, but most are used with the infant standing or using the upper limb. The previously mentioned designs have no dimension and angle of movement required for hypotonic infant therapy, but give the basis for designing a robotic system that is 500 mm in length, supports a weight of approximately 2.5 kg and automatically develops hip, knee and ankle movements in the supine position.

2

Robot Proposal

For generating flexion/extension movements, a mechanism that considers the dimensions of infants (500 mm in length as the maximum value) is required. R ANTROPROJETO software is used to achieve this, which receives as input the length of the infant and as output returns the anthropometric values. To generate the necessary trajectories, parallel robots work appropriately and are positioned horizontally parallel to the infant’s lower limbs in the sagittal plane. The mobile platform of the parallel robot can be located at the infant’s ankle. In this case, a five-bar mechanism is the basis of the robot. Kinematic

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analysis and simulations that generate flexion/extension movements of the hip, knee and ankle in this robot to achieve one of the infant’s rehabilitation phases are presented. The parallel robot design consists of nine links and ten articulated joints, as shown in Fig. 1. With these elements, three basic mechanisms are formed, two with four bars and one with five. The first four-bar mechanism is created with a configuration to give mobility to the femur, while two links are added to configure a five-bar mechanism thereby giving mobility to the tibia and fibula joints. Finally, there is a four-bar mechanism providing movement to the ankle.

Fig. 1. a) Mechanism for rehabilitation of the lower limbs.

The five-bar mechanism has two degrees of freedom which generates two independent movements for carrying out a previously established trajectory (flexion/extension movement). The infant’s characteristics (500 mm of length), provides the dimensions of the OB and BE links, meaning that the links for BF, FG, GH, and HO can be calculated. For restriction and calculation purposes, the dimensions of the OB and BE links are fixed, but can vary to adapt to an infant’s development. To calculate the five-bar mechanism trajectory synthesis, consider Fig. 2 and the Eqs. (1) and (2): L2 + L3 − L1 − L5 − L4 = 0

(1)

Writing in complex numbers L2 ejθ2 + L3 ejθ3 − L1 ejθ1 − L5 ejθ5 − L4 ejθ4 = 0

(2)

Rewriting and using Euler’s formula for complex numbers L2 cosθ2 + L3 cosθ3 − L1 − L5 cosθ5 − L4 cosθ4 = 0

(3)

L2 sinθ2 + L3 sinθ3 − L5 sinθ5 − L4 sinθ4 = 0

(4)

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Fig. 2. The diagram for obtaining the five-bar mechanism synthesis.

With this proposal, the hip, knee and ankle joints will have movement but they are restricted to a single plane and with only two degrees of freedom. In Table 1 the lengths of links are observed, and the actuators that give the system its mobility are denoted by θ2 and θ5 located in joints O and H in Fig. 2. Table 1. Length of links Link Length (mm)

2.1

L1

60

L2

122

L3

40

L4

122

L5

40

Kinematic of Five-Bar Mechanism

The parallel manipulator 5R, shown in Fig. 2, moves the point F = Pxy in the plane with a fixed link OH, while the actuated joints are the points O, H and are measured as θ2 and θ5 , with regard to the x-axis. For inverse kinematics, the coordinates or trajectory of point Pxy are known and the input angles of the manipulator must be calculated. The output point Pxy in reference frame Oxy can be described by position vector P , and P = (x y)T

(5)

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Considering Oxy , position vectors of bi of points Bi are written as: T  b1 = L1 cosθ2 − L3 L1 sinθ2 T  b2 = L1 cosθ5 − L3 L1 sinθ5

(6) (7)

Therefore, the inverse kinematics problem can be solved by writing the constraint equations |pb1 | = r2 and |pb2 | = r5 , in this way (x − L1 cosθ2 + L3 )2 + (y − L1 sinθ2 )2 = L22

(8)

(x − L4 cosθ5 + L6 )2 + (y − L4 sinθ5 )2 = L25

(9)

By knowing the position of output point Pxy , inputs to reach this point can be obtained as (10) θ2 = 2 arctan(z1 ) θ5 = 2 arctan(z2 ) where



b21 − 4a1 c1 2a1  −b2 ± b22 − 4a2 c2 z2 = 2a2

z1 =

−b1 ±

(11)

(12) (13)

after developing Eqs. (12) and (13), terms are: a1 = L21 + y 2 + (x + L3 )2 − L22 + 2(x + L3 )L1 a2 = L24 + y 2 + (x − L6 )2 − L25 + 2(x − L6 )L4 b1 = −4yL1 b2 = −4yL4 c1 = L21 + y 2 + (x + L3 )2 − L22 − 2(x + L3 )L1 c2 = L24 + y 2 + (x − L6 )2 − L25 − 2(x − L6 )L4 Equations (12) and (13) have four possible solutions for inverse kinematics. The Fig. 2 configuration can obtain the sign (±), while Eq. (12) is positive (+) and Eq. (13) sign is positive (+). The resulting configuration is known as (+−) model. The graphs obtained with MATLAB (Fig. 3) to show inverse kinematics implementation in 5-bars mechanisms are two trajectories, a circumference, given by Eq. (14) and an ellipse given by Eq. (15). P x = 40 + 25cos(φ), P y = 120 + 25sin(φ)

(14)

P x = 40 + 50cos(φ), P y = 120 + 25sin(φ)

(15)

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Fig. 3. Circular and elliptical path.

As can be seen, the mechanism follows specific trajectories, validating the links’ dimensions. The next point is to verify that specific angles for a rehabilitation sequence are generated.

3

Results and Discussion

The kinematics validation is carried out by taking the results of Matlab similarity. With the point F position at each instant of time and the use of polyf it tool, two polynomials are constructed which describe the behavior of angles θ2 and θ5 as shown in Eqs. (16) and (17). These polynomials are the input of the angles θ2 and θ5 of actuators in ADAMS View. θ2 = −0.0003t2 − 0.148t + 78.3768 2

θ5 = −0.0005t + 0.4712t − 0.2529

(16) (17)

The polynomials constructed in Matlab are introduced in the inputs θ2 and θ5 of the model in ADAMS View to generate the movement in the mechanism. The angular displacement is also measured and compared with the results obtained from the mathematical model. Figure 4 shows the graphs of the mathematical model and the Matlab polynomials when comparing with the results of the model in ADAMS View, thereby verifying the mechanism movement.

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Fig. 4. Values of θ2 and θ5 measured in a) the ADAMS View and b) MATLAB simulation.

The results obtained from trajectory of point F = Pxy in Fig. 2 correspond to the proposed trajectory, and similar results were observed in Fig. 5 obtained with MATLAB and ADAMS View software. To perform the velocity analysis, Eqs. (8) and (9) were derived with regard to time and simplified as L1 (ycosθ2 − (x + L3 )sinθ2 )θ˙2 = (x + L3 − L1 cosθ2 )x˙ + (y − L1 sinθ2 )y˙

(18)

L4 (ycosθ5 + (L6 − x)sinθ5 )θ˙5 = (x − L6 − r4 cosθ5 )x˙ + (y − L4 sinθ5 )y˙

(19)

Writing Eqs. (18) and (19) in matrix form. Aθ˙ = B p˙ ⇔ Jq q˙ = Jx x˙

(20)

θ˙ = [θ˙2 θ˙5 ]T ⇔ q˙

(21)

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Fig. 5. Result of the simulation in MATLAB and ADAMS View for Px and Py .

p˙ = [x˙ y] ˙ T ⇔ x˙   [ycosθ2 − (x + L3 )sinθ2 ]L1 0 A= 0 [ycosθ5 + (L6 − x)sinθ5 ]L4   x + L3 − L1 cosθ2 y − L1 sinθ2 B= x − L6 − L4 cosθ5 y − L6 sinθ5

(22) (23) (24)

For the velocity simulation, a circumference trajectory was used, and the test was performed with 10 mm/s with the results shown in Fig. 6. As can be seen, the velocity ω2 of actuator one starts at 0 deg/s and increases as time progresses until it reaches 0.9 deg/s after 180 s. For the case of actuator two, it has an initial velocity of 0.34 deg/s denoted by ω5 and decreases until it reaches 0.14 deg/s over a period of 180 deg/s. With these angular velocities of the actuators, the 10 mm/s at the point F of the 5R mechanism is achieved. As a final result, the simulated mechanism in Adams View (Fig. 7) is shown, and the flexion position of the lower limbs in the sagittal plane can be observed. The symmetry of the infant’s lower limbs compared to the mechanism’s position is also observed.

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Angle (º)

0.4 0.3 0.2 0.1 0

0

50

100

150

Fig. 6. Angular velocity values, to have a velocity of 10 mm/s.

Fig. 7. Similarity between the posture of an infant and the posture of the whole mechanism.

4

Conclusions

A mechanism was presented to be used in the design for children with hypotonia rehabilitation prototype and it was observed that it is necessary for the mechanism to follow the pre-established trajectory in rehabilitation sessions. In this work, the following of a circular trajectory and an ellipse was presented to show that the designed mechanism can follow a specific movement such as flexion/extension generated by the hip, knee and ankle in the sagittal plane. The rehabilitation device is made up of a 5-link parallel robot divided into three parts: two with four bars and one with five, with this last bar the main focus of this study. For the dimensional synthesis of the 5-bar mechanism, only the BF, FG, GH and H0 links were calculated because the rest were used from the length of the infant’s lower limb. For inverse kinematics calculations, the constraint equations were considered for obtaining the behavior in position and velocity. The graphical results were validated with the mathematical model in Matlab which has a similar behavior to the model in ADAMS View compared to velocity and position, which provides the basis for the design and construction of the rehabilitation prototype.

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References 1. Alfonso, I., Papazian, O., Valencia, P.: Hipotonía neonatal generalizada. Rev. Neurol. 37(3), 228–239 (2003) 2. Andrade, R.M., Sapienza, S., Bonato, P.: Development of a “transparent operation mode” for a lower-limb exoskeleton designed for children with cerebral palsy. In: 2019 IEEE 16th International Conference on Rehabilitation Robotics (ICORR), pp. 512–517. IEEE (2019) 3. Bouri, M., Le Gall, B., Clavel, R.: A new concept of parallel robot for rehabilitation and fitness: the lambda. In: 2009 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 2503–2508 (2009). https://doi.org/10.1109/ROBIO. 2009.5420481 4. Chaparro-Rico, B.D., Castillo-Castaneda, E., Maldonado-Echegoyen, R.: Design of a parallel mechanism for knee rehabilitation. In: Ceccarelli, M., Hernández Martinez, E.E. (eds.) Multibody Mechatronic Systems. MMS, vol. 25, pp. 501–510. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09858-6_47 5. Eguren, D., Cestari, M., Luu, T.P., Kilicarslan, A., Steele, A., Contreras-Vidal, J.L.: Design of a customizable, modular pediatric exoskeleton for rehabilitation and mobility. In: 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), pp. 2411–2416. IEEE (2019) 6. Gherman, B., Birlescu, I., Plitea, N., Carbone, G., Tarnita, D., Pisla, D.: On the singularity-free workspace of a parallel robot for lower-limb rehabilitation. Proc. Rom. Acad. 20(4), 383–391 (2019) 7. Giergiel, M., Budziński, A., Piątek, G., Wacławski, M.: Personal lower limb rehabilitation robot for children. In: Awrejcewicz, J., Szewczyk, R., Trojnacki, M., Kaliczyńska, M. (eds.) Mechatronics - Ideas for Industrial Application. AISC, vol. 317, pp. 169–176. Springer, Cham (2015). https://doi.org/10.1007/978-3-31910990-9_17 8. Liu, X.J., Wang, J., Zheng, H.J.: Optimum design of the 5R symmetrical parallel manipulator with a surrounded and good-condition workspace. Robot. Auton. Syst. 54(3), 221–233 (2006) 9. Obinata, G., et al.: Hybrid control of powered orthosis and functional neuromuscular stimulation for restoring gait. In: 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 4879–4882. IEEE (2007) 10. Sancho-Perez, J., Perez, M., Garcia, E., Sanz-Merodio, D., Plaza, A., Cestari, M.: Mechanical description of atlas 2020, a 10-DOF paediatric exoskeleton. In: Advances in Cooperative Robotics, pp. 814–822. World Scientific (2017) 11. Sosa-López, E.D., Arias-Montiel, M., Lugo-Gónzalez, E.: A numerical approach for the inverse and forward kinematic analysis of 5R parallel manipulator. In: 2017 14th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), pp. 1–6. IEEE (2017) 12. Vaida, C., et al.: Systematic design of a parallel robotic system for lower limb rehabilitation. IEEE Access 8, 34522–34537 (2020)

Modelling and Simulation of Multibody Systems

Modeling and Simulation of Frictional Contacts in Multi-rigid-Body Systems Paulo Flores(B) CMEMS-UMinho, Department of Mechanical Engineering, University of Minho, Campus de Azurém, 4804-533 Guimarães, Portugal [email protected]

Abstract. Fictional contacts occur in many mechanical systems, and often affect their dynamic response, since the collisions cause a significant change the systems’ characteristics, namely in terms of velocities. This work describes and compared different formulations to handle frictional contacts in multi-rigid-body dynamics. For that, regularized and non-smooth techniques are revisited. In a simple manner, the regularized methods describe the contact forces as a continuous function of the indentation, while the non-smooth formulations use unilateral constraints to model the contact problems, which prevent the indentation from occurring. The main motivation for the performing this study came from the permanent interest in developing computational models for the dynamic modeling of contact-impact events under the framework of multibody systems methodologies. The problem of modeling and simulating contacts with friction in multibody systems includes several steps, the definition of the contact geometry; the determination of the contact points; the resolution of the contact itself; and the evaluation of the transitions between different contact regimens. The last two aspects are investigated in this work within the context of contact dynamics. In the sequel of this process, an application example is utilized to show the effectiveness of the modelling process of contact problems in multibody systems. Finally, future developments and new perspectives for further developments related to contact-impact problems are presented and discussed in this study. Keywords: Frictional contacts · Contact dynamics · Contact detection · Contact resolution · Regularized methods · Non-smooth techniques · Linear complementarity problem · Multibody dynamics

1 Introduction By and large, frictional contacts involves the problem of the modeling the interaction of colliding bodies in the presence of frictional phenomena. This discipline is often named as contact dynamics and deals with the motion of multibody systems subjected to contact-impact forces/impulses. Contact dynamics is omnipresent in many multibody applications, and in most of the cases the function of the systems depends on contact modeling process. Over the last decades, contact dynamics has been one of the most challenging and demanding areas of research in engineering that play a crucial role in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 77–84, 2022. https://doi.org/10.1007/978-3-030-88751-3_8

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vehicle systems, robotics, railway models, mechanisms and machinery, biomechanics, granular systems, toys, just to mention some examples under the framework of multibody systems [1–4]. Contact-impact events are complex phenomena characterized by short duration and high forces that cause rapid changes or discontinuities in systems’ velocities and eventually energy dissipation [5]. The key ingredients of the modeling process of a frictional contact problem include the several issues that can be condensed in two independent steps, chiefly the determination of the contact points and the evaluation of the contact forces/impulses [6]. The determination of the contact points focuses on the resolution of two main issues: the geometric description of the contacting surfaces, and the detection of the potential contact points. In turn, the resolution task can be performed used two main approaches in dynamical systems: the regularized approaches [7], and the non-smooth models [8]. This work aims at analyzing the main aspects related to the modeling problem of frictional contacts in multibody dynamics. The emphasis of the study in on the regularized approaches and non-smooth formulations, where the fundamental issues associated with each technique are highlighted when treating collisions. Discussion of the extensive literature on computational schemes for contact detection is beyond the scope of this study. Anyway, the evaluation of the geometry of contact and search for contact (contact detection) is the same regardless of the choice of the technique utilized to handle the contact interaction between the colliding bodies (contact resolution), being based on regularized or non-smooth methods.

2 Techniques to Model Frictional Contacts There are two main techniques to solve problems, namely the regularized approaches (continuous methods) and the non-smooth formulations (piecewise methods). In the former techniques, also known as compliance or elastic methods, the contacting bodies are considered to be deformable at the contact zone, and the contact forces can be expressed as a continuous function of the local deformation between the contacting surfaces. In the non-smooth formulations, also called instantaneous or rigid methods, the contacting bodies are assumed to be truly rigid, and the contact dynamics is resolved by applying unilateral constraints in order to avoid the penetration from occurring. The regularized approaches are quite important in the context of multibody dynamics because of their good efficiency and extreme simplicity to be implemented. In some circumstances, numerical problems can arise, resulting from bad conditioned system matrices [9]. The transition between contact and non-contact situations can easily be handled from the system configuration and contact kinematics. With the regularized methods, the contact forces include spring-damper elements to prevent interpenetration from occurring. In the regularized approaches, the location of the contact point does not coincide in the contacting bodies, and a large number of potential contact points exists, being the actual contact point the one associated with the maximum indentation. The pseudo-penetration plays a key role as it is utilized to calculate the contact forces according to an appropriate constitutive law [10]. The existence of friction in the continuous methods can easily be incorporated by considering any regularized friction model [11].

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Assuming that the contacting bodies are absolutely rigid, as opposed to locally deformable bodies as in the regularized approaches, the non-smooth formulations resolve the contact problems using unilateral constraints to determine impulses to avoid penetration from occurring. The central idea of the non-smooth formulations is the nonpenetration condition that prevents bodies from moving toward each other and not apart [12]. A complementarity formulation is used to describe the relation between the contact force and gap distance at the contact point. Such unilateral constraint does not permit the interpenetration of the two colliding bodies, and ensures that either contact force or gap distance is null. When the gap distance is positive (inactive contact), the corresponding contact force is null. Conversely, when the contact force is positive (active contact), the gap distance is null [13]. This formulation leads to a complementarity problem, which constitutes the rule that permits to treat multibody systems with unilateral constraints [14]. Figure 1 shows the graphical representation of the normal and tangential contact forces for the regularized approaches and non-smooth formulations. The regularized approaches and the non-smooth methods, utilized to handle contact-impact events under the framework of multibody dynamics, have inevitably advantages and disadvantages. None of these techniques can be identified as superior. In fact, a particular multibody system with collisions might be easily described by one method, nevertheless, this does not automatically implies a general predominance of that formulation in all multibody applications.

fn

fn

ft

ft

μ fn

μ fn

vt

δ (a)

δ

- μ fn

(b)

vt - μ fn

(c)

(d)

Fig. 1. (a) Regularized normal contact force model; (b) Non-smooth normal contact force model; (c) Regularized tangential contact force model; (d) Non-smooth tangential contact force model

3 Regularized Methods for Contact Dynamics The oldest contact force model is the one associated with Hooke’s theory, which can be used when a contact is active. This regularized force model considers a linear spring to mimic the contact interaction, and can be expressed as [10] fn = kδ

(1)

where k represents the spring stiffness related to the contact materials, and δ is the penetration between the contacting surfaces.

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A more advanced contact force model was developed by Hertz, which considers a nonlinear relation between force and penetration as [10] fn = Kδ n

(2)

where the nonlinear exponent, n, is typically equal to 3/2. The contact stiffness, K, can be determined analytically as function of the material properties and geometry of the contacting surfaces. Hunt and Crossley presented a contact force model that associates as nonlinear spring with a nonlinear damper in parallel to mimic the contact interaction. This force model can be expressed as [15]   3(1 − cr ) δ˙ (3) fn = Kδ n 1 + 2 δ˙(−) where the first term is the nonlinear elastic Hertz’s law, and the second term is the dissipative parcel, being cr the coefficient of restitution, δ˙ represents the contact velocity, and δ˙(−) is the contact normal velocity at the initial instant of impact. The most popular contact force model in the multibody dynamics community is the one proposed by Lankarani and Nikravesh [5], which was developed with basis on the hertzian contact theory and on the damping approach by Hunt and Crossley, and can be is written as   3(1 − cr2 ) δ˙ n (4) fn = Kδ 1 + 4 δ˙(−) in which is valid for collisions with high values of the coefficient of restitution, that is, this model is applicable to elastic impacts. More recently, Flores et al. [16] described a contact force model applicable to the entire domain of the possible values for the coefficient of restitution, which is given by   8(1 − cr ) δ˙ (5) fn = Kδ n 1 + 5cr δ˙(−) Over the last decades, a good number of contact force models have been presented in the literature, being the interested reader in specific information referred to the following references [10]. The most well-known friction force model is undoubtedly the one represented by Coulomb’s law, which can be expressed as [17]  if vt = 0 ≤ μs fn (6) ft = μd fn sgn(vt ) if vt = 0  0 if vt  = 0 sgn(vt ) = (7) vt vt  = 0 if vt  in which μs and μd represent the static and dynamic coefficient of friction, respectively, fn denotes the normal contact force, and vt is the relative contact tangential velocity of the contacting elements.

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Threlfall [18] proposed a regularized friction force model that does not present discontinuities, which can be written as

⎧ 3v − t ⎨ μd fn 1 − e v0 sgn(vt ) if vt ≤ v0 (8) ft = ⎩ 0.95μd fn if vt > v0 where v0 is the threshold velocity. Bengisu and Akay [19] presented an alternative friction force model as   − μvs0fn (vt  − v0 )2 + μs fn sgn(vt ) if vt ≤ v0 ft =  −v −κ(v ) t 0 sgn(v ) μd fn + (μs fn − μd fn )e if vt > v0 t

(9)

where κ is a positive parameter representing the negative slope of the sliding state. Ambrósio [20] proposed another regularized approach for the Coulomb’s law that includes a ramp to avoid the numerical difficulties, which can be expressed as ft = cd μd fn sgn(vt )

cd =

⎧ ⎪ ⎨ 0

vt −v0 v1 −v0

⎪ ⎩ 1

if vt < v0 if v0 ≤ vt ≤ v1 if vt > v1

(10)

(11)

in which the dynamic correction factor, cd , prevents that the friction force changes direction for almost null values of the relative tangential velocity. The use of the friction force models given by Eqs. (8), (9) and (10) has the advantage of allowing the numerical stabilization of the integration algorithm used during the resolution of the equations of motion for multibody systems. It must be noticed that several alternative friction force models have been proposed over last decades, being the interested readers referred to the following references [11].

4 Non-smooth Formulations for Contact Dynamics The equations of motion suitable appropriate to describe multibody systems involving impacts can be expressed at the velocity level as [8] Mdu − hdt − wn dPn − wt dPt = 0

(12)

where M is the positive definite and symmetric mass matrix, h represents the vector of all external and gyroscopic forces acting on the system, wn and wt are the generalized ˙ + (u+ − normal and tangential force directions. The measure for the velocities du = udt − ˙ that is continuous, and the atomic parts, u )dη is split in Lebesgue measurable part udt, which occur at the discontinuity points with the left and right limits u– and u+ and ˙ Similarly, the Dirac point measure dη. For impact free motion it holds that du = udt. the measure for the so-called percussions corresponds to a Lagrangian multiplier which

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gathers both finite contact forces, λ, and impulsive contact forces, , that is, dP = λdt+dη [21]. Let consider a multibody system with a total f of unilateral constraints, which can be represented by f inequalities as gni (q, t) ≥ 0

(i = 1, . . . , f )

(13)

in which the quantities gni are the normal gap functions of the contacts. They are formulated such that, gni > 0 indicates an inactive, gni = 0 corresponds to an active contact, and gni < 0 indicates the forbidden interpenetration between rigid bodies. The normal and tangential relative velocities at the contacts as [13] T γni = wni u + w˜ ni

(14)

γti = wtiT u + w˜ ti

(15)

The equations of motion (12) can be complemented with constitutive laws for normal and tangential contact-impact forces, for that, a unilateral version of the Newton’s impact law is considered for the normal direction with local coefficient of restitution εni . The Coulomb’s friction law is used for the tangential direction with coefficient of friction μi , which is complemented by a tangential coefficient of restitution εti . Normal and tangential impact laws can be stated as inclusions −dPni ∈ Upr(ξni )

(16)

−dPti ∈ μi dPni Sgn(ξti )

(17)

ξni := γni+ + εni γni−

(18)

ξti := γti+ + εti γti−

(19)

with

The complete description of the dynamics of non-smooth system, which accounts for both impact and impact-free phases, is given by Eqs. (12)−(19). This problem can be solved by using the Moreau’s time-stepping method [13].

5 Demonstrative Example of Application Figure 2 shows a hexapod system, which consists of one rigid, load carrying mainframe with six legs, similar and symmetrically distributed. Each leg is composed by four links, interconnected by four revolute joints and attached to the main body by means of a fifth revolute joint. Revolute motors and linear actuators accomplish traction movement and elevation, respectively. Two representative virtual simulations are presented in order to study the behavior of the movement characteristics of the proposed legged robot model. In the first simulation, a straight path on a planar, horizontal and non-rough surface

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is considered. The second one deals with climbing a standard set of stairs (height of 170 mm, deep of 280 mm). In the former simulation it was considered both static and dynamic stability, while in the later only static stability was used to simulate the motion. Figure 2 shows an animation sequence of the virtual simulation that corresponds to the second situation.

Fig. 2. Snapshots of the hexapod robot system climbing a set of stairs

One crucial issue deals with the selection of the time step used to perform the computational analysis, since it plays a key role in the contact detection and, hence, on the contact-impact forces that can be artificially large and affect the outcomes. Secondly, the identification of the contact parameters, namely in terms of restitution and friction coefficients is also of paramount importance in order to properly handle the different contact regimens and the transition between them.

6 Concluding Remarks In this work, the problem of modeling frictional contact problems in dynamical systems was revisited. The regularized and non-smooth approaches were considered. A hexapod system was used as a demonstrative example of application. It was clear the contact dynamics is complex problem, requiring more research to reach better models and approaches. Future research can include the development of new algorithms to deal with contact analysis and systems with contact transition regimens. Investigation on parameters identification and estimation, using the input date from physical experiments to drive the simulation of uncertain model parameters will be also a potential future direction for further research. Acknowledgments. This work has been supported by FCT, under the national support to R&D units grant, with the reference project UIDB/04436/2020 and UIDP/04436/2020.

References 1. Dong, H., Qiu, C., Prasad, D.K., Pan, Y., Dai, J., Chen, I.-M.: Enabling grasp action: generalized quality evaluation of grasp stability via contact stiffness from contact mechanics insight. Mech. Mach. Theory 134, 625–644 (2019)

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2. Marques, F., Magalhães, H., Pombo, J., Ambrósio, J., Flores, P.: A three-dimensional approach for contact detection between realistic wheel and rail surfaces for improved railway dynamic analysis. Mech. Mach. Theory 149, 103825 (2020) 3. Pazouki, A., et al.: Compliant contact versus rigid contact: a comparison in the context of granular dynamics. Phys. Rev. E 96(4), 042905 (2017) 4. Leine, R.I., Van Campen, D.H., Glocker, C.: Nonlinear dynamics and modeling of various wooden toys with impact and friction. J. Vib. Control 9(1–2), 25–78 (2003) 5. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990) 6. Machado, M., Flores, P., Ambrosio, J., Completo, A.: Influence of the contact model on the dynamic response of the human knee joint. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 225(4), 344–358 (2011) 7. Khulief, Y.A.: Modeling of impact in multibody systems: an overview. J. Comput. Nonlinear Dyn. 8(2), 021012 (2013) 8. Pfeiffer, F., Glocker, C.: Contacts in multibody systems. J. Appl. Math. Mech. 64(5), 773–782 (2000) 9. Klisch, T.: Contact mechanics in multibody dynamics. Mech. Mach. Theory 34(5), 665–675 (1999) 10. Alves, J., Peixinho, N., Da Silva, M.T., Flores, P., Lankarani, H.M.: A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mech. Mach. Theory 85, 172–188 (2015) 11. Marques, F., Flores, P., Pimenta Claro, J.C., Lankarani, H.M.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn. 86(3), 1407–1443 (2016). https://doi.org/10.1007/s11071-016-2999-3 12. Pfeiffer, F.: The idea of complementarity in multibody dynamics. Arch. Appl. Mech. 72, 807–816 (2003) 13. Glocker, C., Studer, C.: Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody Sys. Dyn. 13(4), 447–463 (2005) 14. Trinkle, J.C., Tzitzouris, J.A., Pang, J.S.: Dynamic multi-rigid-body systems with concurrent distributed contacts. Philos. Trans. Math. Phys. Eng. Sci. 359(1789), 2575–2593 (2001) 15. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42(2), 440–445 (1975) 16. Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Sys. Dyn. 25, 357–375 (2011) 17. Coulomb, C.A.: The theory of simple machines. Mem. Math. Acad. Sic. 10, 161–331 (1785) 18. Threlfall, D.C.: The inclusion of Coulomb friction in mechanisms programs with particular reference to DRAM. Mech. Mach. Theory 13, 475–483 (1978) 19. Bengisu, M.T., Akay, A.: Stability of friction-induced vibrations in multi-degree-of-freedom systems. J. Sound Vib. 171, 557–570 (1994) 20. Ambrósio, J.A.C.: Impact of rigid and flexible multibody systems: deformation description and contact model. Virtual Nonlinear Multibody Syst. 103, 57–81 (2003) 21. Glocker, C.: On frictionless impact models in rigid-body systems. Philos. Trans. Math. Phys. Eng. Sci. 359(1789), 2385–2404 (2001)

Modelling Spherical Joints in Multibody Systems Mariana Rodrigues da Silva1(B) , Filipe Marques1 , Miguel Tavares da Silva2 , and Paulo Flores1 1 CMEMS-UMinho, Departamento de Engenharia Mecânica, Universidade Do Minho, Campus

de Azurém, 4804-533 Guimarães, Portugal {m.silva,fmarques,pflores}@dem.uminho.pt 2 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisbon, Portugal [email protected]

Abstract. Spherical joints are commonly utilized in many real-world scenarios. From the more simplistic to the more complex perspectives, spherical joints might be modelled considering different cases. Thus, the aim of this study is to analyze and compare the influence of different spherical joint modelling approaches, namely the ideal, dry, lubricated, and bushing models, on the dynamic response of multibody systems. Initially, the kinematic and dynamic aspects of the spherical joint models are comprehensively reviewed. In this process, several approaches are explored and studied considering the normal, tangential, lubrication and bushing forces experienced by the multibody systems in such cases of spherical joints. The application of the spherical joint models in the dynamic modeling and simulation of the spatial four bar mechanism is investigated. From the results obtained, it can be stated that the choice of the spherical joint model can significantly affect the dynamic response of mechanical multibody systems. Keywords: Spherical joints · Dry joints · Lubricated joints · Bushing joints · Contact mechanics · Multibody dynamics

1 Introduction Considering the type of applications in which they are designed to operate, spherical joints may be characterized using different models. The simplest approach is to consider the spherical joint as an ideal joint, that is, with no clearance between the socket and the ball. However, in many applications there is clearance between these components. In this case, the dynamics of the joint is controlled by contact-impact forces that develop on the ball and socket and that result from their collision. In multibody systems, instead of dealing with kinematic constraints as in the ideal joint, the spherical clearance joint deals with force constraints. The contact-impact forces developed can significantly affect the dynamic response of the system [3]. One of the most commonly utilized solutions to avoid or reduce the contact within dry clearance joints and to minimize the energy dissipation, is to add a lubricant fluid in the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 85–93, 2022. https://doi.org/10.1007/978-3-030-88751-3_9

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space between the socket and the ball, that is, in the clearance space. The high pressures that develop in this fluid act to keep the ball and the socket apart, preventing the contact between these two components [4, 5]. In other applications, bushing elements may be utilized. These are usually composed of elastomeric materials and are used to absorb shocks and vibrations, handle misalignments, reduce noise and wear, and decrease the transmissibility of irregularities to the system [1]. With this knowledge in perspective, the aim of this study is to analyze and compare the influence of different spherical joint models on the dynamic response of multibody systems. For this purpose, the kinematic and dynamic aspects of the ideal, dry, lubricated, and bushing models are presented. In the aftermath of this process, the spatial four bar mechanism is considered as a demonstrative example of application. This study provides a simple and direct comparison between different methodologies that can be applied to mechanical systems with spherical joints, allowing a better choice of the model to adopt for specific applications.

2 Kinematics of Spherical Joints This section includes a description of the formulations utilized in multibody systems to model the kinematic aspects of spherical joints. Several cases are presented, namely the ideal, dry, lubricated, and bushing joint models. Ideal spherical joints allow the relative rotations between two adjacent bodies i and j, constraining three relative translations. Consequently, the center of the ideal spherical joint has constant coordinates with respect to any of the local coordinate systems of the connected bodies. This means that point Pi on body i is coincident with point Pj on body j. Points Pi and Pj represent the center of the socket and ball, respectively [2]. The condition of the coincidence of points Pi and Pj is as follows: (s,3) ≡ rjP − riP = rj + sjP − ri − siP = 0

(1)

where rkP represents the global position vector of point P located on body k, rk denotes the position vector of the center of mass of body k described in global coordinates and skP is the global position vector of point P located on body k with respect to local coordinates. The velocity constraint equations for an ideal spherical joint are obtained by taking the first time derivative of Eq. (1), and can be expressed as [2]: ˙ (s,3) = r˙ j + s˙jP − r˙ i − s˙iP = 0 

(2)

in which the dot represents the derivative with respect to time. Considering the following condition: ˜ = −˜sω s˙ = ωs

(3)

where the symbol ~ represents the skew symmetric matrix and ω is the angular velocity, then Eq. (2) can be rewritten as follows [2]. ˙ (s,3) = r˙ j − s˜Pj ωj − r˙ i + s˜iP ωi = 0 

(4)

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In this sense, the time derivative of Eq. (4) yields the acceleration constraint equations of the ideal spherical joint as follows [2]: P ¨ (s,3) = r¨ j − s˜˙Pj ωj − s˜Pj ω ˙ j − r¨ i + s˜˙i ωi + s˜Pi ω ˙i = 0 

(5)

The multibody formulation for the case of ideal spherical joints was presented. However, in real-world application scenarios spherical joints present some level of clearance between the socket and the ball. The condition of coincidence between point Pi and point Pj assumed for the ideal joint case is disregarded in the case of the spherical clearance joint. Thus, the three kinematic constraints shown in Eq. (1) are removed and the two bodies are separated and free to move relative to one another. Contrary to the ideal joint case, the spherical clearance joint does not constrain any degree of freedom from the system [4, 5]. In a spherical clearance joint, a spherical part of body j, the ball, resides inside a spherical part of body i, the socket. The radii of the socket and the ball are Ri and Rj , respectively, and the difference between these parameters defines the size of the radial clearance as [4, 5]: c = Ri − Rj

(6)

The vector connecting point Pi to point Pj is defined as the eccentricity vector, e, which is obtained as: e = rjP − riP The magnitude of the eccentricity vector is given by: √ e = eT e

(7)

(8)

and the time rate of change of the eccentricity in the radial direction, that is, in the direction of the line of centers of the socket and the ball, is written as follows: e˙ =

eT e˙ e

(9)

In spherical clearance joints, the situation in which the socket and the ball are contacting with each other is identified by a relative pseudo-penetration δ. The geometric condition for contact between the socket and the ball is defined as: δ = e−c

(10)

˙ is given by: and the relative normal contact velocity, δ,   Q Q Te δ˙ = r˙ j − r˙ i e

(11)

where Qi and Qj are the contact points on bodies i and j. When the clearance joints are considered dry, normal and friction forces are the only effects present when physical contact is detected between the surfaces. However, in most

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mechanisms and machines, the joints are designed to operate with some lubricant fluid with the purpose of ensuring better performance of the mechanical systems by reducing friction and wear, providing load-carrying capacity, and adding damping to dissipate undesirable vibrations [4, 5]. In the case of a spherical joint with lubrication, the space between the ball and the socket is filled with a lubricant. Under applied load, the ball center is displaced from the socket center and the lubricant is forced into the clearance space, provoking a buildup of pressure. The high pressures generated in this lubricant film act to keep the bodies apart. Lubricated joints are designed so that, even when the maximal load is applied, the socket and the ball do not come in contact [5]. Bushing elements are utilized in many mechanical systems to absorb shocks and vibrations, handle misalignments and decrease the transmissibility of irregularities to the system [1]. The formulation utilized in this study closely follows the methodology presented by Ambrósio and Veríssimo [1], in which the bushing element is modelled in the multibody code as a nonlinear restrain that relates the relative displacements between the bodies connected with the joint reaction forces. The kinematic aspects of spherical clearance joints with lubrication and with bushing elements are similar to those of the dry spherical joint.

3 Dynamics of Spherical Joints This section includes a description of the formulations utilized in multibody systems to model the dynamics of spherical joints. The normal, tangential, lubrication and bushing force models are presented. These forces are introduced in the equations of motion of a multibody system as external generalized forces. In the case of the dry spherical clearance model, the dynamics of the joint is controlled by contact-impact forces arising from the collision between the connected bodies. This type of joint can, thus, be referred to as force joint, since it deals with force effects rather than kinematic constraints [4, 5]. Within the scope of this study, the model developed by Lankarani and Nikravesh [6] is analyzed. The authors proposed a continuous contact force model for the contactimpact analysis of multibody systems using the general trend of Hertz contact law incorporated with a hysteresis damping factor to include energy dissipation in terms of internal damping. The contact force model is expressed as:   3(1 − cr2 ) δ˙ (12) fn = Kδ n 1 + 4 δ˙(−) where K denotes the generalized stiffness parameter, δ is calculated by Eq. (10), n represents the nonlinear exponent factor, cr is the restitution coefficient, δ˙(−) denotes the initial contact velocity and δ˙ is obtained from Eq. (11). In real-world applications of mechanical systems involving contacting surfaces with relative motion, friction forces of complex nature might arise. Thus, a rigorous evaluation of these forces is warranted to obtain an accurate modelling of the dynamic response of the system. In the model proposed by Threlfall [7], the friction force and velocity are related by an exponential function whose purpose is to address the numerical difficulties

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associated with the discontinuity in Coulomb’s law. The model was the basis for other friction force models, such as the continuous function expressed as:   vt vt (13) ft = fc tanh v1 vt where vt is the tangential velocity of the contact point, vt denotes the magnitude of the tangential velocity, v1 represents the tolerance for the velocity and f c is the magnitude of the Coulomb friction represented as follows: fc = μk fn

(14)

in which μk is the kinetic coefficient of friction and f n is the normal contact force. Concerning lubricated spherical joints, the squeeze-film and the wedge-film actions comprise the two main groups in which these joints can be categorized into. The squeezefilm action is associated with situations in which the ball does not rotate significantly about is center, but instead it moves along some path inside the socket boundaries. The wedge-film action refers to situations in which the ball has significant rotation, which this is usually observed in high-speed rotating machinery [5]. In this study, the squeeze-film action is considered. The lubrication force due to squeeze-film action developed between the socket and the ball when there is lubricant fluid between these two components can be modelled using a law developed by Flores and Lankarani [2] as follows:   1 6π ν e˙ Ri 1 1 − (15) ln(1 − ε) + 2 fl = (c/Ri )3 ε3 ε (1 − ε) 2ε where ν represents the dynamic lubricant viscosity, e˙ is given by Eq. (9), and ε denotes the eccentricity ratio given by: ε=

e c

(16)

where c and e are given by Eqs. (6) and (8), respectively. For nonideal spherical joints, vector e given by Eq. (7) can be characterized as the gap between the ball and the socket. However, this vector can be defined as the deformation of the elastomer in a spherical joint with a bushing element [1]. Considering this statement, the force due to the bushing deformation adopted in this study is based on the formulation developed by Ambrósio and Veríssimo [1], and it is represented by the following condition: 

 e fb = k δ + bδ˙ e

(17)

where k is the stiffness of the bushing element, δ denotes the bushing deformation and b represents the stiffness proportional damping parameter. As previously determined, the length of vector e is given by Eq. (8). Assuming that no gap exists between the bushing element and the ball, then e = δ and, thus, the time derivative of δ is given by Eq. (9).

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4 Demonstrative Example of Application The objective of this section is to examine the influence of the different cases for modelling spherical joints on the dynamic response and behavior of multibody systems. To this end, a spatial four-bar mechanism is utilized. The spatial four-bar mechanism is composed by four rigid bodies, namely the ground, crank, coupler, and rocker. The numbers of each body and their corresponding local coordinate systems are shown in Fig. 1. Coupler

z

ζ3 η3 ξ3 ζ4 ξ4 η4

Crank

ζ1 ≡ ζ2

y

η1 ≡η2

Rocker

0.020

Ground

ξ1 ≡ ξ2 0.085

x

0.040

Fig. 1. Schematic representation of the spatial four-bar mechanism

The bodies of the spatial four bar mechanism are kinematically connected to each other by means of two revolute joints, connecting the ground to the crank and the ground to the rocker, and two spherical joints, connecting the crank to the coupler and the coupler to the rocker. Clearance and bushing are introduced in the spherical joint connecting the coupler and the rocker to analyze the dry, lubricated, and bushing models. The remaining joints are considered ideal. The initial configuration of the spatial four-bar mechanism is presented in Fig. 1 and the corresponding initial values are presented in Table 1. The system is released from the initial position with null velocities and under the action of gravitational force, acting on the negative z-direction. For the nonideal joint models, initially the ball and the socket of the spherical clearance joint are concentric. Table 1. Initial configuration of the spatial four-bar mechanism Body Nr

x [m]

y [m]

z [m]

e0

e1

2

0.00000

0.00000

0.00000

1.0000

0.0000

3

−0.03746

−0.04250

0.04262

0.9186

−0.1764

4

−0.05746

−0.08500

0.03262

0.3634

−0.6066

e2 0.0000 0.06747 −0.6066

e3 0.0000 −0.3472 0.3634

The dimensions and inertial properties of each body of the spatial four bar mechanism are presented in Table 2.

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Table 2. Dimensions and inertial properties of the spatial four-bar mechanism Body Nr

Length [m]

Mass [kg]

Moment of inertia [kgm2 ] Iζζ

Iηη

Iζζ

2

0.02

0.0196

0.0000392

0.0000197

0.0000197

3

0.122

0.1416

0.0017743

0.0000351

0.0017743

4

0.074

0.0316

0.0001456

0.0000029

0.0001456

The simulation parameters used in all dynamic simulations and in the numerical methods required to solve the dynamics of the system are displayed in Table 3. Table 3. Common and specific simulation parameters for the four-bar mechanism Common Baumgarte coefficient, α

5

Reporting time step

0.00001s

Baumgarte coefficient, β

5

Integration tolerance

10–10

Integrator algorithm

ode15s

Simulation time

2s

Young’s modulus, E

207 GPa

Velocity tolerance, v1

0.0010 m/s

Poisson’s ratio, v

0.3

Kinetic coefficient of friction, μk

0.1

Nonlinear exponent, n

1.5

Socket radius, Ri

10 mm

Ball radius, Rj

9.9 mm

Dry Model

Restitution Coefficient, cr

0.9

Lubricated Model Dynamic lubricant viscosity, ν – 400 cP Bushing Model Bushing stiffness, k - 2.146 × 107 N/m

Stiffness proportional damping, b - 0.01

The results obtained for the spherical joint models studied are shown in Fig. 2. It can be observed that the dynamic performance of the four-bar mechanism is significantly affected by the model chosen to characterize the spherical joint. In general, the frictionless joint model, exhibits more oscillations and produces significantly larger velocities and accelerations than the other models, suggesting that the response of the system becomes chaotic. In fact, the addition of friction to the dry spherical clearance joint tends to smooth the behavior of the system, leading to a less chaotic behavior. The

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Fig. 2. Influence of the spherical joint model on the response of the spatial four-bar mechanism. a) Position, b) velocity, c) acceleration of the rocker and d) mechanical energy of the system.

observation of Fig. 2 (c) also indicates that the spatial four-bar mechanism produces significantly lower accelerations with the ideal joint model when compared to the other models. Concerning the variation of the mechanical energy, the model with friction presents higher energy dissipation, followed by the frictionless and lubricated models, as observed in Fig. 2 (d). As expected, the bushing model produces positions, velocities and accelerations close to the ideal joint case, which means that the bushing element is, in fact, decreasing the noise associated with clearance and stabilizing the system, making it less chaotic. This model dissipates the least amount of energy comparing to the others.

5 Conclusions The dynamic modeling and analysis of spatial mechanisms with different models for spherical joints has been presented in this work. The main kinematic and dynamic aspects related to these models were described under the framework of multibody systems methodologies. A classic spatial four bar mechanism was considered as a demonstrative application example to study the effect of the joint modeling approaches. Overall, the joint models strongly affect the performance of the system, essentially visible in terms of accelerations and mechanical energy.

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Acknowledgments. This work has been supported by Portuguese Foundation for Science and Technology, under the national support to R&D units grant, with the reference project UIDB/04436/2020 and UIDP/04436/2020, as well as through IDMEC, under LAETA, project UIDB/50022/2020.

References 1. Ambrósio, J., Verissimo, P.: Improved bushing models for general multibody systems and vehicle dynamics. Multibody Sys.Dyn. 22, 341–365 (2009). https://doi.org/10.1007/s11044009-9161-7 2. Flores, P.: Concepts and Formulations for Spatial Multibody Dynamics. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-16190-7 3. Flores, P., Ambrósio, J., Pimenta Claro, J.C., Lankarani, H.M.: Dynamics of multibody systems with spherical clearance joints. ASME J. Comput. Nonlinear Dyn. 1, 240–247 (2006). https:// doi.org/10.1115/1.2198877 4. Flores, P., Ambrósio, P., Pimenta Claro, J.C., Lankarani, H.M.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints - Models and Case Studies. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74361-3 5. Flores, P., Lankarani, H.M.: Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation. Nonlinear Dyn. 60, 99–114 (2010). https://doi.org/10.1007/ s11071-009-9583-z 6. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990). https://doi.org/10.1115/1. 2912617 7. Threlfall, D.C.: The inclusion of Coulomb friction in mechanisms programs with particular reference to DRAM au programme DRAM. Mech. Mach. Theory 13, 475–483 (1978). https:// doi.org/10.1016/0094-114X(78)90020-4

Multibody Dynamics Modeling of Delta Robot with Experimental Validation Mohamed Elshami, Mohamed Shehata(B) , Qingshun Bai, and Xuezeng Zhao Harbin Institute of Technology, Harbin, China {mohamed elshami,qshbai,zhaoxz}@hit.edu.cn, [email protected]

Abstract. Delta robot is one of the most known parallel systems which possesses high stiffness and accuracy. In order to build a system that endows the robot to perform the desired tasks, an accurate and validate the dynamic model is required. In recent years, researchers have been focused on the construction of serial structured robots. However, few researchers tried to evolve the delta robots in such a system. In this work, the multibody system dynamics (MBS) approach is used to study the kinematics and dynamics of delta robots. A systematic approach is developed based on load assumption due to end-effector movements. The multibody model is constructed using Matlab Symbolic Toolbox. Moreover, D3S-800 is utilized in this study to validate the multibody model. The comparison of experimental data and numerical solution shows a very good agreement and consequently, the multibody model obtained is suitable for parameter identification, control and design optimization of a delta robot system.

Keywords: Multibody system dynamics symbolic toolbox · Euler parameters

1

· Delta robot · Matlab

Introduction

Robot structures possess a good variety of architectural designs adapted to the different industrial and non-industrial operations. While serial manipulators are the most common dominant type of robots in the industry because of their high flexibility. The parallel structured robots provide a good solution to some inconvenient situations of the serial manipulators, such as the situations where the robot is required to resist high loads [10]. Delta robot is one common configuration of parallel robots and what makes the delta robot fascinating is that unlike most of robotic applications which are biologically inspired [14]. Delta robot perception was a mechanical structure design that depended completely on the theory of machines and the correlated relation between the mechanical linkages [5]. Recently, researchers made a lot of effort into modeling, design and control of delta robots using traditional dynamics methods [6,8]. On the other hand, the multibody system serves as a basis for c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 94–102, 2022. https://doi.org/10.1007/978-3-030-88751-3_10

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many modern models of complex systems and has been applied in many areas of science. The multibody system approach will be used for the dynamic modeling of delta robot in order to increase the efficiency of the system and enhance system control [2]. Also, the multibody model results were verified using actual delta robot system. The remainder of this study is organized as follows: Sect. 2 introduces the delta robot as a multibody system. In Sect. 3, the constraint’s function is expressed, and the model is constructed. Section 4 presents numerical simulation and experimental validation that are followed by the conclusion in Sect. 5.

2

Delta Robot as a Multibody System

In this study, D3S-800 delta robot used, in which The robot has three degrees of freedom, the end-effector is free to move in three translational motions along XYZ axes, see Fig. 1. In addition, The manipulator can achieve rotational motion about Z-axis by means of an actuator installed to the fixed base and the rotational motion is transmitted mechanically to the end-effector [3]. The computational model is established to determine the kinematic relationship between the system coordinates. We apply the MBS mathematical calculation process to the incremental robot mechanism and for simplification, the rotational degree of freedom about Z-axis is not considered. Delta robot as a multibody system consists of a fixed base, three arms, six forearms, six rods and an end-effector. To define system bodies, local frames are assigned to delta robot bodies and the base frame is considered the reference coordinate. For simplicity of the computations, the global frame is assigned to the projection of the fixed platform in the same plane enclosing the three points ai , which are the positions of revolute joints between the base frame and the active arms [13]. Each single chain shown in Fig. 1b consists of a revolute joint directly actuated by means of an electrical motor. The forearms or the passive arms are connected to the active arms at points bi1 and bi2 , the above two points are connected to the movable platform in points ci2 and ci1 forming the closed loop bi1 , bi2 , ci2 and ci1 . Another closed-loop si1 , si2 , ti2 and ti1 formed by the connecting rods which functions are to maintain the connectivity of the spherical joints and to prevent the forearms from the undesired rotations about their longitudinal axis. At the initial home position, an end-effector frame is collinear with the Z-axis of the reference coordinate.

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(a) CAD Model of delta robot system

(b) Parameters assignment of a general chain

Fig. 1. Multibody model of delta robot system

3

Mathematical Model of Delta Robot

The model of the delta robot shown in Fig. 1a, can be constructed, without loss of generality, as shown in Table 1 for the first chain. Let the contact point P be located on the end-effector frame. The system of generalized coordinates is denoted by q and can define function in Euler angles φ, θ and ψ [4]. To avoid a singularity during simulation, the three Euler angles will convert to four Euler parameters [q01 q11 q21 q31 ] [12]. The generalized coordinates of the fixed base including translation and orientation are defined as:   (1) q1 = x1 y 1 z 1 q01 q11 q21 q31 The global portion vector of that point can be expressed as: ri = Ri + Ai u ¯ ip

(2)

Table 1. Components of Delta robot system Joint number Joint type Body(i)

Body(j)

1

Fixed

Fixed Base Ground

2

Revolute

Arm

3

Spherical

Forearam1 Arm

4

Revolute

Rod1

5

Revolute

Forearam2 Rod1

6

Revolute

Rod2

7

Spherical

Forearam1 End-effector

Fixed Base Forearam1 Forearam2

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where ri , is the global position of an arbitrary point, Ri , is the global position of the origin of the end-effector coordinate system, and Ai is the transformation matrix function on the generalized coordinates. It is clear from Eq. (2) that the global position vector of an arbitrary point on the body coordinate system can be written in terms of the rotational coordinate of the body, as well as the translation of the frame-origin of the body. In order to avoid singularities, Euler parameters are used to describe the orientation of the system bodies and result coordinates can be converted back to cartesian coordinates [11]. The constraints function of delta robot can be obtained using multibody constraints equation of Rigid, Revolute and Spherical joints. Equation 3 illustrates constraints equations of rigid joint between fixed base and ground. ⎡ ⎤ x1 ⎢ ⎥ y1 ⎢ ⎥ 1 ⎢ ⎥ z ⎢ ⎥ 1 1 2 1 1 1 1 1 2⎥ ⎢ (3) C(qg ,q1 ,t) = ⎢ 1 − 2(q2 ) − 2q0 q2 − 2q1 q3 − 2(q1 ) ⎥ = 0 ⎢ 2(q 1 )2 + 2(q 1 )2 − 2q 1 q 1 − 2q 1 q 1 − 1 ⎥ 2 0 2 1 3 ⎢ 11 2 ⎥ ⎣ 2(q ) + 2(q 1 )2 − 2q 1 q 1 − 2q 1 q 1 − 1 ⎦ 1 2 0 2 1 3 (q01 )2 + (q11 )2 + (q21 )2 + (q31 )2 − 1 where qg is generalized coordinate vector for ground and it equal zero and q1 is generalized coordinate vector of the fixed base. Figure 2a shows revolute joint between Arm1 and fixed base. The constraints equations of revolute joints between Arm1 and fixed base can be written as: ⎤ ⎡ x1 − x2 ⎥ ⎢ y1 − y2 ⎥ ⎢ 1 2 ⎥ ⎢ z −z ⎥ (4) C2(q1 ,q2 ,t) = ⎢ ⎢ 1 − 2(q 1 )2 − 2q 1 q 1 − 2q 1 q 1 − 2(q 1 )2 ⎥ = 0 2 0 2 1 3 1 ⎥ ⎢ ⎣ 2(q 1 )2 + 2(q 1 )2 − 2q 1 q 1 − 2q 1 q 1 − 1 ⎦ 1 2 0 2 1 3 (q02 )2 + (q12 )2 + (q22 )2 + (q32 )2 − 1

(a) Revolute joint

(b) Spherical joint

Fig. 2. Multibody joint constraints

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where generalized coordinates of Arm1 is define as q2 = [x2 y 2 z 2 q02 q12 q22 q32 ]. Figure 2b shows spherical joint between the forearm and end-effector. The constraints equation of spherical joints between Forearam1 and Arm can be written function on translation constraints as: ⎤ ⎡ 2 x − x3 (5) C3(q2 ,q3 ,t) = ⎣ y 2 − y 3 ⎦ = 0 z2 − z3 Constraints equations for other joints can be defined similarly to spherical and revolute joints. The dynamic equations that govern the motion of system bodies can be systematically obtained using Lagrange formulation as [1]:

  

M CTq q ¨ Q (6) = λ Qd Cq 0 where M is the system mass matrix, Cq is the system Jacobian matrix Cq = ∂C(q,t) , λ the vector of Lagrange multipliers and Qd is a vector absorb terms ∂q that are quadratic in the velocity and Q is a vector of external applied forces and can be written as:

i 



 QexR Fi Fi Q= = = (7) ¯i ¯ iT M G Qiexθ GiT M i where QiexR is the force associated with the translation coordinates and Qiexθ is ¯ i are the forces the force associated with the orientation coordinates. F i and M and moment vectors defined in the local coordinate system of the body. The ¯ i is define function on Euler parameter generalized coordinates and matrix G can be written as: ⎤ ⎡ −q1 q0 q3 −q2 ¯ i = 2 ∗ ⎣ −q2 −q3 q0 q1 ⎦ G (8) −q3 q2 −q1 q0 Equation of motion Eq. 6 yields a system of differential algebraic equations [7]. A set of initial parameters including positions and velocities from the CAD model are used to start the dynamic simulation [4]. The vector q ¨ can be integrated in order to determine the coordinates and velocities. The vector λ can be used to determine the generalized reaction forces that can be used to establish optimization of the design process. Because the direct numerical solution of differential algebraic equations associated with the constrained dynamics of a multibody system poses several computational difficulties, a post-stabilization process is used to brings the solution back to the invariant manifold.

4

Numerical Simulation and Experimental Validation

In this section, Multibody model results are represented and compared with experimental data. A mathematical model of the delta robot is developed

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by applying multibody dynamics theory based on the Lagrange formulation described in the previous section. Kinematics results include system displacements and dynamic results included reaction forces and torques acting on delta robot bodies are computed. The parameters of the D3S-800 delta system are provided in Table 2. The fixed base radius is the radius of a circle that passes through the three points of the revolute joints of arms, while the radius of the movable platform that carries end-effector is the radius of a circle that passes through the six points of the lower spherical joints between the platform and the forearms. By applying an initial motion of 100 mm/s with 45◦ in XY-plane on the endeffector and keep the distance between the fixed frame and end-effector frame 847 mm in the negative Z direction, the corresponding structural displacement of the delta robot links can be obtained. The simulation is performed using Matlab and Adams-Bashforth-Moulton (ODE113) as the numerical integrator for 5 s. Figure 3a shows the constraints violation due to the revolute joint between the arm and fixed base [9]. The violation does not exceed 3 ∗ 10− 13 which indicates the computational efficiency of the multibody system model. Noted that, the constraint equation C6 is an Euler parameter constraint that must be added in case of using Euler parameter to define system generalized coordinates. By integrating the system accelerations forward, the system bodies velocities and configurations are computed. According to initial input, the end-effector displacement in XY-plane and the distances in Z-direction constant, see Fig. 3b. Reactions forces acting on deferent bodies of delta robot system are computed from the MBS model as a function of generalized coordinates using Lagrange multipliers. Figures 4 show reaction forces acting on a fixed base including translation forces and moments. As shown in Fig. 4a, the reaction force in Z-direction is due to the wight. Likewise, other system body’s reaction forces can be computed. Experimental work is carried out in order to validate the dynamic model of the delta robot. As shown in Fig. 5, the D3S-800 delta robot system consists of robot arms, motors, control unite, encoder and programming unite. Figure 6 shows the comparisons between the output rotor velocity of the MBS model and the experimental data. Since the multibody model and experiment data have similar results, the MBS model is accurate and can be useful for establishing the optimization process of delta robot system design and control. Table 2. Delta robot parameters employed in the numerical simulation Components

Dimensions (mm) Mass (kg) Ixx (Kg · m2 ) Iyy (Kg · m2 ) Izz (Kg · m2 )

Fixed base

R = 125

30

0.52202

0.52202

0.88497

Arm

L = 370

6.2

0.00510

0.12448

0.125448

Forearm

L = 960

1.65

0.13940

0.00006

0.13940

Connecting rod L = 95

0.2

0.0000147

0.000101

0.000101

End-effector

0.9

0.001031

0.001031

0.002019

r = 62

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Fig. 3. Constraints violation and global position of the point P

Fig. 4. Reaction forces due to revolute joint

Fig. 5. D3S-800 delta robot system

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Angular Velocity[deg/sec]

-5 -6 MBS model

-7 Expermintal data

-8 -9 -10 -11 -12 -13 -14 -15 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec]

Fig. 6. Angular speed of motor 1

5

Conclusion

In this paper, an efficient modeling procedure for the delta robot system is developed based on Multibody system dynamics. The symbolic and computational work have been carried out using Matlab. The symbolic derivation is carried out and the explicit equations of motion of the delta robot have been derived. The solution of the equations of motion involves the system coordinates and the associated Lagrange multipliers as well. The paper describes an experimental test-rig of the D3S-800 delta robot system in front of end-effector movement and the measured motors speed is collected and compared with the multibody model. The comparison shows a very good agreement which encourages the enhancement of the model by examining unconventional operating conditions. Moreover, Lagrange multipliers can be used to estimate the generalized reaction forces which can be utilized in the optimization of delta robot design. In going and future work, multibody model will be used for parameters identification and design optimization of the delta robot. Acknowledgement. This research was supported by National Natural Science Foundation of China (Grant No.71175102 and 52075129).

References 1. Bai, Q., Shehata, M., Nada, A.: Efficient modeling procedure of novel grating tiling device using multibody system approach. In: Pucheta, M., Cardona, A., Preidikman, S., Hecker, R. (eds.) MuSMe 2021. MMS, vol. 94, pp. 168–176. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-60372-4 19 2. Barreto, J.P., Corves, B.: Matching the free-vibration response of a delta robot with pick-and-place tasks using multi-body simulation. In: 2018 IEEE 14th International Conference on Automation Science and Engineering (CASE), pp. 1487–1492. IEEE (2018) 3. Chang, Z., Ali, R.A., Ren, P., Zhang, G., Wu, P.: Dynamics and vibration analysis of delta robot. In: 5th International Conference on Information Engineering for Mechanics and Materials, pp. 1408–1417. Atlantis Press (2015)

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4. Flores, P.: Concepts and Formulations for Spatial Multibody Dynamics. Springer, Heidelberg (2015) 5. Fumagalli, A., Masarati, P.: Real-time inverse dynamics control of parallel manipulators using general-purpose multibody software. Multibody Syst. Dyn. 22(1), 47–68 (2009) 6. Hamdoun, O., Bakkali, L.E., Baghli, F.Z.: Analysis and optimum kinematic design of a parallel robot. Procedia Eng. 181, 214–220 (2017). https://doi.org/ 10.1016/j.proeng.2017.02.374. https://www.sciencedirect.com/science/article/pii/ S1877705817309578. 10th International Conference Interdisciplinarity in Engineering, INTER-ENG 2016, Tirgu Mures, Romania, 6–7 October 2016 7. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution, vol. 2. European Mathematical Society (2006) https://doi.org/10. 1016/j.proeng.2017.02.374. URL https://www.sciencedirect.com/science/article/ pii/S1877705817309578 8. Lenarcic, J., Wenger, P.: Advances in Robot Kinematics: Analysis and Design. Springer, Heidelberg (2008) 9. Marques, F., Souto, A.P., Flores, P.: On the constraints violation in forward dynamics of multibody systems. Multibody Syst. Dyn. 39(4), 385–419 (2016). https:// doi.org/10.1007/s11044-016-9530-y 10. Merlet, J.P.: Parallel Robots, vol. 128. Springer, Heidelberg (2005) 11. Pappalardo, C.M., Guida, D.: On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems. Arch. Appl. Mech. 87(10), 1647–1665 (2017) 12. Shabana, A.A.: Euler parameters kinetic singularity. Proc. Inst. Mech. Eng. Part K: J. Multi-body Dyn. 228(3), 307–313 (2014) 13. Stock, M., Miller, K.: Optimal kinematic design of spatial parallel manipulators: application to linear delta robot. J. Mech. Des. 125(2), 292–301 (2003) 14. Yang, X., Feng, Z., Liu, C., Ren, X.: A geometric method for kinematics of delta robot and its path tracking control. In: 2014 14th International Conference on Control, Automation and Systems (ICCAS 2014), pp. 509–514. IEEE (2014)

Railway and Vehicle Dynamics

Railway Dynamics with Curved Contact Patch Filipe Marques1(B) , Hugo Magalhães2,3 , João Pombo2,3 , Jorge Ambrósio3 , and Paulo Flores1 1 CMEMS-UMinho, Departamento de Engenharia Mecânica, Universidade do Minho, Campus

de Azurém, 4804-533 Guimarães, Portugal {fmarques,pflores}@dem.uminho.pt 2 Institute of Railway Research, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK {h.magalhaes,j.pombo}@hud.ac.uk 3 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisbon, Portugal [email protected]

Abstract. The wheel-rail contact modeling is of paramount importance for the dynamics of railway vehicles since it represents the interaction between the vehicle and the track. Although, in most cases, the contact generated occurs between convex surfaces which results in planar contact areas, the contact might take place in concave surfaces when negotiation sharp curves or due to the wear of profiles. In that cases, the resulting contact area is not planar. This work proposes a methodology to determine the shape of the contact patch in a curved surface, where the normal direction varies along its lateral direction. This method is based on a semi-Hertzian approach and discretizes the contact into longitudinal strips. The normal pressure distribution is computed in each strip separately using a non-Hertzian contact model and it is summed in a vector form to obtain the total normal force magnitude. Regarding the tangential forces, a look up table approach is considered. Finally, a trailer vehicle negotiating a curve is used to demonstrate the effectiveness of this methodology. Keywords: Railway dynamics · Contact forces · Conformal contact · Non-Hertzian

1 Introduction The utilization of multibody systems methodologies to model the dynamic behavior of railway vehicles has been gaining relevance in their design and development [1]. In that sense, the wheel-rail contact interaction plays a preponderant role since it represents the interface between vehicles and track system. The accurate modeling of the wheelrail contact is fundamental to analyze the dynamic response of the vehicle, in terms of comfort and safety, for any running conditions, requiring taking into account several complex phenomena that occur during contact [2]. Most of the wheel-rail contact force models available in the literature are limited to their application in planar contact patches, i.e., non-conformal contact cases, or even © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 105–113, 2022. https://doi.org/10.1007/978-3-030-88751-3_11

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to point contact simplification [3, 4]. Although the planar contact assumption covers most of the possible interaction scenarios, in which the rail interacts directly with the tread or flange, when negotiating sharp curves or due to worn profiles, the conformal contact tends to occur in the running profile fillet zone. The common methods of contact search tend to fail in finding a unique solution in a con-formal case, either using elastic or constraint approach. In this work, a methodology to consider a curved contact between wheel and rail elements is proposed. In the case of interaction between a convex body and a concave body, under the assumption of rigid bodies, an interpenetration region exists, and the effective contact area is smaller due to the elastic deformation of the surfaces. However, in the conformal case, the resultant contact patch tends to have a curved shape [5].

2 Curved Wheel-Rail Contact Model In the context of the presented methodology, wheels and rails are mathematically represented by parametrized surfaces, i.e., the location of any point on each of surface can be defined by two parameters. The surface of each rail is obtained through the sweep of its cross-section along a given path, which is represented by a set of nodal points and interpolated with a suitable spline. These nodal points define the position and orientation of the rail as function its arc length, sr . Then, the rail profile is represented by a two-dimensional function, in which the profile vertical coordinate f r is defined as a function of the surface parameter that defines the lateral rail coordinate, ur . In turn, since the wheelset is a body of revolution, the wheel surface can be defined by the rotation of its cross-section about its own axis. Thus, any point in the wheel surface is characterized by an angular position and a lateral coordinate. Similar to the rail, the wheel profile is represented by a two-dimensional function, in which the wheel vertical coordinate, f w , is dependent on the profile lateral position. The schematic representation of this parametrization of the wheel and rail surfaces is given in Fig. 1, in which the superscripts ‘L’ and ‘R’ denote the left and right elements, respectively. The half-space approach is widely employed in the development of most wheel-rail contact theories, for which the only exception is when using the finite element method that is computationally intensive. This concept involves several assumptions, namely (i) the characteristic sizes of the contacting bodies are large compared to the size of the contact patch; (ii) the materials are homogeneous, isotropic and linearly elastic, and (iii) the strains are small, and the inertia effects can be neglected [6]. Having in mind that the size of the contact patch tends to increase due to the conformality between surfaces, the first assumption can be violated. However, the elastic half-space assumption can be kept since it is valid for smaller variations of the contact angle, as it is considered here. The procedure proposed is here to compute the shape of a curved patch is illustrated in Fig. 2, in which the contact dimension is exaggerated for sake of understandability. The first step consists of identifying the interpenetration region which limits are obtained with the methodology provided in [3] and denoted by uw,lower and uw,upper for the wheel lateral parameter, and by ur,lower and ur,upper for the rail lateral parameter. Since the patch is not flat, there is no preferential direction, therefore, the profiles must be parametrized according to their arc length. The wheel potential contact points can

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s

fwL

L w

ζ ws

uwL

107

fwR

ηws

uwR ξ ws

swR

frL

frR

urL

urR

z

srL

x

srR

y

Fig. 1. Schematic representation of the wheel and rail surfaces parametrization

be represented by a three-dimensional curve and the angular parameter, sw , is given as function of the lateral parameter and yaw angle, α. Hence, to perform this length’s parametrization, the wheel has to be projected into the rail profile plane, since the rail is considered an extruded body. This strategy transforms the identification of the contact patch in a two-dimensional problem, which simplifies the process of evaluation the penetration over the patch. Hence, after some mathematical manipulation, the expressions to obtain the arc length for rail and the wheel interpenetration region are  ur  1 + fr dur (1) Lr (ur ) = ur,lower

    uw   −sin2 α fw4 + 2fw fw2 fw + 2fw2 + 2fw fw + cos2 α

 Lw (uw ) = tan2 αfw2 2 α duw 2 + tan2 αf 2 f 2 + f + sin uw,lower w w w 1−tan2 αf 2

(2)

w

It must be noticed that the rail’s arc length just depends on the profile shape and the boundaries of the interpenetration region, while the wheel’s arc length also depends on the yaw angle and, therefore, requires its identification for each wheel-rail configuration. Then, both profiles are discretized in N S equally sized spaces, as schematized

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Rail

ΔLr

ΔLw Wheel

(a)

(b)

r n (s )

(x

Rail

s,i

r t (s )

s

, y s,i ) Δs

Wheel

ys xs

r n ( si )

(s ,n ) i

r,i

(c)

r t ( si )

(d)

s

δcp

(s ,n ) i

δcp ( si )

w,i

s (e)

(f)

Fig. 2. Definition of the penetration along the interference region in the wheel lateral direction for conformal contacts: (a) interaction of wheel and rail in a conformal region; (b) discretization of the interpenetration zone by its arc length; (c) identification of the contact patch’s curved axis; (d) establishment of size and center point of each strip; (e) evaluation of penetration in each strip; (f) representation of the penetration along the interference region

in Fig. 2b. Consequently, the curved surface, s, in which the contact patch is contained can be determined through the evaluation of the middle position between wheel and rail points, as pictured in red in Fig. 2c. Since the points obtained are not necessarily equally spaced in this curve, they have to be resampled and treated in a local coordinate system. Furthermore, (x s,i , ys,i ) expresses the coordinates of the ith strip, and s is the width of each strip, as depicted in Fig. 2d. A curved axis s is established representing the direction along the contact patch for which the normal and tangential directions are variable. Thus, the normal and tangential vectors are found for each strip as T  ns,i = − sin θi cos θi

(3)

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T  ts,i = cos θi sin θi

109

(4)

in which the angle of each strip is defined as    θi = arctan ys,i

(5)

where y’s,i denotes the derivative of ys in x s,i that is calculated from the splines produced. Then, the penetration along the curved contact patch, which is measured in the normal direction of each strip, as represented in Fig. 2e, is evaluated. Both points on rail and wheel which define the limits of the interference of a given strip can be obtained through the intersection between a straight line normal to the patch surface and the rail and wheel profiles, respectively. After determining the intersection points for a given strip, the penetration on that strip, δ cp (si ), is the distance between those points. Since the wheel and rail contact is considered locally elastic, their surfaces tend to deform and, therefore, establish an effective contact area which is smaller than the interpenetration region [7]. Hence, the limits of the contact patch, ss and se , are determined by solving the following equation δcp (s) = (1 − ε)δmax

(6)

where ε is the correction factor, which takes into account the deformation of the contacting surfaces and δ max denotes the maximum penetration in the interference region. This procedure is represented in Fig. 3, and it must be noticed that the first and last strips have a smaller width compared with the remaining ones. For the effective contact zone, the semi-Hertzian approach is considered where the contact pressure distribution is elliptical only for the rolling direction. Thus, the longitudinal size of the i-th strip can be given by the location of the leading edge as

  (7) xL (si ) = 2Rsw (si ) δcp (si ) − (1 − ε)δmax in which RsW is the radius of curvature of the wheel surface in the longitudinal direction for a given strip. δcp

δmax i

j

δmax (1 − ε ) i −1

j +1 ss

< Δs

Δs

< Δs

se

s

Fig. 3. Schematic representation of the identification of the contact patch limits

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Subsequently to the identification of the shape and size of the contact patches, the normal pressure distribution evaluation on each patch is needed. Some of the methods available on the literature depend on the location of the maximum penetration point to calculate the contact pressure [7], which might produce numerical problems when two peaks of penetration are found in the same contact area. A method proposed by Sun et al. [8] is adapted to the curved contact model. In that sense, the contribution of each strip for the normal contact force is computed as

fn,i

⎞−1 ⎛  se  xL  2 xL (η) − ξ 2 π 2 Eδcp (si )xL2 (si )s ⎝   = dξ dη⎠ 4 1 − σ2 ξ 2 + (si − η)2 ss −xL

(8)

where E is the Young’s modulus and σ denotes the Poisson ratio. Since the normal direction varies along the contact patch, the normal force magnitude cannot be summed, thus, it is given by the vector sum of the force originated in each strip as fn =

Ns 

fn,i ns,i

(9)

i=1

This model is purely elastic, and a damping component can be added as fnd = fn cd where the damping factor is given as ⎧ ˙ ⎨ ce  δ ≤ −v0  2 δ˙ + v0 3 cd = ce + (1 − ce ) 3r − 2r −v0 < δ˙ < v0 in which r = ⎩ 2v0 1 δ˙ ≥ v0

(10)

(11)

where ce expresses the coefficient of restitution, δ˙ represents the penetration velocity and v0 is a tolerance velocity. Regarding the evaluation of creep forces and spin moment, a lookup table with a regularization for a simple double-elliptical contact region, based on CONTACT software, is considered [9]. This lookup table requires, as input variables, the parametrized spin creepage, semi-axes ratio, creepage angle, parametrized creepage modulus and shape number. An enhanced version of this lookup table is used, in which its discretization was obtained after minimizing the interpolation error [10].

3 Example of Application A multibody model of trailer vehicle negotiating a left curve is utilized as example of application of the proposed methodology for the wheel-rail contact model. This model includes 11 rigid bodies, namely 4 wheelsets, 4 axleboxes, 2 bogie frames and the carbody. All details of this model can be found in [11]. The vehicle starts the simulation with a forward velocity of 18.3 m/s and a lateral misalignment of 2 mm with respect to the track centerline to promote some hunting motion.

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Fig. 4. Representation of (a) the location of the main contact point for each patch and their shape for (b) t = 1 s (c) t = 6 s (d) t = 20 s and (e) t = 36.4 s for the right wheel of the leading wheelset.

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To analyze the obtained results, the contact in the right wheel of the leading wheelset is examined in detail, since it is external to the curve and, therefore the contact can occur in the wheel transition zone. Figure 4a shows the location of the contact points on the wheel profile during the simulation. From the results, it is concluded that the rail interacts with the wheel transition zone during negotiation where two different contact patches are identified. The different shapes of contact patches determined are displayed in Figs. 4b– e, where four different instants of simulation are considered. Figure 4d exhibits the most non-elliptical contact scenario and coincides with the location in which the wheel presents a concave surface.

4 Conclusions A method for the determination of a curved contact patch on the interaction between wheel and rail surfaces is presented in this work. This methodology considers that normal contact direction might vary along lateral direction of the contact area and determines the local penetration based on that assumption. A non-Hertzian method for the normal pressure evaluation is adapted to be applied in the curved contact. This contact model has been applied to a dynamic simulation and demonstrated to be effective in the determination of the contact patches and corresponding forces. Acknowledgments. The first author is supported by the Portuguese Foundation for Science and Technology (FCT) under grant PD/BD/114154/2016, MIT Portugal Program. This work has been supported by FCT with the reference project POCI-01-0145-FEDER-028424, by FEDER funds through the COMPETE 2020 - Programa Operacional Competitividade e Internacionalização. This work has been also supported by Portuguese Foundation for Science and Technology, under the national support to R&D units grant, with the reference project UIDB/04436/2020 and UIDP/04436/2020, as well as through IDMEC, under LAETA, project UIDB/50022/2020.

References 1. Bruni, S., Meijaard, J.P., Rill, G., Schwab, A.L.: State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches. Multibody Syst. Dyn. 49(1), 1–32 (2020). https://doi.org/10.1007/s11044-020-09735-z 2. Meymand, S.Z., Keylin, A., Ahmadian, M.: A survey of wheel-rail contact models for rail vehicles. Veh. Syst. Dyn. 54(3), 386–428 (2016) 3. Marques, F., Magalhães, H., Pombo, J., Ambrósio, J., Flores, P.: A three-dimensional approach for contact detection between realistic wheel and rail surfaces for improved railway dynamic analysis. Mech. Mach. Theory 149, 103825 (2020) 4. Magalhães, H., et al.: Implementation of a non-Hertzian contact model for railway dynamic application. Multibody Syst. Dyn. 48(1), 41–78 (2019). https://doi.org/10.1007/s11044-01909688-y 5. Vollebregt, E.: Detailed wheel/rail geometry processing with the conformal contact approach. Multibody Syst. Dyn. 52(2), 135–167 (2020). https://doi.org/10.1007/s11044-020-09762-w 6. Vollebregt, E., Segal, G.: Solving conformal wheel-rail rolling contact problems. Veh. Syst. Dyn. 52(S1), 455–468 (2014)

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7. Piotrowski, J., Kik, W.: A simplified model of wheel/rail contact mechanics for non-Hertzian problems and its application in rail vehicle dynamic simulations. Veh. Syst. Dyn. 46(1–2), 27–48 (2008) 8. Sun, Y., Zhai, W., Guo, Y.: A robust non-Hertzian contact method for wheel–rail normal contact analysis. Veh. Syst. Dyn. 56(12), 1899–1921 (2018) 9. Piotrowski, J., Liu, B., Bruni, S.: The Kalker book of tables for non-Hertzian contact of wheel and rail. Veh. Syst. Dyn. 55(6), 875–901 (2017) 10. Marques, F., et al.: On the generation of enhanced lookup tables for wheel-rail contact models. Wear, 434–435, 202993 (2019) 11. Marques, F.: Modeling complex contact mechanics in railway vehicles for dynamic reliability analysis and design. Ph.D. thesis, Universidade do Minho (2020)

On the Utilization of Simplified Methodologies for the Wheel-Rail Contact José Ferreira, Paulo Flores, and Filipe Marques(B) CMEMS-UMinho, Departamento de Engenharia Mecânica, Universidade do Minho, Campus de Azurém, 4804-533 Guimarães, Portugal [email protected], {pflores,fmarques}@dem.uminho.pt

Abstract. The utilization of multibody systems formulation allows to study the railway vehicle dynamics, as well study local damaging phenomena that occur in the wheel-rail interaction. However, the analysis of these phenomena requires long simulations to perform a significant prediction of their evolution. Having that in mind, the utilization of simplified approaches for the different steps of calculation of wheel-rail contact interaction are addressed, in particular, the utilization of planar or spatial approaches for the contact detection. Two parametric surfaces are used to describe both wheel and rail geometries, namely a revolution and an extruded body, respectively. Regarding the normal and tangential contact forces, an elastic approach is considered, therefore, these forces are treated as external forces acting on the multibody system and an Hertzian model with damping is employed for their evaluation. Finally, a trailer vehicle is considered as example of application to study different modeling strategies. Keywords: Wheel-rail contact · Railway dynamics · Surfaces definition · Multibody dynamics

1 Introduction The modeling of wheel-rail contact plays a preponderant role on the dynamic analysis of railway vehicles, since it represents the vehicle-track interaction where the developed forces are responsible for supporting and guiding the vehicle, as well as the traction and braking actions [1]. For an efficient multibody simulation of a railway vehicle, the methodology for assessing the wheel-rail contact forces has a critical importance, since it is often the bottleneck for achieving a lower simulation time [2]. This can be a critical issue when it is needed to perform long time simulations to predict the wheel-rail damaging phenomena, as wear or rolling contact fatigue [3]. There are several issues which might contribute for the improving of the numerical efficiency, namely, the contact detection methods [4], the integration algorithms, the degree of detail of the normal and creep force models [5], just to mention a few.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 114–121, 2022. https://doi.org/10.1007/978-3-030-88751-3_12

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2 Wheel and Rail Geometric Definition The mathematical description of the wheel and rail profiles consists of a two-dimensional representation which can be extended to the three-dimensional space by defining their external surfaces. A bi-dimensional approach for the description of the wheel and rail has the advantage of allowing fast and straightforward numerical models but poses significant limitations regarding an accurate representation of the contact scenarios, as, for instance, in the presence of angular misalignments. Typically, the wheel and rail can be represented by two parametric surfaces [6], as represented in Fig. 1. The surface of each rail can be achieved through the sweep of its cross-section along a given path, in turn, the wheel surface is characterized as a revolution of its cross-section about its own axis. Since both cross-sections are defined as function of their lateral coordinate, each of these of surfaces can be described by two independent parameters, i.e., a point on the rail surface is defined by the lateral (ur ) and longitudinal (sr ) coordinates, and a point located on the wheel surface is given by its lateral (uw ) and angular (sw ) coordinates. Based on the proposed surface parametrization, it is possible to write the equations that allow the determination of a location of given point on the wheel and on the rail as function of the surface parameters. Thus, following the representation of Fig. 1, the position of a point P on the rail surface is calculated as side rPside = rrside + rr,P

(1)

where rrside represents the location of the origin of the rail profile given as function of its side expresses the distance vector from the rail origin and point P, which arclength, and rr,P can be computed as  T side side f side = Aside (2) rr,P 0 ur,P r r,P side represents the ordinate of the rail profile, given as function of its lateral in which fr,P denotes the rail transformation matrix coordinate, as schematized in Fig. 2a, and Aside r and can defined as   (3) = trside nrside bside Aside r r

where trside , nrside and bside are the local tangent, normal and binormal vectors of the rail, r respectively, and also given as function of its arclength. In a similar manner, based on the representation of Fig. 1, the location of an arbitrary point Q on the wheel profile can be calculated as side side side = rws + hw + rw,Q rQ

(4)

side represents a where rws denotes location of the center of mass of the wheelset, hw local position vector from the center of mass of the wheelset until the origin of the wheel side expresses the distance vector profile defined to the left or right side accordingly, and rw,Q from the wheel origin and point Q, given by the following expression  T side side side rw,Q = Aws Aside (5) w,s 0 uw,Q fw,Q

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s

fwL

L w

ζ ws

r hLw

uwL

fwR

r ηws ≡ a ws

rL rw,Q

r hRw

uwR

ξ ws

rR rw,Q

r rws

rL rQ

swR

rR rQ

r bLr ≡ frL rL rL rr,P rL L rL rP nr ≡ ur tr

r bRr ≡ frR

rL rr

z

rR rR r rP r R R r,P rR nr ≡ ur tr rR rr

x

y L r

s

srR

Fig. 1. Parametrization of the wheel and rail surfaces side corresponds to the ordiin which Aws denotes the wheelset transformation matrix, fw,Q nate of the wheel profile which, similarly to the rail, is defined as function of its lateral coordinate, as represented in Fig. 2b, and Aside w,s is a transformation matrix that defines the rotation about the wheel axis, and it is computed differently for both sides as



⎤



⎤ ⎡ ⎡ L L R R 0 sin sw,Q 0 − sin sw,Q cos sw,Q − cos sw,Q ⎥ ⎥ ⎢ ⎢ ⎥ or AR = ⎢ ⎥ ALw,s = ⎢ 0 1 0 0 −1 0 w,s ⎣ ⎣



⎦



⎦ L L R R 0 cos sw,Q 0 cos sw,Q − sin sw,Q − sin sw,Q

(6) The definition of the normal and tangent vectors to the wheel and rail surfaces can be achieved using the derivative of the profiles functions as schematized in Fig. 2. Hence, the contact angle obtained from the profile data, which is typically given by a set of nodal points and, then, approximated by spline functions. These data can also be utilized to calculate the profile’s local curvature which it is utilized to estimate the contact stiffness.

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fw

fr

r nr,P

117

P

uw

γ r,P r tr,u,P

r tw,u,Q

ur

γ w,Q

rQ nw,Q (a)

(b)

Fig. 2. Representation of an arbitrary point on (a) rail and (b) wheel profiles

3 Contact Detection In this work, two methodologies for the identification of contact points are analyzed and compared. It must be noted that the definition of the surfaces’ geometry described in the previous section is of paramount importance for the development and implementation of contact detection methodologies, as well as to enhance their efficiency and accuracy. The first approach consists of a simplified methodology which performs the contact detection by the intersection between the 2D wheel and rail profiles, as illustrated in Fig. 3(a). Although the geometric parametrization of the contacting elements describes spatial surfaces, as shown in Sect. 2, this simplified approach searches the contact points through the comparison of the profiles position. First, the closest rail profile to the wheel is found, then the intersections between the rail profile at that position and the wheel transversal profile are identified and, finally, the maximum penetration in each contact patch is determined. Alternatively, a spatial contact detection methodology is utilized to compare the perks of the utilization of these different procedures. In this method, the contact point can be located outside of the plane in which their planar profiles lie, as schematized in Fig. 3(b). This approach follows the methodologies developed by Marques et al. [7] in which the maximum penetration point is found by analyzing independently the strips of the wheel surface.

4 Contact Force Model In this work, the normal contact force is estimated according to an Hertzian-based model which assumes elliptical contact area but considers a viscoelastic force-displacement behavior. Therefore, the normal force is computed as fn = Kδ n cd

(7)

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(a)

(b)

Fig. 3. Schematic representation of (a) bidimensional contact detection between wheel and rail profiles and (b) three-dimensional contact detection between wheel and rail surfaces

where K expresses the generalized contact stiffness, which depends on the local contact geometry and material properties [8], δ is the pseudo-penetration or indentation between contacting surfaces, n defines the degree of nonlinearity of the model and it is 1.5 for metallic contacts, and cd represents the damping factor calculated according to [9]. After the normal force evaluation, the size of the contact ellipse is determined using the local surfaces curvature. Regarding the tangential or creep forces, a Lookup Table established from a contact patch parametrization with five input variables and computed based on CONTACT software is employed here [10, 11]. Although this approach was developed for simple double elliptical contact area, which is generally a non-Hertzian shape, it is adapted here for the Hertzian case.

5 Multibody Systems Formulation The normal and tangential contact forces determined during the wheel and rail interaction are included in the dynamic equations of motion as external forces. Bearing that in mind, the equations of motion for a constrained multibody mechanical system can be formulated recurring to the Newton-Euler formulation with absolute or Cartesian coordinates together with the standard Lagrange multipliers technique [12]. In order to control the violation of the kinematic constraints, the Baumgarte stabilization technique is considered [13, 14], and the system of equations of motion can be written in the

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following form 

M DT D 0

    v˙ g = ˙ − β2 λ γ − 2α 

(8)

where M is the global mass or inertia matrix of the system, D denotes the Jacobian matrix of the constraints equations, v˙ represents the generalized accelerations vector, λ expresses the Lagrange multipliers vector, which represents the reaction forces and moments on the kinematic joints, g denotes the external generalized forces vector, in which the contact forces are included, γ is the commonly named right-hand side vector of ˙ denote the position and velocity constraints vectors, acceleration constraints,  and  respectively, and α and β are positive constants that represent the feedback control parameters for the velocity and position constraints violation.

6 Example of Application A multibody model of a single wheelset running on a tangent track is utilized as example of application. In the beginning of the simulation, the wheelset has a lateral displacement of 2 mm to promote the hunting motion, and its initial velocity is 20 m/s. These dynamic simulations are performed in MATLAB code for spatial multibody dynamics, the simulation time is 10 s and the ode45 algorithm is used for the time integration of the equations of motion. Figure 4 show the lateral motion of the wheelset using the 2D and 3D contact detection methods, which demonstrate equivalent results. The same interpretation can be made with the analysis of the contact point location on the wheel and rail profiles represented in Fig. 5. In terms of computational efficiency, the 2D detection method took around 90% of the computing time of 3D approach.

Fig. 4. Wheelset lateral position during the dynamic simulation

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Fig. 5. Location of the contact points on the (a) rail and (b) wheel profiles

7 Conclusions This work addresses the utilization of simplified methodologies on the evaluation of the forces developed on the wheel-rail contact interaction. For this purpose, two parametrized surfaces are employed for the definition of the contact geometries, and an Hertzian-based model is adopted for the normal and creep forces calculation. This allowed to study the impact of the modeling approaches on different running conditions. Further studies using different multibody models and running conditions will be addressed in the future to better understand the benefits and limitations of the utilization of simplified approaches on the wheel -rail contact modeling. Acknowledgments. This work has been supported by FCT with the reference project POCI-010145-FEDER-028424, by FEDER funds through the COMPETE 2020 - Programa Operacional Competi-tividade e Internacionalização. This work has been also supported by Portuguese Foundation for Science and Technology, under the national support to R&D units grant, with the reference project UIDB/04436/2020 and UIDP/04436/2020.

References 1. Meymand, S.Z., Keylin, A., Ahmadian, M.: A survey of wheel-rail contact models for rail vehicles. Veh. Syst. Dyn. 54(3), 386–428 (2016) 2. Bruni, S., Meijaard, J.P., Rill, G., Schwab, A.L.: State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches. Multibody Syst. Dyn. 49(1), 1–32 (2020). https://doi.org/10.1007/s11044-020-09735-z 3. Soleimani, H., Moavenian, M.: Tribological aspects of wheel-rail contact: a review of wear mechanisms and effective factors on rolling contact fatigue. Urban Rail Transit 3(4), 227–237 (2017) 4. Shabana, A.A., Tobaa, M., Sugiyama, H., Zaazaa, K.E.: On the computer formulations of the wheel/rail contact problem. Nonlinear Dyn. 40(2), 169–193 (2005) 5. Burgelman, N., Sichani, M.S., Enblom, R., Berg, M., Li, Z., Dollevoet, R.: Influence of wheelrail contact modelling on vehicle dynamic simulation. Veh. Syst. Dyn. 53(8), 1190–1203 (2015)

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6. Pombo, J., Ambrósio, J., Silva, M.: A new wheel-rail contact model for railway dynamics. Veh. Syst. Dyn. 45(2), 165–189 (2007) 7. Marques, F., Magalhães, H., Pombo, J., Ambrósio, J., Flores, P.: A three-dimensional approach for contact detection between realistic wheel and rail surfaces for improved railway dynamic analysis. Mech. Mach. Theory 149, 103825 (2020) 8. Goldsmith, W.: Impact – The Theory and Physical Behaviour of Colliding Solids. Edward Arnold Ltd., London (1960) 9. Ambrósio, J., Pombo, J.: A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems. Multibody Syst. Dyn. 42(3), 317–345 (2018). https://doi.org/10.1007/s11044-018-9613-z 10. Piotrowski, J., Liu, B., Bruni, S.: The Kalker book of tables for non-Hertzian contact of wheel and rail. Veh. Syst. Dyn. 55(6), 875–901 (2017) 11. Marques, F., et al.: On the generation of enhanced lookup tables for wheel-rail contact models. Wear 434–435, 202993 (2019) 12. Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Engewood Cliffes (1988) 13. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972) 14. Marques, F., Souto, A.P., Flores, P.: On the constraints violation in forward dynamics of multibody systems. Multibody Syst. Dyn. 39(4), 385–419 (2016). https://doi.org/10.1007/ s11044-016-9530-y

Optimization Tools Applied in the Design of a Hydraulic Hybrid Powertrain for Minimal Fuel Consumption T´arsis Prado Barbosa1(B) , Aline de Faria Lemos2 , Luiz Ot´ avio Ferreira Gon¸calves1 , Ricardo Poley Martins Ferreira3 , Leonardo Adolpho Rodrigues da Silva1 , and Juan Carlos Horta Guti´errez3 1

Department of Telecommunications and Mechatronics Engineering, Federal University of Sao Joao Del-Rei, MG 443, Km 7, Ouro Branco, MG 36420-000, Brazil {tarsisbarbosa,leonardo}@ufsj.edu.br 2 Department of Mechatronics, Optics and Mechanical Engineering Informatics, Budapest University of Technology and Economics, M˝ uegyetem rakpart 3, Budapest 1111, Hungary [email protected] 3 Department of Mechanics Engineering, Federal University of Minas Gerais, Avenida Presidente Antˆ onio Carlos, 6627 - Pampulha, 31270-901 Belo Horizonte, MG, Brazil [email protected]

Abstract. This work aims to design a hydraulic hybrid vehicle model with a series transmission architecture. The hydraulic hybrid transmission uses hydro-pneumatic accumulators to store energy instead of batteries or capacitors and, although less well known, can offer several advantages compared to hybrid-electric systems. In the implemented model, a pump is connected to an engine and supplies hydraulic power to a hydraulic motor, linked to the vehicle wheels axle. Regenerative braking is enabled with the hydraulic motor, working as a pump, and returning fluid power to the accumulator. Optimization tools were used to select the dimensions of the components and to determine the values of the control variables for activating the pump/accumulator. The model implemented had its consumption reduced by about 50 % in relation to the reference values, and the accumulator volume was reduced about 11 % concerning the calculated analytical values. Keywords: Fluid power · Hydraulic hybrid transmission Fuel-saving · Genetic algorithm · Optimization tools

1

·

Introduction

Automotive vehicles with higher energy efficiency and which cause lower environmental impacts are currently the two main demands of society concerning c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 122–130, 2022. https://doi.org/10.1007/978-3-030-88751-3_13

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the automotive industry. Although less well known, hydraulic hybrid powertrain solutions offer an efficient and high-performance alternative to hybrid-electric systems, which are already in the market [9]. In a hybrid hydraulic vehicle (HHV), at least one reversible hydrostatic motor/pump unit is used, acting as a motor to transmit torque and rotation to the wheels during the acceleration phases and, as a pump, perform the regeneration of kinetic energy during braking events. One of the most used architectures in hybrid vehicles is the series architecture (Fig. 1), in which the combustion engine drives a hydraulic pump that will provide the power to drive the motor/pump, which will be connected to the wheels differential axle. In this configuration, the combustion engine operates in regions of higher efficiency, since it will work decoupled from the wheels and its speed can be set independently [1].

Accumulator

Pump Internal Combustion Engine Motor/Pump

Fig. 1. Hybrid hydraulic vehicle - series architecture (adapted from [10]).

The storage system hole of batteries in HEV is occupied by a hydropneumatic accumulator in HHV. A bladder type accumulator consists of a steel hull with an elastomeric bladder inside that is filled with gas (normally nitrogen). The bladder separates the gas from the oil that externally fills the remaining volume of the tank. Therefore, the accumulator stores potential energy and return it to the circuit on demand [7]. Hydraulic accumulators are characterized by higher power density and lower energy density than electrochemical batteries. They can accept high frequencies, high charge/discharge rates and can recover more efficiently power during regenerative braking under city traffic conditions in a hybrid vehicle. However, the relatively low energy density of the hydraulic accumulators, which could lead to low vehicle autonomy, requires a carefully designed powertrain and energy management strategies to avoid increasing accumulator size and still obtain fuel savings [2]. In this context, this work proposes using the optimization tools from Matlab toolbox - pattern search and genetic algorithm (GA) to evaluate the possibilities of reducing the volume of the hydro-pneumatic accumulator and finding optimized operating pressures. In this case, the pattern search tool was used only to find a vector of initial values to feed the genetic algorithm.

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Methods Vehicle Model

The model of the hydraulic hybrid vehicle was implemented in the present work in the Simulink environment and is shown in Fig. 2. The connections between the hydraulic components, which compose the transmission system, are identified with the color beige. The black lines interconnect blocks related to vehicle control, and may contain signals from sensors (pressure, flow rate) and control commands (control of the pump volumetric displacement, reference speed, etc.).

Fig. 2. Hydraulic hibrid vehicle model implemented on Simulink.

The vehicle was represented as having a total mass M , an effective frontal area given by the product of the drag coefficient and the frontal area (cd · Af ), and dynamic radius of the wheel rw . The model considers movement resistance forces related to aerodynamic drag and rolling resistance (product of the rolling resistance coefficient fr and the vehicle weight −M · g) and neglects the slope of the track and the existence of curves. Thus, by Newton’s second law, the torque (T ) that acts on the vehicle’s axis is given by:   v2 T = rw · M v˙ + M gfr + cd Af ρ (1) 2 where ρ is the air density and v is the vehicle speed at the time considered [7]. The movement resistance forces in Eq. (1) were inserted in the model of Fig. 2 in the block called Road Loads. The characteristics of the vehicle designed in this work follow the same values as those used by [10] and by [11], in order to compare both studies. The vehicle simulation was performed by using the EPA UDDS driving cycle: distance of 12.07 km and total time of 1325 s - about 22 min [4]. The parameters considered in the vehicle model are shown in Table 1.

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Table 1. Parameters used in the HHV computational model Parameter

Value

Total mass (M )

1325 kg

Front area (Af )

2.16 m2

Drag coefficient (cd )

0.26

Wheel radius (rw )

0.3 m

Rolling resistance coefficient (fr )

0.011

Air density (ρ)

1.20 kg/m3

Maximum speed - UDDS cycle (v) 90 km/h Maximum accumulator pressure

35 MPa

Accumulator volume (Vacul )

30 L

In Eq. (1), considering the vehicle starting from rest and neglecting the resistance forces, the minimum torque required to move the car would be 596.25 Nm. Based on this value, the theoretical lower volumetric displacement (D) of the motor/pump can be found after setting the minimum pressure value (Pmin ) of the hydraulic circuit. Stelson et al. [10] considered the minimum working pressure of the hydraulic circuit equal to 60% of the maximum working pressure of the hydro-pneumatic accumulator (Table 1), therefore, equal to 21 MPa. Thus, the calculation of the volumetric displacement of the motor/pump can be performed as follows: D=

2π · T Pmin

(2)

Therefore, the calculated value of the volumetric displacement was 1.8 ·10−4 [m ]. The hydraulic pump connected to the combustion engine was considered to have a constant flow rate. An ideal flow source was chosen for the simulation in such a manner that the insertion of the combustion engine to drive the pump was not necessary for this approach (see Fig. 2). As a simplification of the model, the differential gearbox was also not included in the transmission system. The calculations performed to obtain the flow rate of the pump (Q) connected to the combustion engine were made considering that the hydraulic power provided (Pmin · Q) should be equal to the mechanical power necessary to overcome the movement resistance forces. Then, the following relationship was found:   v2 Pmin · Q = M gfr + cd Af ρ ·v (3) 2 3

Considering the vehicle maximum speed in the urban driving cycle (UDDS) equal to 90 km/h (25 m/s), the pump should provide a flow rate of 5 · 10−4 m3 /s (0.5 liters per second ). To compensate losses due to leaks, viscous and mechanical friction, the flow was considered equal to 1 · 10−3 m3 /s, as adopted by [10].

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The maximum working pressure must be lower than the pressure of the accumulator. Stelson et al. [10] considered the accumulator volume equal to 0.03 m3 (30 l) and the maximum working pressure Pmax equal to 90% of the maximum accumulator pressure (Table 1). These parameters were replicated as initial values of the reference model and the pattern search function. The pre-charge pressure of the gas inside the accumulator was considered to be 0.9 · Pmin [5]. The variation of the volumetric displacement of the motor/pump, which produces the variation of the speed of the wheels, was done by a proportional control in a closed-loop using the UDDS driving cycle speed as reference. The logic blocks related to the controller are also shown in Fig. 2. The hydraulic pump was turned on and off based on the pressure value of the accumulator, similarly to a pressure switch (on-off control). When the pressure of the accumulator was lower than the stipulated minimum pressure, the pump started operating providing a constant flow, remaining on until the stipulated maximum pressure was reached. Based on the average power consumed in the cycle, fuel consumption can be estimated to meet demand based on the specific fuel consumption map value from the used internal combustion engine [7]. In the work of [10] and [11], the used combustion engine was a diesel engine with a power of 13.7 kW at an angular speed of 2200 RP M and with an efficiency of 32.1%. The specific fuel consumption, obtained based on these data and used in this work, was 210 g/kWh. The values returned by the simulation for the optimization routine were, firstly, the absolute value of the difference between the speed achieved by the vehicle and the reference speed, and secondly the fuel consumption in the driving cycle. 2.2

Optimization Tools and Program Routine

The structure of the implemented optimization problem was divided into four hierarchical levels, according to the diagram shown in Fig. 3. The first level consists of a .m file that initializes all HHV parameters and defines the starting values for the variables (u) to be optimized: maximum pressures, accumulator volume, and angular speed of the motor/pump. Subsequently, the parameters of the optimization tools in the Matlab toolbox were configured on the second level: upper and lower bounds for the problem-solution vector, maximum number of iterations, function tolerance, etc. Table 2 shows the initial values of the pattern search function and the adopted bound values. In the third level, the objective function F (u) was defined in a new .m file as the fuel consumption of the model in the UDDS cycle. At this level, a constraint was also defined, in which the absolute value of the difference between the reference speed and the model speed could not be greater than 1 m/s, similarly to the method adopted in the Advanced Vehicle Simulator software – Advisor [3,8].

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Fig. 3. Structure of the optimization problem. Table 2. Initial and boundary values used within the optimization tools Parameters

Initial value Lower bound Upper bound

Pmax [M P a] 31.5

20

33

Vacul [l]

30

10

50

ω [rad/s]

80

50

110

In order to obtain the consumed power, the model created in Simulink was called to perform the driving cycle with the updated values of the variables (u), this model being the fourth level. All variables in the .m files were defined as global variables, thus allowing the calculation of consumption by accessing the results obtained in Simulink. In the implemented routine, in order to maintain the minimum working pressure (Pmin ) coupled with the maximum pressure (Pmax ), the equation described by [6] was used, which relates to the kinetic energy that can be recovered in braking equal to the isothermal work of the accumulator, calculated based on the ratio of accumulator pressure, as follows: Pmin =



Pmax

 (4) M ·v 2 e Pmax ·Vacul In addition, the increase in the volume of the accumulator represents an increase in vehicle mass. This increase was equated based on the density of the hydraulic oil used in the simulation (ISO VG 46, ρ = 855.69 kg/m3 ). Also, one consideration adopted was that the initial mass of a full 30 L accumulator had already been counted in the vehicle’s initial mass (1325 kg). Therefore, the increase in mass with the increase in volume was defined as:

Mk+1 = Mk + ρ · (Vacul − 0.03)

(5)

As an initial condition for optimization using GA, the best values obtained with the pattern search tool were used, as will be shown in the next section. Table 3 shows the parameters of the GA implemented in this work.

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Figure 4 shows the UDDS driving cycle with the reference speed and the speed performed by the hydraulic hybrid vehicle. The reference consumption value obtained in this situation was 7.40 l/100 km (autonomy of 13.51 km/l), which was the same value obtained in the HHV model implemented by [10]. Moreover, the proportional speed control operated satisfactorily, since the error between the reference speed value and the vehicle model speed was low. Table 3. Parameters of the implemented genetic algorithm Parameters

Values

Population size

20

Number of generations

50

Crossover probability (%) 80 Mutation function

Adaptive

Elitism’s individuals

1

Minimum tolerance

1e−6

Stopping criteria

Minimum tolerance

(a)

(b)

Fig. 4. a) Comparison between speed reference and the hybrid vehicle speed in the UDDS cycle. b) Comparison between the flow rate provided by the hydraulic pump before and after optimization.

After 197 iterations, the values obtained by the pattern search tool were 26.00 MPa and 13.18 MPa for the maximum and minimum pressures, respectively, and the accumulator volume was 35.54 l. With these new values, it was already possible to obtain a fuel consumption of 5.34 l/100 km, a reduction of 27.8% to the reference value.

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The values obtained by the pattern search tool were used as initial input values for the genetic algorithm, as mentioned before. In addition, the angular speed of the motor/pump was also used as a variable to be optimized using the GA tool, since this parameter is related to the fluid flow, therefore, it can influence the power consumed in the execution of the selected trajectory. The GA solution process was stopped after completing a total of 34 generations, once the minimum tolerance value stipulated in the function was reached (see Table 3). The lowest fuel consumption value found at this step was 3.60 l/100 km, which corresponds to a reduction of approximately 51.4% to the reference consumption value. The accumulator volume found was 26.62 l, a reduction of 11.3% in volume and weight compared to the initial reference value. Regarding the dimensions of the optimized accumulator, a 26.62 l accumulator has a diameter of approximately 230 mm and 1.4 m length. This component could be installed in the vehicle trunk. The pressures also decreased: 21.62 MPa and 9.55 MPa for the maximum and minimum pressures, respectively. The angular speed (ω) found was 56.0 rad/s. Table 4 shows the comparison between the values of the control variables before and after the optimization with GA. Figure 4b shows the comparison between the flow rate delivered by the pump before (blue line) and after the optimization process (red line). In Fig. 4b, a decrease in the pump operation demand was observed. Since this pump is driven by the output shaft of the combustion engine controlled by an on-off control, the reduction in the number of pump starts explains the reduction in fuel consumption obtained in the optimization. Table 4. Comparison between the values of the reference parameters [10] with the values obtained from the genetic algorithm method; and comparison between the fuel consumption of the implemented series HHV and the work of [10]

The comparison of the fuel consumption values found with the values obtained by [10] is also shown in Table 4. The fuel consumption reduction was very similar to the one obtained by these authors in the UDDS cycle applying different optimization methods. The tools applied proved to be useful for new explorations in the design of hybrid powertrain.

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Conclusion

In this work, a hydraulic hybrid vehicle with a series transmission architecture was modeled and simulated in an urban driving cycle (UDDS). The implemented model and the optimization tools applied allowed to find conditions to reduce the operating time of the hydraulic pump coupled to the combustion engine and also the hydraulic components size, reducing the fuel consumption 50 % about the initial hydraulic hybrid vehicle considered as reference. Acknowledgements. The authors are grateful to PPGMEC- UFMG and NIPEM UFSJ for their support.

References 1. Achten, P., Vael, G., Sokar, M.I., Kohm¨ ascher, T.: Design and fuel economy of a series hydraulic hybrid vehicle. In: Proceedings of the JFPS international symposium on fluid power, vol. 2008, pp. 47–52. The Japan Fluid Power System Society (2008) 2. Barbosa, T.P., Eckert, J.J., Roso, V.R., Pujatti, F.J.P., da Silva, L.A.R., Guti´errez, J.C.H.: Fuel saving and lower pollutants emissions using an ethanol-fueled engine in a hydraulic hybrid passengers vehicle. Energy 121361 (2021) 3. Eckert, J.J., Corrˆea, F.C., Santiciolli, F.M., Costa, E.D.S., Dion´ısio, H.J., Dedini, F.G.: Vehicle gear shifting strategy optimization with respect to performance and fuel consumption. Mech. Based Design Struct. Mach. 44(1–2), 123–136 (2016) 4. EPA: Enviroment Protection Agency – Urban Dynanometer Driving Schedule, 28 November 2017. https://www.epa.gov/emission-standards-reference-guide/ epa-urban-dynamometer-driving-schedule-udds 5. Hydac: Hydraulic Accumulators Catalog, 01 December 2017. https://m.hydac. com/fileadmin/pdb/pdf/PRO0000000000000000000003000000051.pdf 6. Ibrahim, M.S.A.: Investigation of hydraulic transmissions for passenger cars. Shaker (2011) 7. Guzzella, L., Sciarretta, A.: Vehicle Propulsion Systems, Introduction to Modeling and Optimization. Springer, Berlin (2013). https://doi.org/10.1007/978-3-64235913-2 8. Mashadi, B., Amiri-Rad, Y., Afkar, A., Mahmoodi-Kaleybar, M.: Simulation of automobile fuel consumption and emissions for various driver’s manual shifting habits. J. Cent. South Univ. 21(3), 1058–1066 (2014) 9. Rydberg, K.E.: Energy efficient hydraulic hybrid drives. In: 11th Scandinavian International Conference on Fluid Power, SICFP 2009, 2-4 June 2009, Link¨ oping, Sweden (2009) 10. Stelson, K.A., Meyer, J.J., Alleyne, A.G., Hencey, B.: Optimization of a passenger hydraulic hybrid vehicle to improve fuel economy. In: Proceedings of the JFPS International Symposium on Fluid Power, vol. 2008, pp. 143–148. The Japan Fluid Power System Society (2008) 11. Van de Ven, J.D., Olson, M.W., Li, P.Y.: Development of a hydro-mechanical hydraulic hybrid drive train with independent wheel torque control for an urban passenger vehicle. In: Proceedings of the National Conference on Fluid Power, vol. 51, p. 503. Citeseer (2008)

Determination of the Effect of Sloshing on the Railcar-Track Dynamic Behavior Juan Carlos Jauregui-Correa1(B) , Frank Otremba2 , Jose A. Romero-Navarrete2 , and Gerardo Hurtado-Hurtado1 1 Universidad Autonoma de Queretaro, 76010 Querétaro, Mexico

[email protected] 2 Federal Institute for Materials Research and Testing (BAM), Berlin, Germany

Abstract. This paper presents the study of the impact caused by a liquid cargo on a railway infrastructure. The dynamic behavior of a tank car corresponds to a multibody dynamic system with several degrees of freedom. This study’s data were obtained from a scale experimental fixture consisting of a track and a railcar with a tank. The track was instrumented with strain gauges and the railcar with accelerometers. The data showed non-periodic and periodic terms; therefore, the results were analyzed with the Empirical Mode Decomposition method (EMD). It was found that the EMD identified the signal components that were related to the sloshing. These components represent the mode shapes of the original signal. The location of the sloshing in the track was found applying spectrograms to the accelerometer data. This paper’s experimental outputs suggest that the sloshing effect is detectable at the track and in the vehicle dynamics. Keywords: Sloshing effect · Empirical Mode Decomposition · Wheel/track interaction · Multibody dynamics

1 Introduction The dynamic interaction between the track and the train has many safety implications, and it determines the life of rails, wheels, and train components [1, 2]. It also determines the passenger’s comfort and safety, as well as the cargo’s integrity. It is a multibody problem since the train has several cars connected through elastic elements, and each car has different bodies interconnect through elastic elements. All these bodies are linked to the rail at the contact point, and this connection has a complex physical representation that determines the dynamic interaction. The dynamic interaction study requires better analytical models and experimental methods that explain anomalous interactions between the track geometry and the car suspensions. One of the critical vehicles representing a complex dynamic behavior is liquid cargo since the liquid has a dynamic behavior that involves the premature damage of railways and derailment in turning infrastructures [1, 2]. There are few publications about the dynamic behavior of liquid cargos and rail dynamics up to the author’s knowledge. Romero and Otremba [3] reported the effects of sloshing on the force magnitude © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 131–140, 2022. https://doi.org/10.1007/978-3-030-88751-3_14

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on vehicle suspension. There are many experimental and analytical publications about the wheel/track interaction. Fermer and Nielsen [4] presented an experimental work to determine the interaction between a railcar and the track; they estimated the rail parameters from a frequency analysis, and they presented a full multi-body dynamic model. Different publications contribute to theoretical models [5–8]. From a multibody dynamic point of view, it is essential to understand the correlation between the force and the acceleration data sets. With this correlation, it would be possible to estimate the wheel forces on the track on the basis of the acceleration data of the railcar. Davis and Sanayei [9] characterized the effect of vehicles passing over a metallic bridge. They estimated the relationship between forces and accelerations in the frequency domain using a similar approach as [10]. Other researchers have applied the Hilbert transform [11] or the Empirical Mode Decomposition technique to identify the features of the track’s vibration measurements [12]. This paper presents the analysis of the interaction between a liquid cargo and the track. The analysis data came from a scale-down experimental facility [13], the forces on the track were recorded with strain gauges, and the railcar acceleration was recorded with a tri-axial accelerometer and three gyroscopes. As the frequency-domain analysis is valid only for periodic functions, the data were decomposed using the Empirical Mode Decomposition technique. The correlation between the force data and the acceleration data was determined at those modes with similar behavior. Each “mode” presented a characteristic frequency located along the railcar trajectory, with spectrograms produced with the Continuous Wavelet Transform.

2 Experimental Test Facility The scaled down experimental facility consisted of a track, a railcar with a tank, and the instrumentation. 2.1 The Track The track (Fig. 1) has a rectilinear accelerating entrance, a 180° curve, and a rectilinear decelerating ending. The track is instrumented with strain gauges along the curve, and a railcar runs freely on the track. A flat straight surface supports the track, and the rail car moves by the effect of the surface’s inclination. The structure that supports the surface is sufficiently rigid to avoid perturbations on the railcar. The track has an adjustable mechanism that modifies its slope, to further control the speed of the vehicle. The gravity pushes the vehicle downwards, enters the curve, and goes upwards until it stops; then, it goes backwards.

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2.2 The Rail Car The railcar (Fig. 2) was instrumented with an optical sensor for measuring its speed; a tri-axial accelerometer for measuring the accelerations in the orthogonal directions; and three gyroscopes for measuring the rotations of the tank. The vehicle’s design has a scale factor of 1/10 [13], and it has a spring suspension between the bogies and the platform. The rotation between the bogies and the platform is friction-free through a roller bearing, and this design eliminates the yaw stiffness. The liquid cargo consisted of a 110 mm diameter glass tank.

Fig. 1. Sketch representing the track and the support structure

Fig. 2. Description of the railcar with a tank

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2.3 Instrumentation The track has a set of strain gauges at different positions along the curve (Fig. 3). They start at the beginning of the curve, following the rail car’s downward direction, and the last one covers the maximum trajectory before the railcar returns backward. The strain gauges have a resistance of 120 Ohms connected to a one-quarter-of-a-bridge configuration. A data-acquisition system records the strain values and stores the data in ASCII files for further analysis. The velocity marks are located along the straight section (forward direction), with a separation of 200 mm between each other. The sample rate was set a 2400 Hz, and the data were recorded into de database. Figure 3 describes the strain gauge location, the velocity marks, and the junction between two sections of the track (gap). This point creates an impact on the vehicle that is only recorded with the accelerometers. The accelerometers box contains three orthogonal accelerometers and three gyroscopes, all of them are MEM’s, while the data were stored in a portable memory during the entire test. A light battery powers the system for more than 24 h. and the data are recorded continuously at a sample rate of 1594 Hz. For synchronizing the two devices’ measurements, a trigger set an impulse on the track, and both systems identified the beginning of the data acquisition period. The tank has camera that recorded the fluid motion during the test. The natural frequency of the fluid was estimated from these videos.

(a)

(bb)

Fig. 3. (a) Strain gauge instrumentation and velocity marks location. (b) Direction of motion

2.4 Test Procedure The rail car starts moving from the highest position; it travels about 2 m before reaching the curve, continues along the curve, and it stops after passing the location of the strain gauge, Channel 8. Then, it returns to the lower part of the curve and rests at the strain gauge Channel 2 location. The sloshing occurred when the railcar is turning, stops, and returns, and the motion sequence when the sloshing was higher started at strain gauge Channel 3, continued until the rail car stopped at strain gauge Channel 8, and went backwards passing over the strain gauge Channel 3. The glass tank showed the water movement, and a video allowed the estimation of sloshing frequencies. The natural frequencies of the railcar were determined through an impact test, when the railcar was on the track, and the vehicle response was recorded with the accelerometers.

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3 Experimental Results Table 1 includes the principal frequencies obtained with the impact test. When the railcar runs, the excitation forces derive from i) the impulse at the gap; ii) the uncontrolled alignment variations in the track; and iii) the liquid cargo sloshing. The sloshing effect was greater at the end of the railcar trajectory, when the vehicle started to move backwards.

Fig. 4. Strain gauge measurements when sloshing was higher.

The data used for this analysis corresponds to the backward movement. It started when the railcar touches the strain gauge at Channel 3, passed over Channel 2, and rests on Channels 2, 3, 4, and 5. When the rail car returns to Channel 5, the water motion was minimum and the sloshing was negligible, the maximum amplitude occurred from Channel 3 to Channel 8 and backwards. Figure 4 shows the measurements during this period. The events on this period are recorded from the instant when the railcar hits Channel 4, stops at Channel 8 (this is the last sensor in the trajectory), and returns to Channel 3. The low amplitudes at Channel 8 are due to the railcar’s low speed. Table 1. Natural frequencies, impact test. Direction Lateral (x) Longitudinal (y) Vertical (z)

First [Hz]

Second [Hz]

Third [Hz]

26.4

176.0

248.0

157.0

260.0

480.0

16.2

54.7

340.0

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The acceleration data were recorded simultaneously as the strain gauge data; the recording data comprised the entire railcar displacement. Figures 5, 6, and 7 show the acceleration data for the three orthogonal directions (Lateral (x), Longitudinal (y), and Vertical (z)).

4 Data Analysis and Discussion Since the time-series have non-periodic terms related to the vehicle dynamics, it is impossible to apply the spectrum analysis. Therefore, it was decided to separate the non-periodic terms from the periodic terms. This situation leads to the application of the Empirical Mode Decomposition (EMD). Although it has some limitations (as define by [14]), the EMD is an analysis method that adapts the time-space domain to process non-stationary or non-linear time series. The procedure consists of separating the time series into “modes” that have specific characteristics. These modes are computed with the Intrinsic Mode Functions using Hilbert’s transform. In the end, the addition of all the modes recovers the original function.

Fig. 5. Vibrations along the lateral direction (x). (in g’s)

Fig. 6. Vibrations along the longitudinal direction (y). (in g’s)

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The accelerometer in the vertical direction has a compensation value of -1 that corresponds to gravity.

Fig. 7. Vibrations along the vertical direction (z). (in g’s)

The following figures show the results of separating the non-periodic and periodic terms. The term “non-periodic modes” is the addition of those modes without a welldefined periodic function, and the other term is the addition of the remaining modes. Depending on the “shape” of the signal, the EMD produced between 10 to 12 modes. The original signal can be reconstructed by adding all the intrinsic modes. The modes were organized into modes with non-periodic time series and modes with periodic time series. Figure 8 shows the periodic and non-periodic modes of the strain gauge signal (Channel 3). Similarly, Figs. 9, 10 and 11 show the non-periodic and periodic terms of the acceleration data. The non-periodic modes in the x and y direction show how the railcar reduced its speed and then stops. The movement along the z-direction continuously oscillates. The sloshing effect is noticeable in the periodic modes, and the water motion modifies the entire dynamic behavior. The bottom part of Fig. 10 shows the periodic

Fig. 8. Non-periodic and periodic modes. Strain gauge Channel 3.

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Fig. 9. Non-periodic and periodic modes. Lateral direction (x). (in g’s)

modes along the direction of the railcar’s movement. Around t = 1, there is a sudden change in amplitude; this behavior change corresponds to the instant when the water started sloshing in the tank. Table 2 shows the characteristic frequency of each mode. The table is limited to those frequencies below the natural frequency (Table 1) for each direction. The highlighted values correspond to the equivalent sloshing frequency.

Fig. 10. Non-periodic and periodic modes. Longitudinal direction (y). (in g’s)

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Fig. 11. Non-periodic and periodic modes. Vertical direction (z). (in g’s)

The vertical direction shows the sloshing vibration at the first mode. The force signal shows the sloshing at the third mode, whereas the lateral direction and the longitudinal direction show the sloshing vibration at modes 2 and 3. Table 2. Mode frequencies [Hz] Mode

1

2

3

4

5

Lateral (x)

0.44 0.70 3.79 6.22 14.9

6 28.5

Longitudinal (y) 0.22 0.24 0.64 1.82

4.88

9.01

Vertical (z)

0.60 1.26 2.26 7.32

7.50 15.5

Force

0.22 0.46 0.95 3.98

8.77 28.0

5 Conclusions Transporting liquid cargo has a significant impact on the interface wheel-track interaction dynamics. The liquid motion, sloshing, affects the load on the track and the railcar’s dynamic behavior. The experimental work presented in this paper showed that the sloshing effect is detectable at the track level and in the vehicle dynamics behavior. Nevertheless, the characterization of such an effect requires combining different analysis techniques, being the Empirical Mode Decomposition (EMD) a fundamental tool for identifying the sloshing effect. It was found that sloshing occurred when significant changes occurred in the railcar dynamics, particularly when the railcar entered a curve

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and reduced its travel speed. A railcar with a tank is a clear example of a multibody dynamic system with a nonlinear behavior. The slosh motion has a low-frequency response in several directions, and EMD analysis identified those waveforms corresponding to the sloshing frequencies, and this method was the basis for locating the position on the track. The location was found using time-frequency maps constructed at each mode. Further work should involve the determination of the transfer function between the acceleration measurements and the force measurements. This transfer function will help to predict the sloshing cargo effect on the track response from the railcar dynamics perspective.

References 1. F.R.A. FRA: Joseph C. Szabo prepared oral testimony. 10 Apr 2014. Testimony. U.S. Department of transportation. Federal Railroad Administration (2014). https://www.fra.dot.gov/ eLib/details/L05001 2. F. R. A. FRA: Rail moving America forward. Current research projects. In: 2019 TRB Annual Conference (2019). https://railroads.dot.gov/sites/fra.dot.gov/files/fra_net/18429 3. Romero Navarrete, J.A., Otremba, F.: Vehicle’s components damage potentials of liquid cargo. In: Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, vol. 235, no. 2–3, pp. 446–454 (2020) 4. Fermér, M., Nielsen, J.C.O.: Vertical interaction between train and track with soft and stiff railpads—full-scale experiments and theory. In: Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, vol. 209, no. 1, pp. 39–47 (1995). https:// doi.org/10.1243/PIME_PROC_1995_209_253_02 5. Dumitriu, M., Ghe¸ti, M.A.: Cross-correlation analysis of the vertical accelerations of railway vehicle bogie. Procedia Manuf. 32, 114–120 (2019) 6. Chen, Z., Fang, H.: An alternative solution of train-track dynamic interaction. Shock Vib. 2019, 1–14 (2019). https://doi.org/10.1155/2019/1859261 7. Chang, C., Ling, L., Han, Z., Wang, K., Zhai, W.: High-speed train-track-bridge dynamic interaction considering wheel-rail contact nonlinearity due to wheel hollow wear. Shock Vibr. 2019, 1–18 (2019). https://doi.org/10.1155/2019/5874678 8. Ciotlaus, M., Kollo, G., Marusceac, V., Orban, Z.: Rail-wheel interaction and its influence on rail and wheels wear. Procedia Manuf. 32, 895–900 (2019) 9. Davis, N.T., Sanayei, M.: Foundation identification using dynamic strain and acceleration measurements. Eng. Struct. 208, 109811 (2020) 10. De Santiago, O., San Andrés, L.: Field methods for identification of bearng support parameters—part i: identification from transient rotor dynamic response due to impacts. J. Eng. Gas Turb. Power 129(1), 205–212 (2007) 11. Malekjafarian, A., OBrien, E., Quirke, P., Bowe, C.: Railway track monitoring using train measurements: an experimental case study. Appl. Sci. 9(22) (2019) 12. Li, J., Shi, H.: Rail corrugation detection of high-speed railway using wheel dynamic responses. Shock Vib. 2019, 1–12 (2019). https://doi.org/10.1155/2019/2695647 13. Romero, J.A., Navarrete, F.O.: A testing facility to assess railway car infrastructure damage. A conceptual design. Int. J. Transp. Dev. Integr. 4(2), 142–151 (2020). https://doi.org/10. 2495/TDI-V4-N2-142-151 14. Singh, P., Joshi, S.D., Patney, R.K., Saha, K.: The Fourier decomposition method for nonlinear and non-stationary time series analysis. Proc. R. Soc. A: Math. Phys. Eng. Sci. 473(2199), 20160871 (2017). https://doi.org/10.1098/rspa.2016.0871

Mechatronic Systems for Energy Harvesting

A Three-Dimensional Piezoelectric Timoshenko Beam Model Including Torsion Warping Emmanuel Beltramo1(B) , Bruno A. Roccia2 , Mart´ın E. P´erez Segura1 , and Sergio Preidikman1 1

Instituto de Estudios Avanzados en Ingenier´ıa y Tecnolog´ıa (IDIT) - CONICET and Departamento de Estructuras, FCEFyN, UNC, C´ ordoba, Argentina {ebeltramo,mperezsegura}@unc.edu.ar, [email protected] 2 Grupo de Matem´ atica Aplicada, UNRC, Argentina and Instituto de Estudios Avanzados en Ingenier´ıa y Tecnolog´ıa (IDIT) - CONICET, C´ ordoba, Argentina

Abstract. In this article, the authors present a three-dimensional formulation of a Timoshenko beam element intended for energy harvesting. The beam cross-section is idealized as an arbitrary number of laminate piezoelectric transducers embedded within an elastic substrate. In addition, each transducer is connected to three different electronic components: a resistance, a capacitor, and an inductor. Unlike standard Timoshenko-like beam theories, the model presented here is enhanced by including warping displacements due to homogeneous torsion. The resulting governing equations are spatially discretized through the finite element method (FEM) and numerically integrated in the time domain R . The performance, advanusing the ODE solvers available in MATLAB tages and disadvantages of the proposed model are evaluated by performing a mesh convergence analysis and a shear locking test. Finally, a comparative study between the current model and an Euler-Bernoulli model, in terms of harvested output power, is presented for different beam lengths. Keywords: Energy harvesting · Timoshenko beam function · Finite element method

1

· Warping

Introduction

In last decade, the drastic reduction in size and energy requirements of electronic devices has promoted the concept of energy harvesting from environmental sources. Mechanical vibrations of structures like wings or bridges can be turned into electrical signals by means of transducers. This electric signal could be harnessed for powering a wide variety of small-sized and low-power devices such as: wireless sensor networks for data collection, micro electromechanical devices, and low maintenance actuators, among others. Regarding transduction mechanisms, piezoelectricity ranks among the top ones for transforming mechanical c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 143–150, 2022. https://doi.org/10.1007/978-3-030-88751-3_15

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strain energy into usable electrical energy. In addition, once the electrical signal is obtained by piezoelectric transducers, it may be used without post-processing; feature which is often stressed as one of its main advantages. Different structural approaches to evaluate the response of piezoelectric energy harvesters are available in the literature, namely: lumped-parameter models [2], analytical solutions for distributed-parameter models [4], assumed-modes approximations [3], and some models taking into account geometric/material non-linearities. In general, distributed-parameter models represent harvester devices as cantilever beams, bonded by one or two piezoelectric layers, and excited harmonically at their fundamental frequency. However, more recently, a number of finite element formulations have been published. The versatility of FEM-based models have made it possible to impulse the study of built-in harvesting structures [5]. In this article, the authors expose a brief description of the most important modeling aspects of a three-dimensional piezoelectric beam formulation. The Timoshenko beam element presented here is enhanced by the inclusion out-ofplane displacements due to torsion warping. Furthermore, a through-thickness electric field distribution is consistently integrated with Timoshenko’s theory [8]. The numerical stability and precision of the current FEM model is assessed by performing a mesh convergence analysis and a shear locking test (for two different sets of shape functions). Finally, a comparative study between the current model and an Euler-Bernoulli (EB) model is presented.

2

Model Description

The main aspects of the structural model are summarized next. As shown in Fig. 1, the cross-section consist of an elastic substrate and nt laminate piezoelectric transducers that can be embedded in or bonded to its exteriorsurfaces. n t +1 The cross-section A is decomposed in nt + 1 sub-domains Ai , i.e. A = i= 1 Ai , where A nt +1 is the cross-section of the elastic layer, and Ai for i ranging from 1 to nt denote the cross-section of the different piezoelectric layers. Furthermore, each transducer is covered by a pair of electrodes, on its top and bottom surfaces. These electrodes are assumed to be perfectly conductive and to have negligible mechanical properties. Regarding the material properties, the elastic layer is assumed to be homogeneous and isotropic. On the other hand, the piezoelectric layers are idealized as transversely isotropic materials, with their isotropic plane normal to the poling direction. A schematic representation of the k -th electrical circuit for serial and parallel connections is shown in Fig. 1b and 1c. The circuit is composed of a piezoelectric transducer, a resistance Rk , an inductance Lk , and a capacitance Ck . The electric coordinate utilized to represent the configuration space (C -space) of each circuit is connection-dependent. On this basis, the electric flux-linkage λk (t) and the electric charge qkE (t) are used as generalized electric coordinates for parallel and serial connections, respectively. As exposed in the literature, the through-thickness electric potential variation is expressed as the sum of a linear term and a quadratic term. The quadratic

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term, known as induced potential, is proportional to bending and shear strains, and it is often neglected in most works. Nevertheless, when the cross-section area of each transducer is comparable to that of the elastic substrate, the effect of the induced potential is significant and should not be omitted. Furthermore, the overall structure stiffness changes as a consequence of the coupling triggered by this term. To fulfill this requirement, a proper through-thickness electric field distribution consistent with Timoshenko’s beam theory should be derived from the electrostatic equilibrium equation. In this work, a three-dimensional extension of the 2D electric potential term proposed by Sulbhewar and Raveendranath [8] is considered.

Fig. 1. (a) Schematic representation of an arbitrary cross-section composed of an elastic layer and nt piezoelectric layers. (b) Parallel RLC circuit. (c) Serial RLC circuit.

In this effort, a Timoshenko beam element is assumed satisfying the following kinematic hypothesis: i ) the cross-section does not remain orthogonal to the beam axis (X1 ) after deformation; ii ) the cross-section does not suffer in-plane deformations, which implies that in plane shear strains and Poisson’s effect due to axial strains are neglected; iii ) the only out-of-plane deformations come from torsion warping; iv ) warping of the cross-sections remain constant along the beam (Saint-Venant torsion hypothesis); v ) the displacement of beam axis U0 (X1, t) is small compared to the beam length L; vi ) the rotations of the cross-section are small enough so that the rotation tensor Λ(X1, t) may be approximated to first  1, t), where I is the identity tensor and order, that is to say Λ(X1, t) ≈ I + Θ(X  Θ(X1, t) is a skew-symmetric tensor whose components may be interpreted as infinitesimal rotations; vii ) all higher-order strain terms in the Green-Lagrange strain tensor are neglected; viii ) the substrate and the piezoelectric layers are perfectly bonded, which means that the strain field is continuous; and, ix ) the reference frame located at each piezoelectric layer is obtained by a translation of the main reference frame {X1, X2, X3 }. Regarding the stress-strain constitutive relations, both materials are assumed to be linearly elastic and governed by the generalized Hooke’s law.

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The variational formulation of the continuum is obtained from Hamilton’s Principle for electromechanical systems. In this case, both the displacement of  1, t) depend continuously beam axis U0 (X1, t) and the skew-symmetric tensor Θ(X on space and time. On the other hand, the flux-linkage λ(t) and the electric charge q(t) only depend on time. In this regard, inertial and elastic properties of the beam are modeled using a distributed-parameter approach, while electrical modeling is based on a lumped-parameter formulation.

3

Finite Element Model

This section presents the FEM governing equations for the electromechanical model described above. Following the classic approach, the one-dimensional nel beam domain [0, L] is divided into nel sub-domains Ie such that [0, L] = e= 1 Ie and Ii ∩ I j =  for i  j and i, j ∈ {1, 2, ..., nel }. Here, each sub-domain Ie ⊂ [0, le ] denotes a typical element with length le > 0 and nel refers to the total number of elements. In this implementation, a two-node finite element with constant crosssection is adopted. Each node has six degrees of freedom, three translations and three rotations, which are arranged in a column array qe (t) = [q1 (t), ..., q12 (t)]Te . Within each sub-domain Ie , the continuous functions describing the beam axis displacement U0 (X1, t) and the cross-section rotations Θ(X1, t) can be approximated, at the element level, as U0e (ηe, t) = Ne (ηe )qe (t) and Θe (ηe, t) = Pe (ηe )qe (t) for ηe ∈ [0, le ], where Ne (ηe ) and Pe (η) are the so-called shape function matrices. In this work, we adopt two different sets of interpolation functions. The first set of functions (called S1 ) is obtained from the homogeneous general solution for a Timoshenko beam [6]. The second set (called S2 ) is obtained from a threedimensional extension of the linked interpolation procedure proposed by O˜ nate [7]. Finally, the system of electromechanical equations of motion for the e-th finite element can be expressed as follow:   np c ns c   p k s k p Ke − Ke qe (t) + k θ es qk (t) − k θ e λk (t) = fe (t), Me q e (t) + Ke + ns c  k=1 np c  k=1

k=1

[k θ es ]T qe (t) + p

k=1

1 E q (t) + Rk qkE (t) + Lk qkE (t) = 0, Cks k p

[k θ e ]T qe (t) + Ck λk (t) +

1  1 λk (t) + λk (t) = 0, Rk Lk

(1)

where Me is the local mass matrix, Ke is the local elastic stiffness matrix, fe (t) p is the local nodal force vector, k Kes (k Ke ) denotes the local stiffness matrix p induced by the k -th serial (parallel) circuit, k θ es (k θ e ) represents the k -th local electromechanical coupling vector associated with the serial (parallel) circuit, nsc (n pc ) indicates the number of serial (parallel) circuits, and the over-dot stands

Piezoelectric Beam Formulation

147 p

for differentiation with respect to time. It should be mentioned that Cks (Ck ) represents the equivalent capacitance of the k -th serial (parallel) circuit, which collects both the inherent piezoelectric material capacitance and the external capacitance. All matrices (vectors) introduced above belongs to R12×12 (R12 ).

4

Numerical Results

In this section, the authors present two simple case studies. In the first case, the previously introduced beam element is analyzed by conducting a mesh convergence study along with a shear locking test for both sets of shape functions (S1 and S2 ). In the second case, the output power for a typical cantilever energy harvester is estimated. The outcomes are compared against those obtained from a EB-based model. 4.1

Case I

This subsection aims to assess the performance of the beam element developed. For this purpose, a simply supported square cross-section beam is considered. The structural member is entirely made of an aluminum elastic substrate without piezoelectric layers, whose mechanical properties are presented by Beltramo et al. [1]. Figure 2 shows a beam schematic, where b denotes the cross-section dimension and L indicates the beam length.

Fig. 2. (a) Schematic of a beam discretized by nel finite elements and nn nodes. (b) Cross-section of the beam.

The first (1F) and the third (3F) beam vibration natural frequencies are adopted as parameters for both analysis (mesh convergence and shear locking test).

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Fig. 3. (a) Variation of 1F with nel for both sets of shape functions. (b) Variation of 3F with nel for both sets of shape functions. (c) 1F - Relative percentage difference between the current and an Euler-Bernoulli model. (d) 3F - Relative percentage difference between the current and an Euler-Bernoulli model.

Figures 3a and 3b present the evolution of the first and third natural frequencies as a function of the number of elements. For a coarse mesh, it can be observed that the results given by S1 are closer to the analytical frequencies than those given by S2 . However, as the number of elements increases, the frequencies obtained with the set S2 approach the analytical solution faster than those obtained with S1 . In consequence, the set S2 is more accurate than the set S1 for the same number of elements when a fine mesh (≈ nel > 16) is utilized. On the other hand, Figs. 3c and 3d shows the evolution of the relative difference between the current model and an EB-based model [1] for 1F and 3F. Results show that such difference is less than 1% when β > 15 for both frequencies. Clearly, as the parameter β increases, the relative difference between models decreases significantly for both sets of shape functions. Therefore, one can infer that both S1 and S2 can avoid the shear locking phenomenon.

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4.2

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Case II

In this subsection, the output power of a cantilever harvester device harmonically excited at its fundamental natural frequency (ω) is analyzed.

Fig. 4. (a) Cantilever harvester device with two pure resistive loads. (b) Cross-section of the beam (b = 10 mm). Both piezoelectric layers have a thickness t = 0.26 mm.

As it can be observed in Fig. 4, the cross-section of the beam is composed of a square elastic substrate bonded by two piezoelectric layers located on the upper and lower surfaces. As before, b denotes the width and the height of the elastic substrate, t is the piezoelectric layer thickness, and L indicates the beam length. For simplicity, two purely resistive electrical circuits are considered. As the Case I, the mechanical properties for both materials are adopted from Beltramo et al. [1]. Additionally, a proportional damping matrix is added such that the damping ratio ζ of the first two vibration modes is 0.1. To avoid overcrowded the figures, in this subsection only the results obtain using the set of shape functions S2 are presented.

Fig. 5. (a) Maximum output power as a function of β. (b) Relative percentage difference between the estimations of maximum output power obtained using Timoshenko and Euler-Bernoulli theories.

The maximum output power for low values of β shows an exponential growing for both the current model and EB-based model (see Fig. 5a). However, from a certain value of β the output power behavior becomes linear. Furthermore, a discrepancy between these models can be observed for low values of β (see

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Fig. 5b). Such a discrepancy can be attributed, at first, to the influence of shear deformations on the piezoelectric behavior. As it is well-known, as the beam slenderness decreases, beam theories based on Euler-Bernoulli hypothesis become inaccurate. This drawback can be overcome by resorting to more sophisticated theories, such as a Timoshenko formulation. Summarizing, it is convenient to employ a Timoshenko model rather than an EB-based model when β < 10.

5

Conclusions

In this work, a brief summary of the most important aspects of a threedimensional formulation of a piezoelectric Timoshenko beam model for energy harvesting is presented. Some important inferences can be deduced from the preceding sections. Both sets of shape functions showed excellent behavior for a wide range of β values, which means that they can successfully avoid shear locking phenomena. However, the set S1 exhibited a better behavior for coarse meshes, while the set S2 showed a faster convergence than S1 as the number of elements increases. Finally, as the beam slenderness decreases (β < 15) it is convenient to switch to a Timoshenko formulation to better estimate the power harvested by a piezoelectric beam. On the contrary, as β increases, the power calculated by using either Timoshenko’s theory or EB-based models get closer to each other (difference of 1% for β > 15). In this way, for β > 15, the effect of shear deformations (on the harvested electrical power) becomes negligible and, therefore, the use of EB-based models is recommended.

References 1. Beltramo, E., Balachandran, B., Preidikman, S.: Three-dimensional formulation of a strain-based geometrically nonlinear piezoelectric beam for energy harvesting. J. Intell. Mater. Syst. Struct. 1045389X20988792 (2020). https://doi.org/10.1177/ 1045389X20988792 2. Dutoit, N.E., Wardle, B.L., Kim, S.G.: Design considerations for mems-scale piezoelectric mechanical vibration energy harvesters. Integr. Ferroelectr. 71(1), 121–160 (2005) 3. Erturk, A.: Assumed-modes modeling of piezoelectric energy harvesters: EulerBernoulli, Rayleigh, and Timoshenko models with axial deformations. Comput. Struct. 106, 214–227 (2012) 4. Erturk, A., Inman, D.J.: A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J. Vibr. Acoust. 144(3), 15 (2008). Article ID 041002 5. Junior, C.D.M., Erturk, A., Inman, D.J.: An electromechanical finite element model for piezoelectric energy harvester plates. J. Sound Vib. 327(1–2), 9–25 (2009) 6. Luo, Y.: An efficient 3D Timoshenko beam element with consistent shape functions. Adv. Theor. Appl. Mech 1(3), 95–106 (2008) 7. O˜ nate, E.: Structural Analysis with the Finite Element Method. Linear Statics: Volume 2: Beams, Plates and Shells. Springer, Heidelberg (2013) 8. Sulbhewar, L.N., Raveendranath, P.: An accurate novel coupled field Timoshenko piezoelectric beam finite element with induced potential effects. Latin Am. J. Solids Struct. 11(9), 1628–1650 (2014)

On the Effect of Hardening/Softening Structural Non-linearities on an Array of Aerodynamically Coupled Piezoelectric Harvesters Bruno A. Roccia1,2(B) , Marcos L. Verstraete1 , Luis R. Ceballos1 , Grigorios Dimitriadis3 , and Sergio Preidikman2 1

2

3

Grupo de Matem´ atica Aplicada, Universidad Nacional de R´ıo Cuarto, Ruta Nacional 36, Km 601, R´ıo Cuarto, Argentina {broccia,mverstraete,lceballos}@ing.unrc.edu.ar CONICET, Consejo Nacional de Investigaciones Cient´ıficas y Tecnol´ ogicas, Dep. de Estructuras, F.C.E.F.yN., Universidad Nacional de C´ ordoba, C´ ordoba, Argentina [email protected] Department of Aerospace and Mechanical Engineering, University of Li`ege, All´ee de la D´ecouverte 9, 4000 Li`ege, Belgium [email protected]

Abstract. In this work, we study the non-linear dynamic response of two vertically-arranged aero-piezoelastic harvesters. The numerical framework consists of the following: i) an aerodynamic model based on the unsteady vortex-lattice method; ii) a three degree-of-freedom lumped-parameter model for each harvester; iii) an inter-model connection to exchange information between models at each time step; and iv ) a numerical scheme based on Hamming’s fourth-order predictor–corrector method to integrate all the governing equations. Particularly, the effect of nonlinear hardening/softening springs on the harvested output power is investigated. Among the results obtained, an interesting finding is that hardening springs yield larger LCO amplitudes and higher harvested power than softening springs. Keywords: Energy harvesting · Structural nonlinearities Piezoelectricity · Aeroelastic flutter

1

· UVLM ·

Introduction

Recent improvements in the development of small-sized, low-power, portable, and remote devices have promoted the concept of harvesting energy from environmental sources [1]. Particularly, those designs which extract energy from the wind (through aeroelastic phenomena) have become a key option for powering small electronic devices in remote and difficult-to-access areas (Antarctica, suspension bridges, skycrapers, etc.). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 151–158, 2022. https://doi.org/10.1007/978-3-030-88751-3_16

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Among the different aeroelastic phenomena, flutter has been extensively studied over several decades, as this instability has been a key for many catastrophic accidents in aeronautical and civil structures. However, scientists have envisaged the possibility of using this nonlinear phenomenon as a desirable means for electric energy generation. Since the mid-2000s, several works have been published on flutter-based energy harvesters (FEH) [9]. Although these works have provided new insights on the design of harvester devices, the risks and potential of energy harvesting from wing aeroelastic systems can be best evaluated by means of nonlinear analysis [2,3,6]. Actually, an aeroelastic system can exhibit a wide variety of nonlinear responses, such as: limit-cycle oscillations (LCOs) [10], bifurcations [4], internal resonances [11], and chaotic motions [5]. These possibilities make the analysis of FEH much more complex and expensive (from a computational point of view) than linear analysis. Despite the progress made, most works in the literature highlight the low amount of energy produced by aeroelastic harvesters; a fact that has driven different proposals to make this technology sustainable, such as the idea of using harvester arrangements to enhance the power output over a wide range of wind speeds [7,8,15]. Notwithstanding these pioneering studies, the integration of several harvesters as a possible technological solution involves a large number of challenges, among which, the output power dependence on the spatial distribution of harvesters and the effect of structural non-linearities remain poorly understood topics. This work is an extension of the study by Roccia et al. [15], whose main goal is to shed some light on the effect of structural non-linearities on the aeropiezoelastic behavior of harvester arrays. Although the piezoelastic model is similar to that used in [9,15], the 3-degree-of-freedom (3-DOF) lumped-parameter approach presented here is enhanced by including nonlinear springs. The loads coming from the flow field around each harvester are predicted using a twodimensional (2D) version of the unsteady vortex-lattice method (UVLM).

2

Numerical Framework for Aero-Piezoelastic Simulations

In this section, we present a brief description of the aero-piezoelastic framework used to study the dynamic and aerodynamic behavior of harvester arrays. The numerical tool is designed by following a co-simulation strategy, where the entire system is partitioned into two subsystems: i ) the piezoelastic model (called Simulator 1), and ii ) the aerodynamic model (called Simulator 2). Although these subsystems come from different modeling processes, the coupling procedure is strong because information can be bi-directionally exchanged, and the chosen time step advance is the same for both models. The numerical scheme used by Simulator 2 is well-known and can be found in the literature [14]. On the other hand, the numerical procedure adopted for Simulator 1 to solve the equations of motion (EoMs) of the harvester array is based on Hamming’s fourth-order predictor–corrector method.

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Fig. 1. Left: Array of two harvesters, Right: Scheme of a piezo-aeroelastic harvester.

2.1

Piezoelastic Model

The array of harvesters analyzed here consists of two vertically-separated aeropiezoelastic devices; each of them idealized as a two-dimensional airfoil (see Fig. 1). The elastic properties of each harvester are modeled through a lumpedparameter approach, where the torsional and bending stiffnesses are represented by nonlinear springs. Under these assumptions, the configuration space (or cspace) for the k-th harvester can be described by Ck = R2 × SO(2), where SO(2) is the special orthogonal group in two dimensions. On this basis, the T configuration coordinate vector qk (t) ∈ Ck is given by qk (t) = (h(t), θ(t), V (t))k , where h(t) is the plunging displacement, θ(t) is the pitching angle, and V (t) is the voltage generated. Due to the lack of constraint equations in the system, the number of degreesof-freedom for each piezoelastic airfoil is equal to the c-space dimension, i.e., nDOF = dim(Ck ) = 3. It follows that the DOF number for the entire system is NDOF = N nDOF , where N is the number of harvesters. In this work N = 2, and therefore NDOF = 6. It should be stressed that the piezoelectric materials are only activated by the plunge degree of freedom. The aero-piezoelastic equations of the system presented in Fig. 1 are obtained by modifying the linear aeroelastic equations of a typical section [12]. Following the same procedure as in prior studies [9,15], the EoMs for one aero-piezoelastic harvester are obtained in matrix form as,  λ         ¨ dh 0 h h˙ m + mf mxθ b cos θ V − l ¨ + 0 dθ ˙ IP mxθ b cos θ 0 θ θ k k k k       k    3 2 ˙ k k h La (t) cos θ 0 h mxθ b θ sin θ + + 1,h ± 2,h 3 = , (1) k2,θ θ k 0 k1,θ k θ k Ma (t) 0 k k   1 λ h˙ + C eq V˙ + V = 0, R k where m is the airfoil mass per length, mf accounts for the fixture mass per length, IP is the moment of inertia per length about the reference point (spring

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joints), b is the semichord length, l is the span length, xθ is the dimensionless chordwise offset of the reference point from the mass center (mc), dh (dθ ) is the structural damping coefficient in the plunge DOF (pitch DOF), k1,h (k1,θ ) is the linear stiffness per length in the plunge DOF (pitch DOF), k2,h (k2,θ ) is the nonlinear stiffness per length in the plunge DOF (pitch DOF), La is the aerodynamic lift per length, Ma is the aerodynamic pitching moment per length, R is the load resistance, C eq is the equivalent capacitance of the piezoceramic layers, λ is the electromechanical coupling term, and the overdot represents differentiation with respect to time. It should be noted that the spring non-linearities are of cubic order. Consequently, it is possible to account for a hardening/softening behavior according to the sign of the term which collects the stiffness k2,h and k2,θ . Finally, the EoMs for the complete dynamic system shown in Fig. 1 are obtained by assembling the EoMs for each harvester. Due to the uncoupled nature of the system, from a mechanical point of view, the global mass, damping and stiffness matrices are all in block diagonal form. The set of ODEs is only coupled through the aerodynamics forces, which are included on the right-hand side. 2.2

Aerodynamic Model

The main objective of the aerodynamic model is to predict the forces coming from the surrounding flow on the array of harvesters. On this basis, the authors have adopted a two-dimensional version of the well-known UVLM to compute the aerodynamic loads. A significant advantage of the UVLM is the desirable trade-off between a relatively high precision (in terms of capturing all possible aerodynamic interferences) and a moderate computational cost when compared to flow solvers based on computational fluid dynamics (CFD) techniques. Vortex methods are based on the spatial discretization of the continuous vortex sheets into straight elements of concentrated vorticity. As a consequence, the vorticity of each element has associated with it a single point vortex (denoted as vortex point, VP) whose circulation is such that the non-penetration condition is satisfied at the so-called control points (CPs) (see Fig. 2). The imposition of the no-penetration condition at each CP along with Kelvin’s circulation theorem leads to a set of linear algebraic equations for the harvester system, whose solution allows to obtain the circulations Γk associated with all of the aerodynamic elements making up the array of harvesters, as well as, all those vortices located at the trailing edges. Then, the aerodynamic loads acting on the airfoils are computed by the following three steps: i ) for each element, the pressure jump ΔCp at the control point is computed by integrating the unsteady Bernoulli equation [15]; ii ) the force on the j-th aerodynamic element is computed as the product of the pressure jump times the element area times the normal unit vector located at the j-th CP; and iii ) the resultant forces and moments are computed as the vector sum of the forces and their moments about a common point. For the k-th harvester, the external load vector, Fk = (La cos θ, Ma )Tk , is given by,

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Fig. 2. Discretization of bounded-vortex sheets and free-vortex sheets.

⎛ Fk =

ne

1 2 ⎝ ρ∞ V∞ − cos θk (ΔCp )j , 2 j=1

ne

⎞T (ΔCp )j xj ⎠ ,

(2)

j=1

where ρ∞ is the air density, V∞ is the magnitude of the free-stream velocity vector, xj is the distance from the j-th CP to the reference point (here the spring joint), and ne is the number of aerodynamic elements per harvester. Once the loads have been computed, each VP of the wakes is “convected” to its new position by means of a first-order method; for instance, the Euler method. For a detailed description of the UVLM, the reader is referred to [13–15].

3

Numerical Results

In this section, the authors present a postcritical analysis for two verticallyseparated harvesters obtained from the implementation in Fortran 90 of the proposed methodology. Since the most expensive part of the simulation process is associated with the aerodynamic simulator, specifically the wake convection, the code has been explicitly parallelized by using a domain decomposition strategy via an OpenMP* library. The simulation framework presented in this work has been verified and validated by comparing the response of an isolated harvester against the analytical and experimental data reported by Erturk et al. [9]. The experiment carried out by Erturk’s team consists of an airfoil connected to the ground through four steel beams, of which two have two PZT-5A piezoceramic patches attached close to the fixed ends in a symmetrical configuration. For their analytical model, they used a linear lumped-parameter two-dimensional approach along with Theodorsen’s unsteady thin airfoil theory for predicting the aerodynamic loads. The flutter speed VF obtained through numerical simulations, for 30 aerodynamic elements, is 9.451 m/s. Although it is slightly lower than the analytical value of 9.56 m/s reported by Erturk et al., the relative error is only 1.14%. It should be mentioned that the discretization used for the bound vorticity sheet comes from a convergence study carried out in advance, which shows that increasingly dense aerodynamic meshes allows this error to be substantially reduced [15]. On the other hand, the experimental VF reported by Erturk et al.

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is 9.3 m/s, which represents an excellent agreement with the current predicted value (relative error of 1.6%). For more details on the verification and validation process of the numerical framework, the reader is referred to [15]. 3.1

Case Study

In this subsection, we study the effect of nonlinear springs (hardening/softening) on an array of two harvesters, separated at d = b, operating in a postcritical region. Both piezoelectric and structural data used here are the same as those used in [9,15], which are: λ = 1.55 mN/V, C eq = 120 nF, m = 1.7799 Kg/m, mf = 2.8425 Kg/m, Ip = 7.06445 × 10−3 Kgm, b = 0.125 m, xθ = 0.260, l = 0.5 m, k1,h = 4.6808 × 103 N/m2 , k1,θ = 1.67540 N, dh = 9.6110 × 10−1 Ns/m2 , dθ = 1.32504 × 10−2 Ns, and R = 100 kΩ. In this first stage, we adopt the following values for the nonlinear spring stiffness: (k2,h , k2,θ ) = Csp X (k1,h , k1,θ ) for Csp = +/ − 1, +/ − 3, +/ − 5. It should be noted that the optimal resistive load R calculated for a single harvester is not necessarily the optimum value for the array of harvesters like the one proposed in this study. In Fig. 3a we show the Hopf bifurcation associated with the plunging motion for d/b = 1. As can be gathered, this bifurcation is supercritical, and hence, the LCOs occur above the flutter speed. For a hardening spring with Csp = 5, the plunge LCO amplitude initially increases, reaches a maximum, and then begins to decrease. This trend was found to be quite similar to the results considering linear stiffness springs reported by Roccia et al. [15]. Further increments of the wind speed beyond V = 8.5 m/s result in a substantial change in the nature of the plunge-LCO phase portrait (period 2 motion). As the nonlinear stiffness decreases, the shape-change point is shifted to the right. Although for softening springs (with Csp = 3 and Csp = 5) the appearance of period 2 motions seems to disappear, they are expected to arise for higher air velocities (larger than 9.00 m/s). In this work, the LCO amplitudes were determined by means of a standard least-squares fitting procedure for ellipses [15]. Similar behavior was found for the pitching response. Regarding the effect of hardening/softening springs on the aero-piezoelastic response, a clear increase in the LCO amplitudes and mean output power (Fig. 3b) can be observed as we move from Csp = −5 (softening) to Csp = +5 (hardening). In principle, this increase can be attributed to the following fact: the more rigid a structure is, the higher its frequency of oscillation. In consequence, its motion will be characterized by higher velocities. Oversimplifying the situation, mechanical power ultimately depends on the force and velocity, therefore it can be surmised that, in general, hardened spring-based harvester systems will increase energy harvesting. According to [15], the introduction of a second harvester into the system produces the undesired effect of decreasing the output power. However, for two harvesters separated by d/b = 1, the flutter onset speed decreases from 9.451 to 7.78 m/s (about 18% lower than the flutter speed for a single harvester). This fact highlights the tremendous advantage of generating energy, at low wind speeds, by means of an arrangement of harvesters. On this basis, the design of

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Fig. 3. (a) Plunging LCO amplitude versus free-stream speed for d/b = 1. A supercritical Hopf bifurcation occurs at 7.78 m/s, (b) Mean output power. Solid line with circular markers indicates Csp = 5, dotted line indicates Csp = 3, and solid line indicates Csp = 1.

self-configuring harvester arrays with hardening structural members will allow us to increase the harvested energy over a wide range of wind speeds.

4

Conclusions

In this work, a simulation framework for studying the nonlinear aero-piezoelastic behavior of an array of two vertically-separated harvesters has been presented. Some important preliminary conclusions can be drawn from the preceding sections. The use of hardening/softening springs was found to have an interesting influence on the mean output power. Specifically, when the spring stiffness is changed from Csp = −5 (softening) to Csp = +5 (hardening), both the LCO amplitudes and the harvested power show a significant increase. At first, this behavior is attributed to the high frequencies of oscillation that take place as a consequence of the stiffening of the structure. Although more investigations are needed, these findings suggest the strong possibility of using airfoil arrays with hardening features as harvester devices for energy generation over a wide range of wind speeds. Acknowledgements. The authors gratefully acknowledge the partial support received from the Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina.

References 1. Energy harvesting: What is energy harvesting (2014). http://www.energyharv esting.net/

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2. Abdelkefi, A., Ghommem, M., Nuhait, A., Hajj, M.: Nonlinear analysis and enhancement of wing-based piezoaeroelastic energy harvesters. J. Sound Vib. 333(1), 166–177 (2014) 3. Abdelkefi, A., Nayfeh, A.H., Hajj, M.: Modeling and analysis of piezoaeroelastic energy harvesters. Nonlinear Dyn. 67(2), 925–939 (2012). https://doi.org/10.1007/ s11071-011-0035-1 4. Abdelkefi, A., Vasconcellos, R., Marques, F.D., Hajj, M.R.: Bifurcation analysis of an aeroelastic system with concentrated nonlinearities. Nonlinear Dyn. 69(1), 57–70 (2012). https://doi.org/10.1007/s11071-011-0245-6 ´ Lau, F., Suleman, A.: A review on non-linear 5. Afonso, F., Vale, J., Oliveira, E., aeroelasticity of high aspect-ratio wings. Prog. Aerosp. Sci. 89, 40–57 (2017) 6. Bae, J.S., Inman, D.J.: Aeroelastic characteristics of linear and nonlinear piezoaeroelastic energy harvester. J. Intell. Mater. Syst. Struct. 25(4), 401–416 (2014) 7. Beltramo, E., P´erez Segura, M.E., Roccia, B.A., Valdez, M.F., Verstraete, M.L., Preidikman, S.: Constructive aerodynamic interference in a network of weakly coupled flutter-based energy harvesters. Aerospace 7(12), 167 (2020) 8. Bryant, M., Mahtani, R.L., Garcia, E.: Wake synergies enhance performance in aeroelastic vibration energy harvesting. J. Intell. Mater. Syst. Struct. 23(10), 1131– 1141 (2012) 9. Erturk, A., Vieira, W., De Marqui Jr, C., Inman, D.J.: On the energy harvesting potential of piezoaeroelastic systems. Appl. Phys. Lett. 96(18), 184103 (2010) 10. Eskandary, K., Dardel, M., Pashaei, M.H., Kani, A.M.: Effects of aeroelastic nonlinearity on flutter and limit cycle oscillations of high-aspect-ratio wings. In: Applied Mechanics and Materials, vol. 110, pp. 4297–4306. Trans Tech Publ (2012) 11. Gilliatt, H.C., Strganac, T.W., Kurdila, A.J.: An investigation of internal resonance in aeroelastic systems. Nonlinear Dyn. 31(1), 1–22 (2003). https://doi.org/ 10.1023/A:1022174909705 12. Hodges, D.H., Pierce, G.A.: Introduction to Structural Dynamics and Aeroelasticity, vol. 15. Cambridge University Press, Cambridge (2011) 13. Katz, J., Plotkin, A.: Low-Speed Aerodynamics, vol. 13. Cambridge University Press, Cambridge (2001) 14. Preidikman, S.: Numerical simulations of interactions among aerodynamics. Ph.D. dissertation, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA (1998) 15. Roccia, B., Verstraete, M., Ceballos, L., Balachandran, B., Preidikman, S.: Computational study on aerodynamically coupled piezoelectric harvesters. J. Intell. Mater. Syst. Struct. 31(13), 1578–1593 (2020)

Hybrid System Design for Energy Harvesting from Low-Amplitude Ocean Waves Gabriel Gutiérrez-Diaz1 , Arturo Solis-Santome2 and Christopher René Torres-SanMiguel2(B)

,

1 Instituto Politécnico Nacional, Escuela Superior de Ingeniería y Arquitectura,

07738 Mexico City, Mexico [email protected] 2 Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, SEPI Unidad Zacatenco, 07738 Mexico City, Mexico {asoliss,ctorress}@ipn.mx

Abstract. This paper describes a hybrid system design consisting of a marine nozzle and a wave energy converter to enhance energy harvesting from low-amplitude ocean waves. The main goal of the proposed design is to get energy from water sea waves. Laboratory experiments show the efficiency and effects for the energy conversion/harvesting system in the case of three sets of low-amplitude waves against the designed nozzle’s hybrid system with a mechanical wave converter to store the energy in a laboratory tank. The results show a satisfactory operation of the system managing low-amplitude sea waves to produce energy conversion. Keywords: Design · Energy sustainable systems · Energy wave converter · Hybrid marine nozzle system · Low -amplitude sea waves · Lab testbed

1 Introduction The global energy consumption in 2018 was 14,230 Mtoe/165,500 TWh/year [1], the potential energy in oceans is estimated at 80 000 TWh/year [2]. This resource is a challenge and alternative exploitation of ocean waves, tidal movements, gradient thermal or saline at sea [3, 4]. Compared with wind or solar energy, wave energy has more significant advantages due to its availability most of the year and the time of day [5, 6]. A wide variety of harvesting devices for wave energy had built (DCEU) [7]. These still have unfavourable aspects of comprehensive safety, reliability, economics, or maintenance, so there is no mainstream technology for harvesting energy from the sea. [8, 9]. Additionally, there is no consensus for the design of WECs, or else, there is a need to establish protocols for their design and testing in extreme weather conditions to achieve their survival [10]. A wave energy converter (WEC) is a concept used to crop power from ocean waves. As a consensus, WECs work on the principle of an activated body that oscillates relative to a fixed part of the device due to wave action. The WECs are classified into four categories: those that use the ocean waves’ kinetic energy, tidal induced currents, ocean currents, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 159–167, 2022. https://doi.org/10.1007/978-3-030-88751-3_17

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and thermic gradient or saline [11]. WECs, which harness ocean waves, categorize them on their operation principle, size, or how they interact with ocean waves [12]. According to this principle, there are three predominant design types: oscillating body (WAB), oscillating water column (OWC), and overtopping wave (OTD). Currently, WAB WECs include three categories: point extraction, attenuators, and terminators [13], which are classified according to the power take-off (PTO) system, which has the function of converting the kinetic energy of the waves into electrical energy [14]. Mexico has 11 122 km of coastline and is considered a country with high potential for energy production from marine sources, especially the area of California Gulf (Sea of Cortés) [15]. On the southwest coast (Pacific Ocean side), the wave potential is lower, in the order of 10 kW/m [16]. This work aims to design a wave energy harvesting mechanism using a hybrid system of a marine nozzle (MN) and a mechanical WEC. The work organization is as follows: In material and methods, it finds a series of steps leading to a hybrid system design. The results show equipment capable of harvesting energy from waves, and the conclusions show a rich discussion of the energy harvesting parameters evaluated through experimental tests.

2 Design Requirements and Experimental TESTBED Most WECs, are designed to work in deepwater areas since the wave’s energy decreases significantly to the coastline located at latitudes more significant than 30°, resulting in good efficiency in capturing energy. The minimum height wave used by WECs is designed greater than 2m, limiting for locations where the wave height is less than this value [17]. This paper uses the following methodological (Fig. 1) framework to design a hybrid system of harvesting wave energy.

Fig. 1. Block diagram of the design hybrid crop energy

2.1 Hybrid System; Marine Nozzle + Wave Energy Converter The Marine Nozzle (MN) walls were constructed with 22 plastic buckets fixed to the tank’s bottom by gravity for the initial stage. Measures cubes are 300 x 250 x 450 [mm],

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arranged in two rows of vertical walls that form a converging channel with a nozzle shaped, Fig. 2a. The WEC is implemented in the system as an oscillating body (the extraction point). Consider or interact with an incident wave to regulate low amplitude in the system hybrid MN-WEC prototype. The scale used for the wave was 1:10 in a wave tank of 7 m x 11 m. Marine nozzle causes an exchange of waves incident momentum to the walls, causing a flow energy zone more densely, enabling this energy to be harvested by WEC installed. The design criteria of the WECs (Fig. 2b) are based on some general rules [18], as well as design guidelines that consider economic aspects [19].

Fig. 2. Hybrid system arrangement; a) Marine Nozzle (MN), b) Wave Generator (WG)

The wave generator (WG) consists of a steel palette of 0.80 × 7 m, actuated by a transmission with a variable speed of 3 kW and is located at the beginning of the waves tank. The WG speed variation is carried on by finite steps, with regular wave spectra (Fig. 3).

WG = wave generator (flat paddle). θ = angle of oscillation of the blade = elevation above water level h = depth of wave tank =medium level water surface = phase velocity (wave speed)

Fig. 3. Wave generator

The operating was adjusted to reproduce 3 sets of regular wave spectra (Fig. 4). The mathematical model to quantify the incident energy (1) in the DCEU float considers the parameters of the height of the significant wave (H0 ), period (Te ) in addition to the density of the water (ρ°) and the typical gravitational acceleration (g) and for the

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1. Movable trowel generator paddle. 2. Wave train. 3. Symmetrical vertical walls (Nozzle) 4. Energy harvest area.

Fig. 4. MN-WEC location in the wave tank.

wave height above graphical water level (2). P=

ρg 2 2 H Te 64π o

(1)

ηG = ηmax cos(ωt − kx)

(2)

Where: ηG = wave elevation respect of z0 at any interesting point. ηmax = ultimate amplitude. ωt = phase of wave (where t it’s time). (x) = position of wave on + x direction. (k) = wave number. For regular incident waves, an energy analysis was considered. The application of the black box model was chosen to obtain an approximation of the Analysis of the phenomenon (Fig. 5).

F = float of the WEC P = vertical facing crest ho = crop area depth = elevation above water level =medium level water surface

Fig. 5. MN harvesting zone.

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An oscillating body makes the WEC design of point extraction activated by the swell. It consists of a 4-bar mechanism with one degree of freedom, which at the free end has a prismatic float installed to capture the wave energy, and at the other, it transmits the torque received to the PTO’s shaft. The PTO is a multiplier transmission box that receives the torque from the oscillating arm and handover it to an electric generator’s shaft. It has a pair of unidirectional clutches to transfer power in a single direction of rotation to the generator, taking advantage of the arm’s oscillation cycle with float (Fig. 6).

Fig. 6. WEC block diagram

The following table lists the components and specifications of the WEC device (Table 1). Table 1. WEC components Float

A prismatic body made of 3 mm thick acrylic sheet with external dimensions 40 × 35 × 14 cm

Swingarm

Four bars mechanism articulated made with tubular aluminum 10 mm diameter with longitudes L1 = L2 = 10 cm and L3 = L4 = 38 cm

Arm swing axis and transmission input Steel of 8 mm diameter × 1 4 0 mm long, supported on bushings Nylamid self-lubricate Transmission gear train

The number of teeth (Z) of multiplier transmission gears are: Z1 = 50, Z2 = 10, Z3 = 38 and Z4 = 11, made with Nylamid

Transmission output shaft

Steel 8mm diameter ground x length 158mm, supported on self-lubricating Nylamid bushings

Coupling with generator shaft

Flexible silicone. Dimensions D = 18 mm × d = 8 mm × L = 22 mm (continued)

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Electric generator

24 VDC, 2 A

The frame (chassis)

Steel sheet caliber 18. General measures 110 mm × 120 mm × 8 mm of perimeter fold to increase rigidity

Assembly system

Galvanized steel screws d = 5/32 inch x L = 3/8 inch thread 32 unf

Clutches

Double cam clutch with two gear sets output Z3 = 38 and Z4 = 11

Flywheel

It is mounted on the shaft of the electric generator to stabilize shaft rotation

3 Experimental Results The proposed mechanisms were adjusted to perform three series of 5 tests with a duration of 7 min each, varying the wave frequency and the angle of the MN faces, to reproduce 7 cm high waves at a scale of 1:10 at the nozzle inlet. The (h) and (ho ) strains are 0.50 and 0.35 m, respectively, in all tests (Table 2). The results shown in the table establish that for angles between the nozzle faces of 90°, the incident wave train in the MN inlet section tends to produce a considerable reflection within it. Therefore, the waves in that area act destructively against each other, making it impossible for the swell to grow, which may be optimal for harvesting energy within the nozzle. However, in the tests with an angle of 60°, the MN usable energy is significant, so the waves’ caused is not optimal. Test series with the angle of 34° achieved within the nozzle a remarkable growth up to around 3.4 times of wave height, which is at the same input, thanks to the exchange of momentum with the nozzle’s internal walls. The frequency with which the waves affect the nozzle does not reflect significant changes in the wave’s height. MN-WECs systems in marine farms, it is possible to crop energy even on lowamplitude wave areas, and the additional benefit of significantly avoiding coastal erosion [20] is obtained and achieving a more uniform energy production on the farm [21]. It is known that marine waves are mainly composed of distant waves (Swell type), which originate even from thousands of kilometers, and local waves (Sea type), which are generated due to the wind produced within areas at distances relatively close to the coast. In real conditions, the waves are irregular and are the result of complex combinations of different waves. This condition is called the State of the Sea, and it is common for local and distant waves (Sea and Swell types) to predominate most of the time. It is interesting to note that various software has been developed globally in recent years, such as WAVE WATCH III, SWAN, and WAM, which allow to establish predictive models of waves, with excellent approximation or software for flow simulation such as Dual SPHysics, SW. In future work, it is important to consider the control loop that is going to manipulate the energy harvesting contemplating the gravity compensation in the system [22] and various mechanisms that will be more efficient as the Slider-Crank on the layout proposed [23].

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Table 2. Results obtained for each series WG frequency

10Hz

Time: 420 s

(ηo /ηe ) *

Serie

One

Angle MN

ηo ( cm)

ηe ( cm)

L (m)

V (m/s)

(1/10)%

90°

7

8

1.50

1.40

1.14

60°

7

7

1.65

1.49

1.00

34°

7

18.8

1.74

1.55

2.69

Serie

Two

WG frequency

12Hz

Time: 420 s

(ηo /ηe )*

Angle MN

ηo ( cm)

ηe ( cm)

L (m)

V (m/s)

(1/10)%

90°

7

9

1.78

1.60

1.29

60°

7

8

1.75

1.65

1.14

34°

7

21.5

1.74

1.70

3.10

Serie

Three

WG frequency

14 Hz

Time: 420 s

(ηo /ηe )*

Angle MN

ηo ( cm)

ηe ( cm)

L (m)

V (m/s)

(1/10)%

90°

7

9

1.80

1.40

1.29

60°

7

8

1.75

1.72

1.14

34°

7

23.8

1.74

1.76

3.40

(ηo /ηe )* = wave elevation within the MN/wave elevation before entering the MN.

4 Conclusions The biggest challenge in implementing marine farms for harvesting and wave energy is these technologies’ higher cost. The MN’s design contemplates that the elements that set up the same walls may have movement with a degree of freedom (DOF) to take advantage of the possibility of oscillating the same and harvest energy directly from the waves that exchange momentum with them. It is needed to carry out the hybrid system’s tests in irregular and height waves as it is presented in the coastlines of Oaxaca or Chiapas, Mexico. Acknowledgments. The authors are grateful for partial support for this work provided to the Mexican Government by the Consejo Nacional de Ciencia y Tecnología (CONACYT), the Instituto Politécnico Nacional. The authors also acknowledge the support of project 20210282 and an EDI grant, all from SIP/IPN. Finally, the support of M.en C. G. Rendón-Ricardi, Head of the LIH-TV of the ESIA Zacatenco of the IPN, makes it possible using the wave’s laboratory and the support of Ing. S. Rosas-Labastida. and J.E. Olmedo-García for the information provided.

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References 1. International Energy Agency (IEA): Data and statistics (2018). https://www.iea.org/data-andstatistics?country=WORLD&fuel=Energy%20supply&indicator=TPESbySource 2. Alifdini, I., et al.: Technology application of oscillating water column on the sungai suci beach as solutions for make a renewable energy in coastal Bengkulu, Indonesia. In: Proceedings of the 3rd Asian Wave and Tidal Energy Conference (AWTEC), Singapore, vol. 2 (2016) 3. Asian development bank: Wave energy conversion and ocean thermal energy conversion potential in developing member countries. Asian Development Bank, ADB, Mandaluyong City, Philippines, pp. 1–136 (2014) 4. Garcia, M.O., et al.: Generación de Energía con dos tipos de WECs en México: Resultados experimentales y avances en su implementación. In: IX Congreso Internacional de la Asociación Mexicana de Ingeniería Portuaria, Marítima y Costera A.C, pp. 15–17 (2015) 5. Chen, J., et al.: Networks of triboelectric nanogenerators for harvesting water wave energy: a potential approach toward Blue Energy. ACS Nano 9(3), 3324–3331 (2015) 6. Khan, N., et al.: Review of ocean tidal, wave and thermal energy technologies. Renew. Sustain. Energy Rev. 72, 590–604 (2017) 7. Barbosa, A.L., et al.: Overtopping device numerical study: openfoam solution verification and evaluation of curved ramps performances. Int. J. Heat Mass Transf. 131, 411–423 (2019) 8. Gomes, M.N., et al.: Constructal design applied to the geometric evaluation of an oscillating water column wave energy converter considering different real scale wave periods. J. Eng. Thermophys. 27(2), 173–190 (2018). ISSN 1810-2328 9. Mendoza, E., et al.: Hydrodynamic behavior of a new wave energy converter: the blow-jet. Ocean Eng. 106, 252–260 (2015) 10. Coe, R.C., et al.: Extreme conditions modeling Workshop Report. Nrel-sandia national laboratories. Technical report. Nrel/tp-5000–62305, snl/sand 16384r. pp. 1–42 (2014) 11. Hammar, L., et al.: Introducing ocean energy industries to a busy marine environment. Renew. Sustain. Ener. Rev. 74, 178–185 (2017) 12. Gomes, M.D.N., et al.: Analysis of the geometric constraints employed in constructal design for oscillating water column devices submitted to the wave spectrum through a numerical approach. 390, 193–210 (2019) 13. Philen, M., et al.: Wave energy conversion using fluidic flexible matrix composite power take-off pumps. Energy Convers. Manag. 171, 1773–1786 (2018) 14. Sang, Y., et al.: Energy extraction from a slider-crank wave energy under irregular wave conditions. National Renewable Energy Laboratory (NREL). Presented at Oceans Washington, D.C. Conference paper NREL/CP-5D00-64875. pp. 1–9 (2015) 15. Lithgow, D., et al.: Integración de los dispositivos conversores de la energía marina a los ecosistemas costero-marinos en México, pp. 241–254 (2019) 16. Felix, A., et al.: Wave energy in tropical regions: deployment challenges, environmental and social perspectives. J. Mar. Sci. Eng. 7(7), 2–21 (2019) 17. Soares, C.G., et al.: Overview and prospects for development of wave and offshore wind energy. Centre for Marine Technology and Engineering (CENTEC), 65(2) (2014) 18. Pecher, A., Kofoed, J.P. (eds.). Handbook of Ocean Wave Energy. Ocean Engineering and Oceanography, vol. 7 (2017) 19. Neary, V.S., et al.: Methodology for design and economic analysis of marine energy conversion (MEC) technologies. In: Proceedings of the 2nd Marine Energy Technology Symposium. METS2014. SAND2014-3561C, Seattle, WA, 15–18 April 2014 20. Mendoza, E., et al.: Beach response to wave energy converter farms acting as coastal defense. Coast. Eng. 87(2014), 97–111 (2013). https://doi.org/10.1016/j.coastaleng.2013.10. 018.0378-3839

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21. Flocard, F., Finnigan, T.D.: Laboratory experiments on the power capture of pitching vertical cylinders in waves. Ocean Eng. 37(989–997), 0029–8018 (2010). https://doi.org/10.1016/j. oceaneng.2010.03.011 22. Perrusquia, A., Flores-Campos, J.A., Torres-San-Miguel, C.R.: A novel tuning method of PD with Gravity compensation controller for robot manipulators. IEEE Access 8, 114773–114783 (2020). https://doi.org/10.1109/ACCESS.2020.3003842 23. Perrusquia, A., Flores-Campos, J.A., Torres-Sanmiguel, C.R., Gonzalez, N.: Task space position control of slider-crank mechanisms using simple tuning techniques without linearization methods. IEEE Access 8, 58435–58442 (2020). https://doi.org/10.1109/ACCESS.2020.298 1187

Robot Design and Optimization

Modelling of a Hydrodynamically Actuated Manipulator Based on Strip Theory R. Santiesteban Cos1,2(B) , J. A. Carretero1,2 , and J. Sensinger1,2 1

2

Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canada {raul.sncos,juan.carretero,j.sensinger}@unb.ca Department of Electrical and Computer Engineering, University of New Brunswick, Fredericton, NB, Canada

Abstract. In this paper, mathematical modelling based on strip theory of the Hydrodynamically Actuated Manipulator (HAM) is proposed to use it for analysis and control synthesis. A set of Nonlinear Ordinary Differential Equations (ODEs) describes the behaviour of the HAM. First, considering the Euler-Lagrange methodology, the motion of the R ⊥ R  P manipulator is obtained. Then, a Newtonian approach is under study to add the effects of a fluid surrounding the 3-DOF manipulator. In order to describe this interaction, each link of the manipulator affected by hydrodynamic forces is discretized on small elements. Their contribution is added to obtain the net acceleration of each link of the HAM. In order to qualitatively validate the model, a numerical simulation are under study. The experiment consists of a comparison between the Lagrangian of the 3-DOF manipulator and its motion when is affected by each added hydrodynamic force. Keywords: Modelling

1

· Nonlinear systems · Euler-Lagrange

Introduction

Defense Research and Development Canada has been working with the Robotics and Mechanisms (RAM) Laboratory at University of New Brunswick and other partners to develop a reliable method to dock Unmanned Underwater Vehicles (UUVs) to moving submarines or surface vessels. The proposal consists of attaching to the outer hull of the vessel a robotic arm which, using an array of sensors and actuators, will be able to autonomously capture a UUV. In [1] and [3], both undergraduate senior design projects, multiple UUV tracking manipulators were proposed, one of them being a serial manipulator with three degrees of freedom (three joints). As in the wheel of inertia, cart-pendulum, or pendubot [7,11,19], the hydrodynamically actuated manipulator may be used as an interesting test bed for analysis and control synthesis because it can be seen as two interacting dynamic c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 171–180, 2022. https://doi.org/10.1007/978-3-030-88751-3_18

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Fig. 1. Docking envelope in relation to the submarine [17].

Fig. 2. The 3-DOF mechanical manipulator.

systems: a fluid and a solid [12,14,15]. There are some features that make this manipulator an interesting test bed for analysis and control synthesis. First, the manipulator by itself could be analyzed as an inverted pendulum that moves with a force caused by a fluid [16]. Depending on the position of the reference frame, there will be gravity-bounded nonlinear terms affecting the behaviour of the system. Second, considering the effects of fluid on the solid, nonlinear quadratic dynamics will affect the mechanical system (e.g., [2,8]). Third, noncollocated dynamics will also affect the behaviour of the entire system because of the underactuated operation mode, which is very interesting for control design purposes (see [18]). Figures 1 and 2 show a sketch of the robot. The only rotation between the frame (x, y, z) and (x , y  , z  ) is a rotation around axis z. The constant l0 is the distance from the platform of the manipulator to the centre of mass of the hydraulic actuator, and l1 denotes the half of the span of the wing. θ is the angle between the axis x and the axis y  . λ is the distance from the centre of mass of the hydraulic actuator to the center of mass of the wing along axis x . γ is the angle between the axis z and the axis x (see [4,17,20]). At its inception ([1] and [3]), the UUV tracking manipulator was a serial manipulator with four degrees of freedom (four joints). The manipulator’s full scale has three mutually perpendicular revolute (R) joints and one prismatic (P) joint perpendicular to the first two. This results in an architecture that is denoted by R ⊥ R ⊥ R  P (with ⊥ denoting the perpendicularity relation between two consecutive joints). More specifically, the arm is to have the first revolute joint (the base joint) with the axis vertical to bring the arm perpendicular to the submarine’s hull. The base joint is conceptualized to be primarily used when deploying or concealing the arm, allowing it to swing to a more compact or streamlined position when not in use. However, before capture, the arm is brought to be perpendicular to the hull and the base joint is then locked. At that point, the second revolute joint (the shoulder joint) will have its axes aligned with the main axis of the submarine. This second joint will then allow rolling (in other words to elevate) the arm in the vertical direction. The third revolute joint is designed to rotate

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the wing. The fourth joint, the linear or prismatic joint will enable to arm to extend further away or closer to the submarine essentially changing the radius of the circle traced by the end effector relative to the base joint. This conceptual design is illustrated in Fig. 2 (see [17]). The design showed in Fig. 2 and the one analysed in here is meant as a halfscale model. This half-scale model is fixed perpendicular to the hull position during its whole operation (i.e., the first revolute joint is fixed and thus not modelled herein). However, the new architecture of the manipulator still has a R ⊥ R  P configuration, where the last revolute joint is used to actively pitch the wing. An interesting design concept is that the hydraulic cylinder can be either a) directly actuated (using a hydraulic actuator) [10] or b) indirectly actuated via hydrodynamic action produced by the interaction of the incoming water flow with the actively pitching wing placed at the end of the arm (see [4] and [20]). In the latter case, the robot is designed to take advantage of the motion of the support vessel by actively controlling the pitch of a wing. This allows the arm to move much like one’s arm would move when sticking the hand out on a fastmoving vehicle and changing the angle of the hand relative to the direction of the incoming wind. Considering the hydrodynamically actuated mode, the manipulator will be actuated using the hydrodynamic force of the water through a wing. Figure 2 shows a sketch of the robot, where θ is the roll angle or the orientation of the axis of the hydraulic cylinder relative to the vertical (upward) direction. That is, θ is measured around z0 as the axis of the cylinder always lies on in the x0 -y0 plane. In this model, γ denotes the angular position around the cylinder’s axis while λ is associated with the length of the hydraulic cylinder. Using an actuator to change the angle of attack α of the wing and the length λ of the hydraulic cylinder, it is possible to change the unactuated angular position θ of the manipulator. The main contribution of this paper is a proposal of a mathematical model of a hydrodynamically actuated mechanical system.

2

Modelling: R ⊥ R  P Manipulator

Due to the complexity of the 3-DOF manipulator, let us consider the EulerLagrange formulation instead of Newton’s formulation. The Euler–Lagrange equation is given by   d ∂L ∂L = Fi , (1) − dt ∂ q˙i ∂qi where q = [θ λ γ]T , Fi denotes the non-conservative and viscous forces on each link, and (2) L = K − U = K0 + K1 − (U0 + U1 ) . The kinetic energy of the hydraulic cylinder associated to its rotational motion is given by  1 1 K0 = m0 v02 + Iz0 θ˙2 = m0 l02 + Iz0 θ˙2 , (3) 2 2

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with Iz0 as the moment of inertia of the hydraulic cylinder around axis z. In order to calculate the potential energy of the hydraulic cylinder, let us consider that its position of rest (θ = 0) has potential energy equal to zero. U0 = m0 gy0 = m0 gl0 (1 − cos(θ)).

(4)

The wing is rotating with respect the axis z but also is rotating with respect the hydraulic cylinder or axis x . Then, the wing has two degrees of freedom and the kinetic energy associated with its rotational and translational motion is given by 1 1 1 K1 = m1 v12 + Ix1 γ˙ 2 + Iz1 θ˙2 (5) 2 2 2 Since v 2 = x˙ 2 + y˙ 2 + z˙ 2 , the kinetic energy of link 1 can be written as follows  2 2  1 2 2 2 ˙ ˙ ˙ m1 λ − c sin(γ)θ + L0 θ + c cos(γ)γ˙ + c sin (γ)γ˙ K1 = 2 1 1 + Ix1 γ˙ 2 + Iz1 θ˙2 (6) 2 2 where c is the distance of the centre of mass of the wing relative to x (see Fig. 3). Finally, the potential energy of link 1 is given by U1 = m1 gy = gm1 ((l0 + λmax ) − (l0 + λ) cos(θ) + c sin(γ) sin(θ)) 2.1

(7)

The R ⊥ R  P Manipulator Subjected to Hydrodynamic Forces

2.1.1 Preliminaries: Nomenclature and Flow Conditions Several works have been conducted to model underwater manipulator dynamics. For instance, in [13] and [14], the authors estimated the added mass and drag forces using and developing potential flow theory and experimental tests. The effect on the underwater manipulator is modelled in a simplified way assuming an irrotational, incompressible flow [6,8,9]. In this section, for readability some i variables are defined: L(λ) = l0 + λ − l1 and ri (λ) = (L(λ) + 2l1 ) − Li (λ) for T i = 2, 3, 4. Let Fq˙ = [fθ˙ fλ˙ fγ˙ ] denotes a source of friction such as viscous friction, drag friction and added mass but also a source of energy from the aerodynamic profile such as lift force. Then, Fq˙ can be written as ⎤ ⎡ ˙ − fθD (q, θ, ˙ αh ) + fθL (q, ¨ ˙ αh ) + fθB (q) − fθam (q, θ) −fθv (θ) ⎦ ˙ + fλB (θ) ˙ q ¨) = ⎣ F (q, q, −fλv (λ) ˙ αh ) + fγB (γ) − fγam (q, γ¨ ) ˙ − fγD (q, −fγv (γ) (8) where fθv , fλv , and fγv denote the viscous friction at each joint, fθB , fλB , and fγB denote the buoyancy force, fθam , and fγam denote the added mass, fθD , while fγD and fθL denote the drag and lift forces, respectively.

Modelling of a Hydrodynamically Actuated Manipulator

Fig. 3. The lift and drag forces with respect to the flow in the negative z0 direction.

175

Fig. 4. The lift force with respect to the flow.

2.1.2 Buoyancy Force For the 3-DOF manipulator, the buoyancy force can be calculated in two parts: the hydraulic cylinder and the wing. The buoyancy of the hydraulic cylinder depends on the volume displaced and can be calculated as V = πR2 L(λ). The buoyancy of the wing depends on two things: the distance of the wing from axis z and the angle of rotation γ. Then, fB (q) can be written as  ⎤ ⎡  1 2 2 gρ 2 πR L (λ) + V1 L0 (λ) sin(θ) + V1 c sin(γ) cos(θ) ⎦ , (9) −gρV1 cos(θ) fB (q) = ⎣ gρV1 c cos(γ) sin(θ) where L0 = l0 + x2 , L = l0 + x2 − l1 , V0 = πR2 L(λ) denotes the volume of the exposed hydraulic cylinder, V1 = 2l1 Acw denotes the volume of the wing. 2.1.3 Added Mass Force Based on [5], for the 3-DOF manipulator, the added mass force can be calculated in two parts: the hydraulic cylinder and the wing. The added mass of a cylinder in potential flow is equal to the mass of fluid displaced by the body ρπR2 .  2 For a wing with chord c0 , ρπ c20 U (t)γ˙ is a Coriolis-like term affecting the acceleration of the wing, which is due to the added mass when the hydraulic cylinder is accelerating around axis z (see [5], Chap. 7). Due to the motion of the wing around axis x , the effect of the added mass on the wing can be projected ¨ xy , where to the plane x − y. This projection can be calculated as ρ sin(γ)θdV  dVxy is the volume of a thin segment of the wing at a distance z from the axis x and distance L(λ) from axis z. A simplified version of the added mass term can be described as ⎤ ⎡     2  2 ˙ 2 ρπ 13 θ¨ R2 L3 (λ) + c20 cos(γ)r3 (λ) + 12 c20 U (t)γr ⎥ ⎢ ¨) = ⎣ f˜am (q, q ⎦(10) 0 0

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2.1.4 Quasy-Steady Lift and Drag Forces A fluid flowing around the surface of an object exerts a force on it. The lift force is the component of this force that is perpendicular to the oncoming flow direction while the drag is the one aligned with it (see Fig. 3). Although the main flow is parallel to the z axis, one needs to also consider that the wing has a rotational motion around both the z and x axes. Thus, the true velocity of the flow v = U (t) affecting a thin segment of the wing can ˙ 2 where r is the distance from z to the thin be calculated as (U )2 = U2 + (θr) segment of the wing. The effective angle of attack αh also considers the angle between the thin segment of the wing and the incident flow, and thus depends on the flow velocity U(t) and θ˙ (Fig. 3). That is,   1˙ αh = −γ − atan2(V, U ) = −γ − atan2 θl, U (11) 2 where atan2 refers to the quadrant-corrected inverse tangent function. For simplicity, only the quasi-steady lift force is considered here. Considering the value CL for a wing at a specified angle of attack, the lift produced for specific flow conditions can be determined: 1 2 ρv SCL (12) 2 where v is the speed of the wing relative to the flow and S is the planform wing area. In this case, the wing has a planform area S = 2c0 l1 . However, the wing is at a distance L(λ) = l0 + λ − l1 from the axis of rotation z. Moreover, the wing has rotational motion with respect to axis x . Therefore, the projected force with respect to z − x should be considered (see Figs. 4 and 3). Since the effective angle of attack in Eq. (11) depends on the distance on each strip of the wing, to solve analytically the torque Equation (12) comes to be very complex for simulation purposes. To solve this issue, an average effective angle ˙ αh ) can be written as of attack is implemented. Therefore, the lift force fL (q, q, ⎤ ⎡  2 1 1 ˙2 ρc sin(α )C (α ) U r + r θ 0 h L h 2 4 2 ⎥ ⎢4 ˙ αh ) = ⎣ fL (q, q, (13) ⎦ 0 0 L=

where CL (αh ) is the lift coefficient which itself is a function of αh [4]. In the case of the 3-DOF manipulator, the wing has a NACA 0020 profile and a chord lenght of c0 = 0.6 m. The wing cover moves away or towards the z0 axis as it is part of the hydraulic cylinder. Therefore, the exposed cross section area of the cylinder is Ac = 2R(l0 + λ − l1 ) = 2RL(λ) (see Fig. 4). The drag ˙ and the drag force around force around axis z depends on the angular velocity θ,  x depends on the angular velocity γ˙ but also depends on the effective angle of ˙ can be written as attack αh . Then, the drag force fD (q, q) ⎤ ⎡   ˙ θ˙ RCDc L4 (λ) + 1 c0 cos(γ)CD (αh )r4 − 14 ρ|θ| 2 ⎥ ⎢ 0 ˙ αh ) = ⎣ fD (q, q, ⎦   2 1 1 ˙2 0.4112 4 − ρc0 sin(αh )CD (αh ) U r2 + θ r4 − ρl1 |γ| ˙ γc ˙ 4

2

4

0

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Fig. 5. Simulation results: wing elevation (θ)

3

Numerical Simulation

A simulation is considered in order to confirm the behaviour of the mechanical system affected by the hydrodynamic forces. In Figs. 5, 6 and 7, the motion of the mechanical system inside the water is shown. In this simulation, the initial conditions for the 3-DOF system are fixed as: θ(0) = π2 +0.2 rad and λ(0) = 0 m, and γ(0) = 0.5 rad, and the velocities of the joint variables are set equal to zero, with the flow U as a function of time (t) expressed as U (t) = t m/s at t = 0.1 s. Figures 5, 6 and 7 show the Lagrangian (red line) and the Lagrangian affected by hydrodynamic forces (blue line). The first graph in every figure shows the position of the joint manipulator while the second graph shows the rate of change of the corresponding joint variable. The main idea of this simulation is to show how the hydrodynamic forces affect the 3-DOF manipulator when the wing is forced to slow its motion, and it creates a lift force driving the hydraulic cylinder upwards. A high level of viscous friction, around the axis x , will force the wing to slow its motion for a small period of time. When the Lagrangian is affected by hydrodynamic forces, the lift force, created by a positive angle of attack of the wing, overcomes the gravitational, friction, and drag forces affecting the hydraulic cylinder. The hydraulic cylinder starts with a horizontal position and it tends to go upwards while the prismatic joint goes in a negative direction, showing that the hydraulic cylinder is contracting. In the case of the Lagrangian, the cylinder starts to fall from

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Fig. 6. Simulation results: wing extension (λ)

Fig. 7. Simulation results: wing pitch (γ)

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the initial position and oscillates around the position of rest while the prismatic joint grows almost linearly, showing that the hydraulic cylinder is extending. Even high viscous friction is imposed on the wing, it tends to align to the flow U (t). At the time that the wing is aligned with the flow U (t), the generated lift force tends to zero and the hydraulic cylinder falls and starts to oscillate around its rest position θ = 0 rad. In the case of the Lagrangian, the wing remains in the same position due to the high friction.

4

Conclusions

A Euler-Lagrange formulation was considered to obtain a model for a R ⊥ R  P hydrodynamically actuated manipulator (HAM). Based on the potential and kinetic energy, the Lagrangian was calculated considering a dissipative force such as viscous friction. The hydrodynamic forces, affecting the motion of the mechanical system, were added one by one considering a Newtonian approach. Forces such as buoyancy, added mass, drag, and lift were calculated using a strip theory approach. Through a simulation, the system’s model was confirmed by observing its behaviour follows basic principles. Although the fidelity of the model may only be verified through lengthy Computational Fluid Dynamics (CFD) simulations or costly physical experiments, the proposed model is believed to replicate enough of the effects to allow for studies on controls to be performed.

References 1. Aske, T., Barton, M., MacKay, T.: Docking of unmanned underwater vehicles with submarines. Master’s thesis, University of New Brunswick (2011) 2. Caldeira, A.F., Prieur, C., Coutinho, D., Leite, V.J.: Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions. Eur. J. Control 43, 1–11 (2018) 3. Cote, J., Gillis, C., Hilton, K.: Docking autonomous underwater vehicles (UUV) on submarines. Senior Design Project Report, Department of Mechanical Engineering (2011) 4. Currie, J.: Dynamic modelling and control of an active autonomous wing dock for subsurface recovery of AUVs. Master’s thesis, University of New Brunswick (2015) 5. Drela, M.: Flight Vehicle Aerodynamics. MIT Press, Cambridge (2014) 6. Fackrell, S.: Study of the Added Mass of Cylinders and Spheres (2011) 7. Fantoni, I., Lozano, R.: Non-Linear Control for Underactuated Mechanical Systems. Springer, Heidelberg (2002). https://doi.org/10.1007/978-1-4471-0177-2 8. Fossen, T.I.: Guidance and Control of Ocean Vehicles. Wiley, New York (1994) 9. Gianluca, A., Antonelli, G.: Underwater Robots, vol. 3. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-319-02877-4 10. Gillis, C.B.: Dynamic model development and simulation of an autonomous active AUV docking device using a mechanically actuated mechanism to recover AUVs to a submerged slowly moving submarine in waves. Master’s thesis, University of New Brunswick (2014)

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11. Goryanina, K.I., Lukyanov, A.D., Katin, O.I.: Review of robotic manipulators and identification of the main problems. In: MATEC Web of Conferences, vol. 226, pp. 1–7. EDP Sciences (2018) 12. Kolodziejczyk, W.: Preliminary study of hydrodynamic load on an underwater robotic manipulator. J. Autom. Mobile Robot. Intell. Syst. 9, 11–17 (2015) 13. Leabourne, K.N., Rock, S.M.: Model development of an underwater manipulator for coordinated arm-vehicle control. In: IEEE Oceanic Engineering Society. Proceedings of OCEANS’98 Conference, vol. 2, pp. 941–946. IEEE (1998) 14. McLain, T.W., Rock, S.M.: Experiments in the hydrodynamic modeling of an underwater manipulator. In: Proceedings of the 1996 Symposium on Autonomous Underwater Vehicle Technology, pp. 463–469. IEEE (1996) 15. Mueller, T., DeLaurier, J.D.: An overview of micro air vehicle aerodynamics. Prog. Astronaut. Aeronaut. 195, 1–10 (2001) 16. Onishi, H., Kitamoto, T., Maeda, T., Shimohara, H., Tanigawa, H., Hirata, K.: Added-mass and viscous-damping forces acting on various oscillating 3D objects. In: ASME 2014 Pressure Vessels and Piping Conference, pp. 1–10. American Society of Mechanical Engineers (2014) 17. Prosser, T.W., Carretero, J.A., Dubay, R.: Construction and evaluation of an actuated arm for capturing unmanned underwater vehicles from submarines. In: 2016 CCToMM Symposium on Mechanisms, Machines and Mechatronics, pp. 26–29 (2016) 18. Spong, M.W.: Underactuated mechanical systems. In: Siciliano, B., Valavanis, K.P. (eds.) Control Problems in Robotics and Automation, vol. 230, pp. 135–150. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0015081 19. Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control, vol. 3. Wiley, New York (2006) 20. Watt, G.D., et al.: A concept for docking a UUV with a slowly moving submarine under waves. IEEE J. Oceanic Eng. 41(2), 471–498 (2016)

Shaking Force Balancing of the 2RRR PPM Specifying Tool’s Motion Mario Acevedo1(B) 1

and Ramiro Vel´ azquez2

´ Facultad de Ingenier´ıa, Universidad Panamericana, Alvaro del Portillo 49, 45010 Zapopan, Jalisco, Mexico [email protected] 2 Facultad de Ingenier´ıa, Universidad Panamericana, Josemar´ıa Escriv´ a de Balaguer 101, 20290 Aguascalientes, Aguascalientes, Mexico

Abstract. Shaking force balancing of mechanisms usually is achieved by an optimal redistribution of the moving masses. This allows the cancellation or the reduction of the variable dynamic loads on the mechanism’s frame. The procedure generally leads to a considerable increment in the mass. Addition of counterweights and counter-inertias are required, thus the driving torques and the shaking moment are incremented too. Recently the shaking force balancing has been successfully achieved by specifying the acceleration motion of the global Center of Mass (CoM) of the robot. Additional masses are avoided. But this may be difficult to apply in practice. Therefore it has been proposed a combined procedure, based on imposing an acceleration motion planning to the manipulator’s tool, and on the selective mass redistribution. This last concept is resumed here using Natural Coordinates, a direct and easy to automate alternative. The force balancing of a 2RRR Planar Parallel Manipulator (PPM), also known as the five-bar mechanism, is used as an application example. The shaking force and the shaking moment are obtained analytically using the proposed direct approach. Linear relation between the acceleration of the CoM and the acceleration of the tool is shown in a clear and explicit way, making it easy to identify the required mass redistribution. A generic CAD model made in Solidworks is used to show the application of the technique and to validate the balancing’s results, obtained from inverse kinematics simulations, through dynamic simulation. Keywords: Dynamic balancing · Shaking force · Shaking moment Parallel manipulator · Manipulator’s tool motion

1

·

Introduction

Robotic industrial manipulators deal with the problem of vibrations transmitted to the fixed frame during high-speed motion. These vibrations are due to different factors, one of them is the unbalanced inertia forces which finally produce the increment of the shaking force and shaking moment. Thus, a primary objective c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 181–190, 2022. https://doi.org/10.1007/978-3-030-88751-3_19

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in the dynamic balancing of manipulators is to eliminate or reduce the variable dynamic loads transmitted to the base and surrounding structures. A traditional method used for the dynamic balancing of manipulators has been the mass redistribution. In this case the objective is to maintain the total center of mass of the moving links stationary, see [3,6,9]. In practice this is done adding a set of counterweights and counter-inertias increasing considerably the mass. This causes an increment in the overall size of the robot-manipulator and in the efforts at the joints. Another alternative also traditionally explored is the use of auxiliary structures. In the same way the aim is to keep the total center of mass of the mechanism static. The structures proposed and generally used are the parallelogram and the pantograph, [7,8,12]. Although this solution leads to relatively complex mechanical system increasing manipulator’s sizes. Recently a new successful approach has been proposed. It is based on imposing constant acceleration motion to the robot-manipulator global Center of Mass (CoM), [5,11]. But this may be difficult to apply in practice. Therefore a combined procedure based on imposing constant acceleration to the robot’s tool along with a selective optimal mass redistribution has been proposed in [4]. Linear relation between the motion of the tool and the motion of the global CoM is found imposing selected balancing conditions, fulfilled by the addition of some appropriate counterweights. The tool’s trajectory is defined as a straight line imposing a “bang-bang” acceleration profile. This ensures a linear trajectory of the CoM with constant acceleration. Thus leading to a reduction in the shaking force. In this work this approach is presented using fully Cartesian coordinates [10], introducing a model of a 2RRR Planar Parallel Manipulator (2RRR PPM). A complete explanation on the use of this modeling technique along with different practical balancing alternatives for this robot have been presented in detail in [1]. The advantage of using fully Cartesian coordinates is that the expressions relating the motion of the general CoM and the motion of the manipulator’s gripper, the shaking force, and the shaking moment, are obtained in a direct and explicit form. This allows the use of software for symbolic computation, making the method easy to program and faster in execution. In this article the acceleration of the robot’s global CoM and the acceleration of the robot’s gripper are related. The resulting expression helps to identify the appropriate balancing conditions that should be used to redistribute the mass and to relate the accelerations linearly. The practical application is shown through a numerical example. A generic CAD model made in Solidworks is used to show the application of the technique and to validate the balancing’s results, obtained from inverse kinematics simulations, through dynamic simulation.

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2

183

Modeling of a 2RRR-PPM

In Fig. 1 a generic 2RRR-PPM (five-bar mechanism) is presented. All links considered rigid. Kinematic revolute joints are identified with points A through E, and points A and E are fixed to the frame. Link lengths are indicated as li , where i = 1, 2, 3, 4. In this case the centers of mass of the links can be identified with the letter gi , respectively. The masses of the bodies are distributed to the “basic points”. In this case points A through E. Thus the system can be seen as a cloud of moving point masses. Thanks to this change, it is possible to work exclusively with Cartesian coordinates. Angular coordinates can be avoided, as well as the explicit use of the moments of inertia. In this case this facilitates the dynamic analysis.

Fig. 1. Model of the 2RRR Planar Parallel Manipulator, showing the local reference frames orientation. The centers of mass are considered in a general position.

The manipulator has been modeled using the “basic points”, that can be organized in a positions vector q: T  q = rA rB rC rD rE

(1)

where rj = [xj yj ], i = {A, . . . , E}, are the position vectors that locate the basic points A through E with respect to the origin of the reference frame. These dependent coordinates are related by a set of four kinematic constraints, obtained from the rigid body condition of the bodies: ⎡ ⎤ (xB − xA )2 + (yB − yA )2 − l12 ⎢ (xC − xB )2 + (yC − yB )2 − l22 ⎥ ⎥ (2) Φ(q) = ⎢ ⎣(xC − xD )2 + (yC − yD )2 − l32 ⎦ = 0 (xD − xE )2 + (yD − yE )2 − l42

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These coordinates and their time derivatives can be used to calculate the position of the global CoM of the system, the shaking force, and the shaking moment directly with respect to the global reference frame (inertial frame). In this procedure the mass matrix plays a key role. The general procedure to obtain the mass matrix has been described in [10] and to this specific manipulator in [1]. Their terms can be reorganized to form the (10 × 10). Its detailed form can be seen in [2]. This matrix is symmetric and positive definite. 2.1

Calculation of the Global Center of Mass

The global center of mass of the system can be calculated directly as:

4

mt gt = BMq

(3)

where mt = i=1 mi , is the total mass or the system, gt is the position vector of the location of the global center of mass, q is vector of dependent Cartesian coordinates, and M is mass matrix. Matrix B is introduced here to sum x and y components in an automatic way and is composed by (2 × 2) identity matrices, I2×2 , its form is:   B = I2×2 I2×2 I2×2 I2×2 I2×2 2.2

(4)

Calculation of the Shaking Force

The shaking force is the time derivative of the linear momentum. The linear momentum can be calculated directly deriving Eq. (3) with respect to time: mt g˙ t = BMq˙

(5)

where q˙ is the vector of velocities, the time derivative of vector q. The shaking force, fsh , can be calculated as the time derivative of the linear momentum. Thus, taking into account the constant mass matrix and that the velocities of point A and of point E are zero, the shaking force of the manipulator can be expressed in extended form as: ⎡ ⎤ x ¨B ⎡ ⎤ ⎢ y¨B ⎥ ⎥ a b c d e f ⎢ ⎢ ¨C ⎥ ⎥ ⎦ ⎢x ¨t = ⎣ (6) fsh = mt g ⎢ y¨C ⎥ ⎥ −b a −d c −f e ⎢ ⎣x ¨D ⎦ y˙ D

Shaking Force Balancing Specifying Tool’s Motion

where:



m1 x1 m2 x2 + + m2 a= − l2 l1  m2 y 2 m1 y 1 − b= l2 l1  m3 x3 m2 x2 + c= l3 l2  m2 y 2 m3 y3 − d= − l3 l2  m4 x4 m3 x3 − + m3 e= l4 l3  m4 y 4 m3 y 3 − f= l4 l3

185

(7) (8) (9) (10) (11) (12)

and where (xi , yi ) with i = 1 . . . 4 are the coordinates of the corresponding center of mass of body i expressed in the local reference frame, see Fig. 1. The origin of the global reference frame coincides with point A, thus (xA , yA ) = (0, 0). Note that it is possible to relate the global CoM acceleration exclusively to the acceleration of the manipulator’s gripper making some coefficients of the mass matrix zero, particularly a = b = e = f = 0. These are precisely the balancing conditions than lead to the optimal mass redistribution. Thus performing the suggested mass redistribution the shaking force can be espressed as: ⎡ ⎤ c d  ¨C ⎦ x ¨t = ⎣ (13) fsh = mt g y¨C −d c The balancing conditions can be fulfilled in different ways. The most common and the ones used in this work are: first to consider that all the links are in-line, which means that their CoM are in the line defined by their corresponding joints, second to use a counterweight at link 1 and at link 4 respectively. The form, size and distance from the link’s origin to the counterweight are up to the designer election. Infinite number of solutions can be proposed. In this case a generic solution has been used, although the future use of an optimization procedure could come to mind. It can be noted that considering the mass redistribution a new simplified expression for the position of the CoM can be calculated: ⎡ ⎤ c d  ⎦ xC (14) mt gt = ⎣ yC −d c 2.3

Calculation of the Shaking Moment

The shaking moment can be obtained by deriving the angular momentum with respect to time. The angular momentum can be calculated as:

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h = ˜rMq˙

(15)

where ˜r is a row vector of cross products as:   ˜r = ˜rA ˜rB ˜rC ˜rD ˜rE

(16)

where ˜ri ≡ [−yi , xi ], i = {A, . . . , E} is a 1 × 2 row vector. Thus the shaking moment can be calculated by: q τ sh = ˜r˙ Mq˙ + ˜rM¨

(17)

¨ is the vector of accelerations. where q Considering that the mechanism is in-line (two of the balancing conditions), ˜r˙ Mq˙ = 0, thus the shaking moment of the manipulator can be reduced to: τ sh = ˜rM¨ q

(18)

that in extended form can be expressed as: 

 m2 x2 2m2 x2 I2 I1 I2 τ sh = m2 − + 2 + 2 (˜rB ¨rB ) + − 2 (˜rB ¨rC + ˜rC ¨rB ) l2 l l1 l2 l2   2 m3 x3 I2 I3 I3 − 2 (˜rC ¨rD + ˜rD ¨rC ) + 2 + 2 (˜rC ¨rC ) + l2 l l3 l3   3 m4 x4 I4 2m3 x3 I3 I2 ¨ − 2 (˜rE rD ) + m3 − + 2 + 2 (˜rD ¨rD ) + l4 l4 l3 l3 l2 (19) where Ii , i = 1...4, are the moments of inertia with respect to the corresponding local reference frame, no with respect to the corresponding link’s CoM. This is a characteristic of the model obtained using Natural coordinates.

3

Numerical Example

To show the application of the equations developed in the previous sections, a specific 2RRR PPM has been selected. A model with the following geometric parameters have been used: l1 = l4 = 0.36 m, l2 = l3 = 0.3 m, and xE = 0.35 m. To locate the centers of mass the following values where used: x1 = x4 = 0.18 m, x2 = x3 = 0.15 m, and y1 = y2 = y3 = y4 = 0. The corresponding masses are: m1 = m4 = 0.2028823 kg and m2 = m3 = 0.1704823 kg, and moments of inertia with respect to the link’s CoM are: Ig1 = Ig4 = 0.00239421 kg·m2 and Ig2 = Ig3 = 0.1704823 kg·m2 . Note that Ii = Igi . These values are obtained from a generic model of a 2RRR PPM made in Solidworks and presented in Fig. 2.

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3.1

187

Motion Given to the End-Effector

The trajectory of the end-effector at C is a straight line limited between the initial position Pi , with the coordinates xi = −0.15 m, and yi = 0.4 m, and the final position Pf , with the coordinates xf = 0.33296 m, and yf = 0.52941 m. The straight line is traveled following a bang-bang motion profile.

Fig. 2. Model of the 2RRR Planar Parallel Manipulator in Solidworks. Up, the original model compared to the partially balanced model (conditions a and e applied). Bottom, the original inferior link compared to the balanced link (conditions a and e applied). All lengths in mm and angles in degrees.

Mass redistribution was performed using the balancing conditions: a = 0, b = 0, e = 0, and f = 0. A generic useful design was selected, its form and dimensions are shown in Fig. 2. The CoM of bodies 1 and 4 were located at x1 = x4 = −80.4347 mm. Additionally the masses and moments of inertia changed as: m1 = m4 = 0.38128 kg, and Ig1 = Ig4 = 0.0143679 kg·m2 . Equations (3), (14), have been used to calculate the CoM trajectory before and after the mass redistribution. The Fig. 3 shows the comparison of results. Equations (6) and (13) were used to calculate the Shaking Force before and after the mass redistribution. In the same way Eqs. (17) and (19) were used to calculate the Shaking Moment before and after the mass redistribution. Results are shown in Fig. 4. The resulting dynamic reactions presented in Fig. 4, have been obtained solving multiple inverse kinematics problems, then substituting values in the equations developed in Sect. 2. Positions, velocities, and accelerations corresponding to the “bang-bang” profile where assigned to the manipulator’s end-effector

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Fig. 3. Trajectories of the manipulator’s CoM before and after mass redistribution.

Fig. 4. Shaking force and Shaking Moment compared.

Fig. 5. Shaking force and Shaking Moment obtained from the direct dynamics simulation performed in Solidworks.

(point C). The corresponding angular motion generated at links 1 and 4 were recorded as data points and used to solve the direct dynamics problem in Solid-

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works. This could be considered as an preliminary validation of the method before building a prototype. The resulting reactions from this simulation is presented in Fig. 5.

4

Conclusions

A combined method, originally presented in [4], based on specifying the acceleration profile of the tool and the selective mass redistribution is summarized here, using an efficient and direct modeling technique. The balancing procedure is presented, and the analytical expressions of the global CoM, the shaking force, and the shaking moment are presented for a generic 2RRR PPM. These expressions along with multiple inverse kinematics simulations, are taken to calculate the CoM trajectory as well as the shaking force and shaking moment. These results are compared to the ones obtained with a generic not balanced model. A model of the resulting balanced mechanism is made in Solidworks to validate its behavior and performance in direct dynamics.

References 1. Acevedo, M.: Conditions for dynamic balancing of planar parallel manipulators using natural coordinates and their application. In: 14th World Congress in Mechanism and Machine Science, Taipei, Taiwan, 25–30 October 2015 (2015) 2. Acevedo, M., Orva˜ nanos-Guerrero, M.: Force balancing of the 2rrr planar parallel manipulator via center of mass acceleration control using fully cartesian coordinates. In: Hernandez, E., Keshtkar, S., Valdez, S. (eds.) LASIRS 2019. MMS, vol. 86. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45402-9 4 3. Agrawal, S., Fattah, A.: Reactionless space and ground robots: novel design and concept studies. Mech. Mach. Theory 39(1), 25–40 (2004). https://doi.org/10. 1016/S0094-114X(03)00102-2 4. Arakelian, V., Geng, J., Fomin, A.S.: Minimization of inertial loads in planar parallel structure manipulators by means of optimal control. J. Mach. Manuf. Reliab. 47(4), 303–309 (2018). https://doi.org/10.3103/S1052618818040027 5. Briot, S., Arakelian, V., Le Baron, J.P.: Shaking force minimization of high-speed robots via center of mass acceleration control. Mech. Mach. Theory 57, 1–12 (2012). https://doi.org/10.1016/j.mechmachtheory.2012.06.006 6. Buganza, A., Acevedo, M.: Dynamic balancing of a 2-DOF 2RR planar parallel manipulator by optimization. In: 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, 19–25 June 2011 (2011) 7. Fattah, A., Agrawal, S.: Design and modeling of classes of spatial reactionless manipulators. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA 2003), Taipei, Taiwan, pp. 3225–3230 (2003) 8. Fattah, A., Agrawal, S.: Design arm simulation of a class of spatial reactionless manipulators. Robotica 3(1), 75–81 (2005). https://doi.org/10.1017/ S0263574704000670 9. Filaretov, V., Vukobratovic, M.: Static balancing and dynamic decoupling of the motion of manipulation robots. Mech. Mach. Theory 3(6), 767–783 (1993). https:// doi.org/10.1016/0957-4158(93)90062-7

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10. de Jal´ on, J.G., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. The Real-Time Challenge. Springer, New York (1994). https://doi.org/10. 1007/978-1-4612-2600-0 11. Nenchev, D.N.: Reaction null space of a multibody system with applications in robotics. Mech. Sci. 4, 97–112 (2012). https://doi.org/10.5194/ms-4-97-2013 12. van der Wijk, V., Herder, J.L.: Synthesis method for linkages with center of mass at invariant link point – pantograph based mechanisms. Mech. Mach. Theory 48, 15–28 (2012). https://doi.org/10.1016/j.mechmachtheory.2011.09.007

Energy Optimization of a Parallel Robot in Pick and Place Tasks Juan Pablo Mora1(B) , Juan Pablo Barreto2 , and Carlos F. Rodriguez1 1 Department of Mechanical Engineering, Universidad de los Andes, Bogotá, Colombia

{jp.mora807,crodrigu}@uniandes.edu.co 2 Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University,

Aachen, Germany [email protected]

Abstract. The increase in the number of industrial robots and the requirements for sustainable production have drawn attention to the study of energy consumption. The present research work studies the energy optimization of a five-bar linkage when performing pick and place tasks, combining trajectory optimization and the addition of springs in parallel to the actuators. First, a method to reduce the energy consumption for a general task is presented. Second, an algorithm to calculate the free vibration response and the springs parameters for a nominal task is explained. Finally, a new energy optimization method is presented to improve the versatility of the robot when a new task is required, while reducing the necessary changes on the system. The experimental results, regarding the free vibration trajectory, encourage the adequate implementation of the calculated springs in the prototype. Moreover, the results of the simulations, regarding the new method, indicate the versatility of the system if the nominal task changes. Keywords: Five-bar linkage · Natural motion · Free vibration response · Pick and place · Energy optimization

1 Introduction Nowadays, the study of energy consumption of industrial robots is very important due to the increase of installed units [1]. One of the most common operations of robotic manipulators are pick and place tasks, where both serial and parallel robots are used to achieve fast motion cycles [2, 3]. Therefore, some strategies have been studied to reduce the energy consumption of these systems. In [4, 5] comprehensive studies, that compile some important contributions on energy saving strategies, are presented. One of the strategies is based on the natural motion, which considers the dynamic response of the mechanical system due to the conversion of potential energy into kinetic energy [5]. Hence, the addition of elastic elements is required to exploit the free vibration response (FVR) of the system [6]. As a result, an oscillatory system suited for pick and place operations will be obtained. However, the natural motion in parallel robots is a subject

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 191–200, 2022. https://doi.org/10.1007/978-3-030-88751-3_20

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that is still under development [5]. For example, in [7] the natural motion of a five barlinkage is studied using a multibody simulation, while in [8, 9] the same topic is studied for a delta robot. In this contribution energy optimization of a five-bar linkage performing pick and place tasks is studied mainly in five stages. First, a computational model of the five-bar linkage is developed to simulate the motion between several pick and place points and estimate the energy consumption. Second, a method to find a minimum-energy trajectory for a general task is implemented. Third, an algorithm to calculate the FVR for a specific task and the springs parameters, namely the stiffnesses (k  i for i = 1, 4) and the torque exerted by the spring when the proximal link angle is zero T0i for i = 1, 4 , is explained. Fourth, free vibration trajectories are implemented in an experimental prototype. Fifth, a method is presented to reduce the energy consumption when the robot must perform tasks that vary significantly from the nominal task that was used for the design of the springs. Finally, some conclusions and ideas for future work are presented.

2 Prototype and Computational Model The prototype studied (Fig. 1a) is a five-bar linkage designed and manufactured in [10, 11]. The system is a planar parallel robot with two degrees of freedom actuated by two rotational servomotors at the ground joints. The control system is composed of commercial devices and software used in typical industrial systems. This should facilitate the transfer of the proposed methods. The generalized coordinates approach [12, 13] was used for the kinematic and kinetic analysis. A coordinate system is set at each physical link of the mechanism (Fig. 1b) with a defined orientation. When the vector of generalized coordinates q is specified, the −

position of the complete mechanism is known. T  q = x1 y1 φ1 x2 y2 φ2 x3 y3 φ3 x4 y4 φ4 , −

(1)

The angles φ1 and φ4 can be obtained solving the inverse kinematics problem from the workspace coordinates of the end effector that is placed at joint B (Fig. 1b). However, to obtain the whole information stored in Eq. (1), a set of twelve equations is needed. Thus, these equations built up the vector of constraints:    K − q, t =  D =0 − −  −

(2)

K and the The vector of constraints comprises the vector of kinematic constraints  − D . The first vector (K ) consists of ten scalar equations vector of drive constraints  − − D) that are obtained gathering two equations per joint. The vector of drive constraints ( − comprises two scalar equations to describe the motion of the two proximal links. Hence, they correspond to two time functions φi (t) for i = 1, 4. Finally, all equations must be written as shown in Eq. (2) and the kinematic scalar equations must be arranged in order

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a)

193

b)

Fig. 1. a) CAD model of the parallel robot. b) Five-bar linkage with generalized coordinates.

from the joint G1 to the joint G4 , this organizes the relative kinematic equations at the K. middle of  − Velocities and accelerations are obtained when Eq. (2) is differentiated: q˙ = −−1 q t ,

(3)

−

q¨ = −

−−1 q







q q˙ −

q˙ + 2qt q˙ + tt q

−

−

(4)

Where q and t are the partial derivatives of  with respect to the components of q −

and t, respectively. qt corresponds to the partial derivative of q with respect to t, and tt to the partial derivative of t with respect to t. Regarding the kinetic analysis of the whole system of constrained links, the NewtonEuler equations of motion are [12, 13]: M q¨ = Q + Tq λ − −

−

(5)

a

Where Q is the generalized vector of external forces and moments at each link, M the −

a

diagonal matrix of masses and inertias, and λ the vector of Lagrange multipliers used _

to obtain the internal forces. In this way, with Eq. (2) and Eq. (5) the driving torques (Ti for i = 1, 4.) that must be applied to the proximal links are calculated. The scalar time functions of the motion of the proximal links, that are contained D , are made up of polynomial functions of time. First, in order to define in vector  − these functions, N points, χ (i = 1, . . . , N ), by which the end effector must go −

i

through, are selected in the workspace. Second, the N angles of each proximal link are obtained throughout the inverse kinematics solution. Then, between these angles, N − 1 polynomial functions, of order k and equal duration, are generated as explained in [14]:

k fi (t) = aij (t − ti )j , i ∈ [1, N − 1], ti ≤ t ≤ ti+1 (6) j=0

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For the calculation of the (N − 1)(k + 1) coefficients the following conditions must be satisfied: fi (t = ti ) = φi , fi (t = ti+1 ) = φi+1 , fi

(n)

i ∈ [1, N − 1]

(n)

(t = ti+1 ) = fi+1 (t = ti+1 ),

i ∈ [1, N − 2] n ∈ [1, k − 1]

(7) (8)

Additionally, a set of (k − 1) boundary conditions are defined to determine all coefficients. At the end, a piecewise time function for each proximal link is determined. The performance of the system in terms of energy consumption is estimated using Eq. (9) and Eq. (10). The calculation of the power, Eq. (9), assuming that no energy recuperation is possible, is based on the mechanical power and an estimation of the electrical losses. Then, an estimation of the energy consumption is obtained by integrating the power over time, Eq. (10). This will be used later as cost function for the optimization of the trajectory:   

  Ti 2 ˙ max Ti φi , 0 + R (9) P= i=1,4 Gi kT

t E= Pdt (10) 0

Where Ti is the torque at the motor i, kT is the torque constant, R the resistance of the motors and Gi is the gear ratio, which in the prototype is equal to one (direct drive).

3 Results and Discussion In this contribution, to study the natural motion, the following optimization problem was developed for two objective functions (J ): argmin J ,

(11)

ξ ∈R(N −2)×2

In both problems an objective function J is minimized by the iterative calculation of a matrix (ξ ) that comprises the intermediate N − 2 points that define the trajectory in the workspace, as the initial and final rows are fixed by the pick and place points of the task. Both optimization problems were solved using the function fminunc of MATLAB® and the BFGS (Broyden, Fletcher, Goldfarb and Shanno) quasi-Newton algorithm [15]. This function finds the minimum of an unconstrained multivariable nonlinear function [16]. The input arguments were the objective function J and an initial guess ξ 0 . Moreover, the feasibility of the solution was verified in terms of the workspace of the robot as well as the maximum torque, velocity and acceleration of the motors. The first optimization problem, based on [14], deals with the reduction of the estimated energy consumption for a given task using Eq. (10) as the objective function (i.e. J1 = E). This procedure works for both the mechanism with and without springs. Thus,

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on each iteration, the points defining the trajectory are changed and the cost function Eq. (10), is calculated. Figure 2 shows the evolution of the optimization process for the mechanism without springs presenting two intermediate iterations (Tr. 2 and Tr. 4). The method starts with an arbitrary initial guess (Tr. 1) and, as shown in Fig. 2c, the value of the estimated energy consumption is gradually reduced until it converges to a minimum-energy trajectory (Tr. 5). A typical point-to-point trajectory (Tr. 3), namely a fifth-order polynomial [17], is also included for comparison.

b)

a)

c)

Fig. 2. Minimum energy algorithm. Polynomial trajectories with N = 7 and k = 5. Velocities and accelerations at pick and place points are zero. a) Trajectories at workspace. b) Power requirement. c) Energy consumption. The colors in each figure correlate.

The second optimization problem, based on the contributions in [6, 9], calculates the optimal spring parameters of two torsional springs mounted in the ground joints. In this case, the trajectory is modified until the relation between the angle of each proximal link (φi ) and the driving torque (Ti ) is linear. Consequently, on each iteration, the torque is calculated for the trajectory guess and a linear regression is applied between the torque and the angle of the proximal link to obtain an estimator (T i ) for the torque. Once the regression is good enough, the spring stiffness ki will be the negative of the slope (ki∗ ) of the resulting linear function, and the other spring parameter T0i will be the intercept. Consequently, the equilibrium angle of the spring would be φeq i = T0i /ki . 



T i = ki∗ φi + T0i , i = 1, 4

(12)

For this optimization, the objective function, defined by the sum of the mean squared errors (MSE) of both regressions, is given by Eq. (13):

J2 = (13) (MSE)i i = 1, 4 i=1,4

Where each MSE, defined as the average squared difference between the estimated and the actual torque, is given by Eq. (14): MSEi =

n j=1

2 1 ˆ n → Observations and i = 1, 4 Tij − Tij n

(14)

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If a perfect linear relation is obtained, the resulting trajectory would correspond to the FVR of the mechanism with springs. Therefore, in absence of dissipative forces and starting from the pick position, the system would oscillate between the pick and place positions without any intervention from the motors. Figure 3 and Table 1 summarize the results of this optimization. For eight different tasks, the FVR and the corresponding spring parameters were calculated. Figure 3a illustrates the motion of the end effector in the workspace and Fig. 3b, c show the resulting linear relationship between the torque and the angle of the proximal links. Additionally, Table 1 shows the resulting springs parameters and each of the MSE values that were obtained with the linear fits. The low values of the mean squared errors, MSE 1 and MSE 4 , indicate that the behavior between the torque and the angle is very close to be linear and that the algorithm works for several trajectories.

b)

a)

c)

Fig. 3. FVR Trajectories. Polynomial trajectories with N = 7 and k = 5. No detention time and task period of 2 s. Velocities and jerk at pick and place points are zero. a) Trajectories at workspace. b) and c) Relationship between torque and angle. The colors in each figure correlate.

Table 1. Torsional spring parameters for each task and MSE for each regression.   Nm k1 rad

T01 [Nm]

1

0.852

2.124

4.696

1.043

0.997

1.196

2

0.892

2.485

4.399

0.607

0.005

8.292

3

0.819

1.974

9.879

0.951

0.567

9.283

4

0.857

2.235

3.827

1.325

1.728

2.406

5

1.046

1.790

0.908

0.961

0.507

2.239

6

0.547

1.331

2.624

0.547

0.389

2.609

7

0.987

1.455

3.879

1.055

0.291

1.929

8

0.361

0.576

3.399

0.590

-0.327

0.490

Task

MSE 1 ×10−6 [Nm]2

  Nm k4 rad

T04 [Nm]

MSE 4 ×10−6 [Nm]2

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Some experimental tests were performed in order to evaluate the possibility of implementing springs and the free vibration trajectories. The FVR and the springs parameters were calculated for Task 3 using a task period of 2 s and 0.67 s, then the motions were implemented in the experimental prototype (without springs). Figure 4 illustrates the experimental measurements and the simulation results of the torque (Ti ) versus the angle of each proximal link. Furthermore, a linear regression of the experimental data was performed to assess the goodness of fit of the measurements compared to the springs’ parameters obtained in the simulation.

a)

b)

Fig. 4. Experimental (Exp.) and simulated (Sim.) torque and angle for each proximal link. Task period of a) 2 s, b) 0.67 s

The experimental results show a good correspondence with the simulations. The relative error between the experimental and the simulated value of k ∗ and T0 indicates that the required torque for the motion of the links during this trajectory could effectively be balanced by constant-stiffness springs. However, Fig. 4 also shows that the springs can only balance the inertial terms, while friction and other unmodeled forces still must be compensated by the motors. Similarly, it is observed that the reduction of the cycle time significantly increases the effect of the inertial forces, especially because a direct drive was employed (Table 2). The method in [6, 9] is based on the idea of calculating the springs parameters for a nominal task and keeping them fixed independently of the task changes. However, this can lead to an increase in the energy consumption for tasks that differ significantly from the nominal task. Thus, this contribution presents one alternative to improve the versatility of the robot, while reducing the necessary changes on the system. The main idea is that T0i can be modified when a new task, that differs significantly from the nominal one, is required. The new optimization problem in Eq. (15) also takes Eq. (10) as the objective function but uses a new argument. arg min J1 ,

K∈R(N −1)×2

(15)

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Table 2. Slope and intercepts of the experimental and simulated results. The percentage errors are shown. Results 

Task period of 2 s 

Sim.

Exp.

Task period of 0.67 s

PE [%] Sim.

Exp.

PE [%]

−0.82 −0.80 2.44

−7.38 −7.47 1.22

T01 [Nm] 1.97 1.91 3.05   −0.95 −0.93 2.11 k4∗ Nm rad

17.77 17.86 0.51

k1∗ Nm rad

T04 [Nm]

0.57

0.53 7.02

−8.56 −8.65 1.05 5.10

5.32 4.31

Where, K is a (N − 1) × 2 matrix, in which the first row contains the values T01 and T04 , while the remaining rows contain the intermediate points of the trajectory.

a)

b)

Fig. 5. a) Trajectories for tasks 1, 2, 3, 6 and 8. b) Energy consumption of each method and task.

Figure 5 shows the results of this optimization problem. The springs were designed for task number one (nominal), but four additional tasks were simulated. For this tasks, three different methods were employed. Namely, a minimum-energy trajectory without springs (ME), a minimum-energy trajectory with fixed springs parameters (MES) and a minimum-energy trajectory, in which the spring stiffness remains fixed, but T0i is optimized for each task (MET). Figure 5a shows the calculated trajectories for each task and Fig. 5b. shows their corresponding estimated energy consumption. The MES method leads to a reduction of the estimated energy consumption, with respect to the ME method, of 60.7% for task 2, 92.0% for task 3, and 1.2% for task 6. Nevertheless, when the MES method is applied to task 8, no reduction in the energy consumption is obtained due to the big difference between the tasks. Thereby, the MET method is presented as an alternative, in which T0i and the trajectory are changed, to produce a drop in the energy consumption. Energy reductions

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of 33.0%, 90.6%, 59.3% and 98.3%, with respect to the MES method, are obtained for tasks 2, 3, 6 and 8, respectively. Similarly, energy reductions of 73.7%, 99.2%, 59.7% and 51.3%, with respect to the ME method, are obtained for tasks 2, 3, 6 and 8, respectively. This drop in the estimated energy consumption shows the great advantage of the proposed method. However, it is important to consider the feasibility of implementing these ideas, in terms of achieving the required spring parameters and trajectories, since for some tasks high deflections on the springs may be obtained during the motion. This should be examined in future work.

4 Conclusions One strategy that has been proposed to reduce the energy consumption of industrial robots performing pick and place tasks is to use constant-stiffness springs, incorporated in parallel to the motors and designed for a nominal task, and to adjust the trajectory that the robot follows for a general task. The underlying idea is that the springs provide some of the necessary torque for the motion. In absence of dissipative forces, the robot with springs should oscillate between the pick and place points without additional intervention of the actuators. Moreover, the relation between the required torque to move the mechanism without springs, following the free vibration response (FVR), and the angle of the proximal links should be linear. The experimental results of this contribution show a good correspondence between the simulated linear relation and the measured data. This encourages the usage of this strategy to obtain a reduction in the energy consumption. An adequate assembly of the calculated springs and the implementation of optimal trajectories should reduce the energy consumption of robots performing pick and place tasks. However, as the springs parameters are calculated for a nominal task, a significant change in the task could generate a raise in the energy consumption. To study the influence of the task variation, the MES method (optimal trajectory with fixed spring parameters) was implemented for a set of tasks. The results indicate that for some tasks that differ significantly from the nominal, no energy reduction is obtained by using the MES method or the energy consumption is even increased. Consequently, a possible method to solve this issue was presented in this contribution. The proposed method (MET) consists in varying T0i , while keeping the same stiffness that was calculated for the nominal task. The results show that even for tasks that differ significantly from the nominal, the MET method achieves a reduction on the estimated energy consumption with respect to the mechanism without springs following a minimum-energy trajectory (ME). Consequently, this method could be used to increase the versatility of the robot with springs (the possibility to fulfill multiple tasks using a single set of springs) and to compensate eventual variations between the designed and the manufactured springs. The results of the simulations indicate that the implementation of springs and the studied optimization methods should reduce the energy consumption of the five-bar linkage performing pick-and-place tasks. However, a deeper study on the experimental implementation of the springs and the optimal trajectories is required and recommended for future work. Finally, a further study on the variation of the task characteristics (pick and place positions and cycle time) are suggested for future work.

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References 1. International Federation of Robotics. IFR. Executive Summary World Robotics 2020 Industrial Robots (2020) 2. Angeles, J.: Trajectory planning: pick and place operations. In: Angeles, J. (ed.) Fundamentals of Robotic Mechanical Systems, pp. 233–256. Springer, Boston (2007). https://doi.org/10. 1007/978-0-387-34580-2_6 3. SP Automation & Robotics: Pick & Place Automation/Robotic Material Handling. https:// sp-automation.co.uk/pick-place-application/. Accessed 15 Dec 2020 4. Carabin, G., Wehrle, E., Vidoni, R.: A review on energy-saving optimization methods for robotic and automatic systems. Robotics 6(4), 39 (2017) 5. Scalera, L., Palomba, I., Wehrle, E., Gasparetto, A., Vidoni, R.: Natural motion for energy saving in robotic and mechatronic systems. Appl. Sci. 9(17), 3516 (2019) 6. Barreto, J.P.: Exploiting the natural dynamics of parallel robots for energy-efficient pick-andplace tasks. Dissertation/Ph.D. Thesis, Institute of Mechanism Theory, Machine Dynamics and Robotics. RWTH Aachen University, Aachen (2021) 7. Barreto, J.P., Schöler, F. J.-F., Corves, B.: The concept of natural motion for pick and place operations. In: Corves, B., Lovasz, E.-C., Hüsing, M., Maniu, I., Gruescu, C. (eds.) New Advances in Mechanisms, Mechanical Transmissions and Robotics. MMS, vol. 46, pp. 89–98. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-45450-4_9 8. Barreto, J.P., Corves, B.: Matching the free-vibration response of a delta robot with pickand-place tasks using multi-body simulation. In: 2018 IEEE 14th International Conference on Automation Science and Engineering (CASE), Munich, pp. 1487–1492 (2018) 9. Barreto, J.P., Corves, B.: Resonant delta robot for pick-and-place operations. In: Uhl, T. (ed.) Advances in Mechanism and Machine Science IFToMM WC 2019. Mechanisms and Machine Science, vol. 73, pp. 2309–2318. Springer, Cham (2019). https://doi.org/10.1007/978-3-03020131-9_228 10. Gallego Rivera, J.: Prototipo para el estudio de ahorro energético en operaciones pick and place. Bachelor thesis (in Spanish), Universidad de los Andes, Bogotá, Colombia (2019) 11. Hermoza Franco, D.Y.: Mechanical design of a planar parallel kinematic manipulator including elastic elements for energy recuperation. Mini thesis, Institut für Getriebetechnik, Maschinendynamik und Robotik, RWTH Aachen University, Aachen, Germany (2019) 12. Rodríguez Herrera, C.F.: Modelado de mecanismos con coordenadas generalizadas. Departamento de Ingeniería Mecánica, Universidad de los Andes, Bogotá, Colombia 13. Huang, E.J.: Computer-Aided Kinematics and Dynamics of Mechanical Systems, vol. II: Modern Methods, ResearchGate (2020) 14. Lorenz, M., et al.: Energy-efficient trajectory planning for robot manipulators. In: ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 5B: 41st Mechanisms and Robotics Conference, Cleveland (2017) 15. Dai, Y.-H.: Convergence properties of the BFGS algorithm. SIAM J. Optim. 13, 693–701 (2002) 16. MathWorks: fminunc. https://www.mathworks.com/help/optim/ug/fminunc.html#but9rn9-5. Accessed 13 July 2021 17. Biagiotti, L., Melchiorri, C.: Trajectory Planning for Automatic Machines and Robots. Springer, Heidelberg (2008)

Gravity Compensation of Articulated Robots Using Spring Four-Bar Mechanisms Vu Linh Nguyen(B) National Chin-Yi University of Technology, Taichung 411030, Taiwan [email protected]

Abstract. This paper presents a design method for gravity compensation of articulated robots. The method is realized by installing spring four-bar mechanisms onto the robot to eliminate the gravity effect on its joints. This elimination allows reducing the actuation torque and energy consumption of the robot during operation. The design of the spring mechanisms is obtained by solving an optimization problem for minimizing the actuation torque of the robot. A numerical example is provided to demonstrate the proposed design method, by which a significant reduction in the actuation torque over the robot workspace is received. A simulation-based validation showed that the actuation torque of the robot could be reduced by approximately 90% when the design method is applied. Keywords: Gravity compensation · Static balancing · Planar four-bar linkage

1 Introduction Reducing the motor effort and actuation torques of robotic manipulators has become one of the main focuses in the advancement of green robotic systems [1]. The reduction of actuation torque allows decreasing the energy consumption of the robot and operating costs [2]. When robots are performed at a low speed, the gravitational torques are more significant than the dynamic torques [3]. In this situation, the design for decreasing the gravitational torques, i.e., gravity compensation, is beneficial to reduce the actuation torques and energy consumption. Gravity compensation can also increase robot performance, safety, and efficiency [4]. To design a robotic manipulator with gravity compensation, passive energy elements, such as counterweights [5, 6] or mechanical springs [7–9], are attached to the robot to compensate for the variation of the potential energy of the robot. This compensation will lead to decreasing the actuation torques of the robot. There are many gravity compensation methods applied to robotic manipulators in the literature. For example, Lacasse et al. [10] showed a seven-degrees-of-freedom (7-DoF) partially statically balanced robot, where remote counterweights are connected to the robot through a low-pressure hydraulic transmission. Kim et al. [11] presented a gravity compensation mechanism based on a slider-crank mechanism and a spring. The spring mechanism is installed onto a multi-DoF robot arm using a double-parallelogram mechanism and bevel gear units. Nguyen et al. [12] proposed the design concept of a gear-spring module, and a series © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 201–209, 2022. https://doi.org/10.1007/978-3-030-88751-3_21

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of the modules were used for the gravity compensation of multi-DoF planar articulated manipulators. Lee et al. [13] presented a wire-driven gravity compensation mechanism and integrated it with timing belts and pulleys to build a multi-DoF gravity-balanced robotic manipulator. This paper will put forward a gravity compensation method for articulated robots to reduce the actuation torque and energy consumption. A spring four-bar mechanism is first conceptualized for compensating the gravitational torque induced by a single rotating weight. The significance of the proposed mechanism is that it has a simple kinematic structure and provides an excellent balancing performance. The spring mechanism is then installed onto the robot for gravity compensation. A numerical example and a simulation-based validation are provided to exhibit the performance of the proposed design method. The conceptual design for the gravity compensation of articulated robots is presented in Sect. 2. Then, Sect. 3 details the gravity compensation method and design procedure. The performance evaluation of the design method is shown in Sect. 4. Last, Sect. 5 concludes the present paper.

2 Conceptual Design The gravity compensation design of an articulated robot is depicted in Fig. 1. A KUKA KR 210 R3100 industrial robot [14] is taken as an example. Since the gravitational torque of a joint is accumulated to its preceding joints, the last three joints near the end-effector can be neglected in the compensation design. The first joint (between links 0 and 1) aligns with the direction of gravity. Thus, the gravity effect on this joint is theoretically zero and also ignored. For gravity compensation, two spring four-bar mechanisms Oi Ai Bi C i (i = 1, 2) are attached to the robot so that joints O1 and O2 are coincident with the shoulder (2nd) joint (between links 1 and 2) and the elbow (3rd) joint (between links 2 and 3) of the robot, respectively. Links Oi Ai and Oi E i are fixed to links i and i + 1, respectively. An additional link A1 A2 is used to maintain the direction of link O2 A2 when link 2 is rotated. Two tension springs k i are used, and each spring k i is connected from link Bi C i (at point Di ) to link i + 1 (at point H i ).

3 Gravity Compensation Method 3.1 Gravitational Torques Let mci , L i , l ci denote the mass, link length, and location of the mass center of link i + 1 (i = 1, 2 for the robot links; i = 3 for a payload), respectively. Note that mc2 also includes the masses of links 4, 5, and 6. Then, the gravitational torques T gi at joints Oi can be expressed as: Tg1 = M1 LC1 g cos θ1 + M2 LC2 g cos θ2

(1)

Tg2 = M2 LC2 g cos θ2

(2)

Gravity Compensation of Articulated Robots D2

B2

Gravity

C2

k2

O2

A2

H1 k1

E2

θ2

Elbow (3rd) joint

H2

G2

O3

mc2g Link 3

G1

E

mc1g

D1

C1

B1 Link 1

G3 mc3g

L1 = O 1 O 2 Shoulder (2nd) joint

Link 0

Payload

θ1

O1

A1

Link 2

E1

203

L2 = O 2 O 3 L3 = O 3 E lc1 = O1G1

y

lc2 = O2G2

x

Fig. 1. Conceptual design for the gravity compensation of an articulated robot.

where M1 = mc1 + mc2 + mc3 , M2 = mc2 + mc3 , LC1 = LC2 =

mc2 lc2 + mc3 L2 mc2 + mc3

mc1 lc1 + (mc2 + mc3 )L1 mc1 + mc2 + mc3 (3)

In Eqs. (1)–(3), θ i and g represent the joint angle of link i + 1 and the gravitational acceleration (g = 9.81 m/s2 ), respectively; M i and L Ci stand for the equivalent rotating mass applying to joint Oi and its location, respectively. Equation (2) shows that the gravitational torque T g2 is independent of the joint angle θ 1 . This means that the compensation design for this joint can be done without considering the configuration of link 2. If the gravitational torque T g2 is canceled by the elastic forces induced by the spring k 2 , the gravitational torque T g1 in Eq. (1) will also be independent of the joint angle θ 2 . Therefore, the compensation design for these two joints can separately be performed if their equivalent masses (to be gravity compensated) and their locations are known. 3.2 Actuation Torques Figure 2 illustrates the free-body diagram of a spring four-bar mechanism. The mechanism is assumed to counterbalance a weight mc with a distance l c from the mass center G to the rotation joint O. Let Fs = [F sx , F sy ]T denote the spring force vector applied at point D, and it can be derived as:     Fs cos γD Fsx = (4) Fs = Fsy Fs sin γD

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where Fs = k(s1 + s0 ), γD = cos

−1



x • DH xDH

 (5)

In Eqs. (4) and (5), k, s1 , s0 , γ D , and x represent the spring stiffness coefficient, spring displacement, spring pre-extension, angle representing the direction of the spring force Fs , and the unit vector of the x-axis, respectively. H

G

Gravity

Tg

D Fs = Fsμs

Fsy

γD

D

B

C TH

l3

F32

B l2

P4

F32x

T34 l4

m4g

m2g

F34

F34y

-F34

C

F32y

P2

m3g

H

-Fs

Fsx l3b

P3

l3a -F32

mcg

F34x l4b l4a

y O

A

E

θ x

l1

Fig. 2. Free-body diagram of a spring four-bar mechanism.

Recall that the equilibrium of forces for a body link i under static conditions can be written as:   Fij + Fik + mi g = 0 (6) j

k

where Fij and Fik stand for the reaction force vector and external force vector (i.e., the gravitational and spring forces) exerted on the i-th link, respectively. Next, the equilibrium of torques (or moments) for the i-th link is expressed as:    rij × Fij + (7) (rik × Fik ) + mi (rci × g) + Tmi = 0 j

k

where rij , rik , and rci are the position vectors associated with the reaction force Fij , the external force Fik , and the mass center of link i, respectively; and Tmi denotes the actuation torque acting on the i-th link. By applying Eqs. (6) and (7) to links 2, 3, and 4, the actuation torque T m at joint O can be derived as: Tm = T34 + TH − Tg

(8)

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205

where T34 = xC F34y − yC F34x − xP4 m4 g

(9)

TH = xH Fsy + yH Fsx

(10)

Tg = −xG m4 g

(11)

3.3 Design Procedure The design of the spring four-bar mechanism aims to reduce the actuation torque T m . Alternatively, minimizing the torque reduction ratio (TRR), i.e., the actuation torque T m over the gravitational torque –T g , can be considered a design objective. For a range of the joint angle θ, only the peak actuation torque is considered. Therefore, an optimization design problem is established as: Minimize

max{|Tm (X, θ )|}



max −Tg (X, θ )

with subject to

X = [l1 , l2 , l3 , l4 , ls , l3a , l3b , l4a , l4b , k, s0 ], X ∈  θ = [θint , θend ] l1 , l2 , l3 , l4 , ls , l3a , l3b , l4a , l4b , k > 0 s0 ≥ 0

(12)

where θ int and θ end stand for the initial and end joint angles of the spring four-bar mechanism, respectively. These two angles can also be used to define the robot workspace (see Fig. 1). The design procedure for the gravity compensation of an articulated robot using the spring four-bar mechanisms (see Fig. 2) can be detailed as: – Step 1. Determine the equivalent rotating masses impacting the shoulder and elbow joints and their locations using Eq. (3); – Step 2. Define the input parameter vector X of each spring mechanism and the joint angles (θ 1 , θ 2 ); – Step 3. Use the Genetic Algorithm (GA) to solve the optimization design problem in Eq. (12) for each spring mechanism.

4 Performance Evaluation This section shows a case study for the gravity compensation of the KUKA robot in Fig. 1. The link lengths and masses of the robot are estimated from the CAD model of the robot, as listed in Table 1. The joint angles of the robot are limited within θ 1 = [0, π /2] and θ 2 = [–π /3, π /3].

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V. L. Nguyen Table 1. Estimated parameters of the KUKA KR 210 R3100 robot.

L 1 (m)

L 2 (m)

L 3 (m)

l c1 (m)

l c2 (m)

mc1 (kg)

mc2 (kg)

mc3 (kg)

1.35

1.4

0.256

0.525

0.482

349

324

30

Table 2. Optimal parameters of the spring four-bar mechanisms. 4-bar l 1 (m) l 2 (m) l 3 (m) l 4 (m) l s (m) l 3a (m) l 3b (m) l 4a (m) l 4b (m) k (N/mm) s0 (m) 1st

0.10

0.25

0.294

0.284

1.147 0.298

0.197

0.244

0.17

616

0.113

2nd

0.132

0.244

0.237

0.236

0.86

0.13

0.28

0.146

231.8

0.065

0.249

By solving the optimization design problem in Eq. (12) using the GA method, the best objectives of the compensation designs for joints O1 and O2 are illustrated in Fig. 3. It shows that the best objective for joint O1 converges to a minimum value of 0.018529 within 119 iterations. The design for joint O2 provides a smaller best objective, i.e., 0.018418, but it requires 344 iterations for convergence. Based on the obtained results, the optimal parameters of the spring four-bar mechanisms can be found, as presented in Table 2.

Best objective (TRR)

O1 O2

Number of iteration

Fig. 3. Convergence of the best objectives with GA optimization.

By adopting the data given in Table 1 and Table 2, the total actuation torques before and after gravity compensation (called the original and compensated torques) over the considered workspace are illustrated in Fig. 4. There is a significant reduction in the actuation torque at any position in the workspace. The maximum original torque (Fig. 4(a)) is about 9096.4 N-m, which is much higher than the compensated one (Fig. 4(b)), i.e.,

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167.8 N-m. These results indicate that the actuation torque of the robot can be reduced by up to 98.2%.

Fig. 4. Distribution of the total actuation torque: (a) original and (b) compensated torque.

Let us take the KUKA robot into the MSC Adams simulation. The robot is assumed to track a trajectory, as depicted in Fig. 5(a). During the tracking, the original and compensated torques of the robot at the 2-nd and 3-rd actuation joints are measured, and the results are shown in Fig. 5(b). The compensated torques at the two joints are considerably smaller than the original ones. At the 2-nd joint, the maximum actuation torque is about 6204 N-m, and it is reduced to 663 N-m when the compensation design is applied. These results mean that the torque reduction rate at this joint could be up to 89.3%. Similarly, the actuation torque at the 3-rd joint is reduced from 2000 N-m to 82 N-m, equivalent to 95.9%. Note that when the payload mc3 is varied, it is necessary to rerun the optimization algorithm.

5 Conclusion This paper has proposed a gravity compensation method for articulated robots using the spring four-bar mechanisms. The design of the spring mechanisms was realized via minimizing the actuation torques of the robot with GA optimization. A case study was presented where the total torque of the robot was considerably reduced over its workspace. The effectiveness of the design method was also validated using MSC Adams simulation. A simulation example showed that the actuation torque of the robot could be reduced by more or less 90%.

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Fig. 5. Simulation with the KUKA robot: (a) studied trajectory and (b) actuation torques.

Acknowledgments. This paper was supported by the Ministry of Science and Technology (MOST), Taiwan (grant number MOST 110-2222-E-167-004).

References 1. Brossog, M., Bornschlegl, M., Franke, J.: Reducing the energy consumption of industrial robots in manufacturing systems. Int. J. Adv. Manuf. Technol. 78(5–8), 1315–1328 (2015) 2. Richiedei, D., Trevisani, A.: Optimization of the energy consumption through spring balancing of servo-actuated mechanisms. ASME J. Mech. Design 142(1), 012301 (2020) 3. Arakelian, V.: Gravity compensation in robotics. Adv. Robot. 30(2), 79–96 (2016) 4. Carricato, M., Gosselin, C.: A statically balanced Gough/Stewart-type platform: conception, design, and simulation. ASME J. Mech. Robot. 1(3), 031005 (2009) 5. Van der Wijk, V.: Design and analysis of closed-chain principal vector linkages for dynamic balance with a new method for mass equivalent modeling. Mech. Mach. Theory 107, 283–304 (2017) 6. Kuo, C.-H., Nguyen, V.L., Robertson, D., Chou, L.-T., Herder, J.L.: Statically balancing a reconfigurable mechanism by using one passive energy element only: a case study. ASME J. Mech. Robot. 13(4), 040904 (2021) 7. Herder, J.L.: Energy-free systems: theory, conception and design of statically balanced spring mechanisms. Ph.D. thesis, Delft University of Technology, Delft, The Netherlands (2001) 8. Nguyen, V.L., Lin, C.-Y., Kuo, C.-H.: Gravity compensation design of Delta parallel robots using gear-spring modules. Mech. Mach. Theory 154, 104046 (2020) 9. Essomba, T.: Design of a five-degrees of freedom statically balanced mechanism with multidirectional functionality. Robotics 10(1), 11 (2021) 10. Lacasse, M.-A., Lachance, G., Boisclair, J., Ouellet, J., Gosselin, C.: On the design of a statically balanced serial robot using remote counterweights. IEEE International Conference on Robotics and Automation (ICRA), Karlsruhe, Germany, pp. 4189–4194 (2013) 11. Kim, H.-S., Min, J.-K., Song, J.-B.: Multiple-degree-of-freedom counterbalance robot arm based on slider-crank mechanism and bevel gear units. IEEE Trans. Rob. 32(1), 230–235 (2016)

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12. Nguyen, V.L., Lin, C.-Y., Kuo, C.-H.: Gravity compensation design of planar articulated robotic arms using the gear-spring modules. ASME J. Mech. Robot. 12(3), 031014 (2020) 13. Lee, D., Seo, T.: Lightweight multi-DOF manipulator with wire-driven gravity compensation mechanism. IEEE/ASME Trans. Mechatron. 22(3), 1308–1314 (2017) 14. KUKA Robotics: The KR QUANTEC series, KR 210 R3100. KUKA Industrial RoboticsHigh Payloads (2016)

Mechatronic Design

Structural Design and Sliding Mode Control Approach of a 4-DoF Upper-Limb Exoskeleton for Post-stroke Rehabilitation Johan Nu˜ nez-Quispe(B) , Alvaro Figueroa, Daryl Campusano, Johrdan Huamanchumo, Axel Soto, Ebert Chate, Jesus Acu˜ na, Juan Lleren, Jose Albites-Sanabria, Leonardo Paul Mili´ an-Ccopa, Kevin Taipe, and Briggitte Suyo Advanced Biomechatronics Research and Innovation Laboratory, Universidad Nacional de Ingenier´ıa, Lima, Peru {johan.nunez.q,efigueroac}@uni.pe

Abstract. In this research, a 4-DoF upper limb exoskeleton was designed to assist patients while performing cross-pattern movements in a Proprioceptive Neuromuscular Facilitation (PNF) rehabilitation task. The mechanical design included 3 DoF for the shoulder and one for the elbow. The exoskeleton was designed and validated by structural simulation based on finite element analysis. The rehabilitation trajectory was obtained from a healthy person by image processing and a Sliding Mode Control (SMC) strategy was applied for tracking. Results from the structural simulation show that the factor of safety is over 1.3 on average for the proposed exoskeleton design. In addition, results show that the controller is robust to variations in the arm weight, with a tracking error of less than 0.0025 rad. Keywords: Upper-limb exoskeleton · Sliding mode control · Mechanical design · Finite element analysis · Robot-aided rehabilitation

1

Introduction

PNF is an approach to exercise therapy that uses specific movement patterns in diagonal and spiral directions together with specific techniques that facilitate the increase in strength and muscle function. An important principle of PNF is that after a muscle has contracted maximally, it will then relax maximally. Then, its opposite counterpart (antagonist) will relax maximally. This can be used by asking the patient to maximally contract the agonist to the muscle to be mobilized followed by application of a stretch [1,3,16]. Regarding great precision and accuracy for repetitive movements with robotic systems, exoskeletons for assisted therapy have been widely adopted in recent years to improve and evaluate motor skills such as muscle strength, velocity, and proficiency in activity of daily life c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 213–223, 2022. https://doi.org/10.1007/978-3-030-88751-3_22

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[5]. In recent years, different exoskeleton models have been developed to achieve optimal trajectory tracking. The design, modelling and implementation of an upper limb exoskeleton controlled by movement commands was designed in [11]. The authors analyzed the mean error and response time for each joint angle, but they did not reduce the tracking error using control methods. In [9], a fuzzy sliding mode controller was applied to a 5-DoF upper-limb exoskeleton robot for position tracking, although they achieved a chattering-free control and reduced mathematical calculation volume, their motors required a very high torque value to control the system. The application of these technologies seeks a physical configuration of the robotic system that adapts, mainly, to a comfortable position of the patient’s limb on the robotic arm [4,7]. For the mechanical modeling and simulation, the exoskeleton design was simplified in order to reduce simulation time keeping the material properties for each piece, as developed in [2,4], and virtual materials as explained below.

2 2.1

Methodology Mechanical Design

For the mechanical design, the following degrees of freedom were considered: 3 DoF for the shoulder and 1 DoF for the elbow. [2,5]. In contrast with other exoskeleton designs [2,13], the proposed structure allows to be manufactured without cutting and excessive welding and only need manufacturing processes such as folding and milling (see Fig. 1). In order to validate the proposed mechanical design, a mechanical model was prepared with the real boundary conditions, contacts, interactions and materials as shown in Fig. 2a according to [4,7]. Regarding boundary conditions (Fig. 2b), they were reduced to 3 conditions to reduce the simulation time. To have a good mass estimation for these components, two new virtual materials, which represent only the mass properties of the components, were set up. In Eqs. (1) and (2), the mass estimation and density for each component are described, respectively. nc  mi (1) mc = i=1

ρc =

mc Vc

(2)

Where nc is the number of pieces in each component. This means that the mass of complex components were approximated while maintaining general mechanical properties to reduce simulation time. Table 1 shows the properties of the obtained virtual materials for the motor and gearbox.

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Fig. 1. Mechanical design scheme and applied manufacturing processes.

Fig. 2. Upper-limb exoskeleton. (a) Contacts and joints. (b) Boundary conditions.

Table 1. Materials properties used on the exoskeleton design and simulation Material

Elastic module [MPa]

Poisson’s Mass density Elastic limit ratio [kg/m3 ] [MPa]

Duraluminum

73000

0.33

2800

Motor

69000

0.33

3475.421

255

200000

0.26

4653.378

550

Gearbox

480

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After the boundary conditions were set up, the model was fully prepared for the structural simulation after modeling the CAD assembly on Solidworks software, according to the procedure explained above. Results for mechanical model were interpreted by the authors by analyzing the structural simulation, specifically for Von Mises stress defined in Eq. (3), contacts interaction and factor of safety (FOS) determined by Eq. (4) according to [15], given as follows:  1 [(σ1 − σ2 )2 + (σ1 − σ3 )2 + (σ2 − σ3 )2 ] (3) σV m = 2 F OS =

σy σV m

(4)

Where σV m is the stress according to Von Mises failure theory in MPa, σi:1,2,3 are the main stress on a certain point, σy is the tensile yield stress for the material in MPa, and F OS is the factor of safety [15]. As mentioned before, the structural simulation software use mathematical models based on Eq. (3), Eq. (4) to determine the mentioned parameters [4,8]. 2.2

Rehabilitation Trajectory

PNF therapy of the upper limb rehabilitation process was performed by a healthy person for 10 s (see Fig. 3a). To obtain the trajectory, the Kinect for Xbox 360 was employed. This camera is widely used for trajectory tracking [10]. It has two sensors: an infrared depth sensor and an RGB color sensor. The Brekel Kinect Pro V1 software was used to record the elbow and shoulder data. This data was collected in a file with the extension bvh (biovision hierarchy), which provides motion capture data. Twenty four points of the human skeleton were registered, but only the points located in the wrist, elbow and shoulder of the right arm were considered. For the calculation of the rotation angles q1 , q2 , q3 and q4 , shown in Fig. 3b, transformation matrices were used. The vectors that join the shoulderelbow in the front and top plane allows to find q1 and q2 respectively. Similarly, the vectors that join the shoulder-elbow and elbow-wrist in the lateral plane were used to find q3 and q4 respectively. In addition, due to small fluctuations in the movement of the upper limb, a first-order Butterworth low-pass filter was applied to smooth the trajectory.

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Fig. 3. (a) Recording of skeleton coordinates in Brekel Kinect Pro V1 software. (b) Obtaining the rotation angles in the MATLAB IDE.

2.3

Control Design

2.3.1 Dynamic Analysis To define the dynamics of the exoskeleton, the reference frames are first attached according to Denavit-Hartenberg (DH) convention. Table 2 shows the DH parameters of the exoskeleton. The dynamic model of the 4-DoF exoskeleton is defined using the energy method based on the Lagrange Euler equation [6], which is expressed as: D(q)¨ q + N (q, q) ˙ =u (5) N (q, q) ˙ = C(q, q) ˙ q˙ + g(q) + F F (q) ˙

(6)

where q ∈ R4 are the angular positions of each joint, u is the control signal, ˙ ∈ R4x4 is the Coriolis matrix and D(q) ∈ R4x4 is the inertia matrix, C(q, q) 4 ˙ ∈ R4 the viscous friction torque. N (q, q) ˙ g(q) ∈ R is the gravity vector, F F (q) represents the internal forces and torques of the system.

Table 2. DH parameters of the exoskeleton Joint a

d

θ

α

1

0

d1 q 1

2

0

d2 q2 + π/2 π/2

3

a3 0

q3 + π/2 0

4

a4 0

q4

π/2

0

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2.3.2 Sliding Mode Control The sliding surface for each joint is expressed in Eq. (7) as: si = λei + e˙i ,

1 ≤ i ≤ 4 ∧ λi > 0

(7)

where ei is the tracking error for each joint, and is defined in Eq. (8) as: ei = qi − qdi

1≤i≤4

(8)

where qi is the angular position of each joint and qdi the desired angular position. The control law must satisfy the condition defined in Eq. (9), considering the Lyapunov candidate function for each component: Vi =

1 2 S 2 i

(9)

Finally, the control law is expressed as: s u = D.(¨ qd − λe˙ + N − k.sat( )) φ

(10)

where ki are the required control resources and λi are scalar gains to establish the rate of exponential convergence to zero of the tracking error after the system reaches si = 0. ki are defined in Eq. (11) from [12]: + ki > Dii .(λi .|e˙ i | + |¨ qdi | +

n  1 + + .(N + Dij .|¨ qj |)) i − Dii j=1,j=i

(11)

− + Where Dij (q) ≤ Dij (q) ≤ Dij (q) and Ni− (q, q) ˙ ≤ Ni (q, q) ˙ ≤ Ni+ (q, q), ˙ 1≤i≤ 4, 1 ≤ j ≤ 4.

3 3.1

Results Structural Simulation

In this analysis, the critical zones that concentrate the stress are determined (see Fig. 4). This information is used to find the factor of safety (FOS) of the whole assembly [4]. Besides, the stress on every contact and joint was calculated using a Finite Element Analysis (FEA) software to determine if it was necessary to add any additional layer between each contact. For this assembly, a rigid neoprene layer between two pieces made of duraluminum and aluminum was added. After running the simulation, the FOS had a minimum value of 1.3007 as shown in Fig. 4. In order to improve and assure a good material selection and mechanical design, it was necessary to check if any contact or joint had a stress

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219

over the elastic limit. Analyzing the results, it is clear that the maximum equivalent stress is under the limit for each piece in the assembly (see Table 1).

Fig. 4. Structural simulation for exoskeleton assembly, contacts and FOS.

3.2

Controller Performance

In this section, the performance of the Sliding Mode control algorithm for the exoskeleton with and without load is shown. The load is represented by a human arm, whose density and length were extracted from [14]. The values of the parameters that define k (see Eq. (11)) are shown in Table 3. Table 3. Parameters of the control resources ki k1

k2

k3

N1+ + D11 − D11 + D12 + D13 + D14

N2+ + D22 − D22 + D21 + D23 + D24

N3+ + D33 − D33 + D31 + D32 + D34

= 20.384 = 0.622 = 0.225 = 0.622 = 0.622 = 0.622

= 23.570 = 0.502 = 0.201 = 0.147 = 0.157 = 0.003

k4 = 18.019 N4+ = 6.461 = 0.309 = 0.309 = 0.347 = 0.157 = 0.014

+ D44 = 0.012 − D44 = 0.012 + D41 = 0.015 + D42 = 0.003 + D43 = 0.014

∅1 = 0.8

∅2 = 0.8

∅3 = 0.8

∅4 = 0.8

λ1 = 16

λ2 = 16

λ3 = 40

λ4 = 40

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Results of the trajectory tracking simulation applying an SMC controller without load are shown in Fig. 5. The joint angles converge rapidly to the desired path. The convergence time is less than 0.5 s and the tracking error is less than 0.00012 rad.

Fig. 5. (a) Desired and actual trajectory of each joint. (b) Torques and tracking error of each joint.

Results of the trajectory tracking simulation applying an SMC controller with load are shown in Fig. 6. The joint angles converge faster to the desired trajectory despite the applied load. The convergence time is less than 0.5 ss and the tracking error is less than 0.0025 rad.

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Fig. 6. (a) Desired and actual trajectory of each joint. (b) Torques and tracking error of each joint.

4

Conclusions

The structural simulation for the proposed upper-limb exoskeleton design showed a stress concentration on the contacts and joints between the harmonic gearboxes and manufactured parts. The obtained FOS, which indicates safety against failure, is over 1.3007 for the proposed mechanical structure. The structural simulation workflow can be applied to other robotic systems based on exoskeletons in order to validate their minimum requirements. The results of the image processing with the Kinect for Xbox 360 allowed to model the desired trajectory in 3D effectively and the post-processing done in MATLAB for PNF therapy was optimal for the control process, due to the similarities of the captured and simulated motion. A SMC controller was designed for the exoskeleton for trajectory-tracking tasks. According to the error values shown in Fig. 5 and 6, there is a small variation between both tests, which shows the robustness of the SMC algorithm, so it is adaptable to different arm weights. According to the

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obtained results, the required torque to complete the task without load was less than 20 N.m, but there was a small increase with load, and to move the first and third articulations, the highest torque values are needed. In future works, the controller will be implemented using a digital signal processor to verify its performance.

References 1. Aguirre Izurieta, G.I.: Facilitaci´ on neuromuscular propioceptiva en hemiplej´ıa. B.S. thesis, Universidad Nacional de Chimborazo, 2019 (2019) 2. bin Abdul Hamid, M.F., bin Mohd Ramli, M.H., Che Zakaria, N.A., Mohamed, Z.: Conceptual design and FEM analysis of an exoskeleton suit for post-stroke patient: a lower limbs Exo suit. In: Shiraishi, Y., Sakuma, I., Naruse, K., Ueno, A. (eds.) APCMBE 2020. IP, vol. 82, pp. 126–134. Springer, Cham (2021). https://doi.org/ 10.1007/978-3-030-66169-4 17 3. Hsieh, H.C., Chen, D.F., Chien, L., Lan, C.C.: Design of a parallel actuated exoskeleton for adaptive and safe robotic shoulder rehabilitation. IEEE/ASME Trans. Mechatron. 22, 2034–2045 (2017) 4. Johan, N., et al.: Preliminary design of an intention-based semg-controlled 3 dof upper limb exoskeleton for assisted therapy in activities of daily life in patients with hemiparesis. In: 2020 8th IEEE RAS/EMBS International Conference for Biomedical Robotics and Biomechatronics (BioRob), pp. 292–297 (2020). https:// doi.org/10.1109/BioRob49111.2020.9224397 5. Kapsalyamov, A., Hussain, S., Jamwal, P.K.: State-of-the-art assistive powered upper limb exoskeletons for elderly. IEEE Access 8, 178991–179001 (2020) 6. Lorenzo Sciavicco, B.S.: Modelling and Control of Robot Manipulators. Springer, Heidelberg (2000) 7. Quispe, J.N., Solis, D.C., Le´ on, J.H., Ccopa, L.M.: Performance comparison between pd and pid controller in an upper limb exoskeleton by analyzing an arm trajectory modeled with image recognition. In: 2020 IEEE Engineering International Research Conference (EIRCON), pp. 1–4 (2020). https://doi.org/10.1109/ EIRCON51178.2020.9254082 8. Ravindran, M., Aswatha, M., Santhosh, N., Ravichandran, G., Madhusudhan, M.: Effect of heat treatment on fatigue characteristics of en8 steel. IOP Conference Series: Mater. Sci. Eng. 1013, 012009 (2021). https://doi.org/10.1088/1757-899x/ 1013/1/012009 9. Razzaghian, A., Moghaddam, R.K.: Fuzzy sliding mode control of 5 dof upper-limb exoskeleton robot. In: 2015 International Congress on Technology, Communication and Knowledge (ICTCK), pp. 25–32. IEEE (2015) 10. Saha, S., Lahiri, R., Konar, A., Banerjee, B., Nagar, A.K.: Hmm-based gesture recognition system using kinect sensor for improvised human-computer interaction. In: 2017 International Joint Conference on Neural Networks (IJCNN), pp. 2776– 2783 (2017) 11. Morales, M., Mosquera, G., S´ anchez, M., Chamorro, W.: Upper limb exoskeleton design and implementation to control a robotic arm. In: 2017 International Conference on Information Systems and Computer Science (INCISCOS), pp. 73–78 (2017). https://doi.org/10.1109/INCISCOS.2017.51 12. Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems. CRC Press, Boca Raton (2009)

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13. Wang, X., Song, Q., Wang, X., Liu, P.: Kinematics and dynamics analysis of a 3dof upper-limb exoskeleton with an internally rotated elbow joint. Appl. Sci. 8(3), 464 (2018) 14. Winter, D.A.: Biomechanics and Motor Control of Human Movement. Wiley, Hoboken (2009) 15. Zhang, J., Jiang, H., Kang, G., Jiang, C., Lu, F.: A new form of equivalent stress for combined axial-torsional loading considering the tension-compression asymmetry of polymeric materials. RSC Adv. 5, 72780–72784 (2015). https://doi.org/10.1039/ C5RA15230E 16. Zhang, L., Li, J., Su, P., Song, Y., Dong, M., Cao, Q.: Improvement of humanmachine compatibility of upper-limb rehabilitation exoskeleton using passive joints. Robot. Auton. Syst. 112, 22–31 (2019)

Mechatronic Design of a Planar Robot Using Multiobjective Optimization Alejandra Rios Suarez1 , S. Ivvan Valdez2 , and Eusebio E. Hernandez1(B) 1

Instituto Politecnico Nacional ESIME Ticoman, Av. Ticoman 600, Mexico City, Mexico [email protected], [email protected] 2 CONACYT-Centro de Investigaci´ on en Ciencias de Informaci´ on Geoespacial, CENTROGEO, A.C., Parque Tecnol´ ogico San Fandila, C.P. 76703 Mexico City, Quer´etaro, Qro, Mexico [email protected]

Abstract. The concurrent design optimization of robots refers to the problem of optimizing parameters that affects different kinds of features at the same time. For instance, this work presents a study case for the concurrent design optimization of the structure and control of a parallelogram mechanism. The main contribution of this work is the definition of an integrated optimization model that considers two conflicting objectives, defined as the energy and error during a trajectory tracking. In addition, the optimization model considers an error constraint, with the purpose of automatically discarding designs that can not closely follow the trajectory, and the simulation considers a saturation constraint that avoids to deliver torques above a threshold. The multi-objective optimization problem is solved using the Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D), the resulting solutions, named Pareto set, are delivered to a final decision maker, to select the adequate design among those with the best compromise between minimum tracking error and energy consumption. A design is a set of lengths and control gains, hence, notice that the control gains are optimized for the corresponding geometry.

Keywords: Concurrent design input parallelogram mechanism

1

· Multi-objective optimization · Two · Pareto set solution

Introduction

Traditionally, the mechanism to be controlled is designed first and independently of the control scheme. Thus, the control of parallel robots for high performance and high-speed tasks can be a challenge to control engineers. The design process of robotic systems can benefit greatly from the mechatronic design, which is the concurrent optimal design of a mechanical system and its embedded control for a given objective function during the same optimization process [3,16]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 224–231, 2022. https://doi.org/10.1007/978-3-030-88751-3_23

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The optimal design of parallel structures can be stated as a traditional optimization problem, but it can be attracted by local minima, it can be sensitive to initial conditions or involve the computation of gradients of the objective function. In [10], a least-square based scheme for the dimensional optimization of a double-rocker linkage is presented. The optimum dynamic balancing of the four-bar mechanism is reported in [1]. Referring the parallelogram structure: the design of a haptic device by considering size, workspace, inertia, and structural properties is described in [9]; in [11], authors consider the mass balancing to simplify the dynamic model, and so design a controller for tracking path. The simultaneous plant-controller design optimization of mechatronics systems is a complex nonlinear optimization problem due to the presence of non-convex, non-differentiable, multiple, dynamic objectives and constraints. To circumvent the above-described problems of classical approaches, optimization techniques based on metaheuristic methods [5] have been used in the design process of different mechatronic systems. For instance, an evolution strategy has been selected for optimizing both the design parameters and non linear PD controller gains of a two-link planar manipulator [14]. In [13], differential evolution is used to improve the parametric reconfiguration of a five-bar linkage. Others works have used genetic algorithm [2], non-dominated sorting genetic algorithm II [6,8,12] or estimation of distributions algorithms [17] for the optimization process. There are two basic approaches for concurrent optimization; the first is to convert the multiobjective problem to a mono-objective problem via a scalarization function. This approach makes assumptions about the final solution to set a priori weights or the target point. The second approach is to deal with the multiobjective problem by searching for a set of solutions that are considered equally optimal, then a posteriori, a single solution is selected. In this context, some authors have preferred to circumvent the problem of multiobjective optimization, for instance, in [4] the authors convert a multiobjective problem into a single objective problem via a scalarization function that assigns a reward to a combination of distance to the target position and stiffness values. In the same regard, in [15], a task-oriented optimization for the path and geometry of a robot is approached. The authors convert a bi-objective problem into a two-stage optimization problem, optimizing two single objective functions sequentially. In contrast, our approach is to concurrently optimize two conflicting objective functions, defined as the energy and error during a specific path, to find the Pareto front. Thus, both the structure and the controller designs are synthesized. The multi-objective optimization problem is formulated and solved by using the MOEA/D. Notice that the Pareto front provides information about the trade-off between optimal solutions and the best and worst performed solutions in the optimal set; hence, having the whole solution set is essential for making informed engineering decisions.

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Mechanism Description

Fig. 1. A mechanical diagram of the parallelogram mechanism system used as a case of study

2.1

Kinematics

In reference to Fig. 1, Li denotes the element lengths, lci refers to the location of the center of mass of each element and ϕ1,2 is the angular positions for both actuated elements, the direct kinematics is given as follows:     x L cos ϕ1 − L4 cos ϕ2 . (1) = 1 L1 sin ϕ1 − L4 sin ϕ2 y Therefore, the inverse kinematics is computed according to Eqs. (2) and (3),   y L4 sin γ − tan−1 and ϕ2 = ϕ1 + (γ + π), (2) ϕ1 = tan−1 x L1 + L4 cos γ  √ 2 x2 + y 2 − L21 − L24 1 − d − and d = cos γ = where γ = tan−1 (3) d 2L1 L4 Moreover, the Jacobian of this mechanism is defined in terms of the Cartesian velocities x˙ and y˙ and the derivative of the generalized coordinates vector q˙ = ˙ The Jacobian matrix is defined [ϕ˙1 ϕ˙2 ] with the following expression: x˙ = J(q)q. for this problem as in the Eq. (4) below,   −L1 sin ϕ1 L4 sin ϕ2 (4) J(q) = L1 cos ϕ1 −L4 cos ϕ2 2.2

Dynamic Model

The dynamic model without friction consideration is described as follows [17]: d11 ϕ¨1 + dd12 ϕ¨2 + (m3 L2 lc3 − m4 L1 lc4 ) sin ϕ2 − ϕ1 ϕ¨2 2 + φ1 (ϕ1 ) = τ1

(5)

d21 ϕ¨1 + dd22 ϕ¨2 + (m3 L2 lc3 − m4 L1 lc4 ) sin ϕ2 − ϕ1 ϕ¨1 + φ2 (ϕ2 ) = τ2

(6)

2

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with 2 2 d11 = m1 lc1 + m3 lc3 + m4 L21 + I1 + I3 , d12 = (m3 L2 lc3 − m4 L1 lc4 ) cos(ϕ2 − ϕ1 ), φ(ϕ1 ) = g cos ϕ1 (m1 lc1 + m3 Lc3 + m4 L1 ), d21 = d12 , 2 2 + m3 L22 + m4 lc4 + I2 + I4 , φ(ϕ2 ) = g cos ϕ2 (m2 lc2 + m3 L2 + m4 lc4 ). d22 = m2 lc2

I1 , I2 , I3 , I4 are the moments of inertia of the elements, m1 , m2 , m3 , m4 are their masses and g is the acceleration of gravity. Besides, τ1 and τ2 denote the input torques. Notice that assuming the equality (m3 L2 lc3 = m4 L1 lc4 ) the dynamic model is simplified since the centrifugal terms are eliminated from the inertial matrix, yielding the following decoupled form: d11 ϕ¨1 + φ1 (ϕ1 ) = τ1

3

and d22 ϕ¨2 + φ2 (ϕ2 ) = τ2

(7)

Optimization Approach

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) is used for the optimal design task. This optimization looks for the best dimensional and control parameters for the robot to track a pre-defined trajectory. The error in the tracked trajectory and consumed energy are both intended to be minimized. The proposed trajectory is the Pascal snail path, that is described as a parametric equation with polar coordinates r = a cos θ + b. The simulation is configured to have a duration of 30 s, and the dynamic model is adjusted for convenience under the condition L1 = L3 . Elements lengths Li have a constant rectangular cross-section with dimensions 1cm × 3cm. For the sake of simplification, it is also assumed that (m3 L2 lc3 = m4 L1 lc4 ), as it is explained in Sect. 2.2. This consideration implies the compliance of Eq. (8). Notice that the equality lci = Li /2 is fulfilled. In consequence, Eq. (8) can be simplified as follows, m4 lc4 m4 l 4 L2 = L1 , lci = Li /2, → L2 = L1 . (8) m3 lc3 m3 l 3 Nevertheless, this assumption results too straight for the solution of the inverse kinematics causing singular positions in some configurations. Thus, the simulation has been restricted to avoid singularities by adding an additional constraint, with the inverse of the condition number of the Jacobian matrix as κ−1 (J) ≤ 0.2. The limitation of κ−1 (J) helps the robot be protected from singularities during the tracking. Besides, the vector of decision variables has been proposed as in Eq. (9), in terms of the PID gains for actuated elements and the element’s lengths, and the objective function is formalized through Eq. (12) where e denotes the error over the trajectory in the XY plane, and τi · dϕi is the work done (and consequently, the energy expended) by the actuated elements at each step of time. The computation is effected in the SI units. The integration of both terms is effected to evaluate the accumulation of both measures during the simulation time.

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d = [kp1 , ki1 , kd1 , kp2 , ki2 , kd2 , L1 , L4 ] , with 0 ≤ [kp1,2 , ki1,2 , kd1,2 ] ≤ 300, 0.1 ≤ L1 , L4 ≤ 1 error(t, d) ≤ 0.1, τi (t) ≤ |10|,  t 

t f (d ) = |error|dt, |τi · dϕi dt 0

(9) (10) (11) (12)

0

To reduce the computational cost generated by each evaluation, we restrict the model to discard those robot configurations showing a rapid accumulation of the error in the trajectory. This limit is set equal to 0.1 m. Moreover, the signal torque in both actuated elements is saturated in a range of [−10, 10]N · m. This is modeled in Eqs. (11). Thus, the optimization model considers the decision variables in Eq. (9), the constraints in Eq. (8), that permits independent control of the actuated elements, limitation of κ−1 (J), that avoids singularities, Inequalities (10), that are box constraints for bounding the search, Inequalities (11), that limits the maximum error and torque of any solution, and the objectives in Eq. (12), that are the energy and error to minimize. The MOEA/D is adjusted with a population of 80 and 200 maximal iterations, that is to say, a total of 16,000 evaluations.

4

Optimization Results

As a result of this bi-objective optimization we show the Pareto front for the minimization of the error and the expended energy on the left graphic in Fig. 2. This Pareto front shows a set of optimal designs so that the designer can decide the adequate according to the requirements of the application. On the other hand, we chose one of this optimal solutions located in the center of the Pareto front and compared the resulting traced trajectory to the expected one with the related decision variables obtained (see Fig. 2). We also provide the plot for the angular displacement, angular error and torques for the actuated elements for the given traced trajectory (Fig. 3). The decision variables yielding the optimal values were found to be as follows (Table 1). Table 1. Optimization results for the extremes and center of the Pareto front [Error,Energy]

kp1

ki1

kd1

kp2

ki2

kd2

L1

L4

[0.0134,0.0113] 274.62 108.57

62.62

81.72 258.21 8.47 0.51 0.26

[0.0153,0.0106] 219.12 108.58

62.62

81.72 258.21 8.47 0.51 0.26

[0.0242,0.0085] 241.94 166.44 140.93 117.28 154.87 3.22 0.52 0.47

Mechatronic Design Using Multiobjective Optimization Pareto set

0.0115

229

Desired trajectory vs Traced trajectory

0.35 0.34

0.011

0.33

Error 0.0153434 Energy 0.0105767

0.32

0.0105

Y Axis

Energy [J]

0.31 0.01

0.3 0.29

0.0095

0.28 0.27

0.009

0.26 0.0085 0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.25 0.38

0.026

0.4

0.42

Error [m]

0.44

0.46

0.48

X Axis

1.2

2.7

1.15

2.6

Angle [rad]

Angle [rad]

Fig. 2. Obtained Pareto front and trajectory traced from the selected optimal solution in coordinates [0.01534, 0.01057]

1.1 1.05 1

0

10

15

20

25

2.3

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

10 -3

Error [rad]

0

10

0 0

5

10

15

20

25

-5 -10 -15 -20

30

4

0

Torque [N.m]

Torque [N.m]

2.4

10 -3

20

Error [rad]

5

2.5

3 2 1

-0.5 -1 -1.5

0

5

10

15

Time [s]

20

25

30

Time [s]

Fig. 3. Angular displacement, angular error and torques experimented in the actuated elements for the traced trajectory in Fig. 2

5

Conclusions

We introduce an optimization model applied to a parallelogram linkage system, it considers a bi-objective optimization problem with several constraints, in one hand, box constraint are the most common in optimization problems, they limit the search, on the other hand, we introduce an error constraint that limits implicitly the search to those mechanism with a lower tracking error, without this constraint, the algorithm will look for the minimum energy without consid-

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ering the error, such solution is a static mechanism with all control gains equal to zero, it is trivial and undesirable. In the same regard, the torque constraint, is introduced for emulating the maximum physical torque of the actuators, this constraint provides of physical meaning of the control gains, and finally, the constraint on the Jacobian condition number, provides a way of requiring an acceptable dexterity during the task and avoid singularities that are possible poses in numerical simulations but not in a real mechanism. The MOEA/D reports a set of solutions that actually perform adequately the task. The designer can choose among a set of the best conflicting solutions, he/she is in charge of selecting the pay off in energy of a tracking error, or viceversa. Notice that this compromise among the best solutions is not known a priori, so, such selection can not be done before applying this optimization process. Currently, there are novel numerical strategies for the selection of a single solution from the Pareto front [7]. Nevertheless, this selection should not be done a priori, without the knowledge of the assigned task for the mechanism. Design is not only reduced to the parameters included in the optimization, further, they have to accomplish quality standards. For this reason, we leave the selection of the single optimal solution open to the designer, with the conscience of how much each objective could be enhanced or sacrificed for the engineering application thought for the robot. Finally, according to our experiments the torques do not reach saturation, this could be due to the error constraint, that avoid large errors, which reduces the computed τi , and to the gains box constraints. Our experiments demonstrate, how it is possible to introduce and deal with constraints that guide the search to the most useful solutions without complicated procedures, we basically consider most of them as conditions in our numerical simulation. Future work contemplate to apply the very same method to other tasks and robots. Acknowledgements. The authors are grateful to SIP-IPN Mexico for supporting part of this work through project SIP-20210243. S. Ivvan Valdez is supported by Catedra-CONACYT 7795.

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Author Index

A Acevedo, Mario, 181 Acuña, Jesus, 213 Albites-Sanabria, Jose, 213 Alegria, Elvis J., 13, 24 Ambrósio, Jorge, 105 Arakelian, Vigen, 56 Arias-Montiel, Manuel, 64 Arroyo, Nícolas, 13 B Bai, Qingshun, 94 Barreto, Juan Pablo, 191 Beltramo, Emmanuel, 143 Bernardon Machado, Lucas, 37 C Campusano, Daryl, 213 Carretero, J. A., 171 Ceballos, Luis R., 151 Ceccarelli, Marco, 46 Chate, Ebert, 213 D de Faria Lemos, Aline, 122 Díaz, Joseph, 24 Dimitriadis, Grigorios, 151 E Elshami, Mohamed, 94 F Ferreira, José, 114 Ferreira Gonçalves, Luiz Otávio, 122

Figueroa, Alvaro, 213 Figueroa, Yanpierrs, 24 Flores, Del Piero, 13 Flores, Paulo, 77, 85, 105, 114 G Gallardo, Alejandro G., 3 García, Marco, 64 Gutiérrez-Diaz, Gabriel, 159 H Hernandez, Eusebio E., 224 Horta Gutiérrez, Juan Carlos, 122 Huamanchumo, Johrdan, 213 Hurtado-Hurtado, Gerardo, 131 I Irakoze, Vivens, 46 J Jauregui-Correa, Juan Carlos, 131 L Lleren, Juan, 213 Lugo-González, Esther, 64 M Magalhães, Hugo, 105 Marques, Filipe, 85, 105, 114 Martins, Daniel, 37 Martins Ferreira, Ricardo Poley, 122 Milián-Ccopa, Leonardo Paul, 213 Mora, Juan Pablo, 191

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Pucheta et al. (Eds.): MuSMe 2021, MMS 110, pp. 233–234, 2022. https://doi.org/10.1007/978-3-030-88751-3

234 N Nguyen, Vu Linh, 201 Nuñez-Quispe, Johan, 213 O Otremba, Frank, 131 P Palma, Diego, 13 Pérez Segura, Martín E., 143 Pombo, João, 105 Prado Barbosa, Társis, 122 Preidikman, Sergio, 143, 151 Pucheta, Martín A., 3 Q Quino, Gustavo, 24 R Roccia, Bruno A., 143, 151 Rodrigues da Silva, Leonardo Adolpho, 122 Rodrigues da Silva, Mariana, 85 Rodriguez, Carlos F., 191 Romero-Navarrete, Jose A., 131 Russo, Matteo, 46

Author Index S Santiesteban Cos, R., 171 Sensinger, J., 171 Shehata, Mohamed, 94 Simas, Henrique, 37 Solis-Santome, Arturo, 159 Solórzano, Renzo, 13 Soto, Axel, 213 Suarez, Alejandra Rios, 224 Suyo, Briggitte, 213 T Taipe, Kevin, 213 Tapia-Herrera, Ricardo, 64 Tavares da Silva, Miguel, 85 Torres-SanMiguel, Christopher René, 159 V Valdez, S. Ivvan, 224 Valdiero, Antonio Carlos, 37 Velázquez, Ramiro, 181 Verstraete, Marcos L., 151 Z Zhao, Xuezeng, 94