Performance Analysis and Optimization of Parallel Manipulators 9819905419, 9789819905416

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Research on Intelligent Manufacturing

Qinchuan Li · Chao Yang · Lingmin Xu · Wei Ye

Performance Analysis and Optimization of Parallel Manipulators

Research on Intelligent Manufacturing Editors-in-Chief Han Ding, Huazhong University of Science and Technology, Wuhan, Hubei, China Ronglei Sun, Huazhong University of Science and Technology, Wuhan, Hubei, China Series Editors Kok-Meng Lee, Georgia Institute of Technology, Atlanta, GA, USA Cheng’en Wang, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China Yongchun Fang, College of Computer and Control Engineering, Nankai University, Tianjin, China Yusheng Shi, School of Materials Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China Hong Qiao, Institute of Automation, Chinese Academy of Sciences, Beijing, China Shudong Sun, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, China Zhijiang Du, State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, Heilongjiang, China Dinghua Zhang, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, China Xianming Zhang, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, Guangdong, China Dapeng Fan, College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan, China Xinjian Gu, School of Mechanical Engineering, Zhejiang University, Hangzhou, Zhejiang, China Bo Tao, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China Jianda Han, College of Artificial Intelligence, Nankai University, Tianjin, China Yongcheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, Hunan, China Zhenhua Xiong, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

Research on Intelligent Manufacturing (RIM) publishes the latest developments and applications of research in intelligent manufacturing—rapidly, informally and in high quality. It combines theory and practice to analyse related cases in fields including but not limited to: Intelligent design theory and technologies Intelligent manufacturing equipment and technologies Intelligent sensing and control technologies Intelligent manufacturing systems and services This book series aims to address hot technological spots and solve challenging problems in the field of intelligent manufacturing. It brings together scientists and engineers working in all related branches from both East and West, under the support of national strategies like Industry 4.0 and Made in China 2025. With its wide coverage in all related branches, such as Industrial Internet of Things (IoT), Cloud Computing, 3D Printing and Virtual Reality Technology, we hope this book series can provide the researchers with a scientific platform to exchange and share the latest findings, ideas, and advances, and to chart the frontiers of intelligent manufacturing. The series’ scope includes monographs, professional books and graduate textbooks, edited volumes, and reference works intended to support education in related areas at the graduate and post-graduate levels.

Qinchuan Li · Chao Yang · Lingmin Xu · Wei Ye

Performance Analysis and Optimization of Parallel Manipulators

Qinchuan Li School of Mechanical Engineering Zhejiang Sci-Tech University Hangzhou, Zhejiang, China Lingmin Xu School of Mechanical Engineering Shanghai Jiao Tong University Shanghai, China

Chao Yang College of Mechanical and Electrical Engineering Jiaxing University Jiaxing, Zhejiang, China Wei Ye School of Mechanical Engineering Zhejiang Sci-Tech University Hangzhou, Zhejiang, China

ISSN 2523-3386 ISSN 2523-3394 (electronic) Research on Intelligent Manufacturing ISBN 978-981-99-0541-6 ISBN 978-981-99-0542-3 (eBook) https://doi.org/10.1007/978-981-99-0542-3 Jointly published with Huazhong University of Science and Technology Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Huazhong University of Science and Technology Press. © Huazhong University of Science and Technology Press 2023 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 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Brief Introduction

This book investigates in detail the performance analysis and optimization design of parallel manipulators. It includes performance evaluation indices for workspace, kinematic, stiffness, and dynamic performance, single- and multi-objective optimization design methods, and ways to improve the optimization design efficiency of parallel manipulators. This book collects the authors’ research results previously scattered in many journals and conference proceedings and presents them in a unified form after the methodical edition. Numerous performance analyses and optimizations of parallel manipulators are presented, in which readers in the robotics community may have a great deal of interest. More importantly, readers can use the method and tool introduced in this book to carry out performance evaluation and optimization of parallel manipulators independently. The book is intended for undergraduate and graduate students, engineers, and researchers who are interested in performance evaluation and dimensional synthesis. This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51525504 and 51935010.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Performance Evaluation Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Workspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Kinematic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Stiffness Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Dynamic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Traditional Optimization Algorithms . . . . . . . . . . . . . . . . 1.3.2 Intelligent Optimization Algorithms . . . . . . . . . . . . . . . . . 1.3.3 Ways to Improve Optimization Algorithms . . . . . . . . . . . 1.4 Multi-Objective Optimization Methods . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Comprehensive Objective Method . . . . . . . . . . . . . . . . . . . 1.4.2 Pareto Frontier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 PCA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 8 10 15 19 19 21 26 30 30 30 31 32 33

2

Kinematic Performance Analysis and Optimization of Parallel Manipulators Without Actuation Redundancy . . . . . . . . . . . . . . . . . . . 2.1 Basics of Screw Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Condition Number Indices and Applications . . . . . . . . . . . . . . . . . 2.2.1 Condition Number Indices . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Example 1: 6PSS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Example 2: 2-(PRR)2 RH PM . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Example 3: 2PRS-PRRU PM . . . . . . . . . . . . . . . . . . . . . . . 2.3 Motion/Force Transmission Indices and Applications . . . . . . . . . . 2.3.1 Motion/Force Transmission Indices . . . . . . . . . . . . . . . . . 2.3.2 Example 1: 6PSS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Example 2: 2PUR-PRU PM . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Example 3: 2PUR-PSR PM . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 44 45 50 59 65 65 67 70 77

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Contents

2.4

Motion/Force Constraint Indices and Applications . . . . . . . . . . . . 2.4.1 Motion/Force Constraint Indices . . . . . . . . . . . . . . . . . . . . 2.4.2 Example 1: 2PUR-PRU PM . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Example 2: 2PUR-PSR PM . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Motion/Force Transmission Performance Analysis and Optimization of Parallel Manipulators with Actuation Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Motion/Force Transmission Indices of Parallel Manipulators with Actuation Redundancy . . . . . . . . . . . . . . . . . . . 3.2 Example 1: 6PSS-UPS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Example 2: 2UPR-2PRU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Example 3: 2PUR-2PRU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motion/Force Constraint Performance Analysis and Optimization of Overconstrained Parallel Manipulators with Actuation Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Motion/Force Constraint Indices of Overconstrained Parallel Manipulators with Actuation Redundancy . . . . . . . . . . . . 4.2 Example 1: 2UPR-2PRU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Example 2: 2PUR-2PRU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Stiffness Performance Evaluation Index . . . . . . . . . . . . . . . . . . . . . 5.2 Example: 2UPR-RPU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Stiffness Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Stiffness Performance Optimization . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Methodology for Optimal Stiffness Design of Parallel Manipulators Based on the Characteristic Size . . . . . . . . . . . . . . . . . . . 6.1 Methodology for the Optimal Stiffness Performance Design of PMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Example 2: Optimal Stiffness Performance Design of the 2PRU-PSR PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84 93 94 97 98

101 102 106 113 124 128 129

131 132 135 137 140 140 143 143 149 151 157 159 160 161 161 163 170 174 175

Contents

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Multi-objective Optimization of Parallel Manipulators Using Game Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Multi-objective Optimization Game Algorithm . . . . . . . . . . . . . . . 7.2 Example: 2UPR-RPU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Regular Workspace Volume . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Motion/Force Transmissibility . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Stiffness Performance Evaluation . . . . . . . . . . . . . . . . . . . 7.2.4 Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . . . 7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 180 181 183 184 186 203 204

Hybrid Algorithm for Multi-objective Optimization Design of Parallel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Hybrid Algorithm and GPR-Based Mapping Modeling . . . . . . . . 8.1.1 Procedure of the Hybrid Algorithm . . . . . . . . . . . . . . . . . . 8.1.2 GPR-Based Mapping Model . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Example: 2PRU-UPR PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Kinematic Performance Index . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Stiffness Performance Index . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Elastodynamic Performance Index . . . . . . . . . . . . . . . . . . 8.2.4 Regular Workspace Volume . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . . . 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 206 206 207 209 211 211 212 217 218 226 228

Sensitivity Analysis and Multi-objective Optimization Design of Parallel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Sensitivity Analysis and Multi-objective Optimization Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Response Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Multi-objective Optimization Design of PMs . . . . . . . . . 9.2 Example: Delta PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Workspace Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Kinematic Performance Index . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Dynamic Performance Index . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Sensitivity Analysis and Multi-objective Optimization Design of Delta PM . . . . . . . . . . . . . . . . . . . 9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 232 232 233 236 236 238 240 241 247 252 253

10 Multi-objective Optimization Design of Parallel Manipulators Based on the Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . 255 10.1 Multi-objective Optimization of PMs Based on the Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.2 Example 1: 3RPS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

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10.2.1 Performance Indices of the 3RPS PM . . . . . . . . . . . . . . . . 10.2.2 Multi-objective Optimization Design of the 3RPS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Example 2: 6PSS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Performance Indices of the 6PSS PM . . . . . . . . . . . . . . . . 10.3.2 Multi-objective Optimization Design of the 6PSS PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Multi-objective Optimization Design of Parallel Manipulators Based on the Intelligent-Direct Search Algorithm . . . . . . . . . . . . . . . . 11.1 Intelligent-Direct Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Introduction of Pareto Front . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Procedure of the Intelligent-Direct Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Example: 2UPR-RPU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Performance Indices of the 2UPR-RPU PM . . . . . . . . . . 11.2.2 Multi-objective Optimization Design of the 2UPR-RPU PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

260 263 267 268 274 284 285 287 287 287 288 289 290 292 296 296

Chapter 1

Introduction

Performance optimization of parallel manipulators (PMs) has attracted adequate attention in recent years. It mainly concerns performance indices, optimization algorithms, and optimization methods. This chapter presents an in-depth, comprehensive, reasoned overview for the three basic issues. The research status and existing problems of these issues along with the distinctive approaches that have been explored are reported. Advantages, disadvantages, application scenarios, and recommendations of various performance indices, optimization algorithms, and methods are discussed. Focus is also placed on future research trends. The results are useful for researchers and engineers to properly select performance indices, optimization algorithms, and optimization methods when designing a PM.

1.1 Research Background The general approach to the considered problems lies in the fact that robot-related performance indices present scalar functions synthesized from matrix models, and among them, the most important are the Jacobian matrix, stiffness and compliance matrices, and mass matrix separately formulated within the static and dynamic versions. The general specific feature of these matrices is the dimensional inhomogeneity resulting in the absence of matrix eigenvalues. This feature is valid for matrices associated with both parallel- and serial-kinematic mechanisms; thus, this feature allows the application of some results formulated for serial-kinematics robots to be applied to parallel-kinematics robots. Additionally, this is the reason for the widespread application of the matrix determinants when synthesizing performance indices since the determinant presents a unique invariant of a dimensionally inhomogeneous matrix.

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_1

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2

1 Introduction

Researchers and industries have studied parallel manipulators (PMs) over the past two decades due to their excellent dynamic performance, high agility, high loadto-weight ratio, distributed joint error, and simple inverse kinematics. A parallelkinematics robot named the Gough–Stewart platform (GSP) was utilized as the subject for scientific investigations in two fundamental works by Stewart [1] and Gough [2]. In the fundamental work by Merlet [3], more than 600 issues relating to PMs were comprehensively analyzed. PMs for engineering applications have been designed and commercialized. Commercially successful parallel robots mainly include Delta robots [4], which are for high-speed pick-and-place operations that are required in food and pharmaceutical industries; a Tricept robot based on a 3UPS-UP PM [5]; a Sprint Z3 spindle head based on a 3PRS PM [6]; and an Exechon robot based on a 2UPR-SPR PM, which is applied in the automobile and aviation industries [7], where U is the universal joint, S is the spherical joint, R is the revolute joint, and P is the prismatic pair. Comprehensive reviews relating to PM performance indices were considered in works [8] and [9]. The PM design is the primary basis for a parallel robot system. Type synthesis is used to propose an architecture satisfying the requirements of modern processing technology. Then, a dimensional synthesis of the mechanism is completed based on the selection of performance indices, optimization algorithms, and optimization methods. The primary condition of the optimization design of PMs is to establish reliable and computationally efficient performance indices that affect the final working performance and optimal design efficiency of the parallel robot system. Therefore, the following aspects of performance indices are issues of concern in industry and academia: determining a mathematical model between the performance indices and design parameters of a PM, exploring the workspace of a PM [10, 11], kinematic performance [12, 13], stiffness performance [14, 15], dynamic performance evaluation methods [16, 17], and establishing reasonable performance indices with clear physical significance. The optimum selection of a structure for a given application is a capital phase in the typological synthesis of parallel robots [18], and the main components can include a mathematical model of the optimization design containing design parameters, performance indices (objective functions), and constraint functions; optimization algorithms, and an evaluation of the optimized results. Studies have found that there may be conflicts among the performance indices of PMs [19]. Engineering applications often require optimal comprehensive performance of a mechanism, which cannot be satisfied by a traditional single-objective optimization design. Therefore, a multi-objective dimensional optimization design [20–22] can improve the comprehensive performance of a mechanism. Reasonable optimization algorithms and a comprehensive evaluation of the optimization results are vital to determining optimal design parameters. Exploring efficient and global optimization algorithms and methods for multi-objective comprehensive optimal solution evaluations remain challenging issues in industry and academia. There are many studies that address performance measures and optimization design of parallel robots, but there has been no review of the performance indices, optimization algorithms, and optimization methods of PMs. This book studies these

1.2 Performance Evaluation Indices

3

three aspects to provide a reference for researchers in related research. In addition, the existing research problems and challenges are analyzed to promote related research.

1.2 Performance Evaluation Indices Establishing reasonable performance indices is the basis for the performance measures and dimensional design of PMs. Reasonable performance indices can significantly improve the final working state of a parallel robot system. Existing performance indices mainly examine the workspace, kinematic performance, stiffness performance, and dynamic performance.

1.2.1 Workspaces The workspace is a collection of points that the end effector can reach under the constraints of rod length, joint rotation angle, and interference. Workspace-related performance indices are used to analyze the shape and size of the workspace of the mechanism and provide a theoretical basis for determining the operating space and planning the trajectory of the robot. Compared with the serial mechanism, PM has a major disadvantage in a smaller workspace. The work of Merlet [8] and Tlusty et al. [23] demonstrate this disadvantage. Therefore, developing effective tools for determining and enlarging workspaces has become the primary task of PM research. The workspaces of PMs mainly concern reachable workspaces, regular workspaces, and dexterous workspaces. For a PM with mixed degrees of freedom (DOFs), the workspace can be further divided into a position workspace (workspace with a constant posture) and a posture workspace (workspace with a constant position). At present, the commonly used methods for PM workspace analysis mainly concern geometric methods, numerical discrete methods, and modern mathematical methods. The workspace is usually evaluated in terms of its shape and size; the former mainly includes the space utilization index, compactness index, and shape index, and the latter refers to the volume or area of the reachable/regular/dexterous workspace.

1.2.1.1

Reachable Workspace Volume

A reachable workspace refers to the points that a moving platform can reach from at least one direction under the physical constraints of a mechanism. A geometric method can split a multi-closed-loop PM into several subchains, and the workspace can be reduced to the intersection of each subchain trajectory under constraint conditions. Zhang et al. [24, 25] used a geometric method to analyze the reachable workspace of 3-DOF PMs. Regarding an n-DOF PM with more than 3

4

1 Introduction

DOFs, the workspace can generally be obtained by fixing its n-3 DOF. Gosselin [26] considered a driving limit constraint and analyzed the end effector position workspace in the constant posture of a 6-DOF PM based on the geometric method. Merlet [27] further introduced bar interference and joint constraint conditions based on Gosselin’s work. Liu et al. [28] used an AutoCAD platform to analyze the reachable workplace of a 6RTS PM. The advantage of the geometric method is that the boundary of the reachable workspace can be obtained through the intersection of the motions of each open-loop subchain. However, this method requires a separate geometric model for each mechanism, with disadvantages such as having difficulties in discerning holes in workspaces and calculating the analytical expressions of the regular workspaces of complex PMs. The numerical discrete method, known as the binary method, is most commonly used to calculate the workspace of a mechanism by judging whether each discrete point meets the kinematic solution, constraints, and interference conditions one by one. It mainly includes grid, Monte Carlo, and interval analysis methods. The grid method first creates a complete discrete boundary space containing all possible points in the workspace and then checks whether each grid node satisfies the inverse kinematics solution and mechanism constraints. The grid method is easy to use. However, the accuracy of the grid method depends on the number of nodes, as the node density increases, the computational cost exponentially increases, resulting in a low calculation efficiency. Due to the complexity of the direct kinematics of PMs, engineering applications mainly use inverse kinematics for numerical discrete methods [29]. Cheng et al. [30] used the spherical coordinate method and numerical discrete method to search and analyze the reachable workspace of a 3RPS PM layer by layer. Carbon et al. [31], Chi et al. [32], and Huang et al. [33] used the polar coordinate method to calculate the numerical expression of the reachable workspace volume, which was then used as one of the objective functions to be optimized. The Monte Carlo method, similar to the grid method, calculates the volume of the workspace by judging whether each discrete point satisfies the solution of the mechanism under the constraint conditions. The difference is that the Monte Carlo method randomly generates a large number of discrete points in a regular and complete workspace and then calculates the volume of the reachable workspace by calculating the ratio of the number of discrete points satisfying the kinematic solution of the mechanism to the total number of discrete points. Zhu et al. [34] used the Monte Carlo method to analyze the reachable workspace of a spraying robot. Andrioaia et al. [35] analyzed the shape and volume of the workspace of a Delta PM based on the Monte Carlo method. The Monte Carlo method improves the computational cost of grid method to some extent through a statistical simulation method. Notably, this method requires a larger simulation sample to achieve smaller errors, which means a higher computational cost. Therefore, the Monte Carlo method cannot significantly improve the efficiency of calculating the workspace volume. An accumulation of floating-point calculation errors may lead to meaningless calculation results. Therefore, the interval analysis method expands point discretization to interval discretization and the obtained interval contains accurate results, which enhances the stability and computational efficiency of the algorithm [36].

1.2 Performance Evaluation Indices

5

Saputra et al. [37], Kaloorazi et al. [38], and Viegas et al. [39] analyzed the reachable workspace of Gough–Stewart, 3RPR, 6UPS, and 3PUR PMs based on interval analysis. The accuracy of the interval analysis method depends on the size of the block interval, and it is difficult to find small holes in the workspace unless the size of the block interval is smaller than the size of the hole; however, smaller block intervals will significantly increase the computational cost of the workspace volume. In addition to the geometric method and numerical discrete method, many researchers have discussed workspace solutions based on modern mathematical tools; here, these tools are collectively referred to as modern mathematical methods. Huang et al. [40] used differential geometry to analyze the workspace of a Stewart parallel robot. The envelope theory of a single-parameter curved-surface family was used to propose an efficient algorithm for solving the boundary of the workspace with the cross-sectional method. Alp et al. [41] analyzed the posture workspace of a Stewart PM using the neural network algorithm. Saputra et al. [37] used the bee colony algorithm to analyze the 2D, 3D, and 6D reachable workspaces of a Gough–Stewart PM and compared the results with those from interval analysis. These methods provide new ideas for analyzing a PM workspace.

1.2.1.2

Regular Workspace Volume

Irregular workspaces are not conducive to trajectory planning and programming control, and some properties of the mechanism may be "ill-conditioned" at the boundary of the reachable workspace. Therefore, more attention is given to regular workspaces to meet specific applications and to facilitate trajectory planning and control in engineering applications. The main forms of regular workspaces (also called useful workspaces) mainly concern cylinders, cuboids, ellipsoids, and spheres, all of which belong to a subset of reachable workspaces. Babu et al. [22] adopted polar coordinate and grid methods to calculate the maximum inscribed regular cylindrical workspace volume. Bounab et al. [42] calculated the regular spherical workspace volume of a Delta PM based on the grid method. Yang et al. [19, 43, 44] and Zhang et al. [45] used grid and polar coordinate methods to analyze the volume of a regular cylindrical workspace [19], an ellipsoidal workspace [43], and a circular-table workspace [44, 45] for PMs. The numerical discretization method can solve almost any workspace shape and can be combined with programming to obtain the largest inscribed regular workspace. However, the accuracy of the solution depends on the number of discrete points; small holes in the workspace sometimes need to be found with a sufficiently small discrete step length, which leads to a high computational cost. Chablat et al. [46] and Wang et al. [47] used interval analysis and numerical discrete methods to calculate the regular cubic workspace of 3-DOF orthogonal, UraneSX, and Delta PMs. Liu et al. [48] adopted the numerical discrete method to calculate the regular cubic workspace of a Delta PM. Kaloorazi et al. [38] analyzed the nonsingular regular cylindrical and spherical workspaces of 3RPR and 6UPS PMs based on the interval analysis method.

6

1.2.1.3

1 Introduction

Dexterous Workspace Volume

In addition to the evaluation indices of reachable and regular workspaces, many researchers have combined the workspace and condition number of the Jacobian matrix to form special indices of dexterous workspaces. Kelaiaia et al. [20, 49] defined the subspace satisfying the minimum condition number in the reachable workspace as a dexterous workspace. Based on the Monte Carlo method, Chaudhury et al. [50] defined the reachable workspace with good condition numbers as the dexterous workspace volume index and analyzed the dexterous workplace of a 2DOF 5-bar closed-loop mechanism and a 6-DOF Stewart PM. Chablat et al. [46] analyzed the regular dexterous workspace of 3-DOF orthogonal and UraneSX PMs based on the interval analysis method. Arrouk et al. [51] defined a workspace with full rotation capability as a dexterous workspace and analyzed the dexterous workspace of a 3RRR PM based on CAD technology. Regular dexterous workspace volume is a reasonable index for evaluating the workspace of PMs. However, there is still a lack of a corresponding theoretical method with high efficiency and low computational cost.

1.2.1.4

Workspace Shape Indices

The workspace shape indices, bounded between 0 and 1, are usually quantified by the ratio between the workspace size and bounding box size to evaluate the rationality of the mechanism workspace. Lee et al. [52] used the space utilization index, defined as the ratio between the reachable workspace size and bounding box size (the smallest rectangle enclosing the manipulator structure and the workspace size), to analyze the workspace of a 2-DOF PM. Stock et al. [53] adopted the space utilization index to analyze the workspace utilization of a linear Delta robot. Hamida et al. [18] proposed a workspace compactness index defined by the ratio between the volume occupied by the structure and the volume of the desired workspace to evaluate the workspace of four translational parallel robots. Castelli et al. [54] defined the ratio between the volume of the manipulator workspace and the smallest parallelepiped containing the manipulator workspace as the shape index to evaluate the compactness of an industry robot. Enferadi et al. [55] defined the ratio between the volume of a singularity-free workspace and the smallest box containing the workspace as the global workspace conditioning index to evaluate the workspace of a 3(UPS)-S spherical PM. The workspace shape indices can evaluate the compactness and utilization of PMs; however, they do not improve the computational efficiency of the workspace evaluation indices. Table 1.1 shows the physical meaning and application scenarios of the commonly used workspace indices to increase their vividness and intuitiveness. The above discussions show that the workspace shape and size are often calculated based on the numerical discrete method, which has a high computational cost, and the calculation accuracy of the objective function depends on the number of discrete points in the workspace. Therefore, establishing an analytical model of the

Volume of reachable workspace Volume of regular workspace

Utilization and compactness of workspace

Workspace shape indices [18, 52–55]

It is recommended to be used in Evaluates the reasonableness of combination with a regular the PM workspace shape workspace It is recommended to be used with regular and dexterous workspaces

Volume of dexterous regular/reachable workspace

Regular workspace volume [19, Dexterous workspace volume [46, 50, 51] 22, 38, 42–44, 47, 48]

Application scenarios Position/posture or mixed Regular workspace is suitable reachable workspace volume of to path planning a PM with no more than 3 Recommended DOFs Not recommended

Physical meaning

Reachable workspace volume [24, 26–31, 35, 37, 40, 41]

Table 1.1 List of commonly used workspace evaluation indices

1.2 Performance Evaluation Indices 7

8

1 Introduction

workspace, especially the regular workspace, or a numerical method with low computational cost and high accuracy can greatly promote the performance measures of a PM workspace. It can also significantly improve the computational efficiency of other performance indices, such as kinematic performance, stiffness performance, and dynamic performance.

1.2.2 Kinematic Performance Kinematic analysis aims to assess the motion/force transmission efficiency, singularity, dexterity, and manipulability of PMs. A kinematic performance measure is essential but remains a challenging issue when designing PMs. The existing kinematic performance indices are divided into those based on the Jacobian matrix and those based on screw theory.

1.2.2.1

Kinematic Performance Indices Based on the Jacobian Matrix

The Jacobian matrix reflects the mapping of the output speed of a moving platform to the speed of the actuation joints, as well as the mapping of the actuation wrench to the output wrench. The condition number index mainly concerns the 2-norm, Frobenius norm, and weighted Frobenius norm based on the Jacobian matrix is one of the most commonly used kinematic indices. The condition number index was first applied to a serial mechanism [10] and later used to evaluate the dexterity of a PM and the distance to singularity [42, 56–58]. It has been widely used in the kinematic performance measure of PMs. However, there are controversies regarding a nonuniform dimension, relating to the coordinate system, the inconsistent dimension of the Jacobian matrix of a mechanism with mixed DOFs, and an undefined physical meaning. Two specific singularities of a GSP are considered by Hunt [59] and Fichter [60]. Additionally, Grassmann–Cayle algebra was applied for the analysis of PMs’ singularity by Ben-Horin and Shoham [61]. For these problems, one commonly used method is to divide the Jacobian matrix by a characteristic or intrinsic length [62, 63]; however, the geometric interpretation of the characteristic length (CL) is unclear. Khan et al. [64] used uniform space to present a more direct geometric explanation of the CL. Pond et al. [65] defined the dimensionless Jacobian matrix using the velocities of three points on different lines of the moving platform. Fang et al. [66] reduced the dimensionality of the 3 × 6 Jacobian matrix of a 3-DOF 3PRS PM to a 3 × 3 square matrix by quoting constraint equations that are conducive to the subsequent analysis of kinematic performance. In addition to the dexterity and singularity analysis indices based on the Jacobian matrix, the manipulability index of the mechanism is also important for kinematic analysis. It measures the transmission qualities of velocity and force, including the velocity manipulability and force manipulability indices, as well as the distance to

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9

/ singularity. Yoshikawa [67] first defined det( J J T ) as the velocity manipulability index, where J is the Jacobian matrix. Wu et al. [68] studied the velocity manipulability ellipse index. Lee et al. [69] proposed a velocity manipulability polyhedron index considering joint constraints. The force manipulability index defines the ability of the unit actuation force to be enlarged to the operating space. Wu et al. [68] compared the condition number, velocity manipulability, and force manipulability performance indices of a 3-DOF plane mechanism. Singularity is very important since it relates not only to Jacobian matrix, but also stiffness matrix. Since the performance of a mechanism deteriorates when the mechanism is close to a singular configuration. Many of the performance indices such as manipulability, dexterity, and stiffness performance have the ability to measure the distance to singularity, and it is independent of the dimensionally inhomogeneity of matrix elements. Table 1.2 lists some commonly used kinematic evaluation indices based on the Jacobian matrix. It is worth noting that the 2-norm and Frobenius norm indices in Table 1.2 are based on the definition of matrix trace. However, the trace of the Jacobian matrix is either undefined since it presents a sum of nonhomogeneous elements or defined after normalization procedure with application of the CL. The former approach is erroneous and the latter has no clear explanations for the CL choice and its physical meaning. Table 1.2 List of some commonly used kinematic evaluation indices based on the Jacobian matrix 2-norm [56, 57, 64] Physical meaning

σmax ( J )/ σmin ( J)

Application J is a positive scenarios matrix, high computational cost

Frobenius norm [42, 64] || || || A|||| A−1 ||

Weighted Frobenius norm [42] || || ||B|||| B −1 ||

Velocity Force manipulator manipulability index [67–69] index [67–69] / / det( J J T ) det( J −1 ( J −1 )T )

It is an Has the Measures both Measures both the analytical advantages the kinematic kinematic expression of the dexterity and dexterity and force of Frobenius motion transmission manipulator norm and transmission posture and renders the suitable for norm when a suitable to gradient a specific evaluation context is needed / / / || || || || Note || A|| = trace( J J T ), || A−1 || = trace( J −1 ( J −1 )T ), ||B|| = trace( J W J T ), || B −1 || = / trace( J −1 W ( J −1 )T ), W is the weighting matrix

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1.2.2.2

1 Introduction

Kinematic Performance Indices Based on Screw Theory

According to screw theory, the instantaneous power of a wrench acting on a twist screw is called virtual work (VW). Yuan [70] defined the virtual coefficients of a transmission wrench and output twist screws as the transmission factor, which were used to evaluate the kinematic performance of a spatial PM. The value range of the transmission factor is from − ∞ to + ∞. Sutherland et al. [71] defined the normalized form of the transmission factor as the transfer index (TI). Tsai et al. [72] considered the transitivity and manipulability of a mechanism and proposed the total transmission index (TTI) as the product of TI and the manipulator index (MI, defined as the ability of the input that can be effectively absorbed into the rest of the mechanism). Based on the TTI, Chen et al. [73] proposed a generalized transmission index, which is a generalized form of the TI and transmission angle. Liu et al. [74–76] proposed a motion/force transmission performance index (including the input transmission performance and output transmission performance) based on screw theory, which can be used to evaluate the transmission efficiency of energy from the input to the output. The motion/force transmission performance index is dimensionless and independent of the coordinate system with a value ranging from 0 to 1 and allows an evaluation of the distance to singularity. Based on the performance chart-based design methodology and motion/force transmission performance index, Wu et al. [77], Wang et al. [78], and Chen et al. [79] optimized the kinematic performance of various PMs such as a 5R PM, while Li et al. [80–85] optimized the kinematic performance of other PMs such as a 2PUR-PSR PM. Chen et al. [86] proposed an orthogonalized force transmission index independent of the coordinate system based on the commonly used pressure angle/transmission angle of serial mechanisms to reflect the mapping relationship between the external load and constraint force. Note that screw theory is used in these indices that solve the shortcomings of the non-uniform dimension of performance indices based on the Jacobian matrix related to the coordinate system. Table 1.3 lists some typical kinematic evaluation indices based on screw theory. The performance indices mentioned above, which are based on the Jacobian matrix and screw theory, are dedicated to reflecting the static kinematic performance of the mechanism, but they are not dynamic. Therefore, establishing a motion/force transmission performance index that considers the inertial force, Coriolis force, centrifugal force, damping force, and dynamic load force can be closer to the real kinematic performance of PMs.

1.2.3 Stiffness Performance The stiffness performance reflects the ability of a PM to resist deformation under an external load, which reflects the accuracy of the mechanism, especially for PMs bearing heavy loads. Therefore, the stiffness performance measures are of great significance to PMs.

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11

Table 1.3 List of some typical kinematic performance indices based on screw theory

Physical meaning

Virtual coefficient [70]

TI [71]

TTI [72]

Generalized transmission index [73]

Motion/force transmission index [74–76]

Virtual work done by a unit force to the output twist, with a unit of length in the range of [−∞, + ∞]

Ratio of the virtual coefficient to the maximum virtual coefficient; dimensionless with a range of [0, 1]

Defined as TI × MI; dimensionless with a range of [0, 1]

Generalized transmission index based on the virtual coefficient; dimensionless with a range of [0, 1]

Minimum of the input and output transmission performance; dimensionless with a range of [0, 1]

Measures the quality of output motion transmission

Measures the quality of the transmissivity and manipulability of the mechanism

Measures the quality of transmissivity for single-loop spatial linkages

Measures the efficiency of energy from the input to the output

Application Measures the quality of scenarios motion transmission

Stiffness performance indices are proposed based on the overall Cartesian stiffness matrix to quantitatively measure the stiffness performance of PMs. The main difficulty relating to this procedure results from the fact that the eigenvalues of the stiffness matrices are undefined because of the dimensional inhomogeneity of the stiffness (or compliance) matrix elements. Comprehensive reviews relating to the dimensional inhomogeneity of robot stiffness and compliance matrices were developed by Kang et al. [87] and Cardou et al. [88]. The existing stiffness performance indices mainly concern two categories reflecting the magnitude and uniformity of stiffness performance.

1.2.3.1

Indices Reflecting the Magnitude of Stiffness Performance

Evaluation indices reflecting the magnitude of stiffness performance are commonly used today. Fundamental works on solving the stiffness evaluation problem were developed by Patterson and Lipkin [89, 90]. The minimum, maximum, and average eigenvalue indices [91] reflect the minimum, maximum, and average values of the overall stiffness matrix of the mechanism, respectively. However, the approach has non-uniform dimensions and undefined physical meaning. The trace index [92] takes the trace of the overall stiffness matrix as the index for stiffness performance measures, with the disadvantage of non-uniform dimensions. The weighted index of the trace [93] takes the sum of the weights of the overall traces of the stiffness matrix as the index to evaluate the stiffness performance of PMs. However, the determination of the weighting factors is subjective, and the dimensions are nonuniform. The determinant index [92] takes the determinant of the overall stiffness

12

1 Introduction

matrix as the index for stiffness performance measures. The main diagonal stiffness index [94] regards the main diagonal elements of the overall stiffness matrix as the index for stiffness performance measures in the corresponding direction. The abovementioned stiffness indices have a simple calculation procedure but with the relation to the coordinate system, non-uniform dimensions, and undefined physical meaning. Yang et al. [19] proposed a comprehensive stiffness index to solve the disadvantage that the main diagonal stiffness index ignores the coupling effect of offdiagonal elements. The extreme stiffness and corresponding extreme direction are decoupled when the force and couple act to obtain the extremum linear stiffness and angular stiffness performance, respectively. A total of four stiffness performance indices are used to avoid the undefined physical meaning caused by the coupling of the linear and angular stiffness elements. Portman developed three methods to obtain direct solutions of the robot stiffness evaluability problem, resulting in basic performance indices such as the maximal and minimal stiffness values, condition numbers, and stiffness range. The indices are obtained using all of the information constituting the original stiffness matrix, without any additional intuitive parameters. The wrench transformation technique [95] transforms the original stiffness matrix in the direct sum possessing three translational and three rotational eigenvalues, and the Schur complements method [96] generalizes the previous result. The same results are partially obtained through the optimization of the unit VW using the Lagrange multipliers method [97]. In addition to the eigenscrew decomposition index [98], the principal axis decomposition of the spatial stiffness matrix based on the congruent transformation [99], and the decoupled linear and rotational indices of the stiffness matrix [100], Raoofian et al. [101] decoupled the stiffness performance indices into two scalar force indices and two scalar couple indices. These indices decouple the linear and angular stiffness performance of the mechanism with the defined dimension and physical meaning, positively affecting the stiffness performance measures. However, these stiffness indices should be combined to evaluate the stiffness performance of the mechanism due to the lack of comprehensive performance indices of unified linear and angular stiffness. Researchers have proposed some stiffness performance indices that unify the linear and angular stiffness from the point of view of energy to solve the above problem. Yan et al. [102] proposed a VW index based on strain energy to unify the linear and angular stiffness for evaluating the stiffness performance of a mechanism. Cao et al. [12] defined the configuration stiffness index as the minimum strain energy stored by the mechanism under the external force or couple. Lian et al. [103] and Wang et al. [104] defined the reciprocal product of the external wrench and the corresponding infinitesimal elastic deformation as the instantaneous energy index. The proposed stiffness index based on strain energy unifies the linear and angular stiffness units and independent of the coordinate system. However, this index based on the energy dimension cannot reflect the working accuracy of PMs.

1.2 Performance Evaluation Indices

1.2.3.2

13

Index Reflecting the Uniformity of Stiffness Performance

Similar to the kinematic index, the condition number of the stiffness matrix can reflect the isotropy, uniformity, and singularity of the stiffness performance of PMs. The norm index introduced by Carbone [92] takes the norm of the stiffness matrix as the index for stiffness performance measures. The condition number stiffness index proposed by Shin et al. [105] was used to evaluate the isotropy and uniformity of the stiffness performance of the mechanism. Zhao et al. [106] defined the ratio of the square of the maximum stiffness value to the product of the minimum stiffness and average stiffness in the workspace as the uniformity index of stiffness performance to evaluate a mechanism in its workspace. Gosselin et al. [107] and Guo et al. [108] defined the ratio of the maximum eigenvalue to the minimum eigenvalue of the stiffness matrix as the local isotropic index of the stiffness performance and the ratio of the maximum to the minimum isotropic indices in the workspace as the global stiffness uniformity index. Yeo et al. [109] used the ratio of the product of the square of the maximum eigenvalue and the square of the minimum eigenvalue in the stiffness matrix to the sum of the maximum and minimum eigenvalues of the stiffness index to solve the separation of the stiffness performance amplitude index and isotropic index. The index is used to evaluate the amplitude and isotropy of the stiffness performance of a mechanism. This index is more suitable for stiffness matrices of planar PMs rather than spatial PMs. On this basis, Görgülü et al. [110] defined the ratio of the ideal sphere volume to the ellipsoid volume formed by the eigenvalues and eigenvectors of the overall stiffness matrix of the mechanism as the isotropic volume index, which is used to analyze the stiffness performance of an R-CUBE PM. This stiffness index is greater than or equal to 1. However, the non-uniform dimensions and unclear physical meaning of this stiffness index are its main shortcomings. To effectively improve the stiffness performance of PMs, it is necessary to establish the stiffness contribution of each factor to the overall stiffness performance. Yang et al. [43] defined the ratio between the strain energy stored in each component and the PM as the strain energy factor index (SEFI), which is frame-free and bound between 0 and 1. With an external load and trajectory at hand, the SEFI can quantitatively evaluate the contribution of each component to the overall stiffness performance of a PM. Yang et al. [111] decoupled the linear and angular elastic deflection contribution of each component to the PM based on the rigidity principle, which allowed a quantitative evaluation of the contribution of each elastic component to the linear and angular stiffness performances of the mechanism, thus providing a new approach for effectively improving the linear/angular stiffness performance of the mechanism. However, if this method is used to address over constrained PMs, the Moore–Penrose inverse is required to avoid singularity of the compliance contribution matrix. In addition to the above-mentioned stiffness performance indices, some related research on the variable stiffness and joint clearances has also been carried out. Zhao et al. [112] proposed a parallel compliance device that was inspired by a 3UPU PM, and in each limb of the device, there was an electromagnetic variable stiffness spring. The endpoint stiffness performance under multiple combinations of

14

1 Introduction

spring stiffness values was exhibited in the form of stiffness ellipsoids. Li et al. [113] proposed a flexible 3PU PM that employed superelastic nickel–titanium rods to achieve compliant movements beyond conventional rigid-body PMs and carried out circle trajectory tracking and contact force tests by cadaveric trials. Fan et al. [15] considered the flexibility of links and the clearance of joints to establish overall stiffness models of Tricept and A3 PMs and proposed stiffness ellipsoid indices to enhance their stiffness performance by means of the component selection method. The difference between the ideal and combined spherical joints was also presented [114]. Notably, their results showed that the combined spherical joints improved the workspace and undermined the stiffness performance of PMs. Table 1.4 lists some typical indices obtained by non-recommended methods. These indices are obtained by direct mathematical operations on the stiffness matrix without considering the non-uniform dimensions of matrix elements. Table 1.5 lists some stiffness indices using recommended methods that include compliant axes method, matrix decomposition method. The Schur complements-based method, the virtual work optimization method, and strain energy method. It is important to emphasize that stiffness performance indices are dimensionless and independent of a coordinate system with a defined physical meaning. The existing stiffness indices ignore the influence of the gravity and friction of the mechanism on the performance indices. Therefore, establishing the stiffness contribution of each factor, including the gravity, friction, and flexibility of the mechanism, can provide important significance for effectively improving the overall stiffness performance Table 1.4 Some typical stiffness performance indices of PMs using the non-recommended methods Minimum, maximum, and average eigenvalues [91]

Trace, determination, and stiffness ellipsoids [92, 93]

Main diagonal stiffness index [94]

Condition Ratio of number [105] eigenvalues or the volume of an ideal sphere and ellipsoid [107–110]

Physical meaning

Eigenvalues have undefined dimensions, so the physical meaning is ambiguous

Non-uniform dimensions of matrix elements and an unclear physical meaning

Contains definite physical dimensions but ignores the coupling effect of off-diagonal elements

Non-uniform dimensions and an undefined physical meaning

Non-uniform dimensions and an unclear physical meaning

Application scenarios

Minimum, maximum, and average magnitudes of stiffness performance

Magnitude of stiffness performance

Linear/angular stiffness performance in the axial direction of a Cartesian coordinate frame

Isotropy and uniformity of stiffness performance

Isotropy and uniformity of stiffness performance

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15

Table 1.5 Some typical stiffness performance indices of PMs using the recommended methods

Physical meaning

VW index Comprehensive Decoupled SEFI [43] related to stiffness index extreme strain energy [19] linear/angular [12, 102–104] stiffness indices [19, 95–100]

Decoupled linear/angular elastic deflection contribution index [111]

Unifies the units of the linear and angular stiffness and has a clear physical meaning

Application Evaluates the scenarios magnitude of stiffness performance based on the strain energy; this is suitable for cases where the external wrench is known

Contains Contains a definite clear physical physical meaning dimensions and considers the coupling effect of off-diagonal elements

Dimensionless and a clear physical meaning

Uniform dimensions and a clear physical meaning

Linear/angular stiffness performance in an arbitrary direction in space

Contribution of each elastic component to the overall stiffness performance; this is suitable for cases where the external wrench is known

Evaluates the linear and angular elastic deflection contributions of each component to PMs; this is suitable for cases where the external wrench is known

Evaluates the extreme linear/angular stiffness performance of a mechanism against an external force/moment

and error compensation of PMs. The decoupling of the overall stiffness/compliance matrix under the action of simultaneous force and moment is also one of the issues to be addressed.

1.2.4 Dynamic Performance Under the high-speed operating conditions of PMs, excellent dynamic characteristics can guarantee higher positional accuracy and workpiece quality. Therefore, dynamic performance measures and control are vital to developing high-precision robots. Commonly used dynamic evaluation indices mainly concern the inertia index, dynamic dexterity index, dynamic manipulability index, dynamic stiffness index, and natural frequency index. The first three indices are mainly used to evaluate the rigid dynamic performance of PMs while the latter two are used to evaluate the flexible dynamic performance of PMs.

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1.2.4.1

1 Introduction

Rigid Dynamic Indices

A rigid dynamic model is established under the assumption that the mechanism is rigid to obtain the dynamic indices, which reflects the rigid dynamic performance of PMs. The inertia index includes two parts, namely the first mass moment of inertia and the second mass moment of inertia. Compared with a serial mechanism, the actuators of a PM can be installed on the base, thereby reducing the moving mass. The minimization of the mass index reduces the inertial force, which allows increases in the speed and acceleration and improves dynamic performance. However, reducing the mass can decrease the stiffness performance. When the mass is optimized, the stiffness performance of PMs should meet the processing requirements [115]. Wan et al. [116] adopted the diagonally dominant elements of a mass matrix as the dynamic performance index of an 8-SPU parallel walking mechanism. Similar to the kinematic dexterity index, the dynamic dexterity index is defined as the condition number of the inertia matrix. Asada [16] proposed a generalized inertial ellipsoid index based on the generalized inertia tensor. The length of the principal axis of an ellipsoid is equal to the reciprocal of the square root of the corresponding eigenvalue. The minimum principal axis corresponds to the maximum inertia, where the speed in this direction is the slowest and vice versa. On this basis, Kilaru [117] divided the square of the number of DOFs of a mechanism by the condition number of the inertia matrix whose eigenvalues are diagonal elements and defined as a standardized inertial isotropy index with a value of 0–1. Merlet [118] proposed a dynamic manipulability index based on Jacobian and mass matrices. Yoshikawa [119] proposed a dynamic manipulability ellipsoid index, which can be used to evaluate the isotropic ability of a mechanism to change the posture and position of its end effector under a constant actuation force. The condition numbers ˜ J˜ +R and M ˜ J˜ + M T are separately used to evaluate the isotropy of the rotational and ˜ denotes the inertia matrix of translational dynamic manipulability of PMs, where M + + the PM and J˜ R and J˜ T denote the Jacobian matrices of rotational and translational acceleration, respectively. Chai et al. [120, 121] analyzed the dynamic performance of a 2UPR-RPU PM based on the dynamic manipulability ellipsoid index. The abovementioned stiffness indices assume that the mechanism is rigid and are established based on the mass matrix without considering the influence of elasticity on the dynamic performance of PMs; thus, these indices are not comprehensive.

1.2.4.2

Flexible Dynamic Indices

Flexible dynamic indices consider the elasticity of PMs, which can be obtained by establishing elastodynamic or rigid–flexible coupling dynamic models. These indices mainly concern the dynamic stiffness index and natural frequency index. The dynamic stiffness index is used to evaluate the response of a mechanism under a variable-frequency dynamic load, which is a function of the external excitation

1.2 Performance Evaluation Indices

17

frequency. When the external excitation frequency is equal to the natural frequency, the dynamic stiffness performance of a mechanism is at its lowest. The dynamic stiffness index is used to evaluate the ability of a PM to resist dynamic loads. Azulay et al. [122] and Alagheband et al. [123] evaluated the dynamic performance of a 3PPRS PM based on the dynamic stiffness index. Similar to the dynamic stiffness index, the natural frequency index of a mechanism, especially the fundamental frequency, is essential to measure the dynamic performance of a mechanism. It comprehensively considers the influence of the mass matrix, stiffness matrix, and damping matrix on the dynamic performance of PMs. A higher natural frequency, especially a higher fundamental frequency, means a higher control bandwidth, which can reduce the vibration response of a mechanism. Hoevenaars [124] used screw theory to obtain a mapping relationship between an end effector and the motion of a subchain rod and combined the inertia matrices of the moving platform and rods to establish the overall inertia matrix of the mechanism. The overall stiffness matrix of the mechanism was obtained based on an elastostatic stiffness analysis, and the natural frequency of a Heli4 PM was analyzed by combining the stiffness and inertia matrices. Luo et al. [125] and Zhang et al. [126] analyzed the natural frequency distribution of a 3RPS PM based on substructure synthesis. Liu et al. [127] analyzed the frequency characteristic distribution of a 3RRS PM using the Lagrangian equation and deformation compatibility equations to analyze the influence of the dimensional parameters, sectional parameters, and material properties of a mechanism on the natural frequency. Yang et al. [128] analyzed the natural frequency distribution of 2UPR-RPU and 2UPR-2RPU PMs in a workspace based on global independent generalized displacement coordinates and a matrix displacement method. After comparison, they found that the redundant limb can significantly improve the fundamental frequency of the mechanism. The natural frequency index has advantages such as clear physical meaning and being frame-free. However, it cannot directly reflect the dynamic response of PMs under external load.

1.2.4.3

Other Indices

The maximum actuation force index [129], power consumption index [130], maximum speed, and acceleration indices [131] are also commonly used to evaluate the dynamic performance of rigid dynamic, flexible dynamic, and rigid–flexible coupling dynamic models of a mechanism. Corinaldi et al. [132] considered angular acceleration as an objective function and planned a path on the spherical surface by means of Bézier curves. Lou et al. [133] investigated three dynamics-based trajectory planning methods, namely time-optimal planning, cubic polynomial planning, and jerk-bounded planning that considered both kinematic and dynamic constraints. A computed torque control scheme was applied to follow the computed tractor, and the optimal control parameters were investigated by considering the settling time that reflected the response speed as an objective function; in this case, overshooting, the steady error, and the actuation saturation were used as constraints. Liu et al. [134]

18

1 Introduction

Table 1.6 List of some typical dynamic evaluation indices Inertia index [116]

Condition number [16, 117–119]

Dynamic stiffness index [122, 123]

Natural frequency [124]

Maximum actuation force, speed, and acceleration indices [129–131]

Physical meaning

Magnitude of the mass matrix

Norm of the mass matrix or a combination of the mass and Jacobian matrices

Influence of the stiffness and mass matrices on the dynamic performance of PMs

Contains defined dimensions and a clear physical meaning

Measures the maximum actuation force, speed, and acceleration

Application scenarios

Suitable for evaluating the inertia force of a rigid dynamic model

Suitable for evaluating the dexterity and isotropy of a rigid dynamic model

Suitable for evaluating the displacement response of flexible and rigid–flexible coupling dynamic models under known dynamic loads

Measures the ability of PMs to resist the vibration response; suitable for the flexible dynamic model

Suitable for evaluating the dynamic performance of rigid, flexible, and rigid–flexible coupling dynamic models

proposed two dynamics-based performance indices, namely the coordination ratio and coordinate factor; the former was combined with the hybrid control of force and position through a synchronous controller to enhance the actuation coordination of a 6PUS + UPU PM, and the latter was combined with trajectory planning to enhance the control stability of a 3RPR PM. Table 1.6 shows the list of some typical dynamic evaluation indices of PMs to increase the clarity and intuitiveness of the article. The kinematic performance, stiffness performance, and dynamic performance indices of PMs introduced above are considered as local performance indices (LPIs). Future research trends for the performance evaluation of PMs should establish decoupled performance indices with defined physical meaning and consider the influence of friction, dynamic load, and gravity. The average [76] and variance [135] of LPIs in a workspace are usually used as global performance indices (GPIs) to evaluate the overall performance of PMs. Since it is difficult to obtain the analytical expression of a workspace, GPIs are often solved by a numerical discrete method. That is, the accuracy of the GPIs is related to the number of discrete points in the workspace, which consists of time-consuming calculations; thus, GPIs are not conducive for performance evaluation and optimization design of a mechanism. Therefore, the main issues of evaluation of GPIs include the analytical expressions of workspaces (especially regular workspaces) and the availability of high-precision and low-cost numerical calculation methods.

1.3 Optimization Algorithm

19

1.3 Optimization Algorithm An optimal design of PM dimensions after determining the performance indices and constraint conditions is used to maximize the comprehensive performance. Reasonable optimization algorithms are significant for obtaining the ultimate optimal working state and design parameters of a parallel robot system. Currently, commonly used optimization algorithms mainly concern two types: traditional optimization algorithms and intelligent optimization algorithms. Robot optimization design is usually divided into single- and multi-objective optimization designs. Single-objective optimization is used for a performance index, which can be achieved through traditional optimization algorithms or intelligent optimization algorithms. However, conflicts exist among various performance indices, and the optimization of a performance index may cause deficiencies in other performance indices. For example, maximizing the workspace of a robot may result in insufficient stiffness and dynamic performance, thus failing to meet engineering requirements. Therefore, a multi-objective optimization design is often used in engineering applications to achieve comprehensive optimization of multiple performance indices. The multi-objective optimization design of parallel robots is often realized through intelligent optimization algorithms.

1.3.1 Traditional Optimization Algorithms Traditional optimization algorithms generally aim at structured problems such as linear programming and quadratic programming with and without constraints. Most of these approaches belong to the category of convex optimization; in other words, these approaches have only a specific global optimal point. Traditional optimization algorithms are deterministic algorithms compared with intelligent optimization algorithms based on heuristic and probabilistic rules. The computational complexity and convergence of a traditional optimization algorithm can be analyzed theoretically, and its search information involves the derivation of an objective function. Commonly used traditional optimization algorithms for the optimization design of PMs mainly concern the exhausted search algorithm, the gradient descent method, Newton’s method, quasi-Newton method, the conjugate gradient method, and the sequential quadratic programming (SQP) algorithm. Additionally, the Lagrange multipliers method was applied to solve the stiffness extremum with the analytical objective functions and equality constraints by Portman [97], Yang et al. [19], and Hu [136].

1.3.1.1

Exhausted Search Algorithm

The exhausted search algorithm discretizes the design parameters into a certain number of grid points and calculates the function value of each node to obtain the

20

1 Introduction

global optimal value of the function. Theoretically, the algorithm can calculate the optimal value of any optimization model; however, the accuracy of the global optimal value depends on the node density. As the density of the grid points increases, the computational cost exponentially increases. Thus, this approach is generally suitable for small-scale optimization problems. For example, Xu et al. [82] used an exhausted optimization algorithm to optimize the kinematic performance of a 2UPR-2PRU PM with two design parameters.

1.3.1.2

Gradient Descent Method and Newton’s Method

The negative gradient direction of the current position is adopted for optimization in the gradient descent method. The method is the earliest and simplest and one of the most commonly used methods. Chaudhury et al. [50] used a gradient descent algorithm to optimize the dexterous workspace of a GSP and other PMs. The gradient descent method is simple to implement through the fmincon function in the MATLAB toolbox. When the objective function is convex, the solution of the gradient descent method is global. Generally, the solution is not guaranteed to be optimal on a global scale, and the speed of the gradient descent method may not be the fastest. When the iteration point is closer to the target point, the step size is smaller with a slower convergence speed. Additionally, there may be a zigzag decline, which may require many iterations. Newton’s method, often called the tangent method, is based on the tangent of the current position to determine the next position. Compared with the gradient descent method with first-order convergence, Newton’s method has second-order convergence with a fast convergence speed. However, the inverse of the Hessian matrix should be solved at each iteration, which requires complex calculations. A quasi-Newton method uses a positive definite matrix to approximate the Hessian matrix, thereby simplifying the complex calculations. Botello et al. [137] used the quasi-Newton method to optimize the workspace and dexterity of a Delta PM. The convergence speed of the quasi-Newton method was much faster than that of the gradient descent method. Since the quasi-Newton method does not require the second derivative, it is sometimes more effective than Newton’s method. The quasi-Newton method can be realized by the fmincon function in the MATLAB toolbox. Only the first derivative is used in the conjugate gradient method, which overcomes the shortcomings of the gradient descent method and Newton’s method. Aboulissane et al. [138] used the conjugate gradient algorithm to optimize the workspace of 3RPR and Delta PMs. The conjugate gradient method is one of the most useful methods for solving large-scale linear and nonlinear optimization problems.

1.3.1.3

SQP Algorithm

The SQP method is an algorithm that transforms complex nonlinear constraint optimization into relatively simple quadratic programming. In the iterative process, each

1.3 Optimization Algorithm

21

step needs to solve one or more quadratic programming subproblems. Erylmaz et al. [139] used the SQP method to optimize the workspace and kinematic condition number of a 3 × 3 UPU PM. Pedro et al. [140] divided the parameter space into several parameter subspaces and summarized the optimal value of each subspace as the initial guess value of the entire parameter space. The kinematic performance of a 3UPS-1RPU PM was optimized based on the SQP algorithm to enhance its dexterity without reducing the workspace. The advantages of the SQP algorithm include high computational efficiency, good convergence, and a strong boundary search ability; therefore, this algorithm has received considerable attention and has been applied in engineering. This algorithm can also be achieved through the fmincon function in the MATLAB toolbox. However, multiple quadratic programming should be solved at each iteration. As the scale of an optimization problem increases, the storage volume and computational cost also increase; thus, this approach is generally suitable for small- and medium-sized problems. The traditional optimization algorithm searches in a single-point manner, and the optimization result depends on the selection of the initial value. Only one optimal point can be obtained through one optimization; thus, the Pareto frontier of multi-objective functions cannot be obtained. Lou et al. [141] took Delta and Gough–Stewart PMs to compare the convergence performance of the SQP and intelligent optimization algorithms and concluded that the SQP algorithm with multiple initial points was more effective for small-scale optimization problems. The disadvantages of traditional optimization algorithms are as follows: (a) They usually find only the local optimal solution of an optimization problem; (b) the optimization results strongly depend on the initial value and are only suitable for smalland medium-scale optimization problems; and (c) they are difficult to implement for multi-objective optimization design.

1.3.2 Intelligent Optimization Algorithms Dimensional synthesis is challenged by the time-consuming calculation of the GPIs of PMs, as well as the coupling competitiveness among performance indices. The traditional optimization algorithms search in a single-point manner, which easily results in a local optimum. Moreover, traditional optimization algorithms often require that the objective function is convex, continuous, and differentiable, and that the feasible region is a convex set; therefore, their abilities to process nondeterministic information are poor. These weaknesses limit the use of traditional optimization algorithms in solving many practical optimization problems. Intelligent optimization algorithms provide a new idea for these complex multiextreme and multi-objective optimization models. The intelligent optimization algorithm, also called a heuristic algorithm, is used to solve a problem based on empirical rules. As a random-search evolutionary algorithm based on probability, it searches for a population number of points in parallel, which is conducive to finding the global optimum. Such new algorithms generally do not require continuous and convex objectives and constraint functions and sometimes do not even require analytical

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1 Introduction

expressions. Intelligent algorithms often used in the multi-objective optimization design of PMs mainly concern the genetic algorithm (GA) and particle swarm optimization (PSO) algorithm. In addition, the differential evolution (DE) algorithm, artificial bee colony (ABC) algorithm, ant colony optimization (ACO) algorithm, and hybrid algorithm have been applied to multi-objective optimization models of PMs.

1.3.2.1

GA

The GA is an intelligent algorithm simulating the genetic and evolutionary laws of organisms in nature and the natural selection and genetic mechanism of Darwin’s biological evolution theory. This makes the optimization process similar to the crossover and mutation of chromogens in biological evolution. In each iteration process, the algorithm will evolve to a new population by selection, crossover, and elite retention until the convergence conditions are reached. Compared with traditional optimization algorithms starting from a single solution, the GA searches from a set of solutions that cover a large region, making it possible to jump out of the local optimum and obtain a global optimum. Wu [131, 142] used the GA to optimize the stiffness performance, kinematic performance, workspace, and dynamic performance of biglide and Ragnar PMs and obtained a scatter matrix between the objective functions and design parameters. This provided a theoretical basis for determining optimal design parameters. Bounab et al. [42] and Kelaiaia et al. [49] took the stiffness performance, mass, and workspace as objective functions and obtained the Pareto frontier of the DELTA PM for multiobjective optimization based on the GA. Ferrari [143] optimized the workspace and longitudinal geometric dimensions of a 6-DOF PM using the GA. Song et al. [144] considered the geometric constraints of a mechanism and optimized the kinematic performance of a 1T3R PM based on the GA. Huang et al. [33] optimized the stiffness performance, kinematic performance, and workspace volume of a 3-DOF PM based on the GA. Ganesh et al. [145] adopted a GA to carry out multi-objective optimization design including a workspace index, global conditioning index, and global stiffness index for a 3-DOF star triangle manipulator. The GA has the following advantages: Obtains the global optimal solution of a problem, exhibits strong robustness, solves complex optimization problems, and utilizes distributed calculations. It has been widely used due to its scalability (easy to combine with other algorithms). Its main disadvantage is that operators, such as the crossover rate and mutation rate, have many parameters, and various selections can seriously affect the quality of the solution. At present, most of the selected parameters are based on experience. The GA has poor local search ability, low search efficiency in the later stage of evolution, and slow convergence speed. In practical engineering applications, the GA is prone to premature convergence. Therefore, developing a method that retains good individuals and a diverse group has always been a challenge with the GA used.

1.3 Optimization Algorithm

1.3.2.2

23

PSO

The basic concept of the PSO algorithm is to simulate the foraging behaviors of a flock of birds; for instance, the flock adjusts its path when searching for food based on its own experience and communication among populations. Each particle in the swarm updates its spatial position and flight speed by tracking two extremums. One is the optimal solution particle found by a single particle in its iteration process, called the individual extremum. The other is the optimal solution particle that the population finds in the iterative process. That is, the extremums of all individuals in each iteration process, which is the global extremum. The PSO algorithm is also based on the concepts of population and evolution. However, it determines the search according to its speed instead of the crossover and mutation operations of the GA, making it easier to implement. Wang et al. [146] optimized the workspace, natural frequency, input coupling rate, and stiffness performance of a planar 3-DOF PM based on the PSO algorithm. Sun et al. [115] optimized the mass and stiffness performance of a T5 PM based on the PSO algorithm. Yun et al. [147] compared the GA, PSO, and gradient-based optimization algorithms for the optimization design of a 3PUPU PM and found that the PSO algorithm was better than the other approaches. Wang et al. [104, 146] used the PSO algorithm for the optimization design of robots such as Tricept and verified its calculation efficiency. Lian [148] used the PSO algorithm to optimize the stiffness performance, dynamic performance, and mass of a PaQuad PM. Qi et al. [149] considered the dimensional parameters and section parameters of a mechanism and used the PSO algorithm to optimize the workspace, kinematics, stiffness, and dynamic performance of a 4RSR-SS PM. Farooq et al. [150] considered the conditioning index, workspace volume, and global conditioning index as a weighted comprehensive index and concluded that the PSO algorithm required fewer iterations and had a higher convergence rate than the GA. Similar to the GA, the PSO algorithm often encounters premature convergence and poor convergence performance for complex high-dimensional problems; thus, convergence to the global optimum is not guaranteed.

1.3.2.3

Other Intelligent Optimization Algorithms

In addition to the GA and PSO algorithms, other intelligent algorithms can also be found in the optimization design of PMs. Similar to the GA, the DE algorithm [151] is also a stochastic model simulating biological evolution, including mutation, crossover, and selection operations. It randomly selects the difference vector of two individual vectors from the population as the source of the random variation of the third individual. After weighting the difference vector, the algorithm is summed with the third individual according to certain rules to generate an individual variation. The DE algorithm adopts real number encoding, a simple mutation operation based on difference, and a one-to-one competitive survival strategy, which reduces the

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1 Introduction

Table 1.7 Development history of the intelligent optimization algorithms for the optimization design of PMs GA [157]

PSO algorithm [158]

DE algorithm [159]

ACO algorithm [160]

ABC algorithm [161]

Year

2007

2008

2011

2013

2014

Optimization objects

Regular dexterous workspace

Path planning and kinematic performance

Path planning and stiffness performance

Dynamic performance

Kinematic and dynamic performance

complexity of the genetic operation [152]. Zhang [153] used the DE algorithm to optimize the stiffness performance and regular workspace volume of a 3UPU PM. The ACO algorithm [154] is a probabilistic algorithm inspired by the behavior of ant colonies searching for food in nature. In the algorithm, several ants jointly construct a solution path and improve the solution quality by leaving and exchanging pheromones on the solution path. This algorithm uses a positive feedback mechanism and was originally applied to combinatorial optimization. Due to its high computational cost, the basic ACO algorithm is not suitable for the direct and continuous optimization of PMs. Yang et al. [155] introduced variable neighborhood search and search along the way to adjust the local search, global search, and pheromone updating rules of the basic ACO algorithm. This improved ACO algorithm was used to optimize the dexterity of a 6-DOF parallel platform. The ABC algorithm [156] is an intelligent optimization method proposed by imitating the foraging behaviors of bees. Bees carry out different activities according to their respective division of labor and realize the sharing and communication of bee colony information. Through the local optimal behavior of each artificial bee, the global optimal value is finally highlighted in the group, thus finding the optimal solution of the problem. Mirshekari et al. [57] considered the geometric constraints of the mechanism and used the ABC algorithm to optimize the kinematic and dynamic performance of a 6RUS PM. For intuitive understanding, the development history of the above-mentioned optimization methods applied to the optimization design of PMs and the corresponding optimization objects are listed in Table 1.7. Additionally, Table 1.8 briefly summarizes the advantages, disadvantages, and application scenarios of each intelligent optimization algorithm.

1.3.2.4

Hybrid Algorithm

Many researchers have studied the mixed use of intelligent optimization algorithms and the combination of intelligent optimization and traditional optimization algorithms in the multi-objective optimization design of PMs. Fabio et al. [162] combined the GA and PSO algorithms to optimize the dexterity, stiffness, and motion/force transmission performance of 2- and 3-RRR PMs and compared their optimization results. Mao et al. [163] combined the DE and PSO algorithms to optimize the forward

1.3 Optimization Algorithm

25

Table 1.8 Comparison of the intelligent optimization algorithms in regard to their advantages, disadvantages, and application scenarios GA [131]

PSO algorithm [146]

DE algorithm [151]

ABC algorithm [156]

ACO algorithm [154]

Distributed calculations, scalability, and robustness

Easy to implement, fast convergence at an early stage

Strong population self-regulation ability, few control parameters, and fast convergence

Scalability, robustness, and a cooperative working mechanism

Distributed calculations and robustness

Disadvantages Poor local search ability, low search efficiency at a later stage, prone to premature convergence, and many parameters of the operators need to be optimized

Low convergence speed at a later stage or even stagnates, prone to premature convergence, and poor performance for discrete optimization problems

The selection of control parameters and evolution strategies are usually determined based on experience, premature convergence, and search stagnation of continuous variables multi-objective optimization

Converges to a local optimum, optimization may stagnate in the neighborhood of some local optimal solutions, and a high computational cost

Low convergence speed, poor population diversity, prone to fall into local optimum, difficult to jump out of the local optimum, and a high computational cost

Application scenarios

Versatile

Small and medium-scale optimization

Path planning

Advantages

Versatile

solution of a 3RPS PM. To obtain the comprehensive optimum of multi-objective performance, Rosyid et al. [164] combined the GA and gradient descent algorithm to optimize the workspace and stiffness performance of a 3PRR PM, and Liu [48] applied the GA and SQP algorithm for the optimization design of a Delta PM. For multi-peak problems [165], intelligent optimization algorithms [166] can jump out of the local optimum through their effective design, thereby converging to the global optimum. However, most intelligent optimization algorithms are heuristic algorithms, which can be used for qualitative analysis instead of quantitative analysis. In addition, intelligent optimization algorithms are mostly based on random characteristics, and their convergence is generally in the sense of probability. The actual performance is uncontrollable, with the disadvantages of slow convergence speed and high computational complexity. Thus, a major task in the current optimization design of PMs is to improve model accuracy and accelerate convergence by enhancing the theory of intelligent optimization algorithms or combining traditional algorithms and

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1 Introduction

intelligent optimization algorithms. It is necessary to increase the population size to obtain more non-dominated solutions in actual engineering; however, this leads to an exponential increase in computational cost. Therefore, developing methods that increase the population size without causing a significant increase in computational cost is currently a major task.

1.3.3 Ways to Improve Optimization Algorithms In the optimization design process, GPIs are often obtained by calculating the mean of the LPIs of a limited number of discrete nodes in a workspace. As the number of discrete nodes increases, the computational cost increases exponentially. The optimization process requires repeated calculation of each node, and a large number of iterations and a high computational cost significantly reduce the efficiency of the optimization design. Many researchers have studied methods to improve the optimization design efficiency of PMs. After years of development, these methods mainly concern the Monte Carlo method, performance atlases, mapping model method, and sensitivity analysis method.

1.3.3.1

Monte Carlo Method

A number of discrete points meeting the constraints and inverse kinematics of the mechanism in a complete workspace are randomly allocated in the Monte Carlo method. These discrete points are analyzed to obtain the GPIs of the mechanism, which improves the inefficiency of the grid point method to some extent. Tsai et al. [167] used the Monte Carlo method to randomly allocate a number of discrete points in a complete hemispherical space and optimized the global condition number performance index of a 3UPU PM by an exhausted point-by-point search algorithm. Chaudhury et al. [50] used the Monte Carlo method to calculate the dexterous workspace of a planar 5-bar PM and a spatial 6-DOF Stewart PM. Then, the gradient descent algorithm was combined to carry out the optimization design of PMs. The Monte Carlo method is a statistical simulation method that is a random approximation method. A larger simulation sample means smaller errors and an increase in computational cost. Therefore, the improvement of optimization design efficiency of PMs by Monte Carlo method is limited.

1.3.3.2

Performance Atlases

An optimization design generally requires known internal of design parameters. Performance atlases [168–170] transform the design parameters into dimensionless parameters without knowing in advance the interval of the design parameters. Performance atlases reduce the design parameters from n to n-1 and design a similar

1.3 Optimization Algorithm

27

mechanism with optimal kinematic performance to establish a mapping relationship between the finite and infinite spaces. In 2006, Park et al. [171] considered the kinematic performance and workspace of the mechanism and optimized a 2DOF planar PM using performance atlases. On this basis, Liu et al. [168] gave a further introduction to performance atlases and pointed out that when the number of design parameters exceeded 4, the performance map between the performance indices and characteristic parameters could not be obtained. Additionally, performance atlases have been used to optimize the kinematic performance of PMs such as PVRRRPV [172–175]. Xu et al. [80–85] reduced the design parameters from 3 to 2 based on performance atlases and optimized the kinematic performance of PMs such as 2UPR-2PRU and 2-(PRR)2RH PMs. The greatest advantage of performance atlases is that a group of similar mechanisms with optimal kinematic performance can be obtained without knowing in advance the interval of the structural parameters. Thus, the design parameters and computational cost can be reduced. However, there are few reports about the performance atlases applied to a task in which the number of design parameters is more than 5.

1.3.3.3

Mapping Model Method

A mapping model between a GPI and design parameters is established to improve the high computational cost of GPIs and the optimization design of a mechanism. This method is a major way to improve the optimization efficiency of PMs. Commonly used methods for establishing a mapping model mainly concern multivariate regression (MR), Gaussian process regression (GPR), and backpropagation (BP) neural networks. MR is based on the least square method that gives the function f (x, a1 , a2 , …, an ) and its measured values are y1 , y2 , …, yN at N different observation points of parameters a1 , a2 , …, an are to be determined to minimize the x 1 , x 2 , …, x N . The Σ N [ f (x i ; a1 , a2 , . . . , an ) − yi ]2 . Sun et al. [115], Qi et al. standard deviation i=1 [149], and Lian et al. [148, 176] established analytical mapping models between objective functions and design parameters based on MR and found that the fitting accuracy of the second or third polynomials was optimal; this was then combined with the PSO algorithm for the multi-objective optimization design of a T5 PM. Rahman et al. [177] combined MR and the gradient descent optimization algorithm to optimize the kinematic performance of a 3P-S-S/S PM. Wang et al. [178] and Palmieri et al. [179] combined MR and the GA to optimize the layout of multiple PMs and compared them with the finite element optimization algorithm. The optimization method combining MR and the GA can significantly improve the optimization design efficiency. MR can establish analytical mapping models between design parameters and objective functions to improve the computational efficiency of GPIs. However, the regression equation is only a speculation, and different orders of polynomials need to be tested to find the optimal fitting accuracy, which affects the diversity and uncertainty of factors.

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1 Introduction

GPR is a nonparametric model using the Gaussian process (GP) prior to the regression analysis of data. It is a kernel-based probability model that has been applied to robot mapping modeling and dimensional synthesis [180]. GPR can provide a posterior of the predicted results; when the likelihood is normally distributed, the posterior is in an analytical form. In addition, the GPR method does not need to specify the order of the model. Nguyen et al. [181] constructed GPR mapping models among the natural frequency, stiffness, and damping coefficients and configurations of a 6-DOF industrial robot using experimental sample data and concluded that the accuracy of the GPR-based model was better than that of the MR-based model. Cheng et al. [182] established a GPR proxy model of the dynamic response and combined it with the GA to optimize the dynamic parameters of a Delta robot to reduce the vibration response of the equipment. Wan et al. [183] and Chen et al. [184] established a mapping model for assembly error and performed error compensation based on the GPR model. Chen et al. [185] combined the GPR and PSO algorithms to propose a hybrid algorithm for the multi-objective optimization design of PMs and compared the GPR- and MR-based mapping models. The results showed that the GPR-based model had a higher accuracy than the MR-based model. The advantage of the GPRbased model is that the predicted value is an interpolation of the observed value, which is probabilistic, so the empirical confidence interval can be calculated. Then, the information is used for refitting (online fitting and adaptive fitting) and prediction in a certain region of interest, and different kernels can be specified. The disadvantage is that GPR uses all of the sample/characteristic information for predictions, which is not efficient. Therefore, GPR is suitable for processing a small collection of data rather than high-dimensional spatial models. The BP neural network, proposed in 1986 by a scientific research group headed by Rumelhart [186], is a multilayer feedforward network trained by an error BP algorithm. A BP network has strong nonlinear mapping ability. Its learning rule uses the steepest descent method to adjust the weights and thresholds of the network through backpropagation to minimize the sum of squared errors of the network. Gao et al. [187] combined the neural network algorithm and the GA to optimize the stiffness performance of a 3-DOF PM. Huang et al. [188] and Hu et al. [189] established mapping models between the workspace, stiffness, condition number, first-order natural frequency performance indices, and structural parameters based on the BP neural network. Gao et al. [93] and Zhang et al. [190] established mapping models between the stiffness and dexterous indices and design parameters of a spatial 6-DOF PM based on the BP neural network and combined it with the GA for a multi-objective optimization design. It has been proved that a three-layer neural network can approximate any nonlinear continuous function with arbitrary precision. The BP neural network also has the advantages of self-learning, self-adaptation, generalization ability, fault tolerance, and associative memory. However, after indepth research, the network has the following shortcomings: Easily falls into local minimization, has a slow convergence speed, and high dependence on the sample. Additionally, it is sensitive to the initial network weights, and initializing the network with different weights can converge to different local minima. This is the fundamental reason why different results are obtained after each training [185].

1.3 Optimization Algorithm

29

Establishing mapping models between GPIs and design parameters is now the main way to improve the multi-objective optimization design efficiency of PMs. However, the mapping model has inevitable errors. Thus, it is a major challenge to establish a theoretical model or a highly precise and efficient mapping model between GPI and design parameters for the optimization design of PMs.

1.3.3.4

Sensitivity Analysis

In the optimization design mathematical model, dimensions are always concerned as variables but it is noted that a mechanism especially PM may associate with amount of dimensions. Some of them have great influence on performance but others may not. Therefore, it is necessary to select the main parameters from the large quantity of parameters to be designed by means of parameter sensitivity analysis method, so as to eliminate unimportant parameters and simplify performance analysis and optimization model. Variance-based methods have assessed themselves as versatile and effective among the various available techniques for sensitivity analysis of model output [191]. The most commonly used variance-based Sobol’ sensitivity analysis method can be roughly divided into two categories: Monte Carlo [192] and direct integral methods [193]. Lian et al. [176, 194] investigated the effects of joint stiffness/compliance coefficients and parameters of cross-section to the mass and stiffness performance of T5 PM through parameter sensitivity analysis based on response surface method and Monte Carlo simulation method. Liu et al. [193] conducted sensitivity analysis in structural crashworthiness based on the Sobol’ direct integral method. Tang et al. [195] design a set of optimized dimensional parameters for constructing the six-cable-driven PM of FAST based on the sensitivity design method and tension performance evaluating functions. The purpose of sensitivity analysis is to determine main/subordinate parameters to the performance evaluating functions, so as to improve the optimization design efficiency by reducing the design parameters scale. However, there are few reports in this field, and thus, the application of the sensitivity analysis results in the optimization design should be enriched in the future research. Since traditional optimization algorithms do not have the ability to jump out of the local optima and obtain non-dominated solutions, intelligent optimization algorithms have attracted more attention for the multi-objective optimization design of PMs. However, their premature convergence and high computational cost are their two main shortcomings. The hybrid use of intelligent optimization algorithms and their combination with traditional optimization algorithms were studied to solve the former, and the mapping model was proposed to solve the latter. Therefore, further developing the theory of intelligent optimization algorithms by increasing the population size without significantly increasing the computational cost, establishing the analytical expression of GPIs and design parameters, and achieving a highly precise and efficient mapping model are the main challenges facing today for the multi-objective optimization design of PMs.

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1 Introduction

1.4 Multi-Objective Optimization Methods Due to the complex coupling and competition of multiple performance indices, it is difficult for multi-objective optimization design to obtain a global optimal solution like single-objective optimization design; e.g., the increase of workspace volume may lead to the decline of stiffness and dynamic performance of PMs. At present, there are three commonly used methods for multi-objective optimization design: the comprehensive objective, Pareto frontier, and principal component analysis (PCA) methods.

1.4.1 Comprehensive Objective Method The principle of the comprehensive objective method is to define the sum of the weights of multiple performance indices as a comprehensive index (the sum of the weight factors is equal to 1) [146, 162, 196]. Its advantage is that a global optimum can finally be obtained and is easy to implement. Huang et al. and Xu et al. [33, 197] obtained multiple sets of global optima under different weight factor combinations, providing multiple sets of optimized results for decision makers. The objective function dimension is non-uniform, and different weighting factors should be optimized separately. Furthermore, the determination of the weighting factors is subjective, which requires considerable calculation time and reduces optimization efficiency. This is a prior decision approach that is suitable for situations where the weight factors of each performance index have been clarified before optimization.

1.4.2 Pareto Frontier Method Compared with the comprehensive objective method that could only obtain a set of optimal solutions at one time. The Pareto frontier method obtains non-dominated solution sets for multiple performance indices. For the optimization design of two objective functions, the Pareto frontier is usually a curve; for the optimization design of more than two objective functions, the Pareto frontier is usually a hypersurface. Stan et al. [198] used the GA to optimize the kinematic performance and workspace volume of a PRRRP PM and obtained the Pareto frontier. The greatest advantage of the Pareto frontier is that both the optimal solution for each performance index and multiple sets of non-dominated solutions can be obtained for decision makers, belonging to the after-the-fact decision. However, selecting a set of optimal solutions from a large number of non-dominated solution sets requires decision makers to have extensive engineering experience and certain subjectivity. Sun et al. [115] and Qi et al. [149] defined the minimum sum of squares of the difference between a set of non-dominated solutions and other non-dominated solutions in the Pareto frontier as

1.4 Multi-Objective Optimization Methods

31

the cooperative equilibrium point. This point is regarded as the optimal solution for decision makers. The concept is equivalent to assuming that the objective functions are equally important, which is unreasonable in actual engineering. Therefore, a universal method has not yet been established for selecting the optimal solution according to the different weights of the objective functions on the Pareto frontier.

1.4.3 PCA Method The principle of the PCA method [199–201] is to integrate many original variables into a few comprehensive indices with minimal information loss. It reduces the scale of evaluation functions by recombining original indices Z 1 , Z 2 …, Z m (such as m indices) with certain correlations into a set of fewer and uncorrelated comprehensive indices F p to replace the original indices (p < m). The comprehensive index should reflect the information represented by the original variable Z p to the greatest extent and ensure the independence of the new indices [202]. It can eliminate the correlation among evaluation indices and reduce the effort of index selection and computational cost. The larger the absolute value of the correlation coefficient among objective functions, the more suitable it is for PCA. The closer the absolute value is to 0, the less suitable it is for PCA. In the comprehensive evaluation function, the weight of each principal component is its contribution rate, which reflects the proportion of the information contained in the original data of the principal component to the total information. Such determination of weights is objective and reasonable, thus eliminating the shortcoming of subjectivity for the determination of weight factor in the comprehensive objective method. Li et al. [203] used the PCA method to optimize the local linear velocity transmission performance, the local angular velocity transmission performance, the input motion/force transmission performance, and the output motion/force transmission performance indices of a 5-PSS/UPU PM and the average power, torque fluctuation, and average power-deviation performance index of a 4-DOF redundant hybrid anthropomorphic arm [204]. Based on the PCA method, Zeng et al. [205] considered the global condition number, global isotropic coefficient, and global minimum singular value indices as objective functions and established a multi-objective optimization model of a Delta PM based on the PCA method. Sun et al. [206] combined the BP neural network and PCA method to synthesize the kinematic condition number, minimum singular value, kinematic manipulability, and inertial force performance indices of a planar 2RR PM. However, the PCA method is not suitable for situations where the correlation among indices is not strong. The cumulative contribution rate of the first few extracted principal components should reach a high level (with the recommended value being 85%). The explanation of the principal components is generally somewhat vague, and the physical meaning is not clear and not as defined as the meaning of the original variables. This is the price of the dimensionality reduction of the variables.

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The acquisition of multi-objective optimization methods mainly includes the comprehensive objective, Pareto frontier, and PCA methods. Due to the non-uniform dimensions among the objective functions, the physical meaning of the comprehensive objective method is not clear; furthermore, it needs to be optimized separately for each weight combination, which is time-consuming. The Pareto frontier method can obtain a set of non-dominated solutions; however, it is a challenge to select the optimal solution according to each set of weight combinations. In the PCA method, the weight of each principal component is determined objectively by its contribution rate, but its vague physical interpretation is its main disadvantage. Therefore, establishing a multi-objective optimization method for PMs that can objectively reflect the weight of each objective and has a clear physical meaning is a major challenge.

1.5 Summary Performance indices, optimization design algorithms, and multi-objective optimization methods are important research directions for investigating PMs and are discussed in detail in this work. First, performance indices were discussed. Then, optimization algorithms were analyzed based on the traditional and intelligent optimization algorithms and the ways to improve these algorithms. Finally, the multiobjective optimization methods were discussed. However, there are still avenues for improvement. 1. Although existing theories can obtain the analytical expressions of LPIs, it is difficult to obtain analytical expressions of a workspace in most cases. Therefore, it is difficult to directly obtain the analytical expression of GPIs by analyzing LPIs. Usually, the average or variance of LPIs for the finite discrete points in a workspace is used as the GPI. This method reduces the calculation accuracy and efficiency and greatly increases the computational cost at the optimization design stage. Therefore, establishing an analytical model for a regular workspace is an important challenge. In addition, considering the influence of dynamic parameters such as the inertial force, Coriolis force, and centrifugal force when evaluating the kinematic performance and decoupling the influence of gravity, flexibility, and friction when evaluating the stiffness performance are urgent needs to be addressed in terms of the performance indices of PMs. 2. The multi-parameter, multi-constraints, highly coupled multi-objective functions, and multi-extreme values of the optimization models of PMs have brought considerable challenges to the optimization design of mechanisms. The research focus of optimization algorithms has transitioned from traditional optimization algorithms to intelligent optimization algorithms. However, due to the timeconsuming calculation of GPIs, the shortcomings of intelligent optimization algorithms, such as their low computational efficiency and high computational cost, have become increasingly prominent. Establishing approximate expressions between GPIs and mechanism design parameters through mapping models is an

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important way to improve optimization efficiency. However, the sample data used for response surface modeling are calculated by the numerical method of discrete points, which contains unavoidable errors. On the other hand, the fitting algorithm also has unavoidable errors. Therefore, controlling these errors and improving the accuracy of the mapping model are important challenges to improve the optimization design efficiency. The combination of intelligent optimization algorithms with traditional optimization algorithms to avoid premature convergence and the uncontrollable convergence of intelligent optimization algorithms is another important challenge to overcome for PM optimization algorithms. 3. The Pareto frontier method can be used to obtain a group of Pareto optimal solution sets, thus solving the shortcoming that the comprehensive objective method can only obtain a set of optimal solutions at a time. However, selecting a set of optimal solutions from the optimal solution sets according to the importance of the objective function still relies on the experience of the decision maker, which is subjective. To address the shortcoming of the Pareto frontier, the PCA method takes the contribution rate of each principal component as its weight, which provides a certain objectivity and avoids the defect of artificially determining the weight. Although the PCA method has the above advantages, it is not suitable for situations where the objective functions are not strongly correlated because the physical meaning is unclear. Establishing an objective multi-objective optimization method with a clear physical meaning is another challenge that needs to be overcome for the optimization design of PMs; this challenge will require systematic and in-depth research. 4. It is worth emphasizing that the multi-objective optimization design approaches have to heavily lean on reliable LPIs. In particular, some LPIs have undefined physical meaning caused by the dimensional inhomogeneity of elements of the fundamental matrices, including the Jacobian, stiffness, compliance, and mass matrices. Obviously, these LPIs cannot be recommended for including in the multi-objective methods. Thus, establishing LPIs that have a defined physical meaning with dimensional homogeneity or are dimensionless, decoupled, and frame-free is also an important challenge to overcome for the performance measures of PMs. In addition to performance indices, optimization algorithms, and optimization methods, the design of defect identification and repair research aimed at exploring factors that cause a greater difference between the theoretical performance and actual performance of PMs is also one of the current development directions.

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Chapter 2

Kinematic Performance Analysis and Optimization of Parallel Manipulators Without Actuation Redundancy

Kinematic performance analysis and optimization of PMs are essential for the actual design and control, which have attracted increasing attention in both academia and industry. Many indices have been proposed and widely used in the PMs, such as manipulability [1, 2] and condition number [3] based on the algebraic characteristics of the Jacobian kinematic matrix, and motion/force transmissibility [4] based on the screw theory. This chapter focuses on the PMs without actuation redundancy. The kinematic performance analysis and optimization of this type of PM will be presented and discussed in detail by using three different kinematic indices, including the condition number index, the motion/force transmission index, and the proposed motion/force constraint index. First, the basics of screw theory will be introduced briefly. Then, the definition of the condition number index based on the dimensionally homogeneous Jacobian matrix [5–8] and its applications on three PMs will be presented. Next, the motion/force transmission index proposed by Liu et al. [4, 9–11] will be introduced simply, and three PMs will be selected to show the applicability of this index. Finally, a new motion/force constraint index proposed by the authors, which is useful for the performance evaluation of overconstrained PMs without actuation redundancy, will be introduced in detail, and two PMs will be used to verify the effectiveness of the proposed constraint index.

2.1 Basics of Screw Theory The basic knowledge of screw theory [12, 13] is introduced first. In screw theory, a unit screw $ is defined as $ = (s; s0 ) = (s; r × s + hs),

(2.1)

where s is a unit vector along the direction of the screw axis, r is the position vector of any point on the screw axis, and h is the screw pitch. © Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_2

43

44

2 Kinematic Performance Analysis and Optimization of Parallel …

A screw $C = (sr ; s0r ) that is reciprocal to a set of screws, $1 , $2 , ..., $n , is defined as $i ◦ $C = si · s0r + sr · s0i = 0 i = 1, 2, . . . , n

(2.2)

where ◦ and $i represent the reciprocal product and the ith screw of the screw set, respectively. A screw is called a twist when it is used to represent the instantaneous motion of a rigid body. Additionally, a wrench screw is used to represent a force or a coaxial couple when acting on a rigid body. In the following analysis, $i j denotes the unit twist that is associated with the jth kinematic joint of the ith limb, while $Ci j denotes the jth unit constraint wrench acting on the moving platform because of the ith limb.

2.2 Condition Number Indices and Applications 2.2.1 Condition Number Indices Based on the algebraic characteristics of the Jacobian matrix of a PM, several indices have been proposed and widely used for performance evaluation and optimization, such as manipulability [1] and condition number [3]. However, these indices are not sound [14]; there is much controversy surrounding these Jacobian-based indices. First, they are coordinate-dependent. In addition, the units of the elements in the Jacobian matrix are inconsistent when applied to the PMs with both rotational and translational DOFs, which leads to unclear physical meanings and subsequently causes erroneous interpretations. Some methods have been proposed to avoid the above problems, and one method that is widely used to deal with this inconsistency is to establish a dimensionally homogeneous Jacobian matrix [5–8]. In this method, a characteristic length L is used to homogenize the original Jacobian matrix J. The new homogeneous matrix J h is then formulated as follows: ( Jh = A

1 L

) B ,

(2.3)

where matrices A and B denote the corresponding translational and rotational mapping Jacobian matrices in the original Jacobian matrix J. To obtain L, the isotropic design of the mechanism should be used in the Jacobian matrix [5, 6]. The Jacobian matrix is isotropic if it satisfies ( J Th J h =

AT A 1 B AT L

1 T A B L 1 BT B L2

) ) ( 2 α I p 0 p×q , = 0q× p α 2 I q

(2.4)

2.2 Condition Number Indices and Applications

45

where p and q denote the DOFs of translation and rotation, respectively. I p and I q denote the p × p and q × q identity matrices, respectively. 0q× p and 0 p×q denote the q × p and p × q zero matrices, respectively. Dexterity is defined as the capability of a robot to arbitrarily change its pose, or apply forces and torques in arbitrary directions [1]. The condition number of the Jacobian matrix is an important local dexterity index that is used to describe the kinematic performance of PM. The condition number c of the homogeneous Jacobian matrix J h is given by c = || J h || · || J −1 h ||,

(2.5)

where the notation ||·|| is defined as a 2-norm of the matrix. Here, we use the reciprocal / of the condition number, i.e., C = 1 c. The reciprocal of the condition number ranges from zero to one. And the closer the reciprocal of the condition number is to unity, the better the kinematic performance of this configuration will be. To provide a global description of the mechanism performance, the global conditioning index (GCI) that was proposed by Gosselin et al. [15] is used in this book. The purpose of calculating the GCI is to obtain the condition number distribution over the whole workspace. The GCI is defined as { GCI =

W

CdW , W

(2.6)

where W denotes the reachable workspace. The GCI ranges from zero to unity, and a GCI value that is closer to unity indicates better dexterity. In this section, the condition number indices based on the dimensionally homogeneous Jacobian matrix are used to evaluate and optimize the kinematic performance of three PMs, namely the 6PSS PM, the 2-(PRR)2 RH PM [16], and the 2PRS-PRRU PM [17].

2.2.2 Example 1: 6PSS PM • Structure description The computer-aided design (CAD) model of the 6PSS PM is shown in Fig. 2.1, which can be intuitively seen as a 6-DOF PM. The moving platform is connected to the fixed base through six identical kinematic PSS limbs. Through actuating six P joints, the 6PSS PM can achieve three translational and three rotational outputs. For each PSS limb, the first and second S joints are connected to the slider and moving platform, respectively. Additionally, the directions of guides of limbs 1 and 2 are parallel to each other, as well as limbs 3 and 4, and limbs 5 and 6. It should be noted that the angle θb between guides 2 and 3 (or guides 4 and 5, guides 6 and 1) is always equal to 120◦ . Let Ai (i = 1, 2, 3, 4, 5, 6) denote the central points of the first S joint

46

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.1 CAD model of the 6PSS PM

in each limb, while the central point of the second S joint in each limb is denoted by Bi (i = 1, 2, 3, 4, 5, 6). Two coordinate frames are established as shown in Fig. 2.1. A fixed reference coordinate frame O − x yz is attached to the fixed base, and the origin O is at the midpoint of the fixed base. Let the x-axis always be perpendicular to the direction of guide 5 or 6, and the y-axis always point along the direction of guide 5 or 6. The z-axis points upward vertically. A moving platform coordinate frame o − uvw is attached to the moving platform, as shown in Fig. 2.1. The u-axis is always perpendicular to the direction of the angular bisector of ∠B2 oB3 , the v-axis always points along the direction of the angular bisector of ∠B2 oB3 , and the w-axis points upward vertically with respect to the moving platform. The architectural parameters of the 6PSS PM are defined as follows: oBi = Rm , Ai Bi = l and ∠B1 oB6 = ∠B2 oB3 = ∠B4 oB5 = θm . Additionally, the distance between two parallel guides is defined as 2lb here. • Inverse kinematics The inverse kinematics should be developed first for the performance evaluation and optimization of the 6PSS PM by using the condition number index. For the 6PSS PM, the inverse kinematics is to find the actuated displacements qi (i = 1, 2, 3, 4, 5, 6), given the locations and orientations of the moving platform, i.e., x, y, z, α, β, and γ . The angles α, β, and γ denote the rotation angle around the x-, y-, and z-axes, respectively. The rotation matrix between the moving coordinate frame o − uvw and the fixed coordinate frame O − x yz can be written as ⎛

⎞ cβcγ sαsβcγ − cαsγ cαsβcγ + sαsγ O Ro = ⎝ cβsγ sαsβsγ + cαcγ cαsβsγ − sαcγ ⎠, −sβ sαcβ cαcβ

(2.7)

where s and c denote sine and cosine functions, respectively. As shown in Fig. 2.1, the position vectors Ai Bi (i = 1, 2, 3, 4, 5, 6) with respect ( )T to the fixed coordinate frame are defined as l i , and vector p = x y z denotes the position vector of the point o with respect to the fixed coordinate frame O − x yz. The position vectors O Ai and oBi (i = 1, 2, 3, 4, 5, 6) with respect to the fixed

2.2 Condition Number Indices and Applications

47

coordinate frame are denoted as ai and bi , respectively. Using the rotation matrix O Ro and angle θm , the vectors bi can be directly obtained as ⎧ ⎪ ⎪ ⎪ b1 ⎪ ⎪ ⎪ b2 ⎪ ⎪ ⎪ ⎨b 3 ⎪ b ⎪ 4 ⎪ ⎪ ⎪ ⎪ b5 ⎪ ⎪ ⎪ ⎩b 6

= = = = = =

( ( ( / ) / ) )T Ro Rm c 30◦ − θm 2 −Rm s 30◦ − θm 2 0 ) ( ( / ) ( / ) T O Ro Rm s θm 2 Rm c θm 2 0 ( / ) ( ( / ) )T O Ro −Rm s θm 2 Rm c θm 2 0 ( / ) / ) )T . ( ( O Ro −Rm c 30◦ − θm 2 −Rm s 30◦ − θm 2 0 ( / ) / ) )T ( ( O Ro −Rm c 30◦ + θm 2 −Rm s 30◦ + θm 2 0 ( / ) / ) )T ( ( O Ro Rm c 30◦ + θm 2 −Rm s 30◦ + θm 2 0 O

(2.8)

Using the distance lb and angle θb , the vectors ai also can be deduced, which are the expressions about actuated displacement qi . As shown in Fig. 2.1, the closed-loop kinematic equation for each limb (i = 1, 2, 3, 4, 5, 6) can be written in the following form as ai + l i = p + bi .

(2.9)

Through solving Eq. (2.9), we can obtain the relationships between the vectors ai and locations and orientations of the moving platform (x, y, z, α, β, γ ), from which the actuated displacement qi can be obtained. • Velocity analysis The Jacobian matrix can describe the mapping relationship between the velocities of inputs and outputs. There are some approaches to obtaining the Jacobian matrix J, including the derivative method, velocity projection method, and so on. Here, the velocity projection method is used for the example of the 6PSS PM, and the direct derivative method will be applied to examples 2 and 3 in this chapter. For the second S joint connected to the moving platform, the velocity of the central point, v Bi (i = 1, 2, 3, 4, 5, 6), can be expressed as v Bi = vo + ωo × bi ,

(2.10)

)T x˙ y˙ z˙ denotes the linear velocity of the origin point of moving ( )T platform, and ωo = α˙ β˙ γ˙ denotes the angular velocity of the moving platform. In addition, the relationship between the velocity of the first S joint, v Ai (i = 1, 2, 3, 4, 5, 6), and actuated velocity q˙i can be expressed as where vo =

(

v Ai = q˙i d i ,

(2.11)

where d i denotes the direction vector of guide i with respect to the fixed frame O − x yz.

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2 Kinematic Performance Analysis and Optimization of Parallel …

According to the velocity projection method, the projection velocity of two S joints on the link should be the same, i.e., vTAi · l i = vTBi · l i . Therefore, the mapping )T ( between the actuated velocity q˙ = q˙1 q˙2 q˙3 q˙4 q˙5 q˙6 and output velocity V˙ = ( )T x˙ y˙ z˙ α˙ β˙ γ˙ can be obtained by using Eqs. (2.10)–(2.11) as q˙ = J V˙ ,

(2.12)

where the kinematic Jacobian matrix J of the 6PSS PM can be expressed as ( J=

l2 l3 l4 l5 l6 l1 l T1 ·d 1 l T2 ·d 2 l T3 ·d 3 l T4 ·d 4 l T5 ·d 5 l T6 ·d 6 (b1 ×l 1 ) (b2 ×l 2 )T (b3 ×l 3 )T (b4 ×l 4 )T (b5 ×l 5 )T (b6 ×l 6 )T l T1 ·d 1 l T2 ·d 2 l T3 ·d 3 l T4 ·d 4 l T5 ·d 5 l T6 ·d 6

)T .

(2.13)

• Homogeneous Jacobian matrix Here, the method proposed by Angeles et al. [5, 6] is used to generate a homogeneous Jacobian matrix of the 6PSS PM. The new homogeneous matrix J h can be formulated as follows: ( J h = Jdiag 1 1 1

1 1 1 L L L

)

( = A

1 L

) B ,

(2.14)

where L is the characteristic length. A and B are two 6 × 3 matrices, in which the A is the translational mapping matrix, and the B is the rotational mapping matrix. Based on Eq. (2.4), the homogeneous Jacobian matrix J h is isotropic if it satisfies ( J Th J h =

AT A 1 B AT L

1 T A B L 1 BT B L2

) ) ( 2 α I 3 03 , = 03 α 2 I 3

(2.15)

where I 3 denotes a 3 × 3 identity matrix and 03 denotes a 3 × 3 zero matrix. However, there is no solution for Eq. (2.15), which means that the isotropic architecture for the 6PSS PM is not feasible [5]. Based on the above situation, we can find an architecture that is close to isotropy. Here the length of the moving platform, Rm , is used as characteristic link lengths L to generate a new homogeneous Jacobian matrix, and the homogeneous Jacobian matrix can thus be written as Jh =

( A

1 Rm

) B .

(2.16)

• Local and global dexterity performances Here, the reciprocal of the condition number, i.e., C = 1/c (C ∈ [0, 1]), as introduced in Sect. 2.2.1, is taken as the local dexterity index to describe the kinematic performance of the 6PSS PM. Because the 6PSS PM has three position outputs and three orientation outputs, the local dexterity performance C in the different positions

2.2 Condition Number Indices and Applications

49

and orientations is discussed separately here. In this section, the architecture parameters of the 6PSS PM are set as follows: Rm = 250 mm, l = 440 mm, lb = 60 mm, and θm = 30◦ . The distribution variations of C in the three-dimensional position workspace are presented first, as shown in Fig. 2.2a, in which the ranges of positions and orientations are set as −50 mm ≤ x ≤ 50 mm, −50 mm ≤ y ≤ 50 mm, 225 mm ≤ z ≤ 275 mm, and α = β = γ = 0°. For clarity, the distribution variations of C when z = 250 mm are also presented, as shown in Fig. 2.2b, from which it is obvious that the distribution is symmetrical about the plane x = 0, and the kinematic performances near the central region are better than others. Similarly, the distribution variations of C in the three-dimensional orientation workspace are also presented in Fig. 2.3a, in which the ranges of positions and orientations are set as x = y = z = 0, −20° ≤ α ≤ −20°, −20° ≤ β ≤ −20°, and − 20° ≤ γ ≤ −20°. The distribution variations of C when γ = 0° are also presented, as shown in Fig. 2.3b, from which it is also obvious that the distribution is symmetrical about the plane β = 0°. The kinematic performances near the central region are better than others, which are similar to those in the three-dimensional position workspace. The GCI is necessary to provide a global performance description of the 6PSS PM. Here, the purpose of calculating the GCI is to obtain the condition number distribution over the whole V and is defined as { CdV , (2.17) GCI = V V where V denotes the volume of position workspace or orientation workspace. Based on the results in Figs. 2.2a and 2.3a, the corresponding GCI can be calculated as 0.4443 and 0.2632, respectively, and designers can improve the kinematic performance by optimizing link parameters.

Fig. 2.2 Distribution of the C for the 6PSS PM in the three-dimensional position workspace: a three-dimensional position distribution and b xy plane distribution when z = 250 mm

50

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.3 Distribution of the C for the 6PSS PM in the three-dimensional orientation workspace: a three-dimensional orientation distribution and b αβ plane distribution when γ = 0°

• Optimization of design parameters Here, the iterative search method is used for the dimensional optimization of the 6PSS PM. Since the influence of design parameters on the position and orientation characteristics may not be the same, these two situations are considered simultaneously during the optimization, and the Rm and l are defined as design parameters here. Considering the actual applications, the variation ranges of two design parameters are respectively defined as 230 mm ≤ Rm ≤ 270 mm, 420 mm ≤ l ≤ 460 mm. Similar to the above section, the position ranges are set as −50 mm ≤ x ≤ 50 mm, −50 mm ≤ y ≤ 50 mm, and 225 mm ≤ z ≤ 275 mm, and the orientation ranges are set as − 20° ≤ α ≤ −20°, −20° ≤ β ≤ −20°, and −20° ≤ γ ≤ −20°. Through iterative search and calculation, the distributions of GCI with different combinations of design parameters can be obtained, as shown in Figs. 2.4a, b. It can be found that even with the same combination of design parameters, the distributions of GCI in the position workspace are different from that in the orientation workspace. In the case of optimization regarding position workspace, the maximum of GCI occurs when Rm = 270 mm, l = 420 mm, as shown in Fig. 2.5a, and the value of GCI is 0.4737. However, in the case of optimization regarding orientation workspace, the maximum of GCI occurs when Rm = 230 mm, l = 420 mm, as shown in Fig. 2.5b, and the value of GCI is 0.2714. Designers can choose one of these combinations for different applications.

2.2.3 Example 2: 2-(PRR)2 RH PM • Structure description As shown in Fig. 2.6a, the CAD model of the 2-(PRR)2 RH PM is composed of a fixed base, two identical actuated limbs, and a moving platform, which has been proven to be a 3T1R PM [16]. Beginning from the fixed base, each limb is a hybrid kinematic limb that consists of a 2-PRR planar parallel subchain and an RH serial subchain. In

2.2 Condition Number Indices and Applications

51

Fig. 2.4 GCI distributions with different parameter combinations: a optimization regarding position workspace and b optimization regarding orientation workspace

Fig. 2.5 Performance distributions of the 6PSS PM with optimized design parameters: a threedimensional position distribution with Rm = 270 mm and l = 420 mm and b three-dimensional orientation distribution with Rm = 230 mm and l = 420 mm

the RH subchain, the axes of the R joint and the H joint are parallel to each other, and they are perpendicular to the axes of the R joints of the 2-PRR subchain. The thread pitches of the two H joints in the two limbs are equal and opposite. Let Ai j for i, j = 1, 2, 3, 4 denote the central point of the R joint, and Bi for i = 1, 2 denotes the midpoint of the lines A12 A22 and A32 A42 , respectively. The two sliders intersect the XY plane at points A1 and A2 , which lie on the triangle O A1 A2 , as shown in Fig. 2.6b. Several coordinate frames are established as shown in Fig. 2.6b. The origin of a fixed reference frame O-XYZ is coincident with the point O of the fixed triangle base ΔA1 O A2 . The Y-axis points in the direction of the angular bisector of ∠A1 O A2 , and the Z-axis points downwards. A moving coordinate frame o-xyz is attached to the central point o of the moving platform. In addition, a local coordinate frame oi − xi yi z i for i = 1, 2, is attached to the middle R joint in the ith PRR subchain, as shown in Fig. 2.6b. Note that the motion of the ith 2-PRR subchain is limited within a plane Li for i = 1, 2, which is perpendicular to O Ai . A plane L3 coincides with the

52

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.6 2-(PRR)2 RH PM: a CAD model and b schematic representation

YZ plane. The angle 2α is measured from plane L1 to plane L2 , in which the angle α is defined as the layout angle. The distance from the origin of the fixed coordinate frame to the sliders is defined as E. The displacement of the ith actuated P joints is defined as di for i = 1, 2, 3, 4. The thicknesses of the moving platform and the link l4 are 2e1 and 2e2 , respectively, and h denotes the thread pitch of the moving platform. The architectural parameters of the 2-(PRR)2 RH PM are set as follows: A11 A12 = A21 A22 = A31 A32 = A41 A42 = l1 = 100 mm, A12 A22 = A32 A42 = l2 = 100 mm, A13 B1 = A33 B2 = l3 = 20 mm, A13 A14 = A33 A34 = l4 = 400 mm, E = 420 mm, e1 = e2 = 10 mm, and h = 5 mm. • Inverse kinematics The inverse kinematics should be developed first for the performance evaluation and optimization of the 2-(PRR)2 RH PM by using the condition number index. For the 2-(PRR)2 RH PM, the inverse kinematics is to find the actuated displacements d1 , d2 , d3 , and d4 , given the locations and orientations of the moving platform, i.e., X o , Yo , Z o , and ϕ. ( ) Let O p = X o Yo Z o be the position vector of the origin of the moving platform o with respect to the fixed frame. 1 p1 is the position vector between the origin of o1 − x1 y1 z 1 and the origin of the moving platform o, and 2 p2 is the position vector between the origin of o2 − x2 y2 z 2 and the origin of the moving platform o, and these vectors are given by

2.2 Condition Number Indices and Applications

53

⎧ ) ( ⎨ 1 p = −l4 (cϕ1 cα − sϕ1 sα) l1 sθ1 + l3 + l4 (sϕ1 cα + cϕ1 sα) l1 cθ1 + l2 − e3 T 1 2 , ⎩ 2 p = ( l sθ + l + l (sϕ cα − cϕ sα) l (cϕ cα + sϕ sα) l cθ + l2 + e )T 2

1

2

3

4

2

2

4

2

2

1

2

2

3

(2.18) where θi is the angle between the ith kinematic limb for i = 1, 2 and the slider of the fixed base, as shown in Fig. 2.6a. ϕ1 is the angle between the end link A13 A14 of limb 1 and the direction of the X-axis, and ϕ2 is the angle between link l4 of limb 2 and the direction of the X-axis. e3 denotes the distance from link l4 of either limb 1 or limb 2 to the point o, as shown in Fig. 2.6b. e3 is then expressed in the following forms ⎧ ⎨ e = e1 + e2 . (2.19) ⎩ e3 = e + ϕh 2π The angles θ 1 , θ 2 , ϕ1 , and ϕ2 satisfy the following two constraint equations {

X o = Ecα − l1 sθ1 sα − l3 sα − l4 cϕ1 , Yo = Esα + l1 sθ1 cα + l3 cα + l4 sϕ1

{

X o = −Ecα + l1 sθ2 sα + l3 sα − l4 cϕ2 . Yo = Esα + l1 sθ2 cα + l3 cα + l4 sϕ2 (2.20)

/( ) )/ ( ) ( ( / ) 1 + ti2 (ti = tan ϕi 2 , i Substituting sϕi = 2ti 1 + ti2 and cϕi = 1 − ti2 = 1, 2) into Eq. (2.20) yields }

( ) ( ) ℛ1 (1 + t12 ) = 2l4 t1 tanα − l4 (1 − t12 ) , ℛ2 1 + t22 = 2l4 t2 tanα + l4 1 − t22

(2.21)

where tan denotes the tangent function. ℛ1 = X o + Yo tanα − E(cα + sαtanα) and ℛ2 = Yo tanα − X o − E(cα + sαtanα). The analytic expressions for the angles ϕ1 and ϕ2 can be obtained as }

ϕ1 = 2 arctan(t1 ) , ϕ2 = 2 arctan(t2 )

(2.22)

where t1 =

l4 tanα −

/ l42 tan2 α − ℛ21 + l42 ℛ1 − l4

, t2 =

l4 tanα +

/ l42 tan2 α − ℛ22 + l42 ℛ2 + l4

. (2.23)

Subtracting Eq. (2.23) from (2.20), the angles θ1 and θ2 can be obtained as

54

2 Kinematic Performance Analysis and Optimization of Parallel …

( θ1 = arcsin

) ) ( Yo − Esα − l3 cα − l4 sϕ1 Yo − Esα − l3 cα − l4 sϕ2 , θ2 = arcsin l1 cα l1 cα (2.24)

Note that O p is given by O

p=

O

R1 1 p1 + O T o1 =

O

R2 2 p2 + O T o2 .

(2.25)

From Eq. (2.25), we have Zo =

d1 + d2 d3 + d4 − e3 = + e3 . 2 2

(2.26)

Then, using Eqs. (2.18)–(2.26), the inverse kinematics of the 2-(PRR)2 RH PM is given by ⎧ ϕh ⎪ +e− ⎨ d1 = Z o + 2π ⎪ ⎩ d = Z − ϕh − e − 3 o 2π

l2 − l1 cθ1 , d2 = Z o + 2 l2 − l1 cθ2 , d4 = Z o − 2

ϕh +e+ 2π ϕh −e+ 2π

l2 + l1 cθ1 2 . l2 + l1 cθ2 2

(2.27)

• Velocity analysis The Jacobian matrix transforms the velocities of the actuated joints d˙ = )T )T ( ( d˙1 d˙2 d˙3 d˙4 into the moving platform velocities X˙ = X˙ o Y˙o Z˙ o ϕ˙ . By taking the derivative of Eq. (2.27) with respect to time, we obtain J x X˙ = J q d˙ ⎛

Jq11 ⎜ 0 Jq = ⎜ ⎝ 0 0 ⎛ Jx11 ⎜ Jx21 Jx = ⎜ ⎝ Jx 31 Jx41

(2.28) ⎞ 0 0 ⎟ ⎟ 0 ⎠ Jq44 ⎞ Jx14 Jx24 ⎟ ⎟, Jx ⎠

0 Jq22 0 0

0 0 Jq33 0

Jx12 Jx22 Jx32 Jx42

Jx13 Jx23 Jx33 34 Jx43 Jx44

(2.29)

(2.30)

1 cα 1 , Jq33 = Jq44 = tanθ , Jx11 = −Jx21 = − tan(α+ϕ − where Jq11 = Jq22 = tanθ 1 2 1) 1 1 sα sα, Jx12 = −Jx22 = − tan(α+ϕ1 ) + cα, Jx13 = Jx23 = tanθ1 , Jx33 = Jx43 = tanθ2 , 1 1 h h Jx14 = Jx24 = 2π , Jx34 = Jx44 = − 2π , Jx31 = −Jx41 = − tan(ϕcα2 −α) + sα, tanθ1 tanθ2 Jx32 = −Jx42 = tan(ϕsα2 −α) + cα.

2.2 Condition Number Indices and Applications

55

Then, the velocity equation of the 2-(PRR)2 RH PM can be established ˙ d˙ = J X,

(2.31)

where J = J q−1 J x is a 4 × 4 Jacobian matrix. • Homogeneous Jacobian matrix Considering the method proposed by Angeles et al. [5, 6], a characteristic length is used to homogenize the original Jacobian matrix of the 2-(PRR)2 RH PM. The new homogeneous matrix J h is then formulated as follows: ( J h = Jdiag 1 1 1

1 L

)

( = A

1 L

) B ,

(2.32)

where L is the characteristic length. Matrices A and B denote a 4 × 3 matrix and a 4 × 1 matrix, which correspond to the translational and rotational mapping matrices, respectively. To obtain L, the isotropic design of the mechanism should be used in the Jacobian matrix [5, 6]. The Jacobian matrix is isotropic if it satisfies ( J Th J h =

AT A 1 B AT L

1 T A B L 1 BT B L2

) ) ( 2 α I3 0 , = 0T α 2

(2.33)

where I 3 is a 3 × 3 identity matrix and 0 is a three-dimensional zero vector. By solving Eq. (2.33), the characteristic length can then be obtained ( ) ( ) tan α + π4 π 3π h ) ( , ϕ1 = , ϕ2 = , θ1 = θ2 = arctan L= 2π 4 4 tan α + π4 sα + cα

(2.34)

Finally, the homogeneous Jacobian matrix J h can be obtained by substituting Eq. (2.34) into Eq. (2.32). • Local and global dexterity performances Here, we also use the reciprocal of the condition number, i.e., C = 1/c (C ∈ [0, 1]) as the local dexterity index to describe the kinematic performance of the 2-(PRR)2 RH PM. For example, Fig. 2.7 shows the mapping of the reciprocal of the condition number of a 2-(PRR)2 RH PM with Z o = 450mm and α = 45◦ . As shown in Fig. 2.7, the distribution of C within the reachable workspace is symmetrical about plane X = 0. And from Fig. 2.7, it can be found that the closer the middle part of the region is, the bigger the reciprocal of condition number will be, which means that the kinematic performance in this region is better than others. Similar to the 6PSS PM, the GCI [15] is also used to evaluate the global dexterity characteristic here. By changing the value of the layout angle α, the corresponding GCI can be calculated, and the results are as shown in Fig. 2.8. From Fig. 2.8, we find that the maximum value of GCI occurs at α = 45◦ , which is equal to 0.1802.

56

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.7 Distribution of the reciprocal of the condition number for the 2-(PRR)2 RH PM in the case when Z o = 450 mm and α = 45°: a three-dimensional distribution and b XY plane distribution

Fig. 2.8 Global conditioning index versus layout angle

• Optimization of design parameters In this section, we focus on the study of optimization of the design parameters when α = 45◦ . The GCI of the 2-(PRR)2 RH PM is highly dependent on the design parameters. Under actual usage conditions, the design parameters of the mechanism cannot be selected arbitrarily because of the effects of assembly and movement. Here, the parameter-finiteness normalization method (PFNM), which was proposed by Liu et al. [4, 18], is used to optimize the parameters. Compared to traditional methods such as the objective-function method [19, 20], PFNM eliminates the influence of infinite parameters. Based on the normalization factor, this method achieves the reduction of the parameter number and the boundary limitation of each normalized parameter and also converts the infinite parameter space into a finite space [18]. However, it is still difficult to describe the performance of the 2-(PRR)2 RH PM using the five parameters E, l1 , l2 , l3 , and l4 . Based on the special structure of the proposed manipulator, the

2.2 Condition Number Indices and Applications

57

parameter l2 has no influence on the kinematic performance through the change of each architecture parameter separately, namely the parameter l2 does not affect the magnitude of GCI at every height. Thus, the parameter l2 is not considered. Without loss of generality, we let l1 = l2 , and l3 = l4 . We can therefore obtain the 2-(PRR)2 RH PM using the three design parameters E, l1 , and l3 , and we define D=

E + l1 + l3 , 3

(2.35)

where D is a normalized factor. The non-dimensional and normalized parameters r1 , r2 , and r3 are obtained r1 =

l3 l1 E , r2 = , r3 = . D D D

(2.36)

Under real usage conditions, the three normalized parameters should satisfy ⎧ ⎨ r1 + r2 + r3 = 3 ( √ ) . ⎩ 1 + 2r2 ≥ r1

(2.37)

Using the PFNM method [18], the parameter design space (PDS) can be obtained as shown in Fig. 2.9. The triangle ABC shown in Fig. 2.9a is the set of all possible points. In the plan view, where the ΔABC is described in another form as shown in Fig. 2.9b, the data points can be observed more directly. The relationship between the parameters in spatial space (r1 , r2 , r3 ) and those in plan space (s, t) can be described as follows: ⎧ r1 = s ⎪ ⎪ ⎪ ⎧ √ ⎪ ⎪ ⎨ ⎨ s = r1 s 3 t− r2 = (2.38) or 3 − r3 + r2 2 √ 2 ⎪ ⎩t = √ ⎪ ⎪ ⎪ 3 ⎪ ⎩ r3 = 3 − 3 t − s 2 2 The procedure of parameter optimization is as follows: Step 1: Determine the design parameters. By considering the actual area of the mechanism, the value of the normalized factor is set as 333 mm. Step 2: Identify an optimal region. The distributions of the GCI can be obtained as shown in Fig. 2.10. By considering the kinematic performance of the mechanism, the region where GCI ≥ 0.2 is chosen as the optimal design region here. Step 3: Select a group of data points from the optimal region. 12 groups of data points that are comparatively better than the other groups are chosen. The non-dimensional parameters r1 , r2 , r3 can be obtained using Eq. (2.38). Using Eqs. (2.36) and (2.38),

58

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.9 Parameter design space of the 2-(PRR)2 RH PM: a spatial view and b plane view

Fig. 2.10 Distributions of GCI

the five design parameters E, l1 , l2 , l3 , and l4 , and the two planar space parameters s and t can be obtained, as listed in Table 2.1. Step 4: Check whether the dimensional parameters obtained in Step 3 are suitable or not for an actual usage condition. If it satisfies the actual assembly condition, the procedure is finished; otherwise, return to Step 3 to choose another group of data from the optimal region, and repeat Steps 3 and 4. The method provides various solutions for the design parameters, and the design parameters in group “*” are the data from the 2-(PRR)2 RH PM proposed at first. All the GCI values of the 12 groups listed in Table 2.1 are better than 0.1802, and designers can choose one of these combinations for different applications.

2.2 Condition Number Indices and Applications

59

Table 2.1 GCI in the optimal design region when α = 45◦ Group

s

t

r1

r2

r3

E

l1 (l2 )

l3

l4

GCI

*

1.26

2.12

1.26

1.20

0.30

420

100.0

20

400.0

0.1802

1

1.61

1.85

1.61

0.80

0.59

537

196.3

266.7

266.7

0.2053

2

1.61

1.91

1.61

0.85

0.54

537

179.0

284.0

284.0

0.2048

3

1.62

1.83

1.62

0.78

0.61

540

201.7

258.3

258.3

0.2059

4

1.62

1.88

1.62

0.81

0.57

540

188.7

271.3

271.3

0.2125

5

1.62

1.92

1.62

0.85

0.53

540

175.7

284.3

284.3

0.2127

6

1.63

1.83

1.63

0.77

0.60

543

200.3

256.7

256.7

0.2026

7

1.63

1.88

1.63

0.81

0.56

543

187.3

269.7

269.7

0.2091

8

1.63

1.92

1.63

0.85

0.52

543

174.3

282.7

282.7

0.2114

9

1.64

1.85

1.64

0.79

0.57

547

191.3

261.7

261.7

0.2036

10

1.64

1.91

1.64

0.84

0.52

547

174.0

279.0

279.0

0.2084

11

1.65

1.87

1.65

0.79

0.56

550

185.3

264.7

264.7

0.2017

12

1.65

1.89

1.65

0.81

0.54

550

179.3

270.7

270.7

0.2018

2.2.4 Example 3: 2PRS-PRRU PM • Structure description The CAD prototype of the 2PRS-PRRU PM is shown in Fig. 2.11, which is a 2R1T PM [17]. The PM is composed of a fixed base, a moving platform, and three actuated limbs. Limbs 1 and 2 are two identical PRS limbs, and limb 3 is a PRRU limb. Limbs 1 and 2 are coplanar and limited to within the plane L1 , which is always perpendicular to the plane L2 . The plane L2 represents the symmetrical plane to which the limb 3 belongs. Beginning from the fixed base, the P joints connected to the sliders are parallel to each other. For limbs 1 and 2, the R joint is connected to the P joint, and the revolute axes of the two R joints are parallel to each other. Meanwhile, the revolute axes of the R joints in limbs 1 and 2 are perpendicular to the P joint. The PRS limbs are connected to the moving platform by an S joint, which the centers are limited in the plane L1 . For limb 3, the revolute axes of the two R joints are parallel to each other, and the revolute axes of the R joints are also perpendicular to the plane L2 . The first revolute axis of the U joint is parallel to the moving platform and limited in the plane L2 , and the second revolute axis is perpendicular to the moving platform. The 2PRS-PRRU PM is actuated by three P joints. Let Bi (i = 1, 2, 3) denote the central points of the first R joints in each limb, while the center of the second R joint in limb 3 is denoted by D3 . The centers of the S joints in limbs 1 and 2 are denoted by C1 and C2 , respectively, and the center of the U joint in limb 3 is denoted by C3 , which lies in the middle of the line C1 C2 . The sliders of limbs interest with the fixed base at points Ai (i = 1, 2, 3), which lie on the plane A1 A2 A3 , as shown in Fig. 2.11.

60

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.11 2PRS-PRRU PM: a CAD model and b schematic representation

Two coordinate frames are established as shown in Fig. 2.11. A fixed reference coordinate frame O-xyz is attached to the fixed base, and the origin O is at the midpoint of line A1 A2 . Let the x-axis always point in the direction of O A2 , the y-axis point along O A3 , and the z-axis point downward vertically. A moving coordinate frame o-uvw is attached to the moving platform, and the origin o coincides with point C3 , as shown in Fig. 2.11. Let the u-axis point in the direction of C2 C3 and the v-axis be perpendicular to the line C2 C3 . The w-axis is acting downward vertically with respect to the moving platform. The architectural parameters of the 2PRS-PRRU PM are defined as follows: O A1 = O A2 = a = 300 mm, O A3 = h = 250 mm, B1 C1 = l1 = 280 mm, B2 C2 = l2 = 280 mm, B3 C3 = l3 = 225 mm, C3 D3 = l4 = 225 mm, C1 o = C2 o = r = 120 mm. • Inverse kinematics The inverse kinematics of the 2PRS-PRRU PM involves the calculations of the actuated joint parameters (d1 , d2 , d3 ) given the orientation and position parameters of the moving platform (α, β, Pz ), where α and β denote the rotation angles around the v- and u-axes of the moving coordinate, respectively, and Pz denotes the distance between the points O and o. Because the rotation about the w-axis is always equal to zero [21, 22], the rotation matrix between the moving coordinate frame o-uvw and the fixed coordinate frame O-xyz is given by ⎛

⎞ cα sαsβ sαcβ O Ro = ⎝ 0 cβ −sβ ⎠. −sα cαsβ cαcβ

(2.39)

Meanwhile, the rotation matrix can also be obtained through the u-v-w Euler angle, and the transformation mapping is given by

2.2 Condition Number Indices and Applications

61

⎛

⎞ −cθ3 sθ4 sθ3 cθ3 cθ4 O Ro = ⎝ sψsθ3 cθ4 + cψsθ4 −sψsθ3 sθ4 + cψcθ4 −sψcθ3 ⎠, −cψsθ3 cθ4 + sψsθ4 cψsθ3 sθ4 + sψcθ4 cψcθ3

(2.40)

where ψ, θ3 , and θ4 denote the rotation angles around the u-, v- and w-axes of the moving coordinate, respectively. Apparently, the rotation matrices in Eqs. (2.39) and (2.40) are two different representations of the same orientations about the moving platform, and thus, the following formulas can be obtained } sαcβ = sθ3 (2.41) −sβ = −sψcθ3 Using Eq. (2.41), the angle ψ can be calculated as follows: (

sβ

)

ψ = arcsin / . 1 − (sαcβ)2

(2.42)

Thus, the inverse kinematics of the 2PRS-PRRU PM can be expressed as follows based on the geometrical conditions ⎧ / ⎪ ⎪ d = P + r sα − l 2 − (a − r cα)2 1 z ⎪ ⎨ /1

d2 = Pz − r sα − l22 − (a − r cα)2 . ⎪ / ⎪ ⎪ ⎩ d = P − l sψ − l 2 − (h − l cψ)2 3

z

4

3

(2.43)

4

• Velocity analysis The Jacobian matrix represents the mapping between the rates of the actuated joints ( )T )T ( q˙ = d˙1 d˙2 d˙3 and the velocities of the moving platform X˙ = α˙ β˙ P˙z . By taking the derivative of Eq. (2.43) with respect to time leads to J q q˙ = J x X˙ ⎛

Jq11 Jq = ⎝ 0 0 ⎛ Jx11 J x = ⎝ Jx21 Jx31

⎞ 0 0 Jq22 0 ⎠ 0 Jq33 ⎞ Jx12 Jx13 Jx22 Jx23 ⎠, Jx32 Jx33

(2.44)

(2.45)

(2.46)

62

2 Kinematic Performance Analysis and Optimization of Parallel …

where Jq11 = Pz + r sα − d1 , Jq22 = Pz − r sα − d2 , Jq33 = Pz − l4 sψ − d3 , Jx11 = (Pz − d1 )r cα + ar sα, Jx12 = 0, Jx13 = Pz + r sα − d1 , Jx21 = ar sα − (Pz − d2 )r cα, Jx22 = 0, Jx23 = Pz − r sα − d2 , Jx31 = AQ + B Q, Jx32 = A R + B R, Jx33 = sαsψcψ Pz − l4 sψ − d3 , A = (d3 − Pz )l4 cψ, B = hl4 sψ, R = 1−scα 2 αc2 ψ , Q = 1−s2 αc2 ψ . Thus, the velocity equation of the 2PRS-PRRU PM can be established, and it is expressed as follows: ˙ q˙ = J q−1 J x X˙ = J X,

(2.47)

where J = J q−1 J x is a 3 × 3 Jacobian matrix. • Homogeneous Jacobian matrix For the 2PRS-PRRU PM, the length of the moving platform, r, is used as the characteristic/natural link length L to generate a new homogeneous Jacobian matrix. The homogeneous Jacobian matrix J h is then formulated as follows: J h = J diag

(

1 1 L L

) ( 1 =

Fr L

) Ft ,

(2.48)

where matrices F r and F t denote a 3 × 2 matrix and a 3 × 1 matrix, respectively. The (matrix F r represents ) the mapping of the linear velocities to angular veloci˙ β˙ , while matrix F t corresponds to the mapping of the linear ties d˙1 , d˙2 , d˙3 |→ α, ( ) velocities to linear velocity d˙1 , d˙2 , d˙3 |→ P˙z . • Local and global dexterity indices For clarity, the reciprocal of the condition number, i.e., C = 1/c (C ∈ [0, 1]) is still used as the local dexterity index for the 2PRS-PRRU PM here. And the rotation ranges of the moving platform are set as follows: −40◦ ≤ α ≤ 40◦ , −40◦ ≤ β ≤ 40◦ . For the 2PRS-PRRU PM, the operating height Pz has no influence on the distribution of condition number. Based on the architecture parameters mentioned above, the distribution variations of the C for the 2PRS-PRRU PM within different orientation workspace are shown in Fig. 2.12. Additionally, through calculation by using Eq. (2.6), the value of the GCI of the 2PRS-PRRU PM with these parameters is equal to 0.426. • Optimization of design parameters Here, the PFNM method [4, 18] is also used to optimize the design parameters by considering the GCI of the 2PRS-PRRU PM. Considering the limitations of rotation angles, we restrict the motion capability as α ∈ [−40◦ , 40◦ ], β ∈ [−40◦ , 40◦ ]. However, it is difficult to illustrate the performance of the 2PRS-PRRU PM using the seven design parameters. Without loss of generality, we let the design parameters l1 = l2 = l3 = l4 , a = h be used as an example to illustrate the influence of the parameters. We can therefore obtain the 2PRS-PRRU PM using the three design parameters a, r and l1 , and they are normalized as

2.2 Condition Number Indices and Applications

63

Fig. 2.12 Distribution of the reciprocal of condition number for the 2PRS-PRRU PM

D=

l1 + r + a . 3

(2.49)

Then the non-dimensional and normalized parameters e1 , e2 , and e3 are deduced as e1 =

l1 r a , e2 = , e3 = . D D D

(2.50)

Considering the real application, the three normalized parameters should satisfy 0 < e2 ≤ e3 , 0 < e3 < 3/2, 0 < e1 , e2 , e3 < 3.

(2.51)

Using the PFNM method [18], the PDS can be obtained as shown in Fig. 2.13. The shaded area shown in Fig. 2.13a is the set of all possible points. For convenience, the chosen area can be transformed into a plan view, as shown in Fig. 2.13b. The relationship between the parameters in spatial space (r1 , r2 , r3 ) and those in plan space (s, t) can be described as follows: ⎧ e1 = s ⎪ ⎪ ⎪ ⎧ √ ⎪ ⎪ ⎨ ⎨ s = e1 s 3 3 e2 = − t − or e −e 2 √2 2 ⎪ ⎩ t = 3√ 2 ⎪ ⎪ ⎪ 3 ⎪ ⎩ e3 = 3 + 3 t − s 2 2 2 The design steps required for performance optimization are as follows:

(2.52)

64

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.13 Parameter design space of 2PRS-PRRU PM: a spatial view and b plane view

Fig. 2.14 Optimal design regions of GCI

Step 1: Identify the optimal regions. The distributions of the GCI can be obtained, as shown in Fig. 2.14. Considering the dexterity of the mechanism, the region where GCI ≥ 0.45 is regarded as having ideal dexterity. Step 2: Select 9 groups of data points from the optimal region in this study. And the chosen parameters should be as far as possible from the bounds of the optimal region. Through Eq. (2.52), the non-dimensional parameters e1 , e2 , and e3 can be obtained. Using Eqs. (2.50) and (2.52), we can then obtain the three design parameters a, r, and l1 , and the two plan space parameters s and t, as listed in Table 2.2. Step 3: Determine the normalized factor D and dimensional parameters a, r, l1 . The normalized factor D can be determined considering practical conditions, and D is chosen as 250 mm in this study. For example, the design parameters in group “7” are chosen as the optimal results, i.e., e1 = 1.10, e2 = 0.83, e3 = 1.07. Then the values of design parameters can be obtained using Eq. (2.52), i.e., l1 = l2 = l3 = l4 = 275 mm, a = h = 268 mm, r = 208 mm. Step 4: Check whether the dimensional parameters obtained in Step 3 are suitable or not for an actual usage condition. If it satisfies the actual assembly condition, the procedure is finished; otherwise, return to Step 3 to choose another group of data from the optimal region, and repeat Steps 3 and 4.

2.3 Motion/Force Transmission Indices and Applications

65

Table 2.2 GCI in the optimal design regions Group

s

t

e1

e2

e3

l1

r

a

GCI

1

1.14

0.36

1.14

0.62

1.24

285

155

310

0.442

2

1.24

0.40

1.24

0.53

1.23

310

133

308

0.415

3

1.32

0.40

1.32

0.49

1.19

330

123

298

0.390

4

1.70

0.44

1.70

0.27

1.03

425

68

258

0.196

5

1.74

0.46

1.74

0.23

1.03

435

58

258

0.166

6

1.68

0.40

1.68

0.31

1.01

420

78

253

0.231

7

1.10

0.14

1.10

0.83

1.07

275

208

268

0.458

8

1.40

0.12

1.40

0.70

0.90

350

175

225

0.474

9

1.32

0.16

1.32

0.70

0.98

330

175

245

0.480

Figure 2.15 shows the comparisons of distributions of dexterity, where the design parameters in groups “3”, “6”, and “9” are chosen as examples. From Figs. 2.15a, b, we can find that the GCIs in groups “3”, “6”, and “9” are 0.390, 0.231, and 0.480, respectively, in which the global performance of the 2PRS-PRRU PM with design parameters in group “9” is better than the original result. Designers can choose one of these combinations for different applications.

2.3 Motion/Force Transmission Indices and Applications 2.3.1 Motion/Force Transmission Indices Motion/force transmission indices [4, 9–11] reflect the capability of transmission between the actuators and the moving platform. It can also be regarded as the capability to sustain the payload applied to the end effector. Different from the above condition number indices that are based on the kinematic Jacobian matrix, the motion/force transmission indices avoid the problems caused by the inconsistencies among the units of the elements of the Jacobian matrix, and they are coordinate independent. Motion/force transmission performance in a configuration can be divided into two parts: the input transmission performance and output transmission performance. The input transmission performance represents the efficiency of power transmitted from the actuated joints to the limbs, while the output transmission performance represents the efficiency of power transmitted from the limbs to the moving platform. Considering the input transmission performance and output transmission performance simultaneously, the local transmission index (LTI) is defined as | | |$Ai ◦ $Ti | | λi = | (2.53) |$Ai ◦ $Ti | max

66

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.15 Distributions of optimized results: a l1 = l2 = l3 = l4 = 330 mm, r = 123 mm, a = h = 298 mm, b l1 = l2 = l3 = l4 = 420 mm, r = 78 mm, a = h = 253 mm, and c l1 = l2 = l3 = l4 = 330 mm, r = 175 mm, a = h = 245 mm

| | |$Oi ◦ $Ti | | ηi = | |$Oi ◦ $Ti |

(2.54)

Γ = min{λi , ηi },

(2.55)

max

and

where λi and ηi denote the input transmission index and output transmission index of the ith limb, respectively. $Ai denotes the input twist screw (ITS) of the ith limb, $Oi denotes the output twist screw (OTS) of the moving platform when locking all the input limbs except the ith limb, and $Ti denotes the transmission wrench screw (TWS) of the ith limb. The range of Γ is from zero to unity, and a larger Γ indicates better motion/force transmissibility. Since the LTI only represents the performance of the motion/force transmission in a single configuration, which hardly represents the global transmission performance

2.3 Motion/Force Transmission Indices and Applications

67

of the robot, it is necessary to define a global index that describes the performance in a set of poses. According to the actual situation and the definition of the transmission angle [23], it is assumed that the mechanism has good motion/force transmissibility when Γ is equal to or bigger than one given value, and these regions are referred to as the good transmission workspace (GTW). The index can thus measure the global motion/force transmissibility of a PM and is defined as { S σ = {G S

dW dW

,

(2.56)

where W is the reachable orientation workspace and S G and S denote the areas of the GTW and overall reachable workspace, respectively. Here, the overall reachable workspace is defined as the curved surface at an operating height with given rotational ranges. Apparently, σ ranges from zero to unity. The PM has better transmission performance when the σ is closer to unity. Here the motion/force transmission indices based on the screw theory are used to evaluate and optimize the kinematic performance of three PMs, namely the 6PSS PM, the 2PUR-PRU PM [24], and the 2PUR-PSR PM [25].

2.3.2 Example 1: 6PSS PM • Local and global transmission performances Before evaluating the motion/force transmission performance of the 6PSS PM, it is necessary to obtain the TWSs and OTSs first. Without loss of generality, we take limb 1 as an example. The TWS should be linearly independent of passive twists and constraint wrenches simultaneously [26]. Since the actuated joint of limb 1 is a P joint, the TWS of limb 1 is reciprocal to the passive joints and is given by ( / ) / $T1 = l 1 |l 1 | a1 × l 1 |l 1 | .

(2.57)

Similarly, the TWSs of the other five limbs can be derived as ⎧ $T2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ $T3 $T4 ⎪ ⎪ ⎪ $T5 ⎪ ⎪ ⎩ $T6

( / = l 2 |l 2 | ( / = l 3 |l 3 | ( / = l 4 |l 4 | ( / = l 5 |l 5 | ( / = l 6 |l 6 |

) / a2 × l 2 |l 2 | ) / a3 × l 3 |l 3 | ) / a4 × l 4 |l 4 | ) . / a5 × l 5 |l 5 | ) / a6 × l 6 |l 6 |

(2.58)

For the 6PSS, there is no constraint wrench inside the manipulator. Therefore, when limb 1 is actuated and the other five limbs are locked, the wrench system exerting on the moving platform can be written as

68

2 Kinematic Performance Analysis and Optimization of Parallel …

] [ U 1 = $T2 $T3 $T4 $T5 $T6 .

(2.59)

Accordingly, the instantaneous 1-DOF twist of the moving platform, which is denoted $O1 , can be obtained using reciprocal screw theory as $O1 ◦ U 1 = 0.

(2.60)

The other five output twist screws, $Oi (i = 2, 3, 4, 5, 6), can be derived in the same way. The architectural parameters of the 6PSS PM here are the same as used above, i.e., Rm = 250mm, l = 440mm, lb = 120mm, and θm = 30◦ . In addition, the LTI distributions in the different positions and orientations are also discussed separately here. The LTI distributions in the three-dimensional position workspace are presented first, as shown in Fig. 2.16a, in which the ranges of positions and orientations are set as −50 mm ≤ x ≤ 50 mm, −50 mm ≤ y ≤ 50 mm, 225 mm ≤ z ≤ 275 mm, and α = β = γ = 0°. For clarity, the LTI distributions when z = 250 mm are also presented, as shown in Fig. 2.16b, from which it is obvious that the distribution is symmetrical about the plane x = 0, and the kinematic performances near the central region are better than others. Similarly, the LTI distributions in the three-dimensional orientation workspace are also presented in Fig. 2.17a, in which the ranges of positions and orientations are set as x = y = z = 0, −20° ≤ α ≤ −20°, −20° ≤ β ≤ −20°, and −20° ≤ γ ≤ − 20°. The LTI distributions when γ = 0° are also presented, as shown in Fig. 2.17b, from which it is obvious that the distribution is symmetrical about the plane β = 0°, and the kinematic performances near the central region are better than others. Here, the motion/force transmission performance of the 6PSS PM in a configuration is regarded as good when the value of LTI is equal to or bigger than 0.7. According to the results in Figs. 2.16a and 2.17a, the GTW in the position region of the 6PSS PM, σ , is approximately 0.8965, and the GTW in the orientation region of

Fig. 2.16 LTI distribution of the 6PSS PM in the three-dimensional position workspace: a threedimensional position distribution and b xy plane distribution when z = 250 mm

2.3 Motion/Force Transmission Indices and Applications

69

Fig. 2.17 LTI distribution of the 6PSS PM in the three-dimensional orientation workspace: a threedimensional orientation distribution and b αβ plane distribution when γ = 0°

the 6PSS PM, σ , is approximately 0.1165. The above results mean that in the position workspace, the motion/force transmission capability with these architectural parameters is reasonably good. However, in the orientation workspace, the global performance of the 6PSS PM with these given parameters is not ideal. • Optimization of design parameters Similar to the above optimization of the 6PSS PM with respect to the GCI, the iterative search method is also used here for the dimensional optimization of the 6PSS PM with respect to the GTW, and the Rm and l are also defined as design parameters. In addition, the variation ranges of two design parameters are respectively defined as 230 mm ≤ Rm ≤ 270 mm, 420 mm ≤ l ≤ 460 mm. Similar to the above section, the position ranges are set as −50 mm ≤ x ≤ 50 mm, −50 mm ≤ y ≤ 50 mm, and 225 mm ≤ z ≤ 275 mm, and the orientation ranges are set as −20° ≤ α ≤ −20°, − 20° ≤ β ≤ −20°, and −20° ≤ γ ≤ −20°. Through iterative search and calculation, the distributions of GTW with different combinations of design parameters can be obtained, as shown in Figs. 2.18a, b. Similar to the optimization results of GCI, the distributions of GTW in the position workspace are also different from that in the orientation workspace. In the case of optimization regarding position workspace, the maximum of GTW occurs when Rm = 230 mm, l = 460 mm, as shown in Fig. 2.19a, and the value of GTW is unity. However, in the case of optimization regarding orientation workspace, the maximum of GTW occurs when Rm = 230 mm, l = 450 mm, as shown in Fig. 2.19b, and the value of GTW is 0.1845. Designers can choose one of these combinations for different applications.

70

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.18 GTW distributions with different parameter combinations: a optimization regarding position workspace and b optimization regarding orientation workspace

Fig. 2.19 Performance distributions of the 6PSS PM with optimized design parameters: a threedimensional position distribution with Rm = 230 mm and l = 460 mm and b three-dimensional orientation distribution with Rm = 230 mm and l = 450 mm

2.3.3 Example 2: 2PUR-PRU PM • Structure description The CAD model of the 2PUR-PRU PM is shown in Fig. 2.20, which has been proven to be a 2R1T PM [24]. The moving platform is connected to a fixed base through three limbs. Two identical limbs are PUR limbs. The third limb is a PRU limb. The position and orientations of the moving platform can be determined by the combination of the three displacements of the P joints. For the two PUR limbs, the first rotational axes of the U joints are coincident with each other. The second rotational axes of the U joints are parallel to each other and the R joints connected to the mobile platform. For the PRU limb, the rotational axis of the R joint is parallel to the first rotational axis of the U joint. The second rotational axis connected to the moving platform is always parallel to the second rotational axes of the U joints of the first two limbs.

2.3 Motion/Force Transmission Indices and Applications

71

Fig. 2.20 2PUR-PRU PM: a CAD model and b schematic representation

We let A1 and A2 denote the centers of the U joints in limbs 1 and 2 and A3 denote the central point of the R joint in limb 3. The central points of the R joints in limbs 1 and 2 are denoted B1 and B2 , and the center of the U joints in limb 3 is denoted B3 . Several coordinate frames are established as shown in Fig. 2.20a. A fixed reference frame O-xyz is attached to the fixed base. We let the x-axis always point in the direction of O A1 and the z-axis point downward vertically. A moving coordinate frame Puvw is attached to the moving platform (Fig. 2.20a). We let the u-axis point in the direction of P B1 and the v-axis point along P B3 . The w-axis is directed downward vertically with respect to the moving platform. In addition, three local frames denoted Ai − xi yi z i (i = 1, 2, 3) are attached to limb i. For A1 − x1 y1 z 1 and A2 − x2 y2 z 2 , the x-axis and y-axis always point along the rotational axes of the U joints of limbs 1 and 2, respectively. For A3 − x3 y3 z 3 , the directions of the axes of the coordinate frame are the same as those of O-xyz. The architectural parameters of the 2PUR-PRU PM are defined as P B1 = P B2 = l1 , P B3 = l2 , A1 B1 = A2 B2 = l3 , A3 B3 = l4 , and Po =H . The coordinates of points A1 , A2 , and A3 with respect to the O-xyz are )T ( )T )T ( ( defined as q1 0 0 , −q2 0 0 , and 0 q3 0 , in which qi (i = 1, 2, 3) denotes the distance between the point O and the P joint of limb i. The coordinates of points ( )T ( )T ( )T B1 , B2 , and B3 are defined as x B1 y B1 z B1 , x B2 y B2 z B2 , and x B3 y B3 z B3 , respectively. Through adding an articulated RR serial head to the moving platform or integrating with two translational gantries, the 2PUR-PRU PM can achieve 5-DOF outputs and be used for five-face milling and large-scale workpiece machining, as shown in Fig. 2.21. • Inverse kinematics Obtaining the inverse kinematics of the 2PUR-PRU PM is to find the actuated displacements (q1 , q2 , q3 ), given the orientations and position of the moving platform (α, β, z o ), where α, β, and z o denote the rotational angle around the x-axis and v-axis, and the distance between the points O and P, respectively.

72

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.21 Applications of the 2PUR-PRU PM: a five-face milling and b large-scale workpiece machining

The rotation matrix between the moving frame P-uvw and the fixed frame O-xyz can be written as ⎛ ⎞⎛ ⎞ 1 0 0 cβ 0 sβ O Ro = R x (α)Rv (β) = ⎝ 0 cα sα ⎠⎝ 0 1 0 ⎠ 0 −sα cα −sβ 0 cβ ⎛ ⎞ cβ 0 sβ = ⎝ −sαsβ cα sαcβ ⎠, (2.61) −cαsβ −sα cαcβ where R x (α) and Rv (β) represent the rotational matrices around x-axis and v-axis, respectively. As shown in Fig. 2.20, the position vectors O Ai with respect to the fixed frame )T ( are denoted q i (i = 1, 2, 3), and the vector p = 0 z o sα z o cα denotes the position vector of the point P with respect to the fixed frame. The position vectors Ai Bi and P Bi with respect to the fixed frame are denoted ai and bi , respectively. We thus obtain ⎧ ⎪ ⎪ a1 = ⎨

( )T R1 −l3 cθ1 0 l3 sθ1 ( )T a2 = O R2 l3 cθ2 0 l3 sθ2 ⎪ ⎪ ( )T ⎩ a3 = O R3 0 −l4 cθ3 l4 sθ3 O

(2.62)

and ⎧ b = ⎪ ⎪ ⎨ 1

)T ( Ro l1 0 0 )T ( b2 = O Ro −l1 0 0 , ⎪ ⎪ )T ( ⎩ b3 = O Ro 0 l2 0 O

(2.63)

2.3 Motion/Force Transmission Indices and Applications

73

where θi denotes the angle between limb i (i = 1, 2, 3) and the fixed base. The rotation matrix O Ri , which is the transformation mapping between the limb frame Ai − xi yi z i and the fixed frame O-xyz, is given by ⎛

⎛ ⎞ ⎞ 1 0 0 100 O R1 = O R2 = ⎝ 0 cα sα ⎠, O R3 = ⎝ 0 1 0 ⎠ 0 −sα cα 001

(2.64)

As shown in Fig. 2.20a, the position vectors q i (i = 1, 2, 3) can be written in the form as q i = p + bi − ai .

(2.65)

We thus obtain the inverse kinematics of the 2PUR-PRU PM through Eqs. (2.61)– (2.65) as ⎧ ⎨ q1 = c1 + l1 cβ , q = c2 + l1 cβ ⎩ 2 q3 = c3 + z o sα + l2 cα / l32 − (z o − l1 sβ)2 , c2 where c1 = / l42 − (z o cα − l2 sα)2 .

=

(2.66)

/ l32 − (z o + l1 sβ)2 , c3

=

• Local and global transmission performances Before evaluating the motion/force transmission performance of the 2PUR-PRU PM, it is necessary to obtain the TWSs and OTSs first. Without loss of generality, we take limb 1 as an example. The TWS should be linearly independent of passive twists and constraint wrenches simultaneously [26]. Since the actuated joint of limb 1 is a P joint, the TWS of limb 1 is reciprocal to the passive joints and is given by ( ) $T1 = x B1 − q1 y B1 z B1 ; 0 −q1 z B1 q1 y B1 .

(2.67)

Similarly, the TWSs of the other two limbs can be derived as {

) ( $T2 = q2 + x B2 y B2 z B2 ; 0 q2 z B2 −q2 y B2 . ) ( $T3 = 0 y B3 − q3 z B3 ; q3 z B3 0 0

(2.68)

When limb 1 is actuated and the other two limbs are locked, the new wrench system exerting on the moving platform can be written as ] [ U 1 = $C1 $C2 $C3 $T2 $T3 ,

(2.69)

74

2 Kinematic Performance Analysis and Optimization of Parallel …

where the constraint wrench screws (CWSs) of the 2PUR-PRU PM can be expressed as ⎧ ( ) ⎪ $ = 0 0 0; 0 sα cα ⎪ ⎨ C1 ( ) . (2.70) $C2 = 0 cα −sα; 0 0 0 ⎪ ⎪ ⎩ $ = ( 1 0 0; 0 z + y tanα 0 ) C3

B3

B3

Accordingly, the instantaneous 1-DOF twist of the moving platform, which is denoted $O1 , can be obtained using reciprocal screw theory as $O1 ◦ U 1 = 0.

(2.71)

The other two output twist screws, $O2 and $O3 , can be derived in the same way. The architectural parameters used here are set as l1 = l2 = 130 mm, l3 = l4 = 340 mm, z o = 260 mm, α ∈ [−20◦ , 20◦ ], and β ∈ [−20◦ , 20◦ ]. The LTI distribution of the 2PUR-PRU PM can thus be obtained by substituting the TWSs and OTSs into Eqs. (2.53)–(2.55), as shown in Fig. 2.22. With the length of the tool head H = 20 mm, the LTI distribution in the three-dimensional workspace is shown in Fig. 2.22a to be symmetrical about the plane x = 0. The LTI distribution in the orientation workspace of the PM is completely symmetrical with respect to angle β, as shown in Fig. 2.22b. Here, the motion/force transmission performance of 2PUR-PRU PM in a configuration is regarded as good when the value of LTI is equal to or bigger than 0.5. According to the results in Fig. 2.22, the GTW of the 2PUR-PRU PM, σ , is approximately 0.74, which means that the motion/force transmission capability with these architectural parameters is reasonably good. • Optimization of design parameters

Fig. 2.22 LTI distribution of the 2PUR-PRU PM: a three-dimensional workspace and b orientation workspace

2.3 Motion/Force Transmission Indices and Applications

75

In this study, optimization of the design parameters of the 2PUR-PRU PM is based on σ . The architectural parameters, rotation angles, and operating √ height are set as follows: l3 = l4 , α ∈ [−20◦ , 20◦ ], β ∈ [−20◦ , 20◦ ], and z o = 2l3 /2. Using the PFNM proposed by Liu et al. [18], the design parameters are normalized as ⎧ l1 + l2 + l3 ⎪ ⎨D = 3 . l ⎪ 1 ⎩ r = , r = l2 , r = l3 1 2 3 D D D

(2.72)

Considering practical applications, the normalized parameters should satisfy {

r1 , r2 ≤ r3 0 < r1 , r2 , r3 < 3

.

(2.73)

As shown in Fig. 2.23, the parameter design space [18] includes all possible points. The relationship between the parameters in three-dimensional space (r1 , r2 , r3 ) and those in plan space (s, t) is described as follows: ⎧ √ s 3 3 ⎪ ⎪ ⎧ ⎪ t− ⎪ r1 = + ⎪ 2 √2 2 ⎨ ⎨ s = r3 r −r s or 3 3 ⎪ ⎩ t = 1√ 2 t− r2 = − ⎪ ⎪ 2 2 2 ⎪ 3 ⎪ ⎩ r3 = s

(2.74)

The design steps for performance optimization are as follows. Step 1: Identify the regions. The distributions of σ can be obtained as shown in Fig. 2.24. Here, region I—in which σ ∈ (0, 0.6)—is regarded as having poor global

Fig. 2.23 Parameter design space of the 2PUR-PRU PM: a spatial view and b plane view

76

2 Kinematic Performance Analysis and Optimization of Parallel …

motion/force transmission, region II—in which σ ∈ [0.6, 0.8) —is regarded as having medium global motion/force transmission, and region III—in which σ ≥ 0.8—is regarded as having good global motion/force transmission. Step 2: Select three groups of data points randomly from each region; altogether, nine groups of data points are chosen in this study. For each group, the non-dimensional parameters r1 , r2 , and r3 can be obtained using Eq. (2.74). Using Eqs. (2.72) and (2.74), one can then obtain the design parameters l1 , l2 , and l3 and the two plan space parameters s and t, as listed in Table 2.3. Step 3: Determine the normalized factor D and dimensional parameters l1 , l2 , and l3 . In view of the actual operating environment, the normalized factor D is determined as 400 mm. For example, the design parameters in group 1 are chosen as results,

Fig. 2.24 Optimization of σ for the 2PUR-PRU PM: a Distribution of σ and b three design regions I, II, and III

Table 2.3 GTWs in the three design regions Region I

II

III

Group

s

t

r1

r2

r3

l1

l2

l3

σ

1

1.68

0.19

0.82

0.50

1.68

328

200

672

0.571

2

1.78

0.43

0.98

0.24

1.78

392

96

712

0.491

3

1.56

0.49

1.14

0.30

1.56

456

120

624

0.430

4

2.06

0.23

0.67

0.27

2.06

268

108

824

0.623

5

2.16

0.15

0.55

0.29

2.16

220

116

864

0.713

6

1.78

− 0.13

0.50

0.72

1.78

200

288

712

0.777

7

1.98

− 0.15

0.38

0.64

1.98

152

256

792

0.874

8

1.92

− 0.29

0.29

0.79

1.92

116

316

768

0.938

9

1.80

− 0.45

0.21

0.99

1.80

84

396

720

0.980

2.3 Motion/Force Transmission Indices and Applications

77

i.e., r1 = 0.82, r2 = 0.50, and r3 = 1.68. The values of li can then be obtained using Eq. (2.72), i.e., l1 = 328 mm, l2 = 200 mm, and l3 = 672 mm. Step 4: Check whether the dimensional parameters obtained in Step 3 are suitable for actual applications. If the actual assembly conditions are satisfied, the procedure is finished; otherwise, return to Step 3, choose another group of data from the regions, and repeat Steps 3 and 4. Based on the dimensional parameters obtained from the previous step (i.e., l1 = 328 mm, l2 = 200 mm, and l3 = 672 mm), one can deduce that a manipulator with these parameters would be suitable for actual use and could work normally. Here, Fig. 2.25 shows comparisons of the motion/force transmissibility in regions I, II, and III, for which the design parameters in groups 3, 6, and 9 are chosen as examples. As shown in Fig. 2.25c, the value of σ in region III is 0.980, which is better than that in either of the other two examples. The results demonstrate that region III could be selected as the optimal region. Designers can choose one of these combinations for different applications.

2.3.4 Example 3: 2PUR-PSR PM • Structure description As shown in Fig. 2.26, the 2PUR-PSR PM [25] is composed of a moving platform, a fixed base, two PUR limbs with identical structures, and one PSR limb. For the two PUR limbs, the first rotational axes of the two U joints are coincident with each other, and also parallel to the movement direction of the P joints. The second rotational axes of the two U joints are perpendicular to the direction of the P joints and also parallel to each other and the R joints connected to the mobile platform. For the PSR limb, it is perpendicular to the rotational axis of the R joint. Let A1 and A2 denote the centers of the U joints in limbs 1 and 2 and A3 denote the central point of the S joint in limb 3. The central points of the R joints in three limbs are denoted as B1 , B2 , and B3 . The coordinate frames are established as shown in Fig. 2.26. A fixed reference frame O-xyz is attached to the fixed base. We let the x-axis always point in the direction of O A3 , the y-axis point in the direction of O A1 , and the z-axis point downward vertically. A moving coordinate frame o-uvw is attached to the moving platform. Let the u-axis point in the direction of oB3 and the v-axis point along oB1 . The w-axis is directed downward vertically with respect to the moving platform. The architectural parameters of the 2PUR-PSR PM are defined as oB1 = oB2 = l1 , oB3 = l2 , A1 B1 = A2 B2 = l3 , A3 B3 = l4 , and l1 · l4 = l2 · l3 .

78

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.25 Comparisons of motion/force transmissibility: a region I with l1 = 456 mm, l2 = 120 mm, l3 = 624 mm (group 3), b region II with l1 = 200 mm, l2 = 288 mm, l3 = 712 mm (group 6), and c region III with l1 = 84 mm, l2 = 396 mm, l3 = 720 mm (group 9)

• Inverse kinematics The inverse kinematics of the 2PUR-PSR PM is to determine the displacement of the P joint in each limb, qi (i = 1, 2, 3), given the architectural parameters of the PM ( )T and the position vector of the end point o p = x y z . For the 2PUR-PSR PM, the orientation variation between the moving frame o-uvw and the fixed frame O-xyz can be written by a rotation matrix as

2.3 Motion/Force Transmission Indices and Applications

79

Fig. 2.26 CAD model of 2PUR-PSR PM

R = R z (ϕ)R y (θ )R x (ψ) ⎛ ⎞ cθ cϕ sψsθ cϕ − cψsϕ cψsθ cϕ + sψsϕ = ⎝ cθ sϕ sψsθ sϕ + cψcϕ cψsθ sϕ − sψcϕ ⎠, −sθ sψcθ cψcθ

(2.75)

where R z (ϕ), R y (θ ), and R x (ψ) denote the rotation matrices about the z-, y-, and x-axes, respectively. According to Fig. 2.26, the closed-loop kinematic equations of the 2PUR-PSR PM can be obtained as q i = p+bi − ai ,

(2.76)

where q i denotes the displacement vector of the actuator in each limb. ai and bi = Rbio denote the vectors from point Ai to Bi and point o to Bi , respectively, in which the vector bio is the displacement vector bi with respect to the moving coordinate frame and can be expressed as ⎧ )T ( ⎪ ⎨ b1o = ( 0 l1 0 ) T b2o = 0 −l1 0 . ⎪ ) ( ⎩ T b3o = l2 0 0

(2.77)

In addition, the axis direction of the R joint in the limb i can be expressed as the vector ci = Rcio , in which the vector cio is the directional vector ci with respect to the moving coordinate frame and can be expressed as ⎧ ( )T ⎪ ⎨ c1o = ( 1 0 0 ) T c2o = 1 0 0 . ⎪ ( )T ⎩ c3o = 0 1 0

(2.78)

80

2 Kinematic Performance Analysis and Optimization of Parallel …

Due to the fact that the axis direction of the R joint of each limb is always perpendicular to q i − p during the motion, the relationship can be written as ( ) ciT q i − p = 0.

(2.79)

Therefore, Eq. (2.79) can be expanded as ⎧ ⎨ xcθ cϕ − (q1 − y)cθ sϕ − zsθ = 0 . xcθ cϕ + (q2 + y)cθ sϕ − zsθ = 0 ⎩ y(cψcϕ + sψsθ sϕ) + (q3 − x)(cψsϕ − sψsθ cϕ) + zsψcθ = 0

(2.80)

According to the above equation, it can be obtained as (q1 + q2 )cθ sϕ = 0.

(2.81)

Because q1 + q2 cannot be equal to zero during the motion, it can be deduced as cθ sϕ = 0.

(2.82)

Combining with Eqs. (2.80) and (2.82), the relationship between the coordinates x and z can be obtained as xcθ cϕ − zsθ = 0.

(2.83)

From Eq. (2.82), it can be inferred that sϕ = 0, from which the angle ϕ is equal to 0 or π . Based on the structural characteristics of the 2PUR-PSR PM, the angle ϕ cannot be equal to π ; that is, the angle ϕ is always equal to 0. Therefore, the third equation in Eq. (2.80) can be rewritten as ycψ − (q3 − x)sψsθ + zsψcθ = 0.

(2.84)

Combined Eqs. (2.83) and (2.84), the coordinates x and y in the position vector ( )T p = x y z can be expressed as }

x = z tan θ . y = (q3 − z tan θ )tanψsθ − z tan ψcθ

(2.85)

Substituting Eq. (2.85) into Eq. (2.76), the analytical expressions of inverse kinematics of the 2PUR-PSR PM can be obtained. • Local and global transmission performances Using the same procedure as listed in the example of the 2PUR-PRU PM, the ITSs, TWSs, CWSs, and OTSs of the 2PUR-PSR PM also can be derived. For example, the TWSs and CWSs of the 2PUR-PSR PM can be obtained as

2.3 Motion/Force Transmission Indices and Applications

( ) ⎧ ⎨ $T1 = (a1 /|a1 |; q 1 × a1 /|a1 | ) $ = a /|a |; q 2 × a2 /|a2 | ) ⎩ T2 ( 2 2 $T3 = a3 /|a3 |; q 3 × a3 /|a3 |

81

(2.86)

and ⎧ ( ) ⎪ $ = −cθ 0 sθ ; 0 0 0 ⎪ ⎨ C1 ( ) . $C2 = 0 0 0; sθ 0 cθ ⎪ ⎪ ⎩ $ = ( 0 cψ sψ; 0 −q sψ q cψ ) C3 3 3

(2.87)

Based on these results, the OTSs of the 2PUR-PSR PM can be calculated. Substituting these screws into Eqs. (2.53)–(2.55), the LTI of the 2PUR-PSR PM can be obtained. Taking the interference between the limbs of the 2PUR-PSR PM into account, the architectural parameters are set as follows: l1 = 200 mm, l2 = 200 mm, l3 = 400 mm, l4 = 400 mm, z = 250mm and the ranges of motion orientation are restricted as follows: ψ ∈ [−30◦ , 30◦ ] and θ ∈ [−30◦ , 30◦ ]. The LTI distributions of the 2PUR-PSR PM are shown in Fig. 2.27. It is clear that the distribution is symmetrical about the ψ = 0◦ , which is consistent with the structure of the 2PURPSR PM. Furthermore, Fig. 2.27 shows that the closer the configuration is to the initial situation, the better the motion/force transmission performance will be. Different from the example of the 2PUR-PRU PM, the region where Γ ≥ 0.7 is defined as the GTW here. According to Fig. 2.27, the ratio of the region where Γ ≥ 0.7 for the 2PUR-PSR PM is only 0.136, which means that the motion/force transmission capability with these link parameters is not good, and the optimization is necessary. Fig. 2.27 LTI distribution of the 2PUR-PSR PM in the orientation workspace

82

2 Kinematic Performance Analysis and Optimization of Parallel …

• Optimization of design parameters The GTW of the 2PUR-PSR PM is highly dependent on the design parameters l1 , l2 , l3 , and l4 . Considering the rotation angles and operating height limitations, the range of orientation capability is restricted as follows: ψ ∈ [−30◦ , 30◦ ], θ ∈ [−30◦ , 30◦ ], and z = 0.667 min[l3 , l4 ]. The PFNM method proposed by Liu et al. [18] is used to optimize these parameters relative to the GTW. Here, l1 , l2 , and l3 are defined as design parameters and are normalized as ⎧ ⎪ ⎪ l4 = (l2 · l3 )/l1 ⎪ ⎪ ⎨ l1 + l2 + l3 D= . 3 ⎪ ⎪ ⎪ l ⎪ ⎩ ri = i D

(2.88)

Considering practical application, the three normalized parameters should satisfy {

r1 , r2 ≤ r3 , r4 0 < r1 , r2 , r3 , r4 < 3

(2.89)

The PDS of the 2PUR-PSR PM can be obtained through Eq. (2.88), as shown in Fig. 2.28. The shade area of the PDS represents the set of all possible points. Each possible point means that the 2PUR-PSR PM with these parameters has the region where Γ ≥ 0.7 in the reachable workspace. The relationship between the parameters in three-dimensional space (r1 , r2 , r3 ) and those in planar space (s, t) is given as follows: ⎧ √ s 3 3 ⎪ ⎪ ⎪ ⎧ r1 = + t− ⎪ ⎪ 2 √2 2 ⎨ ⎨ s = r3 (2.90) r −r s or 3 3 ⎪ r2 = − ⎩ t = 1√ 2 t− ⎪ ⎪ 2 2 2 ⎪ 3 ⎪ ⎩ r3 = s The steps for parameter optimization are as follows: Step 1: Identify the regions. According to the distribution shown in Fig. 2.29a, the whole region is divided into three design subregions for better understanding of optimization, as shown in Fig. 2.29b. Region I where σ ∈ [ 0.1, 0.3) is regarded as having poor global motion/force transmission, which means that the design parameters in this region should be avoided to select for actual manufacturing. Region II—in which σ ∈ [0.3, 0.6) —is regarded as having medium global motion/force transmission, and region III—in which σ ≥ 0.6—is regarded as having good global motion/force transmission.

2.3 Motion/Force Transmission Indices and Applications

83

Fig. 2.28 Parameter design space of the 2PUR-PSR PM: a spatial view and b plane view

Fig. 2.29 Optimization of GTW for the 2PUR-PSR PM: a distribution of σ and b three design regions I, II, and III

Step 2: Determine the design parameters selected from the three design regions. Three groups of data points are selected randomly from each region; nine groups of data points are chosen here, as shown in Table 2.4. Considering the manufacturing error in actual machining, the parameter selection should be far away from the boundary of the optimized region. Using Eq. (2.90), the non-dimensional parameters r1 , r2 , and r3 can be obtained, as well as parameters s and t. Step 3: Determine the normalized factor D and design parameters l1 , l2 , and l3 . Considering the actual size of the PM, the D is determined as 300 mm. For example, the design parameters in group 7 are chosen as results, i.e., r1 = 0.06, r2 = 0.76, and r3 = 2.18. The values of dimensional parameters can then be obtained using Eq. (2.88), i.e., l 1 = 18 mm, l 2 = 228 mm, and l 3 = 654 mm. In the same way, nine groups of data points can be obtained, as listed in Table 2.4. Step 4: Check whether the dimensional parameters obtained in Step 2 are suitable or not for actual usage conditions. If the actual assembly conditions are satisfied, the procedure is finished; otherwise, return to Step 2 to choose another group of data from the regions.

84

2 Kinematic Performance Analysis and Optimization of Parallel …

Table 2.4 GTWs in the design regions Region

Group

s

t

r1

r2

r3

l2

l3

GTW

I

1

1.96

− 0.24

0.31

0.73

1.96

93

219

588

0.249

2

1.92

− 0.18

0.38

0.70

1.92

114

210

576

0.201

3

1.82

− 0.10

0.50

0.68

1.82

150

204

546

0.149

4

2.16

− 0.24

0.21

0.63

2.16

63

189

648

0.394

5

2.36

− 0.20

0.15

0.49

2.36

45

147

708

0.506

6

2.18

0.04

0.44

0.38

2.18

133

113

654

0.376

7

2.18

− 0.40

0.06

0.76

2.18

18

228

654

0.610

8

2.38

− 0.30

0.05

0.57

2.38

15

171

714

0.634

9

2.52

− 0.26

0.02

0.47

2.51

6

141

753

0.669

II

III

l1

In order to show the comparisons of the GTW in regions I, II, and III, the design parameters in groups 3, 6, and 9 are chosen as examples, and the corresponding atlases about the motion/force transmissibility are shown in Fig. 2.30. Compared with the other two examples, the value of σ is 0.669, which means that the overall LTI distribution based on the design parameters in group 9 is better. The results demonstrate that region III could be selected as the optimal region. Designers can choose one of these combinations for different applications.

2.4 Motion/Force Constraint Indices and Applications 2.4.1 Motion/Force Constraint Indices Unlike the motion/force transmissibility that describes the efficiency of energy transmission, the motion/force constrainability reflects the capability of PMs to resist external loads [27], which is of great significance to the application of PMs in the field requiring high rigidity and precision. There are some works examining the constrainability of PMs [27–33]. Han et al. [29] observed the constraint singularity from experiments on a 3-UPU design, because an unexpected DOF occurred when the actuated joints were locked. Then, according to the variation of constraint wrenches in space, Zlatanov et al. [30] proposed the concept of constraint singularity to stress the importance of constrainability. Recently, based on screw theory, Liu and his colleagues [27] proposed three motion/force constraint indices of PMs to evaluate the constrainability of lower-mobility PMs and presented an approach for constraint singularity analysis. Meng et al. [28] concentrated on redundantly actuated and overconstrained PMs with closed-loop subchains. Due to the particularity of redundantly actuated and overconstrained PMs with closed-loop subchains, they addressed a local interaction index with the analysis of motion/force interaction performance that considered the TWSs and CWSs together.

2.4 Motion/Force Constraint Indices and Applications

85

Fig. 2.30 Comparisons of motion/force transmissibility: a region I, b region II, and c region III

The above-mentioned research on constraint performance is mainly aimed at PMs without redundancy actuation and over constraints and rarely involves overconstrained PMs without/with actuation redundancy. This chapter mainly studies the motion/force constraint performance of overconstrained PMs without actuation redundancy, and the research on the motion/force constraint performance of redundantly actuated and overconstrained PMs will be presented in Chap. 4. For the overconstrained PMs without actuation redundancy, they have one obvious feature: the number of constraint wrenches is greater than the limited DOFs. Based on this factor, it can be found that the analysis of the output constraint index (OCI) [27] is complex because the relationship between the output virtual twist and the constraint wrench is not one-to-one, which complicates the analysis of constrainability. To solve this problem, a new evaluation approach and corresponding constraint indices using the screw theory [13] and power coefficients [11] are proposed. • Notations To facilitate subsequent derivation, the notations employed here are presented as follows. A PM contains four wrench and twist screw systems: wrenches of constraint

86

2 Kinematic Performance Analysis and Optimization of Parallel …

{

{ } } { } Wc , wrenches of transmission Wn , twists of restriction Tr , and twists of { } permission Tp . They can be expressed as /\

/\

/\

/\

⎧{ } { } ⎪ Wc = span $C1 , $C2 , . . . , $C(6−m) ⎪ ⎪ ⎪ ⎪ { } ⎪ { } ⎪ ⎪ ⎨ Wn = span $T1 , $T2 , . . . , $Tm { } { }, ⎪ ⎪ = span $ T , $ , . . . , $ ⎪ r Tr1 Tr2 Tr(6−m) ⎪ ⎪ ⎪ { } ⎪ { } ⎪ ⎩ Tp = span $p1 , $p2 , . . . , $pm /\

/\

(2.91)

/\

/\

where $C , $T , $Tr , $p represent CWS, { } TWS, restricted twist screw, and permission twist screw, respectively. Here, Tp can be obtained from the joints of each limb. { } { } { } { } Wc and Wn are the reciprocal and dual of Tp , respectively. Tr is acquired { } from the reciprocal of Wn . More details can be found in research by Huang et al. [31]. Without loss of generality, a general m-DOF overconstrained PM without actuation redundancy is shown in Fig. 2.31, which is composed of a fixed base, a moving platform, and m limbs. Each limb is assumed with only one actuated joint here. Suppose the DOF of the jth limb is q j , so that the limb twist system, U P j , can be written in the form of q j -order /\

/\

/\

/\

/\

/\

] [ U P j = $ j1 $ j2 · · · $ jq j ,

(2.92)

where $ ji represents the ith unit kinematic joint twist of the jth limb. Using screw theory [13], the corresponding jth limb constraint system, U C j , can be expressed as ] [ U C j = $C j1 $C j2 · · · $C j (6−q j ) ,

(2.93)

where $C jk denotes the kth unit CWS of the jth limb. $C jk and $ ji are reciprocal to each other and can be written as $ ji ◦ $C jk = 0 i = 1, 2, . . . , q j ; k = 1, 2, . . . , (6 − q j ).

(2.94)

) Σm ( In total, there are g = j=1 6 − q j CWSs acting on the moving platform, which are expressed as $Ci (i = 1, 2, …, g). Meanwhile, each limb has one TWS in this model, and thus there are m TWSs in the PM, which are expressed as $T j (j = 1, 2, …, m), as shown in Fig. 2.31. Therefore, the constraint wrench system U C and transmission wrench system U T of an overconstrained PM without actuation redundancy can be expressed as

2.4 Motion/Force Constraint Indices and Applications

87

Fig. 2.31 General representation of an m-DOF overconstrained PM without actuation redundancy

] [ ] [ U C = U C1 U C2 · · · U Cn = $C1 $C2 . . . $Cg , ] [ U T = $T1 $T2 · · · $Tm .

(2.95)

• Local constraint performance index For an m-DOF overconstrained PM without actuation redundancy in a certain nonsingular configuration, as shown in Fig. 2.31, it keeps balanced with (m + g) wrenches, which includes m TWSs and g (g > 6 − m) CWSs. Among these (m + g) wrenches, there are six independent wrenches, and others are linearly dependent on these six wrenches. For the overconstrained PMs without actuation redundancy, the influences of these (m + g − 6) linearly dependent wrenches on the motion/force constrainability should be considered. The initial method used for the constraint evaluation is mainly focused on the non-redundant actuated PMs without over constraints [27, 28]. However, they did not consider the comprehensive influence of the (m + g − 6) linearly dependent wrenches. Inspired by the above work, a new method for the constraint performance evaluation of non-redundant actuated PMs with over constraints is presented here, which takes the effects of (m + g − 6) linearly dependent wrenches into account reasonably. First, by locking all the TWSs, they are converted into m new CWSs, and then the non-redundant actuated PM with over constraints becomes a statically indeterminate structure with (m + g) CWSs. Then, through locking m TWSs and (5 − m) linearly independent CWSs and “losing” (m + g − 5) CWSs, the non-redundant actuated PM with over constraints effectively becomes an instantaneous 1-DOF mechanism. The unique instantaneous motion of the moving platform is defined as virtual OTS [27]. Considering the force characteristics of PMs, at least k (k ≥ 1) of the “lost” (m

88

2 Kinematic Performance Analysis and Optimization of Parallel …

+ g − 5) CWSs are linearly independent with the five locked wrenches, which can be used to prevent the virtual instantaneous motion. Next, through ergodic analysis, m+g−5 cases need to be considered. Here, the instantaneous power there are at most C g between the k linearly independent constraint wrenches in the (m + g − 5) CWSs and the unique instantaneous twist is defined as the output constraint performance of the 1-DOF mechanism. For different k, the calculation method is not the same, which will be discussed later in detail. Averaging the values of the output constraint indices of these 1-DOF mechanisms gives the overall output constraint performance of the non-redundant actuated PM with over constraints. Finally, the total constraint performance is calculated by considering both the input and output constraint performances. The purpose of this method is to describe the motion/force constrainability of the non-redundant actuated PM with over constraints by evaluating the constraint performance of several 1-DOF mechanisms with multiple wrenches. A procedure for evaluating a general m-DOF overconstrained PM without actuation redundancy is presented in Fig. 2.32, which is described as follows. Step 1: Determine the transmission wrench system U T and constraint wrench system U C of the non-redundant actuated PM with over constraints firstly, and then lock all the TWSs. The new constraint wrench system U with (m + g) wrenches can be expressed as ] [ U = [U T , U C ] = $T1 $T2 · · · $Tm $C1 $C2 · · · $Cg

(2.96)

Step 2: Randomly choose (5 − m) linearly independent CWSs and m TWSs to be locked, and “lose” other (m + g − 5) CWSs. In different “lost” cases, the linear independent numbers and characteristics in the “lost” CWSs are different, which is numbered as the ith case. The platform wrench system U i of the non-redundant actuated PM with over constraints can be updated as ] [ U i = [U iT , U iC ] = $T1 $T2 · · · $T f $C1 $C2 · · · $C f , f + f , = 5,

(2.97)

where the U i is composed of f locked TWSs ($T f ) and f , CWSs ($C f , ), and the dimension is equal to 5. In this case, the non-redundant actuated PM with over constraints becomes a 1DOF mechanism with a 5-order wrench system U i . At this time, only some wrenches in the “lost” CWSs can do work on the moving platform of the PM, while the locked TWSs and other CWSs cannot do work on the moving platform. Therefore, the instantaneous unit motion of the moving platform, virtual OTS $Oi , can be obtained by the reciprocal condition in screw theory as $Oi ◦ U i = 0.

(2.98)

2.4 Motion/Force Constraint Indices and Applications

Fig. 2.32 Procedure of evaluating a general non-redundant actuated PM with over constraints

89

90

2 Kinematic Performance Analysis and Optimization of Parallel …

Additionally, the “lost” (m + g − 5) CWSs are named as $Wck (k = 1, 2, …, m + , g − 5) for convenience, and they are classified as U i , where ] [ , U i = $Wc1 $Wc2 · · · $Wc(m+g−5) .

(2.99)

However, only k CWSs in the “lost” (m + g − 5) ones, which are linearly independent of the 5 locked wrenches, can counterbalance the external loads along the axis of $Oi . The number k will determine whether the relationship between the $Oi and the applied wrenches is one-to-one or one-to-multiple. It results in the difficulty of calculating the OCI. Therefore, it is necessary to determine the values of k and divide them into two cases to obtain the OCI comprehensively. Case 1: If k = 1, only one wrench remains after eliminating the linear correlation. It means that only one CWS of the “lost” (m + g − 5) CWSs is linearly independent of the locked 5 wrenches. In this case, the output constraint performance index of the 1-DOF mechanism between the ith “lost” wrench $Wci and the virtual output twist $Oi is defined. It is the same as that presented in Liu’s work [27], which can be written as | | |$Wci ◦ $Oi | | , (2.100) ζOi = | |$Wci ◦ $Oi | max

| | | | where |$Wci ◦ $Oi | and |$Wci ◦ $Oi |max denote the instantaneous and maximum power coefficients, respectively. Case 2: If k ≥ 2, k CWSs remain after eliminating the linear correlation. It means that in the “lost” (m + g − 5) CWSs, the number of CWSs that are linearly independent of the locked 5 wrenches is k. In this case, how to evaluate the constraint performance of a 1-DOF mechanism with multiple wrenches is a key issue. Obviously, it is impossible that the effects between these wrenches and the virtual output twist of 1-DOF mechanism can reach their maximum simultaneously. Thus, the output constraint performance index that reflects the power coefficient between the ith “lost” wrenches and the virtual output twist of the 1-DOF mechanism can be written by using a compound formula as ζOi

| | | | | | |$Wc1 ◦ $O1 | + |$Wc2 ◦ $O1 | + · · · + |$Wck ◦ $O1 | | | | | |, = min{ | |$Wc1 ◦ $O1 | + |$Wc2 ◦ $O1 | + · · · + |$Wck ◦ $O1 | max | | | | | | |$Wc1 ◦ $Ok | + |$Wc2 ◦ $Ok | + · · · + |$Wck ◦ $Ok | | | | | | }. ··· , | |$Wc1 ◦ $Ok | + |$Wc2 ◦ $Ok | + · · · + |$Wck ◦ $Ok | max

(2.101)

In Eq. (2.101), the output constraint performance of a 1-DOF mechanism in this case is defined as the combined effect of k CWSs. It should be noted that if any |$ ◦$ |+|$ ◦$ |+···+|$ ◦$ | of |$ Wc1◦$ Ok|+|$ Wc2◦$ Ok|+···+|$ Wck◦$ Ok| is equal to zero, the wrenches cannot counterWc1

Ok

Wc2

Ok

Wck

Ok max

balance the external loads along the axis of $Oi , which means that the constraint

2.4 Motion/Force Constraint Indices and Applications

91

singularity occurs in this configuration. In other words, if the minimum of them is larger than zero, there is no constraint singularity. Thus, ζOi is defined as the minimum of all the situations here. Step 3: Considering the above two cases, together with the reality that the number of selections of the locked 5 linearly independent wrenches (N) must be smaller than m+g−5 m+g−5 Cg , N < Cg times must be studied. It can be imagined that when all the cases are considered in a scenario where the locked wrenches are transformed one by one, the output constraint coefficient of the non-redundant actuated PM with over constraints will always be limited between the maximum and the minimum of those results. Thus, the OCI of a PM can be defined as ζ=

N 1 Σ ζOi , N i=1

(2.102)

where the range of ζ is from zero to unity. At the same time, the input constraint index (ICI) of each limb [34] is determined by j ηI

{ | | } |$Ci ◦ $Tri | | = min{ηi } = min | i = 1, 2, . . . , f, |$Ci ◦ $Tri | i i max

(2.103)

j

where ηI is the input constraint performance of the jth limb. Inspired by Liu’s work [27], a virtual joint is employed to further understand the meaning of the constraint index, where the virtual joint corresponds to an input restricted twist screw. When the index ηi is equal to unity, the ith constraint wrench exerts in the direction of the input restricted twist screw of the ith virtual joint, which means that the ith virtual joint is constrained greatly. Besides, when the index, ηi , is equal to zero, it can be said that the ith virtual joint cannot be constrained by the ith constraint wrench. In other words, the virtual joint becomes active. Thus, the ICI of the non-redundant actuated PM with over constraints in a certain configuration can be written as { { | | }} { } |$C ji ◦ $Tr ji | i = 1, 2, . . . , f j | η = min ηI = min min | , |$C ji ◦ $Tr ji | j j i j = 1, 2, . . . , m max

(2.104)

where the range of η is from zero to unity. Step 4: On account of the influences of the input and output constraint performance on the whole constrainability, the total constraint index (TCoI) of a non-redundant actuated PM with over constraints in a certain configuration can be defined in the form as κ=

/ η ∗ ζ.

(2.105)

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2 Kinematic Performance Analysis and Optimization of Parallel …

A larger value of κ means a better motion/force constrainability of a non-redundant actuated PM with over constraints, which refers to the better effectiveness of internal constraint wrenches resisting external wrenches. Only when both the input and output constraint performances are good will the PM has better motion/force constrainability. In contrast, if one of them is close to zero, the TCoI will also be close to zero. For example, if the ICI and OCI in a certain configuration are 0.01 and 1, respectively, then the TCoI can be obtained as 0.1, which means the constraint performance of the non-redundant actuated PM with over constraints in this configuration is poor. As a result, the PM has poor motion/force constrainability as long as one of the ICI and OCI is bad. In addition, it can be found that the ICI, OCI, and TCoI based on the power coefficient are coordinate-free indices, which avoids the inconsistency of the units of the elements in the Jacobian matrix [14]. • Global constraint performance index The local constraint performance index proposed above is used to evaluate the motion/force constraint performance of an overconstrained PM without actuation redundancy under a certain configuration, rather than the entire workspace of PMs. For the design of overconstrained PMs without redundancy actuation, it is not enough to rely on the local performance index, and the global constraint performance index is also needed as a reference. Here, the motion/force constraint performance of PMs in a configuration is regarded as good when the value of κ is equal to or bigger than the given value, and these regions are referred to as the good constraint workspace (GCoW). The index can thus measure the global motion/force constrainability of a PM and is defined as { dW S , (2.106) σGCoW = {GCoW S dW where W is the reachable orientation workspace and SGCoW and S denote the areas of the GCoW and overall reachable workspace, respectively. Apparently, σGCoW ranges from zero to unity. The PM has better constraint performance when the σGCoW is closer to unity. It should be noted again that our study is totally different from the work of Liu et al. [27]. The constraint performance indices proposed by Liu et al. are mainly used for the non-redundant actuated PMs without over constraints, such as the 6PSS PM that is analyzed in Sect. 2.2.2, which is a PM without over constraints and actuation redundancy. In the following sections, the motion/force constraint indices based on the screw theory will be used to evaluate and optimize the performance of two overconstrained PMs without actuation redundancy, namely the 2PUR-PRU PM [24], and the 2PUR-PSR PM [25], and the effectiveness of the proposed constraint index will be verified.

2.4 Motion/Force Constraint Indices and Applications

93

2.4.2 Example 1: 2PUR-PRU PM • Local and global constraint performance Before evaluating the motion/force constraint performance of the 2PUR-PRU PM, it is necessary to obtain the TWSs and CWSs of limbs first. The TWSs of limbs have been listed in Eqs. (2.67) and (2.68). For the CWSs of each limb in the 2PUR-PRU PM, they can be derived by using the reciprocal screw theory, which can be expressed as { ( ) $C1 = 0 0 0; 0 sα cα (2.107) ) ( $C2 = 0 cα −sα; 0 q1 sα q1 cα { ( ) $C3 = 0 0 0; 0 sα cα (2.108) ) ( $C4 = 0 cα −sα; 0 −q2 sα −q2 cα and {

( ) $C5 = 0 0 0; 0 sα cα ). ( $C6 = 1 0 0; 0 z B3 + y B3 tanα 0

(2.109)

Based on these results, the OTSs of the 2PUR-PRU PM in different cases can be calculated. Substituting these screws into Eqs. (2.100)–(2.105), the TCoI of the 2PUR-PRU PM can be obtained. Taking the interference between the limbs of the 2PUR-PRU PM into account, the architectural parameters are set as follows: l1 = l2 = 130 mm, l3 = l4 = 340 mm, z o = 260 mm, α ∈ [−20◦ , 20◦ ], and β ∈ [−20◦ , 20◦ ]. The TCoI distribution of the 2PUR-PRU PM is shown in Fig. 2.33. It is clear that the distribution is symmetrical about the β = 0◦ , which is consistent with the structure of the 2PUR-PRU PM. Based on the global index defined in Eq. (2.106), the global constraint performance of the 2PUR-PRU PM can be derived. Here, the motion/force constraint performance of the 2PUR-PRU PM in a configuration is regarded as good when the value of TCoI is equal to or bigger than 0.7. According to the distributions in Fig. 2.33, it can be obtained that the ratio of the region where κ ≥ 0.7 for the 2PUR-PRU PM is approximately equal to unity, which means that the motion/force constraint capability with these architectural parameters is good. • Optimization of design parameters Although the motion/force constraint performance of the 2PUR-PRU PM in the workspace is great, the optimization of the design parameters of the 2PUR-PRU PM based on σGCoW is also needed for more different choices of design parameters. The PFNM method also can be used here, and the optimization process is similar to that of the motion/force transmission performances of the 2PUR-PRU PM. The distribution of σGCoW in the PDS is shown in Fig. 2.34, from which one can find that global

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2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.33 TCoI distribution of the 2PUR-PRU PM with original parameters

performance in the most region of the available area is equal to unity, which means that the designers can choose many combinations in the parameter design space for different applications. For example, the designers can choose the combination with s = 2.00 and t = 0.25. By substituting s and t into Eq. (2.74), the non-dimensional parameters can be obtained as r1 = 0.72, r2 = 0.28, and r3 = 2.00. Considering the actual operation environment, the normalized factor D is determined as 400 mm, and the values of li can then be obtained using Eq. (2.72), i.e., l1 = 288 mm, l2 = 112 mm, and l3 = 800 mm. The distribution of TCoI with the above parameters is shown in Fig. 2.35, from which one can find that the 2PUR-PRU PM has good constraint performance by using these parameters, and the value of σGCoW is equal to 1. Designers can choose different combinations of design parameters for the actual applications.

2.4.3 Example 2: 2PUR-PSR PM • Local and global constraint performance Using the same procedure as listed in the example of the 2PUR-PRU PM, the CWSs, and OTSs of the 2PUR-PSR PM also can be derived. For example, the CWSs of each limb in the 2PUR-PSR PM can be obtained as { ( ) $C1 = 0 0 0; sθ 0 cθ (2.110) ) ( $C2 = −cθ 0 sθ ; q1 sθ 0 q1 cθ

2.4 Motion/Force Constraint Indices and Applications

95

Fig. 2.34 Optimization regions of GCoW for the 2PUR-PRU PM

Fig. 2.35 TCoI distribution of the 2PUR-PRU PM with new parameters (l1 = 288 mm, l2 = 112 mm, and l3 = 800 mm)

{

( ) $C3 = 0 0 0; sθ 0 cθ ) ( $C4 = −cθ 0 sθ ; −q2 sθ 0 −q2 cθ

(2.111)

) ( $C5 = 0 cψ sψ; 0 −q3 sψ q3 cψ .

(2.112)

and

Based on these results, the OTSs of the 2PUR-PSR PM in different cases can be calculated. Substituting these screws into Eqs. (2.100)–(2.105), the TCoI of the

96

2 Kinematic Performance Analysis and Optimization of Parallel …

Fig. 2.36 TCoI distribution of the 2PUR-PSR PM with original parameters

2PUR-PSR PM can be obtained. Taking the interference between the limbs of the 2PUR-PSR PM into account, the architectural parameters are set as follows: l1 = 200 mm, l2 = 200 mm, l3 = 400 mm, l4 = 400 mm, z = 250 mm and the ranges of motion orientation are restricted as follows: ψ ∈ [−30◦ , 30◦ ] and θ ∈ [−30◦ , 30◦ ]. The TCoI distribution of the 2PUR-PSR PM is shown in Fig. 2.36. It is clear that the distribution is symmetrical about the ψ = 0◦ , which is consistent with the structure of the 2PUR-PSR PM. Based on the global index defined in Eq. (2.106), the global constraint performance of the 2PUR-PSR PM can be derived. According to the distributions in Fig. 2.36, it can be obtained that the ratio of the region where κ ≥ 0.7 for the 2PUR-PSR PM is approximately equal to 0.87, which means that the motion/force constraint capability with these architectural parameters is reasonably good. • Optimization of design parameters Based on the motion/force constraint performance, the optimization of the design parameters of the 2PUR-PSR PM based on σGCoW also can be presented. The PFNM method also can be used here, and the optimization process is the same as that of the motion/force transmission performances of the 2PUR-PSR PM. The distribution of σGCoW in the PDS is shown in Fig. 2.37, from which one can find that global performance in the most region of the available area is good, which means that the designers can choose many combinations in the parameter design space for different applications. For example, the designers can choose the combination with s = 2.50 and t = 0.05. By substituting s and t into Eq. (2.90), the non-dimensional parameters can be obtained as r1 = 0.29, r2 = 0.21, and r3 = 2.50. Considering the actual operation environment, the normalized factor D is determined as 300 mm, and the values of li can then be obtained using Eq. (2.88), i.e., l1 = 87 mm, l2 = 63 mm, and l3 = 750 mm. The distribution of TCoI with the above parameters is shown

2.5 Summary

97

Fig. 2.37 Optimization regions of GCoW for the 2PUR-PSR PM

Fig. 2.38 TCoI distribution of the 2PUR-PSR PM with new parameters (l1 = 87 mm, l2 = 63 mm, and l3 = 750 mm)

in Fig. 2.38, from which one can find that the 2PUR-PSR PM has good constraint performance (σGCoW = 1) by using these parameters.

2.5 Summary This chapter presents the kinematic performance analysis and optimization of PMs without actuation redundancy. Three different kinematic indices, including the condition number index, motion/force transmission index, and motion/force constraint

98

2 Kinematic Performance Analysis and Optimization of Parallel …

index, are used here. In these indices, the condition number and motion/force transmission indices are common and traditional indices, and the motion/force constraint indices are the new indices proposed in this chapter. In the proposed indices, by locking all m TWSs and (5 − m) linearly independent CWSs and “losing” (m + g − 5) CWSs, the non-redundant actuated PM with over constraints effectively becomes an instantaneous 1-DOF mechanism. The output constraint performance of these 1-DOF mechanisms is the instantaneous power of all the linearly independent wrenches in the (m + g − 5) wrenches that are devoted to motion. Different calculation processes are used for different numbers k of linearly independent wrenches. After that, the output motion/force constrainability of a PM can be described with the mean value of instantaneous power of these 1-DOF mechanisms. Based on the proposed OCI and TCoI, the global index, GCoW, is established. Based on the analysis and dimension optimizations of PMs, it can be found that these types of indices can be effectively used for the kinematic performance evaluation and optimization and also provide important references for the actual applications of PMs.

References 1. R.S. Stoughton, T. Arai, A modified Stewart platform manipulator with improved dexterity. IEEE Trans. Robot. Autom. 9(2), 166–173 (1993) 2. T. Yoshikawa, Manipulability of robotic mechanisms. Int. J. Robot. Res. 4(2), 3–9 (1985) 3. J. Angeles, C.S. López-Cajún, Kinematic isotropy and the conditioning index of serial robotic manipulators. Int. J. Robot. Res. 11(6), 560–571 (1992) 4. X.J. Liu, L.P. Wang, F.G. Xie et al., Design of a three-axis articulated tool head with parallel kinematics achieving desired motion/force transmission characteristics. J. Manuf. Sci. Eng. 132(2), 021009 (2010) 5. O. Ma, J. Angeles, Optimum architecture design of platform manipulators, in The fifth International Conference on Advanced Robotics (1991), pp. 1130–1135. 6. J. Angeles, The design of isotropic manipulator architectures in the presence of redundancies. Int. J. Robot. Res. 11(3), 196–201 (1992) 7. C. Gosselin, Dexterity indices for planar and spatial robotic manipulators, in Proceedings of IEEE International Conference on Robotics and Automation (1990), pp. 650–655. 8. S.-G. Kim, J. Ryu, New dimensionally homogeneous Jacobian matrix formulation by three end-effector points for optimal design of parallel manipulators. IEEE Trans. Robot. Autom. 19(4), 731–736 (2003) 9. F.G. Xie, X.J. Liu, J.S. Wang, A 3-DOF parallel manufacturing module and its kinematic optimization. Robot. Comput.-Integr. Manuf. 28(3), 334–343 (2012) 10. F.G. Xie, X.J. Liu, J.S. Wang et al., Kinematic optimization of a five degrees-of-freedom spatial parallel mechanism with large orientational workspace. J. Mech. Robot. 9(5), 051005 (2017) 11. J.S. Wang, C. Wu, X.J. Liu, Performance evaluation of parallel manipulators: motion/force transmissibility and its index. Mech. Mach. Theory 45(10), 1462–1476 (2010) 12. Z. Huang, Q.C. Li, General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. Int. J. Robot. Res. 21(2), 131–145 (2002) 13. Z. Huang, Q.C. Li, Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method. Int. J. Robot. Res. 22(1), 59–79 (2003) 14. J.P. Merlet, Jacobian, manipulability, condition number and accuracy of parallel robots. J. Mech. Des. 128(1), 199–206 (2006)

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15. C. Gosselin, J. Angeles, A global performance index for the kinematic optimization of robotic manipulators. J. Mech. Des. 113(3), 220–226 (1991) 16. L.M. Xu, Q.H. Chen, L.Y. He et al., Kinematic analysis and design of a novel 3T1R 2-(PRR)2RH hybrid manipulator. Mech. Mach. Theory 112, 105–122 (2017) 17. L.M. Xu, Q.H. Chen, J.H. Tong et al., Dimensional synthesis of a 2-PRS-PRRU parallel manipulator, in IFToMM Asian conference on Mechanism and Machine Science and International Conference on Mechanism and Machine Science (2017), pp. 341–355 18. X.J. Liu, J.S. Wang, A new methodology for optimal kinematic design of parallel mechanisms. Mech. Mach. Theory 42(9), 1210–1224 (2007) 19. M. Stock, K. Miller, Optimal kinematic design of spatial parallel manipulators: application to linear delta robot. J. Mech. Des. 125(2), 292–301 (2003) 20. R. Jeha, C. Jongeun, Volumetric error analysis and architecture optimization for accuracy of HexaSlide type parallel manipulators. Mech. Mach. Theory 38(3), 227–240 (2003) 21. Q.C. Li, J.M. Hervé, Type synthesis of 3-DOF RPR-equivalent parallel mechanisms. IEEE Trans. Rob. 30(6), 1333–1343 (2014) 22. Q.C. Li, X.D. Sun, Q.H. Chen, et al. Kinematics and singularity analysis of 2-PRS-PRRU parallel mechanism. J. Mechan. Eng. 47(3), 21–27 (2011) 23. D.C. Tao, Applied linkage synthesis (Addison-Wesley Publishing Company, Reading, Massachusetts, 1964) 24. L.M. Xu, Q.C. Li, J.H. Tong et al., Tex3: an 2R1T parallel manipulator with minimum DOF of joints and fixed linear actuators. Int. J. Precis. Eng. Manuf. 19(2), 227–238 (2018) 25. W.Z. Zhang, L.M. Xu, J.H. Tong et al., Kinematic analysis and dimensional synthesis of 2-PUR-PSR parallel manipulator. Journal of Mechanical Engineering 54(7), 45–53 (2018) 26. X.J. Liu, C. Wu, J.S. Wang, A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators. J. Mech. Robot. 4(4), 041001 (2012) 27. X.J. Liu, X. Chen, M. Nahon, Motion/Force constrainability analysis of lower-mobility parallel manipulators. J. Mech. Robot. 6(3), 031006 (2014) 28. Q.Z. Meng, F.G. Xie, X.J. Liu, Motion-force interaction performance analyses of redundantly actuated and overconstrained parallel robots with closed-loop subchains. J. Mech. Des. 142(10), 103304 (2020) 29. C. Han, J. Kim, J. Kim et al., Kinematic sensitivity analysis of the 3-UPU parallel mechanism. Mech. Mach. Theory 37(8), 787–798 (2002) 30. D. Zlatanov, I.A. Bonev, C. Gosselin, Constraint singularities of parallel mechanisms, in Proceedings of the 2002 IEEE International Conference on Robotics and Automation, 496–502 (2002) 31. T. Huang, H.T. Liu, D.G. Chetwynd, Generalized Jacobian analysis of lower mobility manipulators. Mech. Mach. Theory 46(6), 831–844 (2011) 32. J. Brinker, B. Corves, Y. Takeda, Kinematic performance evaluation of high-speed Delta parallel robots based on motion/force transmission indices. Mech. Mach. Theory 125, 111–125 (2018) 33. Q.Z. Meng, F.G. Xie, X.J. Liu et al., An evaluation approach for motion-force interaction performance of parallel manipulators with closed-loop passive limbs. Mech. Mach. Theory 149, 103844 (2020) 34. M.J. Tsai, H.W. Lee, Generalized evaluation for the transmission performance of mechanisms. Mech. Mach. Theory 29(4), 607–618 (1994)

Chapter 3

Motion/Force Transmission Performance Analysis and Optimization of Parallel Manipulators with Actuation Redundancy

Over the last few decades, PMs with actuation redundancy have attracted increasing attention in both academia and industry [1–7]. When compared with PMs with non-redundant actuation, PMs with actuation redundancy offer several advantages, including the elimination of singularities and improved stiffness and dexterity [6–10]. There are two categories of PMs with actuation redundancy [6, 7]: the replacement of passive joints with active joints and the addition of active limbs to the PM without changing its mobility. Many studies have focused on the latter category because it offers the advantages of higher stiffness and improved force distribution. This chapter also focuses on the latter type of redundantly actuated PM. Performance evaluation and optimal design are essential for PMs with actuation redundancy. Considering the inconsistencies of Jacobian-based indices when applied to PMs with mixed-DOF [11], the motion/force transmission index that can quantify the transmissibility between the actuators and the moving platform is used in this chapter. A local minimized transmission index (LMTI) [12–15] for transmission evaluation of redundantly actuated PMs has been proposed that separated the redundantly actuated PM into a set of non-redundant PMs by removing the redundant limbs from the redundantly actuated PM in turn. Despite the coordinate independence of this index, the influences of the redundant actuator and limb are not considered. In this chapter, a new index proposed by the authors will be used for the performance evaluation and dimensional optimization of redundantly actuated PMs [16]. First, by locking some actuators in an ergodic manner, the targeted redundantly actuated PM is separated into several subsidiary 1-DOF PMs that are actuated by two or more actuators. Then, the index of output transmission performance is proposed by calculating the mean value of the instantaneous power produced by these 1-DOF PMs. Finally, the LTI is defined as the minimum value of the index of output and input transmission performance. The LTI is coordinate-free and dimensionless. Compared with other indices, this one considers the influences of redundant actuators and limbs and describes the intrinsic characteristics of a redundantly actuated PM more precisely.

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_3

101

102

3 Motion/Force Transmission Performance Analysis and Optimization …

In this chapter, the motion/force transmission performance analysis and optimization of PMs with actuation redundancy will be presented. Three PMs are taken as examples here for analysis and discussion.

3.1 Motion/Force Transmission Indices of Parallel Manipulators with Actuation Redundancy Indices for the evaluation of the transmissibility of the redundantly actuated PMs should be coordinate-free and have a clear physical interpretation. Since this type of PM has interference from different actuators, the one-to-one corresponding transmission between the input actuation twist and output twist of the moving platform disappears. In the following, an evaluation approach and a transmission index are proposed to solve this problem [16]. • Motion/force transmission of a 1-DOF four-bar mechanism with two actuators A 1-DOF planar four-bar linkage mechanism is taken as an example first to illustrate the proposed index, as shown in Fig. 3.1. The mechanism has 1-DOF and is redundantly actuated by two motors fixed on the base. Let the two actuation wrenches be denoted as $T1 and $T2 . Let the motion of the output link be represented by a twist $O . The redundantly actuated mechanism can be regarded as a mechanism consisting of two limbs, a moving platform, and a fixed base. The limb twist systems of the mechanism are written as Fig. 3.1 Redundantly actuated four-bar mechanism

3.1 Motion/Force Transmission Indices of Parallel Manipulators …

⎧ ⎪ ⎪ $11 ⎪ ⎨$ 21 ⎪ $12 ⎪ ⎪ ⎩$ 22

103

= ( 0 0 1 0 0 0 )T = ( 0 0 1 l1 sθ1 −l1 cθ1 0 )T . = ( 0 0 1 0 −l0 0 )T = ( 0 0 1 l2 sθ2 −l0 − l2 cθ2 0 )T

(3.1)

Based on Eqs. (2.92)–(2.95), the TWS of each limb and CWSs of this system can be expressed as {

$T1 = ( 0 1 0 0 0 l1 cθ1 )T $T2 = ( 0 1 0 0 0 l0 + l2 cθ2 )T

,

(3.2)

and ⎛

10000

⎜ ⎜0 ⎜ UC = ⎜ ⎜0 ⎜ ⎝0 0

1000

l0 sθ1 sθ2 sθ1 cθ2 −cθ1 sθ2 −l0 cθ1 sθ2 sθ1 cθ2 −cθ1 sθ2

01000 00100 00010

⎞T ⎟ ⎟ ⎟ ⎟ . ⎟ ⎟ ⎠

(3.3)

Based on the reciprocity between twists and wrenches, we have the output twist as ( $O = 0 0 1

−l0 sθ1 sθ2 l0 cθ1 sθ2 sθ1 cθ2 −cθ1 sθ2 sθ1 cθ2 −cθ1 sθ2

0

)T

,

(3.4)

which indicates the moving platform has an instantaneous rotation around an axis pointing along the paper outward. The output motion/force transmissibility of the four-bar linkage mechanism can be described by the instantaneous power coefficient that $T1 and $T2 act on $O , i.e., | | | | |$O ◦ $T1 | + |$O ◦ $T2 | | | | | η= |$O ◦ $T1 | + |$O ◦ $T2 | max

,

(3.5)

max

where $T1 , $T2 , and $O are normalized. Assume that the planar four-bar linkage mechanism is completely rigid and perfectly manufactured. When a perfect motion/force transmission occurs, the value of η should be 1, that is, energy is transmitted from the actuators to the moving platform without loss. However, even in such an idealized situation, the actual value of η is always less than 1 due to internal forces caused by redundant actuation. Hence, η can be used as an index for performance analysis and optimal design for the planar four-bar linkage mechanism. The planar four-bar linkage mechanism is a 1-DOF single-loop mechanism while a PM is a multi-DOF mechanism with n loops, n being an integer. Thus, we can

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3 Motion/Force Transmission Performance Analysis and Optimization …

separate a redundantly actuated PM with an n-loop into several 1-DOF mechanisms. Evaluating motion/force transmission of the redundantly actuated PM is then reduced to evaluating motion/force transmission of these subsidiary 1-DOF mechanisms. This section provides an explicit approach to generating the 1-DOF mechanisms. It is worth mentioning that this idea is inspired by Liu’s work [17] in which he separated an n-DOF non-redundantly actuated PM into n 1-DOF non-redundantly actuated PMs to calculate the local motion/force transmission. • Local transmission index for redundantly actuated PMs In a non-singular configuration, a six-order wrench system is exerted on the moving platform of a PM after all actuators are locked. The six-order wrench system can be divided into two parts: the m-order transmission wrench system providing the actuation force and the 6-m order constraint wrench system restraining its motion. Although the elements in a single system may be linearly dependent in the redundant actuation case or over constrained case, those from different systems are linearly independent of each other [18]. In addition, any selected m-1 transmission wrenches are linearly independent of each other in the transmission wrench system. This section focuses on the redundantly actuated PMs that work in non-singular configurations. When a PM works in a singular configuration, the linear dependency varies from case to case and should be discussed specifically. When the actuators installed in any m − 1 limbs are locked, the corresponding transmission wrenches that form a (m − 1)-order system turn into extra constraint wrenches of the moving platform. Thus, the moving platform is subjected to (6 − m + m − 1 = 5) independent constraint wrenches. In other words, there is only an instantaneous 1-DOF twist that is redundantly actuated by the other n − m + 1 limbs. For the redundantly actuated PM with n limbs, there are N = Cnm−1 combinations of selection of the locked m − 1 limbs. Let g denote a combination case. The overall constraint wrenches exerted on the moving platform after locking m-1 actuators consist of two parts } [ U g = U C U Tg ,

(3.6)

where } [ g U T = υ1g $T1 υ2g $T2 · · · υng $Tn (g = 1, 2, · · · , N ), and { g υi

=

0 actuated limb . 1 locked limb

The instantaneous 1-DOF twist of the mobile platform is denoted as $Og , which is obtained by using reciprocity between twists and wrenches

3.1 Motion/Force Transmission Indices of Parallel Manipulators …

$Og ◦ U g = 0.

105

(3.7)

Since only the transmission wrenches of the active limb do work in the direction of the output twist and the locked limbs contribute no active transmission wrenches to it, the output transmission performance index of the ith limb reflecting the power coefficient between the ith transmission wrench and the output twist can be written as | Σm g | | | i=1 τi $Og ◦ $Ti g | , (3.8) η = Σm g | τ |$ ◦ $Ti | i=1 i

Og

max

{

1 actuated limb . 0 locked limb The output transmission performance of the moving platform is the coupled effect of the n − m + 1 active limbs. Due to the redundantly actuated limb(s), the poor output transmission coefficient of an active limb may be compensated by the high output transmission coefficient of other active limbs. Since the locked limbs are randomly selected from n limbs, we have to study N = Cnm−1 cases. One can imagine that when the locked limbs are released one by one until all the actuators are activated, the output transmission coefficient of the PM will neither exceed the maximum nor the minimum of those of the 1-DOF mechanisms. Thus, the output transmission performance of the redundantly actuated PM is given by g

where τi =

η=

N 1 Σ g η . N g=1

(3.9)

On the other hand, the input transmission performance of each limb is evaluated by the index | | |$Ai ◦ $Ti | | | λi = (i= 1, 2, . . . , n), λ = min{λ1 ,λ2 , . . . , λn }, |$Ai ◦ $Ti | max

(3.10)

where $Ai denotes the twist of the actuated joint of the ith limb and λ the input transmission performance. Since the motion/force transmission takes both the input and output transmission performance into consideration, the LTI for a redundantly actuated PM is defined as Γ = min{λ, η}

(3.11)

A flowchart that describes the procedure for calculating this LTI is presented in Fig. 3.2. It should be noted that some kinematic performance indices, such as manipulability [19, 20] and dexterity [21], are to measure the sensitivity of the response of

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3 Motion/Force Transmission Performance Analysis and Optimization …

the end effector to the actuators and have to be used very carefully when concerning PMs with both rotational and translational DOFs due to the inconsistency of the units of the elements in a Jacobian matrix. Otherwise, erroneous interpretations may be obtained [11]. To overcome this, some researchers also proposed an LTI-based evaluation method [17, 22]. In this method, a redundantly actuated PM was converted into a group of non-redundantly actuated PMs with equal kinematic constraints, and the LTI of the target PM was regarded to be the maximum value of the LTI of all the non-redundantly actuated PMs. However, the effect generated by internal forces caused by redundant actuations is not considered during this conversion. • Global transmission index for redundantly actuated PMs Since the proposed LTI represents the motion/force transmission performance in a single configuration, it cannot be used to evaluate the global transmission performance of the manipulator. It is therefore necessary to define an index that can describe the manipulator’s performance in a set of poses. Here, the motion/force transmission performance of PMs in a configuration is regarded as good when the value of Γ is equal to or bigger than the given value, and these regions are referred to as the good transmission workspace (GTW). The index can thus measure the global motion/force transmissibility of a PM and is defined as { S σ = {G S

dW dW

,

(3.12)

where W is the reachable orientation workspace and S G and S denote the areas of the GTW and overall reachable workspace, respectively. Here, the overall reachable workspace is defined as the curved surface at an operating height with given rotational ranges. Apparently, σ ranges from zero to unity. The PM has better transmission performance when the σ is closer to unity. In the following sections, the motion/force transmission indices based on the screw theory will be used to evaluate and optimize the kinematic performance of three PMs, namely the 6PSS-UPS PM, the 2UPR-2PRU PM [23], and the 2PUR2PRU PM [24], and the effectiveness of the proposed transmission index will be verified.

3.2 Example 1: 6PSS-UPS PM • Local and global transmission performances For the 6PSS-UPS PM, as shown in the CAD model in Fig. 3.3, which is similar to the structure of the 6PSS PM in Chap. 2. The main difference between these two PMs is the seventh redundant actuated limb UPS, which connects to the origin point of the moving coordinate frame by using the S joint and connects to the origin point of the fixed coordinate frame by using the U joint. Therefore, the structure description

3.2 Example 1: 6PSS-UPS PM

107

Fig. 3.2 Procedure of evaluating a general redundantly actuated PM

and inverse kinematics of the 6PSS-UPS are similar to the 6PSS, and they are omitted here. Before evaluating the motion/force transmission performance of the 6PSS-UPS PM, it is necessary to obtain the TWSs and OTSs first, and then calculate the LTI of this PM. The procedure is described as follows: Step 1: Determine the TWS in each limb of the 6PSS-UPS PM. Without loss of generality, limb 1 is used as an example. The TWS should be linearly independent of passive twists and constraint wrenches simultaneously [25]. Since the actuated joint

108

3 Motion/Force Transmission Performance Analysis and Optimization …

Fig. 3.3 6PSS-UPS PM

of limb 1 is a P joint, the TWS of limb 1 is reciprocal of the passive joints and is given by ( / ) / $T1 = l 1 |l 1 | a1 × l 1 |l 1 | .

(3.13)

Similarly, the TWSs of the other six limbs can be derived as ⎧ $T2 ⎪ ⎪ ⎪ ⎪ ⎪ $T3 ⎪ ⎪ ⎨$ T4 ⎪ $T5 ⎪ ⎪ ⎪ ⎪ $T6 ⎪ ⎪ ⎩ $T7

( / ) / = l 2 |l 2 | a2 × l 2 |l 2 | ( / ) / = l 3 |l 3 | a3 × l 3 |l 3 | ( / ) / = l 4 |l 4 | a4 × l 4 |l 4 | ( / ). / = l 5 |l 5 | a5 × l 5 |l 5 | ( / ) / = l 6 |l 6 | a6 × l 6 |l 6 | ( / ) = p | p| 0

(3.14)

Step 2: List the locked/active TWSs and form a new wrench system for the 6PSSUPS PM, from which the OTS is then obtained. For the 6PSS-UPS PM, it is a 6-DOF redundantly actuated PM with seven limbs. When actuators that are installed in five arbitrarily chosen limbs are locked, the corresponding TWSs form a five-order system and turn into extra constraint wrenches for the moving platform, and it sustains five independent constraint wrenches in total. Using the screw theory, it can be shown that there is only one instantaneous 1-DOF twist that is actuated redundantly by two limbs. For the 6PSS-UPS PM, there are N = C72 = 21 combinations that can be used to select the two locked limbs. The corresponding locked/active limb combinations are listed in Table 3.1. The PM of case 1 is selected as an example here. As listed in Table 3.1, the new wrench system acting on the moving platform after locking two actuators can be written as } [ U 1 = $T3 $T4 $T5 $T6 $T7 .

(3.15)

Accordingly, the instantaneous 1-DOF twist of the moving platform, which is denoted by $O1 , can be obtained using reciprocal screw theory

3.2 Example 1: 6PSS-UPS PM

109

Table 3.1 Locked/active limb combinations of the 6PSS-UPS PM Case (g)

Locked limbs

Active limbs

1

3, 4, 5, 6, 7

1, 2

2

2, 4, 5, 6, 7

1, 3

3

2, 3, 5, 6, 7

1, 4

4

2, 3, 4, 6, 7

1, 5

5

2, 3, 4, 5, 7

1, 6

6

2, 3, 4, 5, 6

1, 7

7

1, 4, 5, 6, 7

2, 3

8

1, 3, 5, 6, 7

2, 4

9

1, 3, 4, 6, 7

2, 5

10

1, 3, 4, 5, 7

2, 6

11

1, 3, 4, 5, 6

2, 7

12

1, 2, 5, 6, 7

3, 4

13

1, 2, 4, 6, 7

3, 5

14

1, 2, 4, 5, 7

3, 6

15

1, 2, 4, 5, 6

3, 7

16

1, 2, 3, 6, 7

4, 5

17

1, 2, 3, 5, 7

4, 6

18

1, 2, 3, 5, 6

4, 7

19

1, 2, 3, 4, 7

5, 6

20

1, 2, 3, 4, 6

5, 7

21

1, 2, 3, 4, 5

6, 7

Wrench system of moving platform ] [ U 1 = $T3 $T 4 $T 5 $T 6 $T 7 [ ] U 2 = $T 2 $T 4 $T 5 $T 6 $T 7 ] [ U 3 = $T 2 $T 3 $T 5 $T 6 $T 7 ] [ U 4 = $T 2 $T 3 $T 4 $T 6 $T 7 [ ] U 5 = $T 2 $T 3 $T 4 $T 5 $T 7 ] [ U 6 = $T 2 $T 3 $T 4 $T 5 $T 6 ] [ U 7 = $T 1 $T 4 $T 5 $T 6 $T 7 [ ] U 8 = $T 1 $T 3 $T 5 $T 6 $T 7 ] [ U 9 = $T 1 $T 3 $T 4 $T 6 $T 7 ] [ U 10 = $T 1 $T 3 $T 4 $T 5 $T 7 [ ] U 11 = $T 1 $T 3 $T 4 $T 5 $T 6 ] [ U 12 = $T 1 $T 2 $T 5 $T 6 $T 7 ] [ U 13 = $T 1 $T 2 $T 4 $T 6 $T 7 [ ] U 14 = $T 1 $T 2 $T 4 $T 5 $T 7 ] [ U 15 = $T 1 $T 2 $T 4 $T 5 $T 6 ] [ U 16 = $T 1 $T 2 $T 3 $T 6 $T 7 [ ] U 17 = $T 1 $T 2 $T 3 $T 5 $T 7 ] [ U 18 = $T 1 $T 2 $T 3 $T 5 $T 6 ] [ U 19 = $T 1 $T 2 $T 3 $T 4 $T 7 [ ] U 20 = $T 1 $T 2 $T 3 $T 4 $T 6 ] [ U 21 = $T 1 $T 2 $T 3 $T 4 $T 5

$O1 ◦ U 1 = 0.

(3.16)

There are 21 cases for the 6PSS-UPS PM. If these cases have been completed, proceed to Step 3; otherwise, return to the start of Step 2 and select the next case.

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3 Motion/Force Transmission Performance Analysis and Optimization …

Step 3: Calculate the OTI and ITI. Because only the transmission wrenches of the active limbs contribute in the output twist direction, the output transmission performance of case 1, which reflects the power coefficient between the active transmission wrenches and the output twist, can be written as | | | | |$ ◦ $T1 | + |$ ◦ $T2 | O1 O1 | | | . η =| |$ ◦ $T1 | + |$ ◦ $T2 | O1 O1 max max 1

(3.17)

In each case, the output transmissibility of the 1-DOF mechanism is the coupled effect of the two active limbs. In a redundantly actuated PM, the output transmission performance will not exceed either the maximum or minimum performance values of the 1-DOF mechanisms. Thus, when considering all cases, the OTI of the 6PSS-UPS redundantly actuated PM in this study is defined as 1 Σ g η . 21 g=1 21

η=

(3.18)

Additionally, the ITI of the ith limb (i = 1, 2, 3, 4, 5, 6, 7) can be obtained using Eq. (3.10). Step 4: Obtain the LTI for a specific configuration. Taking both the input and output transmission performances into account, the LTI for the 6PSS-UPS PM is defined as Γ = min{λi , η}.

(3.19)

For the 6PSS-UPS PM, the LTI in the different positions and orientations is discussed separately here. In this section, the architecture parameters used here are the same of those in the 6PSS PM, and set as follows: Rm = 250 mm, l = 440 mm, lb = 120 mm, and θm = 30◦ . The LTI distributions in the three-dimensional position workspace are presented first, and are shown in Fig. 3.4(a), in which the ranges of positions and orientations are set as −50 mm ≤ x ≤ 50 mm, −50 mm ≤ y ≤ 50 mm, 225 mm ≤ z ≤ 275 mm, and α = β = γ = 0°. For clarity, the LTI distributions when z = 250 mm are also presented, as shown in Fig. 3.4b, from which it is obvious that the distribution is symmetrical about the plane x = 0, and the kinematic performances near the central region are better than others. Similarly, the LTI distributions in the three-dimensional orientation workspace are also presented in Fig. 3.5a, in which the ranges of positions and orientations are set as x = y = z = 0, −20° ≤ α ≤ −20°, −20° ≤ β ≤ −20°, and −20° ≤ γ ≤ −20°. The LTI distributions when γ = 0° are also presented, as shown in Fig. 3.5b, from which it is obvious that the distribution is symmetrical about the plane β = 0°, and the kinematic performances near the central region are better than others. Here, the motion/force transmission performance of the 6PSS-UPS redundantly actuated PM in a configuration is regarded as good when the value of LTI is equal to or bigger than 0.7. According to the results in Figs. 3.4a and 3.5a, the GTW in the

3.2 Example 1: 6PSS-UPS PM

111

Fig. 3.4 LTI distribution of the 6PSS-UPS PM in the three-dimensional position workspace: a three-dimensional position distribution and b xy plane distribution when z = 250 mm

Fig. 3.5 LTI distribution of the 6PSS-UPS PM in the three-dimensional orientation workspace: a three-dimensional orientation distribution and b αβ plane distribution when γ = 0°

position region of the 6PSS-UPS PM, σ , is approximately 0.8965, and the GTW in the orientation region of the 6PSS-UPS PM, σ , is approximately 0.1292. The above results mean that in the position workspace, the motion/force transmission capability with these architectural parameters is reasonably good. However, in the orientation workspace, the global performance of the 6PSS-UPS PM with these given parameters is not ideal. • Optimization of design parameters Similar to the above optimization of the 6PSS PM with respect to the GTW, the iterative search method is also used here for the dimensional optimization of the 6PSSUPS redundantly actuated PM with respect to the GTW, and the Rm and l are also defined as design parameters here. In addition, the variation ranges of two design parameters are, respectively, defined as 230 mm ≤ Rm ≤ 270 mm, 420 mm ≤ l ≤ 460 mm. Similar to the above section, the position ranges are set as −50 mm ≤ x ≤

112

3 Motion/Force Transmission Performance Analysis and Optimization …

50 mm, −50 mm ≤ y ≤ 50 mm, and 225 mm ≤ z ≤ 275 mm, and the orientation ranges are set as −20° ≤ α ≤ −20°, −20° ≤ β ≤ −20°, and −20° ≤ γ ≤ −20°. Through iterative search and calculation, the distributions of GTW with different combinations of design parameters can be obtained, as shown in Figs. 3.6a, b. For the 6PSS-UPS PM, although the distributions of GTW in the position workspace are different from those in the orientation workspace, the maximum value of GTW occurs with the same combination of design parameters, i.e., Rm = 230 mm, l = 460 mm. In the case of optimization regarding position workspace, the maximum of GTW is unity and occurs when Rm = 230 mm, l = 460 mm, as shown in Fig. 3.7a. In the case of optimization regarding orientation workspace, the maximum of GTW occurs when Rm = 230 mm, l = 460 mm, as shown in Fig. 3.7b, and the value of GTW is 0.4966. Designers can choose one of these combinations for different applications.

Fig. 3.6 GTW distributions with different parameter combinations: a optimization regarding position workspace and b optimization regarding orientation workspace

Fig. 3.7 Performance distributions of the 6PSS-UPS PM with optimized design parameters: a three-dimensional position distribution with Rm = 230 mm and l = 460 mm and b three-dimensional orientation distribution with Rm = 230 mm and l = 460 mm

3.3 Example 2: 2UPR-2PRU PM

113

3.3 Example 2: 2UPR-2PRU PM • Structure description The CAD model and a schematic of the 2UPR-2PRU PM with actuation redundancy are shown in Fig. 3.8, which is a 2R1T PM [23]. The PM is composed of a fixed base, a moving platform, and 4-DOF actuated limbs that are arranged symmetrically. The first limb (B1 A1 ) and the second limb (B2 A2 ) are UPR kinematic limbs. Counting from the base, the first revolute axes of the U joints of the two limbs coincide with each other. The second revolute axes of the U joints are parallel to the axis of the R joint that is connected to the mobile platform. Additionally, the R joint axis is perpendicular to the P joint. The third and fourth limbs (B3 A3 and B4 A4 , respectively) are PRU kinematic limbs. The P joints are perpendicular to the fixed base. The R-joint axes in limbs 3 and 4 are perpendicular to the P joints and are also parallel to the first revolute axes of the U joints in limbs 3 and 4. The second revolute axes of the two U joints of limbs 3 and 4, which are connected to the moving platform, coincide with each other. The A1 and A2 denote the central points of the R joints in limbs 1 and 2, respectively, while the central points of the U joints in limbs 3 and 4 are denoted by A3 and A4 , respectively. The centers of the U joints in limbs 1 and 2 are denoted by B1 and B2 , respectively, while the centers of the R joints in limbs 3 and 4 are denoted by B3 and B4 , respectively. The sliders of limbs 3 and 4 intersect with the fixed base at points B3, and B4, , respectively, as shown in Fig. 3.8. The coordinate frames are established as shown in Fig. 3.8a. A fixed coordinate frame, O-xyz, is attached to the fixed base, and the origin O is located at the midpoint of line B1 B2 . The x-axis always points in the direction of O B4, , the y-axis points along O B2 , and the z-axis points vertically downwards. A moving coordinate frame, o-uvw, is attached to the moving platform. The u-axis points in the direction of o A4 , and the v-axis points along o A2 . The w-axis points vertically downward with

Fig. 3.8 2UPR-2PRU PM with actuation redundancy: a CAD model and b schematic representation

114

3 Motion/Force Transmission Performance Analysis and Optimization …

respect to the moving platform. Additionally, limb coordinate frames Bi − xi yi z i (i = 1, 2, 3, 4) are attached to each limb i. For B1 − x1 y1 z 1 and B2 − x2 y2 z 2 , the x- and y-axes always point along the revolute axes of the U joints in limbs 1 and 2, respectively. For B3 − x3 y3 z 3 and B4 − x4 y4 z 4 , the coordinate frame axes have the same directions as those of O-xyz. The linkage lengths in the 2UPR-2PRU PM are defined as follows: B3 A3 = B4 A4 = l, oo, = H , o A1 = o A2 = l1 , O B1 = O B2 = l2 , o A3 = o A4 = l3 , O B3, = O B4, = l4 and l2 · l3 = l1 · l4 . The coordinates of points A1 , A2 , A3 , and A4 with respect to O-xyz are defined as ( )T ( )T ( )T ( )T x A1 y A1 z A1 , x A2 y A2 z A2 , x A3 y A3 z A3 , and x A4 y A4 z A4 , respectively. )T ( )T ( The coordinates of points B1 , B2 , B3, , and B4, are defined as 0 −l2 0 , 0 l2 0 , )T )T ( ( −l4 0 0 , and l4 0 0 , respectively. The coordinates of B3 and B4 are defined ( )T ( )T as −l4 0 q3 and l4 0 q4 , respectively, where q3 and q4 denote the distances between the fixed base and the actuated P joints of limbs 3 and 4, respectively. In Fig. 3.9, it is shown that the 2UPR-2PRU PM can be used to machine a workpiece with a curved surface when mounted on an x–y table. • Inverse kinematics The inverse displacement solution for the 2UPR-2PRU PM requires calculation of the actuated joint parameters (q1 , q2 , q3 , q4 ) when the position and orientation parameters (β, γ , z o ) of the moving platform are given. The rotation matrix between o-uvw and O-xyz can be written as ⎛

⎞ cβ sβsγ sβcγ O Ro = ⎝ 0 cγ −sγ ⎠. −sβ cβsγ cβcγ Fig. 3.9 Application of the 2UPR-2PRU PM

(3.20)

3.3 Example 2: 2UPR-2PRU PM

115

where β and γ denote the rotation angles around the y- and u-axes, respectively. The position vectors of points Ai (i = 1, 2, 3, 4) with respect to the moving and fixed coordinate frames are denoted by o ai and O ai , respectively, and are given by ⎧ o ⎪ a1 ⎪ ⎪ ⎪ ⎪ ⎪ o ⎨ a2 o ⎪ ⎪ a3 ⎪ ⎪ ⎪ ⎪ ⎩ oa 4

)T ( = 0 −l1 0 )T ( = 0 l1 0 )T , ( = −l3 0 0 )T ( = l3 0 0

(3.21)

and O

ai =

O

Ro o ai + P i = 1, 2, 3, 4,

(3.22)

)T ( where the vector P = z o tan β 0 z o denotes the position vector of a point o with respect to O-xyz. Additionally, O ai can be obtained using the rotation matrix O Ri , which is defined as a transformation mapping of Bi − xi yi z i relative to O-xyz, and is given by ⎛

⎞ cβ 0 sβ O R1 = O R2 = ⎝ 0 1 0 ⎠, −sβ 0 cβ

⎛

⎞ 100 O R3 = O R4 = ⎝ 0 1 0 ⎠. 001

(3.23)

The origin Bi of O Ri relative to O-xyz is defined as ⎧ )T ( ⎪ p1 = 0 −l2 0 ⎪ ⎪ )T ⎪ ⎨p =( 0l 0 2 )T . ( 2 ⎪ p3 = −l4 0 q3 ⎪ ⎪ )T ⎪ ⎩p =( l 4 0 q4 4

(3.24)

The position vectors of points Ai (i = 1, 2, 3, 4) when expressed in Bi − xi yi z i are given by ⎧ 1 ⎪ a1 ⎪ ⎪ ⎪ ⎨ 2a 2 3 ⎪ a ⎪ 3 ⎪ ⎪ ⎩ 4a 4

)T ( = 0 q1 cθ1 q1 sθ1 ( )T = 0 q2 cθ2 q2 sθ2 )T , ( = lcθ3 0 lsθ3 )T ( = lcθ4 0 lsθ4

(3.25)

where θi (i = 1, 2, 3, 4) denotes the angle between the direction of vector Bi Ai and the fixed base. Alternatively, the position vectors of points Ai (i = 1, 2, 3, 4) can be written in the form

116

3 Motion/Force Transmission Performance Analysis and Optimization … O

ai =

O

Ri i ai + pi i = 1, 2, 3, 4.

(3.26)

Because Eqs. (3.22) and (3.26) are two different representations of the same position vector, the following can be obtained based on the geometrical conditions ⎧ q1 sθ1 = a1 , q1 cθ1 = c1 ⎪ ⎪ ⎨ q2 sθ2 = a2 , q2 cθ2 = c2 , ⎪ lsθ3 = a3 , lcθ3 = c3 ⎪ ⎩ lsθ4 = a4 , lcθ4 = c4

(3.27)

where ⎧ ⎪ a ⎪ ⎪ 1 ⎨ a2 ⎪ a 3 ⎪ ⎪ ⎩ a4

= z o sec β − l1 sγ , c1 = l2 − l1 cγ =/ z o sec β + l1 sγ , c2 = l1 cγ − l2 . = /l 2 − (z o tan β + l4 − l3 cβ)2 , c3 = z o tan β + l4 − l3 cβ = l 2 − (z o tan β − l4 + l3 cβ)2 , c4 = z o tan β − l4 + l3 cβ

Finally, the inverse kinematics expression of this PM can be written as ⎧ ⎪ q1 ⎪ ⎪ ⎪ ⎨ q2 ⎪ ⎪ ⎪ q3 ⎪ ⎩ q4

/ a 2 + c12 / 1 = a22 + c22

=

= z o + l3 sβ − a3 = z o − l 3 sβ − a 4

,

(3.28)

where/a1 = z o sec β − l1 sγ , c1 = l2 − l1 cγ /, a2 = z o sec β + l1 sγ , c2 = l1 cγ − l2 , a3 = l 2 − (z o tan β + l4 − l3 cβ)2 , a4 = l 2 − (z o tan β − l4 + l3 cβ)2 . • Local and global transmission performances Before evaluating the motion/force transmission performance of the 2UPR-2PRU PM, it is necessary to obtain the TWSs and OTSs first, and then calculate the LTI of this PM. The procedure is described as follows: Step 1: Determine the TWS in each limb of the 2UPR-2PRU PM. Without loss of generality, limb 1 is used as an example. The TWS should be linearly independent of passive twists and constraint wrenches simultaneously [25]. Since the actuated [joint of limb } 1 is a P joint, the TWS of limb 1 is reciprocal of the passive joints $11 $12 $14 and is given by ) ( $T1 = −sβ m 13 −cβ; l2 cβ 0 −l2 sβ . Similarly, the TWSs for the other three limbs can be derived as follows:

(3.29)

3.3 Example 2: 2UPR-2PRU PM

117

⎧ ) ( ⎪ ⎪ $T2 = −sβ m 23 −cβ; −l2 cβ 0 l2 sβ ⎨ ) ( $T3 = lT3 0 n T3 ; 0 lT3 z A3 − n T3 x A3 0 , ⎪ ⎪ ⎩$ = (l 0 n ; 0 l z − n x 0) T4 T4 T4 T4 A4 T4 A 4

(3.30)

where m 13 and m 23 denote the direction components along the y-axis, and others x A3 +l4 z A3 −q3 , n T3 = / , l T4 = can be expressed as lT3 = / (x A3 +l4 )2 +(z A3 −q3 )2 (x A3 +l4 )2 +(z A3 −q3 )2 z A4 −q4 x A4 −l4 / , and n T4 = / . 2 2 x −l + z −q x −l ( A4 4 ) ( A4 4 ) ( A4 4 )2 +(z A4 −q4 )2 Step 2: List the locked/active TWSs and form a new wrench system for the 2UPR2PRU PM, from which the OTS is then obtained. Consider a 3DOF 2UPR-2PRU PM with four limbs. When actuators that are installed in two arbitrarily chosen limbs are locked, the corresponding TWSs form a two-order system and turn into extra constraint wrenches for the moving platform. When combined with the three original constraint wrenches, the PM’s moving platform sustains five independent constraint wrenches in total. Using the screw theory, it can be shown that there is only one instantaneous 1-DOF twist that is actuated redundantly by two limbs. For the 2UPR-2PRU PM, there are N = C42 = 6 combinations that can be used to select the two locked limbs. The corresponding locked/active limb combinations are listed in Table 3.2. The PM of case 1 is selected as an example here. As listed in Table 3.2, the new wrench system acting on the moving platform after locking two actuators can be written as } [ U 1 = $C1 $C2 $C3 $T1 $T2 ,

(3.31)

where the CWSs of the 2UPR-2PRU PM can be expressed as ⎧ ( ) ⎪ ⎨ $C1 = ( 0 0 0; sβ 0 cβ ) . $C2 = cβ 0 −sβ; 0 0 0 ⎪ ⎩ $ = ( 0 1 0; 0 −z − x tanβ 0 ) C3 A3 A3

(3.32)

Table 3.2 Locked/active limb combinations of the 2UPR-2PRU PM Case (g)

Locked limbs

Active limbs

1

1, 2

3, 4

2

1, 3

2, 4

3

1, 4

2, 3

4

2, 3

1, 4

5

2, 4

1, 3

6

3, 4

1, 2

Wrench system of moving platform } [ U 1 = $C1 $C2 $C3 $T 1 $T 2 } [ U 2 = $C1 $C2 $C3 $T 1 $T 3 } [ U 3 = $C1 $C2 $C3 $T 1 $T 4 [ } U 4 = $C1 $C2 $C3 $T 2 $T 3 } [ U 5 = $C1 $C2 $C3 $T 2 $T 4 [ } U 6 = $C1 $C2 $C3 $T 3 $T 4

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3 Motion/Force Transmission Performance Analysis and Optimization …

Accordingly, the instantaneous 1-DOF twist of the moving platform, which is denoted by $O1 , can be obtained using reciprocal screw theory: $O1 ◦ U 1 = 0.

(3.33)

There are six cases for the 2UPR-2PRU PM. If these cases have been completed, proceed to Step 3; otherwise, return to the start of Step 2 and select the next case. Step 3: Calculate the OTI and ITI. Because only the transmission wrenches of the active limbs contribute in the output twist direction, the output transmission performance of case 1, which reflects the power coefficient between the active transmission wrenches and the output twist, can be written as | | | | |$ ◦ $T3 | + |$ ◦ $T4 | O1 O1 | | | . η =| |$ ◦ $T3 | + |$ ◦ $T4 | O1 O1 max max 1

(3.34)

In each case, the output transmissibility of the 1-DOF mechanism is the coupled effect of the two active limbs. In a redundantly actuated PM, the output transmission performance will not exceed either the maximum or minimum performance values of the 1-DOF mechanisms. Thus, when considering all cases, the OTI of the 2UPR2PRU redundantly actuated PM in this study is defined as 1Σ g η . 6 g=1 6

η=

(3.35)

Additionally, the ITI of the ith limb (i = 1, 2, 3, 4) can be obtained using Eq. (3.10). Step 4: Obtain the LTI for a specific configuration. Taking both the input and output transmission performances into account, the LTI for the 2UPR-2PRU PM is defined as Γ = min{λi , η}.

(3.36)

For the 2UPR-2PRU PM, the architectural parameters are set as follows: l1 = 250 mm, ◦ ◦ ◦ ◦ l2 = 400 [ mm, l3 = 200 mm, } l4 = 320 mm, −40 ≤ β ≤ 40 , −40 ≤ γ ≤ 40 , and z o ∈ 400 mm 800 mm . The link parameter l is defined as { } l = max |z o tan βmax + l4 − l3 cβmax |, |z o tan βmax − l4 + l3 cβmax | .

(3.37)

The LTI distribution for the 2UPR-2PRU PM can then be obtained as shown in Fig. 3.10. The workspace volume of the end effector for a tool head length H of 50 mm is shown in Fig. 3.10a. Additionally, the LTI distribution when z o = 600 mm is shown in Fig. 3.10b and is symmetrical about the planes x = 0 and y = 0. The PM’s orientation workspace is completely symmetrical with respect to both angles, as shown in Fig. 3.10c. In Fig. 3.10, it shows that having a configuration that is

3.3 Example 2: 2UPR-2PRU PM

119

Fig. 3.10 LTI distribution for the 2UPR-2PRU PM: a workspace volume, b workspace when z o = 600 mm, and c orientation workspace

closer to the initial situation leads to better motion/force transmission performance. Additionally, in contrast with the angle γ , the motion/force transmission performance decreases more rapidly when the angle β deviates from 0◦ . Here, the region where Γ ≥ 0.7 is defined as the GTW of the 2UPR-2PRU PM. The effects of the architectural parameters, l1 , l2 , l3 , and l4 on σ are shown in Fig. 3.11, in which Figs. 3.11a–d show that higher l1 , l3 , and l4 values produce better σ values for the 2UPR-2PRU PM, and Fig. 3.11b shows the relationship between σ and l2 , where a higher l2 produces a reduced σ . Additionally, the figures show that the effects of l2 , l3 , and l4 on σ are much stronger than that of l1 . • Optimization of design parameters The GTW of the 2UPR-2PRU PM is highly dependent on the design parameters l1 , l2 , l3 , and l4 . By considering the rotation angles and operating height limitations, the ranges of motion are set as follows: β ∈ [−40◦ , 40◦ ], γ ∈ [−40◦ , 40◦ ], and z o = 1.5l2 . Here, the PFNM is used to optimize the design parameters. For a 2UPR-2PRU PM, l1 , l2 , and l3 are defined as design parameters and are normalized as

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3 Motion/Force Transmission Performance Analysis and Optimization …

Fig. 3.11 Trends in σ for different architectural parameters

{

l4 = (l2 · l3 )/l1 . D = l1 +l32 +l3

(3.38)

The non-dimensional and normalized parameters r1 , r2 , r3 , and r4 are then deduced as ri =

li (i = 1, 2, 3, 4). D

(3.39)

In real applications, the normalized parameters should satisfy {

r1 , r3 ≤ r2 , r4 . 0 < r1 , r2 , r3 , r4 < 3

(3.40)

Using the PFNM method [26], the PDS can be obtained as shown in Fig. 3.12, where the shaded area represents the set of all possible points. For convenience, the chosen area can be transformed into a plan view, as shown in Fig. 3.12b. The relationship between the parameters in three-dimensional space (r1 , r2 , r3 ) and those in plan space (s, t) is given as follows

3.3 Example 2: 2UPR-2PRU PM

121

Fig. 3.12 Parameter design space of the 2UPR-2PRU PM: a spatial view and b plane view

⎧ ⎪ ⎨ r1 = s√ s r2 = 23 t − √ 2 ⎪ ⎩ r3 = 3 − 23 t −

{ or s 2

s = r1 . t = 3−r√33+r2

(3.41)

The design steps required for performance optimization are as follows: Step 1: Identify the optimal regions. The GTW distributions can be obtained as shown in Fig. 3.13. When the parameters (s, t) are closer to point B, the σ value is higher. Here, region I (where σ ∈ [0.3, 0.6) ) is regarded as having poor global motion/force transmission, region II (where σ ∈ [0.6, 0.7) ) is regarded as having medium global motion/force transmission, and region III (where σ ≥ 0.7) is regarded as having good global motion/force transmission. Step 2: Select three groups of data points randomly from each of optimal regions I, II, and III; nine groups of data points are selected in this study. The non-dimensional parameters r1 , r2 , and r3 can be obtained for each group using Eq. (3.41). The four design parameters l1 , l2 , l3 , and l4 and the two plan space parameters s and t can then be obtained using Eqs. (3.39) and (3.41), and are listed in Table 3.3. Step 3: Determine the normalized factor D and dimensional parameters l1 , l2 , and l3 . By considering the occupied area and the non-dimensional parameters that are selected from the optimal design regions, the normalized factor D is determined to be 500 mm in this study. For example, the design parameters in group 7 are selected as the optimal results, where r1 = 0.95, r2 = 1.21, and r3 = 0.84. The li values can then be obtained using Eq. (3.39), giving l1 = 475 mm, l2 = 605 mm, l3 = 420 mm, and l4 = 535 mm. Step 4: Check whether the dimensional parameters obtained in Step 3 are suitable for actual use. If the actual assembly conditions are satisfied, then the procedure is

122

3 Motion/Force Transmission Performance Analysis and Optimization …

Fig. 3.13 Optimization of GTW for the 2UPR-2PRU PM: a Distribution of σ and b three design regions I, II, and III

Table 3.3 GTWs in the optimal design regions Region

Group

s

t

r1

r2

r3

l1

l2

l3

GTW

I

1

0.26

2.82

0.26

2.31

0.43

130

1155

215

0.366

2

0.28

2.76

0.28

2.25

0.47

140

1125

235

0.394

3

0.34

2.54

0.34

2.03

0.63

170

1015

315

0.468

4

0.42

2.46

0.42

1.92

0.66

210

960

330

0.606

5

0.62

2.28

0.62

1.66

0.72

310

830

360

0.659

6

0.73

1.94

0.73

1.32

0.95

365

660

475

0.683

7

0.95

1.94

0.95

1.21

0.84

475

605

420

0.719

8

0.96

1.92

0.96

1.18

0.86

480

590

430

0.747

9

0.97

1.85

0.97

1.12

0.91

485

560

455

0.756

II

III

complete; otherwise, return to Step 3, select another group of data from the optimal region, and repeat Steps 3 and 4. Based on the dimensional parameters obtained from the previous step (i.e., l1 = 475 mm, l2 = 605 mm, l3 = 420 mm, and l4 = 535 mm), analysis of the kinematics and performances indicates that a manipulator with these parameters would be suitable for actual use and should work normally. In Fig. 3.14, it compares the motion/force transmissions in regions I, II, and III, where the design parameters in groups 3, 6, and 9 of Table 3.3 are chosen as examples. As shown in Fig. 3.14c, the GTW value in region III is 0.756, which is better than the corresponding values for the other two examples. This shows that region III should be selected as the optimal region. Designers can choose one of these combinations for different applications.

3.3 Example 2: 2UPR-2PRU PM

123

Fig. 3.14 Comparisons of motion/force transmission: a Region I with l1 = 130 mm, l2 = 1155 mm, l3 = 215 mm, and l4 = 1910 mm (group 3), b Region II with l1 = 365 mm, l2 = 650 mm, l3 = 475 mm, and l4 = 859 mm (group 6), and c Region III with l1 = 485 mm, l2 = 560 mm, l3 = 455 mm, and l4 = 525 mm (group 9)

124

3 Motion/Force Transmission Performance Analysis and Optimization …

3.4 Example 3: 2PUR-2PRU PM • Structure description The CAD model and kinematic scheme of the 2R1T 2PUR-2PRU PM [24] are shown in Fig. 3.15, where it can be seen that the moving platform is connected to the fixed base through four kinematic limbs arranged symmetrically. The first limb A1 B1 and the second limb A2 B2 are identical PUR kinematic limbs. The first revolute axes of the U joints coincide with each other. The second revolute axes of the U joints are parallel to the axis of the R joint connected to the moving platform and perpendicular to the P joints. The third limb A3 B3 and fourth limb A4 B4 are PRU kinematic limbs. The axes of the R joints are perpendicular to the P joints and parallel to the first revolute axes of the U joints. Moreover, the second revolute axes of the U joints connected to the moving platform coincide with each other. Because of the special structure, points B1 , B2 , B3 , and B4 are not in the same plane, while B1 C1 and B2 C2 are perpendicular to the moving platform and intersect at points C1 and C2 . Actuated by the four fixed serve motors, the moving platform of the 2PUR-2PRU PM can achieve the required motions, namely 2R1T. As shown in Fig. 3.15b, the fixed frame O-XYZ is attached to the fixed base. The origin O is at the intersection point of A1 A2 and A3 A4 , and the X- and Y-axes pass through points A1 and A3 , respectively. A moving frame P-uvw is attached to the moving platform with the u-axis along PC2 and the v-axis along P B3 . The dimensional parameters of the 2PUR-2PRU PM are defined as follows: PC1 = PC2 = l1 , P B3 = P B4 = l2 , A1 B1 = A2 B2 = l3 , A3 B3 = A4 B4 = l4 , and B1 C1 = B2 C2 = e. The coordinates of points A1 , A2 , A3 , and A4 with respect to O)T ( )T ( )T )T ( ( xyz are defined as q1 0 0 , −q2 0 0 , 0 q3 0 , and 0 −q4 0 , respectively, where qi denotes the distance between the fixed base and the actuated P joints of limb i, respectively. The coordinates of points B1 , B2 , B3 , and B4 are defined as ( )T ( )T ( )T ( )T x B1 y B1 z B1 , x B2 y B2 z B2 , x B3 y B3 z B3 , and x B4 y B4 z B4 , respectively.

(a) Fig. 3.15 2PUR-2PRU PM: a CAD model and b kinematic scheme

(b)

3.4 Example 3: 2PUR-2PRU PM

125

• Inverse kinematics The inverse displacement solution for the 2PUR-2PRU PM requires calculation of the actuated joint parameters (q1 , q2 , q3 , q4 ) when the position and orientation parameters (α, β, z o ) of the moving platform are given. The rotation matrix of the moving frame P-uvw with respect to the fixed frame O-XYZ can be described by successively rotating around the X-axis with the α angle, and around the v-axis with the β angle, as follows ⎛ ⎞⎛ ⎞ ⎛ ⎞ 10 0 cβ 0 −sβ cβ 0 −sβ O R P = R X (α)Rv (β) = ⎝ 0 cα sα ⎠⎝ 0 1 0 ⎠ = ⎝ sαsβ cα sαcβ ⎠, 0 −sα cα sβ 0 cβ cαsβ −sα cαcβ (3.42) where R X (α) and Rv (β) represent the rotational matrices around the X-axis and v-axis, respectively. By considering the structural constraints, the position vector p of point P relative to )T ( the O-XYZ is always limited into the X = 0 plane and defined as p = 0 z o sα z o cα , where z o denotes the operation distance between points O and P. Through this geometric relationship, the position vectors q i of point Ai with respect to O-XYZ can be expressed as follows: q i = p − ai + bi ,

(3.43)

where ai and bi represent the position vectors of Ai Bi and P Bi in O-XYZ, respectively. Additionally, the position vectors bi can be expressed as follows: ⎧ ⎪ b1 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ b2 =

)T ( R P −l1 0 −e )T ( O R P l1 0 −e . )T ( ⎪ ⎪ b3 = O R P 0 l2 0 ⎪ ⎪ ⎪ )T ⎪ ⎩b = OR ( 4 P 0 −l 2 0 O

(3.44)

The inverse kinematic solution for the 2PUR-2PRU PM can be obtained by using Eqs. (3.42)–(3.44) ⎧ / ⎪ ⎪ q = l32 − e12 − l1 cβ + esβ 1 ⎪ ⎪ ⎪ / ⎪ ⎪ ⎪ ⎨ q2 = l32 − e22 − l1 cβ − esβ , / ⎪ 2 2 ⎪ ⎪ q = l − e + l cα + z sα 2 o 3 4 3 ⎪ ⎪ ⎪ / ⎪ ⎪ ⎩ q = l 2 − e2 + l cα − z sα 4 2 o 4 4

(3.45)

126

3 Motion/Force Transmission Performance Analysis and Optimization …

where e1 = z o −ecβ −l1 sβ, e2 = z o −ecβ +l1 sβ, e3 = z o cα−l2 sα, e4 = z o cα+l2 sα. • Motion/Force transmission performance Using the same procedure as listed in the example of the 2UPR-2PRU PM, the ITSs, TWSs, CWSs, and OTSs of the 2PUR-2PRU PM also can be derived. For example, the TWSs and CWSs of the 2PUR-2PRU PM can be obtained as ⎧ ( ( ) ( ) ) ⎪ $T1 = cα q1 − x B1 0 −z B1 cα − y B1 sα; 0 q1 z B1 cα + y B1 sα 0 ⎪ ) ( ⎪ ) ( ) ( ⎨ $T2 = cα q2 + x B2 0 z B2 cα + y B2 sα; 0 q2 z B2 cα + y B2 sα 0 ) ( , (3.46) ⎪ $T3 = 0 y B3 − q3 z B3 ; q3 z B3 0 0 ⎪ ) ( ⎪ ⎩$ = 0 y B4 + q4 z B4 ; −q4 z B4 0 0 T4 and ⎧ ( ) ⎪ ⎨ $C1 = ( 0 −cα sα; 0 0 )0 . $C2 = 0 0 0; 0 sα cα ⎪ ⎩ $ = ( 1 0 0; 0 z + y tan α 0 ) C3 B3 B3

(3.47)

Based on these results, the OTSs of the 2PUR-2PRU PM can be calculated. Substituting these screws into Eqs. (3.8)–(3.11), the LTI of the 2PUR-2PRU PM can be obtained. Using the original parameters (l1 = 120 mm, l2 = 120 mm, l3 = 475 mm, l4 = 375 mm, and e = 80 mm), the LTI distribution of the 2PUR-2PRU PM is shown in Fig. 3.16, wherein the red points marking Γ = 0.7 form two curves. The rotation angles and operating distance are set as α ∈ [−30◦ , 30◦ ], β ∈ [−30◦ , 30◦ ], and z o = 250 mm. With a tool head length of H = 50 mm, the LTI distribution in a threedimensional workspace shown in Fig. 3.16a is symmetrical about the X = 0 and Y = 0 planes, which means that the distribution atlas in the orientation workspace is also symmetrical with respect to both angles, as shown in Fig. 3.16b. In this section, it is assumed here that the region for which Γ ≥ 0.7 can be considered as the GTW. However, the ratio of the region where Γ ≥ 0.7 with these link parameters is only 34.4%, which means that the motion/force transmission capability is not satisfactory. • Optimization of design parameters The dimensional parameters of the 2PUR-2PRU PM are optimized with respect to the GTW. The rotation angles and√operating distance are set as follows: α ∈ [−30◦ , 30◦ ], β ∈ [−30◦ , 30◦ ],and z o = 2/2l3 . Without loss of generality, we let parameter e = 2l2 /3. Using the PFNM method [26], the design parameters are normalized as follows: ⎧ ⎨ l4 = (l1 · l3 )/l2 . (3.48) D = l1 +l32 +l3 ⎩ ri = Dli (i = 1, 2, 3, 4)

3.4 Example 3: 2PUR-2PRU PM

127

Fig. 3.16 LTI distribution of the 2PUR-2PRU PM with original parameters: a three-dimensional workspace and b orientation workspace

Considering practical applications, the normalized parameters should satisfy the following relationship: {

r1 , r2 ≤ r3 , r4 . 0 < r1 , r2 , r3 , r4 < 3

(3.49)

As shown in Fig. 3.17, the parameter design space includes all possible points, and the mapping relationship between (r1 , r2 , r3 ) and (s, t) is described as follows: ⎧ ⎪ ⎨ r1 = s √ r2 = 3√− 23 t − ⎪ ⎩ r3 = 23 t − 2s

{ s 2

or

s = r1 . t = 3+r√33−r2

(3.50)

The distribution of σ in the PDS is shown in Fig. 3.17b, and the global motion/force transmissibility is best at s = 0.52 and t = 2.6. By substituting s and t into Eq. (3.50), the non-dimensional parameters can be obtained as r1 = 0.52, r2 = 0.49, and r3 = 1.99. Considering the actual operation environment, the normalized factor D is determined as 200 mm, and the link parameters can be obtained using Eq. (3.48), i.e., l1 = 104 mm, l2 = 98 mm, l3 = 398 mm, l4 = 422 mm, and e = 65.4 mm. The distribution of LTI with optimized parameters is shown in Fig. 3.18. In comparison with the results obtained using the original parameters, which are presented in Fig. 3.16, the GTW ratio is enlarged from 34.4% to 95.8%, which means that these dimensional parameters are suitable for the design of the 2PUR-2PRU PM. Designers can choose one of these combinations in the parameter design space for different applications.

128

3 Motion/Force Transmission Performance Analysis and Optimization …

Fig. 3.17 Parameter design space of the 2PUR-2PRU PM: a spatial view and b plan view with distribution of σ

Fig. 3.18 LTI distribution of the 2PUR-2PRU PM with optimized parameters: a three-dimensional workspace and b orientation workspace

3.5 Summary This chapter presents the motion/force transmission performance analysis and optimization of PMs with actuation redundancy. To evaluate the motion/force transmission of a redundantly actuated PM accurately, it is essential to establish an index that is consistent with the nature of actuation redundancy. New indices for measuring the motion/force transmission of redundantly actuated PMs are present in this chapter. In the proposed indices, a redundantly actuated PM is divided into multiple subsidiary redundantly actuated 1-DOF PMs. Such a strategy guarantees the consistency of the extracted 1-DOF PMs with the parent PM. The mean value of instantaneous power produced by the multiple actuation wrenches and one twist of the moving platform of these subsidiary 1-DOF PMs is used to describe the output motion/force transmission of the parent PM. Based on this index of output transmission, coordinate-free

References

129

LTI and GTW with clear physical interpretation are established. Three examples, 6PSS-UPS PM, 2UPR-2PRU PM, and 2PUR-2PRU PM, demonstrate that these new indices are effective and suitable to be used in the optimal design of redundantly actuated PMs.

References 1. M. Gouttefarde, C. Gosselin, Wrench-closure workspace of six-DOF parallel mechanisms driven by 7 cables. Trans. Canad. Soc. Mech. Eng. 29(4), 541–552 (2005) 2. S. Bouchard, C. Gosselin, B. Moore, On the ability of a cable-driven robot to generate a prescribed set of wrenches. J. Mech. Robot. 2(1), 011010 (2010) 3. V. Garg, J.A. Carretero, S.B. Nokleby, A new method to calculate the force and moment workspaces of actuation redundant spatial parallel manipulators. J. Mech. Robot. 1(3). 031004 (2009) 4. L.P. Wang, J. Wu, J.S. Wang et al., An experimental study of a redundantly actuated parallel manipulator for a 5-DOF hybrid machine tool. IEEE/ASME Trans. Mechatron. 14(1), 72–81 (2009) 5. J. Kim, F.C. Park, J.R. Sun et al., Design and analysis of a redundantly actuated parallel mechanism for rapid machining. IEEE Trans. Robot. Autom. 17(4), 423–434 (2001) 6. J.P. Merlet, Redundant parallel manipulators. Lab. Rob. Autom. 8(1), 17–24 (1996) 7. S. Kim, Operational quality analysis of parallel manipulators with actuation redundancy, in Proceedings of the 1997 IEEE International Conference on Robotics and Automation (1997), pp. 2651–2656 8. J.S. Wang, J. Wu, T.M. Li et al., Workspace and singularity analysis of a 3-DOF planar parallel manipulator with actuation redundancy. Robotica 27(1), 51–57 (2009) 9. J.A. Saglia, J.S. Dai, D.G. Caldwell, Geometry and kinematic analysis of a redundantly actuated parallel mechanism that eliminates singularities and improves dexterity. J. Mech. Des. 130(12), 124501 (2008) 10. S.H. Kim, D. Jeon, H.P. Shin et al., Design and analysis of decoupled parallel mechanism with redundant actuator. Int. J. Precis. Eng. Manuf. 10(4), 93–99 (2009) 11. J.P. Merlet, Jacobian, manipulability, condition number and accuracy of parallel robots. J. Mech. Des. 128(1), 199–206 (2006) 12. F.G. Xie, X.J. Liu, Y.H. Zhou, Optimization of a redundantly actuated parallel kinematic mechanism for a 5-degree-of-freedom hybrid machine tool. Proc. Inst. Mech. Eng. Part B-J. Eng. Manuf. 228(12), 1630–1641 (2014) 13. F.G. Xie, X.J. Liu, Y.H. Zhou, Development and experimental study of a redundant hybrid machine with five-face milling capability in one setup. Int. J. Precis. Eng. Manuf. 15(1), 13–21 (2014) 14. F.G. Xie, X.J. Liu, X. Chen, et al., Optimum kinematic design of a 3-DOF parallel kinematic manipulator with actuation redundancy, in International Conference on Intelligent Robotics and Applications (2011), pp. 250–259 15. F.G. Xie, X.J. Liu, J.S. Wang, Performance evaluation of redundant parallel manipulators assimilating motion/force transmissibility. Int. J. Adv. Rob. Syst. 8(5), 113–124 (2011) 16. Q.C. Li, N.B. Zhang, F.B. Wang, New indices for optimal design of redundantly actuated parallel manipulators. J. Mech. Robot. 9(1), 011007 (2017) 17. J.S. Wang, C. Wu, X.J. Liu, Performance evaluation of parallel manipulators: motion/force transmissibility and its index. Mech. Mach. Theory 45(10), 1462–1476 (2010) 18. S.A. Joshi, L.-W. Tsai, Jacobian analysis of limited-DOF parallel manipulators. J. Mech. Des. 124(2), 254–258 (2002) 19. T. Yoshikawa, Manipulability of robotic mechanisms. Int. J. Robot. Res. 4(2), 3–9 (1985)

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20. R.S. Stoughton, T. Arai, A modified Stewart platform manipulator with improved dexterity. IEEE Trans. Robot. Autom. 9(2), 166–173 (1993) 21. C. Gosselin, J. Angeles, A global performance index for the kinematic optimization of robotic manipulators. J. Mech. Des. 113(3), 220–226 (1991) 22. X.J. Liu, L.P. Wang, F.G. Xie et al., Design of a three-axis articulated tool head with parallel kinematics achieving desired motion/force transmission characteristics. J. Manuf. Sci. Eng. 132(2), 021009 (2010) 23. L.M. Xu, Q.C. Li, N.B. Zhang et al., Mobility, kinematic analysis, and dimensional optimization of new three-degrees-of-freedom parallel manipulator with actuation redundancy. J. Mech. Robot. 9(4), 041008 (2017) 24. L.M. Xu, X.X. Chai, Q.C. Li et al., Design and experimental investigation of a new 2R1T overconstrained parallel kinematic machine with actuation redundancy. J. Mech. Robot. 11(3), 031016 (2019) 25. X.J. Liu, C. Wu, J.S. Wang, A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators. J. Mech. Robot. 4(4), 041001 (2012) 26. X.J. Liu, J.S. Wang, A new methodology for optimal kinematic design of parallel mechanisms. Mech. Mach. Theory 42(9), 1210–1224 (2007)

Chapter 4

Motion/Force Constraint Performance Analysis and Optimization of Overconstrained Parallel Manipulators with Actuation Redundancy

In Sect. 2.4, the motion/force constraint performance of overconstrained PMs without actuation redundancy [1–3] has been studied. The research on motion/force constraint performance of redundantly actuated and overconstrained PMs will be presented here. For redundantly actuated and overconstrained PMs, they have two obvious features: (1) the number of actuated wrenches is greater than the required DOFs; (2) the number of constraint wrenches is greater than the limited DOFs. With these inherent factors, it can be found that the constraint performance analysis of OCI [4] is complex because the relationship between the output virtual twist and the constraint wrench is not one-to-one, which complicates the analysis of constrainability. To the best of our knowledge, there is still a lack of a general method to evaluate the motion/force constrainability of overconstrained PMs with actuation redundancy. The key to establishing the new method is to properly deal with the influence of multiple wrenches on the constraint performance of redundantly actuated and overconstrained PMs. Inspired by some work [4–11], this chapter proposes a series of new motion/force constraint indices to evaluate the motion/force constrainability of redundantly actuated and overconstrained PMs [12], including the OCI, TCoI, and GCoW. All of these indices are dimensionless and finite that vary from zero to unit. In this chapter, the motion/force constraint performance analysis and optimization of redundantly actuated and overconstrained PMs will be presented. Because the 2UPR-2PRU PM and 2PUR-2PRU PM studied in the above chapter are redundantly actuated and overconstrained PMs, they are taken as examples here for analysis and discussion.

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_4

131

132

4 Motion/Force Constraint Performance Analysis and Optimization …

4.1 Motion/Force Constraint Indices of Overconstrained Parallel Manipulators with Actuation Redundancy • Constraint performance index Without loss of generality, a general redundantly actuated and overconstrained PM with m-DOF is shown in Fig. 4.1, which is composed of one fixed base, one moving platform, and n limbs (n > m). Each limb is assumed with only one actuated joint here. The definitions of constraint wrench system U C and transmission wrench system U T of a redundantly actuated and overconstrained PM are the same as those of the overconstrained PMs without redundancy actuation, which can be found from Eqs. 2.92–2.95. Therefore, there are n TWSs in the redundantly actuated  over-  and constrained PM, which are expressed as $Tj (j = 1, 2, …, n), and g = nj=1 6 − q j CWSs acting on the moving platform, which are expressed as $Ci (i = 1, 2, …, g). For an m-DOF redundantly actuated and overconstrained PM in a certain nonsingular configuration, as shown in Fig. 4.1, it keeps balanced with (n + g) wrenches, which includes n (n > m) TWSs and g (g > 6 − m) CWSs. Among these (n + g) wrenches, there are six independent wrenches, and others are linearly dependent on these six wrenches. For the redundantly actuated and overconstrained PMs, the influences of these (n + g − 6) linearly dependent wrenches on the motion/force constrainability should be considered. Recently, by considering several six-dimensional subsystems separately by the way of removing other (n + g − 6) wrenches, Meng et al. [6] analyzed the motion/force interaction performance of redundantly actuated and overconstrained PMs with closed-loop subchains. Fig. 4.1 General representation of an m-DOF redundantly actuated and overconstrained PM

4.1 Motion/Force Constraint Indices of Overconstrained Parallel …

133

Inspired by the above work, a new method for the constraint evaluation of redundantly actuated and overconstrained PMs is presented here, which takes the effects of (n + g − 6) linearly dependent wrenches into account reasonably. First, by locking all the TWSs, they are converted into n new CWSs, and then the redundantly actuated and overconstrained PM becomes a statically indeterminate structure with (n + g) CWSs. Then, “lose” (n + g − 5) wrenches while remaining five linearly independent wrenches locked so that the redundantly actuated and overconstrained PM effectively becomes an instantaneous 1-DOF mechanism. The unique instantaneous motion of the moving platform is defined as virtual OTS [4]. Considering the force characteristics of redundantly actuated and overconstrained PMs, at least k (k ≥ 1) of the “lost” (n + g − 5) wrenches are linearly independent with the five locked wrenches, which can be used to prevent the virtual instantaneous motion. Next, through ergodic 5 cases need to be considered. Here, the instantaneous analysis, there are at most Cn+g power between the k linearly independent wrenches in the (n + g − 5) wrenches and the unique instantaneous twist is defined as the output constraint performance of the 1-DOF mechanism. For different k, the calculation method is not the same, which is similar to the evaluation process in Sect. 2.4. Averaging the values of the output constraint indices of these 1-DOF mechanisms gives the overall output constraint performance of the redundantly actuated and overconstrained PM, and the OCI is also defined as ζ . Finally, the total constraint performance is calculated by considering both √ the input and output constraint performances, and the TCoI is also defined as κ= η ∗ ζ . Similar to the method in Sect. 2.4, the global constraint performance index of overconstrained PMs with actuation redundancy also can be evaluated, and the GCoW is also defined as σGCoW . The purpose of this method is to describe the motion/force constrainability of the redundantly actuated and overconstrained PM by evaluating the constraint performance of several 1-DOF mechanisms with multiple wrenches. A procedure for evaluating a general redundantly actuated and overconstrained PM is presented in Fig. 4.2. Because the evaluation process of the constraint performance of redundantly actuated and overconstrained PMs is similar to that of overconstrained PMs without actuation redundancy, the detailed steps are not listed here. By combining Sect. 2.4 and Fig. 4.2, the readers can quickly understand this evaluation method. It should be noted again that although the evaluation methods used for the constraint performance in Sect. 2.4 and this chapter are similar, they are completely different in the losing strategy of wrenches. For the overconstrained PMs without actuation redundancy, the constraint performance in each configuration is analyzed by locking m TWSs and “losing” (m + g − 5) CWSs. For the overconstrained PMs with actuation redundancy, however, the constraint performance in each configuration is analyzed by locking n TWSs and “losing” (n + g − 5) wrenches, in which the “losing” (n + g − 5) wrenches are composed of TWSs and CWSs. Therefore, there are more situations that should be considered in the evaluation of redundantly actuated and overconstrained PMs.

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4 Motion/Force Constraint Performance Analysis and Optimization …

Fig. 4.2 Procedure of evaluating a general redundantly actuated and overconstrained PM

4.2 Example 1: 2UPR-2PRU PM

135

In the following sections, the motion/force constraint indices based on the screw theory will be used to evaluate and optimize the performance of two overconstrained PMs with actuation redundancy, namely 2UPR-2PRU [2] and 2PUR-2PRU [3], and the effectiveness of the proposed constraint index will be verified.

4.2 Example 1: 2UPR-2PRU PM • Local and global constraint performances Before evaluating the motion/force constraint performance of the 2UPR-2PRU PM [2], it is necessary to obtain the TWSs and CWSs of limbs first. The TWSs of limbs have been listed in Eqs. 3.29 and 3.30. For the CWSs of each limb in the 2UPR-2PRU PM, they can be derived by using the reciprocal screw theory, which are expressed as    $C1 = 0 0 0; sβ 0 cβ (4.1)   $C2 = cβ 0 −sβ; l2 sβ 0 l2 cβ    $C3 = 0 0 0; sβ 0 cβ (4.2)   $C4 = cβ 0 −sβ; −l2 sβ 0 −l2 cβ    $C5 = 0 0 0; sβ 0 cβ (4.3)   $C6 = 0 1 0; −z A3 − y A3 tanβ 0 0 and 

  $C7 = 0 0 0; sβ 0 cβ   $C8 = 0 1 0; −z A4 − y A4 tanβ 0 0

(4.4)

Based on these results, the OTSs of the 2UPR-2PRU PM in different cases can be calculated, and the TCoI of the 2UPR-2PRU PM can be obtained. Taking the interference between the limbs of the 2UPR-2PRU PM into account, the design parameters of the 2UPR-2PRU PM are set as follows: l1 = 250 mm, l2 = 400 mm, l3 = 200 mm, l4 = 320 mm, −40◦ ≤ β ≤ 40◦ , −40◦ ≤ γ ≤ 40◦ . Here, we consider the TCoI distribution at a constant operation distance, z 0 = 600 mm. The TCoI distribution of the 2UPR-2PRU PM is shown in Fig. 4.3. It is clear that the distribution is symmetrical about the β = 0◦ and γ = 0◦ , which is consistent with the structure of the 2UPR-2PRU PM. The global constraint performance of the 2UPR-2PRU PM can be derived. Here, the motion/force constraint performance of the 2UPR-2PRU PM in a configuration is regarded as good when the value of TCoI is equal to or bigger than 0.7. According

136

4 Motion/Force Constraint Performance Analysis and Optimization …

Fig. 4.3 TCoI distribution of the 2UPR-2PRU PM with original parameters

to the distributions in Fig. 4.3, the ratio of the region where κ ≥ 0.7 for the 2UPR2PRU PM is approximately equal to 1, which means that the motion/force constraint capability with these architectural parameters is reasonably good. • Optimization of design parameters Although the motion/force constraint performance of the 2UPR-2PRU PM in the workspace is great, the optimization of the design parameters of the 2UPR-2PRU PM that based on σGCoW is also needed for more choices. The PFNM method also can be used here, and the optimization process is similar to that of the motion/force transmission performance of the 2UPR-2PRU PM. The distribution of σGCoW in the PDS [13] is shown in Fig. 4.4, from which one can find that global performance in the most region of the available area is equal to unity, which means that the designers can choose many combinations in the parameter design space for different applications. For example, the designers can choose the combination with s = 0.95 and t = 1.94. By substituting s and t into Eq. 3.41, the non-dimensional parameters can be obtained as r1 = 0.95, r2 = 1.21, and r3 = 0.84. Considering the actual operation environment, the normalized factor D is determined as 500 mm, and the values of li can then be obtained using Eq. 3.39, i.e., l1 = 475 mm, l2 = 605 mm, l3 = 420 mm, and l4 = 535 mm. The distribution of TCoI with the above parameters is shown in Fig. 4.5, from which one can find that the 2UPR-2PRU PM has good constraint performance by using these parameters.

4.3 Example 2: 2PUR-2PRU PM

137

Fig. 4.4 Optimization regions of GCoW for the 2UPR-2PRU PM

Fig. 4.5 TCoI distribution of the 2UPR-2PRU PM with new parameters (l1 = 475 mm, l2 = 605 mm, l3 = 420 mm, and l4 = 535 mm)

4.3 Example 2: 2PUR-2PRU PM • Local and global constraint performances Using the same procedure as listed in the example of the 2UPR-2PRU PM, the CWSs, and OTSs of the 2PUR-2PRU PM [3] also can be derived. For example, the CWSs of each limb in the 2PUR-2PRU PM can be obtained as

138

4 Motion/Force Constraint Performance Analysis and Optimization …



  $C1 = 0 0 0; 0 sα cα   $C2 = 0 cα −sα; 0 q1 sα q1 cα    $C3 = 0 0 0; 0 sα cα   $C4 = 0 cα −sα; 0 −q2 sα −q2 cα    $C5 = 0 0 0; 0 sα cα   $C6 = 1 0 0; 0 z B3 + y B3 tanα 0

(4.5)

(4.6)

(4.7)

and 

  $C7 = 0 0 0; 0 sα cα   $C8 = 1 0 0; 0 z B4 + y B4 tanα 0

(4.8)

Based on these results, the OTSs of the 2PUR-2PRU PM in different cases can be calculated, and the TCoI of the 2PUR-2PRU PM can be obtained. Taking the interference between the limbs of the 2PUR-2PRU PM into account, the design parameters of the 2PUR-2PRU PM are set as follows: l1 = 120 mm, l2 = 120 mm, l3 = 475 mm, l 4 = 375 mm, and e = 80 mm. The rotation angles and operating distance are set as α ∈ [−30◦ , 30◦ ], β ∈ [−30◦ , 30◦ ], and z 0 = 250 mm. The TCoI distribution of the 2PUR-2PRU PM is shown in Fig. 4.6. It is clear that the distribution is symmetrical about the α = 0° and β = 0° , which is consistent with the structure of the 2PUR-2PRU PM. Fig. 4.6 TCoI distribution of the 2PUR-2PRU PM with original parameters

4.3 Example 2: 2PUR-2PRU PM

139

The global constraint performance of the 2PUR-2PRU PM can be derived. According to the distributions in Fig. 4.6, the ratio of the region where κ ≥ 0.7 for the 2PUR-2PRU PM is approximately equal to 1, which means that the motion/force constraint capability with these architectural parameters is reasonably good. • Optimization of design parameters Based on the motion/force constraint performance, the optimization of the design parameters of the 2PUR-2PRU PM that based on σGCoW also can be presented. The PFNM method also can be used here, and the optimization process is similar to that of the motion/force transmission performances of the 2PUR-2PRU PM. The distribution of σGCoW in the PDS [13] is shown in Fig. 4.7, from which one can find that global performance in the all regions is equal to unity, which means that the designers can choose many combinations in the parameter design space for different applications. For example, the designers can choose the combination with s = 0.52 and t = 2.60. By substituting s and t into Eq. 3.50, the non-dimensional parameters can be obtained as r1 = 0.52, r2 = 0.49, and r3 = 1.99. Considering the actual operation environment, the normalized factor D is determined as 200 mm, and the values of li can then be obtained using Eq. 3.48, i.e., l1 = 104 mm, l2 = 98 mm, l3 = 398 mm, l4 = 422 mm, and e = 65.4 mm. The distribution of TCoI with the above parameters is shown in Fig. 4.8, from which one can find that the 2PUR-2PRU PM has good constraint performance by using these parameters. Fig. 4.7 Optimization regions of GCoW for the 2PUR-2PRU PM

140

4 Motion/Force Constraint Performance Analysis and Optimization …

Fig. 4.8 TCoI distribution of the 2PUR-2PRU PM with new parameters (l1 = 104 mm, l2 = 98 mm, l3 = 398 mm, l4 = 422 mm, and e = 65.4 mm)

4.4 Summary This chapter presents new constraint indices for measuring the motion/force constrainability of redundantly actuated and overconstrained PMs, which are frameinvariant and range from zero to unity. The key to establishing the new method is to properly deal with the influence of multiple wrenches on the constraint performance of redundantly actuated and overconstrained PMs. In the proposed indices, locking all n transmission wrenches leads to an overconstrained PM with actuation redundancy becoming an indeterminate structure with (n + g) constraint wrenches. Then (n + g − 5) wrenches are “lost” while only the remaining five linearly independent wrenches are locked. So that multiple 1-DOF mechanisms are produced from the redundantly actuated and overconstrained PM. The output constraint performance of these 1-DOF mechanisms is the instantaneous power of all the linearly independent wrenches in the (n + g − 5) wrenches that are devoted to motion. Different calculation processes are used for different number k of linearly independent wrenches. After that, the output motion/force constrainability of an overconstrained PM with actuation redundancy can be described with the mean value of the instantaneous power of these 1-DOF mechanisms. Based on the proposed OCI and TCoI, the global index, GCoW, is established. Two examples demonstrate that these new indices are effective and suitable to be used in the optimal design of redundantly actuated and overconstrained PMs.

References 1. Q.C. Li, N.B. Zhang, F.B. Wang, New indices for optimal design of redundantly actuated parallel manipulators. J. Mech. Robot. 9(1), 011007 (2017)

References

141

2. L.M. Xu, Q.C. Li, N.B. Zhang et al., Mobility, kinematic analysis, and dimensional optimization of new three-degrees-of-freedom parallel manipulator with actuation redundancy. J. Mech. Robot. 9(4), 041008 (2017) 3. L.M. Xu, X.X. Chai, Q.C. Li et al., Design and experimental investigation of a new 2R1T overconstrained parallel kinematic machine with actuation redundancy. J. Mech. Robot. 11(3), 031016 (2019) 4. X.J. Liu, X. Chen, M. Nahon, Motion/Force constrainability analysis of lower-mobility parallel manipulators. J. Mech. Robot. 6(3), 031006 (2014) 5. X.J. Liu, C. Wu, J.S. Wang, A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators. J. Mech. Robot. 4(4), 041001 (2012) 6. Q.Z. Meng, F.G. Xie, X.J. Liu, Motion-force interaction performance analyses of redundantly actuated and overconstrained parallel robots with closed-loop subchains. J. Mech. Des. 142(10), 103304 (2020) 7. C. Han, J. Kim, J. Kim et al., Kinematic sensitivity analysis of the 3-UPU parallel mechanism. Mech. Mach. Theory 37(8), 787–798 (2002) 8. D. Zlatanov, I.A. Bonev, C. Gosselin, Constraint singularities of parallel mechanisms, in Proceedings of the 2002 IEEE International Conference on Robotics and Automation, 496–502 (2002) 9. T. Huang, H.T. Liu, D.G. Chetwynd, Generalized Jacobian analysis of lower mobility manipulators. Mech. Mach. Theory 46(6), 831–844 (2011) 10. J. Brinker, B. Corves, Y. Takeda, Kinematic performance evaluation of high-speed Delta parallel robots based on motion/force transmission indices. Mech. Mach. Theory 125, 111–125 (2018) 11. Q.Z. Meng, F.G. Xie, X.J. Liu et al., An evaluation approach for motion-force interaction performance of parallel manipulators with closed-loop passive limbs. Mech. Mach. Theory 149, 103844 (2020) 12. X.D. Shen, L.M. Xu, Q.C. Li, Motion/Force constraint indices of redundantly actuated parallel manipulators with over constraints. Mech. Mach. Theory 165, 104427 (2021) 13. X.J. Liu, J.S. Wang, A new methodology for optimal kinematic design of parallel mechanisms. Mech. Mach. Theory 42(9), 1210–1224 (2007)

Chapter 5

Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

In this chapter, we focus on the stiffness performance evaluation and optimization design of parallel manipulators (PMs), the comprehensive stiffness index (CSI) that separates the linear stiffness and angular stiffness and considers the effect of the coupling of the non-diagonal elements is introduced, as well as the extreme stiffness index. The 2UPR-RPU PM is taken as an example here for analysis and discussion, and the comparison of optimization results from different optimization algorithms is presented.

5.1 Stiffness Performance Evaluation Index The purpose of stiffness analysis is to obtain a 6 × 6 stiffness matrix that reflects the mapping between the external wrench and infinitesimal twist. To evaluate whether the stiffness performance of the mechanism meets the engineering requirements, it is necessary to transform the stiffness matrix into a quantifiable stiffness index. To achieve this, many researchers have defined multiple stiffness indices for evaluating the stiffness performance of mechanisms, such as the eigenvalue index, which uses the minimum and maximum eigenvalues of the overall stiffness matrix as stiffness indices, PDSI, which uses the six elements of the principal diagonal of the overall stiffness matrix as stiffness indices, the determinant index, which uses the value of the determinant of the overall stiffness matrix as a stiffness index, the trace index, which is defined as the trace of the overall stiffness matrix, the VW index, which is defined as the reciprocal of virtual work done by a unit external wrench, and the instantaneous energy index, which is based on the instantaneous energy defined by the reciprocal product of the external wrench and corresponding deflection [1–9]. Finally, the configuration stiffness index is defined as the minimum strain energy under all possible external unit forces or moments [10]. The above indices can evaluate the relative stiffness of the same mechanism under a given configuration; however, they

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_5

143

144

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

cannot specify whether the stiffness performance meets the accuracy requirements. In this chapter, the CSI is proposed to separate the linear stiffness and angular stiffness that indicate the resistance of the mechanism to a force and couple, it also considers the effect of the coupling of the non-diagonal elements of the compliance matrix on stiffness performance. Compared with the minimum and maximum eigenvalue index and decoupling stiffness index, the CSI has a clear physical meaning and defined dimensions and does not require a complicated decoupling process. The detailed definition of the CSI is as follows. The mapping relationship between an external wrench and the infinitesimal deformation of the mechanism can be expressed as δ = CW,

(5.1)

where δ = [δ dx , δ dy , δ dz ; δ ϕ x , δ ϕ y , δ ϕ z ]T is the infinitesimal deflection, δ dx , δ dy , and δ dz represent translational deflection along the x-, y, and z-axes, respectively, and δ ϕ x , δ ϕ y , and δ ϕ z represent rotational deflection along the x-, y-, and z-axes, respectively. In addition, C is the compliance matrix, w = [f x , f y , f z ; mx , my , mz ]T is the external wrench, f x , f y , and f z represent the forces along the x-, y-, and z-axes, respectively, and mx , my , and mz represent the moment along the x-, y-, and z-axes, respectively. Generally, the overall compliance matrix of a mechanism can be expressed as

(5.2)

where C 11 , C 12 , C 21 , and C 22 represent the four 3 × 3 submatrices. The units of the upper left corner of the compliance matrix in Eq. (5.2) are m/N, reflecting the influence of a force on the linear displacement. The units of the upper right corner are m/N m, reflecting the influence of a couple on the linear displacement. The units of the bottom left corner are rad/N, reflecting the influence of a force on the angular displacement. Finally, the units of the bottom right corner are rad/N m, reflecting the influence of a couple on the angular displacement. To measure the stiffness performance of the mechanism for a force along the along the direction of the x-axis, w = [1 N·m, 0, 0; 0, 0, 0]T is substituted into Eq. (5.1) to yield the following: {

δd x = c11 , δdy = c21 , δdz = c31 . δϕx = c41 , δϕy = c51 , δϕz = c61

(5.3)

Therefore, the resultant translational and rotational deflection of the mechanism when a unit force is applied along the direction of the x-axis can be used as the

5.1 Stiffness Performance Evaluation Index

145

compliance indices of the x-axis direction for an external force as follows: / ⎧ / ⎨ cd x f = δd = δ 2 + δ 2 + δ 2 = c2 + c2 + c2 (m/N) 21 31 dx dy dz / 11 / , ⎩ cϕx f = δϕ = δ 2 + δ 2 + δ 2 = c2 + c2 + c2 (rad/N) ϕx ϕy ϕz 41 61 51

(5.4)

where cdxf denotes the linear compliance index for an external force along the x-axis and cϕ xf denotes the angular compliance index for an external force along the x-axis. Similarly, the compliance indices for external forces in the directions of the y- and z-axes and the couples in the directions of the x-, y-, and z-axes can be obtained as follows: ⎧ / / ⎨ cdy f = c2 + c2 + c2 , cdz f = c2 + c2 + c2 (m/N) 12 22 32 23 33 / / 13 , (5.5) 2 2 2 2 2 ⎩c = c + c + c , c = c + c + c2 (rad/N) ϕy f

42

52

62

ϕz f

43

53

63

⎧ / / 2 2 2 2 2 ⎪ ⎪ c = c2 + c25 + c35 , c + c + c , c = dym d xm ⎪ 14 24 34 ⎪ / 15 ⎪ ⎪ ⎨ 2 2 2 + c26 + c36 (m/N m) cdzm = c16 / / , ⎪ 2 2 2 2 2 2 ⎪ ⎪ cϕxm = c44 + c54 + c64 , cϕym = / c45 + c55 + c65 , ⎪ ⎪ ⎪ ⎩ 2 2 2 + c56 + c66 (rad/N m) cϕzm = c46

(5.6)

where cdyf and cdzf denote the linear compliance indices of the y- and z-axes directions for an external force, respectively, cϕ yf and cϕ zf denote the angular compliance indices of the y- and z-axes directions for an external force, respectively, cdxm , cdym , and cdzm denote the linear compliance indices of the x-, y- and z-axes directions for an external moment, respectively, cϕ xm , cϕ ym , and cϕ zm denote the angular compliance indices of the x-, y- and z-axes directions for an external moment, respectively. Suppose w = [nTf ; nTm ]T , where nf and nm represent the unit vectors of the external force and external couple, respectively, that is, w is a random external wrench. We can project w onto a Cartesian coordinate frame as follows: w = [n f · nx , n f · n y , n f · nz ; nm · nx , nm · n y , nm · nz ]T ,

(5.7)

where nx , ny , and nz are the unit vectors along the x-, y-, and z-axes of the Cartesian coordinate system, respectively. The linear and regular displacements along the axes of the Cartesian coordinate system can be expressed as ⎧ ⎪ ⎪ ⎨ δd x = c11 n f · n x + c12 n f · n y + c13 n f · nz + c14 nm · n x + c15 nm · n y + c16 nm · nz .. . . ⎪ ⎪ ⎩δ = c n ·n +c n ·n +c n ·n +c n ·n +c n ·n +c n ·n ϕz x y z x y z 61 f 62 f 63 f 64 m 66 m 65 m

(5.8)

146

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

Therefore, the compliance index that reflects a general external wrench can be further expressed as ⎧ / ⎨ cd = δd = δ 2 + δ 2 + δ 2 (m/N) dy dz / dx . ⎩ cϕ = δϕ = δ 2 + δ 2 + δ 2 (rad/N) ϕx ϕy ϕz

(5.9)

If w contains only f x , then Eq. (5.8) degenerates to Eq. (5.3). If w is a general force in a random direction, then each formula in Eq. (5.8) contains only the first three terms. In the same manner, if w is a general couple in a random direction, then each formula in Eq. (5.8) contains only the last three terms. Equations (5.8) and (5.9) show that the CSI considers the influence of the non-diagonal elements of the compliance matrix; this makes the evaluation index more objective and comprehensive. Additionally, the CSI has a clear physical meaning and defined dimensional units, which makes it potentially useful in engineering applications. Combining Eqs. (5.8) and (5.9) enables the evaluation of the compliance performance of the mechanism in any direction. Next, we discuss how to find the directions of extreme compliance. We first discuss the maximum and minimum compliance and its corresponding directions for an external force, that is, w = [nTf ; 0, 0, 0]T . Equation (5.8) can now be expressed as ⎧ ⎪ ⎨ δd x = c11 n f x + c12 n f y + c13 n f z .. , . ⎪ ⎩ δϕz = c61 n f x + c62 n f y + c63 n f z

(5.10)

where n f x = n f · nx , n f y = n f · n y , and n f z = n f · nz . To make the calculation more convenient, both sides of Eq. (5.9) are squared: ⎧ 2 c = (c11 n f x + c12 n f y + c13 n f z )2 + (c21 n f x + c22 n f y + c23 n f z )2 ⎪ ⎪ ⎨ df +(c31 n f x + c32 n f y + c33 n f z )2 2 2 2 . ⎪ ⎪ cϕ f = (c41 n f z + c42 n f y + c43 n f z ) + (c51 n f x + c52 n f y + c53 n f z ) ⎩ +(c61 n f x + c62 n f y + c63 n f z )2

(5.11)

The constraint equations can be expressed as ψ(n f x , n f y , n f z ) = n 2f x + n 2f y + n 2f z = 1,

(5.12)

where ψ denotes the constraint function. To find the possible extreme points of Eq. (5.11) under the constraint in Eq. (5.12), the following Lagrangian function is constructed: {

L d (n f x , n f y , n f z ) = cd2 (n f x , n f y , n f z ) + λd f ψ(n f x , n f y , n f z ) , L ϕ (n f x , n f y , n f z ) = cϕ2 (n f x , n f y , n f z ) + λϕ f ψ(n f x , n f y , n f z )

(5.13)

5.1 Stiffness Performance Evaluation Index

147

where L d and L ϕ are Lagrangian functions, and λd and λϕ are Lagrangian multipliers. The partial derivatives of Eq. (5.13) is calculated for n f x , n f y , and n f z . They are then set equal to zero and combined with Eq. (5.12) to obtain ⎧ ∂ Ld ⎪ ⎪ = c11 (c11 n f X + c12 n f Y + c13 n f z ) + c21 (c21 n f x + c22 n f y + c23 n f z ) ⎪ ⎪ ∂n ⎪ fX ⎪ ⎪ ⎪ ⎪ ⎪ + c31 (c31 n f x + c32 n f y + c33 n f z ) + λd f n f x = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ Ld ⎪ ⎪ = c12 (c11 n f X + c12 n f Y + c13 n f z ) + c22 (c21 n f x + c22 n f y + c23 n f z ) ⎪ ⎨ ∂n f Y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ Ld ⎪ ⎪ ⎪ ⎪ ⎪ ∂n f Z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(n

+ c32 (c31 n f x + c32 n f y + c33 n f z ) + λd f n f y = 0

.

= c13 (c11 n f X + c12 n f Y + c13 n f z ) + c23 (c21 n f x + c22 n f y + c23 n f z ) + c33 (c31 n f x + c32 n f y + c33 n f z ) + λd f n f z = 0

f x,

n f y , n f z ) = n 2f x + n 2f y + n 2f z − 1 = 0

(5.14) ⎧ ∂ Lϕ ⎪ ⎪ = c41 (c41 n f x + c42 n f y + c43 n f z ) + c51 (c51 n f x + c52 n f y + c53 n f z ) ⎪ ⎪ ∂n ⎪ fx ⎪ ⎪ ⎪ ⎪ ⎪ + c61 (c61 n f x + c62 n f y + c63 n f z ) + λϕ f n f x = 0 ⎪ ⎪ ⎪ ⎪ ∂ Lϕ ⎪ ⎪ ⎪ = c42 (c41 n f x + c42 n f y + c43 n f z ) + c52 (c51 n f x + c52 n f y + c53 n f z ) ⎪ ⎨ ∂n f y . + c62 (c61 n f x + c62 n f y + c63 n f z ) + λϕ f n f y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ L ϕ = c43 (c41 n f x + c42 n f y + c43 n f z ) + c53 (c51 n f x + c52 n f y + c53 n f z ) ⎪ ⎪ ⎪ ∂n f z ⎪ ⎪ ⎪ ⎪ ⎪ + c63 (c61 n f x + c62 n f y + c63 n f z ) + λϕ f n f z = 0 ⎪ ⎪ ⎪ ⎩ ψ(n , n , n ) = n 2 + n 2 + n 2 − 1 = 0 fx fy fz fx fy fz (5.15) Equations (5.14) and (5.15) can then be expressed as {

{

C T11 C 11 n f + λd f n f = 0 , nTf n f − 1 = 0

(5.16)

C T21 C 21 n f + λϕ f n f = 0 . nTf n f − 1 = 0

(5.17)

Equations (5.16) and (5.17) reveal that λdf (λϕ f ) is an eigenvalue of matrix C T11 C 11 (C T21 C 21 ), and Eq. (5.11) can be expressed as

148

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

{

( )T cd2 f = C 11 n f C 11 n f = nTf C T11 C 11 n f = −λd f )T ( . cϕ2 f = C 21 n f C 21 n f = nTf C T21 C 21 n f = −λϕ f

(5.18)

Thus, the maximum and minimum compliances for an external force are ⎧ /| /| | | ⎨ cd f max = |−λd f | , cd f min = |−λd f | max min /| /| | | . ⎩ cϕ f max = |−λϕ f | , cϕ f min = |−λϕ f | max min

(5.19)

Similarly, the maximum and minimum compliances for an external couple are {

√ √ cdm max = /|−λdm |max , cdm min = /|−λdm |min | | | | , cϕm max = |−λϕm |max , cϕm min = |−λϕm |min

(5.20)

where cdmmax and cdmmin denote the maximum and minimum linear compliances for an external couple, respectively; cϕ mmax and cϕ mmin denote the maximum and minimum angular compliances for an external couple, respectively; and λdm (λϕ m ) denotes the eigenvalue of matrix CT 12C 12 (CT 22C 22 ). If w = [nTf ; nTm ]T is a general wrench, the linear and angular displacements of the mechanism are given by Eq. (5.8). The Lagrangian function is constructed as ⎧ ⎪ L d (n f x , n f y , n f z , n mx , n my , n mz ) = cd2 (n f x , n f y , n f z , n mx , n my , n mz ) ⎪ ⎪ ⎨ +λd ψ(n f x , n f y , n f z , n mx , n my , n mz ) + μd ζ (n f x , n f y , n f z , n mx , n my , n mz ) , 2 ⎪ L ϕ (n f x , n f y , n f z , n mx , n my , n mz ) = cϕ (n f x , n f y , n f z , n mx , n my , n mz ) ⎪ ⎪ ⎩ +λϕ ψ(n f x , n f y , n f z , n mx , n my , n mz ) + μϕ ζ (n f x , n f y , n f z , n mx , n my , n mz )

(5.21) where ζ denotes the constraint functions: {

ψ(n f x , n f y , n f z ) = n 2f x + n 2f y + n 2f z = 1 . ζ (n mx , n my , n mz ) = n 2mx + n 2my + n 2mz = 1

(5.22)

The partial derivatives of Eq. (5.21) are found for n f x , n f y , n f z , n mx , n my , and n mz and set equal to zero. They are then combined with Eq. (5.22), which leads to ⎧ ∂ Ld ∂ Ld ∂ Ld ∂ Ld ∂ Ld ⎪ ⎨ ∂n f x = 0, ∂n f y = 0, ∂n f z = 0, ∂n mx = 0, ∂n my = 0, ψ(n f x , n f y , n f z ) = n 2f x + n 2f y + n 2f z − 1 = 0, ⎪ ⎩ ζ (n mx , n my , n mz ) = n 2mx + n 2my + n 2mz − 1 = 0 ⎧ ∂ Lϕ ∂ Lϕ ∂ Lϕ ∂ Lϕ ∂ Lϕ ⎪ ⎨ ∂n f x = 0, ∂n f y = 0, ∂n f z = 0, ∂n mx = 0, ∂n my = 0, 2 2 2 ψ(n f x , n f y , n f z ) = n f x + n f y + n f z − 1 = 0, ⎪ ⎩ ζ (n mx , n my , n mz ) = n 2mx + n 2my + n 2mz − 1 = 0

∂ Ld ∂n mz

= 0, , (5.23)

∂ Lϕ ∂n mz

= 0, . (5.24)

5.2 Example: 2UPR-RPU PM

149

Simultaneous Eqs. (5.23) and (5.24) are used to determine the directions of extreme compliance, and then Eq. (5.9) is used to obtain the corresponding extreme compliance values. Both Eqs. (5.23) and (5.24) are nonlinear equations, so numerical algorithms should be adopted to solve them. In this chapter, The PSO and genetic algorithms are used to solve them. Compared with the maximum and minimum eigenvalue index, the CSI proposed in this chapter decouples the extreme compliance values of the linear displacement and angular displacement and has a clear physical meaning and defined dimension. The maximum and minimum eigenvalue index does not decouple the linear displacement compliance and angular displacement compliance, and the units of the eigenvalue as well as the physical meaning are not clear. The local CSI (LCSI) can be obtained by directly taking the reciprocal of the compliance index as follows: kd = 1/cd

(N/m), kϕ = 1/cϕ (N/rad),

(5.25)

where cd and cϕ denote the linear and angular compliance indices, respectively. In addition, k d and k ϕ denote the linear and angular stiffness index, respectively. The LCSI only indicates the stiffness performance of the mechanism for a certain position. To obtain the stiffness performance of the mechanism throughout the entire prescribed workspace, the global CSI (GCSI) is defined. Similar to the calculation of the GTI, discrete points are often used to calculate the GCSI: ⎧ ⎪ ⎪ ⎨ Kd = ⎪ ⎪ ⎩ Kϕ =

{

Vr{ kd dVr

{

dVr

Vr{ kϕ dVr

dVr

or K d = or K ϕ =

1 n 1 n

n Σ i=1 n Σ

kdi

.

(5.26)

kϕi

i=1

5.2 Example: 2UPR-RPU PM As we know, the deformation compatibility relationships of the sphere joints are easy to achieve, in order to illustrate the versatility of the proposed modeling, the 2UPRRPU overconstrained PM developed in our laboratory is selected as the example to verify the effectiveness of the modeling. The schematic diagram of the 2UPR-RPU PM [11, 12] is shown in Fig. 5.1. The moving platform is connected to the fixed platform by two UPR limbs and one RPU limb. The UPR limbs connect to the fixed platform at Bi (i = 1, 2) through a universal joint and the moving platform at Ai (i = 1, 2) through an R joint. The axes of the two R joints are parallel with each other. The first rotation axes of the two U joints are collinear with B1 B2 , and the other axis is parallel to the axis of the R joint in the same limb. The RPU limb connects to the fixed platform at B3 through an R joint and the moving platform at A3 through a U joint. The two axes of the U joint of the RPU limb are parallel with the two axes

150

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

Fig. 5.1 2UPR-RPU overconstrained PM. a schematic diagram and b schematic representation

of the U joints of the UPR limb. The axes of the prismatic joints extend along Bi Ai and are perpendicular to the axis of the R joints. With this specific arrangement, the 2UPR-RPU PM has three DOFs, namely a rotation β about the X-axis, rotation γ about the y-axis, and translation along the Oo direction. Moreover, {Ai -x i yi zi } (i = 1, 2) and {B3 -x 3 y3 z3 } represent the local coordinate system of the rods wherein they are located. The coordinate frames of the three R joints are the same as the three local coordinate frames of the rods, and {B1 -uvw}is the coordinate frame of the U joint in the first limb. Because the coordinate frames of the three U joints are identical, to keep the graph clean, only the coordinate system of one U joint is shown in the figure. The geometrical parameters are denoted as follows: OB1 = OB2 = OB3 = r 1 ; oA1 = oA2 = oA3 = r 2 ; A1 B1 = l 1 ; A2 B2 = l 2 ; A3 B3 = l 3 . The inverse kinematic solution of the mechanism must be obtained before the stiffness analysis. From Fig. 5.1, the closed vector-loop equations are obtained (for more details, see [12]) B i Ai = p + O Ro o oAi − O B i (i = 1, 2, 3)

(5.27)

The coordinate system is defined as follows to facilitate the elastostatic stiffness modeling of the 2UPR-RPU PM. ⎡

cγ O Ro = ⎣ sβsγ −cβsγ ⎡ ⎡ ⎤ cϕ1 0 −sϕ1 cϕ2 o R1 = ⎣ 0 1 0 ⎦; o R2 = ⎣ 0 sϕ1 0 cϕ1 −sϕ2

⎤ 0 sγ (5.28) cβ −cγ sβ ⎦ sβ cβcγ ⎡ ⎤ ⎤ 0 −sϕ2 1 0 0 1 0 ⎦; O R3 = ⎣ 0 cϕ3 −sϕ3 ⎦ 0 cϕ2 0 sϕ3 cϕ3 (5.29)

5.2 Example: 2UPR-RPU PM

151

where O Ro is the transform matrix from frames {o} to {O}, o R1 and o R2 denote the transform matrices from limb coordinate systems {A1 } and {A2 } to frame {o}, respectively, O R3 is the transform matrices from limb coordinate system {A3 } to frame {O}. ϕ1 and ϕ2 are the angles between the 1st and 2nd limb with the z-axis of frame {o}, respectively, ϕ3 is the angle the 3rd limb with the Z-axis of frame {O}.

5.2.1 Stiffness Modeling The force analyses of the UPR limb, RPU limb, and mobile platform are shown in Figs. 5.2, 5.3, and 5.4, respectively. For the UPR limb, as shown in Fig. 5.2, the limb coordinate system Ai -x i yi zi is established as follows: The yi axis is parallel to the R axis, the zi axis is along the direction of the limb, and the x i axis is determined by the right-hand rule. Based on screw theory, there exists a driving force f i1 that passes through the U-joint center Bi and points to the R-joint center Ai , a constraint force f i2 that passes through the U-joint center Bi and is parallel to the R-joint axis, and a constraint couple T i1 that is perpendicular to the two axes of the U joint. For the RPU limb, as shown in Fig. 5.3, the x i axis is parallel to the R axis, the zi axis is along the direction of bar Bi Ai , and the yi axis is determined by the right-hand rule. Furthermore, there is a driving force f i1 that passes through U-joint center Ai and points to R-joint center Ai . There is also a constraint force f i2 that passes through U-joint center Bi and is parallel to the R-joint axis. Finally, there is a constraint couple T i1 that is perpendicular to the two axes of the U joint. The unit vectors of f i1 , f i2 , and T i1 are given below: f i1 = l i , τ i1 = X × yi (i = 1, 2, 3), f i2 = yi (i = 1, 2), f i2 = x i (i = 3), (5.30) Fig. 5.2 UPR limb. a free-body diagram and b section view

Bi

fiz

fi 2

fiy

M iz

f i1

M ix

Ti1

zi Ti1 Ai

(a)

yi

fi 2

fi1 (b)

Ti 2

xi

152

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

fiz

Fig. 5.3 Free-body diagram and section view of the RPU limb

M iz

M iy

fix Ti1

zi

fi1

yi fi 2

xi

T31 $33r

Fig. 5.4 Free-body diagram of the moving platform

r 11 11

f $

T11 $13r f12 $12r T12

f 32 $32r T21 $23r f $ r

F$ z

21 21

y

o x

f 22 $22r T22

where X is the unit vector along the X-axis. For the UPR limb, according to the translation of a force from theoretical mechanics, the force acting on a rigid body can be equivalently translated to any point on the rigid body, but a couple must be added to the plane determined by the force and the point. This moment is equal to the moment of the force on the point. Hence, a force f i2 and a couple T i2 at the R-joint center are equivalent to the force f i2 acting at the U-joint center, where T i2 = Ti2 τ i2 = L i z i × f i2 yi = −L i f i2 x i =m i2 (−x i ). Because the flexibility matrix C i is independent of the coordinate systems, for convenience of calculation, we express the above vectors in the limb coordinate system. Based on the mechanics of materials and screw theory [11], the internal forces at any cross-section of the UPR and RPU limbs can be expressed as follows: ⎧ ⎪ f i z = f i1 ⎪ ⎪ ⎪ ⎪ ⎨ f i y = f i2 Mi x = Ti1 τ i1 · x i + Ti2 τ i2 · x i + [vi (−z i ) × f 12 f i1 ] · x i ⎪ ⎪ ⎪ = Ti1 τ i1 · x i − L i f i2 + vi f i2 ⎪ ⎪ ⎩M = T τ · z +T τ ·z = T τ · z iz i2 i2 i i2 i2 i i1 i1 i

(i = 1, 2), (5.31)

5.2 Example: 2UPR-RPU PM

153

⎧ f i z = f i1 ⎪ ⎪ ⎨ f i x = f i2 ⎪ Mi y = Ti1 τ i1 · yi + [vi (−z i ) × f i2 x i ] · yi = Ti1 τ i1 · yi − vi f i2 ⎪ ⎩ Mi z = Ti1 τ i1 · z i = Ti1 τ i1 · z i

(i = 3). (5.32)

The strain energies of the UPR and RPU limbs are given as {L i (

f i2z

f i2y

Mi2x

Mi2z

)

Li Li f2 + f2 2E i Ai i1 2G i Ai y i2 (i = 1, 2) , 0 2 2L 2L · x L τ L i3 · x ) · z ) (τ (τ i1 i i i 2 i 2 f2 − f i2 Ti1 + i1 i Ti1 + i1 i Ti1 + 6E i Ii x i2 2E i Ii x 2E i Ii x 2G i Ii

Ui =

2E i Ai

+

2G i Ai y

+

2E i Ii x

+

2G i Ii

dvi =

(5.33)

{L i ( Ui =

) Mi2y Mi2z f i2z f i2x + + + dvi 2E i Ai 2G i Ai x 2E i Ii y 2G i Ii

0

Li Li L i3 τ i1 · yi L i2 (i = 3). f i12 + f i22 + f i22 − f i2 Ti1 2E i Ai 2G i Ai x 6E i Ii y 2E i Ii y (τ i1 · yi )2 L i 2 (τ i1 · z i )2 L 1 2 + Ti1 + Ti1 2E i Ii y 2G i Ii

=

(5.34)

According to the Castigliano’s second theorem, the elastic deformations of the UPR and RPU limbs at the constraint reaction point along the axes of the constraint wrenches are given as δi1 = δi2 = δi3 =

∂Ui ∂ f i1 ∂Ui ∂ f i2 ∂Ui ∂ Ti1

δi1 = δi2 = δi3 =

= = =

∂Ui ∂ f i1 ∂Ui ∂ f i2 ∂Ui ∂ Ti1

Li f E i Ai i1 Li τ i1 ·x i L i2 L3 f + 3Ei 1Ii x f i2 − 2E Ti1 G i Aiy i2 i Ii x τ i1 ·x i L i2 (τ i1 ·x i )2 L i (τ ·z )2 L − 2Ei Ii x f i2 + Ei Ii x Ti1 + i1G i iIi i

= = =

Li , c12i = E i Ai (τ ·x )2 L c33i = i1Ei Iii x i i i

c23i = c32i = c11 = i

τ i1 ·x i L i2 − 2E , i Ii x i i

Li , c12 E i Ai (τ i1 · yi )2 L i E i Iiy

= c13 = c21 = c31 = (τ ·z )2 L

(5.35)

(i = 3).

(5.36)

Ti1

Li f E i Ai i1 Li L3 τ i1 · yi L i2 f + 3EiiIiy f i2 − 2E Ti1 G i Ai x i2 i Iiy τ i1 · yi L i2 (τ i1 · yi )2 L i (τ ·z )2 L − 2Ei Iiy f i2 + Ei Iiy Ti1 + i1G i iIi i

For the UPR limb, c11i =

(i = 1, 2),

Ti1

c13i = c21i = c31i = 0, c22i =

(τ ·z )2 L + i1G i iIi i (i = 1, 2). L3 0, c22i = G iLAi i x + 3EiiIiy , c23i

L3 Li + 3Ei iIi x G i Aiy

,

For the RPU limb, = c32i = −

+ i1G i iIi i (i = 3). c33i = The limb compliance matrices can thus be obtained as follows:

τ i1 · yi L i2 , 2E i Iiy

154

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

⎡

Li E i Ai

⎢ Ci = ⎢ ⎣ 0 0 ⎡ L

0

⎤

0

Li L3 τ i1 ·x i L i2 + 3Ei iIi x − 2E G i Aiy i Ii x τ i1 ·x i L i2 (τ i1 ·x i )2 L i (τ i1 ·z i )2 L i − 2E + I E I G i Ii i ix i ix i

⎢ Ci = ⎢ ⎣ 0 0

E i Ai

0

⎥ ⎥ (i = 1, 2), ⎦

0

L3 τ i1 · yi L i2 Li + 3Ei iIiy − 2E G i Ai x i Iiy τ i1 · yi L i2 (τ i1 · yi )2 L i (τ i1 ·z i )2 L i + − 2E E i Iiy G i Ii i Iiy

(5.37)

⎤ ⎥ ⎥ (i = 3). ⎦

(5.38)

After C i is solved for, K i can be solved by C i−1 . In Eqs. (5.37) and (5.38), the flexibility matrix C i is a 3 × 3 symmetric matrix rather than the more familiar 6 × 6 matrix and its expression is relatively concise. This indicates that the method of Castigliano’s second theorem can reduce the dimensions of the matrix. The fact that c23i and c32i are not zero means that a coupling relationship exists between δi2 and δi3 . Equations 5.37 and 5.38 shows that flexibility matrix C i is independent with respect to the limb coordinate frame. Hence, the local coordinate system is selected only to facilitate the expression of the vectors. An overconstrained PM is a statically indeterminate structure because of the overconstrained wrenches in each limb. That is, there are more unknowns than equilibrium equations. To complete the elastostatic analysis of an overconstrained PM, as many supplementary equations as overconstraints must be introduced. The detailed process for this is given below. The equilibrium equation of the mobile platform can be obtained using the following equation, defined in the fixed coordinate system: W = ( J 1 J 2 · · · J n ) f = G Ff f ,

(5.39)

where J i = [$ri1 $ri2 $ri3 ], G Ff = [ J 1 J 2 J 3 ], f = [ f 1 f 2 f 3 ]T , W = [F M] is an external wrench acting on the mobile platform, and $irj is a unit screw of the jth constraint force that the ith limb exerts on the mobile platform. The VW principle of rigid bodies yields W T D = [ f 1 f 2 f 2 ]T [δ 1 δ 2 δ 3 ] = f 1T δ 1 + f 2T δ 2 + f 3T δ 3 .

(5.40)

Transposing Eq. (5.39) and then post-multiplying by D on both sides, we have W T D = f 1T J 1T D + · · · + f nT J nT D.

(5.41)

Equations (5.40) and (5.41) lead to δ i = J iT D (i = 1, 2, 3).

(5.42)

where D is the infinitesimal twist of the geometric center point of the mobile platform. Equation (5.42) represents the DCEs between the elastic deformation of the ith

5.2 Example: 2UPR-RPU PM

155

limb end along the axis of the constraint wrenches and the infinitesimal twist of the geometric center point of the mobile platform. After the DCEs are obtained from Eq. (5.42), according to material mechanics, we have f i =K i δ i .

(5.43)

Then, by substituting Eqs. (5.42) and (5.43) into Eq. (5.39), we have W=

n Σ

[[$ri1 . . . $ri3 ] · [ f i1 · · · f i3 ]T ] =

i=1

3 Σ

Ji K i JiT D = K D,

(5.44)

i=1

K=

n Σ

J i K i J iT .

(5.45)

i=1

Transposing Eq. (5.45) on both sides, we have KT=

n Σ

( J iT )T K iT ( J i )T =

i=1

n Σ

J i K iT J iT = K .

(5.46)

i=1

Equation (5.45) is a general and relatively concise algebraic expression for the overall stiffness matrix. When the mechanism is far from the singular position, the stiffness matrix K is full rank, which means the stiffness matrix K is invertible. Equation (5.46) shows that the overall stiffness matrix K is a symmetric matrix whose symmetry is not affected by the coordinate system. In this chapter, we derive an algebraic expression for the overall stiffness matrix from the perspective of the strain energy and Castigliano’s second theorem. The model proposed in this chapter is universally applicable to overconstrained PMs and the expression is relatively concise. From Eq. (5.44), we have D=K −1 W .

(5.47)

Substituting Eqs. (5.42) and (5.47) into Eq. (5.43) gives f iT =K i J iT K −1 W (i = 1, 2, 3).

(5.48)

The driving forces f d can be expressed as f d = [ f 11 f 21 . . . f n1 ]T .

(5.49)

To verify the correctness and feasibility of the method presented in this chapter, finite element commercial software ANSYS was used to develop a unified FEA

156

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

model. In this model, the material and structural parameters are given according to the parameters used in the theoretical model. The compliant limbs are modeled using the beam188 element based on Timoshenko beam theory, which can simulate the spatial composite elastic deformation of a beam. In this chapter, the joints are assumed to be rigid and connect the rigid mobile platform to the flexible limbs. This constraint ensures the deformation compatibility relations and allows force transmission between the two objects. The mpc184 element is used to define this constraint, which can model R joints, U joints, S joints, and rigid beams, among others. Two coincidence points are selected, one attached to the mobile platform and the other attached to the flexible limb [13]. For different joints, the settings of the mpc184 element are different. For different configurations and topologies, we simply change the coordinates of the vertexes of the PMs and the settings of the mpc184 element in the command stream. The mobile platform is assumed to be rigid in this chapter, which is modeled by the rigid beam mpc184 element. The stiffness performance is related to the material, structural parameters, and configuration. Three examples are considered in this chapter. Physical and architectural parameter values of the 2UPR-RPU overconstrained PM are given first, followed by three positions, and then the length of the limbs is given according to the inverse kinematics. In the given applied external wrench, the deformation of the geometric center point of the mobile platform is considered. We suppose that the external wrenches are defined in the fixed coordinate frame {O} and are applied on the geometric center o. The physical and structural parameters of the 2UPR-RPU overconstrained PM are listed in Table 5.1, where R = R1 = R2 = R3 , r = r 1 = r 2 = r 3 , and d denotes the diameter of the limb cross-section. Three configurations of the 2UPR-RPU overconstrained PM and the corresponding external wrenches imposed on the mobile platform are considered, and the inverse kinematics are solved. The results are listed in Table 5.2, where h = Oo. For the three examples, the FEA simulation models with both deformed and un-deformed shapes of the 2UPR-RPU overconstrained PM can be found in Ref. [11]. The results based on the FEA model and the theoretical model are given in Table 5.3. The results in Table 5.3 show that the relative error is within 0.8%. The results obtained using the theoretical model agree well with those obtained by the FEA model, verifying that the theoretical model presented in this chapter is acceptable for stiffness analysis. Table 5.1 Physical and architectural parameter values of the 2UPR-RPU overconstrained PM

Parameters

Units

Values

R

mm

390

r

mm

260

d

mm

100

E

GPa

210

G

GPa

80

5.2 Example: 2UPR-RPU PM

157

Table 5.2 Configurations of the 2UPR-RPU overconstrained PM and its applied wrenches Parameters

Units

Example 1

Example 2

Example 3

h

mm

800

800

600

β

rad

0

0

− 0.524

γ

rad

0

0

0.524

L1

mm

810.494

810. 494

498.066

L2

mm

810. 494

810. 494

748.378

L3

mm

810. 494

810. 494

798.793

Fx

N

0

100

− 20

Fy

N

0

− 100

200

Fz

N

− 100

100

100

Mx

Nm

0

− 70

10

My

Nm

0

− 70

100

Mz

Nm

0

70

150

Table 5.3 Deformation of the geometric center point of the mobile platform of the 2UPR-RPU overconstrained PM (comparison of the FEA model and theoretical method results) δϕ(rad)

δψ(rad)

− 2.522 × 6.466 × 10–8 10–8

0

0

FEA results − 2.474 × 5.177 × 10–22 10–8

− 2.524 × 6.472 × 10–8 10–8

− 1.059 × − 5.294 × 10–22 10–23

Error (%)

0.1

Ex Method 1

2

3

δx(m)

Theoretical 0

–

δy(m)

δz(m)

5.173 × 10–8

0.1

–

–

Theoretical 1.594 × 10–5

− 2.785 × 1.222 × 10–6 10–7

5.244 × 10–7

9.704 × 10–6

2.043 × 10–5

FEA results 1.599 × 10–5

− 2.793 × 1.223 × 10–6 10–7

5.282 × 10–7

9.733 × 10–6

2.048 × 10–5

Error (%)

0.7

0.3

0.2

0.3

0.1

δθ(rad)

0.3

0.1

Theoretical 6.938 × 10–6

5.394 × 10–6

− 2.232 × − 4.618 × 2.376 × 10–6 10–6 10–5

2.263 × 10–5

FEA results 6.956 × 10–6

5.408 × 10–6

− 2.239 × − 4.631 × 2.383 × 10–6 10–6 10–5

2.269 × 10–5

Error (%)

0.3

0.3

0.3

0.3

0.3

0.3

5.2.2 Stiffness Performance Optimization This section adopts the comprehensive stiffness index as the optimization objective. Since the manipulator has different stiffness performance in different directions and bears different loads in different machining modes, so it is impossible to describe the stiffness performance of the mechanism by the stiffness performance of one or several specific directions. In more cases, engineers pay more attention to the linear stiffness

158

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

performance of the mechanism resistance of force load. Without loss of generality, this section mainly evaluates the linear stiffness performance K zf of the mechanism against external forces along the Z-axis of the Cartesian coordinate system. The mathematical model of stiffness performance optimization of the 2UPR-RPU PM can be expressed as follows: objective function : f (x) = K z f design variables: x = [r1 r2 r3 ] |αi1 |max , |αi2 |max ≤ 50◦ constraint: 0.3 m ≤ li ≤ 1.0 m 0.3 m ≤ r1 , r2 ≤ 0.6 m , 0.2 m ≤ r3 ≤ r1

(5.50)

Step size of the design parameters is set as 0.01 m, and the slice distribution of the K zf in the complete parameter space is shown in Fig. 5.5. The maximum value of the objective function is obtained as follows: K zf min = 4.657 × 108 N/m, K zf max = 9.940 × 108 N/m. In this section, genetic algorithm (GA) and particle swarm optimization (PSO) algorithm are adopted to optimize the stiffness performance of the 2UPR-RPU PM. The parameter settings of GA and PSO algorithm are given in Tables 5.4 and 5.5, respectively. The iteration graphs of GA and PSO algorithm are shown in Fig. 5.6a, b, respectively. The results of K zf before and after optimization are given in Table 5.6. The error of the results calculated by the three algorithms does not exceed 0.3%, which shows the effectiveness and feasibility of the optimization results. Compared with the results of before optimization, the stiffness performance after optimization of the 2UPR-RPU PM was greatly improved. The results show that compared with GA, PSO algorithm obtained better results with fewer iterations, and the stiffness performance was improved by 47.93% compared with before optimization. Fig. 5.5 Distribution of the K zf in the complete parameter space

5.3 Summary

159

Table 5.4 Parameter settings of GA Population size

Maximum number of iterations

Number of stop iterations

The number of variables

Cross-proportion

The number of elite

50

300

50

3

0.8

2

Table 5.5 Parameter settings of PSO algorithm Population size

Maximum number of iterations

Inertial weight range

Learning factor c1

Learning factor c2

100

600

[0.2 1.1]

1.49

1.49

Fig. 5.6 Iterations of the K zf . a GA, b PSO algorithm

Table 5.6 Comparison of the K zf before and after optimization K zf (N/m)

r 1 /m

r 2 /m

r 3 /m

Before optimization

6.737 × 108

0.390

0.260

0.260

Numerical discretization method

9.940 ×

108

0.330

0.600

0.200

After optimization-GA

9.959 × 108

0.328

0.600

0.200

After optimization-PSO

9.966 × 108

0.301

0.600

0.200

5.3 Summary In this chapter, a CSI that decouple the linear and angular stiffness is introduced, then, extreme stiffness index and corresponding extreme direction are further obtained based on the Lagrangian equation. Finally, the stiffness performance along the zaxis and optimization of the 2UPR-RPU PM is taken as an example for analysis and discussion. The errors of optimization results of three algorithms that include

160

5 Elastostatic Stiffness Evaluation and Optimization of Parallel Manipulators

numerical discretization method, GA, and PSO are within 0.3%, and compared with GA, PSO algorithm obtained better fitness with fewer iterations, and the stiffness performance along z-axis was improved by 47.93% compared with before optimization.

References 1. E. Courteille, D. Deblaise, P. Maurine, Design optimization of a Delta-like parallel robot through global stiffness performance evaluation, in Proceedings—IEEE/RSJ International Conference on Intelligent Robots and Systems (2009), pp. 5159–5166. 2. B. Hu, Z. Huang, Kinetostatic model of overconstrained lower mobility parallel manipulators. Nonlinear Dyn. 86, 309–322 (2016) 3. C. Gosselin, A global performance index for the kinematic optimization of robotic manipulators. J. Mech. Des. 113, 220–226 (1991) 4. B.S. El-Khasawneh, P.M. Ferreira, Computation of stiffness and stiffness bounds for parallel link manipulators. Int. J. Mach. Tools Manuf 39, 321–342 (1999) 5. G. Carbone, M. Ceccarelli, Comparison of indices for stiffness performance evaluation. Front. Mech. Eng. 5(3), 270–278 (2010) 6. G. Carbone, Stiffness analysis and experimental validation of robotic systems. Front. Mech. Eng. 6(2), 182–196 (2011) 7. S.J. Yan, S.K. Ong, A.Y.C. Nee, Stiffness analysis of parallelogram-type parallel manipulators using a strain energy method. Robot. Comput.-Integr. Manuf. 37, 13–22 (2016) 8. B. Lian, T. Sun, Y. Song, Y. Jin, M. Price, Stiffness analysis and experiment of a novel 5-DoF parallel kinematic machine considering gravitational effects. Int. J. Mach. Tools Manuf. 95, 82–96 (2015) 9. H. Wang, L. Zhang, G. Chen, S. Huang, Parameter optimization of heavy-load parallel manipulator by introducing stiffness distribution evaluation index. Mech. Mach. Theory 108, 244–259 (2017) 10. W.-A. Cao, D. Yang, H. Ding, A method for stiffness analysis of overconstrained parallel robotic mechanisms with Scara motion. Robot. Comput.-Integr. Manuf. 49, 426–435 (2018) 11. C. Yang, Q. Li, Q. Chen, L. Xu, Elastostatic stiffness modeling of overconstrained parallel manipulators. Mech. Mach. Theory 122, 58–74 (2018) 12. X.X. Chai, J.N. Xiang, Q.C. Li, Singularity Analysis of a 2-UPR-RPU Parallel Mechanism. Chin. J. Mech. Eng-En. 51(13), 144–151 (2015) 13. C. Yang, Q.H. Chen, J.H. Tong, Q.C. Li, Elastostatic stiffness analysis of a 2PUR-PSR overconstrained parallel mechanism. Int. J. Precis. Eng. Manuf. 20, 569–581 (2019)

Chapter 6

A Methodology for Optimal Stiffness Design of Parallel Manipulators Based on the Characteristic Size

It is important to design the optimal structural size ratio for the stiffness performance of parallel manipulators (PMs), so that the mechanism can be scaled according to the engineering environment requirements for the stiffness performance to design different specifications. This chapter presents a methodology for optimal stiffness performance design of PMs based on the characteristic size. The dimensional and sectional parameters of the mechanism can be expressed as the function of the characteristic size and non-dimensional scale factors. The 2UPR-RPU and 2PRU-PSR PMs were taken as two examples for analysis and discussion. The influence of the scaling of elasticity modulus and characteristic size on the stiffness performance of PMs is investigated.

6.1 Methodology for the Optimal Stiffness Performance Design of PMs The assumptions of the stiffness modeling for the proposed stiffness performance optimization design method of PMs are given as following: (1) The limbs are modeled by rods, the deformation of rods satisfy Hooke’s law, that is, in the linear elastic range; (2) all rods are made of the same material; (3) the base and moving platform are assumed to be rigid because they are much stiffer than the rod; (4) flexibility, friction, and clearance of joint are ignored; (5) the effect of gravity is ignored, wherein, assumptions (1)–(4) are reasonable for the stiffness analysis of PMs, while assumptions (5) and (6) are to better present the stiffness design method introduced in this paper. The overall stiffness matrix [1] can be expressed as follows: K=

n Σ

J i K i J iT ,

(6.1)

i=1

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_6

161

162

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

where K is the overall stiffness matrix, J i is the constraint screw system of the ith limb, and K i is the stiffness matrix of the ith limb. So far, researchers have proposed many stiffness performance indices, such as determinant index [2], trace index, eigenvalue index [3], condition number index [4], and extreme stiffness index [5], etc. However, how to use these indices to design a group of mechanisms with similar stiffness performance is always a challenge for PM stiffness design. In this chapter, the characteristic size of the mechanism is firstly defined, and then the proportional factor between structure parameters and the characteristic size of the mechanism is defined as non-dimensional design parameters. Theoretically, a reasonable stiffness performance index not only meets the definite physical meaning, but also meets the requirement that the corresponding structure size ratio of the optimal stiffness performance of the mechanism under different characteristic sizes should be consistent, that is to say, the stiffness performance of the mechanism under different characteristic sizes is similar. Therefore, one can scale the characteristic size of the mechanism to design the mechanism size that meets the stiffness performance requirements of the engineering environment. The first and foremost for the optimal stiffness performance design of PMs is to design a set of non-dimensional design parameters. In this chapter, the distance between the base and the moving platform at the initial installation position is defined as the characteristic size h, the design parameters of the mechanism is defined by x ∗ = [a1 , . . . , am , ε1 , . . . , εn ],

(6.2)

where x* is the non-dimensional design parameters, ai (i = 1, 2, … m) is the ratio between the structure parameters consist of moving platform size, base size, and rod length, and the characteristic size h; εi is the length-to-diameter ratio, namely the ration of rod length and rod diameter. Stiffness optimization design is to identify the optimal geometry arrangement consists of finding the array x * opt that maximizes the objective optimization F. Here, F represents the stiffness index attained based on the overall stiffness matrix K. The limb stiffness matrix is generally obtained by inverting the compliance matrix C i that is the function of elasticity modulus E, shear modulus G, rod diameter d i , and G , the limb stiffrod length L i . According to the definition in Eq. (6.2) and E = 2(1+μ) ∗ ness matrix C i based on the non-dimensional design parameters and characteristic size can be rewritten as C i∗ =

εi N ∗ (εi , h, a1 , . . . , am ), Ehci (a1 , . . . , am ) i

(6.3)

where ci (·) denotes coefficient ci is the function of argument (·), N i∗ (·) represents that matrix N i∗ is the function of argument (·). It should be noted that the flexibility matrix C i∗ can be decomposed into two parts: matrix external variables and matrix internal variables. The former includes E, h, εi , a1 , …, am , and the latter includes h, εi , a1 , …, am . The elasticity modulus achieves

6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM

163

the decoupling from the compliance matrix, while h cannot be separated from the compliance matrix. Since the constraint screw system J i is the function of the non-dimensional design parameters and characteristic size, thus, the overall stiffness matrix K can be rewritten as K ∗ = EhΩ ∗ (ε1 , . . . , εn , a1 , . . . , am , h),

(6.4)

where Ω ∗ (·) denotes that matrix Ω ∗ is the function of argument (·). Similarly, K* also consists of two parts, the external variables and the internal variables of the matrix, the former includes E and h, and the latter includes am , ε1 , …, εn , h. According to Eqs. (6.3) and (6.4), the following conclusions can be drawn: (1) The elasticity modulus is decoupled from the overall stiffness matrix, and its influence on the stiffness performance of the mechanism varies with the stiffness evaluation index, e.g., it presents an exponential influence on the determinant index and a linear influence on the eigenvalue index. (2) The stiffness performance of the mechanism increases with the increase of the elasticity modulus and decreases with the length-to-diameter ratio. (3) It is worth noting that for a mechanism composed of the UPS/SPS limb with equal cross-section, such as the 6SPS PM, the limb stiffness matrix K i = EA/L i , where E, εi , a1 , h can be decoupled from the limb stiffness matrix. It can be seen from Eq. (6.1) that the first three rows and the first three columns of the overall stiffness matrix are independent of the characteristic size, that is to say, in this case, the optimal stiffness performance design of the mechanism for the linear displacement stiffness performance under the force load is decoupled from the characteristic size. (4) In general, it is difficult to decouple the characteristic size from the overall stiffness matrix, and its influence on the stiffness performance of the mechanism should be determined in combination with numerical simulation. Next, this paper introduces the stiffness performance design method of the mechanism in detail through two examples of three degree-of-freedom (DOF) asymmetrical PM and identifies the suitable performance evaluation index for the optimal stiffness performance design of the mechanism. Finally, a group of similar mechanisms with the same optimal structural size ratio are designed.

6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM Figure 6.1 shows a 3-DOF 2UPR-RPU PM, which is made up of a fixed base, a moving platform, two extensible UPR limbs, and one RPU limb. The upper end of each UPR limb is connected to the fixed base by a universal joint, while the lower

164

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

Fig. 6.1 Structure of the 2UPR-RPU PM

end is connected to the moving platform by a revolute joint. The upper end of the RPU limb is connected to the fixed base by a revolute joint, while the lower end is connected to the moving platform by a universal joint. The mechanism motion is obtained by moving the three translational motors. Two Cartesian coordinates, O(X, Y, Z) and o(x, y, z), are, respectively, attached to the fixed base and moving platform. The origin of the (X, Y, Z) is located at the midpoint of the line B1 B2 , Xand Y-axes along the direction of OB2 and OB3 , respectively. The origin of frame (x, y, z) is located at the midpoint of A1 A2 , x- and y-axes along the direction of oA2 and oA3 , respectively. The location of the moving platform relative to the fixed base is described by the position vector p and a rotation matrix R. The 2UPR-RPU PM has three DOFs, namely a translation along the Oo, a rotation γ about the y-axis, and a rotation β about the X-axis. OB1 = OB2 = r 1 , OB3 = r 2 , oA1 = oA2 = r 3 , oA3 = r 4 = r 2 r 3 /r 1 . The inverse kinematics of this mechanism can be found in [6]. Consider the distance between the base and moving platform at the initial position as the characteristic size, the geometrical parameters of the 2UPR-RPU PM is defined by ⎧ ⎨ r1 = a1 h; r2 = a2 h a2 a3 . ⎩ r3 = a3 h; r4 = h a1

(6.5)

In order to limit the ratio of the overall size to the characteristic size of the mechanism, without loss of generality, the inequality constraint equation for the scale factor ai in Eq. (6.5) in is defined as follows: 2a1 + a2 + 2a3 +

a2 a3 ≤ 2.5. a1

(6.6)

6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM

165

The length of rods in the initial position can be expressed as the function of the characteristic size and scale factors. ⎧ / / ⎪ ⎨ L 10 = L 20 = h 2 + (r1 − r3 )2 = h 1 + (a1 − a3 )2 / / , (6.7) a2 a3 2 2 2 ⎪ ) ⎩ L 30 = h + (r2 − r4 ) = h 1 + (a2 − a1 where L io is the rod length in the initial installation position. Considering that the cross-section of the rods are circular, ε1 = ε2 is considered due to the one-sided symmetrical structure of the mechanism, and the diameter of section parameters can be expressed as follows: ⎧ / ⎪ h 1 + (a1 − a3 )2 ⎪ ⎪ ⎪ ⎨ d1 = d2 = ε1 / . ⎪ h 1 + (a2 − aa2 a1 3 )2 ⎪ ⎪ ⎪ ⎩ d3 = ε3

(6.8)

So far, all structural parameters of the mechanism are expressed as functions of the characteristic size and scale factors. The elastostatic modeling based on the screw theory is adopted in this work to establish the overall stiffness matrix. The process is briefly reviewed below. Consider the rod cross-section is circular. Limb compliance matrix based on the strain energy and screw theory is given as follows [1]: ⎡

Li E Ai

⎢ Ci = ⎢ ⎣ 0 0 ⎡ L

0

⎤

0

L i3 τ i ·x i L i2 − 2E 3E Ii x Ii x τ i ·x i L i2 (τ i ·x i )2 L i (τ i ·z i )2 L i − 2E + Ii x E Ii x G Ii

Li G Aiy

+

0

i

⎢ E Ai Ci = ⎢ ⎣ 0 0

⎥ ⎥ (i = 1, 2), ⎦

0

L i3 τ i · yi L i2 − 2E 3E Iiy Iiy τ i · yi L i2 (τ i · yi )2 L i (τ i ·z i )2 L i + − 2E Iiy E Iiy G Ii

Li G Ai x

+

(6.9)

⎤ ⎥ ⎥ ⎦

(i = 3),

(6.10)

where τ i is unit vector perpendicular to the two axes of the U joint, xi , yi , and zi denote the unit vectors of the x i -, yi -, and zi -axes, respectively. L i and Ai are the length and cross-sectional area of the ith rod, I ix and I iy are the moment of inertia about x- and y-axes, respectively, I i is the polar moment of inertia. Substituting Eqs. (6.5)–(6.8) into Eqs. (6.9) and (6.10), one can have ⎡ Ci =

1 afi

4εi2 ti ⎢ ⎢ 0 π Eh ⎣ 0

0

0

16ε2 t 2 8τ ·x ε2 t 2(1+μ) + 3ai2 i − i1ha 2i i i afi fi fi 8τ ·x ε2 t 16εi2 [(τ i1 ·x i )2 +(1+μ)(τ i1 ·z i )2 ] − i1ha 2i i i 2 2 h afi fi

⎤ ⎥ ⎥ (i = 1, 2), (6.11) ⎦

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6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

⎡ Ci =

1 afi

⎢ ⎢ 0 π Eh ⎣ 0 4εi2 ti

0

⎤

0

16ε2 t 2 8τ · y ε2 t 2(1+μ) + 3ai2 i − i1ha 2i i i afi fi fi 8τ · y ε2 t 16εi2 [(τ i1 · yi )2 +(1+μ)(τ i1 ·z i )2 ] − i1ha 2i i i 2 h2 a f i fi

⎥ ⎥ (i = 3), ⎦

(6.12)

where t1 = [0, sβ, cβ] − a3 Re1 + a1 e1 , t2 = [0, sβ, cβ] + a3 Re1 − a1 e1 , t3 = [0, sβ, cβ] + aa2 a1 3 Re2 −a2 e2 , e1 = [1, 0, 0]T , e2 = [0, 1, 0]T , e3 = [0, 0, 1]T . a f 1 = a f 2 = 1 + (a1 − a3 )2 , a f 3 = 1 + (a2 − aa2 a1 3 )2 . Finally, the overall stiffness matrix can be obtained by Eq. (6.1). Equations (6.11–6.12) show that the characteristic size h cannot be completely separated from the limb compliance matrix, and it exhibits a coupling relationship with the overall stiffness matrix. It is difficult to conclude from the analytical expression whether the same optimal design parameters exist under different characteristic sizes. Without loss of generality, this paper will select the determinant index, trace index, condition number index, eigenvalue index, and the extreme linear displacement index under the action of force load with a clear physical explanation to carry out the stiffness performance optimization design analysis. The influence of the characteristic size on the optimal design parameters and stiffness performance of the mechanism is investigated, including identification of suitable indices for stiffness performance design. Since the 2UPR-RPU PM is unilateral symmetric mechanism, the optimum length-to-diameter ratio of UPR link and RPU link may be different under the constraint of equal mass, as a result, it is necessary to redesign its slenderness ratio. Considering that the link slenderness ratio of the mechanism before optimization is ε0 = 9, the mass constraint equation of the mechanism is defined as follows: m = m0,

(6.13)

where m = 2ρ L 10 A21 + ρ L 30 A23 and m0, respectively, denote the mass of rods in the initial position of the mechanism before and after optimization. Substituting Eqs. (6.7) and (6.8) into Eq. (6.13) leads to [ | | ε3 = | | |

3/2

af3

( 3/2 af1

2 ε02

+

3/2

af3

3/2

ε02 a f 1

), −

(6.14)

2 ε12

where a f 1 = 1 + (a1 − a3 )2 and a f 3 = 1 + (a2 − a2 a3 /a1 )2 . Therefore, the mathematical model of the optimal stiffness performance design of the mechanism is defined by

6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM

167

⎧ max (stiffness index) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ design parameters x = [a1 a2 a3 a4 ε1 ε3 ] ⎪ ⎪ ⎪ a1 , a2 , a3 , a4 ∈ [0.2, 1.2]; ε1 , ε3 ∈ [6, 12] ⎪ ⎪ ⎪ [ ⎪ ⎪ | ⎪ 3/2 ⎨ | af3 | ( ) ε3 = | , 3/2 af3 ⎪ | 3/2 2 2 ⎪ ⎪ a + − 3/2 ⎪ f1 ε02 ε12 ε02 a f 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a4 = a2 a3 /a1 ⎪ ⎪ ⎪ ⎪ a a ⎪ ⎩ a1 ≥ a3 ; 2a1 + a2 + 2a3 + 2 3 ≤ 2.5 a1

(6.15)

where the stiffness index is the average value of the node function values in the h-plane [β, γ ]–[− 10°, 10°] interval. In order to study the influence of the characteristic size of the mechanism on the optimal design parameters of the stiffness performance and identify which indices suitable for the stiffness performance design, 30 groups of data from the interval 0.1– 5 m for characteristic size h were uniformly generated to carry out stiffness optimization design, respectively. Table 6.1 shows the representative optimal non-dimensional design parameters for multiple stiffness performance indices of the 2UPR-RPU PM under seven characteristic factors. The results show that the robustness of the trace index, condition number index, and eigenvalue index are poor, that is to say, the optimal structure size ratio of the mechanism is different for different characteristic sizes. Obviously, it is not conducive to the stiffness performance design of the mechanism. The main reason is that these indices have vague physical meaning due to the non-uniform dimension of the overall stiffness matrix or do not consider the influence of all elements in the overall stiffness matrix, e.g., the condition number index and eigenvalue index only consider the eigenvalues of the overall stiffness matrix, and the trace index only the diagonal elements of the matrix are considered. The extreme linear displacement stiffness index and linear displacement homogeneous index under force load [5], as well as the determinant index show strong robustness, that is to say, these indices are suitable for similar stiffness performance design of PMs. It is worth noting that the determinant index takes into account the influence of all the elements of the overall stiffness matrix and shows the comprehensive performance of the mechanism. Although it shows strong robustness in the stiffness performance design, it is not recommended to be used for the optimal stiffness performance design of the mechanism due to its unclear clear physical meaning. Figure 6.2 shows the influence curve of elasticity modulus on the extreme linear displacement stiffness performance and linear displacement homogeneous performance under the condition of characteristic size h = 1 m (these influence curves for different characteristic sizes are similar). The elasticity modulus is linearly related to the linear displacement stiffness performance, and one can design a mechanism that meets the stiffness performance requirements in the engineering environment by adjusting the elasticity modulus of the material, e.g., a material with an elasticity

168

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

Table 6.1 Optimal non-dimensional design parameters for multiple stiffness performance indices of the 2UPR-RPU PM under different characteristic sizes Trace index Condition number index

Minimum eigenvalue

ε1

ε3

a1

a2

a3

a4

h = 0.3 m

0.630

0.638

0.200

0.202

8.160

h = 0.8 m

0.200

0.850

0.200

0.850

12.000

6.573

h = 0.1 m

0.560

0.239

0.470

0.201

8.160

11.997

h = 0.5 m

0.540

0.218

0.500

0.202

8.380

10.815

h=1m

0.600

0.315

0.390

0.205

12.000

6.501

h=3m

0.630

0.434

0.300

0.206

11.980

6.456

h = 0.1 m

0.560

0.239

0.470

0.201

8.160

11.997

h = 0.5 m

0.540

0.218

0.500

0.202

8.380

10.815

h=1m

0.600

0.315

0.390

0.205

12.000

6.501

11.926

h=3m

0.630

0.434

0.300

0.206

11.980

6.456

Determinant index

h = 0.1–5 m

0.350

0.606

0.320

0.554

9.740

7.919

kdf min

h = 0.1–5 m

0.640

0.459

0.280

0.201

12.000

6.439

kdf max

h = 0.1–5 m

0.54

0.218

0.500

0.202

8.160

11.961

δ

h = 0.1–5 m

0.640 0.459 0.280 0.201 12.000 6.439 / / Note kd f min = 1/ max(λd f ), kd f max = 1/ min(λd f ), λdf is the eigenvalue of matrix C T11 C 11 , C 11 is the submatrix representing the first three rows and the first three columns of compliance matrix C. linear displacement homogeneous index δ = k df min /k df max

modulus of 140 GPa can be selected to achieve a minimum linear displacement stiffness of 1 × 107 N/m. Figure 6.2c is approximately a straight line (fluctuations are caused by calculation errors), indicating that the linear displacement homogeneous performance of the mechanism is independent of the elasticity modulus of the material, which is consistent with the theoretical results in Eq. (6.4). The analysis results is conducive to the stiffness performance design of PMs. Figure 6.3 shows the influence curve of the characteristic size on the stiffness performance of the mechanism while keeping the elasticity modulus E = 200 GPa unchanged (these influence curves for different elasticity modulus are similar). It is worth noting that the characteristic size of the 2UPR-RPU PM is also linearly related to the extreme linear displacement stiffness performance of the mechanism. One can adjust the characteristic size of the mechanism to design the size of the mechanism that meets the engineering requirements for stiffness performance, e.g., the characteristic size of the mechanism can be designed as h = 2 m to make the minimum linear displacement stiffness performance of the mechanism reach k df min = 2.9 × 107 N/m. Similarly, the scaling of the mechanism characteristic size does not affect the linear displacement homogeneous index of the mechanism. It is worth noting that the optimal structure size proportions of the mechanism corresponding to k df min and k df max are different, while those corresponding to k df min and δ are the same, indicating that the linear displacement homogenous performance of the 2UPR-RPU PM is mainly controlled by the minimum linear displacement

6.2 Example 1: Optimal Stiffness Performance Design of the 2UPR-RPU PM

169

Fig. 6.2 Influence curve of elasticity modulus E on stiffness performance of the 2UPR-RPU PM. a k df min , b k df max , and c δ

Fig. 6.3 Influence curve of characteristic size h on stiffness performance of the 2UPR-RPU PM. a k df min , b k df max , and c δ

stiffness performance. The optimal structure size ratio of the mechanism should be designed according to the engineering requirements of maximizing the minimum or maximum linear displacement stiffness performance. Figure 6.4 shows the stiffness performance comparison of the 2UPR-RPU PM before and after optimization under the characteristic size h = 1 m and elasticity modulus E = 210 GPa (the design parameters before optimization are a1 = a2 = 0.500, a3 = a4 = 0.333, ε1 = ε3 = 9 [1]), where η represents the ratio of the performance index after optimization to that of before optimization. The results show that the minimum, maximum, and homogeneous linear displacement stiffness performance of the 2UPR-RPU PM are all improved, especially the linear displacement homogeneous index, which is 6 times higher than that of before optimization. The optimization results show the effectiveness of the proposed stiffness optimization design method.

170

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

Fig. 6.4 Stiffness indices comparison of the 2UPR-RPU PM before and after optimization. a ηdf min , b ηdf max , and c ηδ

6.3 Example 2: Optimal Stiffness Performance Design of the 2PRU-PSR PM Figure 6.5 shows a 3-DOF 2PRU-PSR PM, the moving platform connected to the three sliders by two PRU limbs and one PSR limb. The one end of each PRU limb is connected to the slider by a revolute joint, while the other end is connected to the moving platform by a universal joint, wherein the R joints perpendicular to the sliders and parallel to the x-axis, the first axes of two U joints parallel to the R-joint axis at the other end, and the second axes collinear. The one end of the PSR limb is connected to the slider by a spherical joint, while the other end is connected to the moving platform by a revolute joint, wherein the R-joint axis parallel to second axis of U joints in PRU limbs. The mechanism motion is obtained by moving the three translational motors mounted on the three sliders. The location of the moving platform relative to the fixed base is described by the position vector p and a rotation matrix R. The 2PRU-PSR PM has three DOFs, namely a translation along the z-axis, a rotation α about the line that pass point A3 and parallel to the x-axis, a rotation β about the v-axis. OC 1 = OC 2 = r 1 , OC 3 = r 2 , oB1 = oB2 = r 3 , oB3 = r 4 = r 2 r 3 /r 1 , A1 B1 = A2 B2 = L 1 , and A3 B. = L 2 . The inverse kinematics of this mechanism can be found in [7]. Consider the distance between Ai and the moving platform as the characteristic size h, the geometric parameters of the mechanism can be expressed based on the characteristic size as Fig. 6.5 Schematic diagram of the 2PRU-PSR PM

6.3 Example 2: Optimal Stiffness Performance Design of the 2PRU-PSR PM

{

r1 = a1 h; r2 = a2 h; r3 = a3 h; r4 = a4 h L 1 = a5 h;

L 2 = a6 h

.

171

(6.16)

It should be noted that the links length are not independent. Once the characteristic size and the radius of the platform and base are specified, the links length are also uniquely determined. Therefore, the following equality constraints should be satisfied {

/ 1 + (a1 − a3 )2 . / a6 = 1 + (a2 − a4 )2 a5 =

(6.17)

As in Example 1, the constraint that the total mass is equal is applied to the rods, considering the cross-section of rods are circular, and the initial slenderness ratio of the rods is ε0 = 9. The slenderness ratio of rods should satisfy the following equality constraint: [ | a3 | ). ε3 = | ( 3 3 6 (6.18) 2a5 +a6 2a53 − 2 2 ε ε 0

1

The inequality constraints for the overall size of the mechanism is given as follows: 2a1 + a2 + 2a3 + a4 + 2a5 + a6 ≤ 7.

(6.19)

So far, all the structural parameters of the mechanism are expressed as functions of the characteristic size and non-dimensional scale factors. The elastostatic compliance matrix of the PSR limb obtained on the screw theory and strain energy can be expressed as follows [8]: [ C3 =

L2 G A3

+ 0

L 32 3E I3

] 0 L2 E A3

.

(6.20)

Substituting Eqs. (6.16)–(6.18) into Eq. (6.20), one can have C3 =

[ ] 4ε23 2(1 + μ) + 16ε23 0 . 0 1 π Ea6 h

(6.21)

The elastostatic compliance matrices of PRU limbs refer to Eq. (6.12). The overall stiffness matrix can be established by Eq. (6.1). The mathematical modeling of stiffness optimization design of the 2PRU-PSR PM can thus be established as follows:

172

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

⎧ max (stiffness index) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ design parameters x = [a1 a2 a3 a4 a5 a6 ε1 ε3 ] ⎪ ⎪ ⎪ a1 , a2 , a3 , a4 ∈ [0.2, 1.5]; ε1 , ε3 ∈ [6, 12] ⎪ ⎪ ⎪ / / ⎨ a4 = a2 a3 /a1 ; a5 = 1 + (a1 − a3 )2 ; a6 = 1 + (a2 − a4 )2 , [ ⎪ ⎪ | ⎪ a3 ⎪ | ⎪ ) ⎪ |( 3 3 6 ε = 3 ⎪ 2a5 +a6 2a53 ⎪ ⎪ − ⎪ 2 2 ε0 ε1 ⎪ ⎪ ⎪ ⎩ a1 ≥ a3 ; 2a1 + a2 + 2a3 + a4 + 2a5 + a6 ≤ 7

(6.22)

where stiffness index is the determinant index, condition number index, minimum eigenvalue index, trace index, extreme linear displacement index, or linear displacement homogeneous index. These indices are obtained by averaging the function ◦ ◦ values of discrete nodes in the plane z = 1.5 h and [α, β,] ∈ [−10 , 10 ]. Similarly, the characteristic size h is uniformly discretized into 30 sets of data in the intervals of h = 0.1–5 m to study the influence of the characteristic size on the optimal structure parameter ratio, so as to identify which index has strong robustness and is suitable for optimal stiffness performance design of the mechanism. Similar to the optimization results of the 2UPR-RPU PM, Table 6.2 shows that the optimal design parameters for the condition number index, minimum eigenvalue index, or trace index of the mechanism are inconsistent for different characteristic sizes, the robustness of these indices is poor, this further illustrates that these indices are not suitable for the optimal stiffness performance design of the mechanism due to its unclear physical meaning; the optimal design parameters for the determinant index, extreme linear displacement index, or linear displacement homogeneous index are consistent under different characteristic sizes, which illustrates the strong robustness of these indices again. Figure 6.6 shows the influence curve of the elasticity modulus E on the extreme linear displacement stiffness performance and linear displacement homogeneous performance under characteristic size h = 0.5 m. The elasticity modulus is linearly related to the extreme linear displacement stiffness performance; there is a horizontal linear relationship between elasticity modulus and linear displacement homogeneous stiffness performance. This is consistent with the optimization results of the 2UPRRPU PM, which shows the strong robustness of the proposed stiffness design method. Figure 6.7 shows the influence curve of the characteristic size h on the extreme linear displacement stiffness and linear displacement homogeneous performance under the elasticity modulus E = 180 GPa. The characteristic size is linearly related and horizontal linear related to the extreme linear displacement stiffness performance and linear displacement homogeneous performance, respectively, e.g., the characteristic size of the mechanism can be designed as h = 1.9 m to make the maximum linear displacement stiffness performance of the mechanism reach k df max = 9.219 × 109 N/m. The homogeneous index deviates greatly from the maximum linear displacement index but is close to the minimum linear displacement index, which indicates that

6.3 Example 2: Optimal Stiffness Performance Design of the 2PRU-PSR PM

173

Table 6.2 Optimal design parameters for multiple performance indices of the 2PRU-PSR PM under different characteristic sizes Trace index

Condition number index

Minimum eigenvalue

ε1

ε3

a1

a2

a3

a4

a5

a6

h= 0.3 m

0.600

1.400

0.200

0.467

1.077

1.368

7.480

11.996

h= 0.8 m

0.240

1.480

0.240

4.180

1.000

1.000

12.000

6.573

h= 0.3 m

0.880

0.200

0.880

0.200

1.000

1.000

8.160

11.952

h= 0.5 m

0.960

0.240

0.800

0.200

1.013

1.001

8.520

10.324

h= 0.8 m

1.000

0.320

0.640

0.205

1.063

1.007

8.920

9.197

h=3m

0.960

0.640

0.320

0.213

1.187

1.087

8.440

11.224

h=5m

0.760

1.160

0.200

0.305

1.146

1.316

7.840

11.821

h= 0.3 m

0.880

0.200

0.880

0.200

1.000

1.000

8.160

11.952

h= 0.5 m

0.960

0.240

0.800

0.200

1.013

1.001

8.520

10.324

h= 0.8 m

1.000

0.320

0.640

0.205

1.063

1.007

8.960

9.096

h=3m

1.000

0.560

0.400

0.224

1.166

1.055

8.600

10.442

h=5m

0.760

1.160

0.200

0.305

1.146

1.316

7.840

11.821

Determinant index

h= 0.1–5 m

0.600

0.800

0.600

0.800

1.000

1.000

9.000

9.000

kdf min

h= 0.1–5 m

0.760

1.160

0.200

0.305

1.146

1.316

7.850

11.776

kdf max

h= 0.1–5 m

0.52

1.480

0.200

0.569

1.050

1.353

7.450

11.918

δ

h= 0.1–5 m

0.880

0.920

0.200

0.209

1.209

1.227

8.150

11.820

Fig. 6.6 Influence curve of the elasticity modulus E on the stiffness performance of the 2PRU-PSR PM. a k df min , b k df max , and c δ

174

6 A Methodology for Optimal Stiffness Design of Parallel Manipulators …

Fig. 6.7 Influence curve of the characteristic size h on the stiffness performance of the 2PRU-PSR PM. a k df min , b k df max , and c δ

Fig. 6.8 Stiffness indices comparison of the 2PRU-PSR PM before and after optimization. a ηdf min , b ηdf max , and c ηδ

the homogeneous index is mainly affected by the minimum linear displacement stiffness performance. The design parameters of the 2PRU-PSR mechanism needs to make a trade-off between the maximum linear displacement and the minimum linear displacement stiffness performance according to the requirements of the engineering task. Taking the design parameters x = [0.751, 0.751, 0.150, 0.150, 1.166, 1.166, 9, 9] proposed in Ref. [7] as the structure parameters of the mechanism before optimization, Fig. 6.8 shows the comparison of the stiffness performance of the 2PRU-PSR PM before and after optimization under the characteristic size h = 0.333 m and elasticity modulus E = 180 GPa. The extreme linear displacement performance and linear displacement homogeneous performance are all improved, which verifies the effectiveness of the proposed stiffness design method in this work.

6.4 Summary This paper proposed a methodology for optimal stiffness performance design of PMs based on the characteristic size, in which the scale factors between the structural parameters and the characteristic size is used as the non-dimensional design parameters. From the perspective of whether the structure size ratio corresponding

References

175

to the optimal stiffness performance of the mechanism is consistent, which stiffness performance indices are suitable for the optimal stiffness performance design of the mechanism and which indices are not applicable were identified. The influence of the characteristic size and elasticity modulus scaling on the stiffness performance of the mechanism was studied, which lays a foundation for determining the mechanism size that meets the engineering environment requirements on the stiffness performance. To our knowledge, this is the first study in which the optimal structure size ratio for the stiffness performance optimization of PMs is presented. The numerical analysis for the optimal stiffness performance design of the 2UPRRPU and 2PRU-PSR PMs were taken as two examples to implement the proposed method. The results show that the robustness of the determinant index and linear displacement index under the action of force is strong, that is, the optimal design parameters of the mechanism are consistent even for different characteristic sizes, which is crucial for the optimal stiffness performance design of PMs; the eigenvalue index, trace index, and condition number index have poor robustness, that is, the optimum structure size ratio of the mechanism is inconsistent under different characteristic sizes, which is not conducive to the optimal stiffness design of PMs. Since the physical explanation for the determinant index is unclear, the extreme linear displacement index is recommended as the stiffness performance design index. The results showed that the elasticity modulus and characteristic size are linearly proportional to the linear displacement performance of the mechanism, but the linear displacement homogeneous performance index is not affected by them. Compared with the structure size ratio of the mechanism in the literature, the stiffness performance of the mechanism obtained by the proposed method in this work was significantly improved. The method presented in this paper provides a new idea for the optimal stiffness performance design and size determination of PMs. In our future work, the decoupling of characteristic size and stiffness performance indices will be studied.

References 1. C. Yang, Q.C. Li, Q.H. Chen et al., Elastostatic stiffness modeling of overconstrained parallel manipulators. Mech. Mach. Theory 122, 58–74 (2018) 2. G. Carbone, M. Ceccarelli, Comparison of indices for stiffness performance evaluation. J. Mechan. Eng. 5(003), 270–278 (2010) 3. H. Shin, S. Lee, J.I. Jeong, et al. Kinematic optimization for isotropic stiffness of redundantly actuated parallel manipulators, in 2011 IEEE International Conference on Robotics and Automation (2011) 4. Y. Li, Q. Xu, Stiffness analysis for a 3-PUU parallel kinematic machine. Mech. Mach. Theory 43(2), 186–200 (2008) 5. E. Courteille, D. Deblaise, P. Maurine, et al., Design Optimization of a Delta-Like Parallel Robot through Global Stiffness Performance Evaluation (2009), pp. 5159–5166

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6. F. Wang, Q.H. Chen, Q.C. Li, Optimal design of a 2-UPR-RPU parallel manipulator. J. Mech. Des. 137(5), 054501 (2015) 7. L.M. Xu, W. Ye, Q.C. Li, Design, analysis, and experiment of a new parallel manipulator with two rotational and one translational motion. Mech. Mach. Theory 177, 105064 (2022) 8. C. Yang, Q.H. Chen, J.H. Tong et al., Elastostatic stiffness analysis of a 2PUR-PSR overconstrained parallel mechanism. Int. J. Precis. Eng. Manuf. 20(4), 569–581 (2019)

Chapter 7

Multi-objective Optimization of Parallel Manipulators Using Game Algorithm

In this chapter, we focus on a multi-objective optimization game algorithm (MOOGA) for parallel manipulators (PMs). First, the distributions of the objective functions in the complete parameter space are calculated and sorted by its importance. Second, game weighting factors and lower bound values are assigned to different objective functions according to the engineering requirements. Finally, after multiple rounds of gaming according to the weighting factors and lower bound values, the objective functions reach an optimal balance point and obtain a balance intersection subspace. In addition, a comprehensive stiffness index (CSI) is introduced, that takes the coupling of the non-diagonal elements into consideration. This index decouples the linear and angular stiffness and has definite physical dimensions as well as a clear physical meaning. A Lagrangian function is used to obtain the maximum and minimum stiffness at a given position along with their corresponding directions. The distributions of extreme stiffness and corresponding directions expressed using spherical coordinates, which are useful for trajectory planning, are introduced. To compare the difference between the CSI and the principal diagonal stiffness index (PDSI), a divergence index κ is introduced. 2UPR-RPU PM is taken as an examples here for analysis and discussion.

7.1 Multi-objective Optimization Game Algorithm The goal of multi-objective optimization is to design reasonable geometric parameters to achieve better multi-objective performance of a mechanism. The main tasks of PM designs are to obtain an excellent regular workspace, motion/force transmission performance, carrying capacity, and stiffness performance. Thus, the above types of performance can be considered as the objective functions to be optimized at the design stage. Generally, the mathematical model of the multi-objective optimization of PMs can be expressed as

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_7

177

178

7 Multi-objective Optimization of Parallel Manipulators Using Game …

max {ωi f i (x)},

(7.1)

subject to ⎧ G(x) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ H (x) < 0

x min ≤ x ≤ x max , ⎪ ⎪ Σ ⎪ ⎪ ⎪ ωi = 1 ⎩

(7.2)

i

where max denotes the maximum; f and ω represent the scalar objective functions and their corresponding weighting factors, respectively; x denotes the design variables; G(x) denotes the equality constraint functions; and H(x) denotes the inequality constraint functions. The multi-objective optimization of a PM is a highly non-linear problem. In recent years, intelligent algorithms including genetic, PSO, simulated annealing, and ant colony algorithms have been widely used. However, the main problems with these algorithms are that they are time-consuming, convergence can be difficult, and the results often depend on the preferences given to the algorithm, which is challenging for engineers to understand. Game theory was established by the book Theory of Games and Economic Behavior, which was published in 1944 [1]. Today, game theory has been successfully applied to problems in various fields such as the allocation of economic benefits, labor markets, and social resources as well as round table negotiation [2–5]. In the MOOGA proposed in this chapter, the balance intersection subspace or optimal balance point is obtained through a multi-round game according to the game weighting factors of the objective functions. The balance intersection subspace and optimal balance point are different from the Pareto optimal set, which provides a set of non-dominated solutions. Engineers can choose the optimal balance point or points in the balance intersection subspace as the geometric parameters. A flowchart of the proposed MOOGA is shown in Fig. 7.1. The detailed steps are as follows: Step 1: Evaluate the objective functions. Calculate the distribution of each objective function in the complete parameter space and express it using graphs or tables to prepare for the optimization, which is called pre-processing. This step is the most critical and time-consuming. The step sizes of the design variables can be determined according to the engineering requirements. The main disadvantage of this pre-processing is that it is time-consuming. However, points in the complete parameter space can be allocated to different servers or computers for calculation and then summarized centrally. Hence, this problem can be solved efficiently. Step 2: Prepare the game. Assign weighting factors to each objective function in the game. Higher values of weighting factor indicate more important objective functions. Simultaneously, determine the lower bound value for each objective function, that

7.1 Multi-objective Optimization Game Algorithm

179

Fig. 7.1 Flowchart of the MOOGA

is, in the process of the game, the constraint ensures that each objective function is greater than or equal to its own lower bound. Step 3: Play the multi-objective game. In this step, each objective function plays the game according to its weighting factor. The results, which are the optimal balance point and balance intersection subspace that satisfy the multiple objectives, can be obtained after multiple rounds of gaming. If there is no intersection subspace for the objectives after multiple rounds of games, return to step 2 to redesign the weighting factors or lower bound values until an intersection is generated. Step 4: Select the optimized design variables. The optimal balance point or an appropriate point from the balance intersection subspace is selected as the final geometric parameter according to the results of the last game. A schematic diagram of the MOOGA is shown in Fig. 7.2. Different colors represent different objective functions. The numbers 1, i, and n represent the number of iterations of the game. Without loss of generality, Fig. 7.2 shows a game with four objectives. The size of the circle of each function represents the size of the parameter

180

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.2 Schematic diagram of MOOGA

set that satisfies the requirements during the game. As shown in Fig. 7.2, each objective function is located at its own optimal position in the first game; however, there is no overlap among the multiple objectives at this time. To obtain results that satisfy multiple objective functions, each objective function plays the game based on its respective weighting factor. Each objective makes concessions to reach the optimal balance point. In the process of the game, the acceptable parameter space of each objective is gradually increased. The optimal balance point of the multi-objective game is finally reached after multiple rounds of games. The main advantages of a MOOGA are that it can be calculated using distributed computing, analyze big data, and perform discontinuous optimization. The preprocessing is critical and time-consuming because it directly affects the reliability of the data analysis and optimization results. However, the discrete points can be allocated to different computers for calculation. The different optimization objectives need to be optimized separately using traditional optimization methods. For the MOOGA, once the data of the discrete points are obtained, we can obtain the optimal solution quickly using big data analysis. Discontinuous optimization is difficult for traditional optimization methods. In contrast, in the game optimization, we can set the step size of the design parameters according to engineering requirements and use design parameters that include both integer and non-integer values, which is sometimes very useful for engineering problems.

7.2 Example: 2UPR-RPU PM Multi-objective optimization design of the 2UPR-RPU overconstrained PM including workspace volume, motion/force transmission performance, and stiffness performance is taken as an example to implement the MOOGA introduced in Sect. 7.1. Structural description and kinematic analysis refer to Sect. 6.2.

7.2 Example: 2UPR-RPU PM

181

7.2.1 Regular Workspace Volume The shape and size of the workspace have an important impact on the engineering application. It is necessary to study the reachable workspace before studying the regular workspace. The reachable workspace contains the points that the moving platform can reach from at least one direction under the physical constraints of the mechanism. The physical constraints include the stroke of the actuator and angular limit of the passive joints. Carbone et al. [6] obtained a numerical expression of the volume of the reachable workspace using a binary method. The shape and size of the reachable workspace are irregular, which is not conducive to the trajectory planning and algorithm control. In practical engineering applications, the regular workspace is generally adopted. Babu et al. [7] obtained the expression of the volume of the maximum inscribed cylinder of the regular workspace using the polar coordinate method. Bounab [8] calculated the volume of the regular sphere workspace of the Delta PM. The polar coordinate method is adopted in this chapter. First, the shape and size of the reachable workspace are obtained using the space search method before calculating the regular workspace. Then, the operating platform height is divided into n equal layers and the maximum inscribed circle radius at each layer is calculated. A schematic diagram and the flowchart of the calculation of the maximum inscribed circle radius for each layer are shown in Fig. 7.3. We can calculate the volume of the regular cylinder or frustum of a cone after obtaining the maximum inscribed circle radius at each layer. To quantify the volume of the regular workspace, the volume of the regular cylinder can be defined as Vr = πρ 2 h,

(7.3)

where V r is the volume of the regular workspace, ρ is the radius of the bottom circle, and h is the height of the cylinder. Note that Eq. (7.3) can calculate both the position and orientation workspace, that is, the units of ρ can be both m and rad.

Fig. 7.3 Calculation of the maximum inscribed circle radius using the polar coordinate method: a schematic diagram and b flowchart

182

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Similarly, the volume of the frustum of a cone can be obtained using numerical methods as follows: Vr =

n−1 Σ 1 i=1

2

(Ai + Ai+1 ) ∗ Δz,

(7.4)

where Ai denotes the area of the maximum inscribed circle at each layer and Δz denotes the height interval. The physical constraints of the 2UPR-RPU PM include the strokes of the prismatic pairs and angular range of the passive joints. The range of the strokes can be expressed as lmin ≤ li ≤ lmax (i = 1, 2, 3),

(7.5)

where li is the stroke, and lmin = 0.3 m and l max = 1.0 m are the maximum and minimum strokes, respectively. The limit of the angle range of the R joints are given by {

αi1 = cos−1 (nli · n pi ) or αi1 = cos−1 (nli · nbi ) (i = 1, 2, 3) |αi1 |max ≤ 50◦

(7.6)

where nli is the unit vector along the direction of the ith limb, and npi and nbi are the unit vectors perpendicular to the moving platform and base, respectively. In addition, α i1 denotes the angle between nli and npi or nli and nbi . The limits of the angle range of the U joints are given by {

αi2 = cos−1 (nli · nbi ) or αi2 = cos−1 (nli · n pi ) (i = 1, 2, 3). |αi2 |max ≤ 50◦

(7.7)

where r 1 = r 2 = 0.39 m and r 3 = r 4 = 0.26 m are the initial geometric parameters before optimization [9, 10]. In this chapter, the space search method is used to obtain the discrete workspace of the 2UPR-RPU PM as shown in Fig. 7.4a. The reachable workspace is symmetric about γ = 0, which is consistent with the structure of the 2UPR-RPU PM. Without loss of generality, the maximum inscribed cylinder with a given moving platform height of 500–800 mm is adopted as the regular workspace of the 2UPR-RPU PM in this study. The radius of the bottom circle of the regular cylindrical workspace can be calculated using the method shown in Fig. 7.3, and the volume can be calculated by Eq. (7.3). Without loss of generality, to ensure the reference point of the end effector can be reached with the desired orientation, the minimum radius of the cylindrical section is set to 10°.

7.2 Example: 2UPR-RPU PM

183

Fig. 7.4 Workspace: a reachable and regular workspace, b regular workspace

7.2.2 Motion/Force Transmissibility The main disadvantages of Jacobian matrix-based kinematic indices [6–8, 11] are that it is related to the coordinate frame and the non-uniform dimensions. In this chapter, the motion/force transmission index proposed by Liu et al. [12, 13] is adopted as a performance evaluation index; it measures the efficiency of the mechanism from input to output. This index is frame-free and relative to the singularity. As shown in Fig. 5.1, the twist screws $i = ($i1 , $i2 , $i3 , $i4 ) and constraint wrenches of each limb can be obtained using screw theory (details can be found in [9, 10]). Considering the first limb as an example, the reciprocal product between TWS N T1 and the twist screws of the passive joints are set equal to zero, as follows: N T1 ◦ $1i = 0 i = 1, 2, 3.

(7.8)

The expression of N T1 can be obtained by N T1 = (B 1 A1 /|B 1 A1 |; oA1 × B 1 A1 /|B 1 A1 |),

(7.9)

where N T1 denotes the wrench screw passing through point A1 in the direction of B1 A1 . Because the actuator joint of the first limb is a P pair, the ITS $A1 can be expressed as $A1 = $13 = (0, 0, 0; B 1 A1 /|B 1 A1 |).

(7.10)

Similarly, the TWSs and unit ITSs of the other two limbs can be obtained as {

N T2 = (B 2 A2 /|B 2 A2 |; oA2 × B 2 A2 /|B 2 A2 |) , N T3 = (B 3 A3 /|B 3 A3 |; oA3 × B 3 A3 /|B 3 A3 |)

(7.11)

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7 Multi-objective Optimization of Parallel Manipulators Using Game …

and {

$A2 = $23 = (0, 0, 0; B 2 A2 /|B 2 A2 |) , $A3 = $33 = (0, 0, 0; B 3 A3 /|B 3 A3 |)

(7.12)

where N T2 denotes the wrench screw passing through point A2 in the direction of B2 A2 and N T3 denotes the wrench screw passing through point A3 and along the direction of B3 A3 . Substituting Eqs. (7.9) and (7.10) into Eq. (4.10) obtains λ1 = λ2 = λ3 = 1.

(7.13)

Locking all actuator joints except the first limb, the TWSs of the other two limbs are transformed into constraint wrenches. Now, the new wrench system acting on the moving platform after locking the other two limbs can be obtained as U 1 = [$r1 , $r2 , $r3 , N T2 , N T3 ]

(7.14)

The dimension of matrix U 1 is 5 and the instantaneous 1-DOF twist of the moving platform is denoted by unit OTS $O1 . In this case, only N T1 can contribute to the moving platform, whereas N T2 and N T3 apply no work to the moving platform, expressed as follows: $O1 ◦ U 1 = 0.

(7.15)

ITSs $O2 and $O3 can be found in the same manner; the LTI can be obtained using Eq. (4.11). LTI only represents the motion/force transmission performance of a mechanism under a given configuration. To evaluate the performance of the mechanism over the entire prescribed workspace, the global transmission index (GTI) is defined. Discrete points are often used for calculation as follows: { GTI =

{

Vr

ς dVr dVr

n 1Σ or GTI = ςi , n i=1

(7.16)

where n denotes the number of discrete points in the prescribed workspace. A higher value of the GTI indicates a better motion/force transmission performance.

7.2.3 Stiffness Performance Evaluation In this chapter, the CSI is adopted as the stiffness performance evaluation index. Because the stiffness performance of the mechanism is related to the direction of

7.2 Example: 2UPR-RPU PM

185

external loading, and the mechanism is subjected to different load directions in different processing modes. In most cases, engineers pay more attention to the linear stiffness performance of the mechanism for a force. To make the stiffness performance of different mechanisms comparable, here K xf , K yf , and K zf are considered as stiffness performance evaluation indices. Indices K xf , K yf , and K zf are linear stiffness indices indicating the mechanism’s resistance to a unit force in directions x, y, and z, which is a reciprocal of the corresponding compliance indices. To more intuitively observe the influence of the CSI proposed in this chapter on the stiffness performance of the mechanism, the PDSI is selected as a comparative object to study the difference between the two. Generally, the stiffness matrix can be expressed as ⎡

k11 ⎢k ⎢ 21 ⎢ ⎢ k31 K =⎢ ⎢k ⎢ 41 ⎢ ⎣ k51 k61

⎤ k12 k13 k14 k15 k16 k22 k23 k24 k25 k26 ⎥ ⎥ ⎥ k32 k33 k34 k35 k36 ⎥ ⎥. k42 k43 k44 k45 k46 ⎥ ⎥ ⎥ k52 k53 k54 k55 k56 ⎦

(7.17)

k62 k63 k64 k65 k66

The PDSI uses the principal diagonal elements of the stiffness matrix as an indicator of stiffness performance and the non-diagonal elements are ignored, which is expressed as follows {

k x, f = k11 , k ,y f = k22 , k z, f = k33 (N/m) , , = k44 , k ,ym = k55 , k zm = k66 (N/rad) k xm

.

(7.18)

As Eq. (7.18) shows, the PDSI can only evaluate the stiffness performance along the axis of the reference coordinate system, and the effect of non-diagonal elements on the stiffness performance of the mechanism is ignored. The following equation defines the degree of difference between the CSI proposed in this chapter and the PDSI, which reflects the effect of non-diagonal elements on stiffness performance: ⎧ k x, f − k x f ⎪ ⎪ κ = ⎪ ⎨ xf k x, f

κy f =

k ,y f − k y f k ,y f

κz f =

k z, f − k z f k z, f

, ⎪ k ,ym − k ym − k zm k zm k , − k xm ⎪ ⎪ κ ym = κ = ⎩ κxm = xm , zm , , k xm k ym k zm

,

(7.19)

where κ xf , κ yf , and κ zf denote the degree of difference in the linear CSI and PDSI stiffness indices along the x-, y-, and z-axis directions for an external force, respectively; κ xm , κ ym , and κ zm denote the degree of difference in the angular CSI and PDSI stiffness indices along the x-, y-, and z-axis directions for an external moment, respectively.

186

7 Multi-objective Optimization of Parallel Manipulators Using Game …

From Eq. (7.19), κ varies from zero to one. If the value of κ is equal to zero, this indicates that the CSI proposed in this chapter is the same as the PDSI. A smaller value of κ leads to a smaller reduction in the CSI with respect to the PDSI, and when these indices are closer in value, the coupling effect of the non-diagonal elements of the compliance matrix on stiffness performance is smaller. Likewise, a larger value of κ indicates a larger influence of the coupling effect of the non-diagonal compliance matrix elements on stiffness performance, and the two indices have a larger deviation. Note that the value of κ is less[than one. ]T If a general wrench w = nTf ; nTm is exerted on a mechanism, according to the principle of small deformation superposition, the deflection of the mechanism is equal to the sum of the deflections of unit force nf and unit moment nm acting on the mechanism. To determine whether the mechanism is more sensitive to the force or moment, the sensitivity factor is defined as Δst =

δd f δϕ f , Δsr = , δdm δϕm

(7.20)

where Δst and Δsr represent the translational and rotational deflection sensitivity factors, respectively; δ df and δ dm denote the translational deflections caused by the unit force and unit moment, respectively; and δ ϕ f and δ ϕ m denote the rotational deflection caused by the unit force and unit moment, respectively. To compare the sensitivity factor of different mechanisms, the directions of unit force and moment can each be assigned along the x-, y- and z-axes of a Cartesian coordinate system, respectively. If the sensitivity factor is equal to one, the deflection is equally sensitive to force and moment. If sensitivity factor is greater than one, the deflection is more sensitive to force and vice versa.

7.2.4 Multi-objective Optimization The mathematical model of the multi-objective optimization design of the 2UPRRPU PM can be expressed as objective function :

f 1 (x) = Vr , f 2 (x) = GTI

f 3 (x) = K x f , f 4 (x) = K y f , f 5 (x) = K z f design parameter : x = [r1 r2 r3 ] subject to : |αi1 |max , |αi2 |max ≤ 50◦ 0.3 m ≤ li ≤ 1.0 m 0.3 m ≤ r1 , r2 ≤ 0.6 m , 0.2 m ≤ r3 ≤ r1

(7.21)

where K xf , K yf y, and K zf denote the linear GCSI resistant of the force along the x-, y- and z-axes, respectively, which can be obtained by Eq. (5.26). Without loss of

7.2 Example: 2UPR-RPU PM

187

generality, the height z of the moving platform is known to be 500–800 mm. In this section, reasonable geometric parameters are redesigned to achieve multi-objective game optimization. Using the constraint conditions given by Eq. (7.21), slice graphs of the distribution of the objective functions in the complete parameter space can be obtained, as shown in Fig. 7.5. Figure 7.5a shows that the value of the GTI is highly affected by r 2 and less affected by r 1 and r 3 . A large r 2 leads to a large GTI. Figure 7.5d shows that the optimal values of K yf are mainly distributed in the region with small r 3 and large r 1 and r 2 ; that is, GTI and K yf simultaneously achieve optimal values in the region where r 3 is small and r 1 and r 2 are large. Figure 7.5b shows that workspace V r reaches optimal values in the region where r 2 is small and r 1 and r 3 are both large and small. Figure 7.5c shows that optimal stiffness K xf is mainly distributed in the region where r 2 is small and r 1 and r 3 are both large or small; that is, in this region, V r and K xf can simultaneously achieve optimal values, and there is an obvious intersection subspace between the two. Finally, Fig. 7.5e shows that K zf achieves the optimal values in the region with large r 2 and small r 1 and r 3 . Figure 7.5a, d, e shows that K yf and K zf can obtain a relatively optimal intersection subspace in the region with large r 2 , small r 3 , and medium r 1 . It can be seen from Fig. 7.5a–e that it is impossible to obtain an intersection subspace for which all the objective functions are optimal. To obtain a value that satisfies multiple objectives, there must be a game among the parties in which one or more parties make a sacrifice. Engineers can assign different weighting factors to different objectives and determine the lower bound values according to the engineering requirements. A lower bound value may not be assigned when it is initially unknown. The design parameters can be determined according to the result of the game, whether or not they satisfy the engineering requirements. Figure 7.6 shows the parameter subspace that corresponds to top 10% of the best values of each objective. The distribution is consistent with that of Fig. 7.5. It can be seen from Fig. 7.6 that there is an obvious intersection among the regions for K yf , K zf , and GTI. Additionally, there is also an obvious intersection between V r and K xf . To obtain a feasible region in which all the objective functions are satisfied, the performance requirements must be reduced for one or more objectives. For example, the performances for V r and K xf can be reduced to obtain a feasible intersection that satisfies all the objective requirements under the premise of ensuring the performance of K yf , K zf , and GTI.

7.2.4.1

Optimization of Three Objective Functions

Considering the multi-objective optimization of GTI, K yf , and K zf as an example, reasonable geometric parameters are explored to maximize each objective as much as possible. In this chapter, a MOOGA is proposed to obtain a balance intersection subspace for multiple objectives as well as the optimal balance point of the game in the subspace; that is, all objectives must be compromised to obtain an intersection subspace. First, the values of the objective functions are sorted from maximum to

188

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.5 Distribution of the objective functions in the complete parameter space: a GTI, b V r , c K xf , d K yf , and e K zf

7.2 Example: 2UPR-RPU PM

189

Fig. 7.6 Distribution of the top 10% values of the objective functions

minimum in the complete parameter space. Without loss of generality, three objectives are considered equally important. That is ωKy : ωKz : ωGTI = 1/3: 1/3: 1/3, and each objective simultaneously compromises to obtain a balance intersection subspace for all three. Meanwhile, the lower bound values are set as follows: K y0 = 8.7 × 107 N/m, K z0 = 8.6 × 108 N/m, and GTI = 0.735. Without loss of generality, r 1 , r 2 , and r 3 are divided into 100 data points, namely the step size is 0.004 m. Each objective is calculated for a total of 1,030,301 points in the complete parameter space. The optimal balance point of the three objectives, [r 1 , r 2 , r 3 ] = [0.452, 0.600, 0.200], can be reached after multiple rounds of the game. At the optimal balance point, none of the three are at their maximum value, but reach a balance position. The multi-objective game stops when the lower bound value of 0.735 for the GTI is reached. Figure 7.7 shows the balance intersection subspace of the three, where all points meet the requirements. Here, K yf , K zf , and GTI create a balance intersection subspace that contains large r 2 , small r 3 , and medium r 1 . A comparison before and after optimization is given in Table 7.1. The optimized values of K yf , K zf , and GTI are significantly improved. At the optimal balance point, GTI, K zf , and K yf increased by 38.05%, 36.28%, and 100.85%, respectively, which indicates the effectiveness of the MOOGA proposed in this chapter (Fig. 7.7). Table 7.1 Comparison of K yf , K zf , and GTI values before and after optimization r 1 (m)

r 2 (m)

r 3 (m)

K yf (N/m)

K zf (N/m) 6.737 ×

GTI

Before optimization

0.392

0.392

0.260

4.832 ×

Optimized

0.420

0.600

0.200

8.961 × 107

9.387 × 108

0.756

0.452

0.584

0.212

8.826 × 107

8.727 × 108

0.743

8.837 ×

Optimal balance point

107

0.512

0.600

0.200

1.085 ×

0.452

0.600

0.200

9.705 × 107

108

108

0.544

108

0.737

9.181 × 108

0.751

190

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.7 Intersection of K yf , K zf , and GTI in the complete parameter space

A genetic algorithm is used as a comparison method for the proposed MOOGA. Without loss of generality, the comprehensive optimality of the regular workspace volume, motion/force transmission performance, and stiffness performance is used to form the objective function, which is expressed mathematically as max

f C O = w1 K z, f + w2 Vr, + w3 GTI, , ⎧ K z f − K z f min ⎪ ⎪ K z, f = ⎪ ⎪ K z f max − K z f min ⎪ ⎨ Vr − Vr min , Vr, = ⎪ ⎪ ⎪ Vr max − Vr min ⎪ ⎪ ⎩ GTI, = GTI

(7.22)

(7.23)

where wi (i = 1, 2, 3) is the weight factor corresponding to the objectives and Eq. (7.23) is a dimensionless operation on K zf , V r , and GTI. In addition, V rmax = 0.1654 rad2 m, V rmin = 0.0273 rad2 ·m, K zf max = 9.9404 × 108 N/m and K zf min = 4.6572 × 108 N/m are obtained from the results of Fig. 7.5. Table 7.2 gives the results of genetic algorithm and MOOGA for six weight combinations, where the error represents the error of f CO between the genetic algorithm and proposed MOOGA. The calculation time of pre-processing phase for the MOOGA is 31,862.37 s on a desk computer with a 3.00 GHz CPU. This is the most timeconsuming phase for the MOOGA. Table 7.2 also shows that the error of the optimal solution is less than 1%, which is mainly caused by the use of discrete points in the MOOGA. The total calculation time for the six examples of the genetic algorithm is 55,149 s. Only the results for the three objectives K zf , V r , and GTI are given in the Table 7.2; if other combinations of different objective functions and weighting factors are considered, the calculation time would substantially increase. The above analysis

7.2 Example: 2UPR-RPU PM

191

Table 7.2 Comparison of the genetic algorithm and MOOGA results Example

w1

w2

w3

Algorithm

f CO

1

0.6

0.2

0.2

Game

0.794

0.01

Genetic

0.801

9216.66

2

0.2

0.6

0.2

Game

0.752

0.01

Genetic

0.753

12,686.45

Game

0.698

0.01

Genetic

0.700

4146.81

Game

0.666

0.01

Genetic

0.668

15,718.89

Game

0.879

0.01

Genetic

0.881

9745.31

Game

0.715

0.01

Genetic

0.715

3638.79

3

0.2

0.2

0.6

4

0.5

0.5

0

5

0.5

0

0.5

6

0

0.5

0.5

Time (s)

Error (%) 0.87 0.13 0.29 0.30 0.23 0.00

shows that the cost of the MOOGA algorithm in the pre-processing stage is worthwhile, it greatly reduces the next optimization time and facilitates the subsequent big data analysis.

7.2.4.2

Optimization of Five Objective Functions

The distribution of the objective functions in Fig. 7.5 shows that it is impossible to achieve a balanced intersection subspace in which K xf , K yf , K zf , V r , and GTI all have excellent values. Hence, there should be a game among them to improve the solution. To more intuitively express the volume of the regular cylindrical workspace, we convert it into the radius of the bottom circle represented by an angle, that is, the range of orientation of the moving platform: / ρ=

Vr 180 · (deg). πh π

(7.24)

Multiple schemes can be adopted to obtain the balance intersection subspace. For example, the performances for V r and K xf could be reduced while those of K yf , K zf , and GTI remain unchanged. In contrast, the performances for K yf , K zf , and GTI could be reduced while those of V r and K xf remain unchanged, or the performances of the five objective functions could all be reduced simultaneously. Without loss of generality, the ratios of the weighting factors are assigned as ωKz : ωGTI : ωKx : ωKy : ωVr : = 6/19: 4/19: 3/19: 3/19: 3/19; that is, K zf is dominant in the process of the game. When K zf takes a step back, GTI takes 1.5 steps back, and K xf , K yf , and V r each take two steps back. Suppose the lower bound values of the objective functions are K x0 = 9.682 × 106 N/m, K y0 = 3.500 × 107 N/m, K z0 = 7.371 × 108 N/m, ρ =

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7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.8 Intersection of V r , K xf , K yf , K zf , and GTI in the complete parameter space

Table 7.3 Comparison of ρ, K xf , K yf , K zf , and GTI values before and after optimization r 1 (m)

r 2 (m)

r 3 (m)

ρ (°)

K xf

K yf

K zf

GTI

Before optimization

0.392

0.392

0.260

20.496

1.043

4.832

67.373

0.544

Optimized

0.360

0.452

0.216

18.500

0.970

5.986

78.465

0.616

0.392

0.464

0.256

18.000

0.972

5.111

75.382

0.628

0.404

0.452

0.220

18.000

0.978

6.841

75.492

0.618

Optimal balance point

15.000°, and GTI = 0.600. After multiple rounds of a game, we obtain the optimal balance point [r 1 , r 2 , r 3 ] = [0.404, 0.452, 0.220]. Meanwhile, the game stops when the lower bound of K xf is reached. The balance intersection subspace that meets the requirements of all the objective functions and the optimal balance point of the game are shown in Fig. 7.8. Table 7.3 shows the comparison before and after optimization. For the optimal balance point, the optimized V r and K xf are decreased by 12.18% and 6.23%, respectively, and K yf , K zf , and GTI are increased by 41.58%, 12.05%, and 13.60%, respectively. Annotation: The units of K xf , K yf , and K zf are107 (N/m) and V r is converted to express with ρ by Eq. (7.24).

7.2.4.3

Distribution of Local Indices

Without loss of generality, a set of geometric parameters is selected from the optimized parameter sets as an example: [r 1 , r 2 , r 3 ] = [0.45, 0.6, 0.2]. The slice graphs in Fig. 7.9 show the distribution of the local evaluation indices in the regular workspace. It can be seen from Fig. 7.9 that each local index decreases with the increase in height of the moving platform, which is particularly noticeable for K xf and K yf . Figure 7.10 shows the distribution of the local indices in the plane z = 0.6 m. Each local index is symmetric about γ = 0, which is consistent with the symmetry of

7.2 Example: 2UPR-RPU PM

193

Fig. 7.9 Distribution of objective functions in the regular workspace: a k xf , b k yf , c k zf , and d LTI

the mechanism. Additionally, the rationality of the CSI proposed in this chapter is preliminarily verified. Figure 7.10c shows that the maximum k zf (the best stiffness performance in the plane z = 0.6 m) occurs at the initial position and rapidly decreases as the orientation of the mechanism changes, that is, the position with the best stiffness performance produces the minimum deflection and the position with the worst stiffness performance produces the maximum deflection. Figure (a), (b), and (d) show that the maxima of k xf , k yf , and LTI are located around position (γ , β) = (0, 12°) rather the initial position (γ , β) = (0, 0), the minima of k xf , k yf , and LTI (the worst stiffness performance and motion/force transmissibility in the plane z = 0.6 m) are located around position (γ , β) = (0, −12°). The divergence index κ between the CSI and the PDSI is discussed next. Figures 7.11 and 7.12 show the distributions of κ in the regular cylindrical workspace and plane z = 0.6 m, respectively. Because of limited space, only the distributions of κ xf , κ yf , and κ zf are discussed in this chapter. It was found that κ xf , κ xf , and κ zf are all symmetric about the plane γ = 0, which is also consistent with the symmetry of the mechanism. Note that in each plane, the maximum of κ is at the position with the worst stiffness rather than at the position with the best stiffness. For example,

194

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.10 Distribution of objective functions in the plane z = 0.6 m: a k xf , b k yf , c k zf , and d LTI

the maxima of κ xf and κ yf are at position (γ , β) = (0, −12°), which has the worst stiffness, rather than (γ , β) = (0, 12°), which has the best stiffness. The minimum of κ zf is also at position (γ , β) = (0, 0), which has the best stiffness, is consistent with Fig. 7.10 and verifies the rationality of the CSI again. It can be seen from Figs. 7.11 and 7.12 that the PDSI optimistically estimates the stiffness performance of the mechanism. Compared with the PDSI in the x, y, and z directions, the CSI is decreased by 98–99% in the x direction, 76–92% in the y direction, and 30–90% in the z direction. Furthermore, the effect of coupling cannot be ignored in the performance evaluation of the true stiffness. It must be considered to evaluate the true stiffness of the mechanism. The 2UPR-RPU PM is only symmetrical about one coordinate axis, so it is insufficient to determine the universality of the effect of coupling on stiffness performance. Therefore, we further discuss the effect of coupling on the stiffness performance of a mechanism symmetric about two axes. The stiffness model of the 2UPR-2RPU PM was described in detail in our previous work [9]. Figure 7.13 shows the distribution of κ for the 2UPR-2RPU PM in the regular cylindrical workspace, and Fig. 7.14 shows the distribution of κ in the plane z = 0.6 m. The distribution of κ is symmetric about

7.2 Example: 2UPR-RPU PM

195

Fig. 7.11 Distribution of κ for the 2UPR-RPU PM in the regular workspace: a κ xf , b κ yf , and c κ zf

Fig. 7.12 Distribution of κ for the 2UPR-RPU PM in the plane z = 0.6 m: a κ xf , b κ yf , and c κ zf

Fig. 7.13 Distribution of κ of the 2UPR-2RPU PM in the regular workspace: a κ xf , b κ yf , and c κ zf

both planes γ = 0 and β = 0, which is consistent with the symmetry of the mechanism. The effect of coupling on the stiffness performance in the x and y directions is still significant. Coupling has no impact on the stiffness in the z direction only at position (γ , β) = (0, 0), where the maximum stiffness is in the z direction, which is consistent with the symmetry about the two axes of the mechanism. The effect of coupling on the stiffness performance in the z direction rapidly increases as the attitude angle changes. Thus, regardless of whether the mechanism is symmetric or asymmetric, the effect of coupling on the true stiffness performance of the mechanism is not negligible. The above analysis discusses the linear stiffness for external forces along the x, y, and z directions. Next, Eqs. (5.18)–(5.20) are used to further determine the

196

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.14 Distribution of κ for the 2UPR-2RPU PM in the plane z = 0.6 m: a κ xf , b κ yf , and c κ zf

extreme values for the compliance of the mechanism and the corresponding directions. Because of limited space, we only discuss extreme compliance values for an external force. Tables 7.4 and 7.5 show the extreme compliance values and corresponding directions at three positions. It was found that the 2UPR-2RPU PM has minimum linear and angular compliances along the direction of the z-axis, and maximum linear and angular compliances along the direction of the x-axis when the moving platform is at the central position. This is consistent with the symmetry of the mechanism. The minimum angular compliance is zero, that is, the angular deformation is zero when an external force is applied along the z-axis, and the angular stiffness is infinite along the direction of the z-axis. Because the 2UPR-RPU PM is only symmetric about the β-axis, there is a deflection angle of 0.745° between the direction of the minimum compliance of the linear and angular displacements and the direction of the z-axis when the moving platform is at the central position. This is determined by the nature of the asymmetry about the two axes of the mechanism. The maximum linear and angular displacements are also along the direction of the x-axis. Tables 7.4 and 7.5 also show the extreme compliances and their corresponding directions for the 2UPR-RPU and 2UPR-2RPU PMs at two general positions, that is, z = −0.8, β = 14°, and γ = 14°, as well as z = −0.6, β = −10°, and γ = 17°. To obtain an intuitive distribution of the compliances under different directions of unit force, the 2UPR-2RPU PM is taken as an example and the distributions of the deflections of the three configurations corresponding to Table 7.5 under different unit force directions are given in Fig. 7.15. These results verify the correctness of the results obtained from Table 7.5. Without loss of generality, Tables 7.6 and 7.7 show the extreme distributions and the directions of a 2UPR-2RPU PM for a general wrench in three configurations. Because of the limited space, only the linear compliances extreme values are discussed here. It can be seen from Tables 7.6 and 7.7 that the results obtained from the genetic and PSO algorithms are basically consistent, which verifies the reliability of the results. The minimum compliances values in Table 7.6 obtained from the PSO algorithm is superior to that of the genetic algorithm, which shows the superiority of PSO with respect to global optimization. Table 7.6 shows for E#1, the directions of force and moment are close to the negative z- and x-axes, respectively, for E#2, the directions of force and moment are close to the positive z- and x-axes, respectively,

7.2 Example: 2UPR-RPU PM

197

Table 7.4 Distribution of the extreme compliance for force of the 2UPR-RPU PM z = −0.8, β = 0, γ = z = −0.8, β = 14°, γ z = −0.6, β = − 0 = 14° 10°, γ = 17° 2.457 × 10–10

2.537 × 10–10

1.926 × 10–10

[0.020, 0.230, − 0.973]T

[0.046, −0.191, − 0.980]T

1.526 × 10–7

1.200 × 10–7

[1, −0.028,

[0.998, −0.044, 0.055]T

Minimum compliance of displacement

cdmin nf

[0, 0.013,

Maximal compliance of displacement

cdmax

1.933 × 10–7

nf

[1, 0,

Minimum compliance of angular

cϕmin

5.327 × 10–12

2.286 × 10–10

3.810 × 10–10

nϕ

[0, −0.030, 1]T

[0.019, 0.270, − 0.963]T

[0.047, −0.149, − 0.988]T

Maximal compliance of angular

cϕmax

1.409 × 10–7

1.261 × 10–7

1.251 × 10–7

nϕ

[1, 0, 0]T

[0.999, 0.038, 0.030]T

[0.999, 0.009, 0.046]T

1]T

0]T

0.013]T

Table 7.5 Distribution of the extreme compliance for force of the 2UPR-2RPU PM E#1: z = −0.8, β = 0, γ = 0

E#2: z = −0.8, β = 14°, γ = 14°

E#3: z = −0.6, β = −10°, γ = 17°

Minimum compliance of displacement

cdmin

1.261 × 10–10

1.323 × 10–10

9.892 × 10–11

nf

[0, 0, 1]T

[−0.010, −0.233, 0.972]T

[0.023, −0.161, − 0.987]T

Maximal compliance of displacement

cdmax

8.548 × 10–8

8.255 × 10–8

3.074 × 10–8

nf

[1, 0, 0]T

[0.999, −0.044,0]T

[0.998, 0.069, 0.012]T

Minimum compliance of angular

cϕmin

0

1.552 × 10–11

1.071 × 10–11

nϕ

[0, 0,

[0.010, 0.235, − 0.972]T

[0.023, −0.164, − 0.986]T

Maximal compliance of angular

cϕmax

5.341 × 10–8

5.726 × 10–8

3.031 × 10–8

nϕ

[1, 0,

1]T

0]T

[1, 0.016,

0.014]T

[0.999, −0.025, 0.027]T

and for E#3, the directions of force and moment are close to the positive x- and negative y-axes, respectively. Moreover, the results show that the minimum value of compliance is highly correlated with the configurations of the manipulator. Table 7.7 shows that the directions of the forces and moments of the three examples are all close to the positive x- and y-axes, respectively, namely if a force and moment are simultaneously exerted on the mechanism, directions of the force and moment that are simultaneously close to the positive x- and y-axes should be avoided. The sensitivity factors along the x-, y- and z-axes of the Cartesian coordinate system of the 2UPR-2RPU PM in the plane z = −0.6 m are shown in Fig. 7.16. Table 7.5 shows that the rotational deflection is equal to zero, so a circle with a radius of

198

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.15 Distributions of the compliance value of the 2UPR-2RPU PM under different directions of unit force: a E#1: cd , b E#1: cϕ , c E#2: cd , d E#2: cϕ , e: E#3: cd , and f E#3: cϕ

1° is removed in Fig. 7.16c to avoid a singularity. It can be seen from Fig. 7.16 that both Δst and Δsr along the x-axis are greater than one, that is, the translational and rotational deflections are more sensitive to force along the y-axis, and the translational and rotational deflections are more sensitive to moment along the z-axis, respectively. Further, Fig. 7.16c–f shows that the translational and rotational deflections are mainly

7.2 Example: 2UPR-RPU PM

199

Table 7.6 Distribution of the minimum compliance for a general wrench of the 2UPR-2RPU PM Example

Algorithm

Minimum compliance of displacement cdmin

E#1: z = −0.8, β = 0, γ = 0

PSO

1.080 ×

Genetic

1.261 × 10–10

[0, 0, 1; 0, 0, 1]T

5.583 ×

[−0.049, 0.3153,0.949; 0.993, 0.100, −0.067]T

E#2: z = −0.8, β = 14°, γ = 14° PSO

E#3: z = −0.6, β = −10°, γ = 17°

n 10–10

10–11

[−0.178, −0.484,−0.856; − 0.843, 0.285, 0.456]T

Genetic

5.750 × 10–11

[0.015, 0.311,0.950; 0.998, 0.000, −0.056]T

PSO

1.089 × 10–11

[0.9632, −0.003,−0.269; 0.057, −0.984, 0.172]T

Genetic

1.225 × 10–11

[0.961, −0.000,−0.276; 0.058, −0.985, 0.164]T

Table 7.7 Distribution of the maximum compliance for a general wrench of the 2UPR-2RPU PM Example

Algorithm

Maximum compliance of displacement cdmax

E#1: z = −0.8, β = 0, γ = 0

PSO

1.389 ×

Genetic

1.389 ×

10–7

[1, 0, 0; 0, 1, 0]T

1.398 ×

10–7

[0.999, −0.030, 0.003; 0.015, 0.872, 0.489]T

Genetic

1.398 × 10–7

[0.999, −0.030, 0.003; 0.015, 0.872, 0.489]T

PSO

6.100 × 10–8

[0.999, 0.042, 0.017; −0.019, 0.917, −0.399]T

Genetic

6.100 × 10–8

[0.999, 0.042, 0.017; −0.019, 0.917, −0.399]T

E#2: z = −0.8, β = 14°, γ = 14° PSO

E#3: z = −0.6, β = −10°, γ = 17°

n 10–7

[1, 0, 0; 0, 1, 0]T

caused by the moment. The maximum sensitivity factor is 0.025 in Fig. 7.16f, namely the rotational deflection is mainly caused by the moment. The sensitivity factors are related to the direction of the external wrench and configuration of the mechanism. If the force or moment exerted on the mechanism can be determined, we can specify a reasonable direction for the loading and configuration of the mechanism to control the translational and rotational deflection. In practical engineering applications, extreme compliances and their corresponding directions can be calculated in any position so as to avoid the direction of the maximum compliance to the greatest extent, improve the stiffness performance of the mechanism as much as possible and provide guidance for trajectory planning. Equations (5.18)–(5.20) give the analytical expressions for the extreme compliance values and their corresponding directions for external forces and moments. The extreme stiffness indices can be obtained by reciprocating the compliance indices.

200

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.16 Distribution of the sensitivity factors of 2UPR-2RPU PM under the plane z = −0.6 m: a Δst along the x-axis, b Δsr along the x-axis, c Δst along the y-axis, d Δsr along the y-axis, e Δst along the z-axis, and f Δsr along the z-axis

Because of space limitations, only the extreme linear stiffness indices of the 2UPRRPU and 2UPR-2RPU PMs for force are described here. To more intuitively express the direction of extreme stiffness, the cart2sph function of MATLAB software is used to convert the Cartesian coordinates into spherical coordinates. The schematic is shown in Fig. 7.17, where θ denotes the elevation angle measured unit vectors from the x-y plane. The value of the θ is in the range [−π /2, π /2]. In addition, φ denotes the azimuth angle, which is the counterclockwise angle in the x-y plane measured

7.2 Example: 2UPR-RPU PM

201

from the positive x-axis and is in the range [−π, π ]. This notation for spherical coordinates is not standard. Note that if θ = 0, the point is in the x-y plane. If θ = π /2, then the point is on the positive z-axis. Compared with Fig. 7.10, the value of k dmin of Fig. 7.18 is closer to k xf and less than k yf and k zf . The corresponding elevation angle θ is less than 3°, which is influenced by the configurations of the mechanism. The value of azimuth angle φ is divided into two parts, about 180° and 2°, and the corresponding relationship between the configurations can be obtained from the graph. During the machining process, the direction of force corresponding to the minimum stiffness at a given configuration should be avoided to improve the machining accuracy. The maximum stiffness is in the range [4.39 4.44] × 109 N/m. As the configurations changes, the value of maximum stiffness does not change much. The corresponding elevation angle is in the range [75° 90°], which is close to the z-axis. The azimuth angle exhibits a spiral surface distribution. In engineering applications, we can use the direction corresponding to the maximum stiffness to maximize processing accuracy. Figure 7.19 shows the distribution of extreme stiffness and corresponding direction for force of a 2UPR-2RPU PM. If the direction of force is along the z-axis, θ is equal to π /2, and the value of φ can be arbitrary, so the θ and φ corresponding to this direction (which corresponds to β = 0 and γ = 0) is not displayed. It can be seen from Fig. 7.19 that the distribution of the 2UPR-2RPU PM is similar to that of the 2UPRRPU PM. Compared with the results for the 2UPR-RPU PM, the minimum stiffness and maximum stiffness of the 2UPR-2RPU PM are greatly improved because of the addition of an RPU limb, the range of elevation angle corresponds to the minimum stiffness is reduced to [−2° 2°]. Moreover, the azimuth angle corresponding to the minimum stiffness is around 180°, and the value at 2° has disappeared. The above analysis shows that the extreme stiffness values and their corresponding directions can be obtained by mathematical operations on the compliance matrix. Hence, the best loading direction is in the direction corresponding to the maximum stiffness, Fig. 7.17 Schematic diagram of the cart2sph function in MATLAB

202

7 Multi-objective Optimization of Parallel Manipulators Using Game …

Fig. 7.18 Distributions of the extreme linear stiffness indices and the corresponding directions for force for the 2UPR-RPU PM in the plane z = −0.6 m. a k dmin , b θ kmin , c φ kmin , d k dmax , e θ kmax , and f φ kmax

Fig. 7.19 Distributions of the extreme linear stiffness indices and corresponding directions for force for the 2UPR-2RPU PM in the plane z = −0.6 m. a k dmin , b θ kmin , c φ kmin , d k dmax , e θ kmax , and f φ kmax

7.3 Summary

203

and it is best to avoid the direction of minimum stiffness to maximize the machining accuracy. These results can be used to guide trajectory planning.

7.3 Summary A MOOGA, which is based on the game theory, was introduced in this chapter. The procedure of the algorithm was provided and performance evaluation indices including the volume of the regular cylindrical workspace, motion/force transmission efficiency, and stiffness performance were considered. The CSI was introduced and used as one of the objective functions for the subsequent multi-objective optimization. The CSI takes into account the influence of non-diagonal elements in the compliance matrix on stiffness performance and more accurately evaluates the stiffness performance of a mechanism. The CSI has a clear physical meaning and defined dimensions and provides a new concept for the evaluation of the stiffness performance of PMs. The maximum and minimum of the stiffness and their corresponding directions at a given position were obtained by constructing a Lagrangian function. To compare the difference between the CSI proposed in this chapter and the PDSI, divergence index κ was proposed. Using the 2UPR-RPU PM as an example, multi-objective optimization examples of three and five objectives were provided, and the optimal balance point and balance intersection subspace of the corresponding multi-objective optimizations were obtained. The comparative results of the performance evaluation indices before and after optimization demonstrate the effectiveness of the MOOGA. In addition to divergence index κ, one of the optimal balance points was used as an example to show the distribution of the local performance evaluation indices in the regular cylindrical workspace. The distribution of κ of the 2UPR-2RPU PM was also provided to further demonstrate the effect of coupling on the bilateral symmetric mechanism. The results show that the CSI values were up to 99% lower than those of the PDSI. Hence, the effect of coupling of the non-diagonal elements cannot be ignored when evaluating the true stiffness performance of the mechanism. The sensitivity factors along the x-, y-, and z-axes of the Cartesian coordinate system of the 2UPR-2RPU PM in the plane z = −0.6 m were also shown in this chapter. The results show that both the translational and rotational deflections are more sensitive to force in the x-axis direction and more sensitive to moment in the y- and z-axes directions. The extreme linear stiffness indices of the 2UPR-RPU and 2UPR-2RPU PMs for force were presented. The results show that the angle between the directions corresponding to the minimum linear stiffness and the x-y plane is less than 3°, and the angle between the directions corresponding to the maximum linear stiffness and the x-y plane is in the range [75° 90°]. Because the 2UPR-2RPU PM has one more RPU limb than the 2UPR-RPU PM, the minimum and maximum linear stiffness values of the 2UPR-2RPU PM are greatly improved. In an engineering application, the best loading direction should follow the direction corresponding to the maximum stiffness, and the direction of the minimum stiffness should be avoided

204

7 Multi-objective Optimization of Parallel Manipulators Using Game …

to maximize the machining accuracy. These results can assist trajectory planning. The minimum and maximum compliance values and directions for a general wrench of the 2UPR-2RPU PM at three positions were obtained using the genetic and PSO algorithms. The results show the superiority of PSO with respect to global optimization. The extreme compliance values and their corresponding directions can be calculated at any position in an engineering application so as to avoid the direction of maximum compliance to the greatest extent, improve the stiffness performance of the mechanism as much as possible, and provide guidance for trajectory planning.

References 1. M. Oskar, V.N. John, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1953) 2. G. Persiano, Algorithmic Game Theory (Springer, Berlin, Heidelberg, 2009) 3. M. Archetti, I. Scheuring, M. Hoffman, Economic game theory for mutualism and cooperation. Ecol. Lett. 14(12), 1300–1312 (2011) 4. A.E. Roth, The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Econ. 92(6), 991–1016 (1984) 5. J.Y. Halpern, P. Rafael, Game theory with translucent players. Int. J. Game Theory 47(3), 949–976 (2018) 6. G. Carbone, E. Ottaviano, M. Ceccarelli, An optimum design procedure for both serial and parallel manipulators. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 221, 829–843 (2007) 7. S.R. Babu, V.R. Raju, K. Ramji, Design optimization of 3PRS parallel manipulator using global performance indices. J. Mech. Sci. Technol. 30, 4325–4335 (2016) 8. B. Bounab, Multi-objective optimal design based kineto-elastostatic performance for the delta parallel mechanism. Robotica 34, 258–273 (2016) 9. X. Chai, J. Xiang, Q. Li, Singularity analysis of a 2-UPR-RPU parallel mechanism. J. Mech. Eng. 51, 144–151 (2015) 10. C. Yang, Q. Li, Q. Chen, L. Xu, Elastostatic stiffness modeling of overconstrained parallel manipulators. Mech. Mach. Theory 122, 58–74 (2018) 11. E. Mirshekari, A. Ghanbarzadeh, K.H. Shirazia, E. Mirshekari, A. Ghanbarzadeh, Structure comparison and optimal design of 6-RUS parallel manipulator based on kinematic and dynamic performances. Latin Am. J. Solids Structure 13, 2414–2438 (2016) 12. J. Wang, W. Chao, X. Liu, Performance evaluation of parallel manipulators: motion/force transmissibility and its index. Mech. Mach. Theory 45, 1462–1476 (2010) 13. J. Yang, X. Chen, X. Liu, Motion and force transmissibility of a planar 3-DOF parallel manipulator, in Proceedings—IEEE International Conference on Mechatronics and Automation (2012), pp. 761–765

Chapter 8

Hybrid Algorithm for Multi-objective Optimization Design of Parallel Manipulators

Since the parallel manipulators (PMs) have excellent dynamic performance, high agility, high load-to-weight ratio, distributed joint error, and simple inverse kinematics [1, 2], scholars, and enterprises have studied and designed PMs with the engineering application value in the past two decades. The successful commercial applications of PMs include the Delta robot [3], Tricept robot [4], Sprint Z3 [5] spindle head, and Exechon robot [6]. Although having the market value in some fields, PMs are not as successful as industrial serial robots because of expensive equipment provides poor performances [7]. Multi-objective optimization design is crucial for issues to be addressed. With the increasing complexity of engineering design, sophisticated analysis and optimization tools are used to study optimization design. However, global performance indices (GPIs) usually obtained by calculating the average or variance value of the local performance indices (LPIs) of limited discrete grid nodes in the workspace, and the accuracy is related to the node density in the workspace. As the node density increases, the computational cost increases exponentially. Also, the high-dimensional design leads to exponentially increased computational cost. In this chapter, we focus on the hybrid algorithm for the multi-objective optimization design with high efficiency and low computational cost based on the Gaussian process regression and particle swarm optimization (PSO) algorithm. The mapping modeling method based on the Gaussian process regression can significantly reduce the computational cost of the GPIs to improve the optimization design of the PMs. The comparison from the accuracy and robustness for back propagation neural network, multi-variate regression, and Gaussian process regression mapping models are discussed. Finally, the 2PRU-UPR PM is taken as an example here for the analysis and discussion.

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_8

205

206

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

8.1 Hybrid Algorithm and GPR-Based Mapping Modeling 8.1.1 Procedure of the Hybrid Algorithm The optimization design of the PM aims to find reasonable design parameters to optimize multi-objective performance. In general, the mathematical model of the multi-objective optimization design of PMs can be expressed as follows. ⎧ max {F1 (x), F2 (x), . . . , Fn (x)} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to G(x) = 0 , ⎪ ⎪ ⎪ H(x) < 0 ⎪ ⎪ ⎪ ⎩ x min ≤ x ≤ x max

(8.1)

where F i (x) and x denote the objective functions and design parameters, respectively; G(x) and H(x) the equality and inequality constraint functions, respectively. Since it is difficult to obtain the analytical expression for the regular workspace volume, the GPIs are usually obtained by the discrete numerical method, which is time-consuming and dramatically undermines the efficiency of multi-objective optimization design. The work proposes an efficient optimization design method based on the GPR-based mapping model and PSO algorithm. Figure 8.1 shows the flow of the proposed multi-objective optimization design method. The difference from the traditional optimization design method is that the objective functions are calculated through the GPR-based mapping model during optimization design, instead of the traditional discrete numerical method with a high computational cost. The detailed steps of the proposed hybrid algorithm for multi-objective optimization design combined with the GPR-based mapping modeling and PSO algorithm are shown as follows: 1. Establish the design parameters along with the range, objective, and constraint functions of the PMs, and construct the mathematical modeling of multi-objective optimization of the mechanism. 2. Obtain the sample dataset by the design of experiment (DOE). Calculate the values of objective functions corresponding to the design parameters by increasing the node density in the workspace, and provide sufficient information for mapping modeling. The LHD is recommended as the DOE, and the principle of LHD is shown in Sect. 3.6. 3. Use the samples obtained in Step (2) to establish the mapping modeling between the objective functions and design parameters by the GPR method. Then, evaluate the accuracy of the GPR-based mapping modeling by two types of model validations, namely cross-validation and external validation. 4. Execute the PMs’ multi-objective optimization design based on the PSO algorithm and GPR-based mapping modeling.

8.1 Hybrid Algorithm and GPR-Based Mapping Modeling

207

Fig. 8.1 Flow of the proposed hybrid algorithm for multi-objective optimization design

8.1.2 GPR-Based Mapping Model GPR is a non-parametric model using Gaussian process (GP) prior approach for regression analysis, and it is a kernel-based probability method. The hypothesis of the GPR method includes noise (regression residual) and Gaussian process prior, and the solution is based on Bayesian inference and joint probability distribution. If kernel function is not restricted, GPR is theoretically a universal approximator of any continuous function in a compact space. Besides, GPR can provide posteriori prediction. When the likelihood is in a normal distribution, the posterior has an analytical form. Therefore, GPR is a probability model with generality and resolvability. The principle of GPR is illustrated as follows.

208

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

For regression modeling, y = f (X) + ε

(8.2)

If the form is not fixed, f (X) is a latent function. GPR takes the prior of the function space as a GP, and f (X) is a GP where the covariance function (kernel function) is zero-mean. )] [ ( f (X) ∼ N 0, κ X, X , ,

(8.3)

) ( where X is the learning sample; κ X, X , the kernel function; X = [xT 1, xT 2,…, xT n]T ; y = [y1 , y2 ,…, yn ]T ; f (X) = [f (x 1 ), f (x 2 ),…, f (x n )]T . Given N sets of learning samples X = {X 1 , …, X N } and y = {y1 , …, yN }, regression residual (noise) is assumed to obey normal distribution p(ε) = N(ε | 0, σ 2 n). Since noise ε is the white noise and independent of f (X), y obeys Gaussian distribution when f obeys. According to the regression modeling and GP definition, the probability distribution of y and the f * value to be estimated as follows. [ ] y ∼ N 0, K + σn2 I ,

f ∗ ∼ N [0, κ(x∗ , x∗ )]

(8.4)

According to the properties of the Gaussian distribution and the data of test and training sets from the same distribution, the joint distribution of the training data and the test data is in a high-dimensional Gaussian distribution, and the prior of the joint probability distribution can be calculated as follows. [

] y f∗

]] [[ ] [ [[ ] [ ]] 0 0 K + σn2 I K ∗ κ(X, X) + σn2 I κ(X, X ∗ ) =N ∼N , , , κ(X ∗ , X ∗ ) K ∗T K ∗∗ 0 κ(X ∗ , X) 0

(8.5) where I is the N × N identity matrix; K (X, X) the N × N matrix. Based on the Bayesian formula, the posterior probability distribution of p (f * | y) can be obtained as ) ( ( | ) ( ) ( | ) ) ( p f, f∗ p f| f∗ p f∗ | = = N f ∗ | f ∗ , cov( f ∗ ) , p f ∗| f = (8.6) p( f ) p( f ) ( ) = K (X ∗ , X ∗ ) − where f ∗ = K (X ∗ , X)(K + σn2 I )−1 y; cov f ∗ ( ) 2 −1 K (X, X ∗ ). K (X ∗ , X) K + σn I Once the conditional distribution of predicted data f * is obtained, the mean value of the distribution is used as its estimated value. The GPR-based modeling can be implemented by function fitrgp in the MATLAB toolbox. The performance indices should be evaluated through cross-validation and external verification to evaluate the accuracy of mapping modeling. The most commonly used performance indices [7] are the square root error (RMSE), and R squared (R2 ).

8.2 Example: 2PRU-UPR PM

209

/Σ RMSE =

n i=1

(yi − yˆi )2 , n

Σn

(yi − yˆi )2 R = 1 − Σni=1 ( )2 , i=1 yi − y i 2

(8.7)

where yˆi is the predicted value calculated by GPR-based mapping modeling; yi the experimental value of discrete points; y i the mean value of yi . Higher R2 and smaller RMSE are preferred, which denotes the higher accuracy of mapping modeling in the overall design space.

8.2 Example: 2PRU-UPR PM The 2PRU-UPR PM, which can form five-axis hybrid machining with a 2-DOF series head added to its moving platform to realize the machining of complex curved surfaces, is taken as a case in the work. The reasons for choosing the 2PRU-UPR PM are as follows: (1) the 2PRU-UPR PM is a lower-mobility overconstrained PM, and its stiffness modeling and dynamic modeling are representative. (2) Its reachable workspace is irregular, so the GPIs do not have analytical expressions, and need to be solved by numerical discretization method, leading to the high computational cost and reduced optimization design efficiency. Thus, the optimization design of the 2PRU-UPR PM is more suitable to combine the mapping model and optimization algorithm to improve optimization design. The structure description and inverse kinematics have been described in Sect. 3.2.4. The structural parameters are reviewed as follows: elasticity modulus E = 210 GPa; shear modulus G = 77 GPa; material density ρ = 7850 kg/m3 . The moving platform is round; its height h = 20 mm; the diameter of the links d = 80 mm. oA1 = oA2 = r 1 ; oA3 = r 2 ; OB3 = r 3 ; A1 B1 = L 1 = A2 B2 = L 2 . The linear distances of the actuators are denoted as OB1 = q1 , OB2 = q2 , and A3 B3 = q3 . p = [x y z]T is the coordinate vector of midpoint o of the moving platform with the fixed frame. To facilitate the dynamics modeling, the reference coordinate frames, as shown in Fig. 8.2, are given as follows: The origin of the fixed coordinate frame O-XYZ is the midpoint of B1 B2 ; the X-axis is along the direction of line OB3 ; Y-axis is along the direction of line OB1 ; the Z-axis is determined by the right-hand rule. The origin of moving coordinate frame o-xyz attached to the moving platform is the midpoint of line A1 A2 . The x-axis is along the oA3 direction, the y-axis is along the oA1 direction; the z-axis is determined by the right-hand rule. The origin of the link coordinate frame Ai -x i yi zi is point Ai ; x i -axis (i = 1, 2) is parallel to the X-axis; the y3 -axis is parallel to the y-axis; the zi -axis (i = 1, 2, 3) is determined by the right-hand rule. For the PRU limb, u- and v-axes of the U-joints frames run along with the 2nd and 1st axes of its U joints, respectively. For the UPR limb, u- and v-axes of the U-joint frame run along with the 1st and 2nd axes of its U joint, respectively. The R-joints frames of the three limbs are parallel to the corresponding link coordinate frames, respectively.

210

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

Fig. 8.2 A 2PRU-UPR overconstrained PM B3 X

B1 Y

O B2

Z

x3

x2

A2

y2

y3

A3 z3

x

o

z

x1

A1

y

y1

z1

z2

Thus, the transform matrices can be expressed as follows. ⎡

⎡

⎤ cγ 0 sγ R = ⎣ −sβsγ cβ sβcγ ⎦, −cβsγ −sβ cβcγ

(8.8)

⎡ ⎡ ⎤ ⎤ ⎤ 1 0 0 1 0 0 cα3 0 sα3 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ R1 = ⎣ 0 cα1 −sα1 ⎦, R2 = ⎣ 0 cα2 −sα2 ⎦, o R3 = ⎣ 0 1 0 ⎦, R3 = Ro R3 , 0 sα1 cα1 0 sα2 cα2 −sα3 0 cα3

(8.9) ⎧ R = R1 R−1 ⎪ x (ϕ1 ) ⎨ U1 −1 RU 2 = R2 R x (ϕ2 ) , ⎪ ⎩ RU 3 = R3 R−1 y (ϕ3 )

(8.10)

where R is the transform matrix from the moving coordinate frame to the fixed coordinate frame; α i (i = 1, 2) the angle between Bi Ai and the z-axis; α 3 the angle between B3 A3 and the x-axis; Ri (i = 1, 2, 3) the transform matrices from the ith link coordinate frames to the fixed coordinate frame; o R3 the transform matrix from the 3rd link coordinate frame {A3 } to the moving coordinate frame. RUi are transform matrices from the three U-joints frames to the fixed frame. ϕ i (i = 1, 2) and ϕ 3 denote the angles between Bi Ai and y- and X-axes, respectively. Rx and Ry are the rotation matrices about x- and y-axes, respectively. The assumptions are made for the following kinematic, stiffness, and elastodynamic model: The weights of all components of the manipulator are negligible. The mobile platform and all joints are rigid, and all joints are frictionless.

8.2 Example: 2PRU-UPR PM

211

8.2.1 Kinematic Performance Index The motion/force transmission index is adopted in this chapter for kinematic performance evaluation to measure the transmission efficiency of a mechanism from input to output for its frame-free, dimensionless, and distance to singularities. The analysis of the motion/force transmission index of the 2PRU-UPR PM refer to Sect. 3.2.4. The LTI only denotes the kinematic index in a single configuration, thereby defining the global transmission index (GTI). Discrete points are often adopted as follows. { n 1Σ Vr ς dVr { or GTI = ςi , (8.11) GTI = n i=1 dVr where n is the number of discrete points in the workspace.

8.2.2 Stiffness Performance Index The amplitudes of constraint wrenches $r i1, $r i2, and $r i3 are represented by f i1 , mi1 , and f i2 , respectively. According to screw theory, the PRU limb exerts active force f i2 along with link Bi Ai , constraint force f i1 passing through point B1 and parallel to its R-joint axis, and a constraint couple perpendicular to the two axes of its U joint. The UPR limb exerts constraint force f 31 passing through point A3 and parallel to its R-joint axis, active force f 32 along the direction of link B3 A3 , and constraint couple m31 perpendicular to its two axes of the U joint. The amplitude of the constraint wrenches system of the ith limb is represented as follows: wi = [ f i1 f i2 m i1 ]T (i = 1, 2, 3)

(8.12)

According to the strain energy and Castigliano’s second theorem, the limb stiffness matrices [8] are obtained as follows. δi = C i wi , ⎡ ⎢ Ci = ⎢ ⎣

li G i λi Ai

+ 0

n· yi li2 2E i I yi

li3 3E i I yi

0 li E i Ai

0

n· yi li2 2E i I yi

0 (τi1 · yi )2 li E i I yi

+

(τi1 ·z i )2 li G i Ii

(8.13) ⎤ ⎥ ⎥ (i = 1, 2), ⎦

(8.14)

212

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

⎡ ⎢ C iB = ⎢ ⎣

qi G i λi Ai

+

qi3 3E i I xi

0 n·x i qi2 2E i I xi

0 qi E i Ai

0

n·x i qi2 2E i I yi

0 (n·x i )2 qi E i I xi

+

(n·z i )2 qi G i Ii

⎤ ⎥ ⎥ (i = 3), ⎦

(8.15)

where δ i = [δ i1 , δ i2 , δ i3 ]T is the elastic deflection of the ith limb corresponding to wi ; Gi and E i are the elasticity modulus and shear modulus of the ith limb, respectively; Axi and Ayi the effective shear area along the x- and y-axes, respectively; Ai is the area of the cross section; I xi and I yi are the area moments of inertia about the x- and y-axes, respectively; I i is the polar moment of inertia; λi the effective shear factor. The overall stiffness/compliance matrix of the 2PRU-UPR PM can be obtained based on the equilibrium equation and virtual principle [9]. K=

n Σ

J i K i J iT , C = K −1 ,

(8.16)

i=1

where J i = [$r i1, $r i2, $r i3]; K and C denote the overall stiffness and compliance matrices, respectively. The current stiffness indices include the eigenvalue index, determinant index, trace index, and virtual index. In the work, the most used trace index [10], defined as the summing of the diagonal elements of the overall stiffness matrix, is adopted as the global stiffness index (GSI). { GSI =

Vr

LSIdVr { dVr

or GSI =

n 1Σ LSIi , n i=1

(8.17)

where LSI is the abbreviation of the local stiffness index; LSI = tr(K), and the symbol tr means trace.

8.2.3 Elastodynamic Performance Index Considering the links are divided into s elements, the generalized displacement coordinates of the ith link are given as i

ui = [i ui,1 i ui,2 . . . i ui,s+1 ]T6(s+1)×1 (i = 1, 2, 3),

(8.18)

where i ui (i = 1, 2, 3) is the generalized displacement coordinates of the ith link expressed in its link coordinate frame; i ui,j = [i Δi,j , i ϕ i,j ]T the displacement coordinate of the jth node of the ith link; i Δi,j and i ϕ i,j denote the linear and angular displacement coordinates of the jth node of the ith link expressed in its link coordinate frame.

8.2 Example: 2PRU-UPR PM

213

Note that i ui,1 = i u Bi , andi ui,s+1 = i u Ai . Equation 8.18 can be further expressed as i

ui = [i u Bi i ui,in i ui, Ai ]T6(s+1)×1 (i = 1, 2, 3).

(8.19)

where i ui,in = [i ui ,2 ,…, i ui ,s ]T 6(s−1)×1 represents the displacement coordinates of inner nodes except for the two endpoints of the ith link. Therefore, the mapping relationship between the displacement coordinates of the jth element and the link can be obtained as i

ui,e j = T e, j i ui ,

(8.20)

where i ui,e j = [i ui ,j i ui ,j+1 ] is the displacement coordinates of the jth element of the ith link; T i,ej the mapping matrix from i ui to i ui,e j . According to the matrix structure analysis (MSA), the mass and stiffness matrices of the ith link with the local coordinate frame are given as ⎧ s Σ ⎪ ⎪ i T i ⎪ M = T i,e M i,e j T i,e j ⎪ i j ⎪ ⎨ j=1 s ⎪ Σ ⎪ i T i ⎪ ⎪ K = T i,e K i,e j T i,e j ⎪ j ⎩ i

(8.21)

j=1

With the transform matrix, the mass and stiffness matrices of the link expressed in the global coordinate frame can be expressed as {

M i = Di i M i DiT , K i = Di i K i DiT Di = diag[Ri , . . . , Ri ](6s+6)×(6s+6)

(8.22)

Establishing the elastodynamic modeling of the mechanism should extract its independent generalized displacement coordinates (IGDC). The method proposed in Ref. [11] is adopted in this chapter. With the multi-point constraint element theory, the compatibility conditions between the linear displacement of point Ai of link Bi Ai and the displacement coordinate uo of point o of the moving platform are given as ] [ Δ Ai = E 3 [ Ai o×] uo (i = 1, . . . , 6)

(8.23)

where E3 is a 3 × 3 identity matrix; [Ai o×] the skew symmetrical matrix. The PRU limbs connect the rods with the moving platform through the U joints, which can rotate freely around its two axes. Therefore, only one angular displacement coordinate at point Ai of link Bi Ai (i = 1, 2) is independent. With the multi-point constraint element theory, the angular displacement coordinates of the two nodes connected by the U joint should satisfy

214

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel … L

[ ] ϕ Ai z = 0 0 1 RU−1i ϕ o = Q i ϕ o (i = 1, 2)

(8.24)

[ ] where Q i = 0 0 1 RU−1i ; L ϕ Ai z is the angular displacement coordinate about the z-axis of point Ai expressed in its joint coordinate frame. The mapping relationship between the angular displacement coordinate of the point Ai of the link Bi Ai (i = 1, 2) in its joint coordinate frame and the fixed coordinate frame is given as ϕ Ai = RUi L ϕ Ai (i = 1, 2)

(8.25)

Equations 8.24 and 8.25 are combined to obtain ⎧ RUi (2, 2)ϕ Ai x − RUi (1, 2)ϕ Ai y − (RUi (2, 2)RUi (1, 3) − RUi (1, 2)RUi (2, 3)) Q i ϕ o ⎪ Lϕ ⎪ = ⎪ ⎪ ⎨ Ai x RU 11 RU 22 − RU 12 RU 21 i

i

i

i

⎪ RUi (2, 1)ϕ Ai x − RUi (1, 1)ϕ Ai y − (RUi (2, 1)RUi (1, 3) − RUi (1, 1)RUi (2, 3)) Q i ϕ o ⎪ L ⎪ ⎪ ⎩ ϕ Ai y = RUi (1, 2)RUi (2, 1) − RUi (1, 1)RUi (2, 2)

(i = 1, 2)

(8.26)

where RUi (j, k) is the element in the jth row and kth column of the matrix. Substituting Eq. 8.26 into Eq. 8.25, one can have ϕ Ai z = ai x ϕ Ai x + ai y ϕ Ai y + ai p Q i ϕ o ⎡ ⎤ ⎡ ⎤ ϕ Ai x ] ϕ Ai x [ = ai x ai y ai p Q i ⎣ ϕ Ai y ⎦ = N i ⎣ ϕ Ai y ⎦ (i = 1, 2), ϕo ϕo

(8.27)

R (1,1)R (3,2)−R (1,2)R (3,1) RUi (3,1)RUi (2,2)−RUi (3,2)RUi (2,1) ; ai y = RUUi (1,1)RUUi (2,2)−RUUi (1,2)RUUi (2,1) ; RUi (1,1)RUi (2,2)−RUi (1,2)RUi (2,1) i i i i RUi (3,2)(RUi (1,3)RUi (2,1)−RUi (1,1)RUi (2,3))−RUi (3,1)(RUi (1,3)RUi (2,2)−RUi (1,2)RUi (2,3)) RUi (1,1)RUi (2,2)−RUi (1,2)RUi (2,1)

where ai x =

ai p = +RUi (3, 3); N i = [aix aiy aip Qi ]. Therefore, ϕ Aix and ϕ Aiy can be used as two independent angular displacement coordinates of the point Ai of the link Bi Ai (i = 1, 2). If ϕ Aix and ϕ Aiz , or ϕ Aiy and ϕ Aiz are considered as independent angular displacement coordinates, mapping matrix N i will be singular in the working space. The derivation process is consistent with the above method, which is not discussed here. By the above method, the boundary conditions of the PRU limb at point Bi (i = 1, 2) can be obtained as {

Δ Bi = 03×1 (i = 1, 2) ϕ Bi y = ϕ Bi z = 0

(8.28)

Similarly, one can obtain the relationship between ϕ A3 of the UPR limb and uo as follows.

8.2 Example: 2PRU-UPR PM

[

ϕ A3 x ϕ A3 z

]

215

[ = N3

[ ] ϕ A3 y , N3 = ϕo

R3 (1,2) R3 (2,2) R3 (3,2) R3 (2,2)

a3x − a3z −

R3 (1,2)a3y R3 (2,2) R3 (3,2)a3y R3 (2,2)

] ,

(8.29)

[

] 1 0 0 −1 R ; Q3 (i,:) is the ith row of the matrix Q3 , a3x = 001 3 R3 (1, 1) Q 3 (1, :) + R3 (1, 3) Q 3 (2, :); a3y = R3 (2, 1) Q 3 (1, :) + R3 (2, 3) Q 3 (2, :); a3z = R3 (3, 1) Q 3 (1, :) +R3 (3, 3) Q 3 (2, :). Similarly, if ϕ A3 x or ϕ A3 z is considered as the independent angular displacement coordinate of the point A3 of the link B3 A3 , matrix N 3 will be singular in the workspace. Therefore, ϕ A3 y can be used as an independent angular displacement coordinate. The boundary conditions of the UPR limb at the point B3 can be obtained based on the above method as ⎧ ⎪ ⎨ Δ B3 = 03×1 [ ] ], [ (8.30) ϕ B3 x ⎪ , V 3 = b3x b3y ⎩ ϕ B3 z = V 3 ϕ B3 y

where Q 3 =

R

(2,2)R

(3,1)−R

(2,1)R

(3,2)

R

(1,1)R

(3,2)−R

(1,2)R

(3,1)

where b3x = RUU3 (1,1)RUU3 (2,2)−RUU3 (1,2)RUU3 (2,1) ; b3y = RUU3 (1,1)RUU3 (2,2)−RUU3 (1,2)RUU3 (2,1) . 3 3 3 3 3 3 3 3 Similarly, if ϕ B3 x and ϕ B3 z , or ϕ B3 y and ϕ B3 z are considered as the independent angular displacement coordinates of the point B3 of the link B3 A3 , matrix V 3 will be singular in the workspace. According to the above analysis, the global IGDC of the 2PRU-UPR PM is defined as follows: ]T [ U = ϕ B1 x uT1,in ϕ A1 x ϕ A1 y ϕ B2 x uT2,in ϕ A2 x ϕ A2 y ϕ B3 x ϕ B3 y uT3,in ϕ A3 y ΔTo ϕ To (8.31) The mapping relationship between the displacement coordinates of the rods/the moving platform and the global IGDC can be obtained as {

ui = H i U (i = 1, 2, 3) uo = H o U

,

(8.32)

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8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

⎡

⎤

03×(18×s−3)

⎢ ⎥ 1 01×(18s−4) ⎢ ⎥ ⎢0 ⎥ ⎢ 1×((18s−3) ⎥ ⎢0 ⎥ ⎢ 1×((18s−3) ⎥ ⎢ ⎥ H 1 = ⎢ 0(6s−6)×1 E (6s−6) 0(6s−6)×(12s+2) , ⎥ ⎢ ⎥ ⎢ 03×(18s−9) ⎥ E3 [ A1 o×] ⎢ ⎥ ⎢ 01×(6s−5) ⎥ 1 01×(12s+1) ⎢ ⎥ ⎣ 01×(6s−4) ⎦ 1 01×12s 01×(6s−5) a1x a1y 01×(12s−3) a1 p Q 1 (6s+6)×(18s−3) (8.33) ⎡ ⎤ 03×(18s−3) ⎢ 01×(6s−3) ⎥ 1 0 1×(12s−1) ⎢ ⎥ ⎢ 0 ⎥ ⎢ 1×((18s−3) ⎥ ⎢ 0 ⎥ ⎢ 1×((18s−3) ⎥ ⎢ ⎥ H 2 = ⎢ 0(6s−6)×(6s−2) E (6s−6) 0(6s−6)×(6s+5) , (8.34) ⎥ ⎢ ⎥ ⎢ 03×(18s−9) ⎥ E3 [ A2 o×] ⎢ ⎥ ⎢ 01×(12s−8) ⎥ 1 01×(6s+4) ⎢ ⎥ ⎣ 01×(12s−7) ⎦ 1 01×(6s+3) 01×(12s−8) ⎡

a2x

a2y

⎢ 0 1 ⎢ 1×(12s−6) ⎢ 1 ⎢ 01×(12s−5) ⎢ ⎢ 01×(12s−6) b3x ⎢ H3 = ⎢ ⎢ 0(6s−6)×(12s−4) E (6s−6) ⎢ 03×(18s−9) E3 ⎢ R3 (1,2) ⎢ 0 ⎢ 1×(18s−10) R3 (2,2) ⎢ 1 ⎣ 01×(18s−10) R3 (3,2) 01×(18s−10) R3 (2,2)

01×6s a2 p Q 2

(6s+6)×(18s−3)

⎤ 03×(18s−3) ⎥ 01×(6s+2) ⎥ ⎥ 01×(6s+1) ⎥ ⎥ ⎥ b3y 01×(6s+1) ⎥ ⎥ 0(6×s−6)×7 ⎥ ⎥ [ A3 o×] ⎥ R3 (1,2) 01×3 a3x − R3 (2,2) a3y ⎥ ⎥ ⎥ 01×6 ⎦ (3,2) 01×3 a3z − RR33 (2,2) a3y

,

(6s+6)×(18s−3)

(8.35) [

H o = 06×((18s−9) E 6

] 6×(18s−3)

,

(8.36)

where ui = [u Bi ui,in ui, Ai ]T6(s+1)×1 (i = 1, 2, 3); ui,in = T [ui,2 . . . ui,s ]6(s−1)×1 (i = 1, 2, 3). Thus, according to the method proposed in Ref. [11], the overall mass and stiffness matrices of the mechanism are as follows.

8.2 Example: 2PRU-UPR PM

217

⎧ 3 Σ ⎪ ⎪ g g ⎪ M = M i + M og ; M i = H iT M i H i ; M og = H To M o H o ⎪ ⎪ ⎨ i=1

(8.37)

3 ⎪ Σ ⎪ ⎪ g g ⎪ K = K i ; K i = H iT K i H i ⎪ ⎩ i=1

where M o = Do o M o DT o and o M o = diag[mo , mo , mo , mo r 1 2 /4, mo r 1 2 /4, mo r 1 2 /2] denote the mass matrices of the moving platform with the fixed and moving coordinate frames, respectively; mo = πr 1 2 hρ is the mass of the moving platform; Do = diag [R, R]. The natural frequencies of the mechanism can be calculated using the equation as ⎧ ⎨ (K − ωi2 M)Φi = 0 , ω ⎩ fi = i 2π

(8.38)

where ωi and f i are the ith angular frequency in rad/s and natural frequency in Hz, respectively. Φ i is the ith mode shape corresponding to ωi . A structure’s fundamental frequency, namely the first natural frequency, is very important. Higher fundamental frequency means higher control bandwidth, which reduces the mechanisms’ vibration response [11]. Thus, the first natural frequency is considered as the dynamic performance index of the mechanism. The global dynamic index (GDI) is defined as { GDI =

{

Vr

f 1 dVr dVr

or GDI =

n 1Σ f 1i n i=1

(8.39)

8.2.4 Regular Workspace Volume In this chapter, the regular circular truncated cone, composed of the maximum inscribed ellipse (the long axis’s value is twice that of the short axis) in different operating platform heights, is adopted as the regular workspace. The polar coordinate method proposed in Ref. [12] is adopted to calculate the regular workspace volume. The platform height is equally divided into t layers, and the maximum inscribed ellipse of each layer is obtained based on the polar coordinate method. The regular workspace volume can be obtained as Vr =

t−1 Σ 1 i=1

2

( Ai + Ai+1 )∗ Δz,

(8.40)

218

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

where Ai = 2πρ 2 is the area of the maximum inscribed ellipse of the ith layer; ρ the length of the minor axis of the ellipse;Δz the layer interval. Without loss of generality, the minimum length of the minor axis is set to 5° to ensure the reference point of the end effector can reach the desired orientation.

8.2.5 Multi-objective Optimization The mathematical modeling of the multi-objective optimization of the 2PRU-UPR PM can be expressed as ⎧ max{GTI ,GDI,GSI, Vr } ⎪ ⎪ ⎪ ⎪ ⎪ x = [r1 r2 r3 L 1 ] ⎪ ⎪ ⎪ ⎨ 0.15 ≤ r , r ≤ 0.35, 0.3 ≤ r ≤ 0.8, 0.6 ≤ L ≤ 1.0 1 2 3 1 , ⎪ ≤ r , r ≤ r , 2r ≤ L r 3 2 3 1 1 ⎪ 1 ⎪ ⎪ ⎪ ⎪ qi min ≤ qi ≤ qi max (i = 1, 2, 3) ⎪ ⎪ ⎩ θ j ≤ θmax ( j = 1, . . . 6)

(8.41)

The LHD has the following advantages: treating every design variable equally important, ensuring the sampling distributed in a given design space, providing more information within a design space, bearing the systematic error rather than random error, and the controllable sample size. Thus, the LHD is considered as the DOE. The samples obtained by the LHD [13] can be prescribed as follows. It is assumed that there are n observations on each of the d design variables; (F 1 , …, F d ) are the uniform distribution functions of independent input variables (x 1 , …, x d ); x ij is the ith value of the jth design variable x j (i = 1, …, n; j = 1, …, d). P = pij and G = gij are defined to be n × d matrices. Each column of matrix P is an independent random permutation of (1, …, n), and that of G is a uniform [0, 1] random variable independent of P. Design sites x ij of a sample by LHD is expressed by xi j = F −1 [( pi j − gi j )/n],

(8.42)

where pij and gij determine the positions of the “cell” and x ij in the “cell”, respectively. The computer-design experimental points consist of x ij . According to the recommended number of sets of parameters in Ref. [7], 476 sets of parameters of experiment 1 (EX 1) are generated by the LHD. As a standard procedure for building the GPR-based model, cross-validation and external validation should be conducted to prove the model’s accuracy. Cross-validation compares the model with all data points except for one, while external validation compares the model predictions with a test dataset not utilized to build the model. tenfold crossvalidation is adopted to evaluate the model.

8.2 Example: 2PRU-UPR PM

219

Fig. 8.3 Q-Q plots from cross-validation. a GTI, b GDI, c GSI, and d V r

Figure 8.3 shows the quantile–quantile (Q-Q) plot from the cross-validation process. The data plot is linear, indicating that the GPR model is in good agreement with the data. The other two experiments are used to generate 162 sample points to analyze the influence of perturbed data on the accuracy of the GPR-based mapping model. The convergence time of the mapping models established by the three experiments is 2.02, 1.68, and 1.69 s, respectively. Table 8.1 compares the RMSE and R2 values of cross-validation and external validation for the three experiments. The acceptance levels for the RMSE value are below 0.2, and the R2 value is above 0.9 [14]. Thus, even for the perturbed data, the results always show the effectiveness and high accuracy of the established mapping model combined with GPR and LHD methods. The data used in building the mapping model, cross-validation, external validation, and data in Table 8.3 can be found online.1 Two mapping models in Refs. [15, 16], the BP-based and MR-based models, were built to compare with GPR-based modeling. Poor robustness is one of the main disadvantages of the BP-based model. Taking V r as an example, Fig. 8.4 shows the 1

https://pan.baidu.com/s/1FzdQHQU0OUfxdC-6ztKpMw (password: 1121).

220

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

Table 8.1 Comparison of RMSE and R2 values of the GPR-based model of three experiments from cross-validation and external validation Cross-validation

External validation

GTI

GDI

GSI

Vr

RMSE-EX 1

0.008

0.004

0.002

0.015

R2 -EX 1

0.947

1.000

1.000

0.993

RMSE-EX 2

0.010

0.002

0.002

0.023

R2 -EX

0.928

0.999

1.000

0.992

RMSE-EX 3

2

0.009

0.003

0.002

0.026

R2 -EX 3

0.924

0.999

1.000

0.989

RMSE-EX 1

0.014

0.005

0.002

0.019

R2 -EX 1

0.946

0.999

1.000

0.998

RMSE-EX 2

0.012

0.004

0.002

0.034

R2 -EX

0.945

0.999

1.000

0.986

RMSE-EX 3

2

0.014

0.007

0.002

0.036

R2 -EX 3

0.945

0.999

1.000

0.986

variations of RMSE and R2 values obtained by the GPR-based and BP-based models from ten experiments. The GPR-based model has a better robust performance than the BP-based model. RMSE and R2 of the GPR-based model are almost unchanged in different experiments with high accuracy; however, those of the BP-based model fluctuate significantly. An analytical MR-based model is built to explore the mathematical relationships by MR. The linear, quadratic, cubic, and quartic polynomial functions can be obtained as

Fig. 8.4 Comparison of RMSE and R2 values between GPR and BP neural network methods. a RMSE and b R2

8.2 Example: 2PRU-UPR PM

221

⎧ t Σ ⎪ ⎪ 1 ⎪ y (x) = a + bi xi ⎪ 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ t t t t ⎪ Σ Σ Σ Σ ⎪ ⎪ ⎪ y 2 (x) = a0 + bi xi + ci xi2 + di j xi x j ⎪ ⎪ ⎨ i=1 i=1 i=1 j=i+1 t t t t t ⎪ Σ Σ Σ Σ Σ ⎪ ⎪ ⎪ y 3 (x) = a0 + bi xi + ci xi2 + di j xi x j + ei xi3 ⎪ ⎪ ⎪ ⎪ i=1 i=1 i=1 j=i+1 i=1 ⎪ ⎪ ⎪ ⎪ t t t t t t ⎪ Σ Σ Σ Σ Σ Σ ⎪ 2 3 ⎪ y 4 (x) = a + ⎪ b x + c x + d x x + e x + f i xi4 0 i i i i j i j i ⎪ i i ⎩ i=1

i=1

i=1 j=i+1

i=1

,

i=1

(8.43) where a0 , bi , ci , d ij , ei , and f i are the estimated regression coefficients obtained through the least square method; t is the sample size; x i x j the coupling of any two parameters, and that of three or more parameters is ignored here. Sample points of EX 1 are taken as a case, and Table 8.2 lists RMSE and R2 values of MR- and BP-based models from external validation. The GPR-based model has consistently higher R2 and lower RMSE values than MR- and BP-based models. The GPR model does not need to specify the model order, which is indispensable for the MR-based model. The regression equation is assumed to be a polynomial in the MR-based model, which affects the diversity of factors and the immeasurability of some factors. According to the above analysis, the GPR-based model is more suitable than the MR-based and BP-based models. The PSO algorithm is also based on the concepts of “population” and “evolution”. However, each particle determines the search strategy according to its speed, instead Table 8.2 Comparison of RMSE and R2 values from the external validation among the GPR-based, BP-based, and MR-based models RMSE

R2

GTI

GDI

GSI

Vr

MR-linear

0.023

0.055

0.022

0.047

MR-quadratic

0.018

0.014

0.004

0.019

MR-cubic

0.018

0.013

0.004

0.020

MR-quartic

0.020

0.013

0.004

0.022

BP-based

0.015

0.005

0.003

0.033

GPR

0.014

0.005

0.002

0.019

MR-linear

0.674

0.923

0.969

0.910

MR-quadratic

0.842

0.995

0.999

0.985

MR-cubic

0.780

0.996

0.999

0.984

MR-quartic

0.897

0.995

0.999

0.980

BP-based

0.890

0.999

0.999

0.988

GPR

0.946

0.999

1.000

0.998

222

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

of the crossover and mutation operation of the GA. The algorithm is easy to be implemented. Supposing the population is composed of N particles in the D-dimensional search space, the optimal position of the ith particle is recorded as pi = (pi1 , pi2 , …, piD ), and the optimal global position recorded as pg = (pg1 , pg2 , …, pgD ) (i = 1, 2, …, N). Particles update their positions and speeds according to these two optimal values. vi j (t + 1) = ω · vi j (t) + c1r1 (t)[ pi j (t) − xi j (t)] + c2 r2 (t)[ pgi (t) − xi j (t)], (8.44) xi j (t + 1) = xi j (t) + vi j (t + 1),

(8.45)

where ω is the dynamic inertia weight; c1 and c2 are the learning factors representing the weight coefficients of the particle-tracking historical value and optimal global value, respectively; r 1 and r 2 the random numbers distributed in the interval [0 1], respectively. The first part of the Eq. 8.44 represents the previous velocity of the particle used to ensure the algorithm’s global convergence. The second and third parts ensure that the algorithm has the ability of local convergence. The step size of the polar coordinate discrete point method is analyzed as follows: The height interval, polar radius, and polar angle are equally divided into eight parts. The calculation time of a set of GPIs is 185.17 s, and the time of using the GPR-based mapping model is about 2 s. Theoretically, one iteration takes 18,517 s, considering that the population size is 100. If node density is finer with the GPI, the calculation time will increase exponentially. The initial dynamic inertia weight of PSO is set as 0.729, and c1 and c2 are set as 1.49 [7]. The calculation time of the multi-objective optimization problem in Eq. (8.41) using the hybrid algorithm is 1016 s on a desk computer with a 3.00-GHz CPU, while using the PSO algorithm requires 63,4832 s. That is, the optimization method proposed in the work can save 99.84% of the computational cost. The results show that the proposed hybrid algorithm is valid. Figure 8.5 shows the Pareto frontier of the multi-objective optimization problem of the 2PRU-UPR PM. Pareto frontiers are not smooth surfaces or curves, caused by competition and coupling among multiple objective functions. GTI and V r change in the opposite direction with GSI and GDI, respectively, indicating that kinematics performance and workspace volume have a competitive relationship with stiffness performance and natural frequency, respectively. GTI with V r and GSI with GDI change in the positive direction. Without considering GTI and V r , both GSI and GDI can achieve the optimal value on the Pareto frontier simultaneously, and vice versa. When GTI is less than 0.543, increasing GTI will not significantly reduce the GDI and GSI of the mechanism, nor will it significantly increase V r . When the GTI is greater than 0.543, increasing the GTI will weaken the GDI and GSI and significantly improve V r . Therefore, GTI = 0.543 is the essential data for the multi-objective optimization design of the 2PRU-UPR PM.

8.2 Example: 2PRU-UPR PM

223

Fig. 8.5 Pareto frontiers of the multi-objective optimization problem of the 2PRU-UPR PM. a GTI, GDI, and GSI, b GTI, GDI, and V r , c GTI, GSI, and V r , d GDI, GSI, and V r , e GTI and GDI, f GTI and GSI, g GTI and V r , h GDI and GSI, i GDI and V r , and j GSI and V r

224

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

Table 8.3 Geometrical parameters and objective functions before and after optimization r1 (mm) Before optimization Case 1

After optimization Case 2

Case 3

r2 (mm)

r3 (mm)

L1 (mm)

GTI

GDI (Hz)

GSI (×108)

Vr (10-1rad2.m) 0.19

250.00

250.00

500.00

700.00

0.50

62.30

7.82

150.34

323.96

333.77

602.29

0.54

110.57

9.61

0.62

295.30

323.79

339.60

600.51

0.52

77.46

9.81

0.44

252.42

325.26

339.47

600.51

0.49

83.73

9.66

0.36

150.02

254.78

321.60

992.55

0.57

44.06

5.67

1.36

153.99

294.46

462.16

882.93

0.56

53.67

6.36

0.94

151.09

298.11

334.38

875.57

0.56

55.58

6.50

1.10

154.95

271.45

475.18

603.72

0.57

104.44

9.07

0.41

153.36

242.95

408.14

603.02

0.55

106.54

9.15

0.50

291.76

323.80

333.91

601.20

0.52

77.77

9.79

0.43

152.98

272.98

344.22

803.01

0.56

64.82

7.05

0.97

150.96

305.36

327.32

681.73

0.55

88.58

8.43

0.77

160.65

319.97

522.95

693.41

0.57

81.32

8.05

0.54

151.01

260.99

360.82

602.56

0.54

108.96

9.33

0.58

175.06

293.59

324.51

734.89

0.56

72.86

7.80

0.85

155.56

285.39

338.59

827.51

0.56

61.18

6.87

1.00

150.96

267.78

377.71

891.13

0.56

53.54

6.31

1.08

159.70

264.49

332.74

974.60

0.57

44.64

5.79

1.33

152.27

265.10

619.88

989.28

0.57

42.22

5.49

0.80

150.48

294.30

356.30

638.08

0.55

99.47

8.95

0.66

159.15

255.83

660.28

995.11

0.58

40.73

5.40

0.71

261.26

318.90

340.30

602.41

0.50

81.62

9.63

0.36

242.45

324.86

343.21

601.32

0.49

85.45

9.62

0.36

295.25

312.84

374.87

603.04

0.52

76.03

9.68

0.45

150.82

292.75

344.86

956.86

0.56

46.98

5.93

1.31

161.48

323.79

333.93

601.19

0.53

107.35

9.61

0.59

160.32

315.21

334.47

710.36

0.56

80.31

8.11

0.83

159.65

295.90

430.44

632.61

0.55

97.20

8.90

0.55

150.67

264.09

326.12

939.00

0.57

48.71

6.00

1.27

150.53

281.49

333.01

667.37

0.55

91.86

8.54

0.74

153.70

268.96

453.40

624.07

0.57

99.97

8.85

0.48

158.67

309.67

341.45

771.61

0.56

69.20

7.44

0.93

154.38

264.83

340.89

862.62

0.56

56.72

6.55

1.05

157.67

309.99

485.74

667.60

0.56

88.14

8.39

0.52

150.80

317.30

333.89

667.10

0.55

92.29

8.66

0.74

156.87

266.80

338.52

903.05

0.56

51.83

6.25

1.15

163.83

307.28

547.98

723.10

0.58

74.23

7.64

0.56

153.12

275.12

328.84

763.60

0.56

71.16

7.43

0.93

152.79

272.49

359.50

721.04

0.56

78.76

7.84

0.83

171.40

312.99

570.56

728.89

0.59

71.61

7.56

0.53

160.89

255.89

660.93

995.07

0.58

40.60

5.40

0.71

Table 8.3 shows 40 sets of geometrical parameters and objective functions on the Pareto frontier, as well as that of before optimization. The mechanism’s optimized performance indices are improved to a certain extent, especially for the graymarked data. All the performance indices are better than those before optimization, which proves the effectiveness of the proposed optimization method. The followings are taken as the cases: optimized parameters listed in Table 8.3 and non-optimized

8.2 Example: 2PRU-UPR PM

225

parameters r 1 = 250 mm, r 2 = 250 mm, r 3 = 500 mm, and L 1 = 700 mm. After optimization, the optimized kinematic performance of the three cases increases by 8, 10, and 10%, with the dynamic performance of 77.48, 59.66, and 48.14, and the stiffness performance 22.89, 14.45, and 10.74%, respectively. Besides, the volumes of the regular workspace indices increase by 63.16, 73.68, and 94.74%, respectively. Figures 8.6, 8.7, 8.8 and 8.9 show the distribution of the LPIs in the middle plane of the regular workspace of the 2PRU-UPR PM before and after optimization. In Figs. 8.6, 8.7 and 8.8, the LTI, f 1 , and LSI in the middle plane of the regular workspace are significantly improved after optimization, and the results are consistent with those in Table 8.3. The ellipse’s minor axis in the middle plane of the regular workspace increases from 12.5 to 13.5° after optimization. Compared with the regular workspace with a height of 0.2 m before optimization (the interval of each layer in Fig. 8.6 is 0.01 m.), the heights for the three cases are 0.18, 0.19, and 0.20 m, respectively. The maximum minor axis radii of the elliptical section of the regular workspace increase from 16 to 21.5, 21, and 23°, respectively. Thus, the workspace volume of the mechanism is

Fig. 8.6 Distribution of the LTI of the 2PRU-UPR PM. a Before optimization, b Case1, c Case 2, and d Case3

226

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

Fig. 8.7 Distribution of the f 1 of the 2PRU-UPR PM. a Before optimization, b Case 1, c Case 2, and d Case 3

improved, consistent with the data in Table 8.3. The optimization results are verified again. It is worth noting that the optimal position of the fundamental frequency has changed after optimization, indicating that geometrical parameters of the 2PRU-UPR PM also affect the optimal position of the fundamental frequency. The performance indices before and after optimization are all symmetrical about β = 0, which is determined by the symmetrical structure of one side of the mechanism.

8.3 Summary As today’s engineering optimization design becomes complex, intelligent analysis and optimization tools are developed to solve multi-objective and multi-parameter optimization design problems. However, the optimization algorithm’s high calculation cost is due to its time-consuming calculation of GPIs, which makes it challenging to apply the optimization algorithms to the high computation-intensive design

8.3 Summary

227

Fig. 8.8 Distribution of the LSI of the 2PRU-UPR PM. a Before optimization, b Case 1, c Case 2, and d Case 3

in a real-life engineering setting. A hybrid algorithm for multi-objective optimization design combined with the GPR-based mapping model and PSO algorithm was introduced in this chapter for such optimization design problems. The regular workspace volume, motion/force transmission performance, stiffness performance, and the fundamental frequency indices were considered as the objective functions. The mapping models between the GPIs and the design parameters were established by resorting to the GPR method. The mapping model combined with GPR and LHD methods showed high accuracy and robustness to the perturbed data. The MR-based and BP-based models were built to compare the accuracy and robustness with the GPR-based mapping model. The results showed that the GPR-based model had consistently lower RMSE and higher R2 than those of the MR- and BP-based models and better robustness than the BP-based model. A hybrid algorithm for multi-objective optimization design combined with the GPR-based mapping model and PSO algorithm was introduced to improve the high computation-intensive optimization design. The 2PRU-UPR PM was taken as an example to implement the introduced method. The computational cost was saved by 99.84% compared with that using the PSO algorithm. The Pareto frontier was obtained, and 40 non-dominated solution sets on the Pareto frontier were listed. In the three sets of the optimized parameters, the GTI increased by 8, 10, and 10%, GDI

228

8 Hybrid Algorithm for Multi-objective Optimization Design of Parallel …

Fig. 8.9 Regular workspace of the 2PRU-UPR PM. a Before optimization, b Case 1, c Case 2, and d Case 3

increased by 77.48, 59.66, and 48.14%, GSI increased by 22.89, 14.45, and 10.74%, V r increased by 63.16, 73.68, and 94.74%, respectively. It verified the correctness of the hybrid algorithm proposed in the work.

References 1. J.P. Merlet, Parallel Robots (Kluwer Academic Publishers, Dordrecht, 2002) 2. L.W. Tsai, Robot Analysis and Design: The Mechanics of Serial and Parallel Manipulators (Wiley, New York, 1999) 3. R. Cravel, DELTA, a fast robot with parallel geometry, in Proceedings of the 18th International Symposium on Industrial Robots (1988), pp. 91–100. 4. B. Siciliano, The Tricept robot: inverse kinematics, manipulability analysis and closed-loop direct kinematics algorithm. Robotica 17, 437–445 (1999) 5. T. Sun, Y. Song, Y. Li, J. Zhang, Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. J. Mech. Robot.-Trans. ASME 2(3), 031009 (2010) 6. D. Zlatanov, M. Zoppi, R. Molfino, Constraint and singularity analysis of the exechon tripod. Appl. Mech. Mater. 162, 141–150 (2012) 7. T. Sun, B. Lian, Stiffness and mass optimization of parallel kinematic machine. Mech. Mach. Theory 120, 73–88 (2018)

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8. C. Yang, Q. Li, Q. Chen, L. Xu, Elastostatic stiffness modeling of overconstrained parallel manipulators. Mech. Mach. Theory 122, 58–74 (2018) 9. C. Yang, Q. Li, Q. Chen, Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint. Appl. Math. Model. 78, 322–349 (2020) 10. G. Carbone, M. Ceccarelli, Comparison of indices for stiffness performance evaluation. Front. Mech. Eng-Prc. 5, 270–278 (2010) 11. C. Yang, Q. Li, Q. Chen, Natural frequency analysis of parallel manipulators using global independent generalized displacement coordinates. Mech. Mach. Theory 156, 104145 (2020) 12. C. Yang, Q. Li, Q. Chen, Multi-objective optimization of parallel manipulators using a game algorithm. Appl. Math. Model. 74, 217–243 (2019) 13. G. Wang, Adaptive response surface method using inherited latin hypercube design points. J. Mech. Des. 125(2), 210–220 (2003) 14. H. Agarwal, J. Renaud, Reliability based design optimization using response surfaces in application to multidisciplinary systems. Eng. Optim. 36, 291–311 (2004) 15. B. Lian, T. Sun, Y. Song, Parameter sensitivity analysis of a 5-DoF parallel manipulator. Robotics and Computer-Integrated Manufacturing 46, 1–14 (2017) 16. Z. Gao, D. Zhang, X. Hu, Y. Ge, Design, analysis, and stiffness optimization of a three degree of freedom parallel manipulator. Robotica 28, 349–357 (2010)

Chapter 9

Sensitivity Analysis and Multi-objective Optimization Design of Parallel Manipulators

In the optimization design mathematical model, dimensions are always concerned as variables, but it is noted that a mechanism especially parallel manipulator (PM) may associate with amount of dimensions. Some of them have a great influence on performance but others may not. Therefore, it is necessary to select the main parameters from the large quantity of parameters to be designed by means of parameter sensitivity analysis method, so as to eliminate unimportant parameters and simplify performance analysis and optimization model. Variance-based methods have assessed themselves as versatile and effective among the various available techniques for sensitivity analysis of model output [1]. The most commonly used variance-based Sobol’ sensitivity analysis method can be roughly divided into two categories: Monte Carlo [2] and direct integral methods [3]. Lian et al. [4, 5] investigated the effects of joint stiffness/compliance coefficients and parameters of cross section to the mass and stiffness performance of the T5 PM through parameter sensitivity analysis based on response surface method (RSM) and Monte Carlo simulation method. Liu et al. [3] conducted sensitivity analysis in structural crashworthiness based on the Sobol’ direct integral method. Tang and Yao [6] design a set of optimized dimensional parameters for constructing the six-cable driven PM of FAST based on the sensitivity design method and tension performance evaluating functions. Meanwhile, traditional optimization design often adopts numerical discrete method to calculate global performance indicators, and its high computational cost will significantly reduce the efficiency of multi-objective optimization design [1, 4]. In this chapter, we focus on a multi-objective optimization design method of PMs combining RSM, sensitivity analysis, and intelligent optimization algorithm. The analytical mapping model between the global performance index and design parameters can be combined with the variance-based Sobol’ sensitivity method to obtain the analytical solution of the sensitivity index, so that the design parameters that have an important impact on the objective function can be obtained. The optimization design method combining sensitivity analysis results, response surface model, and intelligent optimization algorithm can significantly improve the efficiency of

© Huazhong University of Science and Technology Press 2023 Q. Li et al., Performance Analysis and Optimization of Parallel Manipulators, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-99-0542-3_9

231

232

9 Sensitivity Analysis and Multi-objective Optimization Design …

multi-objective optimization design of PMs. The Delta PM is taken as an example here for the analysis and discussion.

9.1 Sensitivity Analysis and Multi-objective Optimization Design Method 9.1.1 Response Surface Model The mathematical model of the RSM based on multi-variate regression (MR) is defined as follows. ⎧ t Σ ⎪ ⎪ 1 ⎪ y (x) = a + bi xi ⎪ 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ t t t t ⎪ Σ Σ Σ Σ ⎪ 2 2 ⎪ ⎪ y (x) = a + b x + c x + di j xi x j 0 i i i i ⎪ ⎪ ⎨ i=1

i=1

i=1 j=i+1

t t t ⎪ Σ Σ Σ ⎪ 3 2 ⎪ ⎪ y (x) = a + b x + c x + d x x + ei xi3 0 i i i i ij i j ⎪ ⎪ ⎪ ⎪ i=1 i=1 i=1 j=i+1 i=1 ⎪ ⎪ ⎪ ⎪ t t t t t t ⎪ Σ Σ Σ Σ Σ Σ ⎪ 4 2 3 ⎪ ⎪ y (x) = a + b x + c x + d x x + e x + f i xi4 0 i i i i ij i j i i ⎪ ⎩ t Σ

t Σ

i=1

i=1

i=1 j=i+1

i=1

i=1

(9.1) where yi (i = 1, 2, 3, 4) represents the ith fitting polynomial functions; a0 , bi , ci , d ij , ei , and f i represent the undetermined regression coefficients; t represents the number of design parameters; x i x j the coupling of two parameters. The coupling of three-parameters x i x j x k and the higher-order coupling are ignored in the equation. The MR-based RSM based on the principle of least squares estimates the regression coefficient by minimizing the square of the difference between estimated value yˆi and the measured value yi , that is, min εi =

n Σ

(yi − yˆi )2

(9.2)

i=1

For linear regression of a design parameter, only two points are needed to determine a straight line; however, the obtained straight line is not accurate for actual measurements. When the measurement point is greater than 2, the number of equations is greater than their unknown number. They are overdetermined equations, which are solved by the fitting method. According to Eq. 9.2, for the first-order fitting polynomial,

9.1 Sensitivity Analysis and Multi-objective Optimization Design Method

⎧ ⎪ ∂εi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂a0 ⎪ ⎪ ⎪ ⎪ ⎪ ∂εi ⎪ ⎪ ⎪ ⎪ ⎪ ∂b1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ε i ⎪ ∂b 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂εi ⎪ ⎪ ⎪ ⎪ ∂b3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂εi ⎪ ⎪ ⎪ ⎩ ∂b 4

= −2

n Σ

(yi − yˆi )

i=1

=2

n Σ

=2 =2

i=1 n Σ

n t Σ Σ ∂ yˆi = −2 [(yi − a0 − bi xi )x1 ] = 0 ∂b1 i=1 i=1

(yi − yˆi )

n t Σ Σ ∂ yˆi = −2 [(yi − a0 − bi xi )x2 ] = 0 ∂b2 i=1 i=1

(yi − yˆi )

n t Σ Σ ∂ yˆi = −2 [(yi − a0 − bi xi )x3 ] = 0 ∂b3 i=1 i=1

(yi − yˆi )

n t Σ Σ ∂ yˆi = −2 [(yi − a0 − bi xi )x4 ] = 0 ∂b4 i=1 i=1

i=1

=2

n Σ i=1

n t Σ Σ ∂ yˆi = −2 (yi − a0 − bi xi ) = 0 ∂a0 i=1 i=1

(yi − yˆi )

i=1 n Σ

233

(9.3)

Regression parameters a0 and bi can be obtained by solving normal Eq. (9.3). The same method can be used to find the regression coefficients of quadratic, cubic, and quartic polynomial fitting functions. Performance indices of the fitted function are evaluated through cross-validation and external validation to assess the accuracy of the fitting polynomial. The most commonly used performance indices are the relative average absolute error (RAAE), relative maximum absolute error (RMAE), root-mean-squares error (RMSE), and R-squared (R2 ). | | | Σn || ⎧ ˆi | max(| yi − yˆi |) i=1 yi − y ⎪ ⎪ | | RAAE = n | ⎪ ⎪ | , RMAE = Σn (| yi − y |/n) Σ ⎪ i ⎪ i=1 | yi − y i | ⎨ i=1 |Σ ⎪ Σn ⎪ n ⎪ ⎪ (yi − yˆi )2 ˆ i )2 ⎪ i=1 (yi − y 2 ⎪ , R = 1 − Σni=1 ( )2 ⎩ RMSE = n yi − y i

(9.4)

i=1

9.1.2 Sensitivity Analysis Sensitivity analysis is an analytical technique that studies how the uncertainty in the model output is distributed to different sources of uncertainties in the input factors of the model, and it can identify important input variables. At present, traditional calculation methods often use sampling techniques [7, 8] to approximate the sensitivity of calculation parameters for specific engineering problems, and the stability and accuracy of calculation results depend on samples. The combination of the MR-based RSM and Sobol’s sensitivity analysis method can avoid or reduce the difficulties in the solution process and increase the calculation efficiency and accuracy.

234

9 Sensitivity Analysis and Multi-objective Optimization Design …

Given model Y = f (X 1 , X 2 , …, X k ), the output parameter Y is a scalar. Based on Sobol’s sensitivity index definition [7], the variance-based first-order sensitivity index is defined as ⎧ ⎨ S = VXi (E X∼i (Y |X i )) i V (Y ) , (9.5) ⎩ V (Y ) = VXi (E X∼i (Y |X i )) + E Xi (VX∼i (Y |X i )) where X i represents the ith impact factor; X ~i the matrix composed of all impact factors except for X i ; E X~i (·) and V X~i (·) represent the mean and variance of input (·) when all possible values of the parameters except for X ~i are taken, respectively; E X~i (Y|X i ) represents that the mean of Y takes all possible values of X ~i with X i unchanged; E Xi (·) and V Xi (·) represent the mean and variance of input (·) when all possible values of X i are taken; S i is a standardized index with a value range of 0–1; V Xi (E X~i (Y|X i )) evaluates the first-order influence of impact factor X i on the model output; E Xi (V X~i (Y|X i )) is usually called the residual. Therefore, the index is also called the local sensitivity index. Local sensitivity index S i only considers the first-order influence of parameter X i instead of the influence of parameters’ cross-action. The global sensitivity index is defined as follows to evaluate the overall impact of X i on the model output. ST i =

VX∼i (E Xi (Y |X ∼i )) E X∼i (VXi (Y |X ∼i )) =1− V (Y ) V (Y )

(9.6)

An analytical MR-based RSM between the model output and design parameters was established in the work. Since the established MR-based RSM is a polynomial with a simplified structure, the sensitivity results of the model can be obtained quickly through Sobol’s direct integration, thereby evaluating the sensitivity of the structural parameters of complex systems. When the Sobol’s method was adopted to identify and quantitatively evaluate model parameters, n-dimensional model parameters X were transformed into unit hypercube H n , H n = {X|0 ≤ X i ≤ 1, i = 1, 2, …, n}. Based on variance decomposition [2], the response function is decoupled as f (X) = f 0 +

Σ i=1

fi +

ΣΣ

f i j + · · · + f 12...n

(9.7)

i=1 j>i

The total number of subitems in Eq. (9.7) is 2n . The subterms of f i = f i (X i ) and f ij = f ij (X i , X j ) can be obtained by following multiple integrations.

9.1 Sensitivity Analysis and Multi-objective Optimization Design Method

235

{ ⎧ ⎪ f = f (X)dX ⎪ 0 ⎪ ⎪ ⎪ ⎪ n K ⎪ ⎪ { ⎪ Π ⎪ ⎪ ⎪ ⎪ fi = f (X) dX k − f 0 ⎪ ⎪ ⎪ ⎪ k/=i n−1 ⎪ K ⎪ ⎪ { ⎪ Π ⎪ ⎪ ⎪ f (X) dX k − f i − f j − f 0 ⎨ fi j = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

k/={i, j}

K n−2

.. .

{

f i1 ...im =

Π

f (X) K n−m

−...

Σ

(9.8) Σ

dX k −

k/={i 1 ,...,i m }

f j1 ... jm−1

j1 f 1 (a), and f 2 (b) < f 2 (a).

11.1.2 Procedure of the Intelligent-Direct Search Algorithm The intelligent optimization algorithm has the ability of global optimization, however, poor local optimization, low search efficiency, slow convergence speed in the later stage of evolution, and exponential increase in computational cost while increasing the population size are its main disadvantages. To fix these problems, the hybrid algorithm of intelligent-direct search is presented in this chapter. The procedure of the proposed algorithm can be summarized as the following three steps. Step 1. Direct search algorithm: coarse meshing. The design parameter space is coarsely meshed into n discrete points, and the approximate amplitude of objective function corresponding to each point is computed, and the results are then used to transform objective functions to be of same order and dimensionless [2].

11.2 Example: 2UPR-RPU PM

289

fi =

GPIi − GPIi min , GPIi max − GPIi min

(11.2)

where GPIimax and GPIimin are maximum and minimum of the ith global performance index (GPI) obtained from coarse meshing, respectively. Step 2. Intelligent optimization algorithm. After objective functions are dimensionless, the multi-objective intelligent optimization algorithm is used to obtain the Pareto front and Pareto optimal set of the multi-objective optimization problem of PMs. The PSO algorithm is adopted as the optimization algorithm in this work due to its faster convergence [3]. Step 3. Direct search algorithm: fine meshing. Inspired by the concept of Pareto front surface, the neighborhood of the Pareto optimal set is refined to obtain more non-dominated solutions closer to the real Pareto front, which will be discussed in detail in the Sect. 11.2. Figure 11.2 shows the procedure of the calculation of non-dominated solutions. Taking the maximization of objective functions as an example, suppose there are n functions, if Eq. 10.3 is satisfied, we call f (i) = [f 1 (i),…, f n (i)] is dominated by f (j), vice versa. ⎧ f 1 (i ) < f 1 ( j ), f 2 (i ) < f 2 ( j ), . . . , f n (i) < f n ( j ) ⎪ ⎪ ⎪ ⎪ ⎪ f 1 (i ) = f 1 ( j ), f 2 (i ) < f 2 ( j ), . . . , f n (i) < f n ( j ) ⎪ ⎪ ⎨ f 1 (i ) < f 1 ( j ), f 2 (i ) = f 2 ( j ), f 3 (i ) < f 3 ( j ), . . . , f n (i ) < f n ( j ) (i /= j) ⎪ ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎩ f 1 (i ) < f 1 ( j ), . . . , f n−1 (i ) < f n−1 ( j ), f n (i ) = f n ( j ) (11.3) There are n × (n − 1) × … × 1 + 1 equations in Eq. 11.3. All sets of non-dominated solutions obtained by Eq. 11.3 form the Pareto optimal front.

11.2 Example: 2UPR-RPU PM The multi-objective optimization design of the 2UPR-RPU PM is taken as an example to implement the presented hybrid algorithm. Structural description and kinematic analysis refer to Sect. 6.2.

290

11 Multi-objective Optimization Design of Parallel Manipulators Based …

Fig. 11.2 Procedure of the calculation of the Pareto optimal front

11.2.1 Performance Indices of the 2UPR-RPU PM • Performance index 1: regular workspace volume Compared with serial mechanisms, one of the disadvantages of PMs is its smaller workspace. Thus, the regular workspace volume is considered as one of the objective functions in this chapter. Since the shape and size of the reachable workspace are irregular that is not conducive to the trajectory planning and algorithm control, so the regular cylinder volume is adopted as the workspace index. To ensure the reference point of the end effector can reach the desired orientation, the minimum radius of the circular cross section is set to 8° . The geometrical constraints and the procedure of calculating the regular workspace refer to Sect. 7.2.1, the equation of regular workspace volume refer to Eq. 7.3.

11.2 Example: 2UPR-RPU PM

291

• Performance index 2: kinematic performance The Jacobian matrix is often adopted as the kinematic performance evaluation index [4–6]. However, the disadvantages of the Jacobian matrix are non-uniform dimension and related to coordinate frame, which leads to its physical meaning is unclear. The motion/force transmission index proposed by Liu [7] is adopted in this chapter as one of the objective functions, which is defined as the efficiency of the mechanism from input to output, and it is frame-free and relative to singularity. This index is composed of two parts: ⎧ | | | N Ti ◦ $Ai | ⎪ ⎪ ⎪ | | ⎪ ⎨ λi = | N Ti ◦ $Ai | max | , | | | ⎪ N ◦ $ ⎪ Ti Oi ⎪ ηi = | | ⎪ ⎩ | N Ti ◦ $Oi |

(11.4)

max

where λi is the input transmission index (ITI), which measures the efficiency of energy from actuator joints to limbs, ηi is the output transmission index (OTI), which measures efficiency of energy from limbs to the output. N Ti is the unit transmission wrench screw; $Ai the unit input twist screw; $Oi the unit output twist screw. LTI can be obtained using Eq. 4.11. LTI only reflects the kinematic performance of the mechanism under a given configuration; in order to evaluate the kinematic performance in the regular workspace, the global transmission index (GTI) is defined in Eq. 7.16. • Performance index 3: Stiffness Performance Compared with serial mechanisms, the excellent stiffness performance is one of the advantages of PMs. In order to evaluate the stiffness performance of PMs, the overall stiffness matrix of the mechanism should be obtained first and then converted into quantifiable stiffness indices. The overall stiffness matrix of the 2UPR-RPU PM can be expressed as K=

n Σ

J i K i J iT ,

(11.5)

i=1

where K is the overall stiffness matrix; K i the limb stiffness matrix; J i the constraint wrenches of the ith limb. The process of the elastostatic stiffness modeling can be found in Ref. [8]. The comprehensive stiffness index (CSI) [2] decouples the diagonal and nondiagonal elements and separates the linear stiffness and angular stiffness, and the physical meaning is clear; thus, it is adopted in our work. The CSI is briefly reviewed as follows. The mapping relationship between the external wrench and elastic deformation is given by: δ = CW,

(11.6)

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11 Multi-objective Optimization Design of Parallel Manipulators Based …

where δ = [δ d ; δ ϕ ] is the elastic deformation, δ d and δ ϕ are the translational and rotational deflections, respectively. C is the overall compliance matrix; W = [nf ; nm ] the external wrench, nf and nm are the force and moment, respectively. The compliance matrix can be expressed as flows: [ C=

] C 11 C 12 , C 21 C 22

(11.7)

where C 11 , C 12 , C 21 , and C 22 represent the 3 × 3 submatrix; the units are m/N, m/N m, rad/N, and rad/N m, respectively. In most cases, engineers pay more attention to the linear deflection of the mechanism resistant of a force. The translational and rotational deflections resistant of the force can be obtained as follows: { δ d f = C 11 n f , (11.8) δ ϕ f = C 21 n f where δ df and δ ϕ f are the translational and rotational deflections resistant of the force, respectively. The linear stiffness performance along the z-axis of the 2UPR-RPU PM is adopted as one of the objective functions here. kd f z =

1 δd f z

,

(11.9)

where k dfz is the local CSI reflect the linear stiffness resistant a unit force along the z-axis direction in the Cartesian coordinate system. δ dfz the translational deflection resistant of a unit force along the z-axis direction. The global CSI can thus be defined as follows: { Kd f z =

Vr

kd f z dVr { dVr

or K d f z

n 1Σ = kd f zi n i=1

(11.10)

11.2.2 Multi-objective Optimization Design of the 2UPR-RPU PM In order to verify the presented hybrid algorithm, three-objective optimization design of the 2UPR-RPU PM are considered in our work, namely K dfz versus V r versus GTI. Consider x = [r 1 , r 2 , r 3 ] as design parameters. The multi-objective optimization problem of the 2UPR-RPU PM is given as follows:

11.2 Example: 2UPR-RPU PM

293

max { f1 , f2 , f3 } subject to r3 − r1 < 0 0.3 ≤ r1 ≤ 0.6

(11.11)

0.3 ≤ r2 ≤ 0.6 0.2 ≤ r3 ≤ 0.6 The step size of the coarse grid is set to 0.05 m. The maximum and minimum of the GPIs corresponding to the discrete points are obtained as follows: K dfz min = 4.657 × 108 N/m, K dfzmax = 9.940 × 108 N/m, V rmin = 0.027 rad2 m, V rmax = 0.165 rad2 m, GTImin = 0.411, and GTImax = 0.760. According to Eq. 11.2, the objective functions are defined as follows. f1 =

K d f z − K d f z min , K d f z max − K d f z min

f2 =

Vr − Vr min , Vr max − Vr min

f 3 = GTI

(11.12)

Figure 11.3 shows the optimal Pareto front of three objective functions and optimal design parameters. 80 non-dominated solutions are obtained in 57,362 s on a desk computer with a 3.00-GHz CPU considering the number of population size is 200 with paretofraction is 0.4. Figure 11.3a shows the hypersurface stretched by the optimal design parameter space. In order to obtain more non-dominated solutions, the ± 0.1 interval of the optimal design parameter neighborhood is refined in step size 0.005 m. The procedure proposed in Fig. 11.2 is used to obtain a total of 242 sets of nondominated solutions in 18,329 s. Figure 11.4 shows the comparison of Pareto front obtained by the hybrid and PSO algorithms. The presented hybrid algorithm not only got more non-dominated solutions without significantly increasing the computational cost but also closer to the real Pareto front. Taking two cases on the Pareto front shown in Fig. 11.4a as examples, Table 11.1 shows the comparison results of the two cases. Compared with the results of PSO algorithm, for case 1, K dfz and GTI of hybrid algorithm increased by 7.54 and 2.00%,

Fig. 11.3 Pareto solution set and Pareto front obtained by PSO. a Pareto solution set, b Pareto front

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11 Multi-objective Optimization Design of Parallel Manipulators Based …

Fig. 11.4 Comparison of Pareto front obtained by PSO and hybrid algorithms. a K dfz , V r and GTI, b projection of K dfz and V r , c projection of K dfz and GTI, d projection of V r and GTI

respectively, while the regular workspace volume does not decrease; for case 2, K dfz and GTI of hybrid algorithm increased by 25.48 and 0.66%, respectively, while the regular workspace volume does not decrease. For a more intuitive comparison of changes in the optimization results of two optimization algorithms, the change ratio is defined as follows: { k, LPIi, μkz = kdd jf fz ⇒ (11.13) μ= , LPIi μLTI = LTI LTI

where μ is the change ratio, and LPIi ’ and LPIi denote the results of the local performance index obtained by hybrid and PSO algorithms, respectively. Figure 11.5 shows the μ distribution of the case 1 in the middle plane z = − 0.63 m. It can be found that the optimized LTI is improved for all configurations in the middle plane, the maximum of the optimized k dfz increased by 25%. It is worth noting that the stiffness performance corresponding to some points is degraded, which means that the improvement of the global stiffness performance does not mean that the improvement of stiffness performance for all configurations in the regular workspace. However, overall, the optimization results of hybrid algorithm are better than those of PSO algorithm.

11.2 Example: 2UPR-RPU PM

295

Table 11.1 Comparison of two cases of PSO and hybrid algorithms Non-optimized

r 1 /m

r 2 /m

r 3 /m

0.390

0.390

0.260

K dzf /N/m (×108 ) 6.837

Vr rad2 .m

GTI

0.124

0.547

Case1-PSO

0.319

0.314

0.206

6.682

0.152

0.451

Case1-hybrid algorithm

0.3000

0.320

0.178

7.186

0.152

0.460

Case2-PSO

0.416

0.600

0.206

9.212

0.485 × 10–1

0.757

0.485 ×

0.762

Case2-hybrid algorithm

0.300

0.600

0.148

11.559

10–1

Fig. 11.5 The μ distribution of case 1 in the middle plane z = − 0.63 m. a μkz , b μLTI

Figure 11.6 shows the μ distribution of case 2 in the middle plane z = − 0.59 m. It can be found that the optimized LTI and k dfz are improved for all configurations in the middle plane, and the maximum of optimized k dfz and LTI are increased by 80 and 7%, respectively. The above results show the effectiveness of the proposed hybrid algorithm combined with the intelligent optimization and direct search algorithms.

Fig. 11.6 The μ distribution of case 2 in the plane z = − 0.59 m. a μkz , b μLTI

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11 Multi-objective Optimization Design of Parallel Manipulators Based …

11.3 Summary With the wide application of intelligent optimization algorithm in the multi-objective optimization design of PMs, its disadvantages are gradually revealed, such as premature maturity, slow convergence rate in the late stage, and the significant increase in computational cost with the increasing of the population size. To obtain more non-dominated solutions closer to the real Pareto front, a larger population size and iterations are required, which will lead to an exponential increase in computational cost. How to intuitively and efficiently get more non-dominated solutions closer to the real Pareto front without significantly increasing the computational cost is an urgent problem to be solved in engineering. This chapter presented a hybrid algorithm that used a direct search algorithm to further optimize the results by refining the grid of the results obtained by the intelligent optimization algorithm and a procedure for direct search algorithm to identify the non-dominated solutions. With the hybrid algorithm, more non-dominated solutions closer to the real Pareto front can thus be obtained without significantly increasing the computational cost. The multi-objective optimization design of the 2UPR-RPU PM is taken as an example to implement the proposed algorithm. Compared with 80 sets of non-dominated solutions obtained by the PSO algorithm, 242 sets of solutions closer to the real Pareto front were obtained by the hybrid algorithm. In the two cases of non-dominated solutions, compared with that of PSO algorithm, the K dfz increased 7.54 and 25.48%, GTI increased 2.00 and 0.66%, respectively, while the V r did not decrease. It verifies the effectiveness of the presented hybrid algorithm.

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