Advances in Mechanism, Machine Science and Engineering in China: Proceedings of IFToMM CCMMS 2022 9811993971, 9789811993978

Citation preview

Lecture Notes in Mechanical Engineering

Xinjun Liu   Editor

Advances in Mechanism, Machine Science and Engineering in China Proceedings of IFToMM CCMMS 2022

Lecture Notes in Mechanical Engineering Series Editors Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini , Dipartimento di Ingegneria “Enzo Ferrari”, Università di Modena e Reggio Emilia, Modena, Italy Vitalii Ivanov, Department of Manufacturing Engineering, Machines and Tools, Sumy State University, Sumy, Ukraine Editorial Board Francisco Cavas-Martínez , Departamento de Estructuras, Construcción y Expresión Gráfica Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland Jinyang Xu, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. To submit a proposal or request further information, please contact the Springer Editor of your location: Europe, USA, Africa: Leontina Di Cecco at [email protected] China: Ella Zhang at [email protected] India: Priya Vyas at [email protected] Rest of Asia, Australia, New Zealand: Swati Meherishi at [email protected] Topics in the series include: • • • • • • • • • • • • • • • • •

Engineering Design Machinery and Machine Elements Mechanical Structures and Stress Analysis Automotive Engineering Engine Technology Aerospace Technology and Astronautics Nanotechnology and Microengineering Control, Robotics, Mechatronics MEMS Theoretical and Applied Mechanics Dynamical Systems, Control Fluid Mechanics Engineering Thermodynamics, Heat and Mass Transfer Manufacturing Precision Engineering, Instrumentation, Measurement Materials Engineering Tribology and Surface Technology

Indexed by SCOPUS and EI Compendex. All books published in the series are submitted for consideration in Web of Science. To submit a proposal for a monograph, please check our Springer Tracts in Mechanical Engineering at https://link.springer.com/bookseries/11693

Xinjun Liu Editor

Advances in Mechanism, Machine Science and Engineering in China Proceedings of IFToMM CCMMS 2022

Editor Xinjun Liu Department of Mechanical Engineering Tsinghua University Beijing, China

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-19-9397-8 ISBN 978-981-19-9398-5 (eBook) https://doi.org/10.1007/978-981-19-9398-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organization

Steering Committee Members Han Ding, Huazhong University of Science and Technology, China Jianrong Tan, Zhejiang University, China Zongquan Deng, Harbin Institute of Technology, China Huayong Yang, Zhejiang University, China Jiansheng Dai, King’s College, London, UK Hong Liu, Harbin Institute of Technology, China Shuxin Wang, Tianjin University, China Tian Huang, Tianjin University, China Feng Gao, Shanghai Jiao Tong University, China

Program Committee Members Xianmin Zhang, South China University of Technology, China Xiangyang Zhu, Shanghai Jiao Tong University, China Delun Wang, Dalian University of Technology, China Yongsheng Zhao, Yanshan University, China Jie Zhao, Harbin Institute of Technology, China Xilun Ding, Beihang University, China Guilin Yang, CSA (Ningbo), China Yan Chen, Tianjin University, China Bing Xu, Zhejiang University, China Qingkai Han, Northeastern University, China Guimin Chen, Xi’an Jiao Tong University, China Chongfeng Zhang, Shanghai Academy of Spaceflight Technology, China Qinchuan Li, Zhejiang Sci-Tech University, China Bin Zi, Hefei University of Technology, China

v

vi

Organization

Guoying Gu, Shanghai Jiao Tong University, China Bo Tao, Huazhong University of Science and Technology, China Meng Chen, Institute of Aerospace Systems Engineering Shanghai, China Yanan Yao, Beijing Jiao Tong University, China Yimin Song, Tianjin University, China Yunjiang Lou, Harbin Institute of Technology (Shenzhen), China Jingjun Yu, Beihang University, China Jianjun Zhang, Hebei University of Technology, China Huafeng Ding, China University of Geosciences, China Hao Wang, Shanghai Jiao Tong University, China Tao Sun, Tianjin University, China Haitao Liu, Tianjin University, China Xiaofei Ma, Xi’an Institute of Space Radio Technology, China Huichan Zhao, Tsinghua University, China Yunxu Shi, Yantai University, China Ling Zhao, Yantai Huangbohai Sea New Area, China

Sponsors (1) Yantai Huangbohai Sea New Area

(2) State Key Laboratory of Fluid Power and Mechatronic Systems

(3) Institute of Aerospace System Engineering Shanghai (Space Structure and Mechanism Technology Laboratory of China Aerospace Science and Technology Group Co. Ltd.)

Organization

(4) Yantai University

vii

Preface

With the theme “Together for a Shared Future with Machine”, the 23rd IFToMM China International Conference on Mechanism and Machine Science and Engineering (IFToMM CCMMS 2022) was held during July 30–August 1, 2022 in Yantai, China. The conference aimed to encourage advancement in the fields of Drive and Transmission, Mechanical Design and Robot Design, Mechanism and Space Mechanism, and Robot and Machine Application. It offered a unique and constructive platform for scientists and engineers throughout the world to present and share their recent research and innovative ideas. Conference papers were submitted via an online submission system at the conference website: https://www.iftomm-ccmms2022.org/. The submission of full papers started on March 14, 2022, and ended on May 14, 2022. The notification of acceptance was sent to the authors by June 5, 2022, and the final paper submission due date was June 30, 2022. A total of 226 manuscripts were received, and finally, 139 papers were accepted for publication with an acceptance rate of about 61.5%. The quality of the papers was guaranteed by the peer-review process. The submitted manuscripts were sent for review to distinguished researchers in the field of mechanism and machine science. Each paper was assigned to three reviewers. Valuable comments were given regarding contribution and potential problems needed to be clarified. IFToMM CCMMS 2022 was organized by IFToMM (International Federation for the Promotion of Mechanism and Machine Science) MO China-Beijing and the Space Structure Branch of Chinese Mechanical Engineering Society, co-organized by Tsinghua University and Harbin Institute of Technology, and sponsored by Yantai Huangbohai Sea New Area, State Key Laboratory of Fluid Power and Mechatronic Systems, Institute of Aerospace System Engineering Shanghai (Space Structure and Mechanism Technology Laboratory of China Aerospace Science and Technology Group Co. Ltd.), and Yantai University. It was hosted by Tsingke+ Robot Research Institute. On this conference, 6 distinguished plenary speakers and 16 keynote speakers delivered their outstanding research work in various fields of robotics. Participants gave a total of 20 oral presentations and 119 poster presentations, enjoying this excellent opportunity to share their latest research findings.

ix

x

Preface

The Proceedings of IFToMM CCMMS 2022 covered 4 research topics and included a total of 139 papers. Here we would like to express our sincere appreciation to all the authors, participants, and distinguished plenary and keynote speakers. Special thanks are also extended to all members of the Organizing Committee, all reviewers for peer-review, all staffs of the conference affairs group, and all volunteers for their excellent work. Beijing, China

Xinjun Liu

Contents

Drive and Transmission Effect of Thermal and Mechanical Training in Twisted and Coiled Polymer Fiber (TCPF) Artificial Muscle for Improved Actuation Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bo Li, Yakun Zhang, Yu Zhang, Yanjie Wang, Alexey Formin, Guimin Chen, and Yong Zhang

3

Transmission Error Analysis of Planetary Gear Train Based on Probability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peng Liu, Jiale Peng, Zhihui Gao, and Yushu Bian

21

Precision Motion Control of Separate Meter-In and Separate Meter-Out Hydraulic Swing System with State Constraints . . . . . . . . . . . Yong Zhou, Zheng Chen, Ruqi Ding, Min Cheng, and Bin Yao

35

Dynamics Analysis of Simplified Axisymmetric Vectoring Exhaust Nozzle Mechanical System with Joint Clearance and Flexible Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaoyu Wang, Haofeng Wang, Chunyang Xu, Zhong Luo, and Qingkai Han

57

Research on Equivalent SEA for Vibration and Noise Prediction of the Gearbox System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lan Liu, Yinxiao Song, Yingjie Xi, Shanna Ma, Jingyi Gong, and Geng Liu

77

Hybrid Joint-space Control Strategies Analysis for One-redundant Cable Suspended Parallel Robots . . . . . . . . . . . . . . . . Zhiwei Qin, Zhen Liu, and Haibo Gao

97

Research on Coupling Analysis of Factors Influencing Backlash in Gear Train Based on Probability Method . . . . . . . . . . . . . . . . . . . . . . . . . Chunyang Shi, Yushu Bian, and Zhihui Gao

117

xi

xii

Contents

Transmission Principles of One Novel High-Order Phasing Gear and Its Influence of Design Parameters on Dynamic Properties . . . . . . . Jing Wei, Miaofei Cao, Aiqiang Zhang, Bing Pen, and Yujie Zhang

135

Kinematic Calibration of a 2-DOF Over-Constrained Parallel Manipulator Using Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuqing Chen, Yixiao Feng, and Tiemin Li

159

Optimal Study for Multi-field Coupling of the Disc Brake Based on Kriging Agent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meisheng Yang, Wen Jiang, Changwei Zhang, and Jiahan Bao

179

Analysis on Noise of High Pressure Direct Injection System . . . . . . . . . . . Wenqiang Liu, Yongjiang Xu, Xiaolong Deng, Junfeng Hu, Bing Gong, and Zhi Wang Parametric Optimization of a Permanent Magnet Driver for Implantable Intramedullary Nail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ShiKeat Lee, Zhenguo Nie, Handing Xu, Kai Hu, Hanwei Lin, and Xin-Jun Liu

201

209

Design and Analysis of Modified Non-orthogonal Helical Face-Gears with a High-Order Transmission Error . . . . . . . . . . . . . . . . . . Junhong Xu and Chao Jia

227

Integrated Kinematic Modeling of Reconfigurable Parallel Robot with 3T/3R Motion Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hao Li, Yimin Song, Xinming Huo, Wei Xian, and Yang Qi

257

Mechanical Design and Robot Design A Novel 2-DOF Translational Robot with Two Parallel Linkages Synchronous Telescopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhihao Li, Hongzhou Wang, Quanguo Lu, Xiaohui Zou, Xiaohuang Zhan, Yongdong Huang, and Jinfeng Liu Measuring Error Correction Method During Deflection Measurement Process of the Regular Hexagon Section Shaft Based on Lever-Type Measuring Mechanism . . . . . . . . . . . . . . . . . . . . . . . . Qingshun Kong and Zhonghua Yu

279

295

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ze Fu, Hao Xu, Yinghui Li, and Weizhong Guo

315

FBCCD: A Forward and Backward Cyclic Iterative Solver for the Inverse Kinematics of Continuum Robot . . . . . . . . . . . . . . . . . . . . . Haoran Wu, Jingjun Yu, Jie Pan, Guoxin Li, and Xu Pei

329

Contents

xiii

Optimization of Bearing Capacity Parameters of Fully Decoupled Two-Rotation Parallel Mechanism for Vehicle Durability Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sen Wang, Xueyan Han, Xingzhen Su, Haoran Li, Yanxia Shan, and Shihua Li

347

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuejian Ma, Yundou Xu, Yu Wang, Fan Yang, Yongsheng Zhao, Jiantao Yao, and Yulin Zhou

367

Force Modulation Mode Harmonic Atomic Force Microscopy for Enhanced Image Resolution of Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ke Feng, Jiarui Gao, Benliang Zhu, Hongchuan Zhang, and Xianmin Zhang Optimization Design of Buffering and Walking Foot for Planetary Legged Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chu Zhang, Liang Ding, Huaiguang Yang, Haibo Gao, Liyuan Ge, and Zongquan Deng Uncertainty Distribution Estimation Based on Unified Uncertainty Analysis Under Probabilistic, Evidence, Fuzzy and Interval Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiangyun Long, Mengchen Yu, Donglin Mao, and Chao Jiang

381

393

409

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Wang, Jiale Han, Yulei Hou, Haitao Liu, and Ke Xu

427

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit into Human Body Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weilin Lv, Wansui Nie, and Jianjun Zhang

439

Interval Principal Component Analysis of Non-probabilistic Convex Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuofeng Hou, Bingyu Ni, Wanyi Tian, Jinwu Li, and Chao Jiang

451

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator with Scissor Legs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianxun Fu, Rongfu Lin, Chengze Liu, Jiayi Zhou, and Feng Gao

477

An Optimized Cable Layout Method for Cable-Driven Continuum Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zheshuai Yang, Laihao Yang, Dong Yang, Yu Lan, Yu Sun, and Xuefeng Chen Design of a Miniature Three-Dimensional Force Sensor for Force Feedback in Minimally Invasive Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qinjian Zhang, Wu Zhang, Haiyuan Li, and Lutao Yan

491

501

xiv

Contents

Configuration Design of Lower-Mobility Parallel Driving Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jinzhu Zhang, Xinjun Liu, Hanqing Shi, Yangyang Huang, and Qingxue Huang

521

Dimensional Analysis of Transmission Mechanism of Novel Simulated “Soft” Mechanical Adaptive Grasper . . . . . . . . . . . . . . . . . . . . . Haibo Huang, Xinpeng Li, and Rugui Wang

533

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zezheng Qi, Shibing Hao, Qianqian Zhang, Ran Shi, and Yunjiang Lou

549

Design of a 3DOF XYZ Precision Positioning Platform Using Novel Z-Shaped Flexure Hinges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lejin Wan, Jiarong Long, Juncang Zhang, and Jinqiang Gan

569

Design and Experiment of Muck Removal Robot for Tunneling Boring Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lianhui Jia, Lijie Jiang, Yanming Sun, Yuanyang Zhao, Xingjian Zhuo, and Zhiguo Zhang

593

Morphology Design and Dimensional Synthesis of a Hexapod Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huayang Li, Chenkun Qi, Feng Gao, Xianbao Chen, and Meng Chen

609

A Structural Optimization Method for Assigning Resonance Harmonics of Atomic Force Microscope Cantilever . . . . . . . . . . . . . . . . . . Junwen Liang, Benliang Zhu, and Xianmin Zhang

625

Modal Analysis of a Symmetric Micro-displacement Amplification Mechanism: Structural Parameters . . . . . . . . . . . . . . . . . . . Buchuan Ma, Shenyuan Dai, Beiying Liu, and Lifang Qiu

637

Dynamic Modeling and Design Parameter Optimization of a 5-DoF Parallel Machining Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zijian Ma, Fugui Xie, and Xin-Jun Liu

647

A Kinetostatic and Dynamic Modeling Method of Piezo-Actuated Compliant Mechanisms Based on Dynamic Stiffness Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . Jianhao Lai, Xianmin Zhang, Dezhi Song, Lei Yuan, Hai Li, and Benliang Zhu

667

A Fabric-Based Flexible Actuator for Thumb Joints of Soft Anthropomorphic Hands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yi Zhao, Ningbin Zhang, Miao Feng, and Guoying Gu

689

Contents

The Design and Analysis of 3D Braiding Mechanism Based on a Small-Size Additive Manufacturing Technology Reinforced by Continuous Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiakai Wei, Wenpeng Han, Wuxiang Zhang, and Xilun Ding An End-Traction Lower Extremity Rehabilitation Robot: Structural Design, Motion Analysis, and Experimental Validation . . . . . Hui Bian, Fan Yang, Zhaoliang Sun, Jiachen Li, Jiebin Ding, and Shuai Li

xv

699

709

Design and Analysis of Disc-Spring-Based Cable-Driven Variable Stiffness Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hang Xiao, Zengrui Xu, Xilun Ding, and Shengnan Lyu

729

Dynamic Accuracy Analysis of Industrial Robots Considering Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Zhang, Bo Li, Yufei Li, and Wei Tian

747

Design and Experiment of Intelligent Cover Robot for Coal Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wenxiao Guo

759

Design, Implementation and Practice of Novel Rim Propulsion Unmanned Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Donglei Dong, Chuan Liu, Lichun Yang, Zaisi Yuan, Guangzhao Zhou, Jinjiang Li, Xianbo Xiang, and Shaolong Yang Experiment on Motion Compensation Platform with Three-Degree-of-Freedom Mounted on Ship . . . . . . . . . . . . . . . . . . . . Xingwen Hao, Zhipeng Zhou, Jinyi Li, Pinghu Ni, and Zongyu Chang Design of Coaxial Motor Pump with an Embedded Cooling Channel for Hydraulic Quadruped Robots . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaohao Ni, Junhui Zhang, Huaizhi Zong, Kun Zhang, Jun Shen, and Bing Xu

771

783

801

Design and Mode Analysis of Reconfigurable Spatial Closed-Chain Wheel-Leg Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Yang, Jianxu Wu, Mingze Weng, and Yanan Yao

815

A Fully Compliant Circular Beam Bistable Mechanism with Enhanced Pitch Stiffness and Uniformly Distributed Stress . . . . . . Liangliang Yan, Kuiyong Zhou, Shuaishuai Lu, and Pengbo Liu

827

Design and Analysis of a New Compliant Monolithic Motion Reverser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zekui Lyu and Qingsong Xu

841

xvi

Contents

Kinematic Modelling and Workspace Prediction of a Hybrid Kinematic Machining Unit Integrating Redundantly Actuated Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongwei Su, Hanliang Fang, Tengfei Tang, Fufu Yang, and Jun Zhang

855

Design of Quasi-zero Stiffness Vibration Isolator Based on Motion Singularity Characteristics of Parallel Mechanism . . . . . . . . . Dawei Xin, Wenjuan Lu, Daxing Zeng, and Shuai Wang

869

Fault-Tolerant Control of 3-PRS Parallel Robot Based on the Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huang Junjie, Zhang Qinlei, Wang Pengfei, and Zhang Bowen

881

Design and Analysis of Bionic Flapping-Wing Flying Robot Based on Two-Stage Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhenya He, Haolun Yuan, and Xianmin Zhang

895

Design and Analysis of Lower Limb Rehabilitation Robot Based on Virtual Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaohua Shi, Yajing Wang, Ruifa Liu, Pengcheng Jiang, and Dejun Mu

911

Mechanism and Space Mechanism Available Wrench Set and Workspace Analysis of a Cable-Driven Parallel Mechanism for On-Orbit Assembly . . . . . . . . . . . . . . . . . . . . . . . . . Jinshan Yu, Jianguo Tao, Guoxing Wang, Xiao Li, and Haowei Wang

935

On-Orbit Thermal Analysis of Ring Truss-Type Deployable Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiong Zhang, Xi Kang, Hailin Huang, and Bing Li

953

Elastodynamic Modeling and Optimization of Parallel Mechanism with Tube Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Caizhi Zhou and Weizhong Guo

967

Response Analysis and Parameter Optimization of a Fully Enclosed Wave Energy Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xinrui Lu and Yuan Chen

987

Configuration Synthesis of Ground Closed 6-DOF Parallel Posture Adjustment Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuesong Qiu, Dongsheng Li, Ruize Gao, Shuo Tian, Gongxinqi Bi, and Yulin Zhou

997

Determination of the Optimal Original Clearance of Joint Bearing . . . . 1017 Qingkai Han, Xiaoyu Wang, and Shuo Jiang A Novel 3-UPU Parallel Ankle Rehabilitation Mechanism . . . . . . . . . . . . 1027 Xuechan Chen, Chaoyang Ji, Yu Guo, Zheng Zhang, and Ziming Chen

Contents

xvii

An Attempt to Apply the Theory of Biquadratic Curve to Generating Plane-Symmetric Bricard Linkages Network . . . . . . . . . . 1045 Yuehao Zhang, Enjie Zhang, Shuaihu Wang, Jie Xiao, and Guangqiang Fang Structural Design, Analysis of Large-Area Flexible Solar Array for Space Solar Power Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 Li Qin, Yulei Fu, Chao Xie, Xiao Wei, Biao Yan, Zhengai Cheng, and Hanfeng Yin Dynamic Analysis of Under-Actuated Thrust Reverse Mechanism Considering Joint Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Xiaoyu Wang, Jingchao Zhao, Huitao Song, Zhong Luo, and Qingkai Han A Combinational Optimization Algorithm for Inverse Kinematics of an 8-DOF Redundant Manipulators . . . . . . . . . . . . . . . . . . 1089 Yu Guo, Hongchuan Zuo, Xuechan Chen, Jiachen Zhu, Chaoyang Ji, and Ziming Chen Kinetic Simulation of the Variable Stator Vanes Adjustment Mechanism and the Effect of Nonlinear Factors at the Joints . . . . . . . . . 1105 Shuo Jiang, Xiaoyu Wang, and Qingkai Han Nonlinear Analysis of Moderately Large-Stroke Flexure Beams in Compliant Mechanisms Using a Dynamic Beam Constraint Model (DBCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 Mingxiang Ling and Xianmin Zhang Tension Cable Configuration and Dynamic Modeling of Truss Modular Deployable Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 Dake Tian, Zhenwei Guo, Lu Jin, Haiming Gao, Junwei Zhang, Zuwei Shi, Anjun Hu, and Bingfeng Zhao Design and Performance Analysis of a Metamorphic Mechanism with Constrained Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Qiang Yang, Xin Zhao, Ruonan Wang, Hailong Huang, Shujun Li, and Benqi Sun Closed-Form Dynamic Modeling for the 3-RPS Parallel Mechanism: A Udwadia-Kalaba Equation Based Approach . . . . . . . . . . 1165 Duanling Li, Yongkang Wei, Kaijie Dong, Wei Zhang, Gang Xiao, Jin Huang, and Ye-Hwa Chen Dynamics Analysis of a Symmetric 3-R(SRS)RP Multi-loop Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 Chuanyang Li, Zhongbao Qin, Changhua Hu, Rongqiang Liu, Ruiwei Liu, and Xiaoke Song

xviii

Contents

Design and Experiment of Compliant Mechanisms-Based Morphing Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195 Yonghong Zhang, Bo Liu, Wenjie Ge, Xiaopeng Hu, and Hongzhi Zhang Bistable Characteristics and Driving Performance Analysis of Four-Link Compliant Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Shanzeng Liu, Zhaopeng Sun, Guohua Cao, and Aimin Li Kinematic Modeling of 3D Clearance in Revolute Joint and Its Application in Overconstrained Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 Jian Qi, Jun Zhang, Fufu Yang, and Yaqing Song Nonlinear Vibration Control of Large Space Antenna Based on Semi-active Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 Fengchen Fan, Zhihui Gao, Yushu Bian, Yechi Zhang, and Qiang Cong Cable-Driven Redundant Manipulator with Variable Stiffness Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263 Ruijie Tang, Qizhi Meng, Fugui Xie, Xin-Jun Liu, and Jinsong Wang Acceleration Planning of Rectilinear Motion of a Close Chain Five-Bow-Shaped-Bar Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 Tiandu Zhou, Mingzhi Wang, Yong Zhang, and Lianqing Yu Topology Optimization and Performance Prediction for Compliant Mechanism with Composites by Neural Network . . . . . . 1293 Xinxing Tong, Bo Yang, and Wenjie Ge Configuration Synthesis of Tetrahedral Mechanisms Containing P and S Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 Fanchen Kong, Jingfang Liu, and Huafeng Ding Origami/Kirigami-Inspired Reconfigurable 6R Linkages and Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333 Weiqi Liu and Yan Chen Traction Control of Planetary Rovers on Soft Terrain Based on the Actuation Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 Jie Li, Jun He, Yan Xing, and Feng Gao Design and Optimization of a Multi-mode Single-DOF Watt-I Six-Bar Mechanism with One Adjustable Parameter . . . . . . . . . . . . . . . . . 1373 Yating Zhang, Xueting Deng, Bin Zhou, and Ping Zhao Kinematic Calibration of a Hybrid Machine Tool with Constrained Least Square Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 Mengyu Li, Liping Wang, and Guang Yu

Contents

xix

Torque Feedforward Control of the Parallel Spindle Head Feed Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 Liping Wang, Xiangyu Kong, and Guang Yu Delaunay Triangulation Voxelization for Print Orientation-Based Topology Optimization in Additive Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419 Qingfeng Xu, Zhenguo Nie, Yaguan Li, Hongbin Lin, Handing Xu, Fugui Xie, and Xin-Jun Liu Design and Kinematic Analysis of a Single-Degree-of-Freedom Rigidly Foldable Winding Origami Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 1431 Sibo Chai, Jiayao Ma, Kaili Xi, and Yan Chen Design of a Novel Compliant Constant-Force Mechanism with High Lateral Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445 Ruiyu Bai, Zhaolong Wu, and Guimin Chen CGA-Based Displacement Analysis of the Three Seven-Link Baranov Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465 Ying Zhang, Yingqi Shao, Shimin Wei, and Qizheng Liao Mechanism Evolution and Analysis of a Novel Space Manipulator . . . . 1487 Sen Zhang, Xiaofei Ren, Yang Qi, Lin Han, Yu Wang, Dong Liang, and Yan Zhang Power Release and Cinch Branches of Vehicle Side Door Latch Based on Multi-mode Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1501 Zhiyang Qu, Lubin Hang, Chuanlei Zhong, Xiaobo Huang, and Chuanshuo Yin Finite Displacement Screw-Based Mobility Analysis of a New Four-DoF 1T3R Cable-Driven Parallel Mechanism . . . . . . . . . . . . . . . . . . 1529 Shuofei Yang, Ziyan Zhu, Hanyan Wang, and Yangmin Li Multi-objective Optimization of a New Redundantly Actuated Parallel Mechanism for Haptic Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1541 Congzhe Wang and Dewei Yang Non-smooth Model of a Frictionless Spherical Joint with Clearance for Spatial Multibody System Dynamics on SE(3) . . . . . 1559 Long Li, Shengnan Lyu, and Xilun Ding Robot and Machine Application Time-Varying Target Formation Control for Multi-agent Systems Based on Stress-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575 Yingxue Zhang, Meng Chen, Jinbao Chen, Chuanzhi Chen, and Hongzhi Yu

xx

Contents

Dynamic Integral Sliding Mode Trajectory Tracking of Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589 Haifeng Zhang, Yaojun Wang, and Qinchuan Li An Approach for Optimizing the Posture of a Friction Stir Welding Robotized Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611 Wei Yue, Haitao Liu, and Guangxi Li The Improvement of the End Position Accuracy of the Serial Manipulator with the Auxiliary Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 1633 Chen Sheng, Sun Zhangwei, Han Wei, Gao Xiang, and Huang Houran Trajectory Planning of Spray-Painting Robot Based on Thickness Uniformity of Paint Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647 Chaoqun Li, Feng Xu, Bin Zi, Yuan Li, and Jiahao Zhao Multi-modal Natural Man–Machine Interaction Technology and Its Application for Lunar Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667 Meng Chen, Chongfeng Zhang, Xiaofei Qin, Jia Ma, and Wenqi Zhang Unitized Modelling and Reuse Method of Assembly Process Oriented to Product Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687 Wei Zeng, Zhenan Jin, Yanpeng Cao, Guodong Yi, and Chuihui Li Hybrid Motion/Force Control of the Hydraulic Cooperative Manipulator in Constrained Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697 Min Cheng, Bolin Sun, Ruqi Ding, and Bing Xu Trajectory Scheduling for a Five-Axis Hybrid Robot in Flank Milling of the S-shaped Test Piece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713 Guangxi Li, Haitao Liu, and Wei Yue Influence of Different Incoming Flow Directions on the Flow Characteristics of Underwater Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . 1727 Yujun Cheng, Derong Duan, Shanbin Ren, Xia Liu, and Xuefeng Yang A Research on Calibration of Multi-coordinates for Cooperative Robotic Grinding System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1741 Cong Li, Wenzheng Ding, Xingwei Zhao, and Bo Tao Modeling and Analysis of Force-Closure Properties for the Flexible Space Manipulator During Docking and Capturing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755 Xiaolong Ma, Ning Li, Chongfeng Zhang, Meng Chen, Xuemei Ju, Song Wu, and Huaiwu Zou Visual Recognition of Surface Defects in a Robotic Lacquer Removing System for Radome Remanufacturing . . . . . . . . . . . . . . . . . . . . 1769 Hua Shao, Zhi Zhang, and Xiaohang Guo

Contents

xxi

Exact Inverse Solution for a 7R Manipulator Based on a Novel Screw-Based Sub-problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783 Tiantian Liu, Zhifeng Liu, Tianyue Gan, and Jingjing Xu Research on Robot Technology of Pipeline Foreign Body Detection Adaptive to Different Pipe Diameters . . . . . . . . . . . . . . . . . . . . . 1797 Zhi Qian Wang, Pei Lei, Yu Lin Dai, Wen Guo Zhang, and Rui Ke Yang A Construction Robot Based on Mobile Manipulator and Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1813 Junjie Ji, Songtao Wei, and Jing-Shan Zhao General Dynamics Modeling and Simulation Analysis of Multi-degree of Freedom Flexible Manipulators . . . . . . . . . . . . . . . . . . 1831 Xichen Jin, Zhizhong Tong, and Haibo Gao Active Disturbance Rejection-Based Torque Control for a Load Simulator of Robotic Hardware-in-the-Loop Simulation . . . . . . . . . . . . . 1855 Xianlei Shan, Huarong Liu, Haitao Liu, and Hao Yuan P-PI Controller Tuning for Multi-axis Motion Control of Hybrid Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1871 Haitao Liu, Bo Yu, Xianlei Shan, Hao Yuan, and Shaofei Meng A New Self-Calibration Method for the 5-DOF Hybrid Robot Based on Vision Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883 Haitao Liu, Yueting Jia, and Zhibiao Yan Fault-Tolerant Walking and Turning Gait Planning for Quadruped Robots with One Locked Leg on Rough Terrains . . . . . . 1899 Zhijun Chen, Feng Gao, Qingxing Xi, and Yue Zhao Design and Implementation Based on Backstepping for Tracking Control of Nonholonomic Mobile Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1919 Quan Liu, Zhao Gong, Zhenguo Nie, and Xin-Jun Liu Adaptive Reactionless Null Space Planning and Control of Free-Floating Space Manipulator After Capturing Unknown Rotating Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1931 Chenyang Gao, Tao Lin, and Xiaoyan Yu Remote 3D Visual Monitoring System of Kinematically Redundant (6 + 3)-dof Hybrid Parallel Mechanism Based on Digital Twin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1949 Jianqin Zhang, Xianmin Zhang, Fenhua Zhang, Ruida Zheng, Tianyu Xie, and Hai Li

xxii

Contents

Motion/Force Transmissibility and Static Stiffness Based Workspace Identification for a Gantry Hybrid Machining Robot . . . . . . 1961 Jie Wen, Fugui Xie, and Xin-Jun Liu Kinematic and Nonparametric Calibration of a Parallel Machining Robot Based on an Artificial Neural Network . . . . . . . . . . . . . 1977 Lefeng Gu, Fugui Xie, Xin-Jun Liu, Jiakai Chen, and Xuan Luo Application of Multi-Degree-of-Freedom Motion Compensation Device in Marine Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1993 Xingwen Hao, Jinyi Li, Pinghu Ni, Zhipeng Zhou, and Zongyu Chang A Fault Diagnosis Method of Rolling Bearing Based on GRU Convolution Denoising Auto-Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009 Duanling Li, Xingyu Wei, Wei Zhang, Jin Huang, Gang Xiao, and Keqian Wan Foothold Planning and Body Posture Adjustment Strategy of Hexapod Robot in Complex Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023 Kailun Liu, Feng Gao, Zhijun Chen, and Qingxing Xi Semi-Closed-Loop Constant Force Control of a Multi-Modes Rigid-Flexible Underactuated Gripper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037 Weilin Chen, Zihong Feng, Qinghua Lu, Huiling Wei, Xiaolu Qin, Jiacheng Mo, Yu Tang, and Sheng Wei Design of an FF HSE Protocol Fault Analyzer Based on Deep Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2061 Shuwei Ding, Yuanbei Gu, and Weihua Bao Robot Automatic Polishing Technology of Curved Parts Based on Adaptive Impedance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2073 Liwei Yang, Yiming Zhang, Yongnian Han, Banghai Zhang, and Zhanxi Wang Automatic Calibration Method of Leg Joint Angles of Quadruped Robot Based on Machine Vision . . . . . . . . . . . . . . . . . . . . . . 2087 Yaguan Li, Zhenguo Nie, Handing Xu, Shi K. Lee, Qizhi Meng, Fugui Xie, and Xin-Jun Liu Towards Large-Space Manipulation Skills Learning with Mobile Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095 Yuqiang Wu, Zhiwei Liao, Chenwei Gong, and Fei Zhao Path Planning Method of Unmanned Surface Vessel Based on Strategy Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111 Caipei Yang, Yingqi Zhao, Jie Liu, and Kaibo Zhou

Contents

xxiii

On Adaptive LOS Guidance Law Based Path Following of an AUV Using Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . 2123 Guangzhao Zhou, Chuan Liu, and Xianbo Xiang SMP-KK RF Coaxial Connector Automated Assembly Method . . . . . . . 2137 Qinghao Liu, Henan Hu, Shilong Wang, Zichao Chen, Changrui Wang, and Wei Tian Contact Detection by Probabilistic Model Fusion for Hydraulic Quadruped Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2157 Kun Zhang, Junhui Zhang, Huaizhi Zong, Ximeng Wang, Jun Shen, Zhenyu Lu, and Bing Xu Type Synthesis and Analysis of Rigid-Flexible Hybrid Stable Tracking Robot on USV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2179 Li Erwei, Lu Ruijie, and Zhao Tieshi Informative Path Planning for Mobile Robot Adaptive Sampling Using DDQN Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207 Zefeng Bao, Yiqiang Wang, Zhiliang Wu, and Yunfeng Li Flexible Sensors for Hand Rehabilitation Training System . . . . . . . . . . . . 2217 Junwei Li, Junhui Liu, Men Chen, Kaibo Yan, Weibiao Gu, Lei Bao, and Qinghua Xia The Recognition of Ankle Movement Patterns Using LDA . . . . . . . . . . . . 2233 Nianfeng Wang, Xinhao Zhang, Guifeng Lin, and Xianmin Zhang Underwater Motion Analysis of Thin Discs with Non-uniform Mass Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253 Longbin Zi, Leyi Zheng, Yongji Fu, and Yanjie Wang Path Planning and Information Protection of Mobile Robots Based on Deceptive Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . 2271 Qingfeng Xu, Yingnan Shi, Junchao Wang, Tim Miller, Hangding Xu, Tianmu Wang, Hongbin Lin, Xin-Jun Liu, and Zhenguo Nie

Drive and Transmission

Effect of Thermal and Mechanical Training in Twisted and Coiled Polymer Fiber (TCPF) Artificial Muscle for Improved Actuation Consistency Bo Li, Yakun Zhang, Yu Zhang, Yanjie Wang, Alexey Formin, Guimin Chen, and Yong Zhang Abstract Twisted and coiled polymer fiber (TCPF) is a new type of artificial muscles featuring large stroke and high loading capacity that can be benefited in soft robotic actuation. However, due to the multi-step fabrication process with various parameter, the existed TCPF suffers from an inconsistent actuation performance, which hinders its further application. The current paper studies the mechanics of TCPF actuation, revealing the significance of coiling and annealing procedure. Effect of twisting weight, annealing temperature and pre-stretch as mechanical and thermal training, are experimentally investigated, and the contraction forces are characterized. With these mechanical and thermal training, consistent and hysteresis-free contraction force is attain. The findings offer a guidance for the high-performance artificial muscle in compliant structures and tensegrity robot. Keywords Artificial muscles · Soft actuator · Soft robot

B. Li · Y. Zhang · Y. Zhang · G. Chen Shaanxi Key Lab of Intelligent Robots, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China e-mail: [email protected] B. Li Key Laboratory of Shaanxi Province for Craniofacial Precision Medicine Research, College of Stomatology, Xi’an Jiaotong University, Xi’an 710049, China Y. Wang Jiangsu Provincial Key Laboratory of Special Robot Technology, Hohai University, Changzhou Campus, Changzhou 213022, Jiangsu, China A. Formin Mechanisms Theory and Machines Structure Laboratory, Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN), 101000 Moscow, Russia Y. Zhang (B) Department of Thoracic Surgery, The First Affiliated Hospital of Xi’an Jiaotong University, Xi’an 710061, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_1

3

4

B. Li et al.

1 Introduction Twisted and coiled polymer fiber (TCPF) is a new type of artificial muscle. It is in the shape of elastic spring as shown in Fig. 1, and capable of high-loading capacity, ultra-light weight and fast response time under a low voltage activation ( 4 N, nylon fibers suddenly break during next process; 3. When the load is between 1.5 N and 4 N, the TCPF artificial muscle with the standard spiral structure is formed, as shown in Fig. 9. According to Eq. (1), without considering the relevant material constants, T coil should only be related to F twist , as in Table 1. 1 mm

Fig. 7 The materials for fabricating TCPF Nylon fiber

Silver fiber

10

B. Li et al.

1 mm

Fig. 8 The fiber in each step of fabrication. a Original fiber, b after twisting, c after coiling, d after annealing

(a)

(b)

(c)

(d)

Fig. 9 Successfully fabricated TCPF

Table 1 T coil versus the preparation load F twist /N

1.5

2

2.5

3

3.5

4

T coil

193

222

245

260

283

300

Figure 10a shows the effect of different preparation loads on length L, and the results suggest that the longer length is attainable when the preparation load is larger. Figure 10b presents the effect of different preparation loads on the external diameter D, from which it can be seen that the larger the preparation load yields a smaller outer diameter D. Figure 10c is the effect of different preparation loads on the elastic modulus, and the results show that the preparation load can improve a greater elastic modulus. From the above experiments, it can be seen that the preparation load will affect the following parameters, and in turn affect the output capability of artificial muscles: 1. Effect of the number of coil. The larger number in the preparation will induce more laps in coil, thus produces a greater torque in the TCPF artificial muscle output. 2. Effect of length and outer diameter. A larger preparation load yields a longer length and a smaller outer diameter, resulting in more coils. So that the contraction force of the artificial muscle output is also greater. 3. Effect of elastic modulus. This parameter affects the initial helix angle, giving the artificial muscles a different driving starting position and affecting the contraction strain.

Effect of Thermal and Mechanical Training in Twisted and Coiled … Fig. 10 Effect of preparation mechanical weight on the TCPF properties. a The length, b the diameter, c the elastic modulus

11

12

B. Li et al.

3.2 The Effect of Annealing After the coiling, another load, F anneal , is applied in annealing. Through thermoplastic deformation, the TCFF artificial muscle is fixed at the spring-like shape, otherwise it will uncoil immediately. The coiled TCFF are in close contact without shrinkage space, and in order to achieve thermal contraction in actuation, annealing process is necessary. Figure 11a shows the TCFF artificial muscles before heat treatment, there is no obvious gap between the coils, Fig. 11b shows the TCFF artificial muscles after heat treatment, the coil gap is significantly enlarged. F anneal in the annealing is to ensure a sufficient coil gap. In the experiments a 700 g weight was initially selected to maximize the stroke in TCPF. Then the heat treatment temperature was studied. The heat treatment time were set to 30 min, and the heat treatment time can be determined after other parameters are determined. A temperature range 50–180 °C was suggested and we fabricated series of TCPFs by 10 °C increment. The results are shown in Fig. 12. When the annealing temperature was less than or equal to 90 °C, the TCPF artificial muscle undergoes violent untwisting and random deformation after it was dismounted from the preparation bench (Fig. 12a). It means that the temperature is too low to realize the plastic deformation. When the annealing temperature is between 100 and 120 °C, the TCPF artificial muscle shrinks into a wave shape, and there was no obvious movement gap between the coils (Fig. 12b). When the temperature was between 130 and 150 °C, the TCPF maintained a standard spring shape, and there was a clear movement gap between the coils (Fig. 12c). When the temperature exceeded 160 °C, the color of the fiber material changes from transparent to turbidity, and TCPF was brittle in Fig. 12d.

(a)

(b)

1 mm

Fig. 11 Effect of annealing on TCPF. a Before annealing, there is no gap in coils, b after annealing, the gap is enlarged, providing actuation contraction stroke

Effect of Thermal and Mechanical Training in Twisted and Coiled …

13

(a)

(b)

(c)

(d)

Fig. 12 Effect of annealing on TCPF with temperature at a ≤ 90 °C, b 100–120 °C, c 130–150 °C, d ≥ 160 °C

We next characterize the actuation performance of the TCPF after annealing. TCPF samples were mounted to a tensile machine, and its contraction force was recorded with the voltage was increase from 1 to 10 V. Figure 13 plots the contraction force under different annealing time and duration. The annealing time is 30 min. This time not only makes the artificial muscle thermally trained, but also enables the maximum contraction force value. The annealing treatment of TCPF artificial muscles is affected by the load, treatment temperature and time duration. The best conditions, according to the experimental results, are: 54.6 MPa, 140 °C and 30 min. This conclusion will be used in subsequent experiments.

14

B. Li et al.

Fig. 13 Contraction force of TCPF under different annealing, a temperature and b time duration

4 Actuation Consistency Characterization 4.1 Cyclic Loading–Unloading Experiment To characterize the performance, a cyclic loading–unloading experiment was conducted. Figure 14 plots the voltage versus contraction force. A series voltage with 1 V increment was applied, and the contraction force was recorded. In Fig. 14, the TCPF generated a stabilized and consistent output only after the first working cycle. This can be understood by considering the mechanism of actuation. When the TCPF is electrically heated, it uncoils to release the torsion. The first loading cycle works as a training period to orient the fiber for a better performance. Similar behavior is also revealed in hydrogel artificial muscle [27]. A batch of fabricated TCPF artificial muscles with the same preparation parameters should have similar performance. This determines the consistency of TCPF artificial muscles, which is the basis for applications of TCPF artificial muscles. We

Effect of Thermal and Mechanical Training in Twisted and Coiled …

15

Fig. 14 Cyclic voltage versus contraction force

measured the performance of 5 TCPF samples after each training, and the results are presented in Fig. 15. It can be observed that the consistency of contraction force of the 5 TCPF artificial muscle samples is identical. Among them, the difference between sample 3 and sample 5 is the largest at the voltage of 10 V, and the contraction force is 1.11 N and 1.03 N, respectively. The relative error is 7%, which is acceptable. Repeatability characterizes the same measurement target. Given multiple times of the same conditional input, and we observe the closeness of multiple output results. In this experiment, the results of contraction force are used to calculate the repeatability 1.2

Contraction force (N)

Fig. 15 Consistency of 5 TCPF samples

1.0

#1 #2 #3 #4 #5

0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

5

6

Voltage (V)

7

8

9

10 11

16

B. Li et al.

Table 2 Contract force for repeatability Repeat number

Force/N

Repeat number

Force/N

Repeat number

Force/N

1

1.49

11

1.49

21

1.40

2

1.39

12

1.49

22

1.46

3

1.33

13

1.53

23

1.48

4

1.54

14

1.49

24

1.49

5

1.39

15

1.48

25

1.48

6

1.45

16

1.49

26

1.51

7

1.41

17

1.49

27

1.39

8

1.48

18

1.31

28

1.39

9

1.47

19

1.40

29

1.53

10

1.40

20

1.51

30

1.33

of the TCPF artificial muscle. Using 300 mm long original fibers, TCPF artificial muscle sample was prepared under a 3 N load and 15 s duration when a 10 V voltage is applied to ensure that the temperature reaches thermal equilibrium, and it was held for 30 s when voltage was removed to ensure that the temperature was close to room temperature. The 30 sampling results are collected in Table 2. After calculation, the standard deviation of the 30 repeated results is 0.063, and the average value is 1.43 N. Therefore, the repeat relative error (standard deviation/average value) of the TCPF artificial muscle is 4.5%. This performance validates that mechanical training and thermal annealing can improve the TCPF actuation.

4.2 Effect of Pre-stretch in Actuation We next study the effect of pre-stretch on TCPF. First, no pre-stretch was applied, and the contraction force in 10 cycles were recorded in Fig. 16. In the first cycle, the loading–unloading curves do not coincide and the contraction force is greater in loading. There is an obvious residual stress of 0.51 N. The residual stress changes to 0.65 in the following cycle, but the loading and unloading curve are more identical, eliminating the hysteresis. We next consider the residual stress by adjusting the pre-stretch in Fig. 17. As the pre-stretching force increases, the contraction force of the TCPF artificial muscle increases; and the residual stress of the TCPF artificial muscle increases too but at a slow rate. Based on Fig. 17, we investigate the contraction force on the artificial muscle with a pre-stretch force of 0.8 N. The contraction force results of the first cycle are shown in Fig. 18a. Compared with the result of the first cycle under the 0 N pre-stretch in Fig. 16a, the curves are initially coincided, that is, the corresponding contraction force difference between the beginning and the end of the first cycle is 0 N.

Effect of Thermal and Mechanical Training in Twisted and Coiled …

17

Fig. 16 Loading–unloading performance of TCPF artificial without pre-stretch. a 1st, b 2nd, c 3rd, d 4th cycles 2.0

Fig. 17 Effect of pre-stretch force on the contraction force and residual stress Contraction force (N)

1.8 1.6 1.4 1.2 1.0 0.8

1.5N 2 2.5N 3N 3.5N 4N

0.6 0.4 0.2 0.0

0

1

2

3

4

5

6

7

8

9

10 11 12

Voltage (V)

Figure 18 is the result of the contraction force and it is found that the data basically coincides. This suggest that with a pre-stretch, a consistent curve in loading–unloading cycles is available.

18 1.6 1.4

Contraction force (N)

Fig. 18 Loading–unloading performance of TCPF artificial with a pre-stretch of 0.8 N

B. Li et al.

1.2 1.0 0.8 0.6 0.4 Loading Unloading

0.2 0.0

0

1

2

3

4

5

6

7

8

9

10

11

Voltage (V)

5 Conclusions The current study investigates the effect of mechanical and thermal training on TCPF artificial muscle. Guided by a mechanical model, different twisting weight, annealing temperature and pre-stretch are investigated experimentally to reveal the ideal combination for consistent performance. A repeat relative error below 4.5% is attained and a hysteresis is eliminated by these mechanical and thermal training. High-performance TCPF can be obtained following the current fabrication setting. Acknowledgements Supported by the National Key R&D Program of China (2019YFB1311600), Natural Science Foundation of China (Grant No. 52075411), Shaanxi Key Research and Development Program (2020ZDLGY06-11), Key Research and Development Program Projects in Shaanxi Province China (2019SF-014), Opening Research Fund from Key Laboratory of Shaanxi Province for Craniofacial Precision Medicine Research, College of Stomatology, Xi’an Jiaotong University (2021LHM-KFKT004).

References 1. Tawfick S, Tang Y (2019) Stronger artificial muscles, with a twist. Science 365(6449):125–126 2. Swartz AM, Ruiz DRH, Feigenbaum HP, Shafer MW, Browder C (2018) Experimental characterization and model predictions for twisted polymer actuators in free torsion. Smart Mater Struct 27(11):114002 3. Haines CS, Lima MD, Li N, Spinks GM, Foroughi J, Madden JD, Baughman RH (2014) Artificial muscles from fishing line and sewing thread. Science 343(6173):868–872 4. Park J, Yoo JW, Seo HW, Lee Y, Suhr J, Moon H, Nam JD (2017) Electrically controllable twisted-coiled artificial muscle actuators using surface-modified polyester fibers. Smart Mater Struct 26(3):035048 5. Yang SY, Cho KH, Kim Y, Song MG, Jung HS, Yoo JW, Choi HR (2017) High performance twisted and coiled soft actuator with spandex fiber for artificial muscles. Smart Mater Struct 26(10):105025

Effect of Thermal and Mechanical Training in Twisted and Coiled …

19

6. Haines CS, Li N, Spinks GM, Aliev AE, Di J, Baughman RH (2016) New twist on artificial muscles. Proc Natl Acad Sci 113(42):11709–11716 7. Ding H, Yang X, Zheng N, Li M, Lai Y, Wu H (2018) Tri-co robot: a Chinese robotic research initiative for enhanced robot interaction capabilities. Natl Sci Rev 5(6):799–801 8. Hiraoka M, Nakamura K, Arase H, Asai K, Kaneko Y, John SW, Omote A (2016) Powerefficient low-temperature woven coiled fibre actuator for wearable applications. Sci Rep 6(1):1– 9 9. Mirvakili SM et al (2014) Simple and strong: twisted silver painted nylon artificial muscle actuated by Joule heating. In: Electroactive polymer actuators and devices (EAPAD), vol 9056. International Society for Optics and Photonics 10. Yang Y, Tse YA, Zhang Y, Kan Z, Wang MY (2019) Low-cost inchworm-inspired soft robot driven by supercoiled polymer artificial muscle. In: 2019 2nd IEEE international conference on soft robotics (RoboSoft). IEEE, pp 161–166 11. Abbas A, Zhao J (2017) Twisted and coiled sensor for shape estimation of soft robots. In: 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS). IEEE, pp 482–487 12. Li T, Wang Y, Liu K, Liu H, Zhang J, Sheng X, Guo D (2018) Thermal actuation performance modification of coiled artificial muscle by controlling annealing stress. J Polym Sci Part B Polym Phys 56(5):383–390 13. Di J, Fang S, Moura FA, Galvão DS, Bykova J, Aliev A, Baughman RH (2016) Strong, twiststable carbon nanotube yarns and muscles by tension annealing at extreme temperatures. Adv Mater 28(31):6598–6605 14. Almubarak Y, Tadesse Y (2017) Twisted and coiled polymer (TCP) muscles embedded in silicone elastomer for use in soft robot. Int J Intell Robot Appl 1(3):352–368 15. Tang X, Li K, Liu Y, Zhao J (2018) Coiled conductive polymer fiber used in soft manipulator as sensor. IEEE Sens J 18(15):6123–6129 16. Abbas A (2018) Modeling of twisted and coiled artificial muscle for actuation and self-sensing. Doctoral dissertation, Colorado State University 17. Zhao P, Xu B, Zhang Y, Li B, Chen H (2020) Study on the twisted and coiled polymer actuator with strain self-sensing ability. ACS Appl Mater Interfaces 12(13):15716–15725 18. Yu JJ, Dong X, Pei X, Zong GH, Kong X, Qiu Q (2011) Mobility and singularity analysis of a class of 2-DOF rotational parallel mechanisms using a visual graphic approach. In: International design engineering technical conferences and computers and information in engineering conference, vol 54839, pp 1027–1036 19. Ma N, Dong X, Axinte D (2020) Modeling and experimental validation of a compliant underactuated parallel kinematic manipulator. IEEE/ASME Trans Mechatron 25(3):1409–1421 20. Russo M, Raimondi L, Dong X, Axinte D, Kell J (2021) Task-oriented optimal dimensional synthesis of robotic manipulators with limited mobility. Robot Comput Integr Manuf 69:102096 21. Russo M, Dong X (2020) A calibration procedure for reconfigurable Gough-Stewart manipulators. Mech Mach Theory 152:103920 22. Ma N, Dong X, Palmer D, Arreguin JC, Liao Z, Wang M, Axinte D (2019) Parametric vibration analysis and validation for a novel portable hexapod machine tool attached to surfaces with unequal stiffness. J Manuf Process 47:192–201 23. Ba W, Dong X, Mohammad A, Wang M, Axinte D, Norton A (2021) Design and validation of a novel fuzzy-logic-based static feedback controller for tendon-driven continuum robots. IEEE/ASME Trans Mechatron 26(6):3010–3021 24. Chen S, Tan MWM, Gong X, Lee PS (2022) Low-voltage soft actuators for interactive humanmachine interfaces. Adv Intell Syst 4(2):2100075 25. Wang J, Gao D, Lee PS (2021) Recent progress in artificial muscles for interactive soft robotics. Adv Mater 33(19):2003088 26. Zhang Y, Zhang Y, Li B, Chen G (2020) Progress of twisted and coiled polymer fiber artificial muscles and its application in soft robots. Sci Sinica Technol 51(2):119–136 27. Lin S, Liu J, Liu X, Zhao X (2019) Muscle-like fatigue-resistant hydrogels by mechanical training. Proc Natl Acad Sci 116(21):10244–10249

20

B. Li et al.

28. Liu C, Yu X, Li Y, Zhao X, Chen Q, Han Y (2021) Flow-induced crystalline precursors in entangled poly (vinyl alcohol) aqueous solutions. Polymer 229:123960 29. Ma Y, Hua M, Wu S, Du Y, Pei X, Zhu X, He X (2020) Bioinspired high-power-density strong contractile hydrogel by programmable elastic recoil. Sci Adv 6(47):eabd2520 30. Lv Z, Shu T, Ren J, Cao L, Pei Y, Shao Z, Ling S (2021) Mechanism of mechanical training-induced self-reinforced viscoelastic behavior of highly hydrated silk materials. Biomacromolecules 22(5):2189–2196 31. Lu C, Wang C, Zhang D, Wang J, Yong Q, Chu F (2021) Ultra-strong hydroxypropyl cellulose/polyvinyl alcohol composite hydrogel by combination of triple-network and mechanical training. Int J Biol Macromol 184:200–208 32. Li B, Jiang L, Ma W, Zhang Y, Sun W, Chen G (2022) A switchable dual-mode actuator enabled by bistable structure. Adv Intell Syst 2100188 33. Ma W, Li B, Jiang L, Sun Y, Wu Y, Zhao P, Chen G (2022) Bioinspired, electroactive colorable and additive manufactured photonic artificial muscle. Soft Matter 18:1617–1627 34. Li B, Zhao X, Wang H, Jiang L, Chen G (2021) Asymmetric soft Ω-robot utilizing minimized energy structure of unequally pre-stretched dielectric elastomer. In: 2021 27th international conference on mechatronics and machine vision in practice (M2VIP). IEEE, pp 241–246 35. Sun W, Li B, Zhang F, Fang C, Lu Y, Gao X, Wang ZL (2021) TENG-Bot: triboelectric nanogenerator powered soft robot made of uni-directional dielectric elastomer. Nano Energy 85:106012 36. Tang C, Li B, Liu L, Ge SS, Shui L, Chen H (2020) Nonlinear out-of-plane resonation of a circular dielectric elastomer. Smart Mater Struct 29(4):045003 37. Pradere C, Sauder C (2008) Transverse and longitudinal coefficient of thermal expansion of carbon fibers at high temperatures (300–2500 K). Carbon 46(14):1874–1884

Transmission Error Analysis of Planetary Gear Train Based on Probability Calculation Peng Liu, Jiale Peng, Zhihui Gao, and Yushu Bian

Abstract In this paper, the statistical calculation formulas of gear transmission error are put forward to predict total transmission error of a gear train. Gear transmission errors come from a number of complex factors and those factors are hardly considered together on gear models. In order to analyze transmission error considering multiple factors, a three-layer coupling relationship model of each factor influencing the transmission error is investigated. Then, the influence of the third layer factor on the second layer factor and the influence of the single factor in the second layer on the gear transmission error are analyzed. Furthermore, the synthetic expression of transmission error for a gear train and the statistical calculation formula is obtained. Keywords Gear train · Transmission · Probability calculation

1 Introduction The difference between the actual and theoretical Angle of the driven wheel when the driving wheel of a gear mechanism rotates in one direction, which is called transmission error. The transmission error is one of the most important indices used to evaluate the accuracy of the meshing movement of gears. Usually, the transmission error of a gear system results from the inherent position error, the device error, the force-induced deformation error, the thermal-induced deformation error, and the P. Liu · J. Peng · Z. Gao (B) · Y. Bian School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China e-mail: [email protected] P. Liu e-mail: [email protected] J. Peng e-mail: [email protected] Y. Bian e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_2

21

22

P. Liu et al.

friction-induced error. Therefore, the research about transmission error of a gear system usually focuses on those errors. The influence of above factors on transmission error has been studied in theoretical and experimental ways. Yu [1] established the calculation formulas of transmission error caused by a couple of eccentricity errors of gears. Cai [2] researched the influences of different loads on the transmission error, it is concluded that in a reasonable loading range, the decrease of load will reduce the fluctuation range of transmission error. Lin [3] proposed the calculation model of transmission error of the gear transmission system with the coupling of multi factors such as inherent errors and device errors. Lu [4] put forward a method to monitor gear condition and gear quality control. Wink [5] studied on tooth contact deviations caused by circumferential force. Benatar [6] provided a comprehensive set of experimental data on motion transmission error behavior of modified helical gear pairs. Benaïcha [7] proposed a flexible multibody approach through ANSYS Mechanical solver to deal with gear mesh contact. Hu [8] presented a theoretical investigation on the overall loaded motion transmission error of planetary gear sets. Lee [9] used a tooth profile modification for loaded gears to avoid a tooth impact. In conclusion, many researchers carried out lots of research works about the transmission errors of a gear system [10–16]. However, there is almost no calculation method which considers the coupling multi factors, such as inherent position error, device error, force-induced deformation error, thermal-induced deformation error, friction-induced error. Therefore, this work proposes a synthetic expression of transmission error for a gear train. In fact, the manufacturing error of a batch of parts is usually distributed within a given range according to a certain statistical law. In view of this, the probability method is used here to calculate the distribution range of transmission error of gear mechanism. And the statistical calculation formula based on probability distribution is also established, which can estimate the maximum and minimum transmission error of a gear train.

2 Coupling Relationship Model of Factors Influencing Transmission Error For the convenience of analysis, a three-layer coupling relationship model of the factors influencing the transmission error is put forward, as shown in Fig. 1. The first layer is the final transmission error of a gear system, which can be measured. The second layer is the factors which can directly influence the transmission error. All of them belong to the geometric deformation, including the inherent position error, the device error, and the force-induced deformation error. The third layer is the factors which can indirectly influence the transmission error through the factors in the second layer, including the temperature, the friction and the load. When the temperature changes, the dimensions of a gear will expand and

Transmission Error Analysis of Planetary Gear Train Based …

23

shrink accordingly, thereby resulting in the change of the inherent position error, the device error and the force-induced deformation error in the second layer, and finally influencing the transmission error. The friction is mainly determined by the positive pressure and the friction coefficient. The influence of the positive pressure can be incorporated into the analysis on the load. When the friction coefficient changes, the pressure angle changes accordingly, thereby resulting in the change of the conversion coefficient of the projection of each geometric deformation to the reference circle, and finally influencing the transmission error. When the gear system is subjected to loads, complex force–deformation coupling problems occur. This dynamic relationship needs to be described using multi-body dynamics, nonlinear contact mechanics and impact dynamics, thereby creating a force–deformation coupling model.

3 Probability Calculation of Factors Affecting Transmission Error 3.1 Third Layer Factor Analysis on Gear Transmission Error In order to obtain the influence of a single factor on gear transmission error in the second layer, the influence of the third layer factors on the second layer factors is analyzed first. The load affects shaft torsional deformation error, shaft bending deformation error, bearing clearance error and gear deformation error in the second layer, the temperature and the friction affect all factors in the second layer, as shown in Fig. 1. The transmission error can be measured on the meshing line or on the reference circle, and the latter is chosen. When subjected to the torque, radial force and circumferential force, the gear system will deform, including the shaft torsional deformation, the shaft bending deformation, the bearing clearance change and the tooth deformation. The gear will transfer torque T in the transmission process, and the gear shaft will undergo torsional deformation, which will cause the line value error (ΔST ) of the gear on the reference circle. As long as the torque T is unchanged, the line value error ΔST is a constant value affecting the transmission error, which is a large period error. The gear will be subjected to radial and circumferential forces in the transmission process, and the gear shaft will bend and deform along the meshing line of the gear, so that the axis will deviate from the theoretical axis. And the deviation is ΔS F , which is the shaft bending error, as shown in Fig. 2. When the system works, due to the rotation of the gear and the influence of forces, the bearing will produce clearance, which will produce backlash to the gear pair. Considering the limiting position, the bearing inner ring is biased to the position of the maximum internal clearance and moves Δr 0 /2 along the meshing line. They are decomposed into radial deviation Δrr and tangential deviation Δr t . According to the relationship between radial backlash and circumferential backlash, the backlash generated by radial deviation and tangential deviation can be

24

Fig. 1 Coupling relationship of factors influencing gear transmission error Fig. 2 Shaft bending deformation deviation

P. Liu et al.

Transmission Error Analysis of Planetary Gear Train Based …

25

written as follows: Δrr =

Δr 0 sin an 2

(1)

Δr t =

Δr 0 cos an 2

(2)

Radial deviation and tangential deviation can be converted into circumferential backlash, which is the bearing clearance error. In the meshing process, the gear tooth deformation with value Δ f 0 will be generated along the meshing line direction under the influence of the circumferential force. The gear deformation error can be obtained by converting the tooth deformation into circumferential backlash. After the gear is manufactured, there is an inherent error, which is represented by tangential composite deviation and tooth-to-tooth tangential composite deviation at standard temperature (20 °C). The gear may deform under the influence of temperature, including tangential composite deviation at the standard temperature (20 °C), tooth-to-tooth tangential composite deviation and deformation of the gear at the temperature. Therefore, the thermal deformation error of gear and the dimension error of gear manufacturing can be comprehensively expressed by measuring the inherent error of gear at a specific temperature, so that the thermal deformation error of gear can be calculated by directly applying the inherent error analysis formula of gear. As the gear is subjected to the action of friction, the direction of force on the gear changes. Therefore, the pressure Angle changes from the standard pressure Angle αn to α. If the friction coefficient is f , then the pressure Angle is: α = αn − arctan f

(3)

To sum up: when the gear train is affected by the load and the temperature, the clearance of the gear train will change. Therefore, the above mainly discusses the influence of the deformation produced by the gear train under the influence of load and temperature on the second layer factors.

3.2 Second Layer Factor Analysis Model of Gear Transmission Error The second layer of factors directly affect the transmission error, including total tangential composite deviation (Fi' ) of each gear, total tooth-to-tooth tangential composite deviation ( f i' ), the eccentricity (e), shaft torsional deformation error (ΔST ), shaft bending deformation error (ΔS F ), the variation of bearing clearance (Δr ) and gear tooth deformation (Δ f ), as shown in Fig. 2.

26

P. Liu et al.

Fig. 3 The inherent position error curve of gear

To calculate the transmission error by probability method, the following assumptions are made first: (1) Gear error and other errors are continuous random variables. (2) The distributions of errors are independent of each other. (1) Inherent position error The total tangential composite deviation and the tooth-to-tooth tangential composite deviation are a group of dynamic accuracy indices. The tooth-to-tooth tangential composite deviation measured at a certain temperature reflects the inherent position error and the gear tooth deformation error caused by temperature. The total tangential composite deviation reflects the inherent position error of large period gear caused by geometric eccentricity and motion eccentricity during gear machining. The tooth-totooth tangential composite deviation is the result of synthetical action of tooth profile error and base pitch error. The inherent position error curve of gear changes with the coordinate of gear angle θ, as shown in Fig. 3. The relationship between gear’s inherent position error, total tangential composite deviation and tooth-to-tooth tangential composite deviation is as follows: E=

1 1 ' (F − f i' ) sin θ + f i' sin nθ 2 i 2

(4)

where θ is the gear angle; n is the number of circumferential pitch angles turned by a gear, and the variation range is 0−360/z (z is the number of teeth). For random variables satisfying Rayleigh distribution, their mean and variance can be calculated by the following formula: √ uR = √ σR =

π σT 2

(5)

4 − 1 uR π

(6)

Transmission Error Analysis of Planetary Gear Train Based …

27

then the mean and variance of 21 (Fi' − f i' ) can be obtained from Eqs. (5) and (6)  √ π 1 ' (Fi − f i' ) = σT 2 2       1 ' 4 ' 2 1 ' ' −1 M (F − f i ) D (Fi − f i ) = 2 π 2 i 

M

when the confidence probability is 99.7%, substituting σT = (8), one obtains

Fi' − f i' 6

(7) (8) into Eqs. (7) and

 √ 2π ' 1 ' ' F − f i' M (Fi − f i ) = 2 12 i   π ' 1 1 ' ' 1− (Fi − f i' )2 D (Fi − f i ) = 2 18 4 

the mean and variance of

1 ' f 2 i

(9) (10)

can also be obtained

 √ 2π ' 1 ' fi = f M 2 12 i   1 ' π  '2 1 1− f D fi = 2 18 4 i 

(11) (12)

the mean and variance of sin θ are: 2π M(sin θ ) =

2π P(θ ) sin θ dθ =

0

(13)

1 1 sin2 θ dθ = 2π 2

(14)

0

2π D(sin θ ) =

1 sin θ dθ = 0 2π

2π P(θ ) sin θ dθ = 2

0

0

it is the same as Eqs. (13) and (14), the mean and variance of sin nθ are: M(sin nθ ) = 0

(15)

1 2

(16)

D(sin nθ ) =

hence, the mean and variance of gear’s inherent position error are: M(E) = 0

(17)

28

P. Liu et al.

D(E) =

1 [(F ' − f i' )2 + f i'2 ] 36 i

(18)

the maximum value of the gear’s inherent position error is: E max =

√ 1 (Fi' − f i' )2 + f i'2 2

(19)

the minimum value of the gear’s inherent position error is: E min = −

√ 1 (Fi' − f i' )2 + f i'2 2

(20)

the maximum variation of the gear’s inherent position error is: E = E max − E min =

√

(Fi' − f i' )2 + f i'2

(21)

(2) Device error (transmission error of geometric eccentricity) According to Fig. 4, the line value transmission error caused by geometric eccentricity is as follows: Δge =

ege sin(ϕge ± αn ) cos αn

(22)

ΔS 2

(23)

ege =

where ege is the geometric eccentricity during gear installation; ϕge is the phase angle of ege and the reference is the center distance direction; ΔS is the radial sloshing quantity; αn is the pressure angle of reference circle. due to ege obeys the Rayleigh distribution and the probability distribution of ϕge on [0 − 2π] satisfies the uniform distribution, one obtains  sin(ϕge ± α) =0 M cos α   sin(ϕge ± α) 1 = D cos α 2 cos2 α 

(24) (25)

in fact, Δge obeys the normal distribution, one obtains Δge =

ΔS 6 cos α

(26)

since random variables ege and ϕge are independent of each other, the mean and variance of transmission error caused by geometric eccentricity are:

Transmission Error Analysis of Planetary Gear Train Based …

29

Fig. 4 Transmission error of external meshing caused by geometric eccentricity

M(Δge ) = 0

(27)

  ΔS 2 1 D(Δge ) = cos2 α 6

(28)

(3) Force-induced deformation error When subjected to the torque, radial force and circumferential force, the gear system will deform, including the shaft torsional deformation, the shaft bending deformation, the bearing clearance change and the tooth deformation. The influence of each variable on transmission error is discussed below. (a) Shaft torsional deformation The gear will transfer torque in the transmission process, and the gear shaft will undergo torsional deformation, which will cause the line value error ΔST of the gear on the reference circle. As long as the torque T is unchanged, the line value error ΔST is a constant value. obvious, the mean and variance of ΔST are obtained as follows: M(ΔST ) = ΔST

(29)

D(ΔST ) = 0

(30)

30

P. Liu et al.

(b) Shaft bending deformation The gear will be subjected to radial and circumferential forces in the transmission process, too, and the gear shaft will bend and deform along the meshing line of the gear, so that the axis will deviate from the theoretical axis, which is the shaft bending error (ΔS F ). The line value error of reference circle generated by ΔS F is: ΔF =

Δ SF0 cos αn

(31)

hence, the mean and variance of ΔF are obtained as follows: M(ΔS F ) =

ΔS F cos α

D(ΔS F ) = 0

(32) (33)

(c) Bearing clearance change As we can know from Sect. 3.1, radial deviation is Δrr and tangential deviation is Δr t . Circumferential backlash directly affects the line error of gear reference circle, while the influence of radial backlash on line error of gear reference circle is related to pressure Angle. The line error ΔS B of reference circle caused by the change of bearing clearance is: ΔS B = Δr t +

Δrr = Δr0 cos αn tan αn

(34)

therefore, the mean and variance of ΔS B are M(ΔS B ) = ΔS B = Δr cos α

(35)

D(ΔS B ) = 0

(36)

(d) Tooth deformation Under the influence of the circumferential force in the meshing process, gear tooth deformation Δ f 0 may be generated along the meshing line direction, and the line value error (ΔS f ) generated on the reference circle is: ΔS f = the mean and variance of sin nθ are

Δ f 0 · sin nθ cos αn

(37)

Transmission Error Analysis of Planetary Gear Train Based …

31

2π M(sin nθ ) =

P(θ ) sin nθ dθ = 0

(38)

1 2

(39)

0

2π D(sin nθ ) =

P(θ ) sin nθ dθ = 0

the mean and variance of Δ f are M(Δ f ) = Δ f

(40)

D(Δ f ) = 0

(41)

according to Eqs. (37)–(41), the line value error ΔS f caused by gear tooth deformation are M(ΔS f ) = 0  D(ΔS f ) = D

Δ f · sin nθ cos α

(42)

 =

1 Δf 2 cos α 2

(43)

to sum up, a probability formula of transmission error of one gear pair is obtained M(T f ) = ΔST 1 +ΔST 2 +

ΔS F2 ΔS F1 + +Δr1 cos α+Δr2 cos α cos α cos α

1 ' (Fi1 − f i1' )2 + f i1'2 + (Fi2' − f i2' )2 + f i2'2 36  1 18 18 + 2 ((ΔS1 )2 + (ΔS2 )2 )+ 2 Δ f 12 + 2 Δ f 22 cos α cos α cos α

(44)

D(T f ) =

(45)

In this section, the total formula of transmission error of a pair of gears is obtained by analyzing the probability formula of each single factor, which is especially applicable to estimate whether transmission error of a gear system meet the requirements. And the 3K planetary gear train is analyzed by this formula in next section.

4 Example: Probability Analysis of Transmission Error of 3K Planetary Gear Train In this example, all gear shafts are mandrel, the gears accuracy level is 6-6-5, the output torque is 1000 N m and the temperature is 80 °C. The 3K planetary gear system

32

P. Liu et al.

is composed of 5 gears, and planetary gears d1 , c4 and d2 are integrated gears, e1 and e2 are fixed gears, as illustrated in Fig. 5. The parameters of gears are listed in Table 1 (Fig. 5). The mean and variance of total transmission error of 3K planetary gear train are M(α Tc ) = M(α TΔcb ) +

M(α TΔac ) M(α TΔde1 ) M(α TΔde2 ) + + i ab i cb i cb

(46)

D(α Tc ) = D(α TΔcb ) +

D(α TΔde1 ) D(α TΔde2 ) D(α TΔac ) + + 2 2 2 i ab i cb i cb

(47)

Table 1 Parameters of the 3K planetary gear train Gears

Modules

Number of teeth

Tooth width

Transmission ratio i cb = 140.0 i ab = −76.2222 i ac = −0.5444

a

2

90

24

c

2

25

26

b

2

140

24

d1

2

24

18

e1

2

139

16

d2

2

24

15

e2

2

139

14

Fig. 5 3K planetary gear train

Transmission Error Analysis of Planetary Gear Train Based …

33

Table 2 Inherent position errors (μm) Gears

a4

c4

d41

e41

d42

e42

b4

Fi'

54

33

33

72

33

72

72

f i'

12

10.8

10.8

12

10.8

12

12

The maximum value of transmission error of 3K planetary gear train is α Tc max

√ = M(α Tc ) + 3 D(α Tc )

(48)

The minimum transmission error of 3K planetary gear train is α Tc min

√ = M(α Tc ) − 3 D(α Tc )

(49)

Therefore, the transmission error range of 3K gear train is: α Tc

√ = 6 D(α Tc )

(50)

Due to the gear accuracy level is 6-6-5, the inherent position errors can be obtained by means of a look-up table, as shown in Table 2. By substituting the parameters in Tables 1 and 2 into Eqs. (48), (49) and (50), the transmission error of 3K planetary gear train can be obtained. √ = M(α Tc ) + 3 D(α Tc ) = 2.207196049

(51)

√ = M(α Tc ) − 3 D(α Tc ) = −0.170867742

(52)

α Tc max

α Tc min

α Tc

√ = 6 D(α Tc ) = 2.378063791

(53)

5 Conclusions This work presented a Probability method for calculating maximum transmission error and minimum transmission error of a gear train, which can guide us to manufacture the gears meeting requirements of accuracy as cheaply as possible. In fact, each single factor error corresponds to a probability distribution, and it is statistically significant to use probability to analyze the transmission error of the gear system. Acknowledgements Supported by National Key R&D Program of China (No.2019YFB2004601).

34

P. Liu et al.

References 1. Yu L, Wang G, Zou S (2018) The experimental research on gear eccentricity error of backlashcompensation gear device based on transmission error. Int J Precis Eng Manuf 19(1):5–12 2. Cai Y, Yao L, Ding J, Ouyang S, Zhang J (2019) Study on transmission error of double circular arc spiral bevel gears for nutation drive based on assembly errors and different loads. Forschung Im Ingenieurwesen Eng Res 83(3):481–490 3. Lin T, He Z (2017) Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications. Mech Syst Signal Process 91:167–182 4. Lu R, Shahriar MR, Borghesani P, Randall RB, Peng Z (2022) Removal of transfer function effects from transmission error measurements using cepstrum-based operational modal analysis. Mech Syst Sig Process 165, Art. no 108324 5. Wink CH, Serpa AL (2005) Investigation of tooth contact deviations from the plane of action and their effects on gear transmission error. Proc Inst Mech Eng Part C J Mech Eng Sci 219(5):501–509 6. Benatar M, Handschuh M, Kahraman A, Talbot D (2019) Static and dynamic transmission error measurements of helical gear pairs with various tooth modifications. J Mech Des 141(10), Art. no 103301 7. Benaicha Y, Perret-Liaudet J, Beley JD, Rigaud E, Thouverez F (2022) On a flexible multibody modelling approach using FE-based contact formulation for describing gear transmission error. Mech Mach Theory 167, Art. no 104505 8. Hu Y, Ryali L, Talbot D, Kahraman A (2019) A theoretical study of the overall transmission error in planetary gear sets. Proc Inst Mech Eng Part C J Mech Eng Sci 233(21–22):7200–7211 9. Lee HW, Park MW, Joo SH, Park NG, Bae MH (2007) Modeling transmission errors of gear pairs with modified teeth for automotive transmissions. Int J Automot Technol 8(2):225–232 10. Cheng J, Liang M, Fan F (2012) Experiment research and analysis of transmission error of gears. In: 2nd international conference on frontiers of manufacturing science and measuring technology (ICFMM 2012), Xian, Peoples Republic of China, vol 503–504, pp 1074–1077 11. Hu Q, Liu Z, Yang C, Xie F (2021) Research on dynamic transmission error of harmonic drive with uncertain parameters by an interval method. Precis Eng J Int Soc Precis Eng Nanotechnol 68:285–300 12. Lei X, Cai Z, Zhang M (2015) Transmission error of static characteristic for planetary gear transmission in the field of high-power tractor. In: 3rd international conference on material, mechanical and manufacturing engineering (IC3ME), Guangzhou, Peoples Republic of China, vol 27, pp 1542–1545 13. Li K, Bian Y, Liu B (2014) Transmission error analysis of external gear drive based on form and position errors. In: International conference on machinery, electronics and control simulation (ICMECS), Weihai, Peoples Republic of China, vol 614, pp 36–39 14. Pears J, Smith A, Curtis S (2005) A software tool for the prediction of planetary gear transmission error. In: International conference on gears, Garching, Germany, vol 1904, pp 357–372 15. Wang G, Su L, Zou S (2020) Uneven load contact dynamic modelling and transmission error analysis of a 2K-V reducer with eccentricity excitation. Strojniski Vestnik J Mech Eng 66(2):91–104 16. Zhao C, Hong L, Wang J, Liu Z, Qu Y, Tan Y (2019) A virtual model to predict the influence of indexing errors on the transmission error of spur gears. In: 10th IEEE prognostics and system health management conference (PHM-Qingdao), Qingdao, Peoples Republic of China

Precision Motion Control of Separate Meter-In and Separate Meter-Out Hydraulic Swing System with State Constraints Yong Zhou, Zheng Chen, Ruqi Ding, Min Cheng, and Bin Yao

Abstract Traditional independent meter-in and meter-out hydraulic systems have been widely used in industry because of their energy saving characteristics and control flexibility. However, the control accuracy is limited by the strong nonlinearities and parametric uncertainties. In addition, independent metering systems suffer from safety problem due to complex hydraulic mechanical structures. The conventional solutions to guarantee safety are using mechanical mechanisms to limit the system’s states which increase complexity further. This paper proposes a novel two loop control strategy which regards the physical limits as state constraints to alternative mechanical structures. A conventional adaptive robust controller (ARC) is utilized in the inner loop to achieve high tracking accuracy and handle the nonlinearities, uncertainties and disturbances. A third order trajectory algorithm is synthesized to modify the original severe trajectory to a modest one which will make sure that the state constraints are not violated. Simulations are carried out and the results demonstrate the effectiveness of the proposed control strategy. Keywords Adaptive robust control · Separate meter-in and separate meter-out systems · State constraints · Trajectory planning Y. Zhou · Z. Chen (B) State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, Zhejiang, China e-mail: [email protected] Z. Chen Ocean College, Zhejiang University, Zhoushan 316021, Zhejiang, China R. Ding Key Laboratory of Conveyance and Equipment, Ministry of Education, East China Jiaotong University, Nanchang 330013, Jiangxi, China M. Cheng State Key Laboratory of Mechanical Transmissions, College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, China B. Yao School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_3

35

36

Y. Zhou et al.

1 Introduction Electrohydraulic servo systems have been widely used in rotation occasions such as construction machineries [1], aircraft actuators [2], robots and manipulators [3] due to their high power density and high loading capability [4, 5]. And the typical independent meter-in and meter-out technique has shown its superiority in control performance and energy saving by raising the systems’ control freedom [6–8]. However, confronted with its advantages, there still exist two important challenges in such systems: (1) high control accuracy in the presence of uncertainties, nonlinearities and disturbances, such as nonlinear friction, flow discrepancy, and parametric uncertainties [9–11]; (2) safety problems because of the extra freedom which also increases the system’s failure probability. Electrohydraulic systems are hard to control owing to their strong nonlinearities and high order characteristics. Numerous studies published have focus on this issue to improve the inherent nonlinear dynamics of electrohydraulic systems. Traditionally, to achieve good steady-state tracking accuracy, high gain based approaches are often used, such as sliding mode control [12], robust H∞ control [13–15], disturbance observed-based control [16], which have good anti-interference capability. To deal with the parametric uncertainties of the hydraulic systems, adaption law is added to the control structure to compensate the parameters real time. Various kinds of control laws have been proposed based on this this thought process [17–19]. Adaptive robust controllers (ARC) proposed by Yao in the last decades is one of this kind of control strategies to handle nonlinearities, uncertainties and disturbances with a widespread application in the electrohydraulic filed [9, 20, 21]. However, this type of controllers can not deal with the safety problems directly. When facing severe trajectories such as step signal or trajectories with large initial tracking error, it’s difficult to keep the system under constraints. To improve the independent metering hydraulic systems’ safety, kinematic and dynamic constraints such as actual workspace limitations, speed limitations and pressure limitations have to be set. And those constraints can be viewed as state constraints to be considered in the controller design procedure to replace complex mechanical limits commonly used in traditional hydraulic system. However, it’s not an easy task to take state constraints into consideration while designing control strategies. To solve this tough issue, many techniques have been proposed. Blanchini proposed set invariance-based methods [22], based on which barrier functions (BFs) including control barrier functions (CBFs) and barrier Lyapunov functions (BLFs) are developed to achieve safety control of systems [23–25]. A CBF is defined in both safe region and unsafe region, and if a control can be found by solving the optimization problem, then the invariant performance of the constraint region is guaranteed. But because control input obtained by the original controller has been changed, stability can not be guaranteed at the same time. To handle this problem, a quadratic programming (QP) method by combining control Lyapunov functions (CLFs) with CBFs was used [26]. However, feasibility becomes a new challenge as a result of more constraints introduced in solving the QP problem. A BLF guarantee

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

37

the system states in safe region by adding barrier terms which will become infinity when the states reach their bounds to the Lyapunov function [23]. This method can achieve safety and stability simultaneously. But it is still an optimization problem in essential, and in addition to the state constraints, other constraints are introduced in the procedure of finding the feasible solution, resulting in a quite conservative controller [27]. Another approach is based on the model predictive control (MPC), which is able to deal with both equality and inequality directly [28, 29], but the time consumed to get the solution will increase with the constraints’ scale. And with the rapid growth in computing power, lots of works have focused on the nonlinear model predictive control (NMPC), where different techniques are developed to reduce the computation time [30, 31]. However, this kind of method still suffer from poor steady-state control accuracy and can not handle with disturbances. Yuan proposed a two-loop control structure for constrained second-order nonlinear systems, where the trajectory is replanned in outer loop to avoid constraint violation while nonlinearities and uncertainties are handled by ARC controller in inner loop [32]. Yao and Lu developed a performance-oriented multiloop scheme to replan the trajectory and design ARC as an integration [33]. But the two method can only handle second order systems while hydraulic systems are third order. In this paper, a novel two-loop control strategy for third order constrained electrohydraulic swing system is proposed, in which the system can achieve high control accuracy with state constraints satisfied. The paper’s contributions are concluded as follows • A novel two loop control strategy is proposed for third-order electrohydraulic swing system, in which high control accuracy issue and safety issue are handled separately. A third order trajectory replanning algorithm is utilized in the outer loop to generated a new moderate reference trajectory which can make the system avoid unsafe region. • In the inner loop, a backstepping ARC controller is designed to track the replanned trajectory given by the outer loop. And by selecting the parameters of the inner loop controller properly, the invariant set is constructed. Then the boundary conditions of the replanned tra-jectory are obtained according to the invariant, which are the constraints in solving the outer loop planning problem.

2 Problem Formulation As shown in Fig. 1, the dynamics of the hydraulic motor can be modeled as   Jm θ¨m = Dmp (P1 − P2 ) − B f θ˙m − A f S f θ˙m − Te + Dm + D˜ m

(1)

where Jm is the swing inertia converted onto the motor shaft, θm represents the swing angular displacement of the motor, Dmp is a constant coefficient, P1 and P2 are the pressures of the chambers, respectively, B f and A f represent the viscous friction coefficient and the Coulomb friction coefficient, respectively, and Te is the external

38

Y. Zhou et al.

Fig. 1 System configuration

torque acting on the motor shaft. Dm + D˜ m denotes the lumped modeling error, including external disturbance and unmodeled terms with Dm being the nominal value. Because the internal leakage of the cylinder is small, it is neglected in the equation for simplicity. Thus, the pressure dynamics of the two chambers can be described by V1 ˙ P1 = −Dm θ˙m + Q 1d + D1 + d˜1 βe   V2 ˙ P2 = Dm θ˙m − Q2d − D2 + d˜2 βe

(2)

where V1 and V2 are the total compressible volumes of the motor champers. βe is the effective bulk modulus. Q 1d and Q 2d represent the desired flows of the chambers. D1 + d˜1 and D2 + d˜2 denote the modeling errors of the two equations, including terms such as valve flow discrepancies with D1 and D2 being nominal values. T  Let x = [x1 , x2 , x3 , x4 ]T = θm , θ˙m , P1 , P2 be the system states, = and define unknown parameter set θ = [θ1 , θ2 , θ3 , θ4 , θ5 , θ6 , θ7 ]T T  1 B f A f Te −Dm , , , Jm , βe , βe D1 , βe D2 , the state equation can thus be rewritten as Jm Jm Jm x˙1 = x2

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

x˙2 = θ1 Dmp (x3 − x4 ) − θ2 x2 − θ3 S f (x2 ) − θ4 + D˜ m 1 −Dm x2 + Q 1d x˙3 = θ5 + θ6 + D˜ 1 V1 V1 Dm x2 − Q 2d 1 x˙4 = θ5 − θ7 − D˜ 2 V2 V2

39

(3)

˜ ˜ where D˜ 1 = βVe d1 1 and D˜ 2 = βVe d2 2 . In practice, the accurate values of parameter θ cannot be exactly known, but the ranges of uncertainties can be known in advance. And for any controller capable of stabilizing the system, the actuator has to be physically powerful enough to deal with the external disturbance, which means the disturbances have to be bounded. So the following assumption is made.

Assumption 1 The bounds of the parametric uncertainties are known and the lumped disturbances are finite and bounded, i.e., Δ

θ ∈ Ωθ ={θ : θmin ≤ θ ≤ θmax }

  Δ D˜ i ∈ Ω D = D˜ i :  Di  ≤ δi i = m, 1, 2

(4)

 T where θmin = is the lower bound of θ , and θmax = θ1 min . . . θ7 min T  θ1 max . . . θ7 max is the upper bounds of θ , and δi is the bund of D˜ i , which are known in advance. Meanwhile, due to safety consideration, the state constraints are given as x1 min x2 min x3 min x4 min

≤ ≤ ≤ ≤

x1 x2 x3 x4

≤ ≤ ≤ ≤

x1 max x2 max x3 max x4 max

(5)

where the bounds for x1 , x2 denote the working space limitation, angular velocity limitation and the bounds for x3 , x4 represent the pressure limitations of chambers. Let C denotes the set of all states that satisfy the above constraints, and C0 ⊂ C denotes the set of all initial states that system states start with will remain in the set C.

3 Control Structure In this section, the novel control structure is proposed in Fig. 2, such that the following control objectives can be achieved simultaneously: (1) the constraints (5) will never be violated, (2) fast transit response for any initial state x0 ∈ C0 which is as large as

40

Y. Zhou et al.

Fig. 2 The proposed two loop control structure

State Feedback

Trajectory Replanning Algorithm

Backstepping ARC Controller

Hydraulic Swing System

State Feedback Outer Loop

Inner Loop

possible, (3) closed loop stability and high steady-state tracking accuracy for any state starts in C0 , (4) good robust performance against parametric uncertainties, uncertain nonlinearities and external disturbances. The overall control structure consists of two loops: • Inner loop: an adaptive robust controller that takes parametric uncertainties, uncertain nonlinearities and disturbances into account simultaneously is designed by synthesizing a parameter adaption law, a stabilizing feedback term and a robust feedback law. High tracking performance of replanned trajectory xr (t) can be achieved in this way. And the tracking errors with respected to xr can be controlled within the preset invariant set which gives the boundary conditions of the xr . • Outer loop: a time-optimal third-order trajectory replanning algorithm is proposed subject to the constraints given by the inner loop, such that the replanned trajectory xr converges to original trajectory xd as fast as possible. The algorithm guarantees a fast transient response of the system with the states not leaving the safe set C. The design procedure of each loop and the overall control law will be presented in the following sections.

4 Inner Loop In this section, the inner loop controller structure is detailed as shown in Fig. 3. The controller design procedure processes in two parts, including motion control part and the pressure control part. In the motion control part, the backstepping ARC control law is utilized to achieve high tracking accuracy and a parameter adaption law is used to estimate the uncertain parameters online. In the pressure control part, a pressure planning method is used to solve the non uniqueness problem of the independent metering system and an ARC controller is designed to keep the lower-pressure side of the motor in the preset back pressure.

Precision Motion Control of Separate Meter-In and Separate Meter-Out … Fig. 3 Structure of the inner loop ARC control law

41

Replanned Trajectory Adaptive Robust Motion Controller

Pressure Planning

Adaptive Robust Pressure Controller

Valves

State Feedback

Hydraulic Swing Motor

Parameter adaption

Actual Trajectory

4.1 Motion Controller Design Define the tracking error as z 1 = x1 − xr , then define z 2 as z2 = z 1 + k1 z 1 = x2 − x2eq , x2eq = x˙1r − k1 z 1

(6)

where k1 > 0 is any positive feedback gain. Define pressure difference PL = x3 − x4 , noting the system dynamic (3), then the dynamic of z 2 can be written as: z˙ 2 = θ1 Dmp PL − θ2 x2 − θ3 S f (x2 ) − θ4 + D˜ m − x˙ 2eq

(7)

To make x1 tracks xr as accurate as possible, the ARC control law is designed as follows PLd = PLda + PLds1 + PLds2  1  θˆ2 x2 + θˆ3 S f (x2 ) + θˆ4 + x˙2eq PLda = θˆ1 Dmp PLds1 = −

k2 z 2 θ1 min Dmp

42

Y. Zhou et al.

τ2 = ω2 φ2 z 2

(8)

where PLda is the adaptive feedforward compensation parts, PLds1 and PLds2 are the feedback terms, k2 > 0 is the linear feedback gain. ω2 > 0 is a design parameter,  T φ2 = PLda , −x2 , −S f (x2 ), −1, 0, 0, 0 , The nonlinear robust feedback term PLds2 is chosen to satisfy the following conditions z 2 PLds2 ≤ 0   z 2 −φ2T θ˜ + D˜ m + θ1 Dmp PLds2 ≤ ∈2

(9)

 T where ∈2 > 0 is a design parameter, and θ˜ = θ˜1 , θ˜2 , . . . , θ˜7 , in which θ˜i = θˆi − θi is the parameter estimation error. The error between expected pressure difference PLd and the actual pressure difference PL is defined as z 3 = PL − PLd

(10)

Because the desired pressure difference is a virtual input, so the next step is to synthesize the flow rate input Q 1d to make the error z 3 as small as possible. Noting (3) and (8), the dynamic of z 3 can be obtained by differentiating z 3  Dm x 2 Q 1d Q 2d Dm x 2 + θ5 + + θ5 V1 V2 V1 V2   1 1 + θ6 + θ7 + D˜ 1 + D˜ 2 − P˙Ldc + P˙Ldu V1 V2 

z˙ 3 = −θ5

(11)



PLd represents the calculable part of P˙Ld , and where P˙Ldc = ∂∂PxLd1 x2 + ∂∂PxLd2 x˙ 2 + ∂ ∂t P˙Ldu = ∂ PLd  x˙ 2 + ∂ PLd θˆ is the incalculable part of P˙Ld . In addition, x˙ 2 is the estimate 

∂ x2

∂ θˆ



of the x˙2 calculated by x˙ 2 = θˆ1 Dmp (x3 − x4 )− θˆ2 x2 − θˆ3 S f (x2 )− θˆ4 , and the estimate error is  x˙ 2 = x˙2 − x˙ 2 . The calculable part P˙Ldc can be compensated when designing the control law, so the control input Q 1d is given by 

Q 1d = Q 1da + Q 1ds1 + Q 1ds2    Dm x 2 Q 2d V1 Dm x 2 ˆ θ5 − θˆ5 Q 1da = + ˆθ5 V1 V2 V2  1 1 ω2 ˙ ˆ ˆ ˆ θ1 Dmp z 2 − θ7 + PLdc − −θ6 V1 V2 ω3 V1 Q 1ds1 = − k3 z 3 θ5 min τ3 = ω3 φ3 z 3

(12)

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

43

where ω3 > 0 is the weighting factor to be designed, φ3 =  ∂ PLd ω2 D z − ∂ x2 PL , ∂∂PxLd2 x2 , ∂∂PxLd2 S f (x2 ), ∂∂PxLd2 , QV1da1 − DVm1x2 + DVm2x2 − QV2d2 , ω3 mp 2 1 T 1 , 1 , Q 2d is the desired value of Q 2 to be given later. Q 1ds2 is the robust V1 V2 feedback term designed to satisfy the following robust performance conditions z 3 Q 1ds2 ≤ 0   ∂ PLd ˜ θ5 Q 1ds2 ≤ ∈3 Dm + z 3 D˜ 1 + D˜ 2 − φ2T θ˜ − ∂ x2 V1

(13)

where ∈3 is a design parameter.

4.2 Pressure Controller Design The desired pressure difference PLd has been generated by the motion controller. However, considering the relationship PLd ≈ P1 − P2 , the solutions of x3 and x4 are not unique since the two champers pressure dynamics are independent in the meterin meter-out system. To solve the problem, a pressure planning similar to [21] is developed to design a desired back pressure for chamber 2. Then the unique solution of P1 and P2 will be attained. A pressure controller is designed such that the lower pressure one of the two chambers can keep at the pre-set back pressure. Here, the desired pressure is given as  P2d =

PLda1 ≥ 0 Pb Pb − PLda1 PLda1 < 0

(14)

where Pb is the pre-set back pressure. And PLda1 instead of PLda is used in the planner to avoid chattering problem and obtain a smooth and stable trajectory. The next step is to synthesize a flow rate Q 2d to make the actual pressure track P2d as T  close as possible. Define θ p = θ5 θ7 as the linearized parameter vector used in T  the pressure controller design and θˆp = θˆ5 θˆ7 as the estimate of θ p , The pressure discrepancy is defined as z p = P2 − P2d , then according the system dynamic (3), the error dynamic can be written as 2d z˙ p = θ5 Dm xV2 −Q − θ7 V12 − D˜ 2 − P˙2d 2

The desired flow rate Q 2d can be generated by the control law Q 2d = Q 2da + Q 2ds1 + Q 2ds2 Q 2da1 = Dm x2 −

θˆ7 V2 ˙ − P2d ˆθ5 θˆ5

(15)

44

Y. Zhou et al.

Q 2ds1 =

V2

θ5 min τp = φpz p

kpz p (16)

T  1 + D x − where φ p = − QV2da , k p > 0 is the feedback gain, τ p is the adaption m 2 V2 2 function utilized in the adaption law, and Q 2ds2 is the nonlinear feedback term which should be designed to satisfy the following conditions − z p Q 2ds2 ≤ 0   Q 2ds2 T ˜ ˜ ≤ ∈p − z p −φ p θ p − D2 + θ5 V2

(17)

 T where ∈ p is a design parameter, and θ˜p = θ˜5 θ˜7 is the estimation error.

4.3 Parameter Adaption To further improve the tracking accuracy, the gradient type adaption law [34] is used to estimate system parameters online to compensate the system’s dynamic feedforward. And the adaption law can be written in the following form   θ˙ˆ = satθ˙ˆ Pr ojθˆ (┌τ )    θ˙ˆp = satθ˙ˆ Pr ojθˆp ┌ p τ p

(18)

p

τ = τ2 + τ3 where ┌ and ┌ p are positive diagonal adaption rate matrices, and τ and τ p are the adaption functions synthesized in the previous section. The saturation function is defined as ⎧ ˙ ˙ ⎪ ⎨ θˆimax , i f ·i > θˆimax satθ˙ˆ (·i ) = θ˙ˆimin , i f ·i < θ˙ˆimin (19) ⎪ ⎩ ·i , else where θ˙ˆimax and θ˙ˆimin are the upper and lower bounds of the ith element in θ˙ˆ , which can be determined in advance. The projection mapping is defined as ⎧ ◦ ⎪ ⎨ ζ, θˆ ∈ Ωθ or n θTˆ ζ ≤ 0   T Pr ojθˆ (ζ ) = ⎪ I − ┌ nTθˆ n θˆ ζ, θˆ ∈ ∂Ωθ and n T ζ ≥ 0 ⎩ n ┌n θˆ θˆ

θˆ

(20)

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

45 ◦

where Ωθ represents the known bounded-convex set of all feasible θ , Ωθ and ∂Ωθ denote the interior and the boundary of Ωθ , n θˆ represents the outward unit normal vector at the boundary ∂Ωθ , and I is the identity matrix. It is easy to prove that such projection law has the following properties P1. θimin ≤ θˆi ≤ θimax   P2. θ˜ T ┌ −1 Pr ojθˆ (┌τ ) − τ ≤ 0 P3. θ˙ˆimin ≤ θ˙ˆi ≤ θ˙ˆimax

(21)

4.4 Invariant Set Construction We can construct an invariable set for the tracking error z 1 , z 2 and z 3 , where once they enter the set, they will stay in the set thereafter. Substituting the control laws (8) and (12) into the error dynamics (7) and (9), then the dynamics of the errors can be written as z˙ 1 = z 2 − k1 z 1 θ1 z˙ 2 = θ1 Dmp z 3 − k2 z 2 + θ1 Dmp PLds2 − φ1T θ˜ + D˜ m θ1min   Dm x 2 Q 1d Q 2d Dm x 2 + θ5 z˙ 3 = −θ5 + + θ5 V1 V2 V1 V2   1 1 + θ6 + θ7 + D˜ 1 + D˜ 2 − P˙Ldc + P˙Ldu V1 V2

(22)

The following assumptions are made when constructing the invariant set. Assumption 2

−Dm x2 1 + θ6 ≤ φ1u (x, t) V1 V1 +Dm x2 1 φ2l (x, t) ≤ θ5 − θ7 ≤ φ2u (x, t) V2 V2 |φ1l (x, t)|≤ h 1 , |φ1u (x, t)| ≤ h 1

φ1l (x, t) ≤ θ5

|φ2l (x, t)|≤ h 2 , |φ2u (x, t)| ≤ h 2 ∀x ∈ C, θ ∈ Ω

(23)

where φ1l (x, t), φ2l (x, t) φ1u (x, t) and φ2u (x, t) are the known bounding functions, h 1 and h 2 are known constants representing the upper bounds.

46

Y. Zhou et al.

Assumption 3 θ5min QM V1 θ5min h 2 + δ2 ≤ QM V2 h 1 + δ1 ≤

(24)

Theorem 1 With the control law (16), the set Ω = {z||z i | ≤ L i } is a positive invariant set and the constraints (5) are never violated if the following conditions are satisfied, where i = 1, 2, 3 and z i is the ith component of the error vector z (1) L 2 ≤ k1 L 1 (2) θ1max Dmp L 3 − k2 L 2 + η1 ≤ 0   θˆ2 x˙ −Q M Q 2d ... ... ˆ ˆ − (h 1 + h 2 ) − + (3) x r ≥ x r min = θ1 Dmp [θ5 V1 V2 θˆ Dmp     1 θˆ1 θˆ1 1 + k1 + k2 L 3 + k1 k2 + 2 k2 L 2] θ1min θ1min Dmp θ1min Dmp 1 

+ k12 L 2 + k13 L 1 + η2 + ξ

  θˆ2 x˙ 2 QM Q 2d ... ... − (h 1 + h 2 ) − + and x r ≤ x rmax = θˆ1 Dmp [θˆ5 V1 V2 θˆ Dmp     1 θˆ1 1 θˆ1 − k1 + k2 L 3 − k1 k2 + 2 k2 L 2] θ1min θ1min Dmp θ1min Dmp 1 

− k12 L 2 − k13 L 1 − η2 − ξ     (4) η1 ≥ −φ2T θ˜ + D˜ m + θ1 Dmp PLds2  max     (5) η2 ≥ + D˜ 1 + D˜ 2 − P˙Ldu  max

(6) xr ≤ xrmax = x1max − L 1 and xr ≥ xrmin = x1min + L 1 (7) x˙r ≤ x˙rmax = x2max − L 2 − k1 L 1 and x˙r ≥ x˙rmin = x2min + L 2 + k1 L 1 (8) x¨r ≤ x¨rmax = θˆ1 Dmp (+x3max − pb ) − θˆ2 x2 − θˆ3 S f (x2 ) − θˆ4 θ1max k 2 L 2 − η1 − θ1max Dm L 3 − k1 L 2 − k12 L 1 − θ1min and x¨r ≤ x¨rmin = θˆ1 Dmp (−x3max + pb ) − θˆ2 x2 − θˆ3 S f (x2 ) θ1max − θˆ4 + θ1max Dm L 3 + k1 L 2 + k12 L 1 + k 2 L 2 + η1 θ1min

(25)

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

47

where ξ is a positive margin considering parametric uncertainties and disturbances, Q M denotes the maximum flow rate through the valve. Conditions (1) and (2) are easy to be satisfied by choosing the design parameters properly. The bound conditions of the replanned trajectory are given by (3), (6), (7) and (8). It is noticed that the minimum value of x3 and x4 are controlled by the pressure controller to stay around preset back pressure pb , which can be chosen larger than the lower bound. It means the pressure will not goes to the lower bound if pb chosen properly and pressure controller designed well, so the lower constraints of the pressure x3min and x4min are regarded as nonexistence in the invariant set construction.

5 Outer Loop The ARC controller designed in Sect. 4 guarantees tracking performance and robust performance. However, the desired trajectory xd (t) can not be directly used as xr (t) when xd (t) is severe such as step signal. Because it may cause constraints (5) not satisfied. A replanned trajectory xr (t) to make sure the system states are in the safe region C is necessary. Define er = xr (t) − xd (t) as the error between the replanned trajectory and the desired trajectory which can also be seen as a trajectory modification. Then the control objective of the outer loop can be formulated by designing an algorithm such that er converges to zero as fast as possible under conditions (3), (6), (7) and (8) of (25). Noting that the trajectory planner can be treated as a third-order integrator with e¨r (t) being the input, the problem can be solved in discrete-time domain with a sampling period T, then the planner can be summarized a optimization problem as the following form   min tf ... e r ,t∈[kT ,t f ] ˙ s.t. e(kT ) = er (kT ), e(kT ) = e˙r (kT ), e¨(kT ) = er (kT )       e t f = 0, e˙ t f = 0, e¨ f t f = 0 emin (t) ≤ e(t) ≤ emax (t) e˙min (t) ≤ e(t) ˙ ≤ e˙max (t) e¨min (t) ≤ e¨(t) ≤ e¨max (t) ... ... ... e min (t) ≤ e (t) ≤ e max (t)

(26)

... where e(t), e(t), ˙ e¨(t) and e (t) represent the temporary variables in solving the problem, and bound conditions of e(t) are given as emin (t) = xr min − xd (t), emax (t) = xr max − xd (t) e˙min (t) = xr min − x˙d (t), e˙max (t) = xr max − x˙d (t) e¨min (t) = max(x¨r min , x˙2 min ) − x¨d (t)

48

Y. Zhou et al.

e¨max (t) = min(x¨r max , x˙2 max ) − x¨d (t) ... ... ... e min (t) = max( x r min , x¨2 min ) − x d (t) ... ... ... e max (t) = min( x r max , x¨2 max ) − x d (t)

(27)

where x˙2max and x¨2max are the upper bounds of replanned acceleration and jerk respectively in case that they are too large. And x˙2min and x¨2max are the lower bounds of replanned acceleration and jerk. It is known from the optimal control theory the above optimal problem is bang... ... bang type and the optimal solution switches between e max (t), e min (t), 0. At each ...∗ sampling period, the analytical solution e (t) of (26) for the next period can be ˙ and e¨(t) are obtained according to the obtained by the method in [33]. And e(t), e(t) discrete third-order integrator ⎤ ⎡ T3 ⎤ ⎤ ⎡ 2 ⎤⎡ e∗ ((k + 1)T ) e (kT ) 1 T T2 ⎢ 62 ⎥... ⎣ e˙∗ ((k + 1)T ) ⎦ = ⎣ 0 1 T ⎦⎣ e(kT ˙ ) ⎦ + ⎣ T2 ⎦ e ∗ e¨(kT ) e¨∗ ((k + 1)T ) 0 0 1 T ⎡

(28)

... The trajectory error er , e˙r , e¨r and e r of the next period are then taken as e∗ , e˙∗ , e¨∗ ...∗ and e . As a result we can get the replanned trajectory by adding the modification to the desired trajectory xr (t) = xd (t) + er (t) x˙r (t) = x˙d (t) + e˙r (t) x¨r (t) = x¨d (t) + e¨r (t) ... ... ... x r (t) = x d (t) + e r (t)

(29)

6 Simulation Results The overall control system is simulated in the Matlab-Simulink. The following units are used in the simulation: Pressure: bar, Flow rate: L/min, Swing velocity: rad/s, Torque: kN, Length: m, Swing inertia: 1000 kg m2 . The dynamic parameters of the swing motor are given by Jm = 270, B f = 2.84, A f = 8.652. And the swing motors constants are Dmp = 0.8185, Dm = 491.1 respectively. The effective bulk modulus are assumed to be βe = 2625 bar. The constrains are set as x1min = −2π , x1max = 2π , x2min = −0.8, x2max = 0.8, x3max = x4max = 400, x3min = x4min = 0. The following parameters for the proposed controller are selected as follows: L 1 = 1 × 10−3 , L 2 = 8 × 10−3 , L 3 = 15, k1 =!5, k2 = 5, k3 = 100, k p = 100,"and the adaption! rates are chosen as ┌ = " diag 1, 10, 10, 50, 3 × 10−4 , 5 × 10−5 and ┌ p = diag 2, 1 × 10−5 .

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

49

Fig. 4 Desired and actual trajectories of C1 and C2 in Case I

Two simulations were carried out in Simulink to test the effectiveness of propose control method. A step signal is chosen as the reference trajectory in case I, and a high-order filter method commonly used in engineering was carried out at same time as a comparison. A point-to-point trajectory with large initial tracking error was used in case II to test the controller’s ability to deal with large trajectory discrepancy.

6.1 Case I The two controllers used as comparison are as follows C1: The proposed two-loop ARC controller with trajectory planner. C2: The ARC controller with a tradition four-order filter commonly used in engineering. The desired and actual trajectories in the comparative simulation are shown in Fig. 4, from which we can find that C1 has faster transient response than C2. And C1 achieves better steady state performance as shown in Fig. 5. The reason why C1 has better control performance can be explained by Fig. 6, which shows that C1 pushes the states to their bounds without going over the constraints and that means C1 can make full use of the system’s ability. While under the control of C2, the system states can not reach their bounds all, which suggests C2 is quite conservative. Meanwhile, tracking errors with respect to the replanned trajectory are shown in Fig. 7, where z 1 , z 2 and z 3 do stay in the invariant set.

6.2 Case II The case II is carried out to test the proposed controller’s ability to handle with the large initial tracking error. The initial position desired point to point trajectory is

50

Fig. 5 Tracking errors with respect to xd (t) of C1 and C2

Fig. 6 System states in case I

Y. Zhou et al.

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

51

Fig. 7 Invariant set of C1 in case I

0.2 rad, while the actual system starts in zero, as shown in Fig. 8. Figure 9 shows the good performance of the controller with steady-state tracking error less than 0.001 rad. And all the system states are within the safe set C as can be seen in figure in Fig. 10. The constructed invariant set Ω are shown in Fig. 11, suggesting all the errors are in it as expected. Therefore, the simulation results clearly demonstrate that the proposed two-loop control algorithm can achieve high tracking performance with states constraints not violated in the presence of parametric uncertainties and uncertain nonlinearities.

Fig. 8 Desired and actual trajectories in case II

52

Fig. 9 Tracking error with respect to xd (t) in case II

Fig. 10 System states in case II

Y. Zhou et al.

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

53

Fig. 11 Invariant set of C1 in case II

7 Conclusion In this paper, a novel two-loop adaptive robust control algorithm with trajectory planning part is proposed for the third-order hydraulic swing system. In the inner loop, the ARC controller is utilized to achieve high tracking performance in the presence of parameter uncertainties, uncertain nonlinearities and external disturbances. And the invariant set in constructed to give the upper and lower bounds that the replanned trajectory xr (t) should satisfy. In the outer loop, a third-order trajectory planning method is utilized to modify the desired trajectory to keep the system states in the safe set. By the novel two-loop method, high tracking accuracy, fast transient response and constraints satisfaction can be achieved simultaneously. The simulation results clearly confirmed the effectiveness and superiority of the proposed control strategy. Acknowledgements Supported by the National Key Research and Development Program of China under Grant 2020YFB2009703 and Zhejiang Provincial Natural Science Foundation of China under Grant LR23E050001.

References 1. Wang L, Gong G, Yang H, Yang X, Hou D (2013) The development of a high-speed segment erecting system for shield tunneling machine. IEEE/ASME Trans Mechatron 18(6):1713–1723 2. Sente PA, Labrique FM, Alexandre PJ (2012) Efficient control of a piezoelectric linear actuator embedded into a servo-valve for aeronautic applications. IEEE Trans Ind Electron 59(4):1971– 1979

54

Y. Zhou et al.

3. Mattila J, Koivumaki J, Caldwell DG, Semini C (2017) A survey on control of hydraulic robotic manipulators with projection to future trends. IEEE/ASME Trans Mechatron 22(2):669–680 4. Cheng M, Zhang J, Xu B, Ding R, Wei J (2018) Decoupling compensation for damping improvement of the electrohydraulic control system with multiple actuators. IEEE/ASME Trans Mechatron 23(3):1383–1392 5. Ding R, Cheng M, Jiang L, Hu G (2021) Active fault-tolerant control for electro-hydraulic systems with an independent metering valve against valve faults. IEEE Trans Ind Electron 68(8):7221–7232 6. Ge L, Quan L, Zhang X, Zhao B, Yang J (2017) Efficiency improvement and evaluation of electric hydraulic excavator with speed and displacement variable pump. Energy Convers Manag 150:62–71 7. Koivumäki J, Zhu W-H, Mattila J (2019) Energy-efficient and high-precision control of hydraulic robots. Control Eng Pract 85:176–193 8. Ding R, Cheng M, Zheng S, Xu B (2021) Sensor-fault-tolerant operation for the independent metering control system. IEEE/ASME Trans Mechatron 26(5):2558–2569 9. Yao B, Bu FP, Reedy J, Chiu GTC (2000) Adaptive robust motion control of single-rod hydraulic actuators: theory and experiments. IEEE/ASME Trans Mechatron 5(1):79–91 10. Yao J, Deng W (2017) Active disturbance rejection adaptive control of hydraulic servo systems. IEEE Trans Ind Electron 64(10):8023–8032 11. Na J, Li Y, Huang Y, Gao G, Chen Q (2020) Output feedback control of uncertain hydraulic servo systems. IEEE Trans Ind Electron 67(1):490–500 12. Levant A (1993) Sliding order and sliding accuracy in sliding mode control. Int J Control 58(6):1247–1263 13. Milic V, Situm Z, Essert M (2010) Robust H∞ position control synthesis of an electro-hydraulic servo system. ISA Trans 49(4):535–542 14. Guo Q, Yu T, Jiang D (2015) Robust H-infinity positional control of 2-DOF robotic arm driven by electro-hydraulic servo system. ISA Trans 59:55–64 15. Rigotti-Thompson M, Torres-Torriti M, Auat Cheein FA, Troni G (2020) H-infinity-based terrain disturbance rejection for hydraulically actuated mobile manipulators with a nonrigid link. IEEE/ASME Trans Mechatron 25(5):2523–2533 16. Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Ind Electron 61(11):6285–6293 17. Yao J, Deng W, Jiao Z (2015) Adaptive control of hydraulic actuators with Lugre model-based friction compensation. IEEE Trans Ind Electron 62(10):6469–6477 18. Yao J, Deng W, Jiao Z (2017) RISE-based adaptive control of hydraulic systems with asymptotic tracking. IEEE Trans Autom Sci Eng 14(3):1524–1531 19. Guo K, Li M, Shi W, Pan Y (2022) Adaptive tracking control of hydraulic systems with improved parameter convergence. IEEE Trans Ind Electron 69(7):7140–7150 20. Helian B, Chen Z, Yao B, Lyu L, Li C (2021) Accurate motion control of a direct-drive hydraulic system with an adaptive nonlinear pump flow compensation. IEEE/ASME Trans Mechatron 26(5):2593–2603 21. Lyu L, Chen Z, Yao B (2021) Advanced valves and pump coordinated hydraulic control design to simultaneously achieve high accuracy and high efficiency. IEEE Trans Control Syst Technol 29(1):236–248 22. Blanchini F (1999) Set invariance in control. Automatica 35(11):1747–1767 23. Tee KP, Ge SS, Tay EH (2009) Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4):918–927 24. Ames AD, Grizzle JW, Tabuada P (2014) Control barrier function based quadratic programs with application to adaptive cruise control. In: 53rd IEEE conference on decision and control, Los Angeles, CA, USA, pp 6271–6278 25. Cortez WS, Oetomo D, Manzie C, Choong P (2021) Control barrier functions for mechanical systems: theory and application to robotic grasping. IEEE Trans Control Syst Technol 29(2):530–545

Precision Motion Control of Separate Meter-In and Separate Meter-Out …

55

26. Ames AD, Xu XR, Grizzle JW, Tabuada P (2017) Control barrier function based quadratic programs for safety critical systems. IEEE Trans Autom Control 62(8):3861–3876 27. Tee KP, Ge SS (2012) Control of state-constrained nonlinear systems using integral barrier Lyapunov functionals. In: 51st IEEE conference on decision and control, Maui, HI, USA, pp 3239–3244 28. Yan Z, Wang J (2012) Model predictive control of nonlinear systems with unmodeled dynamics based on feedforward and recurrent neural networks. IEEE Trans Ind Inf 8(4):746–756 29. Aguilera RP, Lezana P, Quevedo DE (2015) Switched model predictive control for improved transient and steady-state performance. IEEE Trans Ind Inf 11(4):968–977 30. Diehl M, Bock HG, Schloder JP (2005) A real-time iteration scheme for nonlinear optimization in optimal feedback control. SIAM J Control Optim 43(5):1714–1736 31. Quirynen R, Vukov M, Zanon M, Diehl M (2015) Autogenerating microsecond solvers for nonlinear MPC: a tutorial using ACADO integrators. Optimal Control Appl Methods 36(5):685–704 32. Yuan M, Chen Z, Yao B, Liu X (2021) Fast and accurate motion tracking of a linear motor system under kinematic and dynamic constraints: an integrated planning and control approach. IEEE Trans Control Syst Technol 29(2):804–811 33. Lu L, Yao B (2014) A performance oriented multi-loop constrained adaptive robust tracking control of one-degree-of-freedom mechanical systems: theory and experiments. Automatica 50(4):1143–1150 34. Lu L, Yao B (2014) Energy-saving adaptive robust control of a hydraulic manipulator using five cartridge valves with an accumulator. IEEE Trans Ind Electron 61(12):7046–7054

Dynamics Analysis of Simplified Axisymmetric Vectoring Exhaust Nozzle Mechanical System with Joint Clearance and Flexible Component Xiaoyu Wang, Haofeng Wang, Chunyang Xu, Zhong Luo, and Qingkai Han

Abstract In order to study the influence of joint clearance and flexible component on axisymmetric vectoring exhaust nozzle (AVEN) mechanism, the rigid flexible coupling dynamic equation of simplified AVEN mechanism is established based on the finite element method. Specifically, the Lankarani-Nikravesh contact force model is used to analyze the clearance impact force, and the flexible component is established by the absolute node coordinate method (ANCF). The dynamic equation is derived by Lagrange multiplier method, and solved by EULER IMPLICIT and MINRES combined. The influence of clearance and flexibility on the dynamics of AVEN mechanism are analyzed, which implies that the clearance and flexibility will cause nonlinearity of the AVEN dynamics. With the increase of joint clearance, the accuracy and stability of mechanism motion decrease and the reaction force of joint increases. Moreover, the flexibility would compensate the nonlinearity caused by the clearance. Keywords Clearance joint · Aircraft engine · ANCF · Nonlinear dynamics

X. Wang (B) · H. Wang · Z. Luo · Q. Han School of Mechanical Engineering, Northeastern University, Shenyang 110000, Liaoning, China e-mail: [email protected] H. Wang e-mail: [email protected] Z. Luo e-mail: [email protected] Q. Han e-mail: [email protected] C. Xu AECC Shenyang Engine Research Institute, Shenyang 110015, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_4

57

58

X. Wang et al.

1 Introduction The nonlinear problem is widely recognized in the dynamics of mechanical systems. With the increasing accuracy requirements of mechanical systems, such as aviation and robots, the nonlinear dynamic effect becomes quite apparent. Because each component has manufacturing error and assembly error, the clearance in the joint pair inevitably leads to collision, which has a negative impact on the system dynamics. On the other hand materials are deformable with certain elastic modulus under loading condition. Therefore, the traditional method of rigid bodies dynamics analysis is no longer suitable for high-precision mechanical requirements. The dynamic systems with flexible component and clearance should be investigated to characterize motion more precisely. In the past decades, many scholars have devoted themselves to the influence of clearance joint and flexible parts on the mechanism dynamics. Erkaya [1–7] studied the influence of clearance joint and flexible part on the dynamic system by different finite element methods, and compared with the experiments at different clearance values and driving speeds. Machado et al. [8–11] considered lubrication at the revolute joint, through a hydrodynamic model, and analyzed the influence of clearance and lubricant factors on the dynamic behavior. Hou et al. [12–16] analyzed the influence of flexible parts and clearances on the dynamic behavior of the spatial multibody mechanism, by Poincare Map and Bifurcation Diagram. Li et al. [17, 18] formulated flexible components based on ANCF, and established planar rigid-flexible coupling deployable solar array system with friction and wear at the joint. Generally the joint with clearance and flexible parts will have a negative impact on the dynamic behavior of mechanisms, and the research on coupling between flexibility and clearance is not sufficient. In this paper, AVEN mechanism is simplified to simulate the planar motion of divergent flaps considering rigid-flexible coupling and hinge clearance and its dynamic analysis is carried out.

2 Beam Model Based on ANCF Method The absolute node coordinate formula proposed by Shabana [19–21] is used to model the flexible components. Figure 1 shows two-node ANCF beam, where the position of an arbitrary point P can be expressed in the absolute coordinate system ⎡

⎤ ri ⎢ rix ⎥ ⎥ rp (x) = S(x)⎢ ⎣ rj ⎦

(1)

rjx r = S1 I2 S2 I2 S3 I2 S4 I2 qe 

(2)

Dynamics Analysis of Simplified Axisymmetric Vectoring Exhaust …

59

Fig. 1 ANCF beam element model

S1 = 1 − 2x2 + 3x3 

S2 = l x − 2x2 + 3x3 S3 = 3x2 − 2x3 

S4 = l −x2 + x3

(3)

where, x is the local coordinate in the axial direction; l indicates the length of the beam before deformation; S is the shape function of the beam element; I2 is a second-order identity matrix; qe is a generalized coordinate vector with eight node coordinates, which can be expressed

T qe = e1 e2 e3 e4 e5 e6 e7 e8

T = ri1 ri2 ∂r∂xi1 ∂r∂xi2 rj1 rj2 ∂r∂xj1 ∂r∂xj2

(4)

 T where, the vector rk1 rk2 (k = i, j) indicates the position coordinates of the ANCF k beam element node in the absolute coordinate system; ∂r = i, j) represents the ∂x (k absolute position slope of the beam element in the absolute coordinate system and the tangential direction of the axis. The mass matrix can be established as  M = ρ ST SdV. (5) V

60

X. Wang et al.

where v is the volume of the element; ρ is the density of material. The strain energy of the element can be expressed as 1 U= 2

l

 EAε2x + EIκ 2 dx

(6)

0

The virtual work of elasticity can be defined as  δWe =

[EAεx δεx + EIκδκ]dx

(7)

 1

 1 2 J − 1 = rTx rx − 1 2 2

(8)

L

εx =

|rx × rxx | |rx |3 ∂r = r˙ rx = ∂x κ=

rxx =

∂ 2r ∂x2

(9)

(10)

where E is the elastic modulus; A is the cross-sectional area; I is the second moment of area; εx is the axial Green-Lagrangian strain; κ is the bending curvature. Assuming that ξ = x/l, 0 m, m is the DOFs of the end-effector. In this paper, n = 7 and m = 6, focuses on the condition with one redundant, aiming at discussing the control effect

Hybrid Joint-space Control Strategies Analysis for One-redundant …

.

Fig. 1 Geometry of the ith constraint of a CDPR

99

. .

Ai

li

z

x

Bi

O

y End-effector

bi ai

r,R Z O X

Y

under different control strategies. As shown in Fig. 1, Ai and Bi represent the fixed points and the distal anchor points on the platforms, respectively. O and O' represent the global coordinate and local coordinate system of the end effector, respectively. The vectors ai represents the fixed points Ai with respect to the global coordinate frame O, and the vectors bi represent the position vectors of Bi in the local coordinate system of O' . The vector r is the Cartesian position of the moving platform, and the rotation matrix R represents the orientation of the platform frame O' with respect to the global frame O. The kinematic relationships of the CDPR are given below: l i = a i − r − R · bi

(1)

In the above, li is a cable vector, whose unit vector is: ui =

li a i − r − R · bi =√ . ∥li ∥ (a i − r − R · bi )T · (a i − r − R · bi )

(2)

3 Cable Adjustable Force Definition for Cable-suspended Parallel Robots The static equilibrium of the end-effector can be established using the following equation: WT = F

(3)

where W is the wrench matrix, T represents the cable tension matrix, and F contains forces and moments of the end-effector, which is also include the end-effector gravity.

100

Z. Qin et al.



u2 ... un u1 W= Rb1 × u1 Rb2 × u2 . . . Rbn × un

 (4)

There are infinite number of cable tension solutions T, satisfy the static equilibrium because of the redundant characters. However, the suspended configuration imposes a predictable minimum or maximum limit on the tension in all cables. The maximum cable tension range is defined as the cable adjustable force (CAF). For the cable suspended robots with one redundant, the CAF for the ith cable can be expressed as follows: 0 ≤ Timin ≤ Ti ≤ Timax

(5)

CAFi ≤ Timax − Timin

(6)

where Timin and Timax represents the minimum and maximum cable tension of the ith cable. Notably, the aforementioned two values are not a consistent value, but changes with different locations of the end-effector. We define the pose X* in the area of the robot frame is wrench-feasible if one set of cable tensions greater than Ti-min can be found to satisfy Eq. (3). f ind Ti ≥ Ti- min subject to W(X∗ ) · T = F

(7)

The CAF is effective in the wrench-feasible workspace. An algorithm for calculating the CAF of a target cable is explained as follows 1. The target cable, the pose of the end-effector, the external force F, and the minimum cable tension Ti-min should be predefined. 2. Choose six combinations including the target cable vector to form the 6 × 6 matrix G. 3. Ensure that rank (G) = 6, else skip step 4 and 5. 4. Calculated and record the target cable tension. Suppose the identification of the target cable is j, the target cable tension can be expressed as: T j = (G −1 F) j We can obtain the maximum cable tension of the jth cable at this pose, Tmax = max (T1j , …, Tij ). 5. Choose six combinations except the target cable vector to form the 6 × 6 matrix G1 . If min(G 1 −1 F) > Tmin (G1 -1 F)>Tmin , the minimum cable tension of the jth cable at this pose is the predefined value Tmin , else the minimum cable tension is Tmin = min (T1 j , …, Ti j ). 6. Go to step 3 until all the 6 combinations are examined, if all combinations are not satisfied, the robot reaches a singular configuration, and we set CAF = 0.

Hybrid Joint-space Control Strategies Analysis for One-redundant …

101

7. Output the CAF of the target cable. Notably, CAF ≤ 0, which means that the location currently is not in the wrench-feasible workspace. CAF = Tmax − Tmin Considering the robust characters for the force control, the target value for the force control can be defined as: T∗j =

1 CAF + Tmin 2

(8)

The flow chart of the algorithm is illustrated in Fig. 2. Other cable tensions can be calculated Tres if the cable tension is determined and the rank of the matrix G1 = 6. T  F1 = u j Rb j × u j · T∗j

(9)

Tres = G −1 1 · (F − F1 )

(10)

The cable corresponding the largest CAF, shows smallest force distribution sensitive to other cables for one-redundant CDPRs. As illustrated in Fig. 3, the CAF of cable j is larger than cable i, the cable tension error ΔTj in cable j, is also greater than the corresponding error ΔTi in cable i, caused by the ΔTj . Based on the linear nature of one-redundant CDPRs, we hold the relationships j-CAF/i-CAF=ΔT j /ΔT i .

4 Control Strategies Simulation The inverse kinematic are used to calculated the cable length for the length control, and the force calculated above (Eqs. (8)–(10)) can be selected as an optimal solution for the force control. The aforementioned two control strategies are considered in this study, aiming at comparing the performance with the hybrid joint-space control strategy, where a cable is force control while other cables perform the length control. The redundancy of the force control results will be discussed via the virtual simulation analysis framework using the ABAQUS/Explicit software, where the connection of SLIPRING can be used to model seat belts, pulley systems, and taut cable systems. The end-effector was modeled as a rigid body because the deformation of the end-effector can be ignored. The amplitudes of the simulated cable force Ti + ΔTi and cable length l j + Δl j were input via the amplitude tabular data in ABAQUS, as is shown in Fig. 4. Notably, the cable length and cable force cannot be controlled precisely due to the lagging interactions among single controllers and model uncertainties. Since the poses of the robots is coupled with the cable tensions, we add continuous error signals

102

Fig. 2 Flow chart of the proposed CAF calculation algorithm for one redundant

Z. Qin et al.

Hybrid Joint-space Control Strategies Analysis for One-redundant …

103

Cable j Cable 1

...

1 - CAF

j - Tmax

Cable i Ti

i - CAF

Tj

...

j - Tmin

j - CAF

Cable n

n - CAF

Fig. 3 The CAF information

Fig. 4 The simulation verification process, a the length control, b the force control, c the hybrid joint-space control

104

Z. Qin et al.

Δl i and ΔTi in the simulation process, to analysis the cable tensions information T and position errors ∥dp∥. ∥dp∥ =

√

(Xd − X)T · (Xd − X)

(11)

where Xd and X represent the desired position and the simulation location, respectively. A 7-cable 6-DOF cable suspended robot, controlled by the above three strategies is adapted in this study. The gravity of the end-effector is 50 N, and the layout of the robot is shown in Fig. 5. The position parameters of the cable robot are shown in Table 1. A liner path with a 5th degree polynomial motion is considered. It goes from the initial position p1 = [0 0 0] to the final position p2 = [0.25 0.25 0], during the time t end = 60 s. The trajectory of the end-effector can be defined as follows (Fig. 6): p(t) = p1 + α(t)(p2 − p1 ); t ∈ [0 tend ]

(12)

Fig. 5 The layout of the cable suspended robots; the numbers represent cable identifications

Table 1 Position parameters of the cable robots m The positions of fixed points in the inertial frame 1

A1

The positions of distal points in the inertial frame

X

Y

Z

1.992

−0.174

2

x B1

0.1075

y

z

−0.1687

0

2

A2

1.992

0.174

2

B2

0.1075

0.1687

0

3

A3

−0.845

1.813

2

B3

0.0923

−0.1774

0

4

A4

−1.147

1.638

2

B4

−0.1998

0.0087

0

5

A5

−1.147

−1.638

2

B5

−0.1998

−0.0087

0

6

A6

−0.845

−1.813

2

B6

0.0923

−0.1774

0

7

A7

−2

0

2

B7

−0.1998

−0.0087

0

Hybrid Joint-space Control Strategies Analysis for One-redundant …

105

0.25

Fig. 6 Desired position profiles of the end-effector

Position (m)

0.2

X Y

0.15

Z

0.1 0.05 0

α(t) =

6 5 tend

0

t5 −

10

20

30 Time (s)

15 4 10 t + 3 t 3. 4 tend tend

40

50

60

(13)

4.1 Cable Length Versus Force Control Strategies Firstly, the gravity was increased continuously from 0 to G in 10 s to introduce cable tensions continuously (this setting can be considered an initialization for the simulation analysis). The maximum cable length and cable force control errors, Δli and ΔTi are ± 1 mm and ± 0.1 N, as shown in Figs. 7a and 8a, respectively. The cable tensions and the pose errors ∥dp∥ of the seven cable under the cable length control strategy are shown in Fig. 7b, c. Under this control strategy, the maximum position error is about 1.6 mm, with a high control accuracy. However, the cables are not always kept tensioned, which may have negative influence on the oscillation suppression. The cable tensions and the position errors ∥dp∥ of the seven cable under the cable force control strategy are shown in Fig. 8b. The position errors ∥dp∥ are very sensitive to the cable tension errors, despite the fact that the preset maximum cable tension error value, ± 0.1 N, is relatively small compare the gravity of the end-effector, 50 N. These drawbacks motive us to discuss the hybrid joint-space control strategy.

4.2 Hybrid Joint-space Control Strategy How to choose the cable for the force control one of the core problems for the hybrid joint-space control strategy. This study investigates the relations between the

106 3 Desired cable length

2.95

Cable length (m)

2.9 2.85

Control errors

2.8 2.75 2.7 2.65 Cable 3 Cable 4

Cable 1 Cable 2

2.6 2.55

0

10

20

30 Time (s) (a)

Cable 5 Cable 6

40

Cable 7

50

60

20 18

Cable tension (N)

Cable 3 Cable 4

Cable 1 Cable 2

16

Cable 5 Cable 6

Cable 7

14 12 10 8 6 4 2 0

0

10

20

30 40 Time (s) (b)

50

60

70

-3

1.8

10

1.6 1.4 1.2

||dp|| (m)

Fig. 7 The simulation results of the length control, a the cable tensions, b the cable forces, c the norm of position error ∥dp∥ along the trajectory (dp is the distance between the desired position and the simulation location)

Z. Qin et al.

1 0.8 0.6 0.4 0.2 0

0

10

20

30 Time (s) (c)

40

50

60

Hybrid Joint-space Control Strategies Analysis for One-redundant … 18 Desired cable force

16

Control errors

14 Cable tension (N)

Fig. 8 The simulation results of the force control, a the cable tensions, b the norm of position error ∥dp∥ along the trajectory (dp is the distance between the desired position and the simulation location)

107

12 10 8 6 4 Cable 1 Cable 2

2 0

0

10

20

Cable 3 Cable 4

Cable 5 Cable 6

30 40 Time (s) (a)

50

Cable 7

60

70

0.08 0.07

||d p|| (m)

0.06 0.05 0.04 0.03 0.02 0.01 0

0

10

20

30 Time (s) (b)

40

50

60

CAF and the position errors, considering the cable length and cable force control errors simultaneously. Considering the inertia forces and moments of the desired trajectory, the CAF of the seven cables is shown in Fig. 9, the minimum CAF values of the seven cables along the trajectory are also listed in Table 2. The CAF levels of the seven cable are different, with the cable 7 corresponding to the highest level. Figure 10a–g illustrate the cable tensions under the hybrid joint-space control strategy, with the maximum cable length and cable force control errors ± 1 mm and ± 1 N, respectively. Notably, all cable tensions are plotted after a Butterworth filter.

108

Z. Qin et al.

20

Fig. 9 The CAF along the trajectory

18 16

CAF(N)

14 Cable 3 Cable 4

Cable 1 Cable 2

12

Cable 5 Cable 6

Cable 7

10 8 6 4 2 0

0

10

20

30 40 Time (s)

50

60

Table 2 Minimum CAF values along the trajectory N Cable

1

2

3

4

5

6

7

CAF

1.95

1.73

2.23

6.00

7.19

1.99

13.49

4.3 Discussion of the Results We use a sensitive index R to evaluate the force errors in the target cable, affect the overall distribution of tensions in other cables. The R value of the force-control cable is 1, a value R > 1 (or R < 1) means that the tension in cable i changes more (or less) than the tension in the force-control cable. | | max| yi − yi∗ | , i = 1, 2, . . . , 7, 1 ≤ t ≤ 7 (14) R= max|yt − yt∗ | where yi and y* represents desired cable force and the simulation results, respectively. And yt and y*t represents the desired cable force and the simulation results for the force control. Figure 11 shows the cable tension errors by choosing different cables for the force control in the hybrid joint-space control strategy. Cable 7 shows the highest sensitive index, which means that the CAF is positively correlated with the sensitive index R. In addition, the maximum of the norm of position error ∥dp∥ along the trajectory for the seven cables, is illustrated in Fig. 12, showing a negative (or nearly negative) correlation related to the CAF. It is suggested that the cable with largest CAF values should be chosen for the force control, while other cables perform the length control. Notably, the scaling relationship of R with different cables for the force control were

Hybrid Joint-space Control Strategies Analysis for One-redundant …

20 Cable 1 Cable 2

18

Cable 3 Cable 4

Cable 5 Cable 6

Cable 7

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30 40 Time (s) (a)

50

60

70

20 Cable 3 Cable 4

Cable 1 Cable 2

18

Cable 5 Cable 6

Cable 7

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30 40 Time (s) (b)

50

60

70

20 Cable 3 Cable 4

Cable 1 Cable 2

18

Cable 5 Cable 6

Cable 7

16

Cable tension (N)

Fig. 10 The cable tensions of the hybrid joint-space control simulation results, a–g represent choosing cable 1–7 for the force control, respectively

109

14 12 10 8 6 4 2 0

0

10

20

30

40

Time (s) (c)

50

60

70

110

Z. Qin et al. 20

Fig. 10 (continued)

Cable 3 Cable 4

Cable 1 Cable 2

18

Cable 7

Cable 5 Cable 6

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30

40

50

60

70

Time (s) (d) 20 Cable 1 Cable 2

18

Cable 3 Cable 4

Cable 5 Cable 6

Cable 7

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30

40

50

60

70

Time (s) (e) 20 Cable 1 Cable 2

18

Cable 3 Cable 4

Cable 5 Cable 6

Cable 7

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30

40

Time (s) (f)

50

60

70

Hybrid Joint-space Control Strategies Analysis for One-redundant …

111

20

Fig. 10 (continued)

Cable 1 Cable 2

18

Cable 3 Cable 4

Cable 5 Cable 6

Cable 7

Cable tension (N)

16 14 12 10 8 6 4 2 0

0

10

20

30

40

50

60

70

Time (s) (g)

not exactly the same (Fig. 12 (a)-(g)), which were due to the influence of cable length errors in the simulation. Notably, the interfering force for the force control is not limited to amplitude ± 1 N in the hybrid joint-space control strategy. Take cable 4 and 7 as an example, Fig. 13 illustrate the maximum of the norm of position error ∥dp∥ with different amplitude of interfering force, when the gravity of the end-effector is 50 N. The feed-forward force-controlled force shows a robust character, despite the fact that the length control errors with the range of ± 1 mm are all existence within the simulation process. The position error increase dramatically when the interfering force gradually increases to the minimum of CAF along the trajectory, which is due to the fact that the force equilibrium relationship may not be satisfied at some time, under the effective of cable length and force errors. However, unexpected cable force fluctuations may occur in the length-control cable, if we choose a less sensitive cable for the force control. For example, up to 11 times error amplification can be found if we choose cable 1 for the force control, but all the length-control cable force errors are reduced if cable 7 are chosen for the force-control, in the simulation case. This phenomenon should be considered seriously along with the configuration and the trajectory of the robot, when the elasticity of the cable cannot be ignored, where the elastic pre-compensation technology [18] can be used in the feed-forward hybrid joint-space control strategy, to further improve the length control accuracy. At this time, the CAF can be regard as a perform index for configuration optimization.

112

Z. Qin et al.

12

Fig. 11 The cable tension errors by choosing different cables for the force control in the hybrid joint-space control strategy

10

R

8 6 4 2

R

0

9 8 7 6 5 4 3 2 1 0

1

1

2

3 4 5 6 Cable identifications (a)

7

2

3 4 5 6 Cable identifications (b)

7

3 4 5 6 Cable identifications (c)

7

12 10

R

8 6 4 2 0 1

2

Hybrid Joint-space Control Strategies Analysis for One-redundant …

113

3

Fig. 11 (continued)

2.5

R

2 1.5 1 0.5 0

1

2

3 4 5 6 Cable identifications (d)

7

1

2

3 4 5 6 Cable identifications (e)

7

3 4 5 6 Cable identifications (f)

7

2.5 2

R

1.5 1 0.5 0

12 10

R

8 6 4 2 0

1

2

114

Z. Qin et al.

1

Fig. 11 (continued)

0.8

R

0.6 0.4 0.2 0

1

2 3 4 5 Cable identifications (g)

6

7

6

7

-3

Fig. 12 Comparing the maximum of the norm of position error

×10

max(||dp||)(m)

8 6 4 2 0

Fig. 13 Plot of the maximum of the norm of position error ∥dp∥ with different amplitude of interfering force when G = 50 N

1

2

3

4 5 Time (s)

10-3

11 10 9

max(||dp||)(m)

8

Cable 4: minimum CAF = 6 N

7 6 5 4 3

Cable 7: minimum CAF = 13.49 N

2 1

0

2

4

6 8 10 Interfering force (N)

12

14

Hybrid Joint-space Control Strategies Analysis for One-redundant …

115

5 Conclusions This work presented a method to find the target cable for the force control in the hybrid joint-space control. Compare with the cable length control and cable force control, the hybrid joint-space control has the ability to ensure the poses with a small error, while maintain all cables tensioned at the same time. We also investigate the relationship between the CAF and the overall distribution of tensions in other cables for one-redundant CDPRs. Conclude that the cable corresponding to maximum CAF, should be chosen for the force control, resulting in minimum pose errors at the same time (The cable elastic elongation has the least effect on pose of the end-effector). This study can be valuable in the application of hybrid joint-space controller design for the cable suspended robots with one redundant drives. In the future, we will investage the CAF calculation methods and the relationships between the CAFs and pose errors of end-effector for other redundancies. Acknowledgements Supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51521003), and the 111 Project (B07018).

References 1. Alp AB, Agrawal SK (2002) Cable suspended robots: design, planning and control. In: Proceedings 2002 IEEE international conference on robotics and automation (Cat. No. 02CH37292), vol 4, pp 4275–4280 2. Jung J (2020) Workspace and stiffness analysis of 3D printing cable-driven parallel robot with a retractable beam-type end-effector. Robotics 9(3):65 3. Barbazza L, Oscari F, Minto S, Rosati G (2017) Trajectory planning of a suspended cable driven parallel robot with reconfigurable end effector. Robot Comput Integr Manuf 48:1–11 4. Wu Y, Cheng HH, Fingrut A, Crolla K, Yam Y, Lau D (2018) CU-brick cable-driven robot for automated construction of complex brick structures: from simulation to hardware realization, pp 166–173 5. Qin Z, Liu Z, Liu Y, Gao H, Sun C, Sun G (2022) Workspace analysis and optimal design of dual cable-suspended robots for construction. Mech Mach Theory 171:104763 6. Melenbrink N, Werfel J, Menges A (2020) On-site autonomous construction robots: towards unsupervised building. Autom Constr 119:103312 7. Rodriguez-Barroso A, Saltaren R (2020) Tension planner for cable-driven suspended robots with unbounded upper cable tension and two degrees of redundancy. Mech Mach Theory 144:103675 8. Pott A (2014) An improved force distribution algorithm for over-constrained cable-driven parallel robots. In Proceedings of the computational kinematics. Springer, Dordrecht, pp 139– 146 9. Gouttefarde M, Lamaury J, Reichert C, Bruckmann T (2015) A versatile tension distribution algorithm for n-DOF parallel robots driven by n+2 cables. IEEE Trans Rob 31(6):1444–1457 10. Mikelsons L, Bruckmann T, Hiller M, Schramm D (2008) A real-time capable force calculation algorithm for redundant tendon-based parallel manipulators. In: Proceedings of the IEEE international conference on robotics and automation, ICRA 2008. IEEE, pp 3869–3874 11. Gao H, Sun G, Liu Z, Sun C, Li N, Ding L (2022) Tension distribution algorithm based on graphics with high computational efficiency and robust optimization for two-redundant cable-driven parallel robots. Mech Mach Theory 172:104739

116

Z. Qin et al.

12. Fabritius M, Martin C, Gomez GR, Kraus W, Pott A (2021) A practical force correction method for over-constrained cable-driven parallel robots. In: International conference on cable-driven parallel robots. Springer, Cham, pp 117–128 13. Zhao T, Zi B, Qian S, Zhao J (2020) Algebraic method-based point-to-point trajectory planning of an under-constrained cable-suspended parallel robot with variable angle and height cable mast. Chin J Mech Eng 33(1):1–18 14. Yao R, Tang X, Wang J, Huang P (2010) Dimensional optimization design of the four-cabledriven parallel manipulator in fast. IEEE/ASME Trans Mechatron 15(6):932–941 15. Khalilpour SA, Khorrambakht R, Taghirad HD, Philippe C (2019) Robust cascade control of a deployable cable-driven robot. Mech Syst Signal Process 127:513–530 16. Bruckmann T, Mikelsons L, Hiller M, Schramm D (2007) A new force calculation algorithm for tendon-based parallel manipulators. In: 2007 IEEE/ASME international conference on advanced intelligent mechatronics. IEEE, pp 1–6 17. Mattioni V, Idà E, Carricato M (2021) Force-distribution sensitivity to cable-tension errors: a preliminary investigation. In: International conference on cable-driven parallel robots. Springer, Cham, pp 129–141 18. Baklouti S, Caro S, Courteille E (2019) Vibration reduction of cable-driven parallel robots through elasto-dynamic model-based control. Mech Mach Theory 139:329–345

Research on Coupling Analysis of Factors Influencing Backlash in Gear Train Based on Probability Method Chunyang Shi, Yushu Bian, and Zhihui Gao

Abstract In this study, the gear backlash of the common 3K planetary gear train is studied in depth. After the coupling analysis of various factors affecting the gear backlash of the gear, the calculation formula of the gear backlash under various factors is derived, and the probability statistical method is innovatively introduced into the calculation to analyze the influencing factors of gear backlash in gear train closer to the design and manufacturing process. Finally, the numerical analysis example reveals the importance and influence of some parameters that may make the gear backlash larger. The research has a certain guiding significance for the parameter pre-optimization in the design and manufacturing of the gear. Keywords Gear backlash · Numerical analysis · Factors influencing · Statistical method

1 Introduction The accuracy and torque are mainly included in the reliability of gear transmission, and the transmission error and gear backlash reliability are mainly included in the accuracy reliability. An appropriate amount of gear backlash is necessary for the whole transmission system. However, the gear backlash hysteresis is generated by the excessive gear backlash. It is that the phase angle of the output shaft is lagged, the stability margin of the system is reduced and the dynamic quality of the system is affected by the gear backlash. Therefore, the return error should be minimized to improve the stability of the system. C. Shi · Y. Bian (B) · Z. Gao School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China e-mail: [email protected] C. Shi e-mail: [email protected] Z. Gao e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_7

117

118

C. Shi et al.

There is some theoretical and technical research about the gear backlash before. Chang [1] analyzed the main factors affecting the idle stroke error and calculated the gear backlash of the filtering gear reducer with the method of probability statistics. Zhu [2] created the analytical model and derived the formulas of backlash calculation for the new planetary gearing by using probabilistic theory. Zhang [3] carried out the quantitative analysis of the two-input planetary gear backlash by considering multiple influencing factors and gave the calculation expression of its gear backlash. Zhu [4] developed a mathematical model which described the state of multi-tooth contacts and the load distribution characteristics. Seidl [5, 6] used artificial neural networks to identify and compensate for hysteresis caused by gear backlash in precision position-controlled mechanisms. Dagalakis [7] developed a technique for the precise adjustment of gear backlash of the joints of an industrial robot. Stein [8] proposes a technique for the estimation of clearance in mechanical systems under dynamic conditions with specific application to the estimation of backlash in gear systems of servomechanisms based on a momentum transfer analysis. Chen [9] built the assembly models of harmonic drive with circular-arc tooth profile under different wave generators and obtained gear backlash distribution of engaged teeth profile. In summary, this paper makes a comprehensive coupling analysis of various factors affecting the gear backlash and forms a coupling model compared with the above studies, and introduces the probability method to optimize the parameters in the process of design and manufacturing.

2 Research on Reliable Coupling Model of Backlash with Probability Method 2.1 Influencing Factors of Gear Backlash and Coupling Relationship Between Layers There is a three-layer model of the coupling relationship of the gear backlash influencing factors established to facilitate the research and analysis, as shown in Fig. 1. The first layer is the final output of gear backlash which can be detected. The second layer is the factors that directly affect the gear backlash. They are all specific clearance quantities, including the inherent error, the center distance deviation, the axis parallelism deviation, the eccentricity, and the load-induced deformation error. The third layer is the various factors that indirectly influence the transmission error by affecting the second level, including the temperature, the friction, and the load. The parts will be swelled or shrank if the temperature is changed, resulting in the change of the gear inherent position error, the device error, the force-induced deformation error, and other clearance quantities in the second layer, and affecting the gear backlash indirectly. There are various forms of deformation induced by

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

Temperature

Friction

Load

119 The Third Layer

Forcedeformation coupling model Axis Parallelism Error

Gear-itself Error Cross-bar Distance Error

Radial Total Deviation

Center Distance Deviation

Deviation Deviation in Axis Perpendicular Plane to Axis Plane

Axis Parallelism Error Eccentricity

The Shaft Shaft Bearing Gear Torsional Bending Clearance and Tooth Second Deformation Deformation Variation Deformation Layer Error Error Error Error

Return Difference

The First Layer

Fig. 1 Influencing factors of the gear backlash and the coupling relationship between the layers

force, for example, the tension and compression deformation, the bending deformation, the torsion deformation, etc. Meanwhile, the position, the contact area, and the contact area shape of the parts are changed, resulting in the change of load, and a very complex force–deformation coupling relationship is established. This dynamic relationship needs to be characterized by the multi-body dynamics, the nonlinear contact mechanics, and the impact dynamics to form a force–deformation coupling model. The shaft torsional deformation error, the shaft bending deformation error, the bearing clearance error, and the gear deformation error are all affected by the load through the force–deformation coupling model in the second layer, and the gear backlash is affected. This layer is focused on the influence of the shaft torsional deformation error, the shaft bending deformation error, the bearing clearance error, and the gear deformation error on the gear backlash.

2.2 Analysis of Third Layer Influencing Factors of the Gear Backlash There are three factors influencing the gear backlash in the third layer, including the temperature, the friction, and the load. The temperature is inevitable which influences the deformation during the manufacturing process, resulting in the generated of the inherent error. The inherent error is expressed by the cross-bar distance error and the radial integrated total deviation at the standard temperature (i.e., 20 °C). There are three deformation errors, including the cross-bar distance error, the radial integrated total deviation, and the deformation of the gear at the standard temperature (i.e., 20 °C). Friction is one of the inevitable factors in the transmission process. The forced direction of the gear is changed by the exerted friction, and the pressure angle is changed from the standard pressure angle αn to α.

120

C. Shi et al.

In the action of the torque, the radial force, and the circumferential force, there are four types of deformation-induced, including the shaft torsional deformation, the shaft bending deformation, the bearing clearance change, and the tooth deformation. The influence of the load is mainly discussed in the second layer. In summary, the clearance of the gear train is affected by the influence of the load and the temperature.

2.3 Analysis of Second Layer Influencing Factors of the Gear Backlash 2.3.1

Influence Formula of the Single Factor in the Second Layer on the Final Gear Backlash

According to the basic concept, the idea of analyzing the gear backlash is to convert it into circumferential backlash for calculation. The transmission principle of the 3K planetary gear train is shown in Fig. 2. It can be seen that there are four pairs in the 3K planetary gear train and four angular value backlashes to generate the final gear backlash. The angular value backlash is converted into the circumferential backlash value of the output internal gear b4 for analysis. To facilitate the study, the gear parameters of the 3K planetary gear train are summarized in Table 1 (αn = 20◦ ). The final formula is B=

jϕ12 jϕmn + jϕ23 + i 13 i 23

(1)

where jϕ12 , jϕ23 are the angular backlash of a4 to c4 , c4 to b4 , respectively. There are two categories of inherent error that existed in each gear, namely the cross-bar distance error Md , and the radial integrated total deviation Fi'' . Fig. 2 Principle of 3K planetary gear train transmission

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

121

Table 1 The parameters of the 3K planetary gear train gear No.

Gear

Module

1

Input gear a4

2

2

Middle gear c4 in triple

3

Output internal gear b4

4

Tooth number

Width

Transmission ratio

90

24

2

25

26

2

140

24

i12 = −0.5444 i23 = 140.0 i13 = −76.2222

Left gear d41 in triple

2

24

18

5

Fixed internal gear e41

2

139

16

6

Right gear d42 in triple gears

2

24

15

7

Fixed internal gear e42

2

139

14

The formula of the circumferential backlash jt Md generated Md is jt Md = −Md tan αn  −Md tan αn , (caculating the external gear) = +Md tan αn , (caculating the internal gear)

(2)

The value of Md is usually negative in the external gear so that there is a certain thickness reduction on the gear. It eventually leads to the increase of the circumferential backlash value. Therefore, a minus is added to the formula to make the final result positive. The value of Md is usually positive, so it is a plus in front of the formula, and the circumferential backlash is increased. The formula of Md of the all gears influence the final gear backlash is, B1 = (B Md1 + B Md2 ) + (B Md3 + B Md4 ) + (B Mdm + B Mdn )

2.3.2

(3)

Radial Comprehensive Deviation

The formulas of the circumferential backlash jt Fi '' generated by Fi'' are ⎧ '' ⎪ ⎨ jt Fi '' = ±Fi sin ϕ tan αn = ±Fi'' sin ϕ tan αn , (1, 2, 3, 4, 6) ⎪ ⎩ jt Fi '' = Fi'' tan αn , (5, 7)

(4)

where the signs + are applicable to the external meshing pair, the signs − are applicable to the internal meshing pair.

122

C. Shi et al.

The formula of Fi'' of all the gears influencing the final gear backlash B Fi'' is B2 = (B Fi1'' + B Fi2'' ) + (B Fi3'' + B Fi4'' ) + (B Fim '' + B Fin '' )

(5)

The formulas of the circumferential backlash jt Fi '' generated by Fi'' are ⎧ '' ⎪ ⎨ jt Fi '' = ±Fi sin ϕ tan αn = ±Fi'' sin ϕ tan αn , (1, 2, 3, 4, 6) ⎪ ⎩ jt Fi '' = Fi'' tan αn , (5, 7)

2.3.3

(6)

Other Dimensional Errors

Other dimensional errors include center distance deviation, axis parallelism deviation, and eccentricity. Center distance deviation Δ f a is the difference between the actual center distance and the nominal center distance caused by various factors such as component errors and device errors of the gear pair. The calculation formula of the circumferential backlash jt Md caused by center distance deviation Δ f a is jtΔ f a = ±2Δ f a tan αn

(7)

The influence B3 of center distance deviation Δ f a of all gear pairs on the final gear backlash is calculated as follows, B3 = BΔ f a12 + BΔ f a23 + BΔ f amn

(8)

Axis parallelism deviation is caused by the difference between the real axis and ideal axis, and the non-concentricity of the bearing holes at both ends of the same shaft is caused by the device error. It can be divided into axis in-plane deviation f∑δ and vertical in-plane deviation f∑β . The circumferential backlash caused by the deviation in the axis plane is calculated as follows, jt∑δ = − f ∑δ tan αn

(9)

The influence of in-plane deviation of all gear pairs on the final gear backlash is calculated as follows, B4 = B∑δ12 + B∑δ23 + B∑δmn

(10)

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

123

The calculation formula of the circumferential backlash jt∑β caused by deviation in vertical in-plane f∑β is jt∑β = − f ∑β

(11)

The influence B5 of the vertical in-plane deviation f∑β of all gear pairs on the final gear backlash is calculated as follows, B5 = B∑β12 + B∑β23 + B∑βmn

(12)

The eccentricity error e occurs after the gear train is installed. The main feature is that the side gap generated by eccentricity contains a changing phase angle. The calculation formula of the circumferential backlash caused by eccentric error e is jte = ±2e sin ϕ tan αn

(13)

The influence of the eccentricity error e of all gear pairs on the final return error is calculated as follows, B6 = Be12 + Be23 + Bemn

2.3.4

(14)

Force-Induced Errors

Shaft torsional deformation, shaft bending deformation, bearing clearance, and tooth deformation is caused by the action of torque and circumferential force. Torque T is transmitted in the transmission process, and the torsional deformation is caused by the action of torque T which makes the gear produce a line value error ΔST on the indexing circle. The gear backlash is affected by ΔST directly if the input speed direction is changed. The circumferential backlash generated by ΔST is calculated as follows, jt T = ΔST

(15)

The influence of all gears’ ΔST on the final gear backlash is calculated as follows, B7 = (BT 1 + BT 2 ) + BT m

(16)

The gear is exerted by the effect of the radial force and the circumferential force during transmission, and the shaft is bent deformed along the gear meshing line, resulting in the axis being deviated from the theoretical axis by an offset of ΔS F . It is the shaft torsional deformation error if ΔS F is converted into circumferential backlash.

124

C. Shi et al.

The calculation formula of the circumferential backlash jt F caused by ΔS F is jt F =

ΔS F cos αn

(17)

The influence B8 of all gears’ ΔS F on the final gear backlash is calculated as follows, B8 = (B F1 + B F2 ) + B Fm

(18)

Due to the gear rotation and the influence of the different kinds of force during the system working, the clearance is generated on the bearing, and so is the backlash generated on the gear pair. Since the movement is along the direction of the meshing line, it can be decomposed into two components, namely the radial offset Δrr and the tangential offset Δr t . The radial offset and tangential offset are converted to the circumferential clearance, namely the bearing clearance error. According to the relationship between radial backlash and circumferential backlash, the backlash caused by radial offset and tangential offset is Δr sin αn 2

(19)

Δr t = Δr cos αn

(20)

Δrr =

where the line value error of the gear indexing circle is affected by the circumferential backlash directly, and the influence of radial backlash on the line value error of the indexing circle is related to the pressure angle. The circumferential backlash jt B caused by Δr is the following, jt B = Δr t +

Δr sin αn 3Δr Δrr cos αn = Δr cos αn + = tan αn 2 tan αn 2

(21)

The influence B9 of all gears’ Δr on the final gear backlash is calculated as follows, B9 = (B B1 + B B2 ) + B Bm

(22)

The tooth deformation Δ f , a small periodic error, is generated along the meshing line due to the influence of circumferential force in the meshing process. The circumferential backlash jt f caused by Δ f is as the following, jt f =

Δ f sin nθ cos αn

(23)

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

125

The influence B10 of all gears’ Δ f on the final gear backlash is calculated as follows, B10 = (B f 1 + B f 2 ) + (B f 3 + B f 4 ) + (B f m + B f n )

(24)

3 Study on Coupling Rule Under Typical Working Conditions Since the error of a batch of parts is usually distributed according to a certain statistical law within a given tolerance range. Therefore, it is a more reasonable calculation method to establish the statistical comprehensive formula of 3K gear train gear backlash with the method of probability, and a more practical result can be obtained.

3.1 Digital Characteristics of Gear Backlash Single Factor Based on Probability Method Md is belonged to dimensional error and is obeyed normal distribution. The upper and lower deviations and tolerances are Mds , Mdi , and Mdm respectively. The digital characteristics are ⎧ 1 ⎪ ⎨ E(Md ) = E Md = (Mds + Mdi ) 2 (25) ⎪ ⎩ D(M ) = D = 1 (M − M )2 ds di d Md 36 The corresponding digital characteristics of backlash are 

E( jt Md ) = E jt Md = −E Md tan α D( jt Md ) = D jt Md = D Md (tan α)2

(26)

The influence of radial integrated total deviation on the backlash is related to the phase angle, while Fi'' sin ϕ is obeyed the standard normal distribution, and the numerical characteristics are ⎧ '' ⎨ E(Fi sin ϕ) = E Fi'' sin ϕ = 0 (27) ⎩ D(Fi'' sin ϕ) = D Fi'' sin ϕ = 1 Fi''2 36

126

C. Shi et al.

The corresponding digital characteristics of backlash are 

E( jt Fi '' ) = E jt Fi '' = 0 D( jt Fi '' ) = D jt Fi '' = D Fi '' sin ϕ (tan α)2

(28)

In particular, the effect of radial integrated total deviation of the fixed internal gear ring on backlash is independent of phase angle. F '' is obeyed Rayleigh distribution, and the numerical characteristics are ⎧ ⎨ D(F '' ) = D Fi '' = ( 1 Fi '' )2 = 0.047673F ''2 i i 4.58√ (29) ⎩ '' E(Fi ) = E Fi '' = 1.913 D Fi '' = 0.417686Fi''  E(jt Fi '' ) = E jt Fi '' = tan α E Fi '' (30) D(jt Fi '' ) = Djt Fi '' = D Fi '' (tan α)2 The center distance deviation Δ f a is obeyed the normal distribution, and the symmetric deviation is ±Δ f a . The numerical characteristics are ⎧ ⎨ E(Δ f a ) = E Δ f a = 0 ⎩ D(Δ f a ) = DΔ f a = 1 Δ f a2 9

(31)

The corresponding digital characteristics of backlash are 

E( jtΔ f a ) = E jtΔ f a = 0 D( jtΔ f a ) = D jtΔ f a = 4DΔ f a (tan α)2

(32)

The distribution of axis in-plane deviation f ∑δ and vertical in-plane deviation f ∑β is the same, which is the normal distribution. The digital characteristic distribution is the following, ⎧ 1 ⎪ ⎨ E(f∑δ ) = E f ∑δ = f∑δ 2 1 2 ⎪ ⎩ D(f ) = D f ∑δ f ∑δ = 36 ∑δ ⎧ 1 ⎪ ⎨ E(f∑β ) = E f ∑β = f∑β 2 1 2 ⎪ ⎩ D(f ) = D f ∑β f ∑β = 36 ∑β The corresponding digital characteristics of backlash are

(33)

(34)

Research on Coupling Analysis of Factors Influencing Backlash in Gear …



E( jt∑δ ) = E jt∑δ = −E f ∑δ tan α D( jt∑δ ) = D jt∑δ = D f ∑δ (tan α)2  E( jt∑β ) = E jt∑β = −E f ∑β D( jt∑β ) = D jt∑β = D f ∑β

127

(35)

(36)

The influence of eccentricity on the backlash is also related to the phase angle. e sin ϕ is obeyed the normal distribution. Then, the mean and variance of shaft bending deformation error ΔST , shaft torsional deformation error ΔS F , bearing clearance error Δr , tooth deformation error Δ f ⎧ ⎨ E(e sin ϕ) = E e sin ϕ = 0 ⎩ D(e sin ϕ) = De sin ϕ = 1 (2e)2 = 1 e2 36 4  E(ΔST ) = ΔST D(ΔST ) = 0 

E(ΔS F ) = ΔS F D(ΔS F ) = 0  E(Δr ) = Δr

(37)

(38)

(39)

(40)

D(Δr ) = 0 2π M(sin nθ ) =

P(θ ) sin nθ dθ = 0

(41)

1 2

(42)

0

2π D(sin nθ ) =

P(θ ) sin nθ dθ = 0

M(Δ f ) = Δ f

(43)

D(Δ f ) = 0

(44)

The corresponding digital characteristics of backlash are the follows respectively, 

E(jte ) = E jte = 0 D(jte ) = Djte = 4De sin ϕ (tan α)2  E( jt T ) = ΔST D( jt T ) = 0

(45)

(46)

128

C. Shi et al.

⎧ ⎨ E( j ) = ΔS F tF cos α ⎩ D( jt F ) = 0

(47)

⎧ ⎨ E( j ) = 3Δr cos α tB 2 ⎩ D( jt B ) = 0

(48)

⎧ M( jt f ) = 0 ⎪ ⎪

⎪ ⎪ ⎪ Δ f · sin nθ ⎪ ⎪ ⎨ D( jt f ) = D cos α ⎪ 1 ⎪ ⎪ [D(Δ f )D[sin nθ ] + D(Δ f )[M(sin nθ )]2 = ⎪ ⎪ ⎪ cos α 2 ⎪ ⎩ + [M(Δ f )]2 D[sin nθ ]]

(49)

3.2 Study on the Coupling Law of Gear Backlash The gear backlash value of the 3K planetary gear train can be obtained by inputting the parameters of various error factors in the Excel program. The gear backlash coupling law is studied when the output torque is changed from 0 to 1000 Nm if the gear precision grade is 6, the ambient temperature is 20 °C, and without friction. The variation curve of gear backlash changed by the load is shown in Fig. 3. The changing curve is shown that the expected value, maximum value, and minimum value of gear backlash increase with the increase of load, and the value is changed obviously. With the increase of load, the gear backlash variance increases, but the value is not obvious. The gear backlash coupling law is studied when the output torque is changed from 0 to 1000 Nm with the working conditions of gear precision grade at 6, and ambient temperature at 20 °C, but without friction. The variation curve of gear backlash changed by the load is shown in Fig. 4. The change curve is shown that with the increase of the temperature, the expectation, maximum value, and minimum value of gear backlash have a slight upward trend, and the temperature influence on gear backlash is small. The clearance is included gear error and device error, and the inherent error is selected to study the law of gear backlash with the change of clearance. The inherent error value is determined by the gear precision grade. The greater the gear precision grade value is, the greater the inherent error is. When the gear precision grade changes from 4 to 12, the gear backlash coupling law is studied on the condition of output torque at 1000 Nm, the ambient temperature at 20 °C, and without friction. The

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

Expectation Variance Maximum value Minimum value

Fig. 3 The curve of gear backlash changing with load

Return difference

Expectation Variance Maximum value Minimum value

Temperature

Fig. 4 The curve of gear backlash changing with temperature

129

130

C. Shi et al.

Expectation Variance Maximum value Minimum value

Gear precision grade

Fig. 5 The curve of gear backlash changing with gear precision grade

changing curve of gear backlash with the change of the gear precision grade is shown in Fig. 5. The changing curve is shown that with the increase of the inherent error, the variance, maximum value, and minimum value of gear backlash have an increasing trend. When the gear precision grade is high, the change is not obvious, but if the grade is low, the change is obvious. With the increase of inherent error, the expectation of gear backlash increases, and the change is not obvious. The magnitude of friction can be reflected by the friction coefficient in the constant load. The friction coefficient is selected to study the variation law of gear backlash with friction. The gear backlash coupling law is studied on the condition of gear precision grade at grade 6, the ambient temperature at 20 °C, and the output torque at 1000 Nm if the friction coefficient is changed from 0 to 0.1. The curve of gear backlash with the change of the friction coefficient is shown in Fig. 6. The change curve is shown that the expectation, variance, maximum value and minimum value of gear backlash first decrease and then increase with the increase of friction in the selected range, and the change is not obvious. When the ambient temperature is 20 °C and the gear friction is 0, the back difference coupling law is studied if the output torque changes from 0 to 1000 Nm and the gear precision grade changes from 4 to 10. The expected change of gear backlash is shown in Fig. 7. The changing surface is shown that the trend of return error is changed gently in the action of two factors, and it is obvious that the gear backlash is affected by the load and the gear precision grade.

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

Expectation Variance Maximum value Minimum value

Friction coefficient

Return difference expectation

Fig. 6 The curve of gear backlash changing with the friction coefficient

Gear precision grade

Output torque

Fig. 7 The law of gear backlash changing with gear precision grade and load

131

132

C. Shi et al. Output torque 0Nm Output torque 333Nm Output torque 666Nm Output torque 1000Nm

Gear precision grade

Fig. 8 The law of gear backlash changing with gear precision grade, temperature, and load

If the gear friction is selected as 0, the coupling law of backlash is studied when the output torque is changed from 0 to 1000 Nm, the gear precision grade changes from 4 to 12, and the temperature is changed from −50 to 50 °C. The expected change of gear backlash is shown in Fig. 8. The changing surface is shown that the trend of gear backlash is changed gently in the action of three factors. The influence of load on gear backlash is more obvious compared with temperature change and gear precision grade.

4 Conclusions The gear backlash changes gently with the load, temperature, clearance, and friction, and the magnitude of the backlash is in the classification range. Furthermore, the load and clearance have a greater impact on the backlash, and the temperature and friction have a smaller impact on the backlash. Acknowledgements Supported by National Key R&D Program of China (No.2019YFB2004602).

Research on Coupling Analysis of Factors Influencing Backlash in Gear …

133

References 1. Chang X (2010) Calculation and analysis on gear backlash of filtering gear reducer based on probability theory. J Mech Transm 34(12):1–5 2. Zhu XR (2010) Backlash analysis of a new planetary gearing with internal gear ring. J Chongqing Univ 03:38–45 3. Zhang Y, Bing C, Zhang X (2016) Gear backlash analysis of two-input planetary gear reducer. J Mech Transm 40(1):4 4. Zhu C (2002) Study on contact teeth and load distribution of planetary gear with small tooth number difference. China Mech Eng 13(18):1586–1589 5. Seidl DR, Lam SL, Putman JA et al (2002) Neural network compensation of gear backlash hysteresis in position-controlled mechanisms. IEEE Trans Ind Appl 31(6):1475–1483 6. Seidl DR, Lam SL, Putnam JA et al (1993) Neural network compensation of gear backlash hysteresis in position-controlled mechanisms. In: Industry Applications Society annual meeting. Conference record of the 1993 IEEE. IEEE 7. Dagalakis NG, Myers DR (1985) Adjustment of robot joint gear backlash using the robot joint test excitation technique. Int J Robot Res 4(2):65–79 8. Stein JL, Wang CH (1998) Estimation of gear backlash: theory and simulation. J Dyn Syst Meas Contr 120(1):74–82 9. Chen XX, Lin SZ, Xing JZ et al (2011) Simulation on gear backlash and interference check of harmonic drive with circular-arc teeth profile. Comput Integr Manuf Syst 17(3):643–648

Transmission Principles of One Novel High-Order Phasing Gear and Its Influence of Design Parameters on Dynamic Properties Jing Wei, Miaofei Cao, Aiqiang Zhang, Bing Pen, and Yujie Zhang

Abstract Spur gear has a large impact and low load-carrying capacity and will lead to greater vibration, while helical gear will produce axial force. In order to improve the transmission performance, the transmission principle of one novel high-order phasing gear is proposed in this paper. The staggered phase angle of the high-order phasing gear is defined, the calculating method of the contact ratio of the high-order phasing gear. The influences of design parameters on dynamic properties, such as mesh stiffness and dynamic response as well as transmission error, are studied. The results show that when the staggered phase angle of the second-order phasing gear equals to 1/2 pitch angle, that is φ = π/z, the time-varying mesh stiffness fluctuation of the phasing gear is much smaller than that of the spur gear. Meanwhile, the secondorder phasing gear’s fluctuation of dynamic meshing force is the smallest. This novel phasing gear could reduce vibration and has superior transmission performance, and verified through experiments. Keywords High-order phasing gear · Staggered phase angle · Dynamic properties · Vibration test

1 Introduction Gear transmission is widely used in machine tools, automobiles, aircraft, ships and much other mechanical equipment. However, gear meshing shock caused by the fluctuation of mesh stiffness, directly affects the reliability and running stability of gear transmission. It is important to reduce gear mesh stiffness fluctuations to improve gear meshing shock and reduce transmission system vibration. In recent years, many scholars have studied the influence of meshing phase on vibration reduction. Chen et al. [1] studied the impact characteristics of gear transmission systems using experimental static transmission errors and backlash; Rigaud J. Wei (B) · M. Cao · A. Zhang · B. Pen · Y. Zhang State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_8

135

136

J. Wei et al.

and Perret-Liaudet [2] conducted an experimental study on the cogging noise caused by the vibration between gear teeth, and designed a special experimental device to analyze the nonlinear dynamic behavior of cylindrical gears under the condition of input speed fluctuations; Xu et al. [3] studied the effects of intermediate support stiffness on nonlinear vibration response of transmission system; Liu and Yuan [4] developed a rigid-flexible coupling dynamic model for a flexible gearbox with the supported ball bearings and investigated the effects of the faults on the vibration transmission characteristics; Gao et al. [5] studied the effects of an elliptical gear’s basic parameters on vibration instability; Huang [6] expressed the vibration control equation of the gear pair as an equivalent discrete model of mass, damping and spring elements, and studied the influence of lubricating oil viscosity and applied torque on gear dynamics; Park [7] studied the effect of spur gear backlash on dynamic characteristics; Chen [8] proposed a simple new method of phase-shifting spur gears to reduce the vibration, which reduces the change of mesh stiffness by adding a pair of half-pitch phase-shifting gears; These studies have made great contributions to gear drive vibration reduction, but there are also some shortcomings, such as increased cost and increased size, and lack of universal theoretical basis. In addition, Parker [9] studied the influence of meshing phase on planetary modal response and found that the planetary modal response in the planetary gear dynamics can be suppressed by adjusting the meshing phase; Wang [10] established a spur planetary system’s bending-torsion coupling dynamics model, studied the relationship between the phase tuning factor and the component motion characteristics, and optimized planetary meshing phase to predict and suppress certain harmonic resonances of ring gears; Liu et al. [11] studied an annular ring gear with multiple planetary gears, and the results showed that the phase difference between them can eliminate the large amplitude response under certain conditions; Peng et al. [12, 13] proposed a method of using planetary phases to diagnose faulty planets; Wang et al. [14] built the dynamic model of double helical gears considering axial vibration and backlash and studied the dynamic behaviors of double helical gears; Li et al. [15] considered the phase difference between equidistant planetary gears and studied the vibration modulation sideband problem caused by the meshing of multiple planetary gears; Liu et al. [16] proposed a dynamic model for herringbone planetary gears which can be applied in the dynamic analysis of variable speed processes (including acceleration, deceleration, and large speed fluctuation process, etc.), and gave some advice for the design of planetary gear sets to avoid the phenomena of tooth separation and tooth back contacts and suppress the vibrations and dynamic meshing forces; Ding et al. [17] established a set of rigid-flexible coupling dynamics models of planetary gear systems and studied the vibration characteristics of floating sun gears under different conditions; Luo et al. [18] studied the influence of sliding friction on the vibration of the planetary gear system; Shi et al. [19] proposed a new type of planetary gear transmission with torsional vibration damper; Kang and Kahraman [20] found that the cross angle of herringbone teeth from right to left is the most critical dynamic impulse response parameter by experiments; the influences of the shaft angle, asymmetric transmission error excitations and right-to-left stagger angle of double-helical gear teeth on the dynamic transmission error responses and load sharing features of the

Transmission Principles of One Novel High-Order Phasing Gear and Its …

137

two-path split torque gear transmission system are investigated by Hu et al. [21], and they found that the right-to-left stagger angles of double-helical gear teeth is also an important geometry parameter that can affect the dynamic responses and load sharing characteristics of the system. The above research shows that the meshing phase is an important parameter for studying the vibration reduction of planetary gear trains. However, the meshing phase is rarely used for parallel shaft gear meshing. In order to improve the transmission performance of parallel shaft gears, various gears with stronger carrying capacity and better transmission performance have evolved from spur gears, such as helical gears, and herringbone gears. However, there are also some shortcomings, such as increased cost, generated axial force etc. In view of the fact that the meshing phase is rarely used in the research of spur gears and some shortcomings in the current spur gear vibration reduction research, the principle of high-order phasing gear transmission is proposed in this paper, and the principles are reasonable with the evidences obtained by solving the time-varying mesh stiffness of the phasing gears and dynamic meshing force. Furthermore, the optimal transmission parameters of high-order phasing gears are obtained by studying time-varying mesh stiffness and the dynamic meshing force for n-order phasing gears to second-order phasing gears.

2 Principle of High-Order Phasing Gear Transmission 2.1 The Phasing Gear and Staggered Phase Angle For meshing phase, a pair of traditional spur gears meshing, there is no meshing phase difference. However, by changing the gearing (a) to (b) in Fig. 1, the meshing phase difference of single pair of gears can be made, but this change does not affect the transmission ratio of the gears. Thus, the transmission performance of this gear could be improved by adjusting the meshing phase difference. In this paper, this gear is defined as a phasing gear. The improved gear is defined as a second-order phasing gear, namely the involute cylindrical spur gear along the direction of the tooth part into two paragraphs (two order), the second order turns the angle φ in the direction of rotation relative to the first order, get a new gear, make its meshing engagement phase difference about the two order. Then the transmission performance could be changed by adjusting φ, as shown in Fig. 1b. In Fig. 1, φ is defined as the staggered phase angle. The staggered phase angle can be understood in this way. The first-order gear meshing rotates a period of time Δt earlier than the second-order gear meshing, and the first-order gear rotates more than the second-order by the angle φ, which results in a meshing phase difference ΔΨ between the first and second-order gears. The gear rotates a pitch angle (2π/z) to complete a complete meshing cycle. When the first-order gear turns φ more than the second-order gear, the first-order

138

J. Wei et al. One order

Two order

Φ

O

(a) spur gear

(b) Two-order phasing gear

Fig. 1 Definition of second-order phasing gear

gear completes φ/(2π/z) more meshing cycles than the second-order gear. When the stiffness fluctuates with the meshing frequency as the fundamental frequency, each meshing process means that the stiffness function passes through a cycle and the phase change is 2π . Assuming Φi is the meshing phase, the relationship between the staggered phase angle and the meshing phase is Φi = φi z

(1)

2.2 High-Order Phasing Gear Similar to second-order phasing gear, the gear dividing the gear into n steps along the tooth direction forms the n-order phasing gear. For the n-order phasing gear, assuming a pitch angle is divided into n parts, which means that the staggered phase angle between every two steps is 1/n pitch angle. If n is infinite, and the n order rotates 1/n pitch ahead of the n − 1 order around the axis, the phasing gear evolves into a helical gear, and the axial force is introduced. If n is finite, and the order is reversed, odd order is ahead and even order is lagging, then the phasing gear will not introduce axial force. In this paper, n is a finite natural number, as shown in Fig. 2.

2.3 Calculation of Contact Ratio of Phasing Gear Compared spur gears and phasing gears with the same parameters. For spur gear transmission, the gear teeth enter meshing at B2 B2 along the entire tooth width b, and the entire gear is disengaged at B1 B1 , and the meshing area of the gear teeth between B2 B2 and B1 B1 . For a phasing gear, it is equivalent to a pair of gear teeth that are staggered by an angle and divided into two parts 1, 2. The gear teeth start to

Transmission Principles of One Novel High-Order Phasing Gear and Its …

139

Fig. 2 Schematic diagram of n-order phasing gear

mesh at position 1, and fully mesh at position 2. The gear teeth start to disengage at position 3, at position 4 it is completely disengaged, as shown in Fig. 3b. Obviously, the meshing area of the phasing effect of the phasing gear transmission is 2ΔL, that is, the contact ratio εt of the phasing gear is given, εt = ε0 + nΔε

(2)

where nΔε is defined as the contact ratio of phasing effect, n is the phasing order, and ε0 is the basic contact ratio of the spur gear, namely: ε0 =

n [z 1 (tan αa1 − tan α) + z 2 (tan αa2 − tan α)] 2π

(3)

where α is the meshing angle, αa1 and αa2 are the addendum pressure angles of the driving wheel and the driven wheel, respectively. The staggered phase angle takes 2π/z as one cycle, within one cycle, the meshing phase changes 2π, the contact ratio is ε0 , when the stagger angle is φ, there is, 2π ε0 2π = = zφ Δψ Δε

(4)

140

J. Wei et al. L

Fig. 3 Gear transmission meshing surface

B2

B1

Spur gear b B1

B2

(a) spur gear L

L

B2

B1 b/2

1

3

b/2

Tuning gear

2

4

B2

B1 L

(b) phasing gear where Δψ is the meshing phase difference, then there is, Δε =

z 1 φε0 2π

(5)

According to the actual situation, when φ ∈ [0, π/z], the contact ratio of the phasing effect gradually increases from zero. When φ ∈ (π/z, 2π/z), the contact ratio of the phasing effect gradually decreases to zero. φ ∈ (π/z, 2π/z) can be regarded as an inverse process of φ ∈ [0, π/z], then substituting Eq. (5) into Eq. (2), Eq. (6) is obtained:  εt =

[1 + nzφ/(2π )]ε0 , φ ∈ [0, π/z] [1 + nz(2π/z − φ)/(2π )]ε0 , φ ∈ (π/z, 2π/z]

(6)

where ε0 is the contact ratio of the spur gear, and n is the phasing order.

3 The Influence of High-Order Phasing Parameters on the Dynamic Response of Gears 3.1 The Influence on Mesh Stiffness For the second-order phasing gear, the meshing relationship between the gears adopts the equivalent spring stiffness and damping [22], and establishes the coupling dynamic model of the gear shaft segment. o1 −x1 y1 z 1 is the fixed coordinate system

Transmission Principles of One Novel High-Order Phasing Gear and Its …

141

of the first end face of the pinion, o2 −x2 y2 z 2 is the fixed coordinate system of the first end face of the large gear, in which the mesh stiffness, meshing damping and equivalent meshing error of the first meshing gear pair are represented by the symbol: K m (t), C m , E m (t); the mesh stiffness, damping and equivalent meshing error between the second-order meshing gear pairs are represented by the symbol: K m (t + Δt), C m , E m (t + Δt), as shown in Fig. 4. Using the node finite element method, the modified Euler–Bernoulli beam element is used to establish the dynamic equation of the system [23]. The shafting is divided into 30 nodes, among which 4 meshing nodes are coupled by gear meshing elements, and 4 bearing nodes are coupled with corresponding bearing elements, as shown in Fig. 5. The basic parameters a pair of parallel shaft phasing gears are shown in Table 1. y1

Fig. 4 Phasing gear mesh dynamic model

Cm

φ

Km(t+Δt) Em(t+Δt)

o1

z1

Cm Km(t)

y2

Em(t) x2 o2

Fig. 5 Node finite element model of phasing gear

z2

Axis node Bearing node Axis 2

Meshing node Power node Gear2 y Gear1

z o x

142 Table 1 Basic parameters of gear

J. Wei et al. Gear parameters

Pinion

Wheel

Number of teeth

19

63

Gear modulus (mm)

4

Pressure angle (°)

20

Tooth width (mm)

60

Gear material

45#

Staggered phase angle φ

π π 3π 0/ 2z / z / 2z

Torque (N/m)

–

180

Speed (r/min)

6000

–

60

The dynamic response of the gear transmission system is obtained by solving the overall dynamic equation of the system. According to Eq. (6), the contact ratio change curve of the second-order phasing gear with different staggered phase angle can be obtained. When the staggered phase angle of the second-order phasing gear increases, the contact ratio of the second-order phasing gear first increases and then decreases. When the phase angle φ = π/19. The contact ratio is the largest, that is, εt = 3.34. At this time, the second-order phasing gear is three-four-tooth alternately meshing, the stiffness fluctuation is the smallest, the load sharing effect is the best, and the bearing capacity is the strongest; when φ = 0 or φ = 2π/19, the second-order phasing gear becomes a traditional spur gear. At this time, εt = 1.67, the gears are alternately meshed with single and double (two or four) teeth, which is consistent with Sect. 2.3 as shown in Fig. 6. Contact ratio affects gear mesh stiffness. So far, the square wave method, potential energy method, finite element method and experimental method [24] are commonly used to solve the time-varying mesh stiffness of gears. In the potential energy method [25], the gear is equivalent to an involute cantilever beam, which is solved by the Fig. 6 Change curve of contact ratio of different staggered phase angle

Transmission Principles of One Novel High-Order Phasing Gear and Its …

143

relationship between potential energy and stiffness. In this paper, the time-varying mesh stiffness of the high-order phasing gear is calculated by the time-varying stiffness calculation method of the high-order phasing gear in Sect. 3.2. According to the principle of multi tooth alternate engagement of the high-order phasing gear as shown in Fig. 5, the overall mesh stiffness of the phasing gear is Kt = K1 + K2 + · · · Kn

(7)

where K 1 K 2 K n represents the nth order mesh stiffness of the phasing gear respectively. According to the distribution of single and double teeth meshing of spur gear in a cycle, as shown in Fig. 6, the comprehensive mesh stiffness of the second-order phasing gear with different staggered phase angle and different phasing order can be obtained from Eq. (7), when φ = 0, the phasing gear is two-four teeth alternately meshing, when φ = π/2z he phasing gear is two–three-four alternate meshing, when φ = π/z, the phasing gear is three-four teeth alternately mesh. When the phasing order n = 2, the time-varying mesh stiffness fluctuation is much smaller than that of a spur gear. In addition, as the phasing order increases, the overall average stiffness of the phasing gear increases, as shown in Fig. 7. The fluctuation ηCγ of the time-varying mesh stiffness is defined as ηCγ =

ΔK × 100% K

(8)

where 

ΔK = K max − K min K = K min (2 − ε) + K max (ε − 1)

(9)

Compared to traditional spur gears, phasing gears have smaller fluctuations of mesh stiffness. When φ = π/z, mesh stiffness fluctuations is reduced by 27.03% compared to traditional spur gears; implying that when the phasing gear is secondorder phasing and the staggered phase angle is φ = π/z, the fluctuation of mesh stiffness of the phasing gear is much smaller than that of the spur gear, as shown in Fig. 8.

3.2 The Influence of the Staggered Phase Angle on the Transmission Error Transmission error is an important index representing the flexibility of the gear system. Transmission error is defined as the angular deviation of the driven gear relative to the driving pinion [26]. Comparing the dynamic transmission errors (DTE)

144

J. Wei et al.

(N/m) m) Mesh stiffness(N▪

Fig. 7 Time-varying mesh stiffness diagram

Four tooth meshing Three tooth meshing

Double tooth meshing

(N/m) Mesh stiffness(N▪ m)

(a) Mesh stiffness with different staggered phase angles

(b) Mesh stiffness with different phasing orders corresponding to different staggered phase angles, it is found that when the transmission errors are different with different staggered phase angles. When φ = 0 (traditional spur gear), the transmission error is the largest. When φ = π/2z and φ = 3π/2z, the transmission error of the phasing gear is less than that of the spur gear. When φ = π/z, the transmission error is the smallest, which indicates that the transmission performance is the most stable under this circumstance, as shown in Fig. 9.

Transmission Principles of One Novel High-Order Phasing Gear and Its … Fig. 8 Time-varying mesh stiffness fluctuation diagram

Fig. 9 Transmission error diagram of different staggered phase angle

145

146

J. Wei et al.

3.3 The Influence on Dynamic Meshing Force 3.3.1

The Influence of Staggered Phase Angle

By solving the dynamics, the dynamic meshing force of the second-order phasing gears with different staggered phase angles can be obtained. The single-order meshing force of the phasing gear is less than that of the spur gear, that is, the phasing gear can greatly improve the load-carrying capacity. The overall meshing forces fluctuation of the phasing gear with different staggered phase angles are different. When φ = π/z, the overall meshing force fluctuation of the phasing gear is the smallest, indicating that the meshing impact is the smallest under this circumstance, as shown in Fig. 10. ΔF is the amount of fluctuation, defined as ΔF = Fmax − Fmin

Fig. 10 Dynamic meshing force of the phasing gear with different staggered phase angles

(10)

Transmission Principles of One Novel High-Order Phasing Gear and Its …

147

Fig. 11 Dynamic meshing force fluctuation diagram

Comparing the single-order and overall meshing force fluctuations of the phasing gears with different staggered phase angles, when φ = π/z, the dynamic meshing force fluctuation is the smallest, which is consistent with the theoretical derivation of the second-order phasing gear, as shown in Fig. 11.

3.3.2

The Influence of Phasing Order

The meshing force fluctuation is the smallest in the second-order phasing. From Eq. (10), the meshing force fluctuation of the first-order spur gear is 19463.47 N; the meshing force fluctuation of the second-order phasing gear is 2918.25 N; the meshing force fluctuation of the third-order phasing gear is 4536.73 N, as shown in

148

J. Wei et al.

Fig. 12 Comparison chart of meshing force of different phasing orders

Fig. 12. Here, the overall dynamic meshing force of the n-order phasing gear defaults to the algebraic sum of the dynamic meshing force of the n-order gear meshing pair under this circumstance.

3.4 The Influence on Vibration Response 3.4.1

The Staggered Phase Angle

The vibration displacement of the gear drive wheel is tuned with different staggered phase angle, the vibration displacement of the driving wheel of the phasing gear is significantly lower than that of the spur gear. When the phase angle φ = π/z,

Transmission Principles of One Novel High-Order Phasing Gear and Its …

149

Fig. 13 Vibration displacement diagram of different staggered phase angles

the vibration displacement of the driving wheel is the smallest, indicating that the phasing gear itself has a certain vibration reduction effect, as shown in Fig. 13. In addition, the vibration acceleration of the driving wheel of the phasing gear with different staggered phase angle is obtained, when the phase angle φ = π/z, the vibration acceleration fluctuation of the driving wheel is the smallest, as shown in Fig. 14. Meanwhile, the gear transmission vibration acceleration is the smallest.

3.4.2

The Phasing Order

Keeping the phase angle of the phasing gears at the same time, the vibration displacement and acceleration of the driving wheel with different phasing orders are solved, when the phasing order is the second-order phasing, the vibration displacement and vibration acceleration fluctuations are the smallest. When the third-order phasing, the overall mesh stiffness increases, resulting in larger vibration acceleration fluctuations, as shown in Fig. 15.

150

J. Wei et al.

Fig. 14 Vibration acceleration diagram with different staggered phase angles

In summary, when the phasing order is second order and the staggered phase angle φ = π/z, the phasing gear has the best vibration reduction effect. When the phasing order n = 3, the vibration acceleration becomes significantly larger. This is because as the phasing order increases, although the mesh stiffness fluctuation decreases, the overall mesh stiffness of the phasing gear increases, and greater stiffness causes greater acceleration.

Transmission Principles of One Novel High-Order Phasing Gear and Its …

151

Fig. 15 Vibration response diagram of different phasing orders

4 Experimental Verification of High-Order Phasing Gear Transmission Performance 4.1 Experimental Equipment We use the SP-DAQ test system for testing, and use sensors to measure vibration signals. The test bench and gearbox are shown in Fig. 16. Set the sampling frequency to 10,240 Hz and the sampling length to 100k. The test bench consists of a servo drive motor (1), four clamping diaphragm couplings (2/4/12/14), two torque speed sensors (3/13), two bearing support seats (5/11), two grating sensors (6/10), two bearing seat couplings (7/9), a gear box (8), a servo loading motor (15), drive end adjustment platform (16), high precision stage (17) and loading end adjustment platform (18).

152

J. Wei et al.

Fig. 16 Test bench and gearbox

4.2 Design of Phasing Gear Realization Scheme The specific implementation scheme of the phasing gear is to connect two identical spur gears together through splines, and adjust the spline staggered tooth connection to obtain phasing gear sets with different staggered phase angles. It is required that the number of spline teeth and the number of gear teeth must satisfy a certain number relationship, that is, the number of spline teeth and the number of gear teeth of an n-order phasing gear has a common divisor n. In this experiment, to verify the best misaligned phase angle of the second-order phasing gear, four sets of second-stage phasing gear misaligned phase angles are taken, which are 0°, 1/4 pitch angle, 2/4 pitch angle and 3/4 pitch angle. The number of spline teeth is designed to be 32, and the number of gear teeth is 36. By adjusting the number of spline misaligned teeth, the phasing gears with different staggered phase angles can be obtained as shown in Fig. 17.

4.3 Experimental Results and Analysis 4.3.1

Test Result of Transmission Error

We set the input speed to 60 rpm and the load torque to 14 N m, and measure the transmission error of the above 4 sets of gears. The result is shown in Figs. 18 and 19. Figure 18a shows the results in time domain. The peak-to-peak values of transmission error of the phasing gears with different staggered phase angle are 0.5375°, 0.5622°, 0.3763° and 0.5572° respectively. When the staggered phase angle is π/z, the peak-to-peak value of transmission error is the smallest and reduced by 30% compared with spur gears. Figure 18b shows the results in frequency domain. It can

Transmission Principles of One Novel High-Order Phasing Gear and Its …

153

Fig. 17 Schematic diagram of different staggered phase angles

be seen that the amplitudes of shaft frequency components are minimum when the staggered phase angle is π/z. Changing the experimental working conditions, the results of gear transmission error are shown in Fig. 19. Under four different working conditions, the peak-to-peak transmission error is the smallest when the phase angle is π/z. In addition, when the system power increases, the effect of phasing gear to reduce the transmission error is better.

4.3.2

Test Result of Vibration Displacement

The vibration displacement under the experimental conditions (load torque is 8 N m and input speed is 600 rpm) is shown in Fig. 20. Figure 20a shows the results in time domain. The peak-to-peak values of vibration displacement of the output y at different staggered phase angle are 0.07 mm, 0.0806 mm, 0.0582 mm, 0.0786 mm respectively. When the staggered phase angle is π/z, the peak-to-peak value of vibration displacement is the smallest and reduced by 16.9% compared with the spur gear. Figure 20b shows the results in frequency domain. It can be seen that the amplitudes of shaft frequency components are minimum when the staggered phase angle is π/z. The amplitudes of different staggered phase angle are 0.01363 mm, 0.01193 mm, 0.01046 mm, 0.01355 mm respectively. It can be seen from Fig. 21 that for all measuring points, the peak-to-peak value of vibration displacement Is the smallest when the staggered phase angle ϕ is π/z. In other words, the best staggered phase angle is π/z for two-order phasing gear.

154 Fig. 18 Transmission errors of phasing gears with different staggered phase angle

Fig. 19 Maximum comparison of transmission error

J. Wei et al.

Transmission Principles of One Novel High-Order Phasing Gear and Its … Fig. 20 Vibration displacements with different staggered phase angle

Fig. 21 Maximum comparison of transmission error

155

156

J. Wei et al.

5 Conclusions The transmission principles of the novel high-order phasing gear were proposed in this paper. The staggered phase angle of the phasing gear was defined, and the principle of alternating meshing of multiple tooth pairs of the phasing gear was given. The calculation method of the contact ratio of the phasing gear was defined. The influences of design parameters on dynamic properties were studied, and verified through experiments. Some conclusions are as follows: (1) The contact ratio of the phasing gear is greater than that of the spur gear, and they are difference with different staggered phase angle. As the phasing order increases, the contact ratio of the phasing gear also increases. (2) The optimal phasing order should be n = 2 and optimal staggered phase angle of second-order phasing gear should be φ = π/z. Meanwhile, the mesh stiffness fluctuation is only an odd-order component and the mesh stiffness fluctuation of phasing gears is much smaller than that of spur gears. In this case, phasing gear has the best vibration reduction effect and the transmission error is the smallest. (3) Experimental results from high-order phasing gear show that the peak-to-peak value of transmission error and vibration displacement are the smallest, when the staggered phase angle is φ = π/z of two-order phasing gear, under all experimental conditions. Compared with traditional spur gears, the peak-topeak value of vibration displacement can be reduced significantly and the best staggered phase angle of the n-order phasing gear is 2π/(nz). Acknowledgements Supported by the National Natural Science Foundation of China (Grant No. 51775058/52105051), the National Key R&D Program of China (No.2018YFB2001602/2019YFE0121301) and Innovation Group Science Foundation of Chongqing Natural Science Foundation (No. cstc2019jcyj-cxttX0003).

References 1. Chen S, Tang J, Wu L et al (2014) Dynamics analysis of a crowned gear transmission system with impact damping: based on experimental transmission error. Mech Mach Theory 74:354– 369 2. Rigaud E, Perret-Liaudet J (2020) Investigation of gear rattle noise including visualization of vibro-impact regimes. J Sound Vib 467:115026 3. Xu J, Zhu J, Wan L (2020) Effects of intermediate support stiffness on nonlinear dynamic response of transmission system. J Vib Control 26(9–10):851–862 4. Liu J, Yuan L (2020) Vibration analysis of a flexible gearbox systemconsidering a local fault in the outer ring of the supported ball bearing. J Vib Control 1–14 5. Gao N, Meesap C, Wang S, Zhang D (2020) Parametric vibrations and instabilities of an elliptical gear pair. J Vib Control 26:19–20 6. Huang KJ, Wu MR, Tseng JT (2010) Dynamic analyses of gear pairsincorporating the effect of time-varying lubrication damping. J Vib Control 17(3):355–363

Transmission Principles of One Novel High-Order Phasing Gear and Its …

157

7. Park CI (2020) Dynamic behavior of the spur gear system with time varying stiffness by gear positions in the backlash. J Mech Sci Technol 34(2):565–572 8. Ma R, Chen Y, Cao Q (2012) Research on dynamics and fault mechanism of spur gear pair with spalling defect. J Sound Vib 331(9):2097–2109 9. Ambarisha VK, Parker RG (2006) Suppression of planet mode response in planetary gear dynamics through mesh phasing. J Vib Acoust 128(2):133–142 10. Wang S, Huo M, Zhang C (2011) Effect of mesh phase on wave vibration of spur planetary ring gear. Eur J Mech A Solids 30(6):820–827 11. Liu C, Cooley CG, Parker RG (2017) Parametric instability of spinning elastic rings excited by fluctuating space-fixedstiffnesses. J Sound Vib 400:533–549 12. Peng D, Smith WA, Borghesani P, Randall RB, Peng Z (2019) Comprehensive planet gear diagnostics: use of transmission error and mesh phasing to distinguish localised fault types and identify faulty gears. Mech Syst Sig Process 127:531–550 13. Peng D, Smith WA, Randall RB, Peng Z (2019) Use of mesh phasing to locate faulty planet gears. Mech Syst Sig Process 116:12–26 14. Wang C, Wang SR, Yang B, Wang GQ (2018) Dynamic modeling of double helical gears. J Vib Control 24(17):3989–3999 15. Li Y, Ding K, He G, Yang X (2019) Vibration modulation sidebands mechanisms of equallyspaced planetary gear train with a floating sun gear. Mech Syst Sig Process 129:70–90 16. Liu C, Qin D, Lim TC, Liao Y (2014) Dynamic characteristics of the herringbone planetary gear set during the variable speed process. J Sound Vib 333(24):6498–6515 17. He G, Ding K, Wu X, Yang X (2019) Dynamics modeling and vibration modulation signal analysis of wind turbine planetary gearbox with a floating sun gear. Renew Energy 139:718–729 18. Luo W, Qiao B, Shen Z, Yang Z, Cao H, Chen X (2020) Influence of sliding friction on the dynamic characteristics of a planetary gear set with the improved time-varying mesh stiffness. J Mech Des 142:073302-1 19. Shi X, Sun D, Kan Y, Zhou J, You Y (2020) Dynamic characteristics of a new coupled planetary transmission under unsteady conditions. J Braz Soc Mech Sci Eng 42(S1):280 20. Kang MR, Kahraman A (2015) An experimental and theoretical study of the dynamic behavior of double-helical gear sets. J Sound Vib 350:11–29 21. Hu Z, Tang J, Wang Q, Chen S, Qian L (2020) Investigation of nonlinear dynamics and load sharing characteristics of a two-path split torque transmission system. Mech Mach Theory 152:103955 22. Wei J, Zhang A, Wang G (2018) A study of nonlinear excitation modeling of helical gears with modification: theoretical analysis and experiments. Mech Mach Theory 128:314–335 23. Wei J, Zhang A, Qin D (2017) A coupling dynamics analysis method for a multistage planetary gear system. Mech Mach Theory 110:27–49 24. Liang X, Zuo MJ, Feng Z (2018) Dynamic modeling of gearbox faults: a review. Mech Syst Sig Process 98:852–876 25. Fan L, Wang S, Wang X (2016) Nonlinear dynamic modeling of a helicopter planetary gear train for carrier plate crack fault diagnosis. Chin J Aeronaut 29(3):675–687 26. Cirelli M, Giannini O, Valentini PP, Pennestrì E (2020) Influence of tip relief in spur gears dynamic using multibody models with movable teeth. Mech Mach Theory 152:103948

Kinematic Calibration of a 2-DOF Over-Constrained Parallel Manipulator Using Force Data Shuqing Chen, Yixiao Feng, and Tiemin Li

Abstract The over-constrained parallel manipulators have drawn a lot of attention, and their application potential is constantly being explored. To improve the performance of the over-constrained mechanism, the actual structural parameters need to be calibrated. The calibration methods of the non-over-constrained mechanism cannot be applied to the over-constrained mechanism because of the existence of the over-constrained force. Therefore, the calibration methods which utilize a complex model to calculate the deformation and the over-constrained force are proposed. These methods need complex modeling and calculation, and the calculation errors of the forces and deformation may affect the calibration accuracy. To follow the trend of perceptual intelligence and simplify the calibration process, a kinematic calibration method of the over-constrained mechanism is proposed, utilizing the measured force data to calibrate the actual structural parameters. This method can avoid complex modeling and eliminate the impact of the calculation errors of the forces and deformation on the calibration accuracy. To verify the feasibility of this method, a 2-DOF over-constrained parallel manipulator is designed and developed. Each chain of this mechanism contains a force sensor, which is utilized to measure the over-constrained force. The kinematic error model is established in this paper, and it is identified using the Newton–Raphson iteration, the Regularization method, and the least-square method. Finally, a group of experiments is conducted to verify the validity of this method. The experiment results show that this method can reduce the root mean square error of the over-constrained mechanism from 1.243 to 0.028 mm. Keywords Kinematic calibration · Over-constrained · Parallel manipulator · Force data S. Chen · Y. Feng · T. Li (B) Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected] S. Chen e-mail: [email protected] Y. Feng e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_9

159

160

S. Chen et al.

1 Introduction With the progress of advanced robot technology, the importance of the parallel manipulators in the industry is being recognized increasingly [1, 2]. Compared to the serial manipulators, the parallel manipulators have intrinsic advantages in better stiffness to weight ratio [3], motion accuracy [4, 5], and acceleration ability [6, 7]. Because of these advantages, the parallel manipulators have shown potential in a wide range of engineering applications [8], including machining [9, 10], aerial repair [11], pick-and-place [12, 13], and flight simulation [14, 15]. In order to further improve the operation stability and other performance of the parallel manipulators [16, 17], the over-constrained parallel manipulators are proposed by adding extra kinematic constraints which are redundant without changing the degrees of freedom (DOF) of the mechanism [18]. However, the increase of the extra kinematic constrains changes the mechanical state of the mechanism, which introduces the internal forces (the over-constrained forces) [19]. The existence of the over-constrained force brings great difficulties to the kinematic calibration of the mechanism. Nowadays, kinematic calibration is regarded as an effective way to improve accuracy, whose goal is to find out the actual values of the structural parameters [20, 21]. Under the assumption that all the structural errors are constant, the structural parameters of the non-over-constrained parallel manipulators are derived by the kinematic calibration. The over-constrained forces on the over-constrained mechanism change with the motion of the mechanism, and the deformation caused by the forces changes with the position of the mechanism. The structural parameter errors of the over-constrained mechanism cannot be regarded as a constant when it is in motion. So, the kinematic calibration methods cannot be utilized to calibrate the structural parameter errors of the over-constrained parallel manipulators. To solve this problem, many methods have been proposed. Zeng et al. [22] ignored the over-constrained parts of the over-constrained mechanism and directly simplified it to a non-over-constrained mechanism. The actual structural parameters of the overconstrained mechanism were obtained using the kinematic calibration method of the non-over-constrained mechanism. This method may result in low accuracy when the over-constrained force is large. Jiang et al. [23] utilized a compatibility equation of the over-constrained mechanism to establish the modified calibration model by considering the deformation of the over-constrained mechanism. The validity of this model was proved by simulation on a 2-DOF over-constrained mechanism. Ecorchard et al. [24] also considered the impacts of the mechanism’s deformation and regarded the over-constrained parts as structures. A new kinematic calibration method was proposed by using an elastic geometrical model to calculate the stiffness of the mechanism. These two methods yield the structural parameter errors by considering the deformation, which can significantly improve the over-constrained mechanism’s accuracy. Since the over-constrained forces are the major cause of the deformation, utilizing the forces to calibrate the structural parameter errors is more direct. Li et al.

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

161

[25] replaced the constraints with the forces and regarded the over-constrained mechanism as the non-over-constrained mechanism. The kinematic calibration model was established, and the non-over-constrained mechanism’s deformation was obtained. This method works well in calibrating the over-constrained mechanism though it requires complex modeling and calculation. Perceptual intelligence is one of the mainstream trends in advanced robot technology, utilizing various sensors to eliminate the uncertainty. In order to follow this trend and simplify the calibration process, the over-constrained forces can be directly measured rather than obtained through complex modeling and calculation. According to this idea, the force sensor can be mounted on each chain of the mechanism, which is used to measure the over-constrained force directly. And the over-constrained mechanism can be calibrated using these force data. The kinematic calibration method utilizes force data can avoid complex modeling and eliminate the impact of the calculation errors of the forces and deformation on the calibration accuracy. It is noted that though the over-constrained force is measured by the force sensor in this paper, it can be measured using other sensors; the measured deformation can also be utilized to calibrate the over-constrained mechanism, and the calibration process and algorithm can follow the example of this paper. To verify the feasibility of this method, a 2-DOF over-constrained parallel manipulator is designed and developed in this paper. Each chain of this mechanism contains a force sensor, which is utilized to measure the over-constrained force. Then, the positional errors are measured by the absolute tracker. Finally, the kinematic calibration is carried out using the force data and the measured positional errors. The rest of the paper is structured as follows: the mechanism outline of a 2-DOF over-constrained parallel manipulator is introduced in Sect. 2; in Sect. 3, the kinematic error model of the mechanism is established; in Sect. 4, a group of experiments is designed to derive the structural parameter errors of the mechanism; in Sect. 5, the structural parameter errors are compensated and verified; Sect. 6 presents the conclusion.

2 Mechanism Outline of a 2-DOF Over-Constrained Parallel Manipulator A 2-DOF over-constrained parallel manipulator is designed for this study. The detailed 3D and physical models of the mechanism are shown in Fig. 1. As illustrated in Fig. 2, the kinematic structure based on the 3D model is established. This mechanism includes a fixed base, four chains, two liner feed systems, and a terminal platform. As illustrated in Fig. 1, each chain of the over-constrained mechanism contains a force sensor and a flexure unit. Four force sensors are responsible for measuring the over-constrained forces on four chains. The linear feed system I includes the motor I and the ball screw I. The linear feed system II includes the motor II and the ball screw II. The chain A1 C 1 connects the terminal platform T to the slider P1 . The chain A2 C 2 connects the terminal platform T to the slider P1 .

162

S. Chen et al.

(a) Detailed 3D model

(b) Physical model Fig. 1 The 2-DOF over-constrained parallel manipulator A2

A4 T

A1

A3

Y E1

C2 B1 C1

E2

F1 P1

O

X

P2

C4 B2 C3

F2

Fig. 2 Kinematic structure of the mechanism

The chain A3 C 3 connects the terminal platform T to the slider P2 . The chain A4 C 4 connects the terminal platform to the slider P2 . The slider P1 and P2 are driven by the motor I and the motor II, respectively. The terminal platform can be controlled by driving these chains. The mechanism has two translational DOF because of the rotational limit of the parallelogram structures. Any three of the chains can ensure the mechanism’s two translational DOF in this mechanism. The rest chain is utilized to enhance the mechanism’s operational stability.

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

163

3 Error Modeling In order to investigate the impact of the over-constrained forces on the mechanism, this mechanism is regarded as a non-over-constrained mechanism by replacing the redundant chain with the over-constrained forces. So, the error model of this mechanism can be established according to the error model of the non-over-constrained mechanism and the deformation of the non-over-constrained mechanism under the over-constrained forces.

3.1 Error Modeling of the Non-over-Constrained Mechanism As demonstrated in Fig. 3, the global coordinate frame is expressed as {O-XYZ}, and it is in the middle of two linear feed systems. The moving coordinate frame of this mechanism can be expressed as {T-xyz}, and it is in the center of the terminal platform. And the axes of {O-XYZ} and {T-xyz} are parallel. The local coordinate frames can be donated as {Bj -x Bj yBj zBj } and {C i -x Ci yCi zCi }, which are attached to the points of Bj and C i , respectively. The Equation of the non-over-constrained mechanism’s vector loops can be expressed as: t + O R T Ai = L i l i − O R B j b j e1 + O R B j C i y

A1

(1)

A3 x

T A2

A4

Y α3

α1 C1 yB1 B1

X

xB1 α2

C2 Fig. 3 Coordinate system of the mechanism

O

C3 xB2

α4

B2 yB2 C4

164

S. Chen et al.

where terminal platform’s position can be donated as t = [x, y, 0]T , O R T is rotation matrix which rotates from {O} to {T }, Ai (i = 1, 2, 3) represents the position vector of T Ai , O RBj is rotation matrix which rotates from {O} to {Bj }, L i represents the distance between Ai and C i , bj (j = 1, 2) is the distance between Bj and O, e1 = [1, 0, 0]T is the direction vector of the X-axis of {O-XYZ}, l i is the unit vector of link C i Ai , C i is the position vector of C i expressed in {Bj }, α i is the acute angle between the ith chain and the horizontal line, β j is the acute angle between the vector of OBj and the horizontal line and its ideal value is zero. The real kinematic model is quite different from the ideal one because there are errors in this kinematic model: 1. The structural parameter errors of the terminal platform of Ai , so the actual position of Ai is expressed as Ai e = [x Ai + δx Ai , yAi + δyAi , 0]T . 2. The driving error of the linear feed system of bj , so the actual driving parameter bj is obtained as bj + δbj . 3. The structural errors of the slider of C i , so the position of C i can be expressed as C i e = [x Ci + δx Ci , yCi + δyCi , 0]T . 4. The angular structural error of β j is denoted as δβ j , it is noted that {B1 -x B1 yB1 zB1 } and {B2 -x B2 yB2 zB2 } are different, so the rotation matrix O RBj can be represented by: ⎧ ⎛ ⎞ ⎪ cos δβ1 − sin δβ1 0 ⎪ ⎪ ⎪ O Re = ⎝ sin δβ cos δβ 0 ⎠ ⎪ 1 1 ⎪ B1 ⎪ ⎨ 0 0 1 ⎛ ⎞ ⎪ sin δβ − cos δβ 2 2 0 ⎪ ⎪O e ⎪ ⎪ R B2 = ⎝ − sin δβ2 − cos δβ2 0 ⎠ ⎪ ⎪ ⎩ 0 0 1

(2)

5. Structural error of L i , so the length of each chain L i is expressed as L i + δL i . According to the structural parameter errors listed above, the actual vector loop of the non-over-constrained mechanism can be rewritten as: t e + O ReT Aie = (L i + δL i )(l i + δl i )

 − O ReB j b j + δb j e1 + O ReB j C ie

(3)

By deriving the forward kinematics of the non-over-constrained mechanism, the actual position of the terminal platform t e can be obtained. The corresponding kinematic vector loop is shown in Fig. 4.

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

165

y

Aie

x T

Y

Cie

X

δβj O

bj+δbj

Bj

Fig. 4 The vector loop of the non-over-constrained mechanism

3.2 Error Modeling of the Over-Constrained Mechanism According to the previous introduction, any chain of the over-constrained mechanism can be replaced with the over-constrained forces. Without losing generality, this paper replaces the 4th chain with the forces and yields the non-over-constrained mechanism’s stiffness. As demonstrated in Fig. 5, the stiffness model is built and the procedure for building the error model of this mechanism can be concluded in Fig. 6. According to Fig. 5, the stiffness of the terminal platform can be derived as:

K x = K 1 cos α1 + K 2 cos α2 + K 3 cos α3 K y = K 1 sin α1 + K 2 sin α2 + K 3 sin α3

Fig. 5 The stiffness model of the non-over-constrained mechanism

(4)

Ky A1

A3

K1

K2

K3

A2

A3

A1 Kx

A2

A4 F

(a) the original model

A4 F

(b) the equivalent model

166

S. Chen et al.

Ideal structural Identified parameters structural errors

Actual structural parameters

Input actuator parameters

The kinematic error model of the non-over-constrained parallel manipulator

The stiffness of the non-overconstrained parallel manipulator

Actual position of the terminal platform according to the kinematic error model of the nonover-constrained mechanism

The overconstrained force of the 4th chain measured by the sensor

Deformation of the terminal platform of the non-overconstrained mechanism caused by the over-constrained force

Actual position of the terminal platform of the over-constrained mechanism Fig. 6 Procedure for building the error model

Then the deformation of the terminal platform caused by the over-constrained force can be derived as:

t = d

F4 cos α4 F4 sin α4 ,− ,0 Kx Ky

T (5)

Finally, the position of the over-constrained mechanism can be obtained by combining the position of the non-over-constrained mechanism and the deformation caused by the over-constrained force as: t = te + td

(6)

4 Kinematic Error Identification The error model of this mechanism is derived from the unknown stiffness model and structural parameter errors in Sect. 3. This Section yields the stiffness of the 2-DOF over-constrained mechanism and identifies its kinematic parameter errors.

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

167

4.1 Stiffness Experiment As illustrated in Fig. 1, the flexure unit is mounted on each chain. It is designed to simplify the calculations because the stiffness of the flexure unit is far less than the stiffness of other rigid components. To identify the stiffness of each chain, a stiffness experiment is conducted. The main steps are listed as follows: 1. The terminal platform of the mechanism is controlled to the designated position. 2. The rope is utilized to connect the center of the terminal platform with the loading platform over the fixed pulley. The mass blocks are added to the loading platform, and gravity of the mass blocks is transformed into the tension of the terminal platform along the Y-axis of the global coordinate frame. 3. The dial indicator is installed to record the deformation of the terminal platform along the Y-axis of the global coordinate frame. 4. The stiffness is derived according to the deformation and gravity of the mass blocks. In this experiment, the linear feed system I and linear feed system II are driven to the position where b1 = 226.735 mm and b2 = 226.736 mm, respectively. Then the acute angle between four chains and the horizontal line can be derived as α 1 = 0.777 rad, α 2 = 0.773 rad, α 3 = 0.776 rad, and α 4 = 0.777 rad. The loading platform is preloaded, and the weight of the loading platform and the mass block is 370.26 g. The initial reading of the dial indicator is recorded as 0.496 mm. The force applied on the terminal platform can be obtained by accumulating gravity of the mass blocks, and the dial indicator readings can obtain the deformation of the terminal platform. The relationship between the force and the deformation of the over-constrained mechanism is shown in Fig. 7. In Fig. 7, the blue dots are the measurement data from the experiment, and the red line is fitted according to the blue dots. The slope of the red line can be regarded as the stiffness of the over-constrained mechanism along the Y-axis of the global coordinate frame, which is K Ov = 305.063 N/mm. With the assumption that the stiffness of four chains is the same, the stiffness of each can be obtained, which is K = 108.912 N/mm.

4.2 Error Measurement In order to identify the structural parameter errors, the position of the terminal platform of the over-constrained mechanism t* and the over-constrained force f are measured. And the corresponding input actuator parameter q is given. The MonteCarlo method [26–28] is utilized to determine the ideal workspace of the overconstrained mechanism. Though the accuracy of the mechanism can be significantly improved by increasing the number of the measured points, this paper randomly

168

S. Chen et al. 12

Fig. 7 The relationship between the force and the deformation of the over-constrained mechanism

11 10

Force/N

9 8 7 6 5 4 0

0.005

0.010

0.015

0.020

0.025

Deformation/mm

selects 29 measured points for error identification. The 29 measured points are roughly uniformly distributed in the workspace, shown in Fig. 8. An absolute tracker (Leica Corporation, AT901) is utilized in this paper to obtain the position of the terminal platform. The target ball of the absolute tracker is mounted on the center of the terminal platform, which is measured to obtain the actual position of {T }. As illustrated in Fig. 9, the position of the terminal platform can be obtained according to the measurement results. The coordinate frame of the absolute tracker can be expressed as {Om }. In order to establish the relationship between {Om } and {O}, a set of experiments is designed and conducted as follows: 1. The linear feed system I and linear feed system II are controlled to their origin, and the position of {T } is recorded. Fig. 8 The distribution of the 29 measured points in the workspace

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

169

Fig. 9 The measurement process of the absolute tracker

2. The linear feed system I and linear feed system II are driven to move at the same distance in the same direction, and the trace of the terminal platform is along the X-axis of {O}. The positions of {T } are recorded. 3. The linear feed system I and linear feed system II are driven to move at the same distance in the opposite direction, and the trace of the terminal platform is along the Y-axis of {O}. The positions of {T } are also recorded. 4. The linear feed system I and linear feed system II are driven to approach the 29 points, and the actual positions of {T } are recorded. Rotation matrix O ROm , which rotates from {O} to {Om }, is defined as: O

R Om =

O

x Om , O y Om , O z Om

−1

(7)

where O xOm , O yOm , and O zOm are the unit vectors along the X-axis, the Y-axis, and the Z-axis, respectively. The Singular Value Decomposition (SVD) method [29–31] is utilized to identify O xOm , O yOm, and O zOm to minimize the residual error. According to the SVD method, data from step 2 are vertically assembled one by one to obtain O xOm . Data from step 3 are vertically assembled one by one to obtain O yOm . Data of all measured points are assembled vertically to obtain O zOm . The fitted lines of O xOm and O yOm are illustrated in Fig. 10. It is noted that the normal vector of the fitted plane of all points can be regarded as O zOm . The positions of the measured points t* can be expressed as: t∗ =

O

R Om t Om

where t Om is the position of the measured points expressed in {Om }.

(8)

170

S. Chen et al.

Fig. 10 The fitted lines and the fitted plane of measured points

1.0

z /mm

0.8 0.6 0.4 0.2 0 100

80

60

y /mm

40

20

0

-50

-10

-30

10

30

50

x /mm

220

Fig. 11 The 29 measured points expressed in the global frame

200

y /mm

180 160 140 120 100 -40

-30

-20

-10

0

10

20

30

40

x /mm

The 29 measured points expressed in the global coordinate frame are illustrated in Fig. 11.

4.3 Error Identification The structural parameter errors can be derived by combining the Newton–Raphson iteration, the Regularization method [32], and the least-square method. The specific steps are listed as follows: 1. The mechanism is controlled by driving two motors. And the forward kinematic model of this mechanism can be expressed as: F(t, q, r) = 0

(9)

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

171

where r is the kinematic structural parameters. The partial derivative matrix of Eq. (9) can describe the relationship between the position errors and the structural errors of the mechanism linearly. Differentiating Eq. (9) as: ∂F ∂F ∂F dt + dq + d r= 0 ∂t ∂q ∂r

(10)

Since the calibration method is based on the forward kinematic method and the actuator parameters are the inputs of the forward kinematic method, dq = 0. Equation (10) can be rewritten as:  dt =

∂F ∂t

−1

∂F d r = J(t, q, r)d r ∂r

(11)

According to Eq. (11), the relationship between the position errors dt and the modification values of the kinematic structural parameters dr can be derived as: 

d t = J r (n) − r (n−1) = Jd r

(12)

2. The kinematic structural errors are set to zero in this paper. The force sensors are calibrated before measuring the over-constrained forces, and their initial readings are set to zero when it is no-load. According to Eqs. (9) and (12), the simulated position of the over-constrained mechanism t (n) and the Jacobian matrix J in the kth iteration can be obtained, respectively. Linearly dependent structural errors with the same influence on the terminal platform are merged to ensure that the Jacobian matrix is column full rank. Finally, fourteen independent structural errors are involved with the identity matrix. The fourteen structural errors are δyA1 , δb1 , δβ 1 , δL 1 , δx A2 , δyA2 , δL 2 , δyA3 , δb2 , δβ 2 , δL 3 δx A4 , δyA4 , and δL 4 , respectively. 3. The difference between the simulated position and the measured position of the terminal platform dt can be expressed as: d t = t (n) − t ∗

(13)

In order to avoid the effect of measuring errors on the calibration accuracy, the number of the measured points is generally larger than the quantity of actual structural parameter errors. According to the least-square method, this process can be described as:   min  F(t, q, r) − t ∗ 

δr∈R

(14)

The Newton–Raphson iteration is utilized to obtain the modification value of the kinematic structural parameters:

172

S. Chen et al.



−1 T

dr = JT J J F(t, q, r) − t ∗

(15)

However, the Newton–Raphson iteration can’t converge when J is singular or ill-posed [29]. Equation (15) can be modified by the Regularization method as follows: 

−1 T

d r = J T J + αn J J F(t, q, r) − t ∗

(16)

4. The kinematic structural errors can be updated: δr (n) = δr (n−1) + d r

(17)

5. The average difference between the simulated positions and the measured positions of the terminal platform can be derived as: dt

(n)

=

(n) d t (n) 1 + · · · + d t 29 29

(18)

where dt (n) 1 is the difference between the simulated position and the measured position of the first measured point in the nth iteration, dt (n) 29 is the difference between the simulated position and the measured position of the 29th measured point in the nth iteration. If dt (n) does not converge, repeat from step 2 to step 4. If it converges, the loop is ended, and the actual values of structural error parameters can be obtained. The actual values of the structural parameters rreal can be written as: r real = δr (n) + r

(19)

In summary, the steps of the error identification of this mechanism are shown in Fig. 12. According to the above model, the average difference between the simulated positions and the measured positions of {T-xyz} converges after five iterations. The fast iteration process is illustrated in Fig. 13. After the convergence of the average of dt (n) , the kinematic structural errors can be obtained by Eq. (17). The actual kinematic structural parameter errors of the over-constrained mechanism are listed in Table 1.

5 Error Compensation and Its Verification In order to verify the validity, a group of experiments is designed and conducted in this Section. Firstly, the error model of the over-constrained mechanism is compensated. And then, the accuracy of the error model after compensation is verified.

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

173

Input actuator parameters q

Initial structural error parameters

Ideal structural parameters r

Current structural error parameters

Actual kinematic model of the over-constrained mechanism

The overconstrained force f

Simulated position of the over-constrained mechanism t(n)

Measured position of the over-constrained mechanism t*

Jacobian matrix J

Difference between the simulated position and the measured position in each point ∆t Modifications value of the structural parameters dr Update value of the structural error parameters δr(n)= δr(n-1)+dr

N

Average of differences of position converges? Y Actual structural parameters r+δr(n)

Fig. 12 Steps of the error identification Fig. 13 Procedure of the error identification

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

9

The number of iterations

10

174

S. Chen et al.

Table 1 The structural parameter errors of this mechanism No.

Structural error

Value

Unit

No.

Structural error

Value

Unit

1

δyA1

− 0.766

mm

8

δyA3

− 0.966

mm

2

δb1

1.128

mm

9

δb2

0.950

mm

3

δβ 1

− 0.001

rad

10

δβ 2

− 0.001

rad

4

δL 1

− 0.038

mm

11

δL 3

0.165

mm

5

δx A2

0.226

mm

12

δx A4

− 0.476

mm

6

δyA2

− 1.725

mm

13

δyA4

− 0.967

mm

7

δL 2

0.576

mm

14

δL 4

0.159

mm

A circular trajectory with a radius of 30 mm is utilized for error verification. As illustrated in Fig. 14, the ideal positions of the 12 measured points are selected on the circle, which are with an interval of 30 degrees. Firstly, the input actuator parameters of two linear feed systems can be obtained by the inverse kinematics. Then the terminal platform is driven to the actual positions of the 12 measured points according to the input actuator parameters. At last, the actual positions of the terminal platform are measured and converted to be expressed in the global coordinate frame. The error model with the ideal structural parameters of this over-constrained mechanism is named the error model before calibration. The one with the actual parameters is named the error model after calibration. In order to verify the accuracy of the identified kinematic errors, the predicted positions of the terminal platform are derived according to the error model before and after the calibration, respectively. Then they are compared with the measured positions of the terminal platform to derive the absolute errors of the 12 measured points. The procedure for error identification is illustrated in Fig. 15. The absolute errors of the error model before and after calibration are illustrated in Fig. 16. It can be concluded from Fig. 16 that the kinematic calibration method proposed in this paper can significantly improve the accuracy of the over-constrained mechanism. In order to further verify this kinematic calibration method, the Root Fig. 14 The selected points for error verification

190

y /mm

180 170 160 150 140 -30

-20

-10

0

x /mm

10

20

30

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

175

The ideal positions of the 12 measured points The inverse kinematics according to the kinematic error model The ideal positions of the 12 measured points Input actuator parameters of the 12 measured points The kinematic error model before calibration

The kinematic error model after calibration

The predicted positions The real positions of The predicted positions according to the the terminal platform according to the kinematic error model measured by the kinematic error model before calibration absolute tracker after calibration Absolute errors according to the kinematic error model before calibration

Absolute errors according to the kinematic error model after calibration

Fig. 15 Procedure of the error verification

Mean Square Error (RMSE) of the error model before and after calibration is calculated and illustrated in Fig. 17. It can be concluded from Fig. 17 that the calibration method proposed in this paper can reduce the RMSE of the over-constrained mechanism from 1.243 to 0.028 mm. After calibration Before calibration

Fig. 16 The absolute errors of the error model before and after calibration

Absolute errors/mm

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0

0 1 2 3 4 5 6 7 8 9 10 11 12

Number of points

176

S. Chen et al.

Fig. 17 The RMSE of the error model before and after calibration

1.4

After calibration Before calibration 1.243

1.2

RMSE/mm

1.0 0.8 0.6 0.4 0.2 0.028

0

6 Conclusion To enhance the performance of the over-constrained mechanism, this paper proposes a kinematic calibration method, which utilizes the measured force data to calibrate the actual structural parameters. This method can avoid complex modeling and eliminate the impact of the calculation errors of the forces and deformation on the calibration accuracy. To verify the feasibility of this method, a 2-DOF over-constrained parallel manipulator is designed and developed. Each chain of this mechanism contains a force sensor, which is utilized to measure the over-constrained force. By obtaining the deformation caused by the over-constrained forces and the error model of the non-over-constrained mechanism, the error model of this mechanism is established. The Newton–Raphson iteration, the Regularization method, and the least-square method are utilized to derive the structural parameter errors of this mechanism. The optimal solution for the error identification can be obtained in five iterations. Finally, a group of experiments is designed to verify the validity of this method. The experiment results show that the kinematic calibration method proposed in this paper can significantly improve the accuracy of this mechanism, reducing the RMSE of the over-constrained mechanism from 1.243 to 0.028 mm. Acknowledgements Supported by the National Natural Science Foundation of China (Grant No. 52175017).

References 1. Young E, Kuchenbecker KJ (2019) Implementation of a 6-DOF parallel continuum manipulator for delivering fingertip tactile cues. IEEE Trans Haptics 12(3):295–306 2. Wang LP, Xie FG, Liu XJ et al (2011) Kinematic calibration of the 3-DOF parallel module of a 5-axis hybrid milling machine. Robotica 29:535–546

Kinematic Calibration of a 2-DOF Over-Constrained Parallel …

177

3. Thomas MJ, Joy ML, Sudheer AP (2020) Kinematic and dynamic analysis of a 3-PRUS spatial parallel manipulator. Chin J Mech Eng 33(1):1–17 4. Mostashiri N, Dhupia JS, Verl AW et al (2018) A review of research aspects of redundantly actuated parallel robots for enabling further applications. IEEE ASME Trans Mechatron 23(3):1259–1269 5. Sun T, Liang D, Song Y (2018) Singular-perturbation-based nonlinear hybrid control of redundant parallel robot. IEEE Trans Ind Electron 65(4):3326–3336 6. Merlet J (2000) Parallel robots. Kluwer Academic Publishers, Boston, MA 7. Wang LP, Wang D, Wu J (2019) Dynamic performance analysis of parallel manipulators based on two-inertia-system. Mech Mach Theory 137:237–253 8. Zhang DS, Xu YD, Yao JT et al (2018) Analysis and optimization of a spatial parallel mechanism for a new 5-DOF hybrid serial-parallel manipulator. Chin J Mech Eng 31:1 9. Petko M, Gac K, Gora G et al (2016) CNC system of the 5-axis hybrid robot for milling. Mechatronics 37:89–99 10. Zhang DS, Xu YD, Yao JT et al (2018) Design of a novel 5-DOF hybrid serial-parallel manipulator and theoretical analysis of its parallel part. Robot Comput Integr Manuf 53:228–239 11. Chermprayong P, Zhang KT, Xiao F et al (2019) An integrated delta manipulator for aerial repair a new aerial robotic system. IEEE Robot Autom Mag 26(1):54–66 12. Brinker J, Corves B, Takeda Y (2018) Kinematic performance evaluation of high-speed delta parallel robots based on motion/force transmission indices. Mech Mach Theory 125:111–125 13. Zhang X, Mu DJ, Liu YZ et al (2019) Type synthesis and kinematics analysis of a family of three translational and one rotational pick-and-place parallel mechanisms with high rotational capability. Adv Mech Eng 11(6):1–13 14. Pradipta J, Sawodny O (2016) Actuator constrained motion cueing algorithm for a redundantly actuated Stewart platform. J Dyn Syst Meas Control Trans ASME 138(6):1–10 15. Zhou CC, Fang YF (2018) Design and analysis for a three-rotational-d.o.f. flight simulator of fighter-aircraft. Chin J Mech Eng 31(1):1–12 16. Bi ZM, Kang B (2014) An inverse dynamic model of over-constrained parallel kinematic machine based on Newton–Euler formulation. J Dyn Syst Meas Contr 136(4):041001–041009 17. Wang D, Fan R, Chen WY (2014) Performance enhancement of a three-degree-of-freedom parallel tool head via actuation redundancy. Mech Mach Theory 71:142–162 18. Pashkevich A, Chablat D, Wenger P et al (2008) Stiffness analysis of 3-d.o.f. overconstrained translational parallel manipulators. In: 2008 IEEE international conference on robotics and automation, Pasadena, California, USA, May 19–23, 2008, pp 1562–1576 19. Liu WL, Xu YD, Yao JT et al (2017) Methods for force analysis of overconstrained parallel mechanisms: a review. Chin J Mech Eng 30(6):1460–1472 20. Jeong JI, Kang DS, Cho YM et al (2004) Kinematic calibration for redundantly actuated parallel mechanisms. J Mech Des 126(2):307–318 21. Kim HS (2005) Kinematic calibration of a Cartesian parallel manipulator. Int J Control Autom Syst 3(3):453–460 22. Zeng Q, Ehmann KF, Cao J (2014) Tri-pyramid robot: design and kinematic analysis of a 3-d.o.f. translational parallel manipulator. Robot Comput Integr Manuf 30(6):648–657 23. Jiang Y, Li TM, Wang LP et al (2018) Kinematic error modeling and identification of the over-constrained parallel kinematic machine. Robot Comput Integr Manuf 49:105–119 24. Ecorchard G, Neugebauer R, Maurine P (2010) Elasto-geometrical modeling and calibration of redundantly actuated PKMs. Mech Mach Theory 45(5):795–810 25. Li FC, Zeng Q, Ehmann KF et al (2019) A calibration method for overconstrained spatial translational parallel manipulators. Robot Comput Integr Manuf 57:241–254 26. Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44(247):335–341 27. Cutkosky RE (1951) A Monte-Carlo method for solving a class of integral equations. J Res Natl Bur Stand 47(2):113–115 28. King GW (1951) Monte-Carlo method for solving diffusion problems. Ind Eng Chem 43(11):2475–2478

178

S. Chen et al.

29. Businger PA, Golub GH (1969) Singular value decomposition of a complex matrix. Commun ACM 12:10 30. Golub GH, Reinsch C (1970) Singular value decomposition and least squares solutions. Numer Math 14:5 31. Rosen EM, Provder T (1971) The instrument spreading correction in GPC. III. The general shape function using singular value decomposition with a nonlinear calibration curve. J Appl Polym Sci 15(7):1687–1702 32. Huang P, Wang JS, Wang LP et al (2011) Kinematical calibration of a hybrid machine tool with regularization method. Int J Mach Tools Manuf 51(3):210–220

Optimal Study for Multi-field Coupling of the Disc Brake Based on Kriging Agent Model Meisheng Yang, Wen Jiang, Changwei Zhang, and Jiahan Bao

Abstract In order to alleviate the problem of the temperature rise, thermal deformation and unreasonable local distribution of equivalent stress which will cause fatigue damage and reduction of the service life of the disc brake. In this study, the friction-thermal-mechanical multi-dimensional coupling analysis of the disc brake of belt conveyor is carried out, and the distribution of the transient temperature field and stress field of the disc brake under the emergency braking condition is obtained. The structural parameters of brake disc and brake shoe are taken as design variables, and the maximum temperature and maximum equivalent stress of the brake disc are taken as the optimization objectives. The Kriging response surface proxy model of the maximum temperature and maximum equivalent stress of the brake disc is constructed by using the optimal space filling to collect the data sample points. According to this, the multi-objective genetic algorithm (MOGA) is used to optimize the design, and the optimized structural parameters are obtained, such as thickness M. Yang (B) · W. Jiang · C. Zhang · J. Bao School of Mechanical Engineering, Anhui University of Technology, Maanshan 243002, Anhui, China e-mail: [email protected] W. Jiang e-mail: [email protected] C. Zhang e-mail: [email protected] J. Bao e-mail: [email protected] M. Yang State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, Zhejiang, China M. Yang · J. Bao Engineering Technology Research Center of Hydraulic Vibration Technology, Anhui University of Technology, Maanshan 243002, Anhui, China

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_10

179

180

M. Yang et al.

and diameter of the brake disc, thickness, length and width of the brake shoe. The maximum temperature, maximum equivalent stress and maximum deformation are reduced, and the fatigue life cycle is improved, which provides a theoretical basis and reference basis for the improvement of relevant technologies of the disc brakes. Keywords Disc brake · Friction thermal mechanical coupling · Kriging agent model · Multi objective genetic algorithm (MOGA) · Optimal design

1 Introduction As the main equipment of coal mine production and transportation, belt conveyor includes the braking system, the hydraulic system and the electrical control system. Disc brake is widely used in the braking system of belt conveyor because its braking torque can be adjusted by oil pressure, compact structure and high safety and reliability [1]. Disc brake uses the friction between brake shoe and brake disc to produce braking torque, and converts the system kinetic energy generated under the emergency braking condition of belt conveyor into heat energy to realize braking. As most of the heat energy generated by the friction between the brake disc and the brake shoe is absorbed by the brake disc [2], the temperature on the surface of the brake disc rises, affecting the normal operation of the disc brake. Therefore, frictionthermal-mechanical multi-dimensional coupling analysis of the disc brake and the optimization of its structural parameters have important theoretical and practical significance for improving the reliability and safety of the brake. Scholars at home and abroad have done a lot of research on the disc brakes. Barber [3] studied the “hot spots” of the brake disc, and proved through the test that the braking process of the disc brake belongs to the research scope of thermal mechanical coupling. Akshit [4] used ANSYS to conduct thermal analysis and simulation, obtained the temperature field, stress field distribution and deformation of the disc brake contour, and optimized the distribution angle of the heat dissipation hole of the brake disc. Jaiswal [5] studied the fracture of disc brake due to high temperature caused by multiple braking, and carried out finite element analysis with workbench to select more high temperature resistant materials for disc brake. Sun [6] carried out the finite element analysis of fluid structure coupling for the reliability of disc brake, and proved through experiments that the reliability of disc brake is related to flow field, temperature field and stress field. Rahim [7] simulated the distribution of air flow and temperature during braking by finite element method, and numerically optimized the disc brake with the main structural parameters of the brake disc as the design variables, which greatly reduced the cooling time of the brake disc. Hui [8] carried out in-depth research on the noise reduction of disc brake, introduced the response surface method to replace the finite element model, improved the calculation efficiency, and optimized it by genetic algorithm. The results of numerical examples showed that the proposed optimization method is effective in reducing squeak of the disc brake system.

Optimal Study for Multi-field Coupling of the Disc Brake Based …

181

In this paper, the friction-thermal-mechanical multi-dimensional coupling model of the disc brake is established and the finite element model is carried out. Taking the structural parameters of the disc brake as the design variables and the maximum temperature and maximum equivalent stress of the brake disc as the optimization objectives, the optimal space filling experimental design is used to collect sample points, and the Kriging response surface model of the temperature field and stress field of the brake disc is established. Based on this, the optimal candidate points are obtained in the design area by using the multi-objective genetic algorithm, and the optimization results are verified. The influence of brake structure size on friction heat generation and equivalent stress is further revealed. The organization of this paper is as follows: Sect. 2 expounds the working principle and parameters of disc brake; the mathematical modeling and multi-dimensional coupling simulation of disc brake are carried out in Sect. 3; Sect. 4 carries out multi-objective optimization design; and Sect. 5 has concluding remarks.

2 Overview of the Disc Brake 2.1 Working Principle of the Disc Brake As shown in Fig. 1, the disc brake of belt conveyor is mainly composed of brake disc, disc brake, base frame, dust cover, transmission drum and so on. The brake disc is coaxially connected with the transmission drum of the belt conveyor. As shown in Fig. 2, the disc brake of the braking system is mainly composed of connecting shaft 1, brake shoe 2, butterfly spring 3, cylinder 4, piston 5 and so on.

Fig. 1 Disc brake of the belt conveyor

182

M. Yang et al.

1. Connecting shaft 2. Brake shoe 3. Butterfly spring 4. Cylinder 5. Piston 6. Bolt 7. End cap 8. Screw 9. Adjusting nut 10. Housing 11,15. Motor 12,16. Gear pump 13,17. Coarse filter 14,19. Overflow valve 18,35. Fine filter 20,24. Stop valve 21. Proportional relief valve 22. Oil tank 23. Manual pump 25,27,32. Reversing valve 26. Throttle valve 28. Pressure gauge 29. Pressure sensor 30, 34. Check valve pressure 31. Accumulator 33. Pressure relay Fig. 2 Brake structure and hydraulic system diagram

The control system of the belt conveyor sends a signal, and the hydraulic system can adjust the oil pressure in the brake cylinder according to the command, so as to increase or decrease the braking torque of the disc brake. When the belt conveyor works normally, hydraulic oil is introduced, and the piston drives the connecting axial to move back under the action of hydraulic oil, so as to keep the brake shoe away from the brake disc. When the equipment needs braking, under the control of electro-hydraulic system, the braking high-pressure oil is discharged from the oil cylinder, the oil pressure gradually decreases, the disc spring restores the tension, and the end of the connecting shaft contacts with the brake shoe to make the brake shoe close to the brake disc to produce friction braking.

2.2 Parameters of the Disc Brake In this paper, KPZ-800disc brake is selected as the research object, and the parametric model is established. The geometric dimension parameters of the brake disc and brake shoe are shown in Tables 1 and 2 respectively. In the study, the brake disc material

Optimal Study for Multi-field Coupling of the Disc Brake Based …

183

Table 1 Geometric dimensions of the brake disc External diameter/mm

Internal diameter/mm

Thickness/mm

Diameter of the cooling hole/mm

800

160

25

30

Table 2 Geometric dimensions of the brake shoe

Length/mm

Width/mm

Thickness/mm

250

158

20

Table 3 Material parameters Parts

Parameters

Brake disc (kg/m3 )

Brake shoe

7650

2620

Modulus of elasticity (GPa)

210

2.3

Poisson ratio

0.3

0.29

Density

Thermal conductivity (W/(m K))

47.68

2

Coefficient of thermal expansion (10−6 K−1 )

12.8

11

Specific heat capacity (J/(kg K))

472

1100

is 16Mn steel, and the brake shoe is environment-friendly asbestos free composite material [9]. The material properties are shown in Table 3.

3 Finite Element Simulation of the Disc Brake 3.1 Determination of Braking Conditions and Boundary Conditions The parameters are set according to the actual emergency braking conditions of the disc brake. When speed of the brake disc is 5 rad/s, the braking pressure rise to 7 MPa in 0–0.2 s, the friction coefficient between the brake disc and the brake shoe is 0.3, the braking deceleration of the belt conveyor is 0.3 m/s2 , and the relationship between angular velocity and time in the braking process is: w = w0 −

a t = 5 − 0.75t R

(1)

184

M. Yang et al.

The heat transfer between the brake disc and the brake shoe is mainly in three ways: heat conduction, heat convection and heat radiation. Due to the short braking time and large friction heat, the influence of heat radiation is not considered. Heat conduction and heat convection are two forms that have a great impact on the temperature field [10]. The physical effect of heat conduction is added by defining the thermal conductivity of brake disc and brake shoe. The convective heat transfer coefficient of brake disc is related to its surface shape, temperature and air flow rate [11]. During braking, forced convective heat transfer will occur between brake disc and external atmosphere. Input heat and output heat of the brake disc and the brake shoe are carried out alternately, and the convective heat transfer coefficient of the brake disc changes with the change of speed, The convective heat transfer coefficient [12] can be expressed as:  hc =

0.04(ka /d0 )Re0.8 , 0.7(ka /d0 )Re0.55 ,

Re > 2.4 × 105 ; Re ≤ 2.4 × 105 .

(2)

The Reynolds number in the air is: Re =

u a pa d0 μa

(3)

Combined with Eqs. (1), (2) and (3), the convective heat transfer coefficient varying with the angular velocity of the brake disc is:  hc =

3.644w 0.8 , 5.4371w 0.55 ,

w > 12.676 rad/s; w ≤ 12.676 rad/s.

(4)

According to the theory of heat transfer, the heat conduction differential equation [13] of the brake disc temperature field of the disc brake system without internal heat source in the cylindrical coordinate system is:   ∂T 1 ∂2T λ ∂2T 1 ∂T ∂2T = + + + 2 2 2 2 ρc ∂r r ∂r r ∂θ ∂z ∂t

(5)

The heat flow input of the brake disc and the brake shoe during braking is treated as a boundary condition, and the heat load is loaded on the friction surface of the brake disc in the form of moving heat source. The heat flow density input from the friction surface meets the following requirements: q(r, θ, t) = μ · p(r, θ, t) · v(r, θ, t) = μ · p(r, θ, t) · w(t) · r

(6)

Optimal Study for Multi-field Coupling of the Disc Brake Based …

185

3.2 Thermal Mechanical Coupling Analysis of the Disc Brake Workbench is used to conduct friction-thermal- mechanical multi-dimensional coupling finite element simulation of the disc brake, analyze the simulation results, and obtain the cloud diagram of brake disc temperature field and equivalent stress field under emergency braking conditions, as shown in Figs. 3 and 4.

Fig. 3 Cloud diagram of maximum temperature

Fig. 4 Cloud diagram of maximum equivalent

186

M. Yang et al.

Figure 5 shows the curve of the maximum temperature of the brake disc with time under the emergency braking condition of the disc brake. The maximum temperature curve rises in a “zigzag” shape, which is due to the dynamic fluctuation of the temperature caused by the mutual friction heat between the brake shoe and the brake disc through the two physical effects of heat conduction and heat flow. In 0–0.2 s, the positive pressure of the brake shoe increases gradually, and the temperature of the brake disc increases rapidly. In the middle stage of braking, the temperature rise slows down with the decrease of speed, and reaches the maximum temperature of 160.32 °C in 2.956 s; In the later stage of braking, because the effect of heat conduction and heat convection heat dissipation is greater than that of heat generated by braking friction, the temperature will drop slowly. At the end of braking, the maximum temperature of the brake disc surface is 115.16 °C. In order to analyze the radial temperature distribution in the friction area of the brake disc, three nodes N1 , N2 and N3 are selected as the research. The node N1 is 0.175 m away from the center of the brake disc and is measured in the friction area; Node N2 is 0.2 m away from the center of the brake disc and is located in the friction center area. Node N3 is 0.3 m away from the center of the brake disc, which is outside the friction area. The radial node temperature change of the brake disc surface is shown in Fig. 6. The temperature curves of nodes N1 , N2 and N3 with time form an obvious temperature gradient. The maximum temperature of node N1 measured in the friction area is only 27.118 °C, and there is no obvious change in temperature rise. The maximum temperature of node N2 in the friction center area is 106.32 °C. Node N3 is located outside the friction area, and the maximum temperature reaches 154.32 °C. This phenomenon further confirms formula (6). On the surface of the

Fig. 5 Variation of maximum temperature of brake disc surface with time

Optimal Study for Multi-field Coupling of the Disc Brake Based …

187

N1 N2 N3

160 140

Temperature

120 100 80 60 40 20

0

1

2

3

4

5

6

7

Time (s) Fig. 6 Temperature change diagram of radial node in brake disc friction area

friction area of the brake disc, the farther away from the center of the brake disc, the greater the input heat flux density [14], and the higher the temperature rise. Under emergency braking conditions, the brake disc is deformed under the influence of temperature and local stress concentration. The cloud diagram of brake disc surface deformation is shown in Fig. 7, and the curve of maximum thermal expansion deformation of brake disc with time is shown in Fig. 8. The brake disc surface has thermal expansion deformation under the influence of temperature and extrusion deformation under the influence of local stress concentration. At 0–0.2 s, the temperature rises rapidly due to the increase of pressure, and the thermal expansion deformation curve of the friction area of the brake disc also increases accordingly. At the later stage of braking, the thermal expansion deformation curve of the brake disc surface will continue to rise slowly due to the influence of residual temperature, which reaches 1.29 × 10−2 mm at end of braking. Due to the stress concentration at the inner diameter of the heat dissipation hole of the brake disc and the distance from the friction area, it is less affected by temperature. Its deformation mode is extrusion deformation, and the maximum extrusion deformation is 3.6367 × 10−3 mm. The diagram of fatigue life distribution of the brake disc is shown in Fig. 9. The minimum fatigue life of the brake disc is located at the equivalent stress concentration of inner diameter of the brake disc. When experiencing multiple emergency braking, the thermal expansion and extrusion deformation accumulated by damage will cause fatigue cracks and fatigue fracture of the brake disc. Therefore, optimizing the structural parameters of the brake to reduce the maximum temperature and maximum equivalent stress of the brake disc is of great significance to strengthen the service life of the brake disc and the reliability and safety of the disc brake.

188

M. Yang et al.

Fig. 7 Diagram of brake disc surface deformation

Fig. 8 Variation of brake disc thermal expansion with time

4 Multi-objective Optimization of the Disc Brake 4.1 Design Variables and Optimization Objectives In order to reduce the maximum temperature and maximum equivalent stress of the brake disc under emergency braking conditions, taking the minimum-maximum

Optimal Study for Multi-field Coupling of the Disc Brake Based …

189

Fig. 9 Diagram of fatigue life distribution of brake disc

temperature and maximum equivalent stress of the brake disc as the optimization objectives, the size parameters of the brake disc and the brake shoe as the design variables, and the minimum temperature and equivalent stress as the constraint conditions, the numerical optimization method [15] is used to optimize the structural parameters of the disc brake. The optimization range of design variable diameter of brake disc (X1 ), thickness of brake disc (X2 ), length of brake shoe (X3 ), width of brake shoe (X4 ) and thickness of brake shoe (X5 ) is shown in Table 4. The sample points generated by optimal space filling [16] have good space filling ability. This sampling method can fit Kriging model with high precision through a relatively small number of sample data points. The sample points obtained by sampling are substituted into the original model for finite element analysis. Some sample points and finite element calculation results are shown in Table 5. Table 4 Value range of design variables Design variable

X1 /mm

X2 /mm

X3 /mm

X4 /mm

X5 /mm

Initial value

800

25

250

158

20

Upper limit

770

22.5

225

143.2

18

Lower limit

830

27.5

275

173.8

22

190

M. Yang et al.

Table 5 Data and calculation results of the sampling points 1

X1 /mm

X2 /mm

X3 /mm

X4 /mm

X5 /mm

Temperature/°C

Stress/MPa

805.25

24.18

241.87

171.03

18.35

156.90

281.18

2

797.75

23.81

271.87

163.13

18.45

171.32

308.80

3

775.25

25.68

251.87

160.76

18.15

164.66

267.68

4

820.25

22.68

256.87

165.51

19.75

164.38

304.82

5

784.25

25.43

248.12

167.87

21.95

163.30

270.54

6

776.75

26.06

266.875

147.35

19.55

166.55

255.07

7

799.25

27.31

253.12

164.71

18.55

160.06

251.77

8

817.25

24.93

234.37

144.96

20.95

154.83

232.21

9

790.25

22.93

233.12

157.60

18.65

158.79

267.35

10

770.95

23.06

254.37

159.97

19.35

166.85

298.01

…

…

…

…

…

…

…

…

37

785.75

26.93

246.87

145.75

20.85

155.26

239.47

38

782.75

24.68

236.87

142.59

19.25

158.57

230.15

39

821.75

23.43

235.62

150.49

18.95

159.27

250.88

40

796.25

26.43

228.12

152.86

18.75

153.96

224.66

41

787.25

23.68

274.37

156.02

20.35

178.90

302.67

42

779.75

25.06

264.37

172.61

19.95

167.30

305.86

43

778.25

24.56

226.87

155.23

21.05

150.61

238.57

44

815.75

26.18

249.37

143.38

18.85

162.05

246.80

45

794.75

24.31

259.37

146.54

18.05

163.06

259.77

46

829.25

25.93

250.62

166.29

21.25

162.84

259.50

4.2 Construction and Verification of Kriging Response Surface Model The essence of constructing the response surface is to construct the approximate model between the design variable and the objective function [17]. The response surface provides the estimated value of the output parameters. In the whole area of the analysis design space, the approximate value of the output function can be obtained without complete calculation [18]. The theoretical basis for constructing the response surface is the least square regression method. Generally, the regression equation is used to express the relationship between the response and the term in the Kriging proxy model. The complete polynomial of the fitted second-order Kriging proxy model [19]. y(x) = a0 +

n  i=1

ai xi +

n  i=1

aii xi2 +

n  i=1, j=i

ai j xi x j + ε

(7)

Optimal Study for Multi-field Coupling of the Disc Brake Based …

191

The collected sample points and formula (7) are fitted into a mathematical model related to the design variables x1 , x2 , x3 , x4 and x5 . The Kriging response surface approximate models of the maximum temperature and maximum equivalent stress of the brake disc are obtained as formulas (8) and (9): y1 = 6595 − 8.7x1 − 79.2x2 − 2.49x3 − 7.93x4 − 108.9x5 + 0.0249x1 x2 + 0.00051x1 x3 − 0.00537x1 x4 + 0.0225x1 x5 + 0.0030x2 x3 + 0.0425x2 x4 + 0.243x2 x5 − 0.00195x3 x4 − 0.0098x3 x5 + 0.1501x4 x5 + 0.00521x12 + 0.92x22 + 0.0058x32 + 0.0275x42 + 1.59x52

(8)

y2 = −12,045 + 18.9x1 + 50x2 + 20.8x3 + 12.4x4 + 88x5 + 0.0355x1 x2 + 0.006x1 x3 − 0.00911 x4 + 0.0199x1 x5 − 0.0851x2 x3 + 0.02263x2 x4 + 0.0803x2 x5 + 0.00582x3 x4 + 0.061x3 x5 − 0.078x4 x5 − 0.0108x12 − 0.55x22 − 0.01328x32 − 0.0202x42 − 2.61x52

(9)

The Kriging response surface model of the maximum temperature and maximum equivalent stress of the disc brake of the belt conveyor is established in the workbench, as shown in Figs. 10 and 11. The accuracy of Kriging response surface model obtained by fitting the sampling points is related to the algorithm of collecting sample points and the number of sample

0

17

5

0 16 5

15

isc

(m

m)

23

0

of

ed

Fig. 10 Maximum temperature and dependent variable response surface

m oe (m sh

ak e

240

br

250

br ak

gt h

of

Le n

ss

27

kn e

26

hic

)

270

260

25

( )

0 15 5 4 T1

24

temperature

16

23

Maximum

180 5 17

192

M. Yang et al.

320 Equivalent

300 28

0

0

24

24

m

th

230

of

0

(m m)

Le ng

isc

27

ed

br ak e

250

br ak

26

of

sh oe (m

23

ss

)

270

260

25

20

2 T hic kn e

24

Stress(MPa)

0 26

Fig. 11 Maximum equivalent stress and dependent variable response surface

points in the experimental design. The approximate value obtained by the established response surface model has a certain error with the real finite element results [20]. Therefore, it is necessary to analyze the error of the model. In industrial design, the accuracy of the response surface is usually judged according to the certainty coefficient. When it is greater than 0.9, it indicates that the accuracy of the model is high. If it is lower than 0.9, it indicates that the accuracy of the model is not enough, and the approximate model needs to be reestablished. The certainty coefficient can be expressed as [21]: n (y i − yi )2 R = 1 − i=1 n 2 i=1 (yi − y) 

2

(10)

The ten verification points were regenerated, the error between the real value calculated by the real finite element method and the approximate value obtained according to the response surface were analyzed. It can be seen from Figs. 12 and 13 that the fitting points of the maximum temperature and the maximum equivalent stress of the brake disc are distributed near the fitting line. The values of the certainty coefficient are shown in Table 6. The certainty coefficients of the Kriging proxy model for the maximum temperature and maximum equivalent stress are greater than 0.9. The accuracy of the constructed Kriging proxy model meets the requirements, and the error between the predicted value obtained from the response surface and the real value calculated by the finite element is within the allowable range [22].

Optimal Study for Multi-field Coupling of the Disc Brake Based …

193

185 180 175

True value

170 165 160 155 150 145 145

150

155

160

165

170

175

Predictive value Fig. 12 Error analysis of maximum temperature

Fig. 13 Error analysis of maximum equivalent stress Table 6 Certainty coefficient R2

Response surface

Certainty coefficient R2

Maximum temperature

0.95668

Maximum equivalent stress

0.9897

194

M. Yang et al.

4.3 Optimization Based on Multi-objective Genetic Algorithm In this paper, MOGA (multi-objective genetic algorithm) is used for optimization. While continuing to maintain the advantages of traditional genetic algorithm (GA), the multi-objective genetic algorithm is applied to multi-objective, and its purpose is to obtain the optimal candidate point in the global optimization [23]. The structural parameters of the disc brake are took as the design variables, the temperature and equivalent stress are took as the optimization objectives, the minimum maximum temperature and maximum equivalent stress are took as the constraints, a multi-objective optimization is established. The mathematical model of multi-objective optimization can be expressed as follows: min y1 (x) min y2 (x) ximin ≤ xi ≤ ximax , i = 1, 2, 3, 4, 5

(11)

The multi-objective genetic algorithm [24] is used to solve the parameters with the most design variables. The feature of multi-objective optimization is to obtain multiple groups of candidate points that meet the conditions through iterative solution [25]. The initial number of samples is 5000, the number of samples per iteration is 1000, the convergence criterion is 70%, and the maximum number of iterations is 20. The recommended optimization result parameters obtained by seeking the global optimal solution on the response surface are shown in Table 7. The structural parameters after rounding the disc brake parameters are shown in Table 8. Through the multi-objective optimization of the disc brake, the cloud map of the optimized results is obtained, as shown in Fig. 14: (a) is the cloud map of the optimized maximum temperature distribution, (b) is the curve of the optimized maximum temperature with time, the maximum value is 146.63 °C, which is 8.53% lower than that before optimization, (c) is the optimized the maximum equivalent stress distribution of the brake disc after optimization, the maximum value is 221.6 MPa, which Table 7 Parameters of optimized candidate points Point

X1 /mm

X2 /mm

X3 /mm

X4 /mm

X5 /mm

Point 1

770.6

27.456

225.24

142.28

21.929

Point 2

770.6

27.137

225.02

142.35

21.95

Point 3

770.02

27.083

225.17

142.35

21.894

Table 8 Optimized parameter values Parameters

X1 /mm

X2 /mm

X3 /mm

X4 /mm

X5 /mm

Initial

800

25

250

158

20

Optimized

770.6

27.5

225.2

142.3

22

Optimal Study for Multi-field Coupling of the Disc Brake Based …

195

is 17.36% lower than that before optimization, (d) is the optimized fatigue life of the rear brake disc, the minimum value is 37,689 times, which is 162.7% higher than that before optimization, (e) is the optimized distribution cloud diagram of the maximum thermal expansion deformation in the friction area of the rear brake disc surface, (f) is the curve of the maximum thermal expansion with time after optimization, and the maximum value is 1.186 × 10−2 mm, which is 8.56% lower than that before optimization. The optimization result is significantly improved, the multi-objective optimization of disc brake is realized, and the service life of brake disc is greatly improved.

5 Conclusion (1) In this paper, the finite element simulation analysis of friction-thermal–mechanical multi-dimensional coupling is carried out for the disc brake. In the early stage of braking, the temperature of the brake disc increases rapidly, the effect of heat conduction and heat convection in the later stage of braking is greater than that of friction heat generation, and the temperature decreases gradually. The temperature of the radial node in the friction area of the brake disc changes in a positive gradient. The friction area on the surface of the brake disc expands and deforms under the influence of temperature, and the inner wall of the heat dissipation hole of the brake disc is extruded and deformed under the influence of stress concentration. (2) The Kriging response surface proxy model of maximum temperature and maximum equivalent stress is established by collecting test sample points in the way of optimal hyper Latin cubic sampling. The accuracy of the proxy model is tested, and its certainty coefficients are greater than 0.9, which proves that the accuracy of the Kriging proxy model meets the requirements and can replace the finite element model for multi-objective optimization design. (3) The multi-objective genetic algorithm (MOGA) is used for multi-objective optimization design. After optimizing the size of the brake disc and the brake shoe of the disc brake. It is concluded that the maximum temperature of the brake disc is reduced from 160.32 to 146.63, which decreases by 8.41%; The maximum equivalent stress decreased from 258.6 to 221.6, which decreases by 17.34%; The maximum thermal expansion deformation in the friction area of the brake disc surface is reduced from to, which reduces by 8.52%; The minimum fatigue life of the brake disc is increased from 14,344 times to 37,689 times, which increases by 162.7%, which proves the reliability and effectiveness of the optimal design.

196

M. Yang et al.

Fig. 14 Optimized results

(A) Maximum temperature distribution after optimization

(B) Variation of maximum temperature with time after optimization

(C) Maximum equivalent stress distribution after optimization

Optimal Study for Multi-field Coupling of the Disc Brake Based … Fig. 14 (continued)

(D) Fatigue life distribution after optimization

(E) Maximum deformation distribution after optimization

(F) Variation diagram of thermal expansion deformation with time after optimization

197

198

M. Yang et al.

Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 52105041), Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (Grant No. GZKF-202018), and Key Projects of Natural Science Research in Anhui Universities (Grant No. KJ2020A0258).

Appendix Thermal conductivity of air Outer diameter of brake disc Density of air Dynamic viscosity of air Material thermal conductivity Material density Material specific heat capacity Friction coefficient between brake disc and brake shoe Angular speed of brake disc rotation Contact pressure Approximate values obtained from response surface model Real value obtained by finite element analysis Average of true values Design variables of disc brake Upper and lower limits of design variables Target maximum temperature Target maximum equivalent stress

k a = 0.0276 W/m K d 0 = 0.8 m pa = 1.13 kg/m3 μa = 1.91 × 10− 5 N s/m2 λ (W/m K) ρ (kg/m3 ) c (J/(kg K)) μ w (rad/s) p (kN) yˆi yi y x = (x 1 , x 2 , x 3 , x 4 , x 5 ) ximin , ximax y1 (x) y2 (x)

References 1. Jia Y, Dai Y (2014) Design of disc brake for mine hoist. Mach Tools Hydraulics 42(10):57– 58+71 2. Hou Y, Xie F, Huang F (2011) Control strategy of disc braking systems for downward belt conveyors. Min Sci Technol (China) 21(4):491–494 3. Barber JR (1969) Thermoelastic instabilities in the sliding of conforming solids. Proc R Soc Lond 312(1510):381–394

Optimal Study for Multi-field Coupling of the Disc Brake Based …

199

4. Akshit C, Aryan J, Akshay K, Mohammad Z (2022) Coupled thermal and structural analysis of disc brake rotor with varying angle of rotation of ventilation holes. Mater Today Proc 56(2):834–844 5. Jaiswal R, Jha AR (2016) Structural and thermal analysis of disc brake using solidwork and ANSYS. Int J Mech Eng Technol 7(1):67–77 6. Sun D, Liu S, Hao Y (2019) Thermal reliability analysis of in-wheel wet brake assembly under high-intensity braking with multi-field coupling. Autom Eng 41(2):161–169 7. Rahim J, Recep A (2022) Optimization and thermal analysis of radial ventilated brake disc to enhance the cooling performance. Case Stud Thermal Eng 30(10):173–186 8. Hui L, Yu D (2016) Optimization design of a disc brake system with hybrid uncertainties. Adv Eng Softw 98:112–122 9. Kumar N, Singh A, Singh S, Singh JIP, Kumar S (2021) Napier natural fibre used as reinforcement polymer composite: as asbestos free brake friction material. Mater Today Proc 56:2532–2536. Part 5, ISSN 2214-7853 10. Wang N, Yu Z, Zong C (2020) Simulation analysis and experiment study on instantaneous temperature field of disc brake for mine belt conveyor. Coal Min Mach 41(5):74–76 11. Mcphee AD, Johnson DA (2008) Experimental heat transfer and flow analysis of a vented brake rotor. Int J Therm Sci 47(4):458–467 12. Young J, Ikjin L, Sung JK (2017) Topology optimization of heat sinks in natural convection considering the effect of shape-dependent heat transfer coefficient. Int J Heat Mass Transf 109:123–133 13. Liu Z, Li M, Zhang G (2013) Simulation and analysis on temperature field of disk brake for large belt conveyors in collieries. Min Mach 41(08):64–67 14. Han W (2019) Analysis on temperature distribution of disc brake in mine transportation system based on finite element method. Mech Res Appl 32(01):68–70+73 15. Li W, Yue W, Huang T, Ji N (2021) Optimizing the aerodynamic noise of an automobile claw pole alternator using a numerical method. Appl Acoust 171:45–56 16. Wu J (2020) A new sequential space-filling sampling strategy for elementary effects-based screening method. Appl Math Model 83:419–437 17. Lu H, Yu D, Xie Z, Lu H (2013) Optimization of vehicle disc brakes stability based on response surface method. J Mech Eng 49(09):55–60 18. Wang G, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):415–426 19. Arthur K, Julien M, Michel K (2015) Global extremum seeking by Kriging with a multi-agent system. IFAC-Papers 48(28):526–531 20. Simpson TW, Mauery TM, Korte JJ (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39:2233–2241 21. Falsone G, Impollonia N (2004) About the accuracy of a novel response surface method for the analysis of finite element modeled uncertain structures. Probab Eng Mech 19(1):53–63 22. Duddeck F (2008) Multidisciplinary optimization of car bodies. Struct Multi Optim 35(4):375– 389 23. Wang Y, Chen W, Xie Y, Deng S (2017) Application of multi-objective genetic algorithm to body-in-white dynamic performance optimization. Autom Eng 39(11):1298–1304 24. Deb K, Pratap A, Agarwal S (2002) A fast and elitist multi-objective genetic algorithm: NSGAII. IEEE Trans Evol Comput 6(2):182–197 25. Yang J, Zhang D (2019) Multi-objective optimization of container based on genetic algorithm. Veh Power Technol 04:1–36

Analysis on Noise of High Pressure Direct Injection System Wenqiang Liu, Yongjiang Xu, Xiaolong Deng, Junfeng Hu, Bing Gong, and Zhi Wang

Abstract The identification and solution of NVH problem is the main way to continuously improve the comfort in the mechanical field. With the application of 350 bar high pressure fuel supply system, the solution of noise problem of high-pressure system has become an important work content in the development of sound quality of automobile cab. This paper studies the noise reduction of high-pressure direct injection engine, comprehensively analyzes the key factors of high-pressure direct injection system noise, and provides new ideas and references for the system to solve the high-pressure direct injection noise problem. The main path solutions for high-pressure direct injection noise and vibration problems in the mechanical field are to optimize the structural mode, increase vibration isolation, change the pipeline direction, and optimize the injection strategy. Keywords Harshness (NVH) · Sound quality · High pressure direct injection

1 Introduction With the development of mechanical automation, consumers have higher and higher requirements for product comfort. In the field of mechanical automation, the main indexes to measure product comfort are noise, vibration and acoustic-vibration roughness, referred to as NVH (Noise, Vibration, Harsh-ness). The NVH problem of automobile is transmitted to the vehicle by various excitation such as powertrain, road surface and environment through different transmission paths. Among them, the direct injection system of high pressure direct injection engine is a very important high frequency noise source [1–4]. W. Liu · Z. Wang Tsinghua University, Beijing 100084, China W. Liu · Y. Xu · X. Deng · J. Hu (B) · B. Gong Ningbo Geely Royal Engine Components Co., Ltd., Ningbo 315336, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_11

201

202

W. Liu et al.

In this paper, the difficult high frequency noise problem encountered in the development of NVH for a certain engine is solved. By means of injection pulse width adjustment scanning and modal analysis, the mechanism of the high frequency noise is identified as the coupling between injection pulse excitation and the natural frequency of oil in the branch pipe of high pressure oil rail, resulting in resonance. By optimizing the injection pulse width, the noise source problem is solved. The path transfer is reduced by optimizing the connection relationship between oil rail and engine cylinder head [5–7]. After the implementation of the two schemes, the problems have been effectively solved. In this paper, the theory of high-pressure oil resonance is put forward, which provides an effective solution to the problem that the traditional optimization scheme of high-pressure system can’t completely solve the noise of high-pressure system, and provides a new idea for the analysis and solution of noise and vibration problem.

2 Noise Principle of High Pressure Direct Injection Injection mechanical noise: When the injector works, the armature will hit the connecting rod in the process of opening the injector, and the valve ball will hit the valve seat in the process of closing the injector, both of which will generate vibration and noise, which will radiate and spread outward through the injector shell, thus forming the working noise of the injector. For this noise, we can design a light needle valve or shorten the stroke of the needle valve at the source, and use the vibration isolation unit APM of injector and cylinder head installation point, vibration isolation unit APH of injector and oil rail installation point or suspension fuel injector at the path. The combination of these schemes can effectively solve such problems. Mechanical noise of high pressure oil pump: When the high-pressure oil pump works, the solenoid valve of the high-pressure oil pump opens and closes, and the impact of the needle valve produces the working noise of the high-pressure oil pump. In view of this noise, applying buffer current to the source to slow down the seating speed of the needle valve, lightening the needle valve, adopting thick damping cover plate, and controlling the working cycle and phase of the high-pressure oil pump are all effective schemes. Resonance noise of high pressure tubing: The high-pressure oil pipe is a slender pipe. When the high-pressure oil rail and high-pressure oil pump excite the mode of the high-pressure oil pipe, the noise and vibration will increase obviously. The vibration of the oil pipe is transmitted to the car through the path, resulting in obvious knocking sound. The noise can be effectively solved by adding Damper to the highpressure tubing to absorb vibration energy. In addition, the structural optimization of the tubing and the vibration isolation of the connection path around the tubing can also have certain effects. Fuel pulsation noise: When the injector is opened and closed, the positive water hammer effect will cause the fluctuation of oil pressure. When the opening and closing period is coupled with the period of oil pressure fluctuation, the resonance

Analysis on Noise of High Pressure Direct Injection System

203

of oil in the oil rail will be aroused, which will cause the noise of the high-pressure system to increase sharply. This paper mainly aims at the fuel pulsation noise, combined with the high pressure direct injection noise of a certain model from 1600 to 1900 Hz, and carries out systematic analysis and verification work.

3 Noise Analysis of High Pressure Direct Injection The test vehicle in this paper is a passenger car, and the main parameters are shown in Table 1. The vehicle picture is shown in Fig. 1. There is obvious noise problem of high pressure direct injection in the car during idling and acceleration. By forcibly stopping the operation of the high-pressure oil pump, it can be intuitively judged that this problem is strongly related to the high-pressure common rail system, as shown in Fig. 2. When the high-pressure oil pump stops working, the knocking sound in the car and near field disappears, and when it resumes working, the knocking sound reappears. In order to solve this problem, the mechanism of the high frequency noise is identified as the coupling between the injection pulse excitation and the natural frequency of the oil in the branch pipe of the high pressure oil rail, resulting in resonance by means of injection pulse width adjustment scanning and modal analysis. Figure 3 shows the scanning results of different injection pulse width adjustment. The minimum injection pulse width, the higher the corresponding vibration peak Table 1 Parameters of whole vehicle and engine

Fig. 1 High pressure direct injection noise test prototype vehicle

Vehicle

Servicing weight

kg

1400

Engine

Displacement

L

1.499

Intake mode

–

Turbocharging

Fuel injection mode

–

Direct injection in cylinder

Rated power

kW

120

Rated speed

r/min

5500

Maximum torque

Nm

270

204

W. Liu et al.

Fig. 2 Noise confirmation of high voltage system

frequency, and reaches the highest point at 1551 Hz. It can be judged that the excitation of injection pulse width is coupled with the natural frequency of oil fluctuation at 1551 Hz. In the vicinity of 1551 Hz, there is a dense oil mode calculated by CAE facsimile, as shown in Fig. 4, which is consistent with the fuel injection pulse width scanning results. Fig. 3 Scanning results of fuel injection pulse width adjustment

Analysis on Noise of High Pressure Direct Injection System

205

Fig. 4 Oil modal simulation results

4 Noise Optimization of High Pressure Direct Injection System By loading the oil rail system, the radiation noise of direct injection system to engine cylinder head is analyzed. By cutting off the direct transmission between the injector and the installation point of the oil rail and cutting off the connection between the injector and the boss, the ERP amplitude level of the oil rail in 1700 Hz band is reduced from 29.7 to 28.1 dB (A), which is 1.6 dB (A) lower as shown in Figs. 5 and 6. Taking rail pressure and rotational speed of high-pressure system as variables, the optimal combination is 800 rpm rotational speed and 6 MPa rail pressure. As shown in Figs. 7 and 8, the problem peak value of this combination is reduced from the maximum of 13.3 dB (A) to 5.4 dB (A), which is 7.9 dB (A) lower than that of the original combination.

Fig. 5 Oil rail excitation position and structural adjustment scheme

206

W. Liu et al.

Fig. 6 Comparison of oil rail structure adjustment ERP

Fig. 7 Oil pressure adjustment of high pressure system

The combined scheme of rail pressure adjustment and connecting bolt vibration isolation is adopted, and the noise of the high-voltage system is effectively reduced, and the subjective evaluation reaches more than 7 points. The effect is shown in Table 2.

5 Conclusions Based on the noise problem of the high pressure direct injection system, the test and simulation analysis are carried out in this paper, and the influencing factors of the system structure and the effective scheme are verified. At the same time, a lot of verification work is done on the control strategy. (1) It is proved that the mode and structure of the high pressure direct injection system have influence on the noise, and the interior noise can be reduced by 1.6 dB (A) by changing the structure.

Analysis on Noise of High Pressure Direct Injection System

207

Fig. 8 Oil pressure adjustment effect of high pressure system

Table 2 Comparison of optimization effects

Status description

Driver’s external ear

External ear of co-pilot

Original state

6.1 dB (A)

13.8 dB (A)

After optimization

0.6 dB (A)

3.8 dB (A)

(2) Adjust the rotational speed and the rail pressure of the high-pressure common rail system, and adopt the optimal combination, and the maximum problem peak value is reduced by 7.9 dB (A). (3) By adopting the combination scheme of rail pressure adjustment and connecting bolt vibration isolation, the noise of the high-voltage system is effectively reduced, the maximum peak value of the problem is reduced by 10 dB (A), and the subjective evaluation is higher than 7 points, thus solving the problem.

References 1. Wang Y, Zhang K, Wang Y, Chen Z, Shi L (2021) Simulation design of high pressure tubing pressure control model. Mod Mach (06):48–52. http://doi.org/10.13667/j.cnki.52-1046/th.2021. 06.011 2. Ye H, Shen L, Zhao S, Ke Y (2021) Research on pressure control of high-pressure tubing based on global optimization model. Sci Technol Innov (16):57–59

208

W. Liu et al.

3. Song C, Zhang S, Cui Z, Zhang Y, Shao Y (2021) Pressure control scheme of highpressure tubing. Ind Technol Innov 08(02):132–137. http://doi.org/10.14103/j.issn.2095-8412. 2021.04.023 4. Yu T, Chen Z, Tian C, Jin S, Feng J (2021) Pressure control of high-pressure tubing. Neijiang Sci Technol 42(03):28–31 5. Wang W, Wang J, Chen T, Zhang Z, Shi L, Fan F, Ma M, Lin Y (2014) Simulation and experimental study on NVH performance improvement of turbocharged direct injection gasoline engine. Autom Eng 36(10):1189–1192 6. Kamasamudram K, Yezerets A, Chen X et al. New insights into reaction mechanism of selective catalytic ammonia oxidation technology for diesel after treatment applications. SAE Paper 201101-1314 7. Zou M, Liu Z, Deng P (2020) Dynamic control of one-way valve on high-pressure oil pipe pressure. Paper Equipment Mater 49(06):55–57

Parametric Optimization of a Permanent Magnet Driver for Implantable Intramedullary Nail ShiKeat Lee, Zhenguo Nie, Handing Xu, Kai Hu, Hanwei Lin, and Xin-Jun Liu

Abstract An intramedullary lengthening nail avoids complications frequently found in the external fixator and shows remarkable clinical results. The magnetic drive is the mainstream actuation method for intramedullary lengthening nails which shows good results, but clinicians claim insufficient distraction torque of the intramedullary lengthening nail will occur occasionally. In this paper, we investigate the optimal parameters for magnetic drive intramedullary lengthening nail based on parametric modeling to ensure the rotor magnet is rotated synchronously and the distraction torque is sufficient for bone lengthening. The driver-rotor magnet model is established, and parametric optimization of the angular position of the driver magnet local coordinate system, the angular velocity of the driver magnet, the number of driver magnets, and the number of pole pairs of the driver magnet to determine optimal values. Optimal parameters are selected and verified by experiments. The results show that a pair of single-pole pairs of the driver magnet rotating with an angular speed of 50 rpm can meet the minimal distraction torque of 1.5 N mm, which is around 5.6 N mm. Keywords Limb lengthening · Intramedullary lengthening nail · Magnetic drive · Parametric optimization

S. Lee · Z. Nie (B) · H. Xu · H. Lin · X.-J. Liu The State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected] Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, China K. Hu Jiangsu Key Lab of Special Robot Technology, Hohai University, Changzhou 213022, Jiangshu, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_12

209

210

S. Lee et al.

1 Introduction Ilizarov external fixator, a medical device that is commonly used in orthopedic surgery to lengthen or reshape limb bone, shows inevitable complications such as pin site infection, soft tissue incarceration, etc. [1–3]. Hence, the researchers invented an intramedullary lengthening nail to eliminate the inconvenience caused by Ilizarov external fixator, which achieved outstanding results in clinical treatment [4, 5]. However, precise distraction control of intramedullary lengthening nails has often puzzled the researchers and became the research focus [6–10]. The intramedullary lengthening nail can be categorized into three primary actuation methods, mechanical drive, motorized drive, and magnetic drive. Guichet et al. proposed an Albizzia intramedullary nail, requiring the patient to flex the hip or knee to 45° and rotate the limb internally and externally to initialize the ratchet system [11], yet, it is doomed to transient existence due to patient intolerance of the rotational and mechanical failure [12]. Fitbone, a motorized intramedullary lengthening nail powering via an induction method through the subcutaneous antenna, may malfunction due to tissue fluid corrosion [13]. NuVasive company proposed a computerized magnetic drive intramedullary lengthening nail, PRECICE, which achieved remarkable clinical results. However, some researchers noticed that PRECICE is likely to decelerate or stop distraction in several cases [7–9]. As a result, the stability and precision of the distraction rate for magnetic drive intramedullary lengthening nail is our main concern. Magnetic drive, as an external field driving method, is widely used in the field of precision medicine for motion control of micro-nano robots, drug treatment, and disease diagnosis due to its strong penetrating power, high accuracy, and harmless to bioorganisms properties [14]. The main sources of proceeding rotational magnetic fields for medical devices are permanent magnets or energized Helmholtz coils. Ishiyama et al. proposed an energized triaxial Helmholtz coil placed orthogonally to synthesize a rotary magnetic field for micro-machines [15–17]. Zhang W. et al. proposed a permanent magnet rectangular array with a specific pose angle to generate the external magnetic field for a capsule-like robot [18, 19]. Ye B. et al. designed a capsule endoscope that possesses a permanent magnet that is driven by the external magnetic field [20–22]. Jian X. Y. et al. proposed a combination of energized gradient coil and uniform coil to generate a uniform gradient and dynamically adjustable low field strength region for a magnetic drive [23]. In this paper, we investigate the optimal parameters for magnetic drive intramedullary lengthening nail based on parametric modeling to ensure the rotor magnet is rotated synchronously and the distraction torque is sufficient for bone lengthening. The paper is organized in the following way. Section 2 describes the modeling of the driver magnet and rotor magnet utilized in intramedullary lengthening nails, and Sect. 3 investigates the kinematics model of the magnetic driving system. Section 4 shows the simulation results, follows by the experimental results in Sect. 5. Finally, Sect. 6 concludes.

Parametric Optimization of a Permanent Magnet Driver for Implantable …

211

Fig. 1 a Schematic figure of intramedullary lengthening nail and permanent magnet driver. 1-thigh, 2-femur, 3-bone screws, 4-implantable lengthening nail, 5-permanent magnet driver. b Simplified model of the system. 6-driver magnet, 7-rotor magnet

2 Driver Magnet and Rotor Magnet Modelling in Intramedullary Lengthening Nail In this section, we introduce the driver magnet and rotor magnet model along with their dimensions, materials, and other properties, while the arrangement and orientation of the model are discussed.

2.1 Parameters of Driver Magnet and Rotor Magnet An internal permanent magnet of implantable lengthening nail is coupled to rotate synchronously with the permanent magnet driver, a set of external permanent magnets that spin around its central axis. The schematic diagram of implantable lengthening nail and permanent magnet driver structures is illustrated in Fig. 1. For simplicity and consistency, the external permanent magnets adopted in the permanent magnet driver are denoted as the driver magnet, while the internal permanent magnet adopted in the implantable lengthening nail is denoted as the rotor magnet. The simplified model of the driver magnet and rotor magnet is designed with the design specifications listed in Table 1 and is utilized throughout the simulation, as shown in Fig. 1.

2.2 Configuration of Driver Magnet in the Model Configurations of the driver magnet including driver magnet magnetization direction, layout orientation, position arrangement, and the number of pole pairs (λ), are elaborated before the simulation. As Fig. 2 shown, the magnetization direction is radial, and the axis of rotation passes through the center of the magnet that is perpendicular to the paper. The number of pole pairs may vary in accordance with the simulation result.

212

S. Lee et al.

Table 1 Driver magnet and rotor magnet design specifications

Product/component Driver magnet

Rotor magnet

Represent

Value/parameter

Inner diameter

10 mm

Outer diameter

60 mm

Length

25 mm

Material

NdFeB35

Inner diameter

4 mm

Outer diameter

11 mm

Length

25 mm

Material

NdFeB35

Fig. 2 Schematic of driver magnet magnetization direction, and rotation direction with different numbers of pole pairs

Before position arrangement, a global coordinate system o−x yz is fixed on the middle of the rotor magnet, and a local coordinate system o−xi yi z i is fixed on each center of the driver magnets. Driver magnets distribute around the rotor magnet with the radius of r0 , while θ denotes the angle between the global coordinate system and the local coordinate system. The relations between the global coordinate system and the local coordinate system may be described as follow: ⎧ ⎪ ⎨ xi = r0 cos θi yi = r0 sin θi , ⎪ ⎩ zi = 0

(1)

and  θi =

(−1)i−1 θ + (i − 1)π, i = 1, 2 , (−1)i−1 θ + (i − 2)π, i = 3, 4

(2)

where i is the ith number of driver magnet (Fig. 3). According to medical research, the thigh circumference of Asian men is 53.9 ± 6.3 cm, while Asian women is 52.8 ± 6.0 cm [24]. Therefore, we can assume the radius r0 has a value of 120 mm. Once the position of driver magnets is confirmed, motors are adopted to rotate the driver magnet with an angular speed of ω around its rotation axis.

Parametric Optimization of a Permanent Magnet Driver for Implantable …

213

Fig. 3 Schematic of a pair and two pairs of single-pole pairs driver magnet circumferential distribution

3 Kinematics Analysis of the Model The force analysis of the rotor is carried out to clarify the minimum rotation conditions based on magnetic coupling. The rotational magnetic field provided by the driver magnets must be sufficient to induce the rotor magnet for rotation, where the torque T is written as: T ≥ Tr + J α,

(3)

where Tr is the distraction torque for bone lengthening, J is the moment of inertia and α is the angular acceleration. Lauterburg et al. [25] research showed that the maximum distraction force of body mass is around 14 N/kg. We assume the patient body weight is 70 kg which gives us the total distraction force is nearly 1000 N. Therefore, the distraction torque based on the known distraction force can be derived from the design specification of the intramedullary lengthening nail [26, 27]. Fdm Tr = 2(e · i ) N



 L + π μdm , π dm − μL

(4)

where F is the distraction force, dm is the mean diameter of the lead screw, L is the lead of the screw, μ is the coefficient of friction, e, i, N are the efficiency, reduction ratio, and the number of stages of the gearbox respectively. Based on Eq. (4), we assume the minimal distraction torque is 1.5 N mm. The rotor magnet is a thick-walled cylindrical tube with an inner diameter d and outer diameter D. Therefore, the moment of inertia of the rotor is:

ρπl D 4 − d 4 , (5) J= 32 where ρ is the density of rotor magnet and l is the length of the magnet. The density of the magnet material is 7400 kg m−3 , which is substituted into Eq. (5) along with

214

S. Lee et al.

Table 1 values giving 2.35 × 10−7 kg m2 . Therefore, the J α term can be neglected since the moment of inertia value is too small. On the other hand, the torque endowed by the rotation motion of the driver magnet to the rotor magnet is obtained in terms of Eq. (6): T = ψ B sin Θ,

(6)

where ψ is the magnetic dipole moment, B is the external magnetic field and Θ is the angle between vector ψ and B. Hence, the torque reaches its maximum value when the angle Θ is ±π/2. Based on Eqs. (3) and (6), the inequality relation between angle and distraction torque is derived as: sin Θ ≥

Tr , ψB

(7)

which explains that as the driver magnets rotate, the angle Θ becomes wider until it satisfies the distraction torque, and the rotor magnet starts spinning.

4 Parametric Optimization and Simulation ANSYS Maxwell 2019R3 is utilized to simulate the different configurations of driver magnets, and parametric modeling is built to optimize the model. The optimization parameters are listed in Table 2. The boundary condition is set as Vectorpotential so that the boundary represents infinite distance and the magnetic flux is tangent with the border. Next, the material of the solving domain is set as a vacuum since the permeability of the body tissue between the driver magnet and rotor magnet has a similar value. For instance, when θ = 45◦ , ω = 50 rpm, n = 2 and λ = 1, the distributions of magnetic flux with four different angular displacements (ϑ) of rotor magnets are illustrated in Fig. 4. We notice that the central region has denser and well-distributed magnetic flux. Table 2 Optimization parameters for parametric modeling

Parameter

Represent

θ

Angular position of local coordinate system

ω

Angular velocity of driver magnets

n

Number of driver magnets

λ

Number of pole pairs of driver magnets

Parametric Optimization of a Permanent Magnet Driver for Implantable …

215

Fig. 4 Schematic of magnetic flux under four different angular displacements of rotor magnets

4.1 Optimization of Angular Position of Local Coordinate System An assumption of ω = 50 rpm, n = 2, and λ = 1 is made before parametric optimization for the angular position of the local coordinate system, θ . The range of angular position is set between 15◦ and 75◦ , with an interval of 15◦ . The value cannot be too large or too small because there will have interference between driver magnets (Fig. 5). The torque curves of the rotor magnet vary under θ , as shown in Fig. 6. The torque curve is an oscillation curve, and its amplitude gradually stabilizes to 1.5 N mm as the time passes. The main difference between different θ is that the time required for the torque converges to 1.5 N mm, which 45◦ is the fastest. Figure 7 shows the angular displacement curve of rotor magnets within 0 to 200 ms. Overall, the angular displacement rises with fluctuation, but θ = 45◦ is the mildest among them. By comparing the theoretical and simulated angular displacement of the rotor magnets, we find that θ = 45◦ has the nearest value to the theoretical value, meaning the rotor magnet is rotated more accurately with the driver magnets (Table 3).

216

S. Lee et al.

Fig. 5 Schematic of driver magnets position arrangement when r0 = 120 mm, n = 2, and λ=1

Fig. 6 Torque curve of the rotor magnet vary under θ

4.2 Optimization of Angular Velocity of Driver Magnets An assumption of θ = 45◦ , n = 2, and λ = 1 is made before parametric optimization for the angular velocity of driver magnets, ω. The range of angular velocity is set between 20 and 110 rpm, with an interval of 10 rpm. The torque curves of the rotor magnet vary under ω, as shown in Fig. 8. The torque curve is an oscillation curve, and its amplitude gradually stabilizes to 1.5 N mm as the time passes. The main difference between different ω is that the time required for the torque converges to 1.5 N mm, which 20 rpm is the fastest.

Parametric Optimization of a Permanent Magnet Driver for Implantable …

Fig. 7 Angular displacement curves vary under θ Table 3 Comparison of theoretical and simulated results under different θ within 0 to 200 ms

θ

ϑ

(ϑ − 60°)/ϑ (%)

15°

61.621°

2.63

30°

61.009°

1.65

45°

59.437°

−0.095

60°

57.609°

−4.15

75°

55.332°

−8.44

Fig. 8 Torque curve of the rotor magnet vary under ω

217

218

S. Lee et al.

Fig. 9 Angular displacement curves vary under ω

Table 4 Comparison of theoretical and simulated results under different ω within 0–400 ms

ω (rpm) 20

ϑ Theo. 48°

ϑ Sim. 47.402°

(ϑ Theo. − ϑ Sim. )/ϑ Theo. (%) 1.25

30

72°

71.525°

0.66

50

120°

119.609°

0.33

70

168°

167.661°

0.20

90

216°

215.448°

0.26

110

264°

263.600°

0.15

Figure 9 shows the angular displacement curve of rotor magnets within 0–400 ms. The angular displacement rises gently, and the error between theoretical and simulated results is smaller than 1.5% in any case. Overall, the error value decreases steadily as the angular velocity increase, but it takes a longer time to converge, so there is a trade-off between these variables (Table 4).

4.3 Optimization of Number of Driver Magnets An assumption of ω = 50 rpm, θ = 45◦ , and λ = 1 is made before parametric optimization for the number of pairs of driver magnets, n. The range of number of pairs is set as 2 and 4, as shown in Fig. 10. The torque curves of the rotor magnet vary under n, as shown in Fig. 11. The torque curve is an oscillation curve, and its amplitude gradually stabilizes to 1.5 N mm as the time passes. The main difference between different n is that the time required for the torque converges to 1.5 N mm, which n = 4 takes a lesser time.

Parametric Optimization of a Permanent Magnet Driver for Implantable …

219

Fig. 10 Schematic of driver magnets position arrangement when r0 = 120 mm, ω = 50 rpm, and λ = 1, a n = 2, b n = 4

Fig. 11 Torque curve of the rotor magnet vary under n

Figure 12 shows the angular displacement curve of rotor magnets within 0–400 ms. The angular displacement rises linearly and has a similar error between theoretical and simulated results. Overall, n = 4 has the optimal solution, but it is much more costly (Table 5).

4.4 Optimization of Number of Pole Pairs of Driver Magnets An assumption of ω = 50 rpm, θ = 45◦ , and n = 2 is made before parametric optimization for the number of pole pairs of driver magnets, λ. The range of number of pole pairs is set as 1 and 2, as shown in Fig. 13. The torque curves of the rotor magnet vary under λ, as shown in Fig. 14. The torque curve is an oscillation curve, and its average value is around 1.5 N mm. Moreover,

220

S. Lee et al.

Fig. 12 Angular displacement curves vary under θ

Table 5 Comparison of theoretical and simulated results under different n within 0–400 ms n

ϑ

(ϑ − 120°)/ϑ (%)

2

119.609°

−0.33

4

119.689°

−0.26

Fig. 13 Schematic of driver magnets position arrangement when r0 = 120 mm, ω = 50 rpm, and n = 2, a λ = 1, b λ = 2

the angular displacement curve is approximately linear, where λ = 1 takes 2.385 s to complete one period, while λ = 2 takes 2.385 s. Overall, λ = 2 has a smaller amplitude, and it takes lesser time for the rotor magnet to complete one period, but it will not converge to an average value (Fig. 15).

Parametric Optimization of a Permanent Magnet Driver for Implantable …

221

Fig. 14 Torque curve of the rotor magnet vary under λ

Fig. 15 Angular displacement curves vary under λ

5 Experiment Validation of Rotor Magnet Torque The experiment on the rotor magnet torque is conducted, and the experiment setup is shown in Fig. 16. The cylindrical tube shape of a rotor magnet and driver magnets are manufactured as the design specifications shown in Table 1. A pair of two poles driver magnets are distributed around the rotor magnet with a radius (r0 ) of 120 mm and angular position (θ ) of 45◦ . The driver magnets are rotated with a hand crank, and a timing belt is adopted to ensure they spin synchronously with an angular velocity of 50 rpm. A computer is utilized to collect and monitor real-time data from the torque measurement system. The maximum and minimum values of the torque are approximate 6.86 N mm, and 4.9 N mm, respectively, while its average value is 5.6 N mm, as shown in Fig. 17. The experimental data may be inaccurate due to torque measurement system sensitivity, improper contact between the sensor probe and the axis of the rotor magnet, etc.

222

S. Lee et al.

Fig. 16 Experiment setup to test rotor magnet torque

Fig. 17 Experiment validation of rotor magnet torque

Similar parameters are adopted in ANSYS Maxwell to simulate the torque of the rotor magnet, which gives us an approximate value of 5.6031 N mm. To be concluded, the parameters of ω = 50 rpm, θ = 45◦ , n = 2, and λ = 1 satisfy the minimal value of distraction torque, 1.5 N mm, and the rotor magnet and driver magnets can rotate synchronously (Fig. 18).

6 Conclusion and Future Works In this paper, the configurations of driver magnets are proposed, and the optimization parameters are utilized for parametric modeling, which gives us the conclusion below: (1) The number of pole pairs of the driver magnets and rotor magnet must be consistent. Single pole pairs of driver magnets are recommended because the oscillation curve can converge to an average value compared with other pole pairs.

Parametric Optimization of a Permanent Magnet Driver for Implantable …

223

Fig. 18 Simulation of rotor magnet torque

(2) The number of driver magnets affects the time for the oscillation curve to converge, but it does not affect the synchronization of the rotor magnet. (3) Based on conclusions (1) and (2), the angular position of the local coordinate system θ = 45◦ has the smallest amplitude and takes lesser time for convergence. (4) On the basis of (1) and (2), the angular velocity of the driver magnets affects the precision of the angular displacement of the rotor magnet, but there is a trade-off with the stability of torque curves. The optimization parameters of ω = 50 rpm, θ = 45◦ , n = 2, and λ = 1 are tested and simulated, which gives us an average torque of 5.6 N mm, which is satisfactory for bone distraction. In future work, the permanent magnet driver and the intramedullary lengthening nail will be manufactured, and an experiment will be conducted. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 52175237), ZhongGuanCun (ZGC) open laboratory proof-of-concept (POC) projects (Grant No. 20210421084), and Beijing Municipal Science and Technology Commission (Grant No. Z211100004021022).

References 1. Fragomen AT, Miller AO, Brause BD, Goldman V, Rozbruch SR (2017) Prophylactic postoperative antibiotics may not reduce pin site infections after external fixation. HSS J 13(2):165–170 2. Bhave A, Shabtai L, Woelber E, Apelyan A, Paley D, Herzenberg JE (2017) Muscle strength and knee range of motion after femoral lengthening: 2-to 5-year follow-up. Acta Orthop 88(2):179– 184

224

S. Lee et al.

3. Kazmers NH, Fragomen AT, Rozbruch SR (2016) Prevention of pin site infection in external fixation: a review of the literature. Strat Trauma Limb Reconstr 11(2):75–85 4. Zhang J, Zhang Y, Wang C et al (2021) Research progress of intramedullary lengthening nail technology. Chin J Reparative Reconstr Surg 35(5):642–647 5. Li M, Zeng P, Qing S (2016) Research progress in lower limb lengthening by intramedullary nail. Chin J Reparative Reconstr Surg 30(5):647–650 6. Mahboubian S, Seah M, Fragomen AT, Rozbruch SR (2012) Femoral lengthening with lengthening over a nail has fewer complications than intramedullary skeletal kinetic distraction. Clin Orthop Relat Res 470(4):1221–1231 7. Schiedel FM, Vogt B, Tretow HL, Schuhknecht B, Gosheger G, Horter MJ, Rödl R (2014) How precise is the PRECICE compared to the ISKD in intramedullary limb lengthening? Reliability and safety in 26 procedures. Acta Orthop 85(3):293–298 8. Nasto LA, Coppa V, Riganti et al (2020) Clinical results and complication rates of lower limb lengthening in paediatric patients using the PRECICE 2 intramedullary magnetic nail: a multicentre study. J Pediatr Orthop B 29(6):611–617 9. Iliadis AD, Palloni V, Wright J, Goodier D, Calder P (2021) Pediatric lower limb lengthening using the PRECICE nail: our experience with 50 cases. J Pediatr Orthop 41(1):e44–e49 10. Cole JD, Justin D, Kasparis T, DeVlught D, Knobloch C (2001) The intramedullary skeletal kinetic distractor (ISKD): first clinical results of a new intramedullary nail for lengthening of the femur and tibia. Injury 32:129–139 11. Guichet JM, Deromedis B, Donnan LT, Peretti G, Lascombes P, Bado F (2003) Gradual femoral lengthening with the Albizzia intramedullary nail. JBJS 85(5):838–848 12. Birch JG (2017) A brief history of limb lengthening. J Pediatr Orthop 37:S1–S8 13. Black SR, Kwon MS, Cherkashin AM, Samchukov ML, Birch JG, Jo CH (2015) Lengthening in congenital femoral deficiency: a comparison of circular external fixation and a motorized intramedullary nail. J Bone Joint Surg Am 97(17):1432 14. Koleoso M, Feng X, Xue Y, Li Q, Munshi T, Chen X (2020) Micro/nanoscale magnetic robots for biomedical applications. Mater Today Bio 8:100085 15. Sendoh M, Ishiyama K, Arai KI (2003) Fabrication of magnetic actuator for use in a capsule endoscope. IEEE Trans Magn 39(5):3232–3234 16. Ishiyama K, Sendoh M, Yamazaki A, Inoue M, Arai KI (2001) Swimming of magnetic micro-machines under a very wide-range of Reynolds number conditions. IEEE Trans Magn 37(4):2868–2870 17. Nishimura K, Sendoh M, Ishiyama K, Arai KI, Uchida H, Inoue M (2004) Fabrication and swimming properties of micro-machine coated with magnetite prepared by ferrite plating. Phys Status Solidi (B) 241(7):1686–1688 18. Zhang W, Huang P, Meng YG (2008) Mechanism and experiment research on rotational magnetic field generated by circumferentially arrayed permanent magnets. Chin J Eng Des 15(3):191–197 19. Zhang W, Chen YJ, Huang P (2007) Experimental study on a capsule-like robot driven by an outer rotating magnetic field. Chin J Eng Des 14(2):89–96 20. Ye B, Zhang W, Sun ZJ, Guo L, Deng C, Chen YQ et al (2015) Study on a magnetic spiral-type wireless capsule endoscope controlled by rotational external permanent magnet. J Magn Magn Mater 395:316–323 21. Bo Y, Zhenjun S, Yaqi C, Honghai Z, Sheng L (2016) A new magnetic control method for spiral-type wireless capsule endoscope. J Mech Med Biol 16(03):1650031 22. Sun ZJ, Ye B, Qiu Y, Cheng XG, Zhang HH, Liu S (2014) Preliminary study of a legged capsule robot actuated wirelessly by magnetic torque. IEEE Trans Magn 50(8):1–6 23. Jian XY, Mei T, Wang XH (2005) Driving method of an endoscopic robot capsule by external magnetic field. Robot 27(4):367–372

Parametric Optimization of a Permanent Magnet Driver for Implantable …

225

24. Xu W, Wang M, Jiang CM, Zhang YM (2011) Anthropometric equation for estimation of appendicular skeletal muscle mass in Chinese adults. Asia Pac J Clin Nutr 20(4):551–556 25. Lauterburg MT, Exner GU, Jacob HAC (2006) Forces involved in lower limb lengthening: an in vivo biomechanical study. J Orthop Res 24(9):1815–1822 26. Shigley JE (2011) Shigley’s mechanical engineering design. Tata McGraw-Hill Education 27. Lee S, Nie Z, Xu H et al (2021) Design and optimization of a novel intramedullary robot for limb lengthening. In: International conference on intelligent robotics and applications. Springer, Cham, pp 103–112

Design and Analysis of Modified Non-orthogonal Helical Face-Gears with a High-Order Transmission Error Junhong Xu and Chao Jia

Abstract In order to improve the meshing performance of non-orthogonal helical face-gears, a designation of double-crowned tooth modification with high-order transmission error (HTE) is constructed in this paper. First, the double-crowned tooth modification is designed based on the tooth profile modification of the rack-cutter and the motion relationship between the rack-cutter and the processed pinion, and then, the designation of high-order tooth modification is verified by using tooth contact analysis (TCA). Second, a parameterized 3D grid of modified gears is generated based on MATLAB programming, and a simulation of loaded tooth contact analysis for modified non-orthogonal helical face-gears is carried out by using ABAQUS software. Third, the calculation results of the standard tooth, second-order parabolic modified tooth and high-order parabolic modified tooth are compared with and without the assembly errors. Ultimately, an example is presented and the results show that the geometric transmission error of the newly designed high-order modified non-orthogonal helical face-gears conforms to the expected design. Compared with the second-order parabolic tooth modification, the newly designed high-order tooth modification pair shows a better meshing performance with and without assembly errors. The research of this paper can provide ideas and methods for the tooth modification design of non-orthogonal helical face-gear. Keywords High-order transmission error · Non-orthogonal helical face-gear · Tooth contact analysis · Parameterized 3D grid · Loaded tooth contact analysis

1 Introduction Face-gears has the advantages of large-stage ratio, compact structure and so on. It is of great significance to improve the technical level of power transmission of aviation equipment when it is applied to aeronautical power-split transmission. At present, the face-gears has been successfully applied to the main reducer of the helicopter. J. Xu · C. Jia (B) School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_13

227

228

J. Xu and C. Jia

Under the working conditions of high speed and heavy load, the elastic deformation and thermal deformation of the face-gears will lead to uneven load distribution on the tooth surface, decrease of load bearing capacity, increase of vibration and noise, etc. Therefore, it is necessary to carry out the tooth modification design for face-gears in order to reduce the adverse effects of deformation and error [1–4]. Tooth modification of gears usually refers to removing a small amount of material from the tooth surface of gears. The typical tooth modification of gears is the middleconvex modification as tooth profile modification, tooth axial modification and threedimensional modification and so on [5–11]. These typical methods are called the traditional tooth modification method herein. There is enough research to show that the traditional tooth modification can reduce the effect of errors. However, with the development of science and technology, the requirement of gear transmission in modern industry is higher and higher, and some disadvantages of traditional tooth modification are gradually emerging. For example, the traditional tooth modification may increase the maximum contact stresses of gears. Excessive contact stress can exacerbate the surface wear, reduce the working life and decrease the gear drive efficiency. But it must be pointed out that the errors such as assembly, manufacturing, and so on, are inevitable in practical application. To some extent, tooth modification is necessary. Therefore, researchers have been exploring new tooth modification methods for a long time. It is hoped that the new method can not only maintain the advantages of the traditional one, but also reduce the adverse impact on gears. And high-order tooth modification is such a potential new method [12, 13]. Jiang [14] proposed a high-order transmission error (HTE) modification for cylindrical gears, and determined the coefficients of the high-order polynomial through particle swarm optimization (PSO), and the feasibility of high-order modification is proved by analyzing and comparing the loading transmission error before and after modification. Jia et al. [15] modified the tooth surface of the cylindrical gear through the pre-designed high-order transmission error to reduce the error margin of the loading transmission. Su et al. [16] proposed a seventh-order transmission error modification for helical bevel gears to prove that the bending stress of the tooth surface with the seventh-order transmission error modification is smaller than that of the parabolic tooth surface, while the contact stress is almost equal. The modification of the high-order transmission error can improve the meshing performance of the spiral bevel gear with high coincidence degree [17]. The amplitude of loaded transmission error and the gear vibration of the helical gear with high-order transmission error is reduced compared with standard gear [18]. Daqing et al. [19] derived the tooth-surface equation of the face-gears from the high-order parabolic rack based on the grinding process, and the tooth-surface equation of the pinion is worked out by means of the modification principle of the high-order parabolic curve. And the effect of new tooth-surface equation on the meshing performance of the face-gear pair is analyzed. From above discussion, it can be found that the research of high-order tooth modification mainly focuses on the cylindrical gears and spiral bevel gears, while the research on high-order tooth modification of face-gears is rare. Therefore, to

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

229

Face-gear

Fig. 1 Schematic diagram of face-gears shaping

Cutter

ω2

ωS

O2 , OS

Z2

ZS

γm

improve the meshing performance of non-orthogonal helical face-gears, a designation of double-crowned tooth modification with HTE is constructed in this paper. The double-crowned tooth modification is designed based on the tooth profile modification of the rack-cutter and the motion relationship between the rack-cutter and the processed pinion. The innovation of the new method is that the basic tooth profile of the controllable HTE is designed according to the theoretical contact path of the gear. And the design of high-order modifications is verified by tooth contact analysis (TCA) and the loaded contact analysis (LTCA).

2 Derivation of Tooth Surface Equation for Non-orthogonal Helical Face-Gears 2.1 The Principle of Face-Gear Shaping The process of face gear shaping is based on the movement of the cutter and the face gear. Its processing principle is shown in Fig. 1.

2.2 Generation of Face-Gears Tooth Surface Base on the motion relationship between the cutter and the processed face-gears during the cutting process of the face-gears, establish a non-orthogonal helical facegears machining coordinate system, S2 and Ss are rigidly connected to the face gear and the cutter, where Z 2 and Z s are the axes of the cutter and face-gears respectively. The angle γm is the shaft angle. Convert the position vector of the cutter from the coordinate system Ss to the coordinate system S2 based on the movement of the cutter and the face-gears. And the face-gears tooth surface ∑2 is generated by meshing equation [20] (Figs. 2 and 3).

230

J. Xu and C. Jia

Fig. 2 Generation of coordinate system for face-gears

L0

Xa

ϕs

Og rps − ag O2 , Om

γ m ϕ2 X2

Oa

Ym Y2

Zs , Za

Xm

Ya

Z2 , Zm

Ys

Xc

Fig. 3 Profile of a rack cutter

Xs

Zs

Xa

Zc

β

us

Ys

Oa

Yc

Oc

Ob 0.25π mn

Ob

Yc

Ya

(b)

(a)

The rack profile can be represented by Eq. (1).  r as

= [ −acs u 2cs u cs l g 1 ]

(1)

where acs is modification parameter for tooth profile modification; u cs and l g are the related tooth flank parameters of the rack cutter. The tooth surface of the face-gears can be represented by Eq. (2).  r 2 (u cs , l g , ψs , ϕ2 (ϕ1 ))



= M2a r as (u cs , lg , ψs )

(2)

where M2a is a 4 × 4 matrix from the coordinate system Sa to the coordinate system S2 (Figs. 4 and 5). where ϕ2 is rotation angle of face-gears; ψs is rotation angle of cutter.

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

231

Z2 , ZB

Fig. 4 Coordinate system of face-gears

X2 XB YB Y2

ϕ2

Fig. 5 Coordinate system of cutter

Ys 0 X s X s0

Ys

ψs Zs , Zs0

Base on the motion relationship between the cutter and the processed face-gears, the meshing equation of the cutter and face-gears can be represented by Eq. (3). T vg2 f 2 (u cs , ψs , ϕ2 (ϕ1 )) = n sr

(3)

where n sr is the unit normal vectors in the coordinate system Ss ; vg2 is the relative velocity of cutter and face-gears.

3 A Designation of Double-Crowned Tooth Modification of Pinion 3.1 Derivation of Parabolic Profile Equation of Helical Cylindrical Gears According to the principle of gear meshing processing, the tooth surface of the pinion is formed by the envelopment of the tooth surface of the rack. The coordinate system

232

J. Xu and C. Jia

Fig. 6 Profile of rack

X r1

u0i

uci

Xc

Oc , Or1

α

Yr1 Yc

Sc is connected to the cutter and the coordinate system Sr 1 is connected to the cutter pinion. Profile of rack with second order parabolic can be represented by Fig. 6. The generation process of pinion can be represented by Fig. 7. The fixed coordinate system Sn is connected to the pinion, The rotating coordinate system S1 is connected to the pinion, The moving coordinate system Sr1 is connected to the rack-cutter. When the pinion rotates angle ϕ1 , the rack-cutter moves r p1 ϕ1 . Where r p1 is the radii of pitch circle of pinion. Position vector of rack profile can be represented by Eq. (4).  r ai

= [ −aci u 2ci u ci − u 0i ld 1 ]

(4)

where aci is modification parameter for tooth profile modification; u ci and ld are the related tooth flank parameters of the rack cutter, u 0i is the distance from initial position to rack profile position. The tooth surface of pinion can be represented by Eq. (5). Yn

Yr1

Fig. 7 Roll motion in a pinion cutting process

rp1ϕ1

Y1

X r1

Or1 X1

ϕ1 Xn

On , O1 rp1

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

New tooth

Fig. 8 Tooth profile modification of the pinion

 r 1 (u ci , ld , ϕ1 )



= M1i r ai (u ci , ld )

233

Old tooth

(5)

Base on designing modification parameter for tooth profile modification of the rack-cutter, the tooth profile of the rack-cutter can be modified to a tooth profile with second-order parabolic, so as to achieve the effect of modifying the tooth profile of the pinion (Fig. 8).

3.2 HTE In this paper, the tooth surface of the pinion is produced by rack-cutter. The rackcutter performs a translational motion and the pinion rotates with the rack-cutter. The modified flank of pinion is controlled by the HTE. Based on the meshing theory of gears, Generation of a pinion tooth can be represented by Fig. 9. In Fig. 10, the TE is predesigned as a six-order parabolic function, which is controlled by five points. There are seven unknown parameters in the TE function, and a system of seven equations is provided by the five predesigned points on the TE function. The x-coordinate and y-coordinate of the five given points are the rotation angle of the pinion and the transmission error of the face-gears, respectively (Fig. 11). The TE function of Fig. 10 is written as Eq. (6). δϕ2 = a0 + a1 ϕ1 + a2 ϕ12 + a3 ϕ13 + a4 ϕ14 + a5 ϕ15 + a6 ϕ16 = X Y T

(6)

where ϕ1 and δϕ2 represent the TE angle of the pinion and face-gears, respectively, and a0 − −a6 represent the coefficient of HTE function:

234

J. Xu and C. Jia

Xc

X q1

Fig. 9 Generation of a pinion tooth

rp1 × (θ1 + Δθ1 )

rp1

Yc

Oc

X1

θ1 O1 Oq1

Yq1 Y1

λBTm

Fig. 10 The high-order parabolic transmission error

PB (φ1B,δφ2B)

y

⎧ ⎪ ⎪ ϕ1 ⎪ ⎪ ϕ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ϕ1 ϕ1 ⎪ ⎪ ϕ1 ⎪ ⎪ ⎪ ⎪ ⎪ ϕ1 ⎪ ⎪ ⎩ ϕ1

PA (φ1A, δφ2A) Double tooth Triple tooth

λDTm

PC (φ1C,δφ2C)

Triple tooth

= T1 , δϕ2 = ε1 = Tm − λ1 T , δϕ2 = 0 = 0.5(T1 + T2 ) = Tm , δϕ2 = ε2 = Tm + λ2 T , δϕ2 = ε3 = T2 , δϕ2 = ε4 2 =0 = Tm + λ2 T , ∂δϕ ∂ϕ1 ∂δϕ2 = Tm − λ1 T , ∂ϕ1 = 0

x

PD (φ1D,δφ2D)

PE (φ1E,δφ2E) Double tooth Triple tooth

(7)

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

235

Fig. 11 The traditional second-order parabolic transmission error

x PB (φ1B,δφ2B)

y

PC (φ1C,δφ2C)

PA (φ1A,δφ2A)

Solve the above formulas simultaneously to obtain the solution matrix of the high-order transmission error curve coefficients: ⎧ ⎡ 1 T1 T12 ⎪ ⎪ ⎪ ⎪ ⎢ 1 (T − λ T ) (T − λ T )2 ⎪ m m ⎪ ⎢ 1 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎢0 1 2(Tm − λ1 T ) ⎪ ⎪ ⎢ ⎪ ⎪ ⎨A = ⎢ Tm Tm2 ⎢1 ⎢ ⎢ 1 (Tm + λ2 T ) (Tm + λ2 T )2 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣0 1 2(Tm − λ2 T ) ⎪ ⎪ ⎪ ⎪ ⎪ 1 T2 T22 ⎪ ⎪ ⎪ ⎪

⎪ ⎩ B = ε1 0 0 ε2 ε3 0 ε4

T13 (Tm − λ1 T )3 3(Tm − λ1 T )2 Tm3 (Tm + λ2 T )3 3(Tm − λ2 T )2 T23

T14 (Tm − λ1 T )4 4(Tm − λ1 T )3 Tm4 (Tm + λ2 T )4 4(Tm − λ2 T )3 T24

T15 (Tm − λ1 T )5 5(Tm − λ1 T )4 Tm5 (Tm + λ2 T )5 5(Tm − λ2 T )4 T25

⎤ T16 (Tm − λ1 T )6 ⎥ ⎥ ⎥ 6(Tm − λ1 T )5 ⎥ ⎥ ⎥ Tm6 ⎥ ⎥ 6 (Tm + λ2 T ) ⎥ ⎥ 6(Tm − λ2 T )5 ⎦

(8)

T26

where, 

  X = a0 a1 a2 a3 a4 a5 a6 ,  Y = 1 ϕ1 ϕ12 ϕ13 ϕ14 ϕ15 ϕ16

X = A−1 B

(9)

The higher-order transmission error can be represented by Eq. (10). δϕ2 (ε1 , ε2 , ε3 , ε4 , λ1 , λ2 ) = A−1 BY T

(10)

4 Tooth Contact Analysis of Face-Gears Pair 4.1 Meshing Equation of Tooth Surface The non-orthogonal helical face-gears pair is composed of helical cylindrical gear and non-orthogonal helical face-gears. According to the principle of tooth contact analysis, the coordinate system can be established in Fig. 12. The coordinate systems

236

J. Xu and C. Jia

S1 and S2 are connected the pinion and the face-gear, respectively. In the figure, γ f = γm + Δγ , where γm is the shaft angle, and Δγ is the shaft angle error. 







Transformed the position vectors r 1 , r 2 and unit normal vectors n 1 , n 2 into the coordinate system S f , and then, the tooth surface of the pinion and face-gear can be represented by Eqs. (11), (12), (13) and (14).  ( f )  r 2 u cs , l g , ψs , ϕ2 (ϕ1 ) ( f ) r 1 (u ci , ld , ϕ1 )

  = M f 1 r 1 (u ci , ld )

 ( f )  n 2 u cs , l g , ψs , ϕ2 (ϕ1 ) ( f ) n 1 (u ci , ld , ϕ1 )

    = M f 2 r 2 u cs , l g , ψs

(11) (12)

    = L f 2 n 2 u cs , l g , ψs

  = L f 1 n 1 (u ci , ld )

(13) (14)

∑ ( f ) ( f ) where r 1 and r 2 are position vectors of the tooth surface of the pinion 1 and ∑ ( f ) ( f ) face-gear 2 ; n 1 ∑andn 2 are unit normal vectors of the toothsurface of the pinion ∑ 1 and face-gear   2 ; M f i (i = 1, 2) is a 4 × 4 matrix; L f i (i = 1, 2) is a 3 × 3 submatrix of M f i (i = 1, 2). ∑ The moving coordinate system S f is connected to the pinion. In S f , surfaces 1 ∑ ( f ) and 2 are in tangent, and their position vectors r i (i = 1, 2) and unit normal ( f )

vectors n i (i = 1, 2) considered in S f must be equal, which can be represented by Eq. (15). Fig. 12 Face-gear pair

Y2 O2 O 1 Z2

Z1

γf

Y1 Pinion

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

⎧ f (u cs , l g , ψs ) = 0 ⎪ ⎪ ⎪ ⎨ ( f )   ( f ) r 2 u cs , l g , ψs , ϕ2 (ϕ1 ) = r 1 (u ci , ld , ϕ1 ) ⎪ ⎪ ⎪  ( f ) ⎩ ( f )  n 2 u cs , l g , ψs , ϕ2 (ϕ1 ) = n 1 (u ci , ld , ϕ1 )

237

(15)

where ϕ2 is the rotation angle of face-gear; ϕ1 is the rotation angle of pinion. By setting the value of ϕ2 to solve the nonlinear equation system, the contact path of the face-gear tooth surface can be obtained by bringing the solved parameters into the equation. When edge contact occurs, the edge of crest of the pinion contacts the tooth flank of the face-gear when out of meshing. At this time, in the fixed coordinate system S f , the position vectors of the tip edge of the pinion and the tooth surface of the face-gear must be equal, and the tangent vector of the tip edge of the pinion is perpendicular to the normal vector of the tooth surface of the face-gear. Likewise, when the two gear teeth come into mesh, the edge of crest of the face-gear is in contact with the tooth flank of the pinion. In the fixed coordinate system, position vectors of the edge of crest of the face-gear and the tooth surface of the pinion must be equal, and the tangent vector of the edge of crest of the face-gear is perpendicular to the normal vector of the tooth surface of the pinion. Both cases can be represented by Eqs. (16) and (17). (1) f (u ci , ld , ϕ1 )

∂r

∂ld

(2)

·nf =0

(2) f (u cs , l g , ψs , ϕ2 (ϕ1 ))

∂r

∂l g where

(1) f (u ci ,ld ,ϕ1 )

∂r

∂ld (2) ∂ r f (u cs ,l g ,ψs ,ϕ2 (ϕ1 )) ∂l g

(1)

·nf =0

(16)

(17)

is the tangent vector of the edge of crest of the pinion; (1)

is the tangent vector of the edge of crest of face-gear; n f is (2)

the normal vector of the tooth surface of pinion; n f is the normal vector of the tooth surface of face-gear. By solving the above equations, the contact point of the face gear can be determined.

4.2 Calculation of Hertzian Contact Stress According to the equations of non-orthogonal helical face-gears tooth surface and the contact path, the contact point of each meshing position can be obtained. Based on the principle of contact mechanics, at a certain meshing position, the contact point

238

J. Xu and C. Jia

is deformed due to the action of the load, and the initial contact point expands into an elliptical contact area. The displacement at the center of the ellipse is the largest, and the contact stress is also the largest. The calculation of the elliptical contact area is related to the curvature of the two contact surfaces at the contact point, and a digital program can be written to calculate the corresponding curvature. Based on the Hertzian elastic contact theory [21–23], the two gears in contact are deformed under the action of the normal load, and expand into an elliptical contact area at the initial contact point O. The contact deformation can be represented by Fig. 13.  2 2 a /b E(e) − K (e) B = A K (e) − E(e) Fn =

(18)

2 p0 πab 3

(19)

1/2  b2 e = 1− 2 a

(20)

1 − v12 1 − v22 1 = + E∗ E1 E2

(21)

p0 =

1.5Fn π ab

(22)

where Fn is the normal load of the contact point; a is the semimajor axis of ellipse; b is the semi-minor axis of ellipse; K (e) is elliptic integral of the first kind; E(e) is elliptic integral of the second kind; e is the eccentricity of ellipse; p0 is the maximum contact stress. z

Fig. 13 Hertzian elastic contact

P

Surface 1 Deformed surface 1'

δ

O

x Deformed surface 2' Surface 2

P

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

239

5 Numerical Examples and Discussions 5.1 TCA The calculation results of the standard tooth, second-order parabolic modified tooth and high-order parabolic modified tooth without the assembly errors can be represented by Fig. 14 (Table 1). Among various assembly errors, the shaft angle error is the parameter which is most significant on the meshing performance of the face-gear pairs [24–26], so the shaft angle error is considered in this paper. By comparing Fig. 14a–c with Figs. 15a– c and 16a–c, it can be found that when the shaft angle error Δγ = 1.5' , the paths of contact on face-gear surfaces moves towards the outer end of the face-gear, the contact points of the crest edge are reduced, the contact points of the tooth root edge are increased, and the transmission error is also significantly increased and when the shaft angle error Δγ = −1.5' , the paths of contact on face-gear surfaces moves towards the inner end of the face-gear, the contact points of the crest edge are increased, the contact points of the tooth root edge are reduced, and the transmission error is significantly increased. The meshing effect of the face-gear pair will easily be deteriorated due to the deviation of the contact point.

5.2 Tooth Modification The double-crowned tooth modification with HTE is designed based on the tooth profile modification of the rack-cutter and the motion relationship between the rackcutter and the processed pinion. Figure 17 shows the optimized tooth modification values of the pinion with different tooth modification methods. As shown in Fig. 17a, the pinion tooth is modified with the traditional second-order parabolic TE. The traditional TE is a secondorder parabola; accordingly, the modification value is middle-convex. Figure 17b shows that the pinion tooth is modified with the new HTE. Unlike the tradition second-order TE, the HTE is middle-concave; then, the corresponding modification value is middle-concave. It can be seen clearly that the HTE ensures more tooth modification values in the zone of triple tooth. Meanwhile, the HTE ensures the TEs more even in the middle region on the tooth surface.

5.3 The Establishment of Parametric Mesh Model The theoretical model of the tooth surfaces of the pinion and the face-gear are discretized to obtain the discrete data lattice of the tooth surface, and the lattice of the two tooth surfaces is rotated around their respective axes to obtain the tooth node.

240

J. Xu and C. Jia

Fig. 14 Paths of contact and transmission error on face-gear surfaces without the assembly errors

0

TE (arc sec)

-5 -10 -15 -20 -25 -40 -30 -20 -10 0 10 20 30 Pinion rotation (deg)

40

50

40

50

(a) Standard tooth

0 -5

TE (arc sec)

-10 -15 -20 -25 -30 -35 -40 -40 -30 -20 -10

0

10

20

30

Pinion rotation (deg)

(b) Second-order parabolic modified tooth

0 -5

TE (arc sec)

-10 -15 -20 -25 -30 -35 -40 -30 -20 -10 0 10 20 30 Pinion rotation (deg)

40

(c) High-order parabolic modified tooth

50

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

241

Table 1 Design parameters of face-gears Design parameter

Value

Pinion tooth number

25

Cutter tooth number

28

Face-gear tooth number

160

Normal module (mm)

6.35

Pressure angle (°)

25

Helix angle (°)

15

Shaft angle (°)

100

Inner radius (mm)

510

External radius (mm)

600

Output torque (N m)

1600

Young’s modulus (MPa)

206,800

Poisson’s ratio

0.29

And a parameterized 3D grid of modified gears is generated based on MATLAB programming. Then, according to the writing order of C3D8R in ABAQUS, the nodes of the modified gears are sequentially numbered to construct an eight-node hexahedron element. Finally, the five-tooth model of the face-gear pair is obtained by rotating the generated single-tooth mesh models of the face gear and pinion around their respective axes ϕ1 and ϕ2 , respectively. Due to the difference in the number of teeth participating in the meshing when the tooth surfaces are in contact and considering the deformation conditions during the meshing process of the face-gear, the single-tooth model is used to calculate the contact stress. The results of the comparison between the tooth surface contact stress calculated by ABAQUS and the theoretical Hertzian contact stress can be shown in Table 2. According to the comparison between the results of FEM and the hertzian contact stress, the error of the tooth surface contact stress at the pitch circle is less than 10%, which verifies the correctness of the modeling method and preprocessing settings used in this paper.

5.4 FEM The contact stress distribution of the face gear is shown in Fig. 18a. Before the tooth surface modification, at the crest and tooth root position of the face-gear, the contact area of the tooth surface is reduced, resulting in edge contact and the contact stress value increases sharply; After the modification, the load distribution of the tooth

242

J. Xu and C. Jia

Fig. 15 Paths of contact and transmission error on face-gear surfaces with the assembly errors (Δγ = 1.5' )

0 -5 -10

TE (arc sec)

-15 -20 -25 -30 -35 -40 -45 -40

-30

-20

-10

0

10

20

30

40

50

30

40

50

Pinion rotation (deg)

(a) Standard tooth

0

TE (arc sec)

-10 -20 -30 -40 -50 -60 -40 -30

-20

-10 0 10 20 Pinion rotation (deg)

(b) Second-order parabolic modified tooth

0

TE (arc sec)

-10 -20 -30 -40 -50 -60 -40 -30 -20 -10 0 10 20 Pinion rotation (deg)

30

(c) High-order parabolic modified tooth

40

50

Design and Analysis of Modified Non-orthogonal Helical Face-Gears … Fig. 16 Paths of contact and transmission error on face-gear surfaces with the assembly errors (Δγ = −1.5' )

243

0 -5

TE (arc sec)

-10 -15 -20 -25 -30 -35 -40 -40 -30 -20 -10 0 10 20 30 Pinion rotation (deg)

40

50

(a) Standard tooth

0

TE (arc sec)

-10 -20 -30 -40 -50 -60 -40 -30 -20 -10 0 10 20 30 40 50 Pinion rotation (deg) (b) Second-order parabolic modified tooth

0 -10

TE (arc sec)

-20 -30 -40 -50 -60 -40 -30 -20 -10

0

10

20

30

40

Pinion rotation (deg)

(c) High-order parabolic modified tooth

50

244

J. Xu and C. Jia 0

Fig. 17 Tooth modification values of the pinion

-5

TE (arc sec)

-10 -15 -20 -25 -30 -35 -40 30

20

0 10 20 10 Pinion rotation (deg)

30

40

Modification value (um) 0 20 40 40

30

20

10

0

-10

-20

-30

-40

86

84

82

80

(a) The second-order modification 0 -5

TE (arc sec)

-10 -15 -20 -25 -30 -35 -30

-20

-10 0 10 20 Pinion rotation (deg)

30

40

Modification value(um) 0 10 20 30 40 40 20 0 -20 -40

86

(b) The HTE modification

84

82

80

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

245

Table 2 The comparison between the results of finite element method and the hertzian contact stress Results

Contact stress/MPa

Contact stress of FEM

749.02

Hertzian contact stress

686.3

Error

8.3%

surface is optimized, and the adverse effects of stress concentration are eliminated. For the traditional second-order parabolic modification method, the modification amount of the tooth surface increases gradually from the middle area of the tooth surface to the boundary area. As mentioned earlier, the clearance between the pinion and face-gear surfaces is determined by the tooth modification. Therefore, the load on the tooth surface of the face-gear is transferred from the boundary region to the middle region, so the load in the middle region increases sharply, and the contact stress also increases accordingly. In the new modification method with HTE, the load is transferred from the boundary to the middle region, but due to the concave flank modification, the load distribution in the middle region is more uniform than the traditional second-order parabolic modification. The contact stress distribution of the modified tooth surface is shown in Fig. 18b, c. Compared with the unmodified tooth surface in Fig. 18a, The face-gear with modified tooth surface avoids the influence of edge contact. And this modification method improved tooth surface load distribution, which is very important to reduce tooth surface contact stress. The contact stress and bending stress distribution of the standard tooth, secondorder parabolic modified tooth and high-order parabolic modified tooth of the face gear without the assembly errors are compared in Figs. 18d and 19d. As shown in Figs. 18d and 19d, before the modification, there is edge contact at the meshing of face-gear, and the contact stress of the tooth surface increases significantly. The maximum contact stress is 890.7 MPa, and the maximum bending stress is 51.1 MPa. After the traditional tooth surface modification, the edge contact at the crest and root position is avoided. The maximum contact stress is 739.7 MPa, and the maximum bending stress is 58.5 MPa. After the new high-order tooth surface modification, the concave modification in the middle meshing area makes the tooth surface load distribution more uniform, the maximum contact stress of the face-gear is only 580.4 MPa, and the maximum bending stress is only 44.6 MPa. It is clear that the traditional second-order parabolic modified gear has a much greater contact stress than that of the new high-order parabolic modified one. The maximum contact stress and maximum bending stress without assembly errors are shown in Table 3. There is a 34.83% decrease in maximum contact stress and a 12.72% decrease in maximum bending stress of face-gear based on the new high-order modification method; there is a 16.95% decrease in maximum contact stress, but a 14.48% increase in maximum bending stress of face-gear based on the traditional second-order parabolic tooth modification method.

246

J. Xu and C. Jia

Fig. 18 Comparison of the contact stresses without assembly errors

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Contact stress (MPa)

1000 800 600 400 200 0 -15

-10

-5 0 5 Meshing positions

10

(d) Comparison of the bending stresses

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

247

Fig. 19 Comparison of the bending stresses without assembly errors

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Bending stress (MPa)

60 50 40 30 20 10 0 -15

-10

-5 0 5 Meshing positions

10

(d) Comparison of the bending stresses

248

J. Xu and C. Jia

Table 3 Comparison of the maximum contact stress and maximum bending stresses without assembly errors Results

Contact stress Results (MPa)

Bending stress Variation

Results (MPa)

Variation

Standard tooth

890.7

–

51.1

–

Second order

739.7

− 16.95%

58.5

+ 14.48%

High order

580.4

− 34.83%

44.6

− 12.72%

In the face-gear transmission system, the assembly error of the shaft angle of the face- gear is inevitable. In addition, the higher the transmission ratio of the gear transmission, the greater the influence of the assembly error of the shaft angle. The gear ratio of the face gear pair is large. Therefore, it is of great significance to study the error sensitivity of the gear. Hereinafter, Δγ is the assembly error of the shaft angle. The contact stress and bending stress distribution of the standard tooth, secondorder parabolic modified tooth and high-order parabolic modified tooth of the facegear with the assembly errors (Δγ = 1.5' ) are compared in Figs. 20d and 21d. As shown in Figs. 20d and 21d, before the modification, the maximum contact stress is 858.7 MPa, and the maximum bending stress is 56.4 MPa. After the traditional tooth surface modification, the maximum contact stress is 745 MPa, and the maximum bending stress is 58.5 MPa. After the new high-order tooth surface modification, the maximum contact stress of the face-gear is only 656.2 MPa, and the maximum bending stress is only 47.7 MPa. The simulation results further verify the superiority of the method. The maximum contact stress and maximum bending stress with assembly errors are shown in Table 4. There is a 23.58% decrease in maximum contact stress and a 15.42% decrease in maximum bending stress of face-gear based on the new highorder parabolic tooth modification method; there is a 13.24% decrease in maximum contact stress, but a 3.72% increase in maximum bending stress of face-gear based on the traditional second-order parabolic tooth modification method. The contact stress and bending stress distribution of the standard tooth, secondorder parabolic modified tooth and high-order parabolic modified tooth of the facegear with the assembly errors (Δγ = −1.5' ) are compared in Figs. 22d and 23d. As shown in Figs. 22d and 23d, before the modification, the maximum contact stress is 940.9 MPa, and the maximum bending stress is 63.9 MPa. After the traditional tooth surface modification, the maximum contact stress is 774 MPa, and the maximum bending stress is 80.6 MPa. After the new high-order tooth surface modification, the maximum contact stress of the face-gear is only 694 MPa, and the maximum bending stress is only 58.8 MPa. The simulation results further verify the superiority of the method.

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

249

Fig. 20 Comparison of the contact stresses with assembly errors, Δγ = 1.5'

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Contact stress (MPa)

1000 800 600 400 200 0 -15

-10 -5 0 5 10 Meshing positions (d) Comparison of the contact stresses

250

J. Xu and C. Jia

Fig. 21 Comparison of the bending stresses with assembly errors, Δγ = 1.5'

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Bending stress (MPa)

60 50 40 30 20 10 0 -15

-10 -5 0 Meshing positions

5

(d) Comparison of the bending stresses

10

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

251

Table 4 Comparison of the maximum contact stress and maximum bending stresses with assembly errors, Δγ = 1.5' Results

Contact stress Results (MPa)

Bending stress Variation

Results (MPa)

Variation

Standard

858.7

–

56.4

–

Second order

745

− 13.24%

58.5

+ 3.72%

High order

656.2

− 23.58%

47.7

− 15.42%

The maximum contact stress and maximum bending stress with assembly errors are shown in Table 5. There is a 26.24% decrease in maximum contact stress and a 7.98% decrease in maximum bending stress of face-gear based on the new highorder parabolic tooth modification method; there is a 17.73% decrease in maximum contact stress, but a 26.13% increase in maximum bending stress of face-gear based on the traditional second-order parabolic tooth modification method.

6 Conclusions In order to improve the meshing performance of non-orthogonal helical face-gears, a designation of double-crowned tooth modification with new high-order transmission error is constructed in this paper. And the design of high-order modifications is verified via TCA and LTCA. From the present study, the following conclusions are drawn. (1) Based on the modified tooth profile of the rack cutter and the contact path in the meshing process of the face gear pair, a high-order transmission error is designed. (2) The introduced new modification causes significantly more stress decreases. The service life of face-gear will be further extended compared with the traditional second-order parabolic tooth modification. (3) By comparing the maximum contact stresses and maximum bending stresses of the tooth surfaces under the assembly error condition of the examples, the superiority of the new tooth surface high-order modification method is further verified. (4) The core idea of the new tooth surface high-order modification method is based on the transmission error designation and therefore it can be universal, not limited to face-gear drives, which can be extended to other types of gear drives.

252

J. Xu and C. Jia

Fig. 22 Comparison of the contact stresses with assembly errors, Δγ = −1.5'

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Contact stress (MPa)

1000 800 600 400 200 0 -10

-5

0 5 10 Meshing positions

15

(d) Comparison of the contact stresses

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

(a) Standard tooth

(b) Second-order parabolic modified tooth

(c) High-order parabolic modified tooth Standard Second-order High-order

Bending stress (MPa)

80 60 40 20 0 -10

-5 0 5 10 15 Meshing positions (d) Comparison of the bending stresses

Fig. 23 Comparison of the bending stresses with assembly errors, Δγ = −1.5'

253

254

J. Xu and C. Jia

Table 5 Comparison of the maximum contact stress and maximum bending stresses with assembly errors, Δγ = −1.5' Results

Contact stress

Bending stress

Results (MPa)

Variation

Results (MPa)

Variation

Standard

940.9

–

63.9

–

Second order

774

− 17.73%

80.6

+ 26.13%

High order

694

− 26.24%

58.8

− 7.98%

Acknowledgements Supported by National Natural Science Foundation of China (Project No.52005107), and the Natural Science Foundation of Fujian Province, China (Project No. 2020J05100).

References 1. Litvin FL, Zhang Y, Wang JC et al (1992) Design and geometry of face-gear drives. J Mech Des 114(4):642–647 2. Feng G, Xie Z, Zhou M (2019) Geometric design and analysis of face-gear drive with involute helical pinion. Mech Mach Theory 134:169–196 3. Guan Y, Fa Ng Z, Yang X et al (2018) Tooth contact analysis of crown gear coupling with misalignment. Mech Mach Theory 126:295–311 4. Liu S, Song C, Zhu C et al (2018) Concave and convex modifications analysis for skewed beveloid gears considering misalignments. Mech Mach Theory 133 5. Deng J, Nie S, Deng X et al (2020) Tooth surface mismatch modification method of cycloidal bevel gear based on conjugate tooth surface modification. J Adv Mech Des Syst Manuf 14(1):1– 22 6. Huang KJ, Su CY (2013) A finite element approach to dynamic stresses of helical gear pairs considering tip relief and crowning modifications. Appl Mech Mater 284–287:577–581 7. Sun Z, Chen S, Hu Z et al (2022) Improved mesh stiffness calculation model of comprehensive modification gears considering actual manufacturing. Mech Mach Theory 167:104470 8. Zanzi C, Pedrero JI (2005) Application of modified geometry of face gear drive. Comput Methods Appl Mech Eng 194(27):3047–3066 9. Tran VQ, Wu YR (2019) A novel method for closed-loop topology modification of helical gears using internal-meshing gear honing. Mech Mach Theory 145:103691 10. Li T, An X, Deng X, Li J, Li Y (2020) A new tooth profile modification method of cycloidal gears in precision reducers for robots. Appl Sci 10(4):1266 11. Litvin FL, Jian L, Townsend DP et al (1997) Computerized simulation of meshing of conventional helical involute gears and modification of geometry. Mech Mach Theory 34(1):123–147 12. Zschippang HA, Weikert S, KüüK K et al (2019) Face-gear drive: geometry generation and tooth contact analysis. Mech Mach Theory 142:103576 13. Dongsheng H (2012) Research on numerical analysis modeling method for gear meshing and its applications. Dalian University of Technology (in Chinese) 14. Jiang J, Fang Z (2015) Design and analysis of modified cylindrical gears with a higher-order transmission error. Mech Mach Theory 88 15. Jia C, Fang Z (2019) Design and analysis of double-crowned high-contact-ratio cylindrical gears considering the load sharing of the multi-pair contact. Mech Mach Theory 131:92–114

Design and Analysis of Modified Non-orthogonal Helical Face-Gears …

255

16. Su J, Fang Z, Cai X (2013) Design and analysis of spiral bevel gears with seventh-order function of transmission error. Chin J Aeronaut 26(5):1310–1316 17. Mu Y, Li W, Fang Z et al (2018) A novel tooth surface modification method for spiral bevel gears with higher-order transmission error. Mech Mach Theory 126:49–60 18. Yunbo S (2013) Novel design process for face gear drive with a high order polynomial function of transmission error. J Xi’an Technol Univ 33(12):7 (in Chinese) 19. Daqing L, Suzhen W, Rangqian Z et al (2020) Design of the high-order curve tooth profile and analysis on the face gear’s tooth contact. J Mach Des 37(11):109–114 (in Chinese) 20. Ning Z, Xiaochun Z, Hui G et al (2008) Tooth contact analysis of face gear drive with helical pinion and simulation of the tooth face. J Aerosp Power 23(10):1927–1932 (in Chinese) 21. Jacquin V (2005) Quasi-static analysis of a face gear under torque. Comput Methods Appl Mech Eng 194:4301–4318 22. Vouaillat G, Noyel JP, Ville F et al (2019) From Hertzian contact to spur gears: analyses of stresses and rolling contact fatigue. Mech Ind 20(6):1–16 23. Wang H, Tang L, Zhou C et al (2021) Wear life prediction method of crowned double helical gear drive in point contact mixed elastohydrodynamic lubrication. Wear 484–485:204041 24. Litvin FL, Fuentes A, Zanzi C et al (2002) Face-gear drive with spur involute pinion: geometry, generation by a worm, stress analysis. Comput Methods Appl Mech Eng 191(25–26):2785– 2813 25. Xuezhong F, Zongde F, Long X et al (2018) Assembly errors tolerance and sensitivity of offset face gears. J Harbin Eng Univ 39(7):1227–1232 (in Chinese) 26. Xianlong P, QichaoI X, Xiangying H et al (2020) Finite element analysis of gear bearing transmission performance influenced by misalignment. J Harbin Eng Univ 41(12):1861–1867 (in Chinese)

Integrated Kinematic Modeling of Reconfigurable Parallel Robot with 3T/3R Motion Patterns Hao Li, Yimin Song, Xinming Huo, Wei Xian, and Yang Qi

Abstract Reconfigurable parallel mechanisms have more than one motion pattern and facilitate complex and compound manipulations. However, it is still a challenge to describe all the motion patterns by an integrated expression, which leads to difficulties in the analysis of transformation process. Furthermore, performance model of each motion pattern is always constructed individually in spite of their coupled relationship. In this paper, a reconfigurable parallel robot with spatial translational (3T) and rotational (3R) motion patterns is presented for on-orbit manipulation of cabins, including target capture, racemization, screwing and plugging. The integrated topology description, transformation process and whole kinematic modeling of this robot are investigated in-depth. Firstly, the topologies of two motion patterns are described in finite screw format and combined by defining a transformation factor. In this way, the transformation process is analyzed algebraically. Then taking advantage of the differential mapping of finite and instantaneous screws, the Jacobian matrix is derived from the motion equations directly. Finally, a numerical example is given to show the on-orbit manipulation process of the reconfigurable robot, involving inverse kinematics and workspace analysis. The method proposed in this paper could connect the topologies under different motion patterns algebraically, which facilitates to investigate the motion and performance of reconfigurable parallel mechanisms as a whole. Keywords Reconfigurable mechanism · FIS theory · Integrated topology and kinematic modeling · Workspace analysis

H. Li · Y. Song · X. Huo (B) · W. Xian School of Mechanical Engineering, Tianjin University, Tianjin 300354, China e-mail: [email protected] H. Li e-mail: [email protected] Y. Qi School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin 300222, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_14

257

258

H. Li et al.

1 Introduction Reconfigurable parallel mechanisms combine the merits of both parallel mechanisms and reconfigurable mechanisms. Due to the compact structure, feasible kinematics, high stiffness and multi- motion patterns, they have been applied in mobile robot, 3D-printer and other fields requiring complex and compound manipulations. The first reconfigurable parallel mechanism was invented by Wohlhart in 1996. He found the phenomenon that some parallel mechanisms may have bifurcation and fall in different motion patterns. Then inspired by biological evolution, Dai proposed variable topologies with variable degree of freedom, which are known as metamorphic mechanisms [1]. From then on, reconfigurable parallel mechanisms have attracted much attention from scholars and showed special advantages on multi tasks [2–4]. Motion description is the foundation of kinematic modeling and analysis. Compared with traditional parallel mechanism, it is obvious that reconfigurable parallel mechanisms have no less than two motion patterns. Some researchers list and describe each motion pattern individually. Zlatanov et al. analyzed the each motion pattern of 3-UPU parallel mechanism by using instantaneous screw and constraints [5]. In the similar manner, five alternatives of 3-URU are also discussed [6]. Walter studied the same case from the point view of finite motion, in which dual quaternion was applied and motion patterns were derived by quasi prime decomposition of kinematic polynomial [7]. Applying this method, Stigger explored the motion pattern of 3-RUU parallel mechanism [8]. Dai pointed out that the motion description of reconfigurable parallel mechanisms could not be independent of motion transformation [9]. Wu introduced the adjoint matrix and described the motion transformation by matrix operations [10]. Yang simplified the adjoint matrix and associated topology transformation operations with matrix elements [11]. Zhang proposed screw adjoint matrix method to solve the problem of space parallel mechanisms [12, 13]. These approaches can describe the motion patterns and their transformation conditions However, there are still few methods to describe them in an integrated framework. The individual description leads to the individual kinematic models. Some dimensions may have effect on not one motion patterns but they are analyzed individually. In fact, the motion characteristics of different motion patterns determine the whole performance of the mechanism. Zhang analyzed all the motion patterns of a new 4-URU reconfigurable parallel mechanism and unified them with the help of quaternions [14]. Gan analyzed four topological alternatives of a reconfigurable parallel mechanism 3-rTPS based on screw theory and obtained a unified kinematic model [15, 16]. Furthermore, Baran et al. proposed a unified formula for kinematic analysis of all possible Delta mechanisms with variable angle and variable link length [17]. It could be seen that current work still studies the kinematics of each motion pattern individually and then attempt to connect them, which sometimes depend on observation or experiment.

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

259

In our previous work, a novel 3T/3R reconfigurable parallel mechanism has been proposed as a space hand working for on-orbit manipulation of cabins, such as target capture, racemization, screwing and plugging. This paper will carry out the kinematic analysis of this parallel mechanism. In this paper, the motion pattern description and kinematic modeling would be investigated on the basis of finite and instantaneous screws (FIS). The main structure of this paper is as follows. The mathematic foundation of FIS [18–24], which is a concise and unified framework for topology description and kinematic modeling [25–29], would be introduced in Sect. 2. The integrated description of 3T and 3R motion patterns would be discussed in Sect. 3. Section 4 formulates the integrated kinematic models of the mechanism including the modeling and analysis of displacement, and velocity. Finally, a numerical case is given to verify the integrated method.

2 Finite and Instantaneous Screw Theory As shown in Fig. 1, the finite motion of rigid body moving from State I to State II could be described as     θ sf 0 +t (1) S f = 2 tan sf 2 rf × sf where S f is the finite motion from State I to State II. s f is the unit vector of the finite screw axis of the finite motion and r f is the position vector pointing from the origin to an arbitrary point on the screw axis. θ and t describe the amplitudes of the finite screw, representing respectively the rotation angle about and the translation distance along the finite screw axis. Through the differential mapping between finite and instantaneous screws [18, 30, 31], the expression of the velocity can be solved as: Fig. 1 Rigid body motion. a Finite motion and b instantaneous motion

260

H. Li et al.

    θ˙ θ s˙ f sf + 2 tan 2 r˙ f × s f + r f × s˙ f cos2 θ2 r f × s f     0 0 +t + t˙ s˙ f sf

S˙ f =

(2)

At the initial pose, the parameter of the amplitudes θ and t equal zeros. So, Eq. (2) can be rewritten as      st 0 ˙S f  ˙ =θ (3) + t˙ = St θ =0,t=0 st r t × st where St is famous as the instantaneous screw. The finite motion of an open loop can be described by the composition algorithm of the motions of each joint, which is denoted by the screw triangle product: S f,L,i = S f,1,i Δ . . . ΔS f,k,i . . . ΔS f,n,i

(4)

The null transformation is defined as a screw with both the rotational angle and the translational distance set to be zero:     0 sf 0 +0 (5) S f 0 = 2 tan sf 2 rf × sf The screw triangle product of the null transformation and an arbitrary screw satisfies the communicative law, as S f 0 ΔS f,a = S f,a

(6)

S f,a ΔS f 0 = S f,a

(7)

In parallel robots, the finite motion of the moving platform is the same as that of each limb, therefore, the finite motion model of the parallel manipulators (PM) is formulated as the motion intersection: S f,PM = S f,L,1 ∩ S f,L,2 ∩ · · · ∩ S f,L,m

(8)

where S f,L,i denotes the finite motion of each limb, ‘∩’ denotes the motion intersection of each limb. As the differential mapping shown in Eqs. (2) and (3), the velocity model of the open loop and the parallel robot are respectively described as Eqs. (6) and (7) according to the linear superposition of velocities:

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

St,L,i = S˙ f,L,i = S˙ f,1,i Δ . . . Δ S˙ f,k,i . . . Δ S˙ f,n,i =

261 n ∑

St,k,i

(9)

k=1

St,PM = S˙ f,PM = St,L,1 ∩ St,L,2 ∩ St,L,3

(10)

where the velocity models of PM are not only the same as the ones of each limb, but are the intersection of models of each limb as well.

3 Integrated Motion Description of 3T/3R Parallel Robot As shown in Fig. 2, the 3T/3R parallel robot is composed by fixed platform and moving platform, connected by three identical limbs. Each limb has five common joints which are free in both motion patterns and one transforming joint RTr,i which is locked in transformation. Among the five common joints, the motion axis of the first joint R1,i is passing through the center of fixed platform, which is also assigned as active joint. R2,i and R4,i are parallel with R3,i and R5,i , respectively, as shown in Fig. 3.

Fig. 2 Reconfigurable parallel robot 3Rf R1 R1 R2 R2 in different motion pattern

Fig. 3 Topology of each limb as R1 Rtra R1 * R1 R2 R2 in a motion pattern I and b motion pattern II

262

H. Li et al.

3.1 Multi-motion Patterns and Their Integrated Description According to the finite screw theory mentioned in Sect. 2, the finite motion of 3T/3R parallel robot can be described as: S f ,PM = SIf,PM ∪ SIIf,PM

(11)

where SIf,PM and SIIf,PM are the finite motion of translational and rotational motion patterns, respectively. I I SIf,PM = SIf,L,1 ∩ S f,L,2 ∩ S f,L,3

(12)

SIIf,PM = SIIf,L,1 ∩ SIIf,L,2 ∩ SIIf,L,3

(13)

In motion pattern I, R2,i and R3,i are parallel with R1,i and RTr,i is locked. The finite motion of each limb in pattern I is denoted as: SIf,L,i

  s f,i,4 θ f,i,4 Δ2 tan 2 r f,i,5 × s f,i,5 r f,i,4 × s f,i,4     s f,i,3 s f,i,1 θ f,i,3 θ f,i,2 Δ2 tan Δ2 tan 2 2 r f,i,3 × s f,i,3 r f,i,2 × s f,i,1      str,i s f,i,1 θ f,i,1 θtr,i  Δ2 tan Δ2 tan  2 r tr,i × str,i  2 r f,i,1 × s f,i,1

θ f,i,5 = 2 tan 2



s f,i,5



(14)

θtr =0

where i = 1, 2, 3, θtr,i is the rotational variable of the transforming joint which would be zero during motion pattern. str,i and r tr,i is the direction and position vectors of the transforming joint. In the similar manner, R2,i and R3,i intersect at the center of fixed platform O and RTr,i is locked. The finite motion of each limb in pattern II is denoted as: SIIf,L,i

   s f,i,5 s f,i,4 θ f,i,4 Δ2 tan 2 r f,i,5 × s f,i,5 r f,i,4 × s f,i,4     s f,i,3 s f,i,2 θ f,i,3 θ f,i,2 Δ2 tan Δ2 tan 2 2 r f,i,3 × s f,i,3 r f,i,1 × s f,i,2      str,i s f,i,1 θ f,i,1 θtr,i  Δ2 tan Δ 2 tan  2 r tr,i × str,i  2 r f,i,1 × s f,i,1

θ f,i,5 = 2 tan 2



θtr =0

Comparing Eqs. (11) and (12), it could be found out that

(15)

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

s f,i,2 = Rs f,i,1

263

(16)

⎡

⎤ cos φ − sin φ 0 where R = ⎣ sin φ cos φ 0 ⎦. 0 0 1 If φ = 0, R = E, the 3T/3R parallel robot is in pattern I; if φ /= 0, R /= E, the 3T/3R parallel robot is in pattern II. Substituting Eq. (13) into Eq. (8), the motion of 3T/3R parallel robot could be described in an integrated expression S f,PM = S f,L,1 ∩ S f,L,2 ∩ S f,L,3

(17)

where S f,L,i

   s f,i,5 s f,i,4 θ f,i,4 Δ2 tan 2 r f,i,5 × s f,i,5 r f,i,4 × s f,i,4     s f,i,2 s f,i,3 θ f,i,3 θ f,i,2 T Δ2 tan Δ2 tan 2 2 r f,i,3 × s f,i,3 r f,i,2 × s f,i,2      str,i s f,i,1 θ f,i,1 θtr,i  Δ2 tan Δ 2 tan  2 r tr,i × str,i  2 r f,i,1 × s f,i,1

θ f,i,5 = 2 tan 2



(18)

θtr =0

where ⎧ , S f,PM = SIf,PM ⎨T = E  6×6  R 0  ⎩T = R ⊆ S O(3), S f,PM = SIIf,PM 0 R 

3.2 Transforming Between Two Motion Patterns The motion pattern transformation of the robot could be considered as a finite motion from the switching point Ps to the final point Pf where the robot falls in the next motion pattern. Before the switching point Ps and after the final point Pf , the transforming joint would be released and locked again, respectively. When the robot transforms from pattern I to pattern II, the motion of component limb from Ps turns to be     s s θ θ f,i,5 f,i,4 f,i,5 f,i,4 SI→II Δ2 tan f,L,i = 2 tan 2 2 r f,i,5 × s f,i,5 r f,i,4 × s f,i,4

264

H. Li et al.

    s f,i,3 s f,i,1 θ f,i,3 θ f,i,2 Δ2 tan Δ2 tan 2 2 r f,i,3 × s f,i,3 r f,i,2 × s f,i,1     str,i s f,i,1 θ f,i,1 θtr,i Δ2 tan Δ2 tan 2 r tr,i × str,i 2 r f,i,1 × s f,i,1

(19)

It indicates that the limb has 6 degree-of-freedom. Since the motion of moving platform is determined by the motion intersection of all the limbs, the robot turns to be a 6-DoF mechanism. At any moment, the finite motion of the second common joint could be describe as S f,i,2

 = T φ/=0, R/= E



s f,i,2 r f,i,1 × s f,i,2

 (20)

when r f,i,2 = r f,i,1 , the transformation finite motion from Pf could be written as SII→I f,L,i

  s f,i,4 θ f,i,4 Δ2 tan 2 r f,i,5 × s f,i,5 r f,i,4 × s f,i,4     s f,i,3 s f,i,2 θ f,i,3 θ f,i,2 Δ2 tan Δ2 tan 2 2 r f,i,3 × s f,i,3 r f,i,1 × s f,i,2     str,i s f,i,1 θ f,i,1 θtr,i Δ2 tan Δ2 tan 2 r tr,i × str,i 2 r f,i,1 × s f,i,1

θ f,i,5 = 2 tan 2



s f,i,5



(21)

SII→I f,L,i shows that the robot still has 6-DoF. It would fall in the motion pattern II after locking the transforming joint. It could be seen that Ps and Pf are in the workspace of the motion pattern I and motion pattern II. If the switching point is located at the common workspace of the two motion patterns, it also could be taken as a final point. In this case, Ps and Pf are coincident. It means that there exist at least two set of solutions of the joint variables at point Ps (Pf ) for the 6-DoF parallel mechanisms in the transformation process. If the switching point is selected out of the common workspace of the two motion patterns, the final point would be different with the switching point. In the similar manner, the robot could be transformed from motion pattern II to I, when releasing the switching joint. In fact, this process is the inverse motion of II→I SI→II f,L,i , which could be described as Eq. (18) and assigned as S f,L,i . In this case, Pf and Ps are the switching point and the final point, respectively. They can be coincident if one of them is in the common workspace.

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

265

4 Integrated Kinematic Modeling and Analysis After describing the motion patterns and their transforming process in an integrated manner, this section focuses on the integrated kinematic modeling of the robot, including displacement modeling and Jacobian matrix.

4.1 Integrated Displacement Model As shown in Fig. 4, a fixed reference frame O0 -x 0 y0 z0 is established at the center of the top plane of the fixed base, in which axis x 0 is opposite to the first joint position vector bi , axis z0 is perpendicular to the fixed base and axis y0 is determined by the right-hand principle. For the 3T/3R parallel robot, the motion of moving platform is the same with that of each limb S f,PM = S f,L,1 = S f,L,2 = S f,L,3 where

Fig. 4 Integrated displacement model of reconfigurable parallel robot

(22)

266

H. Li et al.

S f,PM = SIf,PM ∪ SIIf,PM In pattern I,  SIf,PM

= tx

0 sx



 + ty

0 sy



 + tz

0 sz



where s x , s y and s z are direction axes of world coordinate, respectively. Equation (11) can be rewritten as:   s2,i θ21,i + θ22,i I S f,L,i = 2 tan 2 r 22,i × s2,i   0    Δ   exp θ21,i s˜ 2,i − E 3 r A2 ,i − r A1 ,i   s f,i θ f,i + θ12,i + θ13,i Δ2 tan 2 r 13,i × s f,i   0     Δ   exp θ f,i + θ12,i s˜ f,i − E 3 r B3 ,i − r B2 ,i   0   Δ     (23) exp θ f,i s˜ f,i − E 3 r tr,i − r f,i where the rotational motions satisfy: 2 tan 2 tan

 s2,i = 0, r 22,i × s2,i   + θ12,i + θ13,i s f,i =0 r 13,i × s f,i 2

θ21,i + θ22,i 2 θ f,i



Therefore, the amplitudes of the rotational motions: 

θ21,i + θ22,i = 0 θ11,i + θ12,i + θ13,i = 0

Similarly, in pattern II, Eq. (12) can be rewritten as: SIIf,L ,i

  s2,i θ21,i + θ22,i = 2 tan 2 r 22,i × s2,i   0      Δ exp θ21,i s˜ 2,i − E 3 r A2 ,i − r A1 ,i

(24)

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

  θ12,i + θ13,i s f,i Δ2 tan r 13,i × s f,i 2    0   Δ   exp θ f,i + θ12,i s˜ f,i − E 3 r B3 ,i − r B2 ,i   s f,i θ f,i 2 tan 2 r f,i × s f,i

267

(25)

In order to obtain the angular mapping relationship between the moving platform and joints of each limb in pattern II, Eq. (21) can be further rewritten through changing the position of finite motions as: SIIf,PM

 s2,i r 22,i × s2,i   s1,i θ13,i + θ12,i Δ2 tan 2 r 12,i × s1,i   s f,i θ f,i Δ2 tan 2 r f,i × s f,i   0 Δ t3R,i s3R,i

θ21,i + θ22,i = 2 tan 2



(26)



   0 0      = where , herein, ui = t3R,i s3R,i exp θ f,i s˜ f,i exp θ12,i s˜ 1,i (ui − v i )      exp θ21,i s˜ 2,i − E 3 r A2 ,i − r A1 ,i + bi , v i = r B2 ,i − r B1 ,i is related to the angles of the R joints in each limb. There is a certain mapping relationship between the finite motion of moving platform and angular displacements of each R joint:   S f,i = f θ22,i , θ21,i , θ13,i , θ12,i , θ f,i

(27)

α, β and γ are angular displacements of 3T/3R parallel robot, respectively. Combined with Eqs. (22) and (23), the integrated displacement model of 3T/3R parallel robot can be denoted as:   ⎧ tx = f x −θ21,i , θ21,i , −θ12,i − θ f,i , θ12,i , θ f,i ⎪ ⎪ ⎪   ⎪ ⎪ t y = f y −θ21,i , θ21,i , −θ12,i − θ f,i , θ12,i , θ f,i ⎪ ⎪ ⎪ ⎪ ⎨ tz = f z −θ21,i , θ21,i , −θ12,i − θ f,i , θ12,i , θ f,i    ⎪ ⎪ ⎪ α = f α θ22,i + θ21,i , θ13,i + θ12,i , θ f,i  ⎪ ⎪ ⎪ β = f β θ22,i + θ21,i , θ13,i + θ12,i , θ f,i ⎪ ⎪ ⎪   ⎩ γ = f γ θ22,i + θ21,i , θ13,i + θ12,i , θ f,i

(28)

268

H. Li et al.

which can be computed by numerical method.

4.2 Integrated Velocity Model and Jacobian Matrix According to the differential mapping between finite and instantaneous screws, the integrated velocity model can be solved through differentiating Eq. (8), as: St,i = S˙ f,PM = St,1 ∩ St,2 ∩ St,3 =

5 ∑ k=1

St,i,k =

5 ∑

θ˙i,k Sˆ t,i,k , i = 1, 2, 3

(29)

k=1

In parallel robots, the velocity models of terminal motions of each limb and moving platform are equal to:    s1,i s1,i ˙ + θ12,i T r 11,i × s11,i r 12,i × s1,i   s1,i + θ˙13,i T r 13,i × s1,i     s2,i s2,i ˙ ˙ + θ21,i + θ22,i r 21,i × s2,i r 22,i × s2,i 

St,i

= θ˙11,i

(30)

where  ⎧ E 0 3×3 ⎪ ⎪ S f ,PM ⊆ SIf,PM ⎨ 0 E 3×3  T=  ⎪ R 0  ⎪ ⎩ R ⊆ S O(3) S f ,PM ⊆ SIIf,PM 0 R  In each limb of 3T/3R parallel robot, the wrench of each limb is expressed as: Sw,i = f a,i Sˆ wa,i + f c,i Sˆ wc,i , i = 1, 2, 3

(31)

where Sˆ wa,i and Sˆ wc,i stand for the unit actuation wrench of the actuation joint and constraint wrench of each limb. In each limb, there is a constraint wrench reciprocal to all the joints: T Sˆ wc,i St,i = 0, i = 1, 2, 3

(32)

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

269

which can be obtained through computing pair screw system; there is one actuation wrench reciprocal to all the passive joints, as: T T Sˆ wa,1,i St,i = q˙1,i Sˆ wa,1,i Sˆ t,1,i , i = 1, 2, 3

which can be rewritten in Plucker coordinate form as:   r wa,i × swa,i Swa,i = swa,i

(33)

(34)

      where swa,i = s11,i × s12,i × s21,i × s22,i , r wa,i = bi + r A2 ,i − r A1 ,i + T  √   r B3 ,i − r B2 ,i + R B2,2 R z wi . Herein, wi = 0 0 22 L per , R B2,2 and R z are the vector along s3 , transformation matrices of joint R2,i and R3,i , respectively. According to Eqs. (31) and (32), Eq. (30) can be transformed as: J w St = J q q˙

(35)

     T  q˙ a J wa J qa where J w = , Jq = , q˙ = , q˙ a = θ˙ f,1 θ˙ f,2 θ˙ f,3 , 0 J wc 03×1 T T   J wa = Sˆ wa,1 Sˆ wa,2 Sˆ wa,3 , J wc = Sˆ wc,1 Sˆ wc,2 Sˆ wc,3 , J q = ⎤ ⎡ T Sˆ wa,1 Sˆ t, f,1 ⎥ ⎢ T ⎥ ⎢ Sˆ wa,2 Sˆ t, f,2 ⎥. ⎢ T ⎥ ⎢ ˆ ˆ ⎦ ⎣ Swa,3 St, f,3 03×3 Equation (30) can be rewritten as: 

˙ St = J −1 w Jqq ⎡ T Sˆ wa,1 ⎢ T ⎢ Sˆ wa,2 ⎢ T ⎢ ˆ ⎢ S = ⎢ wa,3 ⎢ Sˆ T ⎢ wc,1 ⎢ ˆT ⎣ Swc,2 T Sˆ wc,3 = J q˙

⎤−1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎢ ⎢ Jq⎢ ⎢ ⎢ ⎣

θ˙1,1 θ˙2,1 θ˙3,1 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(36)

where matrix J = J −1 w J q is the Jacobian matrix of 3T/3R parallel robot. The parallel mechanism is in singular configurations when the reciprocal of the conditional number of the Jacobian matrix is near zero.

270

H. Li et al.

5 Kinematic Analysis Based on the kinematic models formulated in Sect. 4, a numerical example would be discussed in this section to analyze the kinematics of the 3T/3R reconfigurable parallel robot. Spinning followed by spatial translation are typical operations of the in-orbit robots. As shown in Fig. 5, a predetermined trajectory is defined by working requirements. In the spinning stage, the moving platform rotates from −20◦ to 0◦ . At the end point, the transforming process would be started, in which the robot has 6-DOF and the moving platform rotates from 0◦ to 0◦ . Then the robot would fall in the spatial translational stage. As shown in Fig. 6, the dimensions of the robot are listed as Table 1. Taking the trajectory of moving platform illustrated in Fig. 5 as target, the motion variables of driving joints including actuation and transforming joints are solved for three stages as shown in Fig. 7. It would be seen that in the transformation stage, the transforming joints are released and in other two motion stage, they are locked.

Fig. 5 The predetermined trajectory by working requirements, a spinning, b transformation process, c three-dimensional translation

Fig. 6 Dimension of each limb

Integrated Kinematic Modeling of Reconfigurable Parallel Robot … Table 1 Dimensional parameters of the 3T/3R reconfigurable parallel robot

Dimension

271 Parameter

Transforming angle

30°

L T1

50 mm

L T2

110.10 mm

L T3

80 mm

L Par

60 mm

L Per

141.42 mm

Radius of fixed base

141.34 mm

Radius of moving platform

76.79 mm

Fig. 7 The angles of the driving joints

According to integrated velocity model mentioned in Sect. 4.2, the angular velocity of the actuation joint of each limb is calculated in spinning, transformation and three-dimensional translation process, respectively, as shown in Fig. 8. The angular velocity of the transforming joint of each limb is also calculated in transformation process. The mutation of the actuation joint of each limb corresponds to the rapid change in the end of transformation process. Workspace of the robot is the working area of the moving platform, which is an important index for performance evaluation. The integrated workspace of the 3T/3R reconfigurable parallel robot could be calculated as: 

Sf

r

 = SIf

r

 + SIIf

r

 − SIf

r

 ∩ SIIf

r

(37)

  r r where SIf and SIIf denote the linear-workspace and orientation-workspace under the 3T and 3R motion patterns, respectively. Therefore, they only have intersection at the initial pose,

272

H. Li et al.

Fig. 8 The angular velocity of the driving joints



SIf

r

 ∩ SIIf

r

= {O}

For 3T and 3R motion patterns, the workspace could be described as 



SIf

r

SIIf

r

 ⎧ ⎫ ⎨   φ= 0 ⎬  = f θk,i  d lT1 , l B3 A1 ≥ d0 ⎩  θ ∈ [θ , θ ], i = 1, 2, 3 ⎭ 1,i L U  ⎧ ⎫ ⎨   φ= π/6  ⎬ = f θk,i  d lT1 , l B3 A1 ≥ d0 ⎩  θ1,i ∈ [θL , θU ], i = 1, 2, 3 ⎭

  where d lT1 , l B3 A1 denotes the distance between the axis of transforming joint and the bar between R2,i and R3,i . θ1,i is the motion variable of the actuated joint of the ith limb, which has lower and upper boundary as θL and θU . Herein, d0 = 10 mm, θL = −π/2, θU = 0. The linear-workspace and orientation-workspace are obtained through the boundary searching method as shown in Figs. 9 and 10, respectively. It could be seen that the linear-workspace is symmetrical around plane z = 30 mm. As for the top and bottom surface, the three vertices of the triangle correspond to the position of actuation joints on the fixed base. The orientation-workspace is symmetrical around plane γ = 0. The upper and 5 5 π and γ = 12 π , respectively. As lower limit of angle about axis z are γ = − 12 for the top and bottom surface, the three vertices of the triangle correspond to the position of actuation joints on the fixed base.

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

273

Fig. 9 Linear-workspace in 3T pattern, a workspace in 3T pattern, b bottom surface z = 0, c top surface z = 60

Fig. 10 Orientation-workspace in 3R pattern, a orientation in 3R pattern, b bottom surface γ = 5 5 − 12 π , c top surface γ = 12 π

6 Conclusions This paper focuses on the description, transformation and kinematic modeling of the reconfigurable parallel robot with 3T and 3R motion patterns for on-orbit manipulation. The conclusions are listed in the following. (1) The topologies of 3T and 3R motion patterns are described and connected algebraically on the basis of finite screw. The transformation process of topology and motion is investigated in-depth. (2) Kinematic modeling of the robot is performed by taking two motion patterns as a whole. Jacobian matrix is derived on the basis of the differential mapping between finite and instantaneous screws. (3) Numerical example is presented to show the motion and kinematic performance of the reconfiguration robot under two motion patterns and their transformation process, including trajectory planning and workspace analysis. The work in this paper helps to analyze the motion and performance of reconfigurable robots in an integrated and algebraic manner. In the future work, trajectory planning of the transforming process would be focused to obtain the smooth motion of both transforming and actuation joints. Acknowledgements Supported by National Natural Science Foundation of China (NSFC) (Grant No. 51905378, 51875391), Natural Science Foundation of Tianjin (Grant No.

274

H. Li et al.

20JCQNJC00360), and Tianjin Enterprise Science and Technology Commissioner Project (Grant No. 20YDTPJC00450), Tianjin Science and Technology Planning Project (Grant No. 18PTLCSY00080, 20201193), the Science and Technology Development Fund Program of Tianjin Colleges and Universities (2020KJ105), Open-end Fund from Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education.

References 1. Dai JS, Jones JR (1998) Mobility in metamorphic mechanisms of foldable/erectable kinds. In: ASME 1998 design engineering technical conferences 2. Kong X (2014) Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method. Mech Mach Theory 74:188–201 3. Kong X (2016) Reconfiguration analysis of a 4-DOF 3-RER parallel manipulator with equilateral triangular base and moving platform. Mech Mach Theory 98:180–189 4. Yu J, Liu K, Kong X (2020) State of the art of multi-mode mechanisms. J Mech Eng 56(19):14– 27 5. Zlatanov D, Bonev IA, Gosselin CM (2002) Constraint singularities of parallel mechanisms. In: Proceedings 2002 IEEE international conference on robotics and automation (Cat. No.02CH37292), vol 1, pp 496–502 6. Zlatanov D, Bonev IA, Gosselin CM (2002) Advances in robot kinematics theory and applications. Kluwer Academic Publishers 7. Walter DR, Husty ML, Pfurner M (2009) A complete kinematic analysis of the SNU 3-UPU parallel robot. In: Interactions of classical and numerical algebraic geometry, pp 331–346 8. Lenarcic J, Parenti-Castelli V (2018) Advances in robot kinematics. Springer, Bologna 9. Li D, Zhang Z, Dai J et al (2010) Overview and prospects of metamorphic mechanism. Chin J Mech Eng 46(13):14–21 10. Wu J, Jin G, Li D, Yang S, Dai JS (2007) Adjacent matrix method describing the structure changing of metamorphic mechanisms. Chin J Mech Eng 43(7):23–26 11. Yang F, Tao J, Deng Z (2011) New method to describe the structure changing of metamorphic mechanisms and its application in structure synthesis. Chin J Mech Eng 47(15):1–8 12. Zhang Z, Li D (2014) Analysis method of screw algebra for motion characteristics of metamorphic mechanisms. J Huazhong Univ Sci Technol Nat Sci 42(4):11–15 13. Zhang Z, Li D (2013) Description of displacement subgroup for structure transformations of metamorphic mechanisms. J Beijing Univ Posts Telecommun 36(3):44–48,59 14. Zhang ZH, Sun J, Wang ZH et al (2016) Quaternion method for the kinematics analysis of parallel metamorphic mechanisms. Adv Reconfigurable Mech Robots II:259–274 15. Gan D, Dai JS, Dias J et al (2013) Reconfigurability and unified kinematics modeling of a 3rTPS metamorphic parallel mechanism with perpendicular constraint screws. Robot Comput Integr Manuf 29(4):121–128 16. Gan D, Dias J, Seneviratne L (2016) Unified kinematics and optimal design of a 3rRPS metamorphic parallel mechanism with a reconfigurable revolute joint. Mech Mach Theory 96:239–254 17. Baran EA, Ozen O, Bilgili D et al (2019) Unified kinematics of prismatically actuated parallel delta robots. Robotica 37(9):1513–1532 18. Sun T, Yang SF (2019) An approach to formulate the hessian matrix for dynamic control of parallel robots. IEEE-ASME Trans Mechatron 24(1):271–281 19. Sun T, Lian BB, Song YM (2016) Stiffness analysis of a 2-DoF over-constrained RPM with an articulated traveling platform. Mech Mach Theory 96:165–178 20. Sun T, Lian BB, Song YM et al (2019) Elastodynamic optimization of a 5-DoF parallel kinematic machine considering parameter uncertainty. IEEE-ASME Trans Mechatron 24(1):315– 325

Integrated Kinematic Modeling of Reconfigurable Parallel Robot …

275

21. Sun T, Song YM, Dong G et al (2012) Optimal design of a parallel mechanism with three rotational degrees of freedom. Robot Comput Integr Manuf 28(4):500–508 22. Qi Y, Sun T, Song YM (2018) Multi-objective optimization of parallel tracking mechanism considering parameter uncertainty. J Mech Robot Trans ASME 10(4) 23. Sun T, Liang D, Song YM (2018) Singular-perturbation-based nonlinear hybrid control of redundant parallel robot. IEEE Trans Ind Electron 65(4):3326–3336 24. Sun T, Yang S, Lian B (2020) Finite and instantaneous screw theory in robotic mechanism. Springer, Singapore 25. Song YM, Zhang JT, Lian BB et al (2016) Kinematic calibration of a 5-DoF parallel kinematic machine. Precis Eng J Int Soc Precis Eng Nanotechnol 45:242–261 26. Song YM, Lian BB, Sun T et al (2014) A novel five-degree-of-freedom parallel manipulator and its kinematic optimization. J Mech Robot Trans ASME 6(4) 27. Sun T, Lian BB, Yang SF et al (2020) Kinematic calibration of serial and parallel robots based on finite and instantaneous screw theory. IEEE Trans Rob 36(3):816–834 28. Sun T, Yang SF, Huang T et al (2018) A generalized and analytical method to solve inverse kinematics of serial and parallel mechanisms using finite screw theory. Comput Kinematics 602–608 29. Sun T, Yang SF, Huang T et al (2018) A finite and instantaneous screw based approach for topology design and kinematic analysis of 5-axis parallel kinematic machines. Chin J Mech Eng 31(1) 30. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 31. Yang SF, Sun T, Huang T et al (2016) A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech Mach Theory 104:405–419

Mechanical Design and Robot Design

A Novel 2-DOF Translational Robot with Two Parallel Linkages Synchronous Telescopic Structures Zhihao Li, Hongzhou Wang, Quanguo Lu, Xiaohui Zou, Xiaohuang Zhan, Yongdong Huang, and Jinfeng Liu

Abstract A parallel robot mechanism with two parallel links is proposed. It can realize the motion characteristics of synchronous telescopic in the swing state by employing a set of constraints of a synchronous belt drive system. Further, the moving platform can have 2-DOF translational motion and no singularity in its workspace by integrating the special synchronous belt drive system into the parallelogram mechanism. The kinematics analysis of the mechanism includes position-forward solution, position-reverse solution, velocity analysis, and workspace analysis. Moreover, the trajectory planning of the robot and one of its trajectory routes are presented. The distribution mode of velocity and acceleration on the trajectory is set, and the position, velocity, and acceleration of the trajectory interpolation points are obtained. Finally, the prototype manufacture is completed, and relevant tests are carried out. Keywords Parallelogram · Synchronization telescope · Two translational · Parallel mechanism

1 Introduction The parallelogram is widely used in the design of a low-DOF parallel mechanism because of its special geometric constraint performance [1, 2]. The famous Delta Z. Li · Q. Lu · Y. Huang · J. Liu Jiangxi Province Key Laboratory of Precision Drive and Control, Nanchang Institute of Technology, Nanchang 330000, China H. Wang (B) · X. Zou · X. Zhan Jiangxi Institute of Mechanical Science, Nanchang 330000, China e-mail: [email protected] H. Wang School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_15

279

280

Z. Li et al.

mechanism [3, 4] is a 3-DOF translational degree-of-freedom parallel mechanism and each limb contains a 4S closed-loop sub-chain, in which the 4S closed loop sub-chain always presents a parallelogram state. As representatives of the three parallelogram parallel mechanism, the Herve star mechanism [5] and Tsai mechanism [6] were also composed of three chains with parallelogram structure. Three-dimensional translational and one-dimensional rotational 4-DOF parallel mechanisms also employ a large number of chains containing parallelogram 4S closed loop sub-chains, such as H4 mechanism [7], Par4 mechanism [8], I4 mechanism [9], and X4 mechanism [10, 11]. Similarly, most 2-DOF translational parallel mechanisms also contain parallelogram structures, such as Diamond mechanism [12–14], V2 mechanism [15], X2 mechanism [16], etc. At the same time, many scholars have studied the synthesis of parallel mechanisms with parallelogram structures. Gogu [17] and Liu respectively introduced a large number of parallel mechanisms with parallelogram structure in their works. As many parallel mechanisms with parallelogram closed loop sub-chains are synthesized [18–21]. This shows the importance of parallelogram structure in a parallel mechanism. It is an essential characteristic of parallelogram structure that two groups of opposite sides always keep parallel in motion, and it is also the geometric condition for parallel mechanism’s branch chain constrained moving platform to keep translational. Some scholars put forward a special parallelogram structure, which can realize the change of link length on the premise of keeping the linkage parallel to each other, on the other hand, that is the parallelogram of variable link length structure. Gogu takes this special structure as the condition of type synthesis and synthesizes the parallel mechanism with a parallelogram of variable link length structure [17]. Yang [22, 23] and Wang [24] also did relevant research and designed 2-DOF translational, 3-DOF translational, and three flat one revolution 4-DOF parallel mechanisms with this structure. At present, the parallelogram structure with variable link length is implemented by adding a group of parallelograms. This structure will increase an instantaneous degree of freedom when it swings into a rectangle, which will divide a complete sector workspace into two parts and seriously affect its application scope. The parallelogram with a variable link length structure has a good application prospect in a parallel mechanism, but the research on this content is not mature at present. In this paper, a new type of parallelogram structure with variable link length is presented by adding a set of the synchronous belt drive system to realize the synchronous expansion of parallel connecting rod. Firstly, the basic principle and method for realizing the parallelogram with variable link length are revealed. Secondly, this new type of parallel connecting rod synchronous telescopic rod length parallelogram structure. Thirdly, the fundamental analysis of the robot is completed, including position analysis, velocity analysis, workspace analysis, trajectory planning, and so on. Finally, the experimental prototype was built and studied to verify the correctness of the analysis content and the rationality of the new structure.

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

281

2 Novel Parallelogram Structure with Variable Link Length A simple diagram of the parallelogram structure with variable link length can be shown in Fig. 1a, where Ri (i = 1, 2, 3, and 4) represents the revolute pair, Pi represents the prismatic pair, A, B, C, and D represent the central position of the revolute pair, and meet AD∥BC, AB∥CD. If AD link is taken as a rack, then the structure becomes a mechanism, and its schematic diagram is shown in Fig. 1b. If each kinematic pair operates independently, the condition of AD∥BC and AB∥CD cannot be satisfied. Therefore, special constraints need to be added to be implemented. Gogu [17], Yang [22, 23] et al. added an EF rod to make AD∥BC and AB∥CD, as shown in Fig. 2a. The role of the EF link is to make R1 and R4 synchronous rotation, so the structure shown in Fig. 2b represents Fig. 2a, and the swing angle of the connecting rod 1 and 2 is always equal. On account of the geometrical constraint of the parallelogram, the BC link can make 2-DOF translational motions. While there exists the problem of singular lines in the workspace in the method of restricting the synchronous motion of two rotating pairs. As shown in Fig. 3a, assuming that R1 (R4) and P1 are active pairs, when α1 = α2 = 90°, the BC rod rotates around point B, and the movement direction of Point C is the same as that of Point P2, and the mechanism appears instantaneous instability. Thus, the workspace of the agency is divided into two parts by a singular line, as shown in Fig. 3b. To solve the problem of discontinuity in the workspace, a parallelogram mechanism with variable link length that restricts the synchronous motion of P1 and P2 is

R2

Fig. 1 Novel parallelogram structure and mechanism of variable link length

B

C P2

P1

D

A

R1

R4

R3

R2

R2 P1 A

R1 R5

B

C P2

D R4 F

E a

R6

C

P1

P2

R1 A

D R4

a

Fig. 2 Parallelogram mechanism of variable link length with synchronous pendulum angle

B

R3

b

R3

R2

C

B

P1

P2

R1 α1 R4 α2 A D b

R3

282 Fig. 3 Singularity of parallelogram mechanism with synchronous pendulum angle

Z. Li et al. line of singularity

R2

B

C

P1

R3 P2

R1 α 1 A

R4 D

α2

Workspace 1

Workspace 2

b

a

Fig. 4 Synchronous telescopic parallelogram mechanism of connecting rod

Workspace R2

B

C

P1 R1

R3 P2

R4 A D a

b

designed in this paper, as shown in Fig. 4a. Assume P1 (P2) and R1 are active pairs. Since there is no internal singularity, their workspace is a complete region, as shown in Fig. 4b. In this paper, the new type of parallel robot with 2-DOF translational motions adopts the synchronous telescopic structure of the connecting rod.

3 Mechanical Structure of a New Type of 2-DOF Translational Robot Figure 5a is the general view of the 3D model of the robot, and Fig. 5b is the general view of the 3D model behind the hidden rack and motor. The 2-DOF translational robot is composed of rack, oscillating arm 1, oscillating arm 2, oscillating rod 1, oscillating rod 2, capstan, driven wheel, rotating wheel 1, rotating wheel 2, adapting piece 1, adapting piece 2, motor 1, motor 2 and moving platform. Oscillating arm 1 is connected to the rack by a revolute pair, and the capstan is also connected to the rack by a revolute pair, and the axis of the two revolute pairs coincide. Motor 1 drives oscillating arm 1 to rotate, and motor 2 drives the capstan to rotate. Oscillating arm 2 is connected to the rack by a revolute pair, and the driven wheel is connected to the rack by a revolute pair. The servo wheel 1 and the servo wheel 2 are respectively mounted on the oscillating arm 1 and oscillating arm 2 by revolute pairs. The capstan drives the driven wheel and the servo wheel through the synchronous belt 1 and

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

283

the synchronous belt 3 respectively, and the driven wheel drives the servo wheel 2 through the synchronous belt 2. The above belt drives are a 1:1 transmission ratio. The oscillating rod 1 is connected to the oscillating arm 1 by moving pair 1 and is consolidated by coupling 1 and synchronous belt 1. The oscillating rod 2 is connected to the oscillating arm 2 by a prismatic pair 2 and is consolidated by a coupling 2 and a synchronous belt 2. The moving platform is connected with oscillating rod 1 and oscillating rod 2 respectively by two revolute pairs. Figure 5c is the elevation view of the 3D model behind the hidden rack and motor. The auxiliary axis of rotation B' on the servo wheel 1 and the auxiliary axis of rotation A on the servo wheel is parallel and coplanar with the auxiliary axis of rotation B on the moving platform. The auxiliary axis of rotation C' on the driven wheel 2 and the auxiliary axis of rotation D on the driven wheel are parallel and coplanar with the auxiliary axis of rotation C on the moving platform. A-axis, B-axis, C-axis, and D-axis are parallel to each other, and the distances between A-axis and B-axis are the same as those between C-axis and D-axis, and the distances between A-axis and D-axis are the same as those between B-axis and C-axis.

Fig. 5 Mechanical structure of 2-DOF translational robot

284

Z. Li et al.

The robot is a planar mechanism, and its schematic diagram is shown in Fig. 5d. AB and CD are parallel connecting rods with synchronous changes in rod length, and AD and BC are parallel to each other.

4 Kinematics Analysis As shown in Fig. 6, take the midpoint of AD as the origin O, the direction of AD as the Y-axis, the vertical Y-axis vertically upward as the Z-axis, and the X-axis determined by the right-hand principle. Establish the coordinate system O-XYZ. Select point P as the midpoint of BC. Due to special geometric relations, AB, OP, and CD are parallel and equal to each other. The position of point P(yp, zp) is taken as the output variable of the mechanism, the angle of Y-axis turning around X-axis to OP is the input angle variable α, and the length of OP is the input lever length variable l.

4.1 Position Analysis Given the output variable P(yp, zp) and the input variables α and l, the inverse solution equation of the mechanism position can be obtained as follows: ⎤ ⎡ y   cos−1 √ 2p 2 α y p +z p ⎦ =⎣ √ l y 2p + z 2p

(1)

Given the input variables α and l and the output variable P(yp, zp), the positive solution equation of the mechanism position can be obtained as follows: Fig. 6 Simplified mechanism model

A Novel 2-DOF Translational Robot with Two Parallel Linkages …



yp zp





l cos α = −l sin α

285

 (2)

4.2 Velocity Analysis Differentiate the position equation of the mechanism to time t respectively, and obtain the velocity equation of the mechanism:

y˙ p = l˙ cos α − l α˙ sin α z˙ p = −l˙ sin α − l α˙ cos α

(3)

The above formula is arranged into a matrix form as follows: 

y˙ p z˙ p





−l sin α cos α = −l cos α − sin α

  α˙ l˙

(4)

From the above equation, the Jacobian matrix of the robot can be written: 

−l sin α cos α J= −l cos α − sin α

 (5)

4.3 Workspace Analysis Given the change range of l is [lmin, lmax] = [90, 240], the range of a given α is [αmin, αmax] = [40°, 140°]. According to the positive solution formula of the position of the mechanism, its workspace can be calculated as a sector, as shown in Fig. 7.

5 Trajectory Planning 5.1 Trajectory Path Trajectory planning means that the robot calculates the expected trajectory according to the task requirements. Reasonable trajectory planning can not only shorten the motion cycle and improve work efficiency but also reduce the structure vibration and prolong the service life.

286 Fig. 7 Mechanism workspace

Z. Li et al.

140

40

r=90

R=240 Workspace

Fig. 8 Gate track

Grab and drop operation is the main working mode of two parallel translational robot. Grasp and release operation generally adopts the curved trajectory of door shape, including vertical motion-horizontal motion-vertical motion composed of three parts. Because of the right-angle transition at the connection between vertical motion and horizontal motion, the velocity and acceleration of motion are discontinuous. The vibration of the mechanism will be caused by the condition of high-speed motion. In this paper, the arc connection is used to optimize the trajectory at the right-angle transition. The track of the grab and drop operation is shown in Fig. 8. The start point of the track is A(YA, ZA), and the end point is H(YH, ZH). Vertical motion distance |AC| = lb = 50 mm, |AB| = lh = 20 mm, |BC| = r = 30 mm, horizontal motion distance |AH| = la = 185 mm, |DE| = lc = 125 mm, the right-angle transition part is replaced by an arc with a radius of r.

5.2 Velocity Analysis After the transitional right-angle region of the arc curve is determined, the position, velocity, and acceleration sequence of interpolation points can be obtained by sampling along the trajectory. To reduce the residual vibration and improve the precision of the landing point, asymmetric acceleration law is used in this paper to reduce the residual vibration of the robot by extending the deceleration time. The displacement function is polynomial of order 6:

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

p(τ ) = aτ 6 + bτ 5 + cτ 4 + dτ 3 + eτ 2 + f τ + g

287

(6)

where τ = t/T, t is the time, T is the cycle, p(τ) = s/S is the displacement, S is the total displacement, and a, b, c, d, e, f, g are coefficients. The first and second derivatives are: p ' (τ ) = 6aτ 5 + 5bτ 4 + 4cτ 3 + 3dτ 2 + 2eτ + f

(7)

p '' (τ ) = 30aτ 4 + 20bτ 3 + 12cτ 2 + 6dτ + 2e

(8)

Set the acceleration and deceleration time as 4:6, and write the boundary conditions: When τ = 0 p(0) = 0; p' (0) = 0; p'' (0) = 0 When τ = 1 p(1) = 1; p' (1) = 0; p'' (1) = 0 When τ = 0.4 p'' (0.4) = 0 According to Eqs. (6), (7), (8) and boundary conditions, it can be calculated as follows: a = −10; b = 36; c = −45; d = 20; e = f = g = 0 Therefore, the polynomial of the displacement function is: p(τ ) = −10τ 6 + 36τ 5 − 45τ 4 + 20τ 3

(9)

Its derivatives are as follows: p ' (τ ) = −60τ 5 + 180τ 4 − 180τ 3 + 60τ 2

(10)

p '' (τ ) = −300τ 4 + 720τ 3 − 540τ 2 + 120τ

(11)

p ''' (τ ) = −1200τ 3 + 2160τ 2 − 1080τ + 120

(12)

The first derivative reaches the maximum value, the second derivative is zero, so when τ = 0.4, p' (τ) is max. So p' max = p' (0.4) = 2.1875 When the third derivative is zero, p'' (τ) is max. So we can solve for that:



√ 1 4 − 6 ≈ 8.135 p'' max = p '' 10

288

Z. Li et al.

5.3 Workspace Analysis Suppose the total motion time of the robot to complete a grab and drop operation is T, and the motion trajectory is shown in Fig. 8. The arc curve length in the trajectory is lr, then the total trajectory length is: lsum = 2 ∗ (h + lr ) + lc , Using the above polynomials, displacement, velocity, and acceleration can be obtained as follows:   s(t) = lsum ∗ −10τ 6 + 36τ 5 − 45τ 4 + 20τ 3 s ' (t) = s '' (t) =

(13)

 lsum  ∗ −60τ 5 + 180τ 4 − 180τ 3 + 60τ 2 T

(14)

 lsum  ∗ −300τ 4 + 720τ 3 − 540τ 2 + 120τ 2 T

(15)

Suppose that the robot end moves from point A at time t = 0, and the whole movement is divided into N parts according to time. According to the time series, ti = T * i/N, i = 0, 1, 2 … N to determine the position coordinates, velocity, and acceleration of the interpolation points of the robot end in the motion trajectory. When 0 ≤ s(ti) < lh, if the end of the robot is in a vertical motion, then the interpolation point S: ⎧ ⎨ S(x, y) = (0,  s(ti ))  S ' (x, y) = 0, s ' (ti )  ⎩ '' S (x, y) = 0, s '' (ti )

(16)

When lh ≤ s(ti) < lh + lr, if the end of the robot is in an arc motion, then the interpolation point S is: ⎧ θ − r, h + r ∗ sin θ )  ⎨ S(x, y) = (r  ∗ cos ' ' S (x, y) = −s (ti ) ∗ sin θ, s ' (ti ) ∗ cos θ  ⎩ '' S (x, y) = −s '' (ti ) ∗ sin θ, s '' (ti ) ∗ cos θ

(17)

In Eq. (17): θ = (s(ti ) − h)/r . When lh + la ≤ s(ti) < lh + lr + lc, If the end of the robot is in a horizontal motion, then the interpolation point S: ⎧ ⎨ S(x, y) = (lh − s(ti ),lh + r ) S ' (x, y) = −s ' (ti ), 0 ⎩ '' S (x, y) = (−s(ti ), 0)

(18)

When lh + lr + lc ≤ s(ti) < lh + 2 * lr + lc, if the end of the robot is in an arc motion, then the interpolation point S is:

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

⎧ ⎨ S(x, y) = (r  − r ∗ sin θ − lr , lh + r ∗ cos θ ) S ' (x, y) = −s ' (ti ) ∗ cos θ, −s ' (ti ) ∗ sin θ  ⎩ '' S (x, y) = −s '' (ti ) ∗ cos θ, −s '' (ti ) ∗ sin θ

289

(19)

In Eq. (19): θ = (s(ti ) − lh − r − lc )/r . When lh + 2 * lr + lc ≤ s(ti) ≤ 2 * lh + 2 * lr + lc, If the end of the robot is in a vertical motion, then the interpolation point S is: ⎧ ⎨ S(x, y) = (−l  r , 2 '∗ lh + 2 ∗ lr + lc − s(ti )) ' S (x, y) = 0, −s (ti )  ⎩ '' S (x, y) = 0, −s '' (ti )

(20)

6 Control System For two parallel translational robot, the motion control system based on PC and Googol GTS-400 motion control card is adopted in this paper. GTS-400 motion control card is a motion controller based on the PCI bus, which can be used to control a stepping system or servo system. Through DSP and FPEG motion planning, it can output pulse or analog quantity instruction. It supports point position and continuous trajectory, multi-axis synchronization, and other motion modes, and can freely set acceleration and deceleration, S-shaped curve smoothness, and other parameters. It provides VC, VB, and other development environment library functions. Users can better realize the controller programming to build a control system. In this paper, the GTS-400 motion control card USES pulse command to control the servo motor. The position and speed of the servo motor are controlled by the number and frequency of the output pulse, so as that realize the high-speed point position control of the servo motor. Based on the VC platform, this paper constructs the control interface and program of 2-DOF translational parallel robot. It controls the movement of the manipulator by calling the control card API library function and combining the kinematics forward and inverse solution algorithm. The basic control flow is shown in Fig. 9.

7 Prototype and Experiments The experimental prototype of 2-DOF translational parallel robot was built, as shown in Fig. 10, including the human–machine interaction interface, control system, and mechanical body. The human–computer interaction interface is a display program based on VC, with functions such as switching machines, single-step control, trajectory control, and executive grip control. The control system takes the GTS-400 motion

290 Fig. 9 Control flow of a new type of two parallel translational robot

Z. Li et al.

Position and attitude of points on the trajectory

Difference compensation algorithm

Inverse solution of position

Motion controller and drive system

The actual position and attitude of an end effector

control card as the core of connecting the mechanical body. The pneumatic chuck gripper is adopted in the actuator of the prototype, which has the advantages of simple installation, low mass, and fast response velocity. To test the robot, 2-DOF translational robot is built, as shown in Fig. 11. A coordinate system O-XYZ and P-sale, based on AD midpoint origin of coordinates, to the AD is in the Y-axis, Y-axis vertical upward direction for Z-axis, X be determined by the right-hand rule, BC midpoint as the point P, V-axis parallel to the Y-axis, W-axis parallel to the Z-axis, U-axis is determined by the righthand rule. Fig. 10 A prototype of the 2-DOF translational robot

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

Fig. 11 Motion trajectories of 2-DOF parallel translational robot

291

292

Z. Li et al.

As shown in Fig. 11b, the initial position of the robot is Py = 0 and Pz = −100. Figure 11c shows that when the moving platform of the robot moves 50 mm along the Y-axis, it reaches the position Py = 50 and Pz = −100. Figure 11d shows that the robot performs the capture task along the door trajectory, where Py = 92.5 and Pz = −150. Figure 11e shows that the robot performs the capture task along the door trajectory, where Py = 92.5 and Pz = −130. Figure 11f shows that the robot performs the capture task along the door trajectory, where Py = 75.5 and Pz = −115. Figure 11g shows that the robot performs the capture task along the door trajectory, where Py = 0 and Pz = −100. Figure 11h shows that the robot performs the capture task along the door trajectory, where Py = −92.5 and PZ = −150. The rationality of the robot mechanism, the feasibility of the control system, and the correctness of the trajectory planning are proved by experiments.

8 Conclusion (1) By adding a set of synchronous wheel drive system, a parallelogram structure of variable rod length with a synchronous telescopic type of parallel connecting rod can be obtained. Compared with the parallelogram structure of synchronous oscillating angle variable link length, this structure has the advantages of no singularity inside the workspace, extensive and complete workspace. (2) The parallelogram structure of variable link length with synchronous telescopic parallel connecting rod can be directly applied to a 2-DOF translational parallel robot. The new parallel robot has the advantages of a simple and compact structure and a small proportion of its volume and workspace. The rationality and superiority of the system are verified by theoretical analysis and prototype tests. Acknowledgements Supported by Science and technology project of Jiangxi Provincial Department of Education (Grant No. GJJ204702), and Science and technology project of Jiangxi Provincial Department of Education (Grant No. GJJ214702).

References 1. Zou Q, Zhang D, Luo X et al (2020) Enumeration and optimum design of a class of translational parallel mechanisms with prismatic and parallelogram joints. Mech Mach Theory 150:103846 2. Liu X-J, Wang J (2003) Some new parallel mechanisms containing the planar four-bar parallelogram. Int J Robot Res 22:717–732 3. Carp-Ciocardia D (2003) Dynamic analysis of Clavel’s delta parallel robot. In: 2003 IEEE international conference on robotics and automation (Cat No 03CH37422) 4. Pierrot F, Reynaud C, Fournier A (1990) DELTA: a simple and efficient parallel robot. Robotica 5. Hervé J (1992) Group mathematics and parallel link mechanisms. In: Proceedings of IMACS/SICE international symposium on robotics, mechatronics and manufacturing systems’ 92, Kobe, Japan

A Novel 2-DOF Translational Robot with Two Parallel Linkages …

293

6. Tsai L-W, Walsh GC, Stamper RE (1996) Kinematics of a novel three DOF translational platform. In: Proceedings of IEEE international conference on robotics and automation 7. Pierrot F, Company O (1999) H4: a new family of 4-dof parallel robots. In: 1999 IEEE/ASME international conference on advanced intelligent mechatronics (Cat No 99TH8399) 8. Nabat V, de la O Rodriguez M, Company O et al (2005) Par4: very high speed parallel robot for pick-and-place. In: 2005 IEEE/RSJ international conference on intelligent robots and systems 9. Krut S, Benoit M, Ota H et al (2003) I4: a new parallel mechanism for Scara motions. In: 2003 IEEE international conference on robotics and automation (Cat No 03CH37422) 10. Xie F, Liu X-J (2016) Analysis of the kinematic characteristics of a high-speed parallel robot with Schönflies motion: mobility, kinematics, and singularity. Front Mech Eng 11:135–143 11. Xie F, Liu X-J (2015) Design and development of a high-speed and high-rotation robot with four identical arms and a single platform. J Mech Robot 7:041015 12. Huang T, Li Z, Li M et al (2004) Conceptual design and dimensional synthesis of a novel 2-DOF translational parallel robot for pick-and-place operations. ASME J Mech Des 126:449–455 13. Huang T, Liu S, Mei J et al (2013) Optimal design of a 2-DOF pick-and-place parallel robot using dynamic performance indices and angular constraints. Mech Mach Theory 70:246–253 14. Yang X, Zhu L, Ni Y et al (2019) Modified robust dynamic control for a diamond parallel robot. IEEE/ASME Trans Mechatron 24:959–968 15. Meng Q, Xie F, Liu X-J (2017) V2: a novel two degree-of-freedom parallel manipulator designed for pick-and-place operations. In: 2017 IEEE international conference on robotics and biomimetics (ROBIO) 16. Zhang J, Chen Q, Wu C et al (2014) Kinematic calibration of a 2-DOF translational parallel manipulator. Adv Robot 28:707–714 17. Gogu G (2004) Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations. Eur J Mech A Solids 23:1021–1039 18. Jin Q, Yang T, Luo Y (2001) Structural synthesis and classification of the 4DOF (3T– 1R) parallel robot mechanisms based on the units of single-opened-chain. China Mech Eng 12:1038–1041 19. Kong X, Gosselin C (2004) Type synthesis of 3-DOF translational parallel manipulators based on screw theory. J Mech Des 126:83–92 20. Li Z, Lou Y, Zhang Y et al (2012) Type synthesis, kinematic analysis, and optimal design of a novel class of Schönflies-motion parallel manipulators. IEEE Trans Autom Sci 10:674–686 21. Shi Q, Gao F, Yu F (2008) Type synthesis of 3T1R parallel mechanisms based on GF set. Mach Des Res 24:31–36 (Chinese) 22. Yang Y, Peng Y, Pu H et al (2018) Design of 2-degrees-of-freedom (DOF) planar translational mechanisms with parallel linear motion elements for an automatic docking device. Mech Mach Theory 121:398–424 (Chinese) 23. Yang Y, Zhang W, Pu H et al (2018) A class of symmetrical 3T, 3T–1R, and 3R mechanisms with parallel linear motion elements. J Mech Robot 10:051016 24. Wang H, Yang R, Kang X et al (2019) A novel family of parallel mechanisms with synchronous telescopic parallelograms. IEEE Access 7:184808–184824

Zhihao Li is currently pursuing the M.S. degree in electronic information with Nanchang Institute of Technology, Nanchang, China. Hongzhou Wang Is currently pursuing the Ph.D. degree with Shanghai University, Shanghai, China. He has 26 patents for invention. His main research interests include mechanism theory, especially configuration synthesis, and analysis of parallel mechanisms.

Measuring Error Correction Method During Deflection Measurement Process of the Regular Hexagon Section Shaft Based on Lever-Type Measuring Mechanism Qingshun Kong and Zhonghua Yu Abstract Straightness measurement needs to be performed before straightening. In this paper, considering the rationality of structural design and protecting the sensor, we used the lever structure to measure the straightness of the regular hexagon section shaft. However, due to lever parameters and the contact mode between the lever and the shaft and between the lever and the sensor, the lever model has measurement errors. Two kinds of lever-type measuring mechanisms are discussed to improve the measuring precision in this paper. The error correction method is proposed through strict mathematical derivation. At the same time, sensitivity analysis of lever parameters was carried out. Finally, the validity of the error correction is proved through experimental verification. The experiments indicate that the error of measurement results of two lever models after error correction is controlled within ± 0.006 mm, which improves the measurement accuracy significantly. At the same time, after correcting the measurement results of deflection of the regular hexagon section shaft, the length error of deflection vector is reduced to 0.01 mm basically. Keywords Lever-type · Straightness · Error correction · Straightening · Deflection

Notations f g Z M Z* H R

The centroid point of the hexagonal shaft The center of rotation of the hexagonal shaft The displacement of the point f in the measured direction The measurement result of the sensor The value deduced by M The width of the shaft The probe radius of left arm of the lever

Q. Kong (B) · Z. Yu State Key Laboratory of Fluid Power and Mechatronic Systems, College of Mechanical Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_16

295

296

A, L, B D r θ O

Q. Kong and Z. Yu

Structural size of the lever Lever ratio Probe radius of the displacement sensor Lever rotation angle The center of rotation of the lever

1 Introduction Precision measurement is the precondition and basis of advanced manufacturing development. The requirement of measurement precision is getting higher and higher with the development of technology. As the pilot technology of advanced manufacturing, the accuracy of the measurement results will affect the subsequent manufacturing process significantly. Regular hexagon section shafts are used in water pumps widely because they can be used as a transmission shaft of the pump and can carry the load. If the workpiece has a large deflection, it will influence the assembly accuracy of the product and the product’s performance and reduce the product’s reliability and safety. Straightening [1–3] is used as an essential means to improve straightness widely. However, we must measure straightness before the straightening process; straightness measurement [4– 7] is vital for straightening. So as the pilot process of straightening, the measurement accuracy of straightness affects straightening accuracy [3, 8]. At present, there are many straightness measurement methods of the shaft. Using traditional manual methods such as V-type block to measure straightness is inefficient, and the measurement accuracy cannot meet the requirements. Moreover it is not consistent with the development trend of automation and intelligence of modern manufacturing. Pei et al. [9] proposed a non-contact method for measuring the bending deflection of inner/outer Circular surfaces with guide rails and displacement sensors. After rigorous mathematical modelling, the author proved the validity of the method by experiments. However, we need to leave enough space for the sensor to be installed when measuring straightness. Chen et al. [10, 11] used the time-domain threepoint method to measure shaft straightness. The experimental results show that the measuring device has good repeatability, and the measurement error is reduced to 0.005 mm. The error influence analysis and optimization of the parameters of the measuring device are carried out, and the accuracy of the measurement results is improved. Gao et al. [12] introduced a scanning multi-probe system by placing two sets of probes on either side of a cylindrical workpiece and then moving the probes to measure the contour of the workpiece. However, this method needs to provide enough space for the probe to move. The image processing technology is also widely used in straightness measurement [13, 14]. Patwari and Ullah [15] considered to check the circularity and straightness

Measuring Error Correction Method During Deflection Measurement …

297

Fig. 1 Structure of self-made straightening machine

of a seamless pipe and thereby present a new machine vision technique proven to be very effective. Li et al. [16] adopt machine vision to measure the steel pipe’s 3-D straightness. Hao et al. [17] proposed a method based on geometric constraints and coding reference. They measure the straightness error of slender shaft by image processing, and the measuring result is accurate and stable. However, the findings of other researchers do not apply to hexagonal shaft straighteners. First of all, the above methods are suitable for the straightness measurement of the shaft with a circular cross-section. Secondly, there is not enough space to install the displacement sensor. As shown in Fig. 1, it is a self-made straightener of a regular hexagon section shaft. The primary working process of the equipment is as follows. (1) Automatically feeding part loads the shaft. (2) The straightness measurement section measures the straightness of the shaft. (3) The straightening part worked according to the straightness measurement results. (4) The straightness measurement part measures the straightness again to check whether the straightening is successful. (5) Then, according to the straightness measurement results to decide whether to continue straightening or uploading. So the straightness measurement and straightening functions are combined in the device; when designing the device, it is necessary to leave enough space for the horizontal movement of the actuator of straightening above the shaft to select a suitable straightening location, and adequate space under the shaft should be left for the loading process of the actuator of straightening. Meanwhile, sufficient space should also be left for automatic loading and unloading of the shaft. Finally, due to the length of the shaft (1.2 m), the measurement accuracy of the image processing method may be insufficient, and because there is much interference in the field of vision, image preprocessing will be more troublesome. So, considering the protection sensor and the space requirements, the lever measuring mechanism is used for straightness measurement. On the premise of considering the cost, Wang et al. [18] proposed a low-cost deflection measurement method, which mainly includes lever and the displacement sensor. Moreover, a data processing formula is proposed to get the deflection value of the shaft. Experimental

298

Q. Kong and Z. Yu

results show that this method has high precision. However, the measurement algorithm proposed by the author is only applicable to a circular rod. Although Kong et al. [19] used the lever mechanism to measure the straightness of the slender shaft with a regular hexagon section, the error caused by the lever measuring mechanism is not directly discussed. In the actual data processing, the author considers the error of lever measurement structure, but the authors did not discuss the error effect of the lever structure in the paper directly. If the measurement mechanism has much measurement error, then the straightness measurement results will be meaningless, and the straightening process will not be carried out successfully. So it is vital to explore the error law of the lever measuring mechanism and the mathematical relationship of data transmission on both sides of the lever. Therefore, literature [19] will be supplemented in this paper. This paper will discuss the error caused by the lever mechanism when measuring the straightness of the regular hexagon section axis by using the lever structure and put forward the error correction method by strict mathematical derivation, which has been verified the effectiveness through experiments. Meanwhile, in view of the measurement accuracy of straightness on the straightening effect and to improve the measurement accuracy, sensitivity analysis of lever parameters was carried out. Finally, the correction effect of the measurement error in the deflection measurement process of the regular hexagon section shaft is verified by experiments.

2 Discussion on Lever Model In this paper, the capacitive displacement sensor was used. This sensor has a measuring rod that has an oriented role. The direction of motion of the measuring rod is the measurement direction of the sensor. The positive and negative signs of Z and M are shown in Fig. 2. When the lever is perpendicular to the sensor’s measurement direction (measuring rod), Z = 0 and M = 0. In this paper, data was collected when the surface of the polygonal shaft is rotated till perpendicular to the measurement direction of the sensor during straightness measurement because of the difference between circular shaft and polygonal shaft Fig. 2 The plus and minus sign stipulations for Z and M

Measuring Error Correction Method During Deflection Measurement …

299

[19], and data is collected six times in total per 360° (shaft have six planes). The following contents are based on this condition. Two sides of lever contact with the shaft and the sensor respectively. Because the polygonal shaft is different from the circular shaft, the side of the lever which in contact with the polygonal shaft (which was called the left arm of the lever in this paper) requires an arc structure. The specific description of the two lever models is in the following paragraphs.

2.1 The First Measurement Structure (Model 1) (1) Model-building Differences between the two structures are the type of contact between the right arm of the lever and the sensor. The contact features of the first structure are as follows. The right arm of the lever adopts the arc-plane contact (the right arm of the lever is plane, the sensor probe is an arc). The structure diagram is shown in Fig. 3a, b are the different states of the measuring mechanism during the rotation of the shaft. L1 is the measurement direction of the sensor and is also the center line of the measuring rod of the sensor. L2 is the straight line that passes the center of the sensor probe arc and is parallel to the lever arm. According to plane geometry and trigonometric functions, the lines L1 and L2 can be expressed as:

Fig. 3 The first structure diagram of lever that measuring deflection

300

Q. Kong and Z. Yu

L1 : x = D × L L2 : y = x tan θ −

B +r cosθ

(1) (2)

The contact between the right arm of the lever and sensor probe can be seen as that the center of the probe’s arc slide on line L2 during the lever rotation. As shown in Fig. 3a, it is assumed that the lever is in a position that is perpendicular to the measurement direction. As shown in Fig. 3b, the lever will tilt at an angle θ when the shaft rotates to that the measurement direction is perpendicular to another plane of shaft. Assuming that the center of shaft drops Z in measurement direction, the relationship between Z and angle θ can be deduced from the geometric relationship: Z = A−

√

A2 + L 2 × sin

   A −θ tan−1 L

(3)

The displacement of the measuring rod in the measuring direction, which is the measured value of the sensor, is the change of the intersection position of line L1 and L2 (point P), the geometric relationship of M and θ can be deduced as follows: M = D × L × tan θ −

B +r + (B + r ) cos θ

(4)

We can also deduce the relationship between Z and M from the geometric relationship: M=

A×D×L×

√

+ B +r

L 2 − Z 2 + 2 × Z × A − D × L 2 × (A − Z ) − (B + r ) × (L 2 + A2 ) √ (A − Z ) × A + L × 2 × A × Z − Z 2 + L 2

(5)

(2) Error analysis Z is the actual changed value of the center of the shaft in the measuring direction of the sensor, M is the measured value of the sensor, and the lever ratio is D, so error e is: e=Z− =Z− −

M D D× A×L×

(B + r ) D

√ 2 × A × Z + L 2 − Z 2 − L 2 × D × (A − Z ) − (B + r ) × (A2 + L 2 )   √ A × (A − Z ) + L × 2 × Z × A + L 2 − Z 2 × D

(6)

To analyze whether the error can be removed by parameter adjustment, we divide formula (4) by D, and then it was subtracted from formula (3); finally, the equation is as follows.

Measuring Error Correction Method During Deflection Measurement …

   A −θ tan−1 L   B +r B +r − − L × tan θ + D × cos θ D

e= A−

√

301

A2 + L 2 × sin

(7)

Taking the first derivative of formula (4), √ ∂e = A2 + L 2 × cos ∂θ

     L + B+r × sin θ −1 A D −θ − tan 2 L cos θ

(8)

∂e It can be seen from formula (8), we cannot make ∂θ /= 0 by adjusting to the parameter. Therefore, we cannot eliminate the error in theory by parameter adjustment.

(3) Building of error correction model In the actual measuring process, we can only get the value of M, so we can only deduce Z by M got by the sensor, and just for the sake of distinction, let us write the Z deduced by M as Z ∗ . In order to deduce Z ∗ , we must establish a mathematical relationship. Wherein the vertical distance between the line L2 and the rotating center of the lever is B+r . The geometric relation of the right arm of the lever during rotating is shown in Fig. 4. As we can see from Figs. 3 and 4, line L2 goes through point P(D × L , M − (B + r )) and is tangent to the circle of radius (B + r ) at point K(x0 , y0 ). According to plane geometry and trigonometric functions, we can get x0 ; to solve θ , we have to figure out what the x0 is. that θ = sin−1 B+r According to plane geometry and trigonometric functions, we can get

Fig. 4 Geometric relations in lever right arm rotation

302

Q. Kong and Z. Yu

⎧ √   2   2   (B+r )2 (B+r )2 D×L D×L ⎪ × M−(B+r − M−(B+r + − M−(B+r (B+r )2 × 1+ − M−(B+r ⎪ ) ) ) ) ⎪ ⎪ ⎪ 2  − M r+B ⎩ D×L − M−(B+r )

(9)

+1

Simplifying formula (9),

x0 =

⎧ ⎪ ⎪ ⎨

√ M−B−r |M−B−r | ×

⎪ ⎪ ⎩ (B+r )2 D×L

  2  2    (B+r )2 (B+r )2 D×L D×L − M−(B+r × M−(B+r − − M−(B+r (B+r )2 × 1+ − M−(B+r ) ) ) ) 2  D×L 1+ − M−(B+r )

M /= r + B M =r+B (10)

So ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

√ 2    2    (B+r )2 (B+r )2 M−B−r × (B+r )2 × 1+ − D×L D×L − M−(B+r × M−(B+r − − M−(B+r |M−B−r | M−(B+r ) ) ) ) −1  sin M /= r + B 2  θ= D×L +1 (r +B)× − M−(B+r ⎪ ) ⎪ ⎪   ⎪ ⎪ ⎩ sin−1 B+r M =r+B D×L

(11) Substituting θ into formula (3), we can get a formula of Z ∗ about M. Z ∗ = f (M) √ ⎧ 2 2 ⎪ A− ⎪  A +L  × sin ⎪⎛ ⎞ ⎪ ⎪ A ⎪ −1 ⎪ tan ⎪ ⎪ ⎟ ⎜ L ⎪ ┌ ⎪ ⎜ ⎪  2 ⎟ | ⎪ ⎟ ⎜ 2    ⎪ | 2 ⎪ ⎟ ⎜ M − B − r D × L + r (B ) ⎪ ⎪⎜ ⎟ ⎪ × √(B + r )2 × 1 + − − ⎨⎜ ⎟ M /= r + B |M − B − r | M − (B + r ) M − (B + r ) ⎟ ⎜ = ⎜ ⎟   2 ⎪⎜ ⎟ + r D × L (B ) ⎪ ⎪ ⎟ ⎜ ⎪ × − − ⎪ ⎟ ⎜ ⎪ M − (B + r ) M − (B + r ) ⎪ −1 ⎟ ⎜ ⎪   − sin ⎪  ⎠ ⎝ ⎪ 2 ⎪ D×L ⎪ ⎪ (r + B) × − M−(B+r ) + 1 ⎪ ⎪ ⎪     √ ⎪ ⎪ ⎩ A − A2 + L 2 × sin tan−1 A − sin−1 B+r M =r+B L D×L

(12) We can get the value of Z ∗ by measuring the value M of the sensor and using formula (12) directly. Theoretically, we can eliminate the error generated by the lever structure entirely.

Measuring Error Correction Method During Deflection Measurement …

303

2.2 The Second Measurement Structure (Model 2) (1) Model-building Contact features of the second structure are as follows. Differences between model 1 and model 2 are that the right side of the lever is the arc and the sensor probe is the plane in model 2. The structure diagram is shown in Fig. 5a, b are the different states of the measuring mechanism during the rotation of the shaft. According to plane geometry and trigonometric functions, we can obtain the following two formulas (13) and (14).    A −θ tan−1 L    √ −1 B 2 2 −θ M = B − B + (D × L) × sin tan DL Z = A−

√

A2 + L 2 × sin

(13) (14)

Like formula (5), the relationship between Z and M can also be deduced from the geometric relationship: √ M = B − (D × L)2 + B 2   √ ⎛ ⎞ (L × B − A × D × L) × 2 × A × Z + L 2 − Z 2 + D × L 2 + A × B × ( A − Z ) ⎝ ⎠ √ ×   B 2 + (D × L)2 × A2 + L 2

Fig. 5 The second structure diagram of lever that measuring deflection

(15)

304

Q. Kong and Z. Yu

(2) Error analysis As we all know, √ (D × L)2 + B 2 M B e=Z− =Z− + D D D   √ ⎛ ⎞ (L × B − A × D × L) × 2 × A × Z + L 2 − Z 2 + D × L 2 + A × B × (A − Z ) ⎝ ⎠ √ ×   B 2 + (D × L)2 × A2 + L 2

(16)

To analyze whether the error can be removed by parameter adjustment, we divide formula (14) by D, and then it was subtracted from formula (13); finally, the equation is as follows.    B √ 2 A M = A− − A + L 2 × sin tan−1 −θ e=Z− D D L √     B 2 B + −θ (17) + L 2 × sin tan−1 D D×L Taking the first derivative of formula (17),    √ ∂e −1 A 2 2 = A + L × sin tan −θ ∂θ L √     B 2 B −1 2 − −θ + L × sin tan D D×L

(18)

∂e = 0 when A = DB . Moreover, e is constant simultaneIt is evident that ∂θ ously. Therefore, we can eliminate the error in theory by parameter adjustment and compensating the constant e in the measurement result.

(3) Building of error correction model However, as far as the overall structure of the equipment is concerned, the structural design according to the condition of A = DB may not be optimal in terms of spatial layout and other factors during structural design, so the condition may not be carried out. Therefore, a mathematical relationship between Z and M when A /= DB still needs to be studied. Like formula (12), the formula Z ∗ = f (M) can be deduced. √ Z ∗ = A − L 2 + A2 √ ⎞ ⎛   (D × A × L − L × B) × (D × L)2 + 2 × B × M − M 2 + A × B + D × L 2 × (B − M) ⎟ ⎜   ×⎝ √ ⎠ L 2 + A2 × B 2 + (D × L)2

(19)

We can directly get the value of Z ∗ by sensor measurement value M and using formula (19).

Measuring Error Correction Method During Deflection Measurement …

305

Table 1 Results of sensitivity analysis ∂e ∂ A | Z =5

∂e ∂ L | Z =5

∂e ∂ D | Z =5

∂e ∂ B | Z =5

Model 1

0.0018

− 4.73e − 04

− 0.011

0.002

Model 2

0.0018

− 6.06e − 04

0.0031

− 0.002

∂e ∂r | Z =5

0.002 –

2.3 Sensitivity Analysis To explore the influence of dimension parameters, we must analyze the sensitivity of lever parameters. In the self-made device, because of the rationality of the equipment structure layout, the parameters of the lever are as follows finally, and we can use these parameters to analyze the influence: A = 15.5 mm, L = 82.5 mm, D = 77/82.5, B = 3.5 mm, r = 1.5 mm. Due to structural design, when we measure the straightness in the self-made device, the shaft surface contacting with the lever was lower than the rotating center of the lever basically, so we select the parameter Z = 5 mm to conduct sensitivity analysis. According to formulas (6) and (16), we can deduce Table 1. Except for the lever ratio, the partial derivatives of all parameters are less than 0.002. The lever ratio is the ratio of two arms. Although the lever arm has machining error, the lever ratio error will be tiny. So we can see from the results that the measuring structure’s machining accuracy and assembly accuracy have little influence on the measuring accuracy of the measuring structure.

3 Experiments 3.1 The First Measurement Structure (Model 1) Experimental facility was made according to model 1. The following parameters are used: A = 15.5 mm, L = 82.5 mm, D = 77/82.5, B = 3.5 mm, r = 1.5 mm. As shown in Fig. 6, to control the lever’s left arm’s height variation, we replace the regular hexagon section shaft with a dial gage to simulate the contact between the shaft’s plane and lever. The experiment of model 2 is the same. Keep the lever perpendicular to the measurement direction and then reset the displacement sensor reading to 0. The dial gage changes 1 mm (Z) each time, and the sensor reading (M) is recorded, then the data are shown in Table 2. Error before correction is e0 = Z − M/D, we have gotten the value of M by the sensor so that we can figure out Z ∗ by formula (12), and error after correction is e∗ = Z − Z ∗ . Except for Z and M, all the numbers are derived from the formula, and there are wireless non-cyclic decimals inevitably, so they all were kept four significant digits after the decimal point. The experiment of model 2 is the same.

306

Q. Kong and Z. Yu

Fig. 6 Replace shaft with dial gage at left arm of lever

Table 2 The relationship of Z and M in structure 1 Z (mm)

M (mm)

M/D (mm)

Z ∗ (mm)

e0 (mm)

e∗ (mm)

0

0

0

0

0

0

1

0.93

0.9964

0.9979

0.0036

0.0021

2

1.86

1.9929

1.9984

0.0071

0.0016

3

2.79

2.9893

3.0010

0.0107

− 0.001

4

3.71

3.9750

3.9946

0.0250

0.0054

5

4.64

4.9714

5.0003

0.0286

− 0.0003

6

5.56

5.9571

5.9960

0.0429

0.004

3.2 The Second Measurement Structure (Model 2) Experimental facility was made according to model 2. The following parameters are utilized: A = 15.5 mm, L = 82.5 mm, D = 77/82.5, B = 1.5 mm, r = 1.5 mm. Keep the lever horizontal and then reset the displacement sensor reading to 0. The dial gage changes 1 mm (Z ) each time, and the sensor reading (M) is recorded. The data are shown in Table 3. Error before correction is e0 = Z − M/D, we have gotten the value of M by the sensor so that we can figure out Z ∗ by formula (12), error after correction is e∗ = Z − Z ∗ . Table 3 The relationship of Z and M in structure 2 Z (mm)

M (mm)

M/D (mm)

Z ∗ (mm)

e0 (mm)

e∗ (mm)

0

0

0

0

0

0

1

0.93

0.9964

0.9974

0.0036

0.0026

2

1.86

1.9929

1.9969

0.0071

0.0031

3

2.79

2.9893

2.9984

0.0107

0.0016

4

3.72

3.9857

4.0019

0.0143

− 0.0019

5

4.64

4.9714

4.9966

0.0286

0.0034

6

5.57

5.9679

6.0042

0.0321

− 0.0042

Measuring Error Correction Method During Deflection Measurement …

307

3.3 Deflection Measurement for Regular Hexagonal Shaft To check the effect of error correction, we measure the deflection of the regular hexagon section shaft. For comparison, the actual deflection at one point on the shaft is first measured. The measurement process is as follows. As shown in Fig. 7, we make three marks around the circumference of the shaft. The edge of the block gauges is always in contact with points 1, 2, and point 3 is in contact with the displacement sensor probe. As shown in Fig. 8, data of point 3 were collected three times and then averaged per shaft. A total of 10 shafts were measured. Then we use model 2 to verify the effect of error correction. Keep the edge of the block gauges in contact with points 1, 2 and keep point 3 in contact with the left arm probe of the lever. Again, data are measured three times and then averaged per shaft. A total of 10 shafts were measured. Processing the measured data by formula (19), we got values of the direction of a, b and c shown in Fig. 9, then the coordinates were converted to rectangular coordinates by Clarke transformation [19], as shown in Fig. 9, point g is the origin of Fig. 7 Three marks on the shaft

Fig. 8 Measuring the true deflection of the shaft

308

Q. Kong and Z. Yu

Fig. 9 Deflection diagram of point 3

Table 4 Actual deflection, deflection before correction and deflection after correction of point 3 True values of f

f before correction

f after correction

x (mm)

x ∗ (mm)

x0 (mm)

y (mm)

2.65581

− 1.81833

2

0.19245

3

− 0.69475

4 5

y ∗ (mm)

2.61045

− 1.78393

0.40333

0.18970

0.23833

− 0.67632

− 0.91606

0.30167

− 0.17898

− 0.60167

6

0.09180

7 8 9

1

10

y0 (mm)

2.64884

− 1.81015

0.39286

0.19232

0.39828

0.23571

− 0.68562

0.23894

− 0.89283

0.29821

− 0.90511

0.30230

− 0.17011

− 0.58661

− 0.17246

− 0.59472

− 0.38350

0.08660

− 0.38036

0.08780

− 0.38562

− 0.11740

0.21833

− 0.10928

0.20893

− 0.11089

0.21182

0.35642

0.21467

0.34229

0.23094

− 0.31667

0.22682

− 0.20592

− 0.59333

− 0.19589

0.210714

0.34704

0.21364

− 0.29643

0.22997

− 0.30054

− 0.58036

− 0.19860

− 0.58841

the rectangular coordinates, then the deflection vector f (x, y) at point 3 was solved. Furthermore, the data of deflection vector f (x, y) were plotted in Table 4. We keep five significant digits after the decimal point.

4 Analysis and Discussion Without loss of generality, we use the structure parameters in experiments to analyze. By formula (12) (model 1) and formula (19) (model 2), we can get the curve of Z ∗ − e (the curve of error correction), which e = Z ∗ − M/D, the curve is shown as follows. As shown in Fig. 10, the more the distance between the left arm and horizontal position, the more measurement errors. Meanwhile, the error growth rate is not constant. In the experiment part, we got the value of M by the sensor, and the errors can

Measuring Error Correction Method During Deflection Measurement …

309

be corrected by figuring out Z ∗ according to the mathematical relationship formulas (12) and (19). From Sect. 2.3, we can know that the measuring structure’s machining accuracy and assembly accuracy of the measuring structure have little influence on the measuring accuracy. So the primary source of measurement errors is mathematic relation indicated by formulas (12) and (19). Due to the lever’s structural characteristics and the contact form between the lever and the sensor and the between the lever and workpieces, there are errors in the measurement results. The data from Tables 2 and 3 are plotted in Fig. 10 to form Fig. 11. As is shown in Fig. 11, the error e0 before correction is consistent with the theoretical curve of Z ∗ − e basically, it indicated that the mathematical relationships of formulas (12) and (19) is correct. So we can correct the error by figuring out Z ∗ using formulas (12) and (19). The error e∗ (Z − Z ∗ ) after Fig. 10 The curve of error correction

310

Q. Kong and Z. Yu

correction fluctuates around 0. Obviously, error correction based on mathematical relations of formulas (12) and (19) has a good correction effect, and the measurement error caused by the lever model is eliminated basically. In the experiment results, the most maximum error is 0.0429 mm in model 1 and 0.0321 mm in model 2, and be reduced to 0.0054 mm and 0.0042 mm after correction by using formulas (12) and (19) respectively. So we have reason to believe that the elimination of errors by formulas (12) and (19) has a good effect. In the actual measurement process, we can only get M, so we can use the formula to calculate Z ∗ to eliminate the measurement error, making the measured value close to the actual change value Z of the lever left arm. To show the correction effect of formula (19) on the hexagonal shaft deflection measurement intuitively, as shown in Table 5, we convert rectangular coordinates into polar coordinates, and we keep five significant digits after the decimal point. Fig. 11 Measurement errors comparison before correction and after correction

Measuring Error Correction Method During Deflection Measurement …

311

Meanwhile, the actual values of f were subtracted from values before correction of f and values after correction of f to obtain the error, and the error was plotted in Table 6. Obviously, the length errors of the deflection were reduced. As shown in Fig. 12, the deflection length errors are reduced within 0.0135 mm in ten sets of data entirely, and 70% of the data were reduced within 0.01 mm. However, there is little difference in angle error before and after correction. As shown in Fig. 13, in point 3, angle errors of the deflection before and after error correction are overlapped; it indicates that the error correction method has little effect on the angle error correction of the deflection. As we can see from Table 6, the maximum angle error is 0.023 rad, which is equal to 1.3°. Table 5 Polar data of true deflection, deflection before correction and deflection after correction of point 3 True values of f

Values before correction of f Values after correction of f

j (mm)

α (rad)

0

3.21864

− 0.6004

1

0.44690

1.12560

j ∗ (mm)

α ∗ (rad)

3.16178 0.43626

j0 (mm)

α0 (rad)

− 0.59948

3.20827

− 0.59948

1.12094

0.44228

1.12094

2

0.73449

2.81112

0.71622

2.80624

0.72607

2.80625

3

0.96446

2.82347

0.94132

2.81923

0.95426

2.81924

4

0.62772

− 1.85993

0.61078

− 1.85305

0.61922

− 1.85304 − 1.34693

5

0.39433

− 1.33585

0.39010

− 1.34693

0.39549

6

0.24789

2.06414

0.23579

2.05273

0.23905

2.05273

7

0.41607

0.54210

0.40195

0.55182

0.40753

0.55182

8

0.39193

− 0.94069

0.37325

− 0.91766

0.37843

− 0.91766

9

0.62805

− 1.90485

0.61252

− 1.89632

0.62102

− 1.89632

Table 6 Error data of before and after correction Number

j ∗ − j (mm)

j0 − j (mm)

0

− 0.05687

− 0.01038

α ∗ − α (rad)

α0 − α (rad)

0.00087

0.00088

1

− 0.01063

− 0.00461

− 0.00465

− 0.00465

2

− 0.01826

− 0.00842

− 0.00489

− 0.00487 − 0.00423

3

− 0.02314

− 0.01020

− 0.00424

4

− 0.01695

− 0.00850

0.00689

0.00689

5

− 0.00424

0.00115

− 0.01108

− 0.01108 − 0.01141

6

− 0.01211

− 0.00884

− 0.01141

7

− 0.01413

− 0.00854

0.00972

0.00972

8

− 0.01868

− 0.01350

0.02302

0.02303

9

− 0.01553

− 0.00703

0.00853

0.00853

312

Q. Kong and Z. Yu

Fig. 12 Length error of the deflection at point 3 after and before correction

Fig. 13 Angle errors of the deflection at point 3 after and before correction

5 Conclusion In this paper, the measurement errors of the two different lever models in measuring the deflection of the regular hexagon section shaft are analyzed by derivation formulas, and mathematic relationships for different models that can correct the errors is provided. Experiments prove the correctness and effectiveness of the correction methods. The conclusions are as follows. 1. The first structure cannot eliminate errors by parameter adjustment, while we can adjust the second structure’s parameters to meet the condition A = DB to eliminate errors directly. 2. Both models can be corrected by the error correction method, which based on a strict mathematical relationship, and there will be no error theoretically. However, the correction results are affected by the resolution of the displacement sensor. 3. Experiments verify the validity of the correction method. The experimental results show that the measurement errors of the two models are controlled within ± 0.006 mm through a strict mathematical relationship.

Measuring Error Correction Method During Deflection Measurement …

313

4. It can be seen from the deflection measurement experiment of regular hexagon section shaft, the length error of deflection can be well reduced after error correction, and length error of deflection is less than 0.01 mm generally. However, it has little effect on reducing the angle error of the deflection, but the angle error itself is small relatively; the maximum is only 1.3°. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 51675481), Basic public welfare research project of Zhejiang province (LGG18E050008).

References 1. Gomes HM, de Almeida FJF (2014) Appl Math Model 38(3):938–948. https://doi.org/10.1016/ j.apm.2013.07.019 2. She H, Li C, Tang Q et al (2018) The investigation of the coupled vibration in a flexible-disk blades system considering the influence of shaft bending vibration. Mech Syst Signal Process 111:545–569. https://doi.org/10.1016/j.ymssp.2018.03.044 3. Pei YC, Wang JW, Tan QC et al (2017) An investigation on the bending straightening process of D-type cross section shaft. Int J Mech Sci 131:1082–1091. https://doi.org/10.1016/j.ijmecsci. 2017.08.032 4. Su BY, Zhang YK, Sun GF et al (2018) 3-D FEM simulation of laser peening straightening for shaft straightness. Key Eng Mater 765:243–248. https://doi.org/10.4028/www.scientific. net/KEM.765.243 5. Muratkin GV, Sarafanova VA (2020) Straightening of shafts by surface plastic deformation with elastic flexure of the workpiece. Russ Eng Res 40(8):637–641. https://doi.org/10.3103/ S1068798X20080183 6. Chen X, Yan N, Wang C et al (2020) Study on straightening quality control for slender rod based on digital twin. J Phys Conf Ser 1633(1):012160. https://doi.org/10.1088/1742-6596/ 1633/1/012160 7. Kim SC, Chung SC (2002) Synthesis of the multi-step straightness control system for shaft straightening processes. Mechatronics 12(1):139–156. https://doi.org/10.1016/S09574158(00)00066-0 8. Wang C, Yu G, Wang W et al (2018) Deflection detection and curve fitting in three-roll continuous straightening process for LSAW pipes. J Mater Process Technol 255:150–160. https:// doi.org/10.1016/j.jmatprotec.2017.11.060 9. Pei YC, Yuan Y, Li H et al (2020) A bending deflection measuring method of inner/outer cylindrical surface parts. Measurement 165:108082. https://doi.org/10.1016/j.measurement. 2020.108082 10. Chen X, Bi M, Xue C (2019) Study on straightness measurement technology of slender rod based on time-domain three-point method. J Phys Conf Ser 1303(1):012140. https://doi.org/ 10.1088/1742-6596/1303/1/012140 11. Chen X, Bi M, Xue C (2019) Study on measuring straightness of piston rod in large oil cylinders. J Phys Conf Ser 1303(1):012137. https://doi.org/10.1088/1742-6596/1303/1/012137 12. Gao W, Yokoyama J, Kojima H et al (2002) Precision measurement of cylinder straightness using a scanning multi-probe system. Precis Eng 26(3):279–288. https://doi.org/10.1016/ S0141-6359(02)00106-X 13. Zatoˇcilová A, Paloušek D, Brandejs J (2016) Image-based measurement of the dimensions and of the axis straightness of hot forgings. Measurement 94:254–264. https://doi.org/10.1016/j. measurement.2016.07.066

314

Q. Kong and Z. Yu

14. Zatoˇcilová A, Polišˇcuk R, Paloušek D et al (2013) Photogrammetry based system for the measurement of cylindrical forgings axis straightness. In: Optical measurement systems for industrial inspection VIII, vol 8788. International Society for Optics and Photonics, p 87881L. https://doi.org/10.1117/12.2020917 15. Patwari AU, Ullah SMT (2013) Development of new computational image analysis technique for measuring the circularity and straightness of seamless pipe. Ann Fac Eng Hunedoara 11(4):293 16. Li X, Du X, Li X et al (2011) Design of machine vision system for steel pipe’s straightness measurement. Opt Techn 3. https://doi.org/10.13741/j.cnki.11-1879/o4.2011.03.010 17. Hao F, Shi J, Meng C et al (2020) Measuring straightness errors of slender shafts based on coded references and geometric constraints. J Eng 2020(6):221–227. https://doi.org/10.1049/ joe.2019.1259 18. Wang CH, Pei YC, Tan QC et al (2017) An improved high precision measuring method for shaft bending deflection. Appl Math Model 48:860–869. https://doi.org/10.1016/j.apm.2017. 02.050 19. Kong Q, Yu Z, Mao X et al (2020) Rotation error modeling and compensation of spindle based on Clarke transformation in straightness error measurement of regular hexagon section shaft. Measurement 166:108233. https://doi.org/10.1016/j.measurement.2020.108233

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable Wheels Ze Fu, Hao Xu, Yinghui Li, and Weizhong Guo

Abstract The wheel-legged hybrid structure has been utilized by ground mobile platforms in recent years to achieve good mobility on both flat surface and rough terrain. However, many designs of the obstacle-crossing part and transformation driving part of this structure is highly coupled, which limits its optimal performance in both aspects. This paper presents a novel wheel-legged robot with rim shape changeable wheels, which has the bi-directional and smooth obstacle-crossing ability. Based on the kinematic model, the geometric parameters of the wheel structure and the design variables of the driving four-bar mechanism are optimized separately. Experiments show that the prototype installed with the novel transformable wheel can overcome steps with the height of 1.52 times of its wheel radius with less fluctuation of its centroid and performs good locomotion capabilities in different environments. Keywords Mobile platform · Transformable wheel-legged robot · Kinematics analysis · Mechanical design · Obstacle crossing

1 Introduction Mobile platforms determine the motion performance of robots. Especially in the fields of military reconnaissance, disaster rescue and extra-terrestrial exploration, the obstacle-crossing ability is highly demanded. At present, ground mobile platforms can be divided into wheel type, leg type and crawler type according to their motion mode. Z. Fu · H. Xu · Y. Li · W. Guo (B) State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: [email protected] Z. Fu e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_17

315

316

Z. Fu et al.

Among them, wheel-based locomotion is simplest and most efficient [1]. However, if the wheeled robot encounters an obstacle with a height greater than the radius of its wheel, it will not be able to pass it. Therefore, the application in the complex terrain of the wheeled robot is limited by its poor obstacle-crossing ability. With the help of bionic walking gaits, the legged robot has good motion performance on uneven surfaces [2]. But the control system of leg-based locomotion is more complex. Moreover, the fluctuation of centroid and the collision between the foot and the ground lead to low energy utilization efficiency [3]. Wheel-legged hybrid robot is another important research direction. The early wheel-legged hybrid structure is fixed. WHEGS’s [4] rimless wheel has three spokes with hooks at the end of them. It mimics the barbs on the insects’ legs and overcomes rocks by hooking on their edges. RHex [5] adopts the half-circular rim as legs to expand the motion range. RoMiRAMT [6] uses rotating wheels with six legs and adds a spine to lift or lower the body, which allows the robot to climb higher obstacles. Over the past few years, in order to make the robot transform between the wheel mode and leg mode so as to combine the advantages of both types, many scientists have studied wheel-legged robots. Quattroped’s [7] wheel can be separated into two semicircular legs with obstacle-crossing capability. TurboQuad [8, 9] is in the same series with quattroped. The difference is that both semicircular legs can extend outward to increase the leg’s motion range. WheeLeR [10] uses the center gear to open or close the wheel rim. When opened, the rim can hook the edge of the obstacle and lift the robot to cross the obstacle. FUHAR [11] uses a four-bar linkage to drive the circular wheel rim into six claws. Dynamic and simulation models are established to verify its obstacle-crossing performance. Most of these wheel-legged hybrid robots adapt structure similar to a claw to hook the edge of obstacles and cross over them, which lead to violent fluctuations of the robot’s centroid. At the same time, due to the asymmetry of the mechanism, most of the wheel-legged structures only have one-directional obstacle-crossing ability, which limits their application in complex terrain. In this paper, a novel wheel-legged robot with rim shape changeable (RSC) wheels is designed, which has the ability of bi-directional obstacle-crossing. During the climbing process, the rim contacts the obstacle’s surface smoothly to reduce the centroid fluctuation of the mobile platform. The kinematic model of the RSC wheel is first established, and the relationship between its geometric parameters and climbing ability is revealed. On this basis, the optimal geometric parameters of the wheel meeting the obstacle-crossing requirements are selected, and then the design variables of the driving four-bar mechanism are optimized to improve the transformation efficiency. Experiments show that the prototype is able to traverse rough terrain steadily. The organization of this paper is as follows: Sect. 2 presents the kinematic model of the RSC wheel when crossing obstacles; The optimal design parameters of the driving four-bar mechanism are formulated in Sect. 3; Sect. 4 conducts the prototype experiment to verify the obstacle-crossing stability and terrain adaptability of the robot; and Sect. 5 has concluding remarks.

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable …

317

2 Kinematic Model of the Transformable Wheel in Obstacle Crossing Process The wheel with the shape changeable rim is essentially an equally divided circle, and the arc after equal division rotates around a point on it. As shown in Fig. 1, take the wheel rim divided into three parts as an example to model the kinematics of the climbing process. The structure of the RSC wheel is determined by r, n, θ and δ, which correspond to the wheel radius, the number of equally divided arcs, the rotation angle of each arc and the central angle of each arc between their endpoint and the rotation point respectively. The RSC wheel rotates periodically on the flat surface, and each period includes two stages: arc rolling stage and tip rotating stage. As shown in Fig. 2a, the former stage is the rotation of the circle in which the arc BC in contact with the ground is located, and the trajectory of any point on the RSC wheel is the cycloid. Assuming that at the beginning, the contact point between BC and the ground is A, and O M1 is vertical. The center angle of the arc rotation point M1 and contact point A is ϕ. Take A as the origin and establish the coordinate system. The rotation angle of circle O1 is α, and the trace of O1 is 



Fig. 1 The RSC wheel and its motion coordinate system

Fig. 2 Movement process of the RSC wheel. Stage of a arc rolling. b Tip rotating. c Contacting the obstacle surface

318

Z. Fu et al.



O1x = r α

(1)

O1y = r And the trace of M1 is cycloid, which has the initial phase angle ϕ 

M1x = r ((α + ϕ) − sin(α + ϕ)) + r ϕ M1y = r (1 − cos(α + ϕ))

(2)

By rotating O1 M1 counterclockwise around M1 at the transform angle θ , the trace of the wheel center O in the first stage can be written as 

Ox



Oy

 =

M1x



 + R1

M1y

O1x − M1x

 (3)

O1y − M1y

Here, R1 is the rotation matrix 

cos θ − sin θ R1 = sin θ cos θ

 (4)

As shown in Fig. 2b, when the rotation angle α = 360◦ /n − δ − ϕ, δ ≤ 180◦ /n

(5)



The arc BC ’s endpoint B contacts the ground, and the wheel enters the tip rotating stage. At this time, all the points on the wheel rotate the angle of β around B. So, the trace of O1 and M1 can be expressed as 

M1x



 =

Bx



 + R2

M1x − Bx

 (6)

M1y − B y       O1x Bx O1x − Bx = + R2 O1y By O1y − B y M1y

By

(7)

Here, R2 is the rotation matrix  R2 =

cos β sin β − sin β cos β

 (8)

Similarly, the trace of the wheel center O at this stage can be obtained from Eq. (3). This paper evaluates the obstacle-crossing ability by taking climbing stairs as an example. As shown in Fig. 2c, the height of the step is H , the arc D E will touch the step surface before contacting the ground. When D E is tangent to the step 



Design of a Novel Wheel-Legged Robot with Rim Shape Changeable …

319

surface, the following condition is satisfied O2y − r − H = 0

(9)

Thus, the height of the step that can be climbed is: H = O2y − r

(10)

If the number of equally divided arcs is n, the center angle of each arc is 360◦ /n. That is, the angle between O O1 and O O2 is 360◦ /n. The coordinate of O2 in Eq. (10) can be described as       O2x Ox O1x − Ox = + R3 (11) O2y Oy O1y − O y Here, R3 is the rotation matrix  ⎤ 2π 2π − sin cos ⎢ n n ⎥  ⎥ R3 = ⎢ ⎣  2π 2π ⎦ cos sin n n ⎡



(12)

By adopting the above obstacle climbing method, the rim can smoothly and rapidly move on the step’s surface once it contacts the step. When H = 15 cm, r = 10 cm, n = 3, δ = 60◦ and θ = 50◦ , the trace of the wheel center O during the process of climbing steps can be derived by combining Eqs. (3) and (10), which is shown in Fig. 3. The RSC wheel experiences the arc rolling stage on the ground and the rotation stage around the end of the arc (P1 to P2 )—contacting the step (at P2 )—the arc rolling stage on the step (P2 to P3 ) as shown in Fig. 3a. Figure 3b shows the change of the trajectory slope. Except at P2 , the slope changes continuously, and the maximum value does not exceed 1.5. The slope change at P2 is 1.45, and it is always positive throughout the process. The centroid of the wheel fluctuates mildly in the whole process, and the motion stability is good.

3 Design of Transformable Wheel Mechanism 3.1 Concept of the Wheel Mechanism The obstacle-crossing capability can be indicated by the ratio of maximum height of obstacle the wheel can overcome Hm to the wheel radius in round shape r [12], and it depends on the geometric parameters of the RSC wheel. The mathematical

320

Z. Fu et al.

(a)

(b) Fig. 3 Obstacle climbing process when H = 15 cm, r = 10 cm, n = 3, δ = 60◦ and θ = 50◦ . a Bi-direction trajectory of the wheel center O. b The change of trajectory slope

expression of Hm /r is analyzed by taking the RSC wheel which has four divided legs as an example. As shown in Fig. 4, when overcoming the obstacle with maximum height, the arc’s endpoint B should be at the intersection of the ground and the vertical surface of the step. The geometric constraint in this position Bx = O2x

(13)

The relationship between Hm /r and n, δ, θ can be derived by substituting Eq. (13) into the kinematic model in Sect. 2, which is shown in Fig. 5. Figure 5 illustrates that regardless of the number of equally divided legs n, Hm /r increases with θ and δ. As mentioned above, the maximum value of δ is 180◦ /n. In Fig. 4 The position of the RSC wheel with four legs when overcoming the obstacle with maximum height

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable …

321

Fig. 5 Relationship between n, θ , δ and the obstacle crossing ability Hm /r

Fig. 6 Relationship between n, θ and the obstacle-crossing ability Hm /r when δ = 180◦ /n

this case, δ is half of the central angle corresponding to each arc and the configuration of the wheel is symmetrical. Thus, δ is determined as 180◦ /n to obtain bi-directional barrier-overcoming capacity. From Fig. 5, the relationship between n, θ and Hm /r can be derived in Fig. 6. Figure 6 illustrates that under the same θ , the obstacle-crossing performance is strongest when n = 3, slightly weaker when n = 2, and gets worse when n > 3 as n increases. Therefore, the structure with three legs after transformation is chosen as shown in Fig. 7. Fig. 7 The RSC wheel driven by the four-bar mechanism. a Schematic diagram of the mechanism. b Free body diagram of the wheel during the transformation process

322

Z. Fu et al.

As shown in Fig. 7a, based on the optimal structure, the transformation driving part will be designed in the following. For the compactness and reliability, three planar four-bar mechanisms are used to drive the transformation of the wheel, in which the rocker l1 provides the driving force.

3.2 Selection of Design Variables Based on Kinematics The design variables, especially the length of the linkage l1 , l2 , l3 , are analyzed and properly selected based on the kinematic model in this part. Figure 5 reveals that the larger the transformation angle θ , the stronger the barrier-crossing capability. Therefore, an optimization problem with the maximum θ as the objective can be constructed. Considering the clockwise transformation of the rim, the relationship between θm and the length of each linkage is: θm = arccos

l3 l32 + r 2 − (l1 + l2 )2 + arcsin − 90◦ 2l3r 2r

(14)

The condition for the existence of the four-bar mechanism: l1 + l2 + l3 > r r + l2 + l3 > l1 l1 + r + l3 > l2

(15)

l1 + l2 + r > l3 The dimension parameters of the linkages are also limited by the size of the structure to avoid motion interference: r < l1 + l2 < r + l3 δ l3 < 2r sin 2 l1 + r > l2 + l3

(16)

Besides, the torque required for transformation should be small to ensure the successful shape change of the rim. After the gear reduction, there is maximum torque TM ≤ 3 Nm. As shown in Fig. 6b, by the principle of virtual work Tm dγ = FN cos μd B



(17)

By combining Eqs. (14)–(17), the optimization variable space can be drawn as Fig. 8.

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable …

323

Fig. 8 Relationship between the θm and the length of linkages under the constraints

The optimal four-bar mechanism parameters satisfying the constraints are obtained from the variable space. l1 = 0.625r, l2 = 0.880r, l3 = 0.735r The maximum height of the step is set as 15 cm. From Fig. 6, the corresponding minimum radius of the wheel rmin = 9.09 cm. Taking the machining error and the stability margin into account, r is designed as 10 cm. Thus, the optimized design variables are l1 = 6.25 cm, l2 = 8.80 cm, l3 = 7.35 cm The mechanism with optimized design variables is shown in Fig. 9. The Maximum clockwise transformation angle θm reaches 50.4° and Hm /r reaches 1.52. Each leg is connected to the center turning disk so that the whole transformation can be driven by just one motor. Active bar l1 rotates clockwise or counterclockwise to change the wheel into forward or backward direction motion mode respectively, thus providing the mobile platform with a bi-directional obstacle-crossing capability.

Fig. 9 The wheel mechanism with optimum design variables

324

Z. Fu et al.

4 Prototype Design and Experiment 4.1 The Design of Mobile Platform The mobile platform installed with the RSC wheel is shown in Fig. 10. It consists of three parts: the RSC wheel, the body and the passive assistant wheel. The prototype with two RSC wheels is developed as shown in Fig. 11. The overall size of the prototype is 46 × 40 × 24 cm3 . The whole system is powered by a lithium battery with a capacity of 9800 mAh, which provides 12 V voltage to the controller and the motor drivers. It can ensure the continuous operation of the robot for one hour. The specific parameters of the prototype are listed in Table 1.

Fig. 10 The schematic diagram of the mobile platform during the obstacle-crossing process

Fig. 11 Prototype of the wheel-legged robot. a Top view of the prototype. b Side view of the RSC wheel

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable … Table 1 Parameters of the robot prototype

325

Items

Features

Total mass

5.2 kg

RSC wheel radius

0.1 m

Assistant wheel equivalent radius

0.13 m

Body length

0.35 m

Body materials

Aluminum alloy

Maximum transformation angle of the 50° wheel Maximum speed

1.67 m/s (wheel mode)

4.2 The Obstacle-Crossing Experiment To verify the obstacle-crossing ability and stability of the prototype and test whether the kinematics model established above is correct, the experiments are carried out in the real environment shown in Fig. 12. In Fig. 12a, the height of the step in front of the robot is 14.6 cm, and the ratio of the step height to the radius of the RSC wheel H/r is 1.46. The equivalent radius of the passive assistant wheel is 13 cm. When encountering an obstacle, the robot will switch from wheel mode to leg mode, in which the transformation angle of the wheel rim is 50°. Figure 12b records the centroid trajectory of the robot (approximately considered to be concentrated at the center of the RSC wheel) during the step-climbing process. As shown in Fig. 13, taking the initial wheel center position as the origin, the height variation range of the centroid is always within 15 cm and the maximum absolute value of track slope is 1.0079 in the whole process. Similar to the kinematic analysis in Sect. 2, before T = 3 s, that is, the RSC wheel completely climbs the obstacle, the trajectory slope is always positive. The fluctuation of the centroid is slight, which proves that the robot has good stability during the obstacle-crossing process. In addition, as shown in Fig. 14, the locomotion ability of the prototype on rough surfaces is also examined. It is shown that the prototype can smoothly pass through the rough terrain with complex features such as stones and pits, and has strong terrain adaptability.

5 Conclusions This work proposes a novel wheel-legged robot with RSC wheels that can switch between wheel mode and leg mode based on a four-bar mechanism. The robot maintains both high mobility in flat ground and bi-directional obstacle-crossing performance in rough roads.

326

Z. Fu et al.

Fig. 12 Obstacle-crossing experiment of the prototype. a Robot switches motion modes in front of the step. b Step-climbing process

Design of a Novel Wheel-Legged Robot with Rim Shape Changeable …

327

Fig. 13 Obstacle-crossing trajectory and the change of slope

Fig. 14 Experiment of traversing rough terrains

The design process gets rid of the high coupling between the geometric structure and the transformation driving mechanism, which can provide a reference for the design of the wheel-legged robot. The stable and efficient obstacle-crossing performance of the mobile platform will expand its application in fields such as detection and rescue, and improve the success rate of the missions in complex terrains. Acknowledgements Supported by the State Key Lab of Mechanical System and Vibration Project (Grant No. MSVZD202008).

References 1. Kim Y, Lee Y, Lee S, Kim J, Kim HS, Seo T (2020) STEP: a new mobile platform with 2-DOF transformable wheels for service robots. IEEE-ASME Trans Mechatron 25(4):1859–1868 2. Siegwart R, Lamon P, Estier T, Lauria M, Piguet R (2002) Innovative design for wheeled locomotion in rough terrain. Robot Auton Syst 40(2–3):151–162 3. Remy CD (2017) Ambiguous collision outcomes and sliding with infinite friction in models of legged systems. Int J Robot Res 36(12):1252–1267 4. Daltorio KA, Wei TE, Wile GD, Southard L, Palmer LR, Gorb SN, Ritzmann RE, Quinn RD (2007) Mini-Whegs (TM) climbing steep surfaces with insect-inspired attachment mechanisms. Int J Robot Res 28(2):285–302

328

Z. Fu et al.

5. Saranli U, Buehler M, Koditschek DE (2001) RHex: a simple and highly mobile hexapod robot. Int J Robot Res 20(7):616–631 6. Kwak B, Bae J (2018) Design and analysis of a rotational leg-type miniature robot with an actuated middle joint and a tail (RoMiRAMT-II). J Bionic Eng 15(2):356–367 7. Chen SC, Huang KJ, Chen WH, Shen SY, Li CH, Lin PC (2014) Quattroped: a leg-wheel transformable robot. IEEE-ASME Trans Mechatron 19(2):730–742 8. Chen WH, Lin HS, Lin PC (2014) TurboQuad: a leg-wheel transformable robot using bioinspired control. In: IEEE international conference on robotics and automation (ICRA), pp 2090–2090 9. Chen WH, Lin HS, Lin YM, Lin PC (2017) TurboQuad: a novel leg-wheel transformable robot with smooth and fast behavioral transitions. IEEE Trans Robot 33(5):1025–1040 10. Zheng CQ, Lee K (2019) WheeLeR: wheel-leg reconfigurable mechanism with passive gears for mobile robot applications. In: International conference on robotics and automation (ICRA), pp 9292–9298 11. Mertyuz I, Tanyildizi AK, Tasar B, Tatar AB, Yakut O (2020) FUHAR: a transformable wheellegged hybrid mobile robot. Robot Auton Syst 133 12. Daltorio KA, Wei TE, Wile GD, Southard L, Palmer LR, Gorb SN, Ritzmann RE, Quinn RD (2007) Mini-Whegs (TM) climbing steep surfaces with insect-inspired attachment mechanisms. In: International conference on intelligent robots and systems, vols 1–9, pp 2562–2562

FBCCD: A Forward and Backward Cyclic Iterative Solver for the Inverse Kinematics of Continuum Robot Haoran Wu, Jingjun Yu, Jie Pan, Guoxin Li, and Xu Pei

Abstract This paper proposes a new numerical approach called Forward and Backward Cyclic Coordinate Descent (FBCCD), which is based on the Cyclic Coordinate Descent (CCD) algorithm. A specific set of solutions can be found from infinite solutions of multi-segment continuum robots using the iterative numerical algorithm. Inspired by the Forward and Backward Reaching Inverse Kinematics (FABRIK) algorithm, the inverse kinematics (IK) of a multi-segment continuum robot is divided into two phases: a forward iteration of end coordinates and a backward iteration of end direction. Forward and backward iterations correct and compensate each other, making the end pose close to the target. By altering the goal function of a single iteration, the FBCCD algorithm can also be applied to the continuum robot with a movable base. The numerical experiment results illustrate that this algorithm is with higher convergence rate and effectiveness compared with some of the most popular IK approaches. The average operating time for a five-segment continuum robot is 361 ms and the average number of iterations is 22.89. Keywords Continuum robot · Inverse kinematics · Cyclic Coordinate Descent

H. Wu · J. Yu · J. Pan · G. Li Robotics Institute, Beihang University, Beijing 100191, China e-mail: [email protected] J. Yu e-mail: [email protected] J. Pan e-mail: [email protected] G. Li e-mail: [email protected] X. Pei (B) Department of Mechanical Design, Beihang University, Beijing 100191, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_18

329

330

H. Wu et al.

1 Introduction As a new sort of bionic robot, continuum robots can change their shape continuously [1]. Due to the flexible deformation ability and strong adaptability to narrow environments [2–4], it is extensively used in expedition [5], aviation manufacturing [6], medical surgery [7, 8] as well as other realms. As shown in Fig. 1, for operation and maintenance, the continuum robot follows a fixed route, avoids obstacles, and enters a small space. At this phase, the continuum robot’s inverse kinematics must be precisely solved. And the efficiency and reliability of inverse kinematics are important factors to guarantee control of a continuum robot in perfect sync. The conventional approach to resolving the continuum robot’s IK mainly includes numerical method, analytical method and other heuristic methods. However, due to the complexity of the multi-segment continuum robot’s kinematic model, the analytical method has some difficulties in practical application. The traditional numerical method for solving the IK of the continuum robot is the Jacobian method [9–11]. The feasible solution of the continuum robot is obtained by Jacobian pseudo-matrix iteration. However, the joint boundary conditions are difficult to implement. Troubles with singularities and non-convergence are unavoidable [12]. The low efficiency of the calculation of the Jacobian matrix’s pseudo-inverse leads to poor real-time performance. Neural network [13–15] is also applied for obtaining the continuum robot’s IK. As a model-free solution strategy [16], the primary use of a neural network is to handle the mapping of configuration space to drive space. Although neural network has higher efficiency and stronger robustness, a larger number of training set is required for multi-segment continuum robot. Based on the 3D tractrix curve, a solution to the continuum robot’s IK was put forward by Sreenivasan et al. [17]. The continuum robot’s movement is simulated by the 3D tractrix curve. Inspired by the following movement of snake and rope, Williams and Mayhew [18] proposed the Follow-the-Leader (FTL) heuristic algorithm. The

Fig. 1 An illustration of the use of a continuum robot in a confined space

FBCCD: A Forward and Backward Cyclic Iterative Solver …

331

continuum robot’s end follows the given collision-free projected path [19–21]. A path-following algorithm is proposed by Gu [22] to reduce the error of trajectory tracking. However, the continuum robot’s potential motion capability is limited by a given trajectory which further reduces the kinematic dexterity. Aristidou and Lasenby [23] put forward the Forward and Backward Reaching Inverse Kinematics (FABRIK) algorithm for obtaining the IK of the hyper-redundant robot. The IK is obtained by forward and backward iteration strategies, which are also applied in this paper. Due to the new joint being situated on the connection between the original and the adjacent joint, the FABRIK has higher computational efficiency [24]. The original developers of the Cyclic Coordinate Descent (CCD) algorithm [25] were Wang and Chen for solving the IK of rigid robots. Due to the CCD algorithm’s clarity and easy operation, it has faster calculation efficiency and better robustness [26, 27]. More natural poses of continuum robot can be generated by CCD. The primary contribution of this study is the improvement of the CCD algorithm for multi-segment continuum robots’ IK. Inspired by the FABRIK algorithm, the multi-segment continuum robot IK problem is divided into two phases: a forward iteration of end coordinates and a backward iteration of end direction. The rest of this essay is outlined: The continuum robot’s kinematics model is established as the foundation of the Piecewise Constant Curvature (PCC) model [28–30]. In the following section, the overall steps of the FBCCD algorithm are put forward. Finally, through numerical experiments and comparison, the FBCCD algorithm’s performance is completely confirmed.

2 Continuum Robot’s Kinematics Model The continuum Robot’s Kinematics Model primarily includes the mapping from the joint space to the driving space and the task space to the joint space. The method suggested in this publication is mainly applied to obtain the mapping between the task space and the joint space. In order to make the calculation faster and more efficient, the continuum robot’s kinematics model is constructed on the foundation of the PCC model. As illustrated in Fig. 1, the single segment is viewed as a circular arc. A continuum robot’s configuration space can be defined by two variables: the direction angle θ and bending angle α. The continuum robot’s joint variable definition is shown as follows: The bending angle: α denotes the continuum robot’s joints’ degree of bending, which is expressed by the central angle of the bending arc. The direction angle: the continuum robot’s joints’ bending direction is represented by θ. As illustrated in Fig. 1, the angle between the YOZ plane and the bending plane. The single-segment of continuum robot has a length of l, and the end coordinate of the robot is P(x, y, z). Equation (1) can be obtained according to geometric relations.

332

H. Wu et al.

⎞ ⎛ li sθ i (1−cαi ) ⎞ − px αi ⎜ li cθ i (1−cα i) ⎟ ⎝ ⎠ P = py = ⎝ ⎠ αi li sα i pz α ⎛

(1)

i

The rotation matrix of the end of a single-segment continuum robot can be solved by homogeneous coordinate transformation. i−1 i T (αi , θi )

= Rot(z i , −θi )Trans(0, li (1 − cαi )/αi , li sαi /αi )Rot(xi , −αi ) ⎞ ⎛ i) cθi −cαi sθi −sθi sαi − li sθi (1−cα αi ⎜ −sθ cα cθ cθ sα li cθi (1−cαi ) ⎟ ⎟ ⎜ i i i i i αi (2) =⎜ ⎟ li sαi ⎠ ⎝ 0 −sαi cαi αi 0 0 0 1

Figure 2 depicts a diagram of a continuum robot with three-segment. The whole continuum robot is formed of multi-segment connected in series. The chain rule can be used to determine the continuum robot’s final pose. n 0T

=

n  i=1

Fig. 2 A single-segment continuum robot’s schematic

i i−1 T

(3)

FBCCD: A Forward and Backward Cyclic Iterative Solver …

333

3 Forward and Backward Cyclic Coordinate Descent The FBCCD method is divided into two stages: an iteration of coordinates that moves forward and an iteration of the final direction that moves backward. The forward iteration process is carried out first, making the end coordinates close to the target. Then the backward iteration is performed to make the end direction close to the target pose. However, in the process of backward iteration, the end coordinate may deviate from the target. Consequently, the next iteration is executed to correct the resulting deviation. Forward and backward iterations correct and compensate each other, making the end pose close to the target.

3.1 FBCCD Algorithm with Fixed Base 3.1.1

Forward Iteration of End Coordinates

In the process of single-step calculation, only the Single-segment parameters are altered. In this case, the continuum robot’s end pose is illustrated as Eq. (4) T = Tbase T (αi , θi )Tend =

R P 1 0

(4)

where: ⎧ i−1  ⎪ ⎪ ⎨ Tbase = ⎪ ⎪ ⎩ Tend =

m=1 n 

m−1 m T

m=i+1

(5) m−1 m T

The direction angle is altered while the bending angle is fixed. The IK of continuum robot is transformed into the maximum value of an interval for a single variable function. And for continuously differentiable functions, extreme values can be obtained by gradient descent. And only the bending angle is changed in the subsequent process of iteration.   ⎧ min  P − Ptarget  ⎪ ⎪ ⎨ s.t. αi = α0 (θ = θ0 ) (6) ⎪ ⎪ θlow ≤ θi ≤ θhigh  ⎩ αlow ≤ αi ≤ αhigh The forward iteration process of the continuum robot with three-segment is shown in Fig. 3. As shown in Fig. 3a, the target is Pt , and the current endpoint is P4 . The continuum robot’s base serves as the starting point for the iterative process. Firstly, the

334

H. Wu et al.

Fig. 3 Multi-segment continuum robot kinematics schematic

second and third segments are fixed, and the first segment’s configuration parameters can be found. In the following iteration progress, the configuration parameters of the second and third segment can be solved in the same way. The iteration process is implemented from the continuum robot’s base to the end, making the end coordinates close to the target.

3.1.2

Backward Iteration of End Direction

After the forward iteration of coordinates, the continuum robot’s end direction diverges from the intended direction. Therefore, the backward iteration of the end direction is carried out to reduce the error. Simultaneously, it is necessary to ensure that the offset of the end coordinates is not too large. The final pose’s direction vector is nc and the direction vector of the desired pose is nt . The included angle of the two vectors is illustrated as Eq. (7). βi = cos

−1



nt · nc n t n c 

(7)

Only the direction angle of the single segment is altered, and the bending angle remains constant. The minimum value of the angle can be obtained using the method described above. Only the bending angle is changed in the subsequent process of iteration.

FBCCD: A Forward and Backward Cyclic Iterative Solver …

335

Fig. 4 Forward iteration of end coordinates

⎧ min βi ⎪ ⎪ ⎨ s.t. αi = α0 (θ = θ0 ) ⎪ ⎪ θlow ≤ θi ≤ θhigh  ⎩ αlow ≤ αi ≤ αhigh

(8)

Adjusting the end segment of a continuum robot has a bigger impact on the final pose than the final coordinate. So, the iteration progress of the end direction starts from the end to the base. Meanwhile, the iteration of the end pose has a greater convergence rate. Therefore, the backward iteration stops when the included angle with the target direction is less than a certain threshold to avoid large deviations of the end coordinates. The backward iteration process of the three-segment continuum robot is shown in Fig. 4. Firstly, the configuration parameters of the third segment can be obtained, while the first and second segments are fixed. In the subsequent iterations, the configuration parameters of the second and first segment can be obtained in the same way. The iteration process is carried out from the continuum robot’s tip to its base.

3.2 FBCCD Algorithm with Movable Base For continuum robot with movable base, the forward and backward iteration strategies are basically consistent with the preceding part. However, the forward iteration objective function should be adjusted. Given the coordinates of the target P(px , py , pz ) and the end of continuum robot T (tx, ty, tz). If the continuum robot’s base is equipped with a guide rail, the z-axis feeding movement can be carried out. The objective function of a single forward iteration is illustrated as Eq. (9).

336

H. Wu et al.

Fig. 5 Backward iteration of end direction

2  f cost = ( px − tx )2 + p y − t y

(9)

Using the same strategy, only one of the configuration parameters for the continuum robot should be changed in order to obtain the objective function’s maximum value. ⎧ min f cost ⎪ ⎪ ⎨ s.t. αi = α0 (θ = θ0 ) (10) ⎪ ⎪ θlow ≤ θi ≤ θhigh  ⎩ αlow ≤ αi ≤ αhigh The robot’s feed can be computed after the iteration progress. z f eed = pz − tz

(11)

Figure 5 depicts the trajectory tracking of the continuum robot with three-segment. Given the tip trajectory of the continuum robot, the sampling points (p1 , p2 , p3 and p4 ) are equidistant from it. The continuum robot’s IK can be solved by the FBCCD algorithm and the feed of the base can be further calculated. According to the end pose obtained at the previous sampling point, the IK of this point is obtained.

3.3 Overall Steps of the FBCCD Algorithm The overall steps of FBCCD algorithm are illustrated in Fig. 6. The posture interpolation is performed first based on the tip pose and the target. The current IK is solved according to the current tip pose and the previous sampling point’s IK. In the specific procedure of each cycle, the forward iteration of the coordinate is carried out first, and then the backward iteration of the end direction is implemented. The

FBCCD: A Forward and Backward Cyclic Iterative Solver …

337

Fig. 6 A schematic diagram of continuum robot’s trajectory tracking

above procedures are carried out repeatedly until the target deviation is below a predetermined threshold.

4 Simulation and Comparation of FBCCD Algorithm 4.1 Efficiency of FBCCD Algorithm To authenticate the performance of the algorithm, experimental verification is carried out on a continuum robot with five-segment as an example. The maximum bending angle of a single segment is 60°, and each segment is 10 in length. The convergence threshold of coordinates is 0.05 mm and the threshold of direction is 0.01 rad. The equipment with Intel i7-10850H (2.3 GHz) processor and 16 GB of memory was used for all tests. And the test software is MATLAB2019b. The relationship between the error of the end direction and end distance of the FBCCD algorithm with iteration is shown in Fig. 7. The average results are obtained by solving 100 random poses in space. At the beginning of the iteration, the error of distance increases with the iteration, while the angle decreased sharply. Since the initial pose and the target are at a greater included angle, there is a stage of direction adjustment. After that, the error of direction and the error of distance decrease with iteration. However, the distance between the target has a greater convergence rate. In the previous iteration, the deviation of the end direction caused by the forward iteration can be compensated by the backward iteration. It takes average 22.89 iterations to reach the target. The time distribution of FBCCD algorithm is illustrated in Fig. 8. The distribution is obtained from 300 sets of arbitrary poses in space. And the mean convergence time of FBCCD algorithm is 361 ms.

338

H. Wu et al.

Fig. 7 Overall steps of the FBCCD algorithm

4.2 Kinematics Simulation Verification 4.2.1

Simulation Verification of FBCCD Algorithm with Fixed Base

The linear trajectory tracking of continuum robot is shown in Fig. 9 and the configuration parameters of the continuum robot change depending on the sampling points are shown in Fig. 10. Additionally to the fluctuation of configuration parameters when the motion direction changes, the continuum robot’s motion is generally steady and free of unpredictable discontinuities. The overall movement of continuum robot is organized and logical. The continuum robot’s joint offset is gradually reduced from the end to the base.

FBCCD: A Forward and Backward Cyclic Iterative Solver …

339

Fig. 8 The relation between the error of the FBCCD algorithm and the iteration numbers (100 runs)

Fig. 9 Time distribution of FBCCD algorithm

The continuum robot’s tracking of arc trajectories is illustrated in Fig. 11 and the configuration parameters of continuum robot vary with sampling points is shown in Fig. 12. The continuum robot’s entire movement is relatively stable and a more natural trajectory can be generated in the process of circular motion. Due to continuous adjustment of the continuum robot’s final posture, the variation of configuration parameters at the end is obviously greater than that at the base.

340

H. Wu et al.

Fig. 10 Continuum robot tracking its linear trajectory continuously

Fig. 11 The continuum robot’s configuration parameters change depending on the sampling points (point to point trajectory tracking)

FBCCD: A Forward and Backward Cyclic Iterative Solver …

341

Fig. 12 Tracking a continuum robot’s round trajectory

4.2.2

Simulation Verification of FBCCD Algorithm with Movable Base

The tip trajectory tracking of the continuum robot with a movable base is illustrated in Fig. 13 and the continuum robot’s configuration parameters vary based on the sampling points is illustrated in Fig. 14. As illustrated in Fig. 15, the continuum robot’s general movement is well-coordinated and realistic.

4.3 Comparison of Different IK Methods As an iterative heuristic algorithm, a set of visually natural poses can be solved from infinite solutions of continuum robots. In order to verify the effectiveness and precision of the FBCCD algorithm, the Jacobian and FABRIK algorithms have been compared to the FBCCD algorithm. The experimental results are shown in Table 1. Compared with the Jacobian method and FABRIK method, FBCCD has a greater convergence rate and higher calculation efficiency. Since each iteration calculates the pseudo-inverse of the Jacobian matrix, the computational efficiency of the Jacobian method is relatively low. At the same time, the Jacobian method converges very slowly and there may be non-convergence results and singularities poses. Gradient descent is applied in each iteration of the FBCCD method, while each iteration of the FABRIK method only requires a simpler linear operation, so the FABRIK has higher computational efficiency in a single iteration. FBCCD ensures a faster convergence rate through forward and backward iteration progress.

342

H. Wu et al.

Fig. 13 The continuum robot’s configuration parameters vary with sampling points (arc trajectory tracking)

Fig. 14 Trajectory tracking simulation of continuum robot

5 Conclusion In this paper, an improved Cyclic Coordinate Descent algorithm is proposed for solving the inverse kinematics of the multi-segment continuum robot. Based on the Piecewise Constant Curvature model, the kinematics model of continuum robot is constructed. Inspired by the FABRIK algorithm, the iteration phase of FBCCD is separated into two stages: a forward iteration of the end coordinate and a backward iteration of the end pose. Forward and backward iterations correct and compensate each other, resulting in the end pose close to the target. Finally, the FBCCD

FBCCD: A Forward and Backward Cyclic Iterative Solver …

343

Fig. 15 The continuum robot’s configuration parameters change depending on the sampling points (trajectory tracking with movable base)

Table 1 Average results (100 runs) for solving the IK using various techniques

FBCCD

FABRIK

Jacobian

Number of iterations

22.89

52.13

493.04

MATLAB exe. time/(s)

0.361

0.4510

83.6689

Interactions per second

63.41

115.6

6.3

Time (per interaction)/(ms)

15.77

8.65

169.71

algorithm’s effectiveness and reliability are thoroughly tested through the trajectory tracking experiment and numerical comparison. The FBCCD algorithm has strong robustness and extensive application due to its simplicity and high efficiency. Thus, it can be applied in the real-time control of the multi-segment continuum robot. However, the algorithm does not perform well when the continuum robot has fewer segments. And further research can be conducted in the future to combine simulated annealing algorithm to avoid local optimality. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. U1813221), and U1813221 (Grant No. 2019YFB1311200).

References 1. Walker ID (2013) Continuous backbone “continuum” robot manipulators. Int Sch Res Notices 2013 2. Webster RJ III, Jones BA (2013) Design and kinematic modeling of constant curvature continuum robots: a review. Int J Robot Res 29(13):1661–1683

344

H. Wu et al.

3. Ding H, Yang X, Zheng N, Li M, Lai Y, Wu H (2018) Tri-Co Robot: a Chinese robotic research initiative for enhanced robot interaction capabilities. Natl Sci Rev 5(6):799–801 4. Liu XJ, Yu JJ, Wang GB, Lai YN, He BY (2016) Research trend and scientific challenge of robotics. Bull Natl Nat Sci Found China 30(5):426–431 5. Buckingham R, Graham A (2012) Nuclear snake-arm robots. Ind Robot Int J 6. Buckingham R, Chitrakaran V, Conkie R et al (2007) Snake-arm robots: a new approach to aircraft assembly. SAE technical paper 7. Chikhaoui MT, Burgner-Kahrs J (2018) Control of continuum robots for medical applications: state of the art. In: ACTUATOR 2018; 16th international conference on new actuators. VDE, pp 1–11 8. Burgner-Kahrs J, Rucker DC, Choset H (2015) Continuum robots for medical applications: a survey. IEEE Trans Robot 31(6):1261–1280 9. Jones BA, Walker ID (2005) A new approach to Jacobian formulation for a class of multi-section continuum robots. In: Proceedings of the 2005 IEEE international conference on robotics and automation. IEEE, pp 3268–3273 10. Dul˛eba I, Opałka M (2013) A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators. Int J Appl Math Comput Sci 23(2):373–382 11. Xu D, Li E, Liang Z et al (2020) Design and tension modeling of a novel cable-driven rigid snake-like manipulator. J Intell Robot Syst 99(2):211–228 12. Cobos-Guzman S, Palmer D, Axinte D (2017) Kinematic model to control the end-effector of a continuum robot for multi-axis processing. Robotica 35(1):224–240 13. Jiang H, Wang Z, Liu X et al (2017) A two-level approach for solving the inverse kinematics of an extensible soft arm considering viscoelastic behavior. In: 2017 IEEE international conference on robotics and automation (ICRA). IEEE, pp 6127–6133 14. Melingui A, Merzouki R, Mbede JB et al (2014) Neural networks based approach for inverse kinematic modeling of a compact bionic handling assistant trunk. In: 2014 IEEE 23rd international symposium on industrial electronics (ISIE). IEEE, pp 1239–1244 15. Thuruthel TG, Shih B, Laschi C et al (2019) Soft robot perception using embedded soft sensors and recurrent neural networks. Sci Robot 4(26):eaav1488 16. George Thuruthel T, Ansari Y, Falotico E et al (2018) Control strategies for soft robotic manipulators: a survey. Soft Robot 5(2):149–163 17. Sreenivasan S, Goel P, Ghosal A (2010) A real-time algorithm for simulation of flexible objects and hyper-redundant manipulators. Mech Mach Theory 45(3):454–466 18. William RL II, Mayhew JB IV (1997) Obstacle-free control of the hyper-redundant NASA inspection manipulator. In: Proceedings of the fifth national conference on applied mechanics and robotics, pp 12–15 19. Cho CN, Jung H, Son J et al (2016) An intuitive control algorithm for a snake-like natural orifice transluminal endoscopic surgery platform: a preliminary simulation study. Biomed Eng Lett 6(1):39–46 20. Choset H, Henning W (1999) A follow-the-leader approach to serpentine robot motion planning. J Aerosp Eng 12(2):65–73 21. Kang B, Kojcev R, Sinibaldi E (2016) The first interlaced continuum robot, devised to intrinsically follow the leader. PLoS ONE 11(2):e0150278 22. Wang JG, Tang L, Gu GY et al (2018) Tip-following path planning and its performance analysis for hyper-redundant manipulators. J Mech Eng 54(3):18–25 23. Aristidou A, Lasenby J (2011) FABRIK: a fast, iterative solver for the inverse kinematics problem. Graph Models 73(5):243–260 24. Ananthanarayanan H, Ordóñez R (2015) Real-time inverse kinematics of (2n + 1) DOF hyperredundant manipulator arm via a combined numerical and analytical approach. Mech Mach Theory 91:209–226 25. Wang LCT, Chen CC (1991) A combined optimization method for solving the inverse kinematics problems of mechanical manipulators. IEEE Trans Robot Autom 7(4):489–499 26. Mukundan R (2009) A robust inverse kinematics algorithm for animating a joint chain. Int J Comput Appl Technol 34(4):303–308

FBCCD: A Forward and Backward Cyclic Iterative Solver …

345

27. Mahmudi M, Kallmann M (2011) Feature-based locomotion with inverse branch kinematics. In: International conference on motion in games. Springer, Berlin, Heidelberg, pp 39–50 28. Jones BA, Walker ID (2006) Kinematics for multisection continuum robots. IEEE Trans Robot 22(1):43–55 29. Mahl T, Hildebrandt A, Sawodny O (2014) A variable curvature continuum kinematics for kinematic control of the bionic handling assistant. IEEE Trans Robot 30(4):935–949 30. Jones BA, Walker ID (2006) Practical kinematics for real-time implementation of continuum robots. IEEE Trans Robot 22(6):1087–1099

Optimization of Bearing Capacity Parameters of Fully Decoupled Two-Rotation Parallel Mechanism for Vehicle Durability Testing Sen Wang, Xueyan Han, Xingzhen Su, Haoran Li, Yanxia Shan, and Shihua Li Abstract The problem of low reproduction accuracy of road spectrum of wheelcoupled vehicle durability test bench needs to be solved. From the perspective of mechanism innovation, a fully decoupled two-rotation parallel mechanism with large load-bearing capacity for vehicle durability testing is proposed in this paper. This kind of parallel mechanism can solve the problem of low-bearing capacity of fully decoupled parallel mechanisms. Based on the requirement of reproduction accuracy of real road spectrum, the degree of freedom required by the mechanism is analyzed. Due to the requirements of fully decoupled and large load-bearing, the configuration design of the parallel mechanism is carried out. In order to further improve the bearing capacity of the mechanism, based on the kinematic analysis results of the mechanism, the size parameters of the mechanism are optimized by using the graph method. The research results show that the bearing capacity of the mechanism is increased by 50% after adding the closed-loop unit and optimizing the size. The finite element simulation verifies that the load-bearing capacity of the optimized fully decoupled two-rotation parallel mechanism satisfies the design index requirements. This research lays a theoretical foundation for the application of fully decoupled parallel mechanisms with large bearing capacity. Keywords Wheel-coupled vehicle durability test bench · Two-rotation · Parallel mechanism · Fully decoupled · Large load-bearing capacity · Optimization

S. Wang · X. Han · H. Li · Y. Shan · S. Li (B) School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China e-mail: [email protected] Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China X. Su CITIC Dicastal Co. Ltd., Qinhuangdao 066004, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_19

347

348

S. Wang et al.

1 Introduction The vehicle durability test bench is the main equipment used for indoor durability testing, and the realization of high-precision road spectrum reproduction is a research hotspot around the world. Although China is a big vehicle manufacturing country, we are still not able to break through the technical bottleneck of high-precision road spectrum reproduction in original vehicle durability test benches. “Made in China 2025” states clearly that the innovative design of high-end mechanical equipment is the key direction of national strategic development. The development of a highprecision vehicle fatigue durability test bench with independent intellectual property rights is a key technology that is extremely challenging and needs to be solved urgently. Therefore, it is of great significance to develop a high-precision vehicle durability test bench. Lots of scholars and vehicle companies have carried out researches on vehicle durability test benches. In 1962, the American MTS company developed the world’s first wheel-coupled road simulation test bench [1]. American Moog company developed a four-column road simulation test bench for chassis and suspension testing [2]. The British Salvo Company has developed 4-channel and 6-channel road simulation system which mainly used to test body and chassis of the vehicle [3]. Instron [4] developed a road simulation test system for light or heavy vehicles, which was used for vehicle fatigue durability test, comfort test, NVH and other common vehicle tests. Gao [5] pointed out that 6-channel and 8-channel wheel-coupling road simulation test systems have been successfully developed in China. Gong and Yao [6] introduced the principle of the bench road simulation test and summarized the key technologies that affecting the bench test. Xiang et al. [7] put the road load spectrum as the driving signal into the wheel-coupled vehicle durability test bench to proceed accurately reproduce of the road spectrum. Ju et al. [8] carried out the fatigue durability test of vehicle cabs by building a CAE simulation model of the vehicle test bench and combining the load spectrum signals collected from the actual vehicle road test. The successful development of the above-mentioned wheel-coupling road simulation test bench has greatly shortened the time for high-precision wheel-coupling durability testing and improved the life of the vehicle. So far, most of the vehicle durability test benches have adopted a four-column mechanical structure. They can only provide the road spectrum in the vertical direction, and can highest restore 70% of the road conditions. Moreover, the real road spectrum in pitch and roll directions is ignored that resulted the low road spectrum reproduction accuracy. Due to the unique advantages of compact structure, strong bearing capacity, small cumulative error and high precision, parallel mechanisms can be used to solve the above problems [9]. The control difficulty will increase because of the strong coupled characteristic of parallel mechanism and the complexity of the road spectrum. Existing decoupled mechanism cannot meet the requirements of the road spectrum reproduction accuracy. Therefore, the decoupled parallel mechanism can be used to solve this problem. At present, many scholars have done a lot of work on the design and analysis of decoupled parallel mechanisms [10–13]. However, the existing fully decoupled parallel

Optimization of Bearing Capacity Parameters of Fully Decoupled …

349

mechanisms have poor bearing capacity and cannot be better applied to vehicle durability tests. In this paper, a fully decoupled two-rotation parallel mechanism with large bearing capacity for vehicle durability test is proposed according to the demand of highprecision road spectrum reproduction of vehicles. It can effectively solve the problem of weak bearing capacity of the existing decoupled mechanism. The decoupled characteristics, degrees of freedom, inverse kinematics and velocity Jacobian of the mechanism are analyzed. The bearing capacity of the decoupled mechanism is calculated based on the velocity Jacobian matrix. The influence of the posture change of the moving platform on the bearing capacity is obtained. Based on the high-precision reproduction requirements of the road spectrum, the motion/force transmission performance of this mechanism is analyzed. The optimal design of its structural parameters is carried out according to the mechanism performance map, then the bearing capacity of the decoupled mechanism is optimized. The research fills the gap of the large load-bearing fully decoupled parallel mechanism configuration. It provides a theoretical foundation for the application of the high-precision vehicle durability test bench mechanism in the field of vehicle testing.

2 Design of Fully Decoupled Two-Rotation Parallel Mechanism 2.1 Mechanism Design Indexes The technical index of the vehicle durability test bench is determined based on the actual operating conditions of the vehicle. The specific technical indexes of the vehicle durability test bench are shown in Tab 1. This paper conducts relevant research on the vehicle durability test bench based on these technical indexes. Table 1 Technical indexes of vehicle durability test bench

Technical parameter

Technical indicators

X-direction corner range

± 30°

Y-direction corner range

± 30°

Bearing capacity

3T

Maximum speed

0.2 m/s

Maximum angular velocity

10°/s

350

S. Wang et al.

2.2 Mechanism Configuration Design In this paper, a new type of fully decoupled 2UPR/2RPU/RR parallel mechanism with large bearing capacity is proposed. It is mainly composed of a moving platform, a fixed platform and five branch chains connecting the two platforms, as shown in Fig 1. The mechanism provides two degrees of freedom of pitch and roll for each wheel in the vehicle, reproduces the road spectrum accurately under different road conditions. The road spectrum of the four wheels is controlled independently. The specific construction process of the mechanism as shown in Fig 1. First, the construction of just-constrained branch chain and the basic decoupled branch chain are proposed according to the fully decoupled criterion of two-rotation parallel mechanism with large bearing capacity [14]. The branch chain 1 is a UPR branch structure. The branch chain 2 is an RPU branch structure. The just-constrained branch chain is an RR branch structure. The X and Y axes of the kinematic pairs of the just-constrained branch chain are the output motion axes of the mechanism. The branch chain 1 controls the rotational freedom of the moving platform around the X axis. The branch chain 2 controls the rotational freedom of the moving platform around the Y axis. The revolute pairs which directly connected to the fixed platform of the branch chain 1 and the just-constrained RR branch are coaxial. The revolute pairs which directly connected to the fixed platform of the branch chain 2 and the just-constrained RR branch are coaxial, as shown in Fig 2. In order to improve the carrying capacity of the mechanism, the redundant branch chain 3 and the redundant branch chain 4 of the loop are constructed, as shown in Fig 3. Branch 1 and branch 3 have the same UPR branch structure, Branch 2 and branch 4 have the same RPU branch structure. The revolute pairs which directly connected to the fixed platform of the branch chain 3 and the just-constrained RR branch are coaxial. The revolute pairs which directly connected to the fixed platform of the branch chain 4 and the just-constrained RR branch are coaxial. Branch 1 and Branch 3 constitute closed loop 1. Branch 2 and branch 4 constitute closed loop 2. Each closed loop is composed of the same structure basic decoupled branch and redundant driving branch. The closed loop does not change the decoupled characteristics of the mechanism, but increases the load-bearing capacity of the mechanism. Fig. 1 2UPR/2RPU/RR parallel mechanism with large bearing capacity

Optimization of Bearing Capacity Parameters of Fully Decoupled … Fig. 2 UPR/RPU/RR parallel mechanism

351

B2

z1 y1 x1 o1

branch2

A2

Moving platform

B1

z y

Fixed platform

o x

branch1 A1

Fig. 3 2UPR/2RPU/RR parallel mechanism

B2

$T2

$24 $23

Branch2 $22 $T1

Branch1 Closed loop1

o x $11

A1

Branch4

z y

$13

Moving platform

A3

B1

$21

Branch3 B4

$14 A2

B3

z1 y1 x1 o1

Fixed platform Closed loop2 A4

$12

The drive displacement of redundant drive of Branch 3 and branch 1 are equal in magnitude and opposite in direction. The drive displacement of redundant drive of Branch 4 and branch 2 are equal in magnitude and opposite in direction. The output motions of the two loops do not interfere with each other, and satisfy two degree of freedom rotation decoupled requirements.

2.3 Analysis of Mechanism Decoupled Characteristics Liu et al. [15] proposed that the force-transferring screw means that when the input joint is locked, the reciprocal product of all other kinematic spinor except the driving screw in the branch is zero. Such a screw represents the generalized transfer force in the branch chain that transfers the motion/force from the input joint to the moving platform. Therefore, the decoupled characteristics of the parallel mechanism are defined based on the transfer force screw theory as follows: the transfer force screw

352

S. Wang et al.

of the branch can only do work on the output degrees of freedom controlled by the branch. In this paper, the force transmission screw $T 1 of the branch chain 1 of the two-rotation parallel mechanism is always coplanar with the y-axis and different from the x-axis. Therefore, it can only output the rotational degree of freedom in the x-axis direction. The force transmission screw $T 2 of the branch chain 2 is always coplanar with the x-axis and different from the y-axis. Therefore, the rotational degrees of freedom in the y-axis direction can be output. It proves that the two rotational degrees of freedom of the mechanism are fully decoupled.

3 Kinematic Analysis of Fully Decoupled Two-Rotation Parallel Mechanism 3.1 Analysis of Degree of Freedom of Mechanism As shown in Fig 3, a fixed coordinate system o-xyz is established. Ai (i = 1, 2, 3, 4) represent the center points of the kinematic pairs connected to the fixed platform from branch chains 1 to 4 of the mechanism, respectively. Bi (i = 1, 2, 3, 4) represent the center points of the kinematic pairs connected to the moving platform from mechanism branch 1 to branch 4, respectively. The center point of the fixed platform is the original point o of the coordinate system. oA4 is the x-axis direction, oA3 is the y-axis direction, and the z-axis is determined by the right-hand spiral rule. The moving coordinate system o1 − x1 y1 z 1 is established. The center point of the moving platform is the original point o1 of the coordinate system. o1 B4 is the direction of the x 1 axis, o1 B3 is the direction of the y1 axis, and the z1 axis is determined by the right-hand rule. The two UPR branches are arranged symmetrically about the x-axis of the fixed coordinate system. The two RPU branches are arranged symmetrically about the y-axis of the fixed coordinate system. It can be known from the screw theory that the mechanism has a total of twelve constraint screws. Among them, four screws are linearly independent, so the mechanism has eight redundant constraints, namely v = 8. The mechanism has no passive degrees of freedom, ς = 0. The number of mechanism components is 11, the number of kinematic pairs is 14, and the total number of degrees of freedom of the mechanism’s kinematic pairs is 18. Applying the modified G-K formula, the degrees of freedom of the mechanism is F = d(n − g − 1) +

g 

fi + v − ς

i=1

= 6(11 − 14 − 1) + 18 + 8 − 0 = 2

(1)

To sum up, the two-rotation over-constrained 2UPR-RR-2RPU parallel mechanism has two degrees of freedom, namely, rotational degrees of freedom around the

Optimization of Bearing Capacity Parameters of Fully Decoupled …

353

x-axis and y-axis. The number of mechanism driving pairs is 4, so it is a redundant parallel mechanism.

3.2 Inverse Solution of the Mechanism The posture of the moving platform can be represented by the intermediate constrained branch. First, the fixed coordinate system o rotates the angle α around the y-axis. Then the new coordinate system rotates the angle β around the x-axis to obtain the moving coordinate system o1 . The rotation transformation matrix can be expressed as ⎡

⎤ cos α sin α sin β sin α cos β O ⎣ 0 cos β − sin β ⎦ O1 R = − sin α cos α sin β cos α cos β

(2)

The origin of the moving platform coordinate system in the fixed coordinate system is expressed as O

 T Po1 = −d sin α 0 d cos α

(3)

The midpoint of each kinematic pair on the fixed platform is expressed in the fixed coordinate system as  T PA1 = 0 −a1 0 T  o PA2 = −a1 0 0 T  o PA3 = 0 a1 0 T  o PA4 = a1 0 0

o

(4)

The midpoint of each kinematic pair on the moving platform is expressed in the moving coordinate system as T  PB1 = 0 −b1 0 T  o1 PB2 = −b1 0 0 T  o1 PB3 = 0 b1 0 T  o1 PB4 = b1 0 0

o1

(5)

The midpoint of each kinematic pair in the moving coordinate system is expressed in the fixed coordinate system as

354

S. Wang et al. o

PBi =

O O1 PBi O1 R

+ O Po1

(6)

Direction vector of each branch, which along the driving pair direction is Ai Bi = o PBi − o PAi

(7)

The inverse solution of the mechanism means that the length of each drive branch is li =

√ |Ai Bi |2 (i = 1, 2, 3, 4)

(8)

3.3 Solution of Mechanism Velocity Jacobian Matrix Taking a derivation to the inverse solution of each branch, the speed relationship  T between the input variable l˙i and the output variable θ˙1 θ˙2 can be obtained. ⎡˙ ⎤ ⎡ l1 J11 ⎢ l˙2 ⎥ ⎢ J21 ⎢ ⎥=⎢ ⎣ l˙3 ⎦ ⎣ J31 J41 l˙4

⎤ J12 ˙ J22 ⎥ ⎥ θ1 J32 ⎦ θ˙2

(9)

J42

Among them, J 11 , J 12 , J 21 , J 22 , J 31 , J 32 , J 41 and J 42 are the expressions about the posture angle of the moving platform and the size parameters of the mechanism.

4 Analysis of the Force Transmission Performance of the Mechanism The motion/force transmission performance of the mechanism describes the external load bearing capacity of the driving force transmitted to the moving platform by each branch. It also describes the motion output ability of the moving platform under the input of a certain branch drive pair. In order to meet the requirements of high-precision reproduction of the road spectrum and bearing capacity, it is necessary to analyze the motion/force transmission performance of the mechanism. Liu et al. [16] proposed three motion/force transmission performance indexes which are input transmission index (ITI), output transmission index (OTI) and local transmission index (LTI). They can describe the efficiency of the transmission force screw from the drive joint to transmit motion/force to the end effector. These indexes have the advantage of being dimensionless and irrelevant to the selection of the origin of the coordinate

Optimization of Bearing Capacity Parameters of Fully Decoupled …

355

system. Therefore, these indexes are used to analyze the performance of motion/force transmission. In order to integrally evaluate the motion and force transmission performance of the input and output ends of the mechanism, the input and output transmission indexes of the mechanism are defined as     $T i ◦ $ I i   γ I = min  (10) $ T i ◦ $ I i  max

    $T i ◦ $ Oi   γ O = min  $T i ◦ $ Oi 

(11)

max

$T i represents the transmission force screw of the ith branch, $ I i represents the input motion screw of the ith branch, $ Oi represents the output motion screw of the ith branch, γ I represents the input transmission index of the ith branch, γ O represents the output delivery index of the ith branch. The larger the γ I is, the higher efficiency of the input motion transmitted by the ith drive joint of the mechanism is. The larger γ O is, the higher transmission efficiency of the ith transmission force screw to the moving platform is. In other words, the stronger the ability of the mechanism to balance the external force in the direction of the output motion axis is, the greater the bearing capacity is. The local transmission index is used to evaluate the overall motion/force transmission performance of the mechanism. The definition of the local transmission index is as follows γ = min{γ I , γ O }

(12)

Since the value of the input and output transmission index is related to the position and posture of the mechanism. Namely, the input and output transmission index of the mechanism is different under different positions and postures. Therefore, the local transmission index is used to describe the motion/force transmission performance of the mechanism. The above performance indexes can not evaluate the motion/force transmission performance of the redundant mechanism accurately. Therefore, the local minimized transmission index (LMTI) proposed in the paper [17] is used to analyze the motion/force transmission performance of the redundant drive mechanism. When the motion/force transmission performance of the redundant drive parallel mechanism is analyzed, the input drives of r redundant drives should be removed first. Namely, make the input drives of the r redundant drives as passive kinematic pairs, then (r is the number of redundant drives, n is the number of drive branches) q = Cnr non-redundant parallel mechanisms can be obtained. According to the definition of the local transmission index in Eq3, the LTI value of each mechanism can be obtained under any given terminal posture. There is a mechanism in q nonredundant parallel mechanisms whose motion/force transmission performance is

356

S. Wang et al.

better than other mechanisms. The LTI value of this mechanism can be taken as the local minimized transmission index of the redundant mechanism. The expression for the local minimized transmission index is M = max{γi } (i = 1, 2, . . . , q)

(13)

Rotation around the y-axis (°)

Rotation around the y-axis (°)

This index reflects the motion/force transmission performance of the redundant parallel mechanism under the specified position and posture. The larger the value of LMTI is, the higher the motion/force transmission performance of the redundant mechanism is. In order to calculate the motion/force transmission performance of the parallel mechanism, the drives in branch 1 to branch 4 are removed respectively. Then four non-redundant parallel mechanisms can be obtained, denoted as mechanism 1, mechanism 2, mechanism 3 and mechanism 4. The LTI value of each non-redundant parallel mechanism under the rotation angle of the moving platform around the xaxis and y-axis in the interval of (− 30, 30°) is studied. Then, the contour maps of LTI which changes with the posture of the moving platform of each non-redundant parallel mechanism are obtained. The figures are shown in Fig 4a–d.

Rotation around the x-axis (°)

Rotation around the x-axis (°)

c) LTI performance map for mechanism 3

Rotation around the y-axis (°)

Rotation around the y-axis (°)

a) LTI performance map for mechanism 1

Rotation around the x-axis (°)

b) LTI performance map for mechanism 2

Rotation around the x-axis (°)

d) LTI performance map for mechanism 4

Fig. 4 LMTI index performance map of the parallel mechanism

Optimization of Bearing Capacity Parameters of Fully Decoupled …

357

By analyzing Fig 4, it can be obtained that the contour line of the LMTI which changes with the posture of the moving platform of the mechanism is similar to the “lip” shape, and it is symmetrical about the x-axis and the y-axis respectively. The LMTI values are all greater than 0.5, and most of the areas are between 0.85 and 1 when the moving platform of the mechanism is at any angle in the workspace. The LMTI value is close to 0.7 only when the moving platform rotates around the x-axis and the y-axis at about 25°. Therefore the motion/force transmission performance of the parallel mechanism is good when it works within 0–25°.

5 Optimization of Two-Rotation Parallel Mechanism 5.1 Optimization Indicator of Mechanism Carrying Capacity The bearing capacity index of the parallel mechanism reflects the ability of the mechanism to balance external forces. The extreme value of the modulus of the output force or moment vector F is the index of the bearing capacity when the modulo length of the driving force f is 1. The extreme value is also the maximum and minimum singular values of the inverse Jacobian matrix. In order to evaluate the bearing capacity of the mechanism in its working space integrally, the global bearing capacity index is used. The expression of the global bearing capacity index is 

ζmax ζmin

  σmax J −1 dW  = W dW   −1  J dW W σmin  = W dW W

(14)

(15)

  Among them, σmax J −1 is the largest singular value of the inverse Jacobian   matrix of the two-rotation mechanisms. σmin J −1 is the smallest singular value of the inverse Jacobian matrix of the two rotating mechanisms. The larger the bearing capacity index is in the whole area, the stronger bearing capacity of the mechanism is.

5.2 Bearing Capacity Analysis of the Mechanism When the global bearing capacity index is used to evaluate the bearing capacity of the mechanism, the dimension parameters of the mechanism need to be dimensionless. The size parameters of the parallel mechanism are set as follows: the distance between the center of each kinematic pair on the moving platform and the moving platform center is r 1 . The distance between the center of each kinematic pair on the fixed

358

S. Wang et al.

platform and the fixed platform center is r 2 . The initial height of the mechanism moving platform is z. The dimension parameters r 1 , r 2 , z of the mechanism need to be dimensionless, parameter D can be expressed as D=

r1 + r2 + z 3

(16)

The dimensionless parameters a, b, and c of the three parameters r 1 , r 2 and z can be expressed as a=

r1 r2 z b= c= D D D

(17)

Based on the dimensionless parameters, the bearing capacity of the two-rotation parallel mechanism is analyzed and optimized. The motion range of the moving platform is set to (0°, 30°). Then, the bearing capacity index of the moving platform is calculated in each posture, respectively. The surface diagram of the bearing capacity which changes with the posture of the moving platform of the non-redundant mechanism UPR/RPU/RR is obtained, as shown in Fig 5. Fig 5a, b are the surface diagram of the maximum bearing capacity index and the minimum bearing capacity index of the mechanism moving platform, respectively. By analyzing Fig 5a, b, it can be obtained that the maximum and minimum bearing capacities of the mechanism changes with the posture and position of the moving platform. The closer it is to the initial position, the better the maximum and minimum bearing capacity of the mechanism is. Furthermore, the larger the rotation range is, the worse the maximum and minimum bearing capacity of the mechanism is.

5.3 Optimization Design of Mechanism Bearing Capacity 5.3.1

Add Redundant Drives to Optimization the Load-Bearing Capacity of the Mechanism

The non-redundant mechanism UPR/RPU/RR can realize the fully decoupled of the two-rotation degrees of freedom, which meets the requirements of road spectrum reproduction accuracy. However, bearing capacity of the decoupled mechanism is weak. It can not bear the weight of the entire vehicle, which is difficult to apply for vehicle fatigue durability test. On the basis of the non-redundant mechanism, the redundant drives with the same displacement and opposite direction as the basic decoupled branch drives are added. Then the load-bearing capacity index of the moving platform in each posture is calculated respectively. The Surface diagrams of the bearing capacity of the redundant mechanism 2UPR/2RPU/RR are showed in Fig 6a, b.

Optimization of Bearing Capacity Parameters of Fully Decoupled …

359

Maximum carrying capacity

a

Minimum bearing capacity

b

Fig. 5 a Surface diagram of the maximum bearing capacity index of the non-redundant mechanism. b Surface diagram of the minimum bearing capacity index of the non-redundant mechanism

By analyzing Fig 6a, b, it can be obtained that by adding redundant drives, the maximum bearing capacity of the mechanism is increased by 40%, and the minimum bearing capacity is increased by 35%. The above results demonstrate that the maximum and minimum load-bearing capacity of the mechanism can be significantly improved by adding redundant drives.

360

S. Wang et al.

Maximum carrying capacity

a

Minimum bearing capacity

b

Fig. 6 a Surface diagram of the maximum bearing capacity index of the redundant mechanism. b Surface diagram of the minimum bearing capacity index of the redundant mechanism

5.3.2

Mechanism Dimension Parameter Optimization

In order to achieve the correct assembly relationship and motion capability of the redundant mechanism, the dimensionless size parameters a, b, and c of the parallel durability test bench must meet the following conditions

Optimization of Bearing Capacity Parameters of Fully Decoupled …

361

⎧ 0 < a, b, c < 3 ⎪ ⎪ ⎨ a+b+c =3 ⎪ b>a ⎪ ⎩ c > 1.25a

(18)

The parameter design space of the mechanism can be represented as the space triangle ABC shown by the shaded part in Fig 7. In order to convert this space triangle into a plane triangle, let ⎧ ⎪ ⎨ a = s√ s b = 23 t − √ 2 ⎪ ⎩ c = 3 − 23 t −

(19) s 2

Then the parameter design space of the mechanism can be transformed into the triangle ABC as shown in Fig 8. Through the calculation, the performance diagram of the global maximum bearing capacity in the two-rotation parallel mechanism can be obtained as shown in Fig 9. c A 3

Fig. 7 Three dimensional parameter design space of two-rotation mechanisms

B 3 C b

O

a

3

b

Fig. 8 Two dimensional parameter design space of two-rotation mechanisms

3

t c

3

C B

A

3

s(a)

362

S. Wang et al.

The area which ζmax = 1.2 in the performance diagram is defined as the good performance area. In order to obtain a good bearing capacity, the parameters of the good performance area are selected as the optimization parameters of the mechanism, such as s = 0.6, t = 2. The dimensionless parameters of the optimized mechanism can be obtained as a = 0.6, b = 1.43, c = 0.9679. The size parameters of the mechanism are restored according to the dimensionless proportional coefficient. Then the final optimized actual size parameters are obtained that r 1 = 300 mm, r 2 = 715 mm, z = 483.95 mm. The surface diagram of the maximum bearing capacity index of the optimized mechanism is shown in Fig 10. It can be found that the maximum bearing capacity of the mechanism is increased by 10% and the minimum value is increased by 20% compared with that before the size optimization. C

Fig. 9 Optimization diagram of global maximum bearing capacity

3

0.8 1.2

t (mm)

2.5 2

B

1.5 1 0.5

Fig. 10 Distribution diagram of maximum bearing capacity after optimization

Maximum carrying capacity

A 00

0.8

0.1

0.2

0.3

0.4 0.5 s(mm)

0.6

0.7

0.8

0.9

Optimization of Bearing Capacity Parameters of Fully Decoupled …

363

Fig. 11 Finite element stress diagram of parallel mechanism after size optimization

6 Verification of Load-Bearing Capacity of Two-Rotation Parallel Mechanism Import the 3D model with the optimized size into the workbench to verify the bearing capacity of the mechanism. The material of the mechanism is given as structural steel. In the initial positon and posture, the vertical downward one-quarter vehicle weight 7500 N is used as the boundary condition. The stress cloud diagram of the mechanism can be obtained by the simulation as shown in Fig 11. The maximum stress is 8.27 MPa, which is far less than the yield stress of structural steel. The optimized configuration size meets the requirements of the bearing capacity index of the vehicle durability test bench.

7 Mechanism Application Outlook At present, most domestic vehicle durability test benches use four-column mechanical structures. They can only provide vertical freedom of motion and cannot reproduce the real road spectrum. Although the multi-degree-of-freedom parallel mechanism can reproduce the road spectrum more accurately, it can not meet the requirements of high bearing capacity and easy control at the same time. Therefore, it is difficult to be applied to vehicle durability test. The fully decoupled parallel mechanism with large load-bearing capacity studied in this paper can accurately reproduce the wheel pitch, roll and other road spectrum of the vehicle under different operating conditions. At the same time it can reduce the difficulty of control, and has a good application prospect in vehicle durability test.

364

S. Wang et al.

8 Conclusion (1) In this paper, a new two-rotation fully decoupled parallel mechanism with large bearing capacity is proposed. (2) Based on the actual operating conditions of the vehicle, the bearing capacity index of the vehicle durability test bench is determined. The decoupled characteristics of the mechanism are analyzed. The degree of freedom of the mechanism is solved based on the screw theory. The kinematic inverse model and the inverse Jacobian matrix of the mechanism are established. The evaluation index of the bearing capacity is given based on the maximum and minimum singular values of the inverse Jacobian matrix. Based on the high-precision reproduction requirement of the road spectrum, the analysis of the motion/force transmission performance of the mechanism is carried out. (3) By adding redundant drive and size optimization, the bearing capacity of the mechanism is optimized based on the graphical approach. The bearing capacity of the optimized mechanism is increased by 50%. It is verified by finite element simulation that the optimized mechanism satisfies the requirements of bearing capacity of the vehicle durability test bench and justified the correctness of the proposed mechanism. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 51775475), the Key Project of Natural Science Foundation of Hebei Province (Grant No. E2022203077) and the Key Research and Development Program of Hebei Province (Grant No. 202230808010057).

References 1. Wang Y (2018) Research on rectangular control strategy of wheel coupling road simulation. Harbin Institute of Technology (in Chinese) 2. Gao YK, Wang GH, Han JP (2017) The study on fatigue test of cab assembly based on 4channel road simulation bench. In: WCX™ 17: SAE world congress experience, Michigan, 4–6 Apr 2017, pp 1–11 3. Zhang LP, Yang ZD, Qu ZY (2015) Modeling and vibration decoupling control of multi-axial shaking table. In: International industrial informatics and computer engineering conference, Xian, China, 10 Jan 2015, pp 1878–1883 4. Mohajer N, Abdi H, Nelson K, Nahavandi S (2018) Vehicle motion simulators, a key step towards road vehicle dynamics improvement. Veh Syst Dyn 53(8):1–23 5. Gao LS (2016) Analysis of road simulation test system. Autom Pract Technol 9:188–190 6. Gong HB, Yao L (2018) Review on the development of indoor bench road simulation test technology. Chin J Automot Eng 8(1):72–78 (in Chinese) 7. Xiang HR, Xie F, Wang WY (2019) Application of road spectrum in vehicle durability bench test. J Chongqing Univ Technol 33(5):40–46 (in Chinese) 8. Ju DJ, Xiao P, Yu RJ, Zheng GF (2021) Research on boundary load spectrum of cab based on virtual test bed. J Chongqing Univ Technol 35(4):55–61+116 (in Chinese)

Optimization of Bearing Capacity Parameters of Fully Decoupled …

365

9. Li YQ, Zhang Y, Guo Y, Li XR, Zhang LJ (2019) Research on a new method for configuration synthesis of two-transfer and one-transfer redundant drive parallel mechanism. Chin J Mech Eng 55(23):25–37 (in Chinese) 10. Qu S, Li R, Ma C (2021) Type synthesis for lower-mobility decoupled parallel mechanism with redundant constraints. J Mech Sci Technol 35(6):2657–2666 11. Wu KY, Zhang F, Cui GH, Sun J, Zheng MH (2021) Kinematics and force transmission analysis of a decoupled remote center of motion mechanism based on intersecting planes. J Mech Robot 13(2):1–24 12. Zhang WZ, Xu ZP, Ren DQ, Jin WB (2020) Kinematics and performance analysis of partially decoupled parallel mechanism with four degrees of freedom. Mach Tool Hydraul 48(11):21–26 13. Tian CX, Fang YF, Ge QJ (2019) Design and analysis of a partially decoupled generalized parallel mechanism for 3T1R motion. Mech Mach Theory 140:211–232 14. Wang S, Han XY, Li HT, Li HR, Li SH (2021) Research on the comprehensive method of loop decoupled two-rotation parallel mechanism with large load capacity. Chin J Mech Eng 57(21):68–77 (in Chinese) 15. Zhou YH, Xie FG, Liu XJ (2015) Configuration and kinematics optimization design of main drive mechanism of servo punch press. Chin J Mech Eng 51(11):1–7 (in Chinese) 16. Chen X, Xie FG, Liu XJ (2014) Evaluation of the maximum value of motion/force transmission power in parallel mechanism. Chin J Mech Eng 50(3):1–9 (in Chinese) 17. Liu XJ, Xie FG, Wang JS (2018) Basics of parallel robot mechanism. Higher Education Press, Beijing (in Chinese)

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P Xuejian Ma, Yundou Xu, Yu Wang, Fan Yang, Yongsheng Zhao, Jiantao Yao, and Yulin Zhou

Abstract This work proposes a hybrid machining robot RPR/RP + RR + P based on a planar parallel mechanism. Based on the screw theory, the characteristics of the degree of freedom of the hybrid robot are analyzed, and then the inverse and forward kinematics are solved by geometric methods. And size optimization is carried out by taking the workspace and driving forces as the objects. Then the working space of the hybrid robot is calculated by the Monte Carlo method. Finally, the three-dimensional model of the mechanism is established, and through ADAMS simulation, the accuracy of the inverse kinematics solution was verified. Keywords Hybrid machining robot · Inverse kinematics · Forward kinematics · Size optimization · Workspace

X. Ma · Y. Xu (B) · Y. Wang · F. Yang · Y. Zhao (B) · J. Yao Laboratory of Parallel Robot and Mechatronic System of Hebei Province, Yanshan University, Qinhuangdao 066004, China e-mail: [email protected] Y. Zhao e-mail: [email protected] J. Yao e-mail: [email protected] Y. Xu · Y. Zhao · J. Yao Key Laboratory of Advanced Forging and Stamping Technology and Science, Yanshan University, Qinhuangdao 066004, China X. Ma · Y. Xu · Y. Wang · F. Yang · Y. Zhao · J. Yao · Y. Zhou School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_20

367

368

X. Ma et al.

1 Introduction With the development of aerospace, aviation, automotive industries, and so on, the demands for high-efficiency machining of large-scale structural parts with high material removal ratio and complex parts with curved surfaces increase rapidly. At the same time, traditional serial machine tools have insufficient flexibility, large inertia, and parallel machine tools have small working space, which fails to meet the requirements. The new types of machining equipment are urgently needed. Compared with serial robots and parallel robots, hybrid robots combine the advantages of these two robots and have attracted the attention of many researchers and companies. Hybrid robots can be divided into planar hybrid robots (formed by connecting a swing head with a certain degree of freedom (DOF) in serial on a planar parallel mechanism) and spatial hybrid robots (formed by connecting a swing head with a certain DOF in serial on a spatial parallel mechanism). Compared with the spatial hybrid mechanism, the planar hybrid mechanism has the advantages of simple structure, simple kinematics and dynamics models and low manufacturing cost. At present, there are many spatial hybrid robots, such as Tricept hybrid robots [1, 2], Exechon hybrid machining center [3, 4], TriMule [5, 6] and TriVariant [7, 8] hybrid robots as well as other robots [9–14] etc. And there are relatively few planar hybrid mechanisms. Yang and Wang [15] proposed to develop the XNZD755 planar hybrid machine tool and analyzed the dynamics of the machine tool. A redundantly actuated 3-DOF planar PM was developed in [16] for a 4-DOF hybrid machine tool and again in [17] for 5-DOF one. This work will propose a new five-DOF planar hybrid robot RPR/RP + RR + P, which is used to machine complex aluminum alloy parts for new energy vehicles. The remainder of this article is organized as follows: Sect. 2 describes the composition of the hybrid robot, and introduces the DOF of the mechanism. Section 3 analyzes the robot’s inverse and forward kinematic. Section 4 carries out the size optimization by taking the workspace and driving forces as the objects, and analyzes the workspace of the hybrid robot under the given mechanism parameters. Section 5 gives the three-dimensional model of the mechanism, and the verifies the correction of inverse kinematics analysis. Finally, conclusions are drawn in Sect. 6.

2 Description of the Five-Axis Hybrid Robot 2.1 Mechanism Composition The RPR/RP + RR hybrid robot proposed in this article is shown in Fig. 1. The robot consists of three parts: a planar parallel mechanism RPR/RP, a serial BC swing head, and a serial P joint. The moving platform of the parallel mechanism RPR/RP is supported by two limbs A1 a1 and A2 a2 . The limb A1 a1 is connected to the fixed platform and moving platform both through the R joint. The limb A2 a2 is connected

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P Fig. 1 Schematic diagram of the hybrid robot

z4 x4

y4 B

369

The end tip

o4(G) K 1

y3

z3 x3 a1(R12)

a2

C o3

l1

x1 A1(R11)

y1

z1

y2

z2

o2 (A2,R21)

x2

o1 P3 h X

The fixed platform

l2

P2

P1

The moving platform

2

Z

Y

The ground platform

O

to the fixed platform through an R joint, which is connected to the moving platform through a prismatic joint directly. The axes of three R joints are parallel to each other, and the axes of the prismatic are perpendicular to the axes of the R joints. As shown in Fig. 1, the fixed platform and the moving platform of the parallel mechanism are equivalent to two straight lines. The length of the fixed platform is 2a, and the length of the moving platform is 2b. Point G is the intersection point of the BC swing head. The straight-line Go3 is perpendicular to the moving platform, the vertical foot is the o2 point, the length of Go3 is k, and the length between the intersection point of the BC swing head and the tip of the tool is f .

2.2 Coordinate System Establishment In order to establish the position model of the mechanism, the global frame O-XYZ is established at point O on the fixed platform, where the X-axis is parallel to the line A2 A1 , the Z-axis is parallel to the axis of the R joint, and the right-hand system rule determines the Y-axis. The local frame o1 -x 1 y1 z1 is established at o1 on the fixed platform, where the x 1 -axis points from point A2 to A1 , the z1 -axis is collinear with the Z-axis, and the right-hand system rule determines the y1 -axis. The local frame o2 -x 2 y2 z2 is established at point A2 , where the x 2 -axis is collinear with A1 A2 , the z2 -axis is collinear with the axis of the R21 , and the right-hand system rule determines the y2 -axis. The local frame o3 -x 3 y3 z3 is established at the midpoint of a1 a2 , where the x 3 -axis is collinear with a1 a2 , the y3 -axis is collinear with the C

370

X. Ma et al.

axis of the BC swing head, and the right-hand system rule determines the z3 -axis. Moreover, the local frame o4 -x 4 y4 z4 is established at the point G, where the y4 -axis is collinear with the axis of the end handle, and the z4 -axis is collinear with the B axis. The right-hand system rule determines the x 4 -axis, as shown in Fig. 1.

2.3 Degree of Freedom Analysis The hybrid robot is composed of a parallel part and two serial parts. The parallel part mechanism is RPR/RP, composed of limbs A1 a1 and A2 a2 . The schematic diagram of the parallel mechanism RPR/RP is shown in Fig. 2. The DOF of the parallel mechanism can be calculated from the revised G-K formula: K = d(n − g − 1) +

g 

fi + v

(1)

i=1

d n g fi v

The order of the parallel mechanism, d = 6 − λ; Number of parallel mechanism components; Number of joints of parallel mechanism; The number of DOFs of the ith joint; Number of redundant constraints of parallel mechanism. The expression of redundant constraint v is: v =l −k

(2)

where l is the number of remaining constraint wrenches after removing the public constraints of the mechanism, and k is the rank of the remaining constraint wrenches. Fig. 2 Schematic diagram of the parallel mechanism

a1(R12)

y3

z3 x3

a2

o3 P2 y2

P1

y1 z1 x1 o1

A1(R11)

z2 x2

A2(R21) o2

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

371

It can be obtained that l is 3, k is 3, the DOF of the RPR/RP mechanism can be obtained as: K = d(n − g − 1) +

g 

fi + v

i=1

= 3 × (5 − 5 − 1) + 5 =2

(3)

The DOF of the parallel mechanism is 2, including the translation along the rod A2 a2 and the rotation around the axis of the R joint at A2 . The mechanism is a parallel mechanism in serial with a serial two-DOF BC swing head and a serial one-DOF P joint, so the hybrid robot is a five-DOF hybrid robot.

3 Inverse and Forward Kinematics Analysis 3.1 Inverse Kinematics of Parallel Mechanism In the inverse solution, it is considered that the coordinates of the tool tip point F = [Fx , Fy , Fz ]T and the unit direction vector S of the end tool rod are known, and the purpose is to calculate the input of each actuator, i.e., h, l1 , l 2 , α, β, which are the inputs of P3 , P1 , P2 , C, B respectively. Because the tool tip point coordinate F and the unit direction vector S are known, the coordinates can be easily obtained from the established coordinate system: G x = Fx − f · [1, 0, 0] · S G y = Fy − f · [0, 1, 0] · S G z = Fz − f · [0, 0, 1] · S

(4)

The expression of point G in the global frame O-XYZ and local frame o2 -x 2 y2 z2 can be obtained: T   T G = Gx , G y, Gz G 2 = G x + a, G y , 0

(5)

Then the input of P3 is Gz , that is: h = Gz . As shown in Fig. 1, the angle θ1 can be obtained by the coordinate relationship of point G in local frame o1 -x 1 y1 z1 : Gx + a |G 2 | Gz cos θ1 = |G 2 |

sin θ1 =

(6)

372

X. Ma et al.

where θ1 is the angle of ∠A2 Go1 , l G is the length of GA2 . As shown in Fig. 1, in the trapezoid G K a2 o2 , ∠1 is known, o3 a2 = b, Go3 = k, then through geometric relations, the length of G K and K a2 can be calculated. In the triangle G K A2 , according to the law of sine sin θ2 =

G K sin(∠1) |G 2 |

(7)

where θ2 is angle of ∠KA2 G. And the length of l2 can be calculated based on sine low: l2 =

G K sin(π − ∠1 − θ2 ) − K a2 sin(θ2 )

(8)

After obtaining θ1 and θ2 , the rotation angle θ3 of the mechanism around the axis of R21 can be obtained. Then the point of 3 ai (i = 1, 2) can be expressed in the local frame o1 -x 1 y1 z1 . 1

ai = R(θ3 )3 ai + 13 P

(9)

where R(θ3 ) is the transformation matrix of local frame o3 -x 3 y3 z3 respect to local frame o1 -x 1 y1 z1 , and 13 P is origin of local frame o3 -x 3 y3 z3 respect to local frame o1 -x 1 y1 z1 , which can be gotten through the closed-loop vector. The length of two limbs of the parallel part can be obtained: li = ∥ai1 − Ai ∥ (i = 1, 2)

(10)

Substituting the coordinates of points a11 , a21 , A1 , and A2 , the length of two limbs can be calculated.

3.2 Solving the Rotational Angles of the Serial Part The unit direction vector of the hilt is S, S = (m, n, q)T , and set two rotating angles of the BC swing head in the local frame O-XYZ: α ' is the rotating angle around the x3 axis, and β ' is the rotating angle around the z4 axis. Then the angle α ' and angle β ' are: α ' = arctan

n

m β ' = arccos(q)

(11)

Then through transformation matrix R(θ3 ), the input angle α of head C and the input angle β of head B can be obtained.

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

373

3.3 Forward Kinematics Analysis In the forward solution, the inputs of h, l1 , l 2 , α and β are known. The purpose is to T  solve the coordinate F = Fx , Fy , Fz of the tool tip point and the unit direction vector S of the hilt. Because the length l2 is known, the length of l G can be obtained based on cosine law: √ l G = K G 2 + K A22 − 2 cos(∠1) · K G · K A2 (12) As shown in Fig. 2, in the quadrilateral A1 A2 a1 a2 , the length of four sides and ∠2 are known. According to the law of cosines, the length a1 A1 and the angle of ∠a1 A2 a2 and ∠a1 A2 A1 can be calculated. According to formula (7), the angle θ2 can be obtained. Then the angle of ∠A1 A2 G and θ3 can be obtained by Eq. (13). ∠A1 A2 G = ∠a1 A2 a2 + ∠a1 A2 A1 − θ2 θ3 = ∠a1 A2 a2 + ∠a1 A2 A1 + ∠1 − π

(13)

The coordinate of G in the frame O-XYZ can be expressed: G = [l G cos(∠A1 A2 G) − a, l G sin(∠A1 A2 G), h]

(14)

And the input angles α and β are known, the unit direction vector S can be obtained by the rotation transformation matrix R(θ3 ). Then the coordinates of the tool tip point F can be obtained. 

Fx = G x + f · [1, 0, 0] · S



Fy = G y + f · [0, 1, 0] · S

(15)



Fz = G z + f · [0, 0, 1] · S

4 Dimensional Optimization and Workspace Analysis Taking the working space and driving force as optimization objective, the robot’s key dimensions are optimized. The parallel mechanism part is optimized as the main optimization object of the hybrid robot. Firstly, the parallel mechanism is determined based on the size of the BC swing head, and the size of its moving platform is 420 mm. Secondly, based on the limit of the drive joint, the change range of the PM’s limb is determined to be [390 mm, 760 mm], and the optimization range of the fixed

374

X. Ma et al. Institution parameter initialization 2b=420,2a=640 Given a set of discrete coordinate values (x,y) Solving for the corresponding rod length No 2a

2a

Whether the requirements of branch rod travel constraints are met Yes, i=i+1 Calculate the driving force of each branch rod and save it

x

x

y

y

No Whether the coordinate cycle ends No

Whether the platform parameter cycle ends

Compare the force of the branch rod and the size of the working space i under the parameters of each platform, and select the corresponding optimal configuration

Fig. 3 Flow chart of platform parameter scaling optimization

platform’s size is [640 mm, 840 mm]. Under the action of external force F = [10 N, 10 N, 10 N, 10 N mm, 10 N mm, 10 N mm]T , finding the optimal size of the fixed platform through optimization. The flow chart of the optimization procedure is shown in Fig. 3. Table 1 lists the force and workspace indexes with different sizes of the fixed platform. Finally, through comparison, it is determined that the fixed platform size of the parallel mechanism is 740 mm. Therefore, the key dimensions of the hybrid robot are shown in Table 2. According to the dimensions given in Table 2, the Monte Carlo method is used to analyze the new workspace of the hybrid robot, and its dexterity space is shown in Fig. 4. Through observation, the robot can translate in a large range in the Z direction, which is convenient for machining narrow and long parts. At the same time, due to the existence of the BC swing head, the flexibility of the robot end tool in the workspace can be ensured, which is convenient for machining various complex curved surfaces, holes, etc. Therefore, it can meet the machining workspace requirements of aluminum alloy parts with complex curved surfaces for new energy vehicles.

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

375

Table 1 The force and workspace indexes with different sizes of the fixed platform Fixed platform size (mm)

The average value of limb 1 driving force (N)

The average value of limb 2 driving force (N)

Average value of the difference in driving force between limbs 1 and 2 (N)

Driving force fluctuation coefficient of limb 1

Driving force fluctuation coefficient of limb 2

Driving Workspace force (points) difference fluctuation coefficient of limbs 1 and 2

640

19.2932

30.6320

11.3388

1.5627

1.9477

0.4247

142,618

660

18.3576

29.6661

11.3085

1.2791

1.7073

0.4689

139,208

680

17.4987

28.7601

11.2613

1.0499

1.5217

0.5154

136,055

700

16.7077

27.9063

11.1986

0.8619

1.3777

0.5647

133,149

720

15.9776

27.0992

11.1216

0.7079

1.2676

0.6172

130,466

740

15.3024

26.3337

11.0313

0.5833

1.1846

0.6734

127,986

760

14.6774

25.6062

10.9288

0.4860

1.1235

0.7329

125,710

780

14.0981

24.9141

10.8160

0.4167

1.0784

0.7933

123,518

800

13.5609

24.2544

10.6936

0.3762

1.0464

0.8551

121,453

820

13.0622

23.6249

10.5627

0.3639

1.0244

0.9174

119,482

840

12.5988

23.0233

10.4244

0.3750

1.0095

0.9792

117,553

Table 2 The key dimensions of hybrid robot RPR/RP + RR

Symbol

Value

Symbol

Value

a

370 mm

l 1max , l 2max

760 mm

b

210 mm

l 1min , l 2min

390 mm

k

424.5 mm

hmin , hmax

[0, 2000] mm

f

200 mm

α

[− 180°, 180°]

∠1

114.567°

β

[− 90°, 90°]

5 Simulation Verification of Inverse Kinematic Analysis Based on the obtained dimensions of the hybrid robot RPR/RP + RR + P shown in Table 2. As shown in Fig. 5, the three-dimensional model of the hybrid robot RPR/RP + RR + P is designed. In ADAMS, a motion trajectory is given for the tool tip of the hybrid robot, as shown in Eq. (16).

376

X. Ma et al.

Fig. 4 The workspace of the five-axis hybrid robot RPR/RP + RR + P

Fig. 5 Three-dimensional model of the hybrid robot

⎧ ⎪ X = 20t (mm) t ⎪ ⎪ ⎪ ⎪ Y = 10(t − 5) (mm) t ⎨ Z = 40t (mm) t ⎪ ⎪ ⎪ α ' = π t/45 (◦ ) t ⎪ ⎪ ⎩ β ' = π t/90 (◦ ) t

∈ [1, 5] ∈ [5, 10] ∈ [1, 5] ∈ [1, 5] ∈ [1, 5]

(16)

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

377

Fig. 6 The coordinates variation curve of end tip measured in the simulation model

After simulation, the lengths l1 , l 2 , h, and angles α, β of the BC swing head can be measured. Then substituting the inputs into forward kinematics model, the coordinate curve of the tool tip can also be obtained again, as shown in Fig. 6. It can be found that the results are consistent with the values given by Eq. (16), indicating the kinematics analysis is correct.

6 Conclusions and Future Works A new type of five-degree-of-freedom hybrid robot RPR/RP + RR + P is proposed for machining of aluminum alloy plates for new energy vehicles. The modified hybrid robot includes a two-degree-of-freedom parallel mechanism and two series parts, which are the translation along the limb A2 a2 and the rotation around the axis of the R joint. Through the screw theory, the degree of freedom of the robot is analyzed, and the result shows that the robot has five DOFs, and can achieve five-axis hybrid processing. Through the geometric relationship, the forward and inverse position solutions of the hybrid robot are analyzed. Compared with the general hybrid robot, the forward and inverse position model are simple, thus the hybrid robot is easy to control. Taking the limb driving forces and working space as the optimization objective, the dimensions of the mechanism is optimized, and the fixed platform size is determined

378

X. Ma et al.

to be 370 mm. After that, the working space of the robot is obtained by the Monte Carlo method, which is a cylindrical body with an approximate triangular section. Finally, the three-dimensional model of the mechanism is also constructed through Solidworks software, and the forward and inverse kinematic models of the hybrid robot are verified by using the software ADAMS and MATLAB. The force and stiffness analysis of the five-axis hybrid machining robot will be carried out in future work. And the prototype will be developed, the prototype experiments will be carried out. Acknowledgements Supported by the National Natural Science Foundation of China (Grant No. 51875495), and the National Natural Science Foundation of China (Grant No. U2037202), and Hebei Science and Technology Project (Grant No. 206Z1805G).

References 1. Caccavale F, Siciliano B, Villani L (2003) The Tricept robot: dynamics and impedance control. IEEE/ASME Trans Mechatron 8(2):263–268 2. Neumann KE (2004) Next generation Tricept—a true revolution in parallel kinematics. In: Proceedings of the 4th Chemnitz parallel kinematics seminar, pp 591–594 3. Zhao YQ, Jin Y, Zhang J (2016) Kinetostatic modeling and analysis of an Exechon parallel kinematic machine (PKM) module. Chin J Mech Eng 29(1):33–44 4. Jin Y, Bi ZM, Liu HT et al (2015) Kinematic analysis and dimensional synthesis of Exechon parallel kinematic machine for large volume machining. J Mech Robot 7(4):041004 5. Huang T, Dong C, Liu H et al (2019) A simple and visually orientated approach for type synthesis of over-constrained 1T2R parallel mechanisms. Robotica 37(7):1161–1173 6. Huang T, Cheng LD, Liu H et al (2018) Five-degree-of-freedom hybrid robot with rotational supports. U.S. Patent, 17 Apr 2018 7. Li M, Huang T, Chetwynd DG et al (2006) Forward position analysis of the 3DOF module of the TriVariant: a 5DOF reconfigurable hybrid robot. J Mech Des 128(1):319–322 8. Wang YY, Huang T, Zhao XM et al (2007) Finite element analysis and comparison of two hybrid robots—the Tricept and the TriVariant. In: IEEE/RSJ international conference on intelligent robots and systems, pp 490–495 9. Liu XJ, Wang LP, Xie F, Bonev IA (2010) Design of a three-axis articulated tool head with parallel kinematics achieving desired motion/force transmission characteristics. ASME J Manuf Sci Eng 132(2):021009 10. Xie FG, Liu XJ, Wang JS (2012) A 3-DOF parallel manufacturing module and its kinematic optimization. Robot Comput Integr Manuf 28:334–343 11. Li QC, Hervé JM (2014) Type synthesis of 3-DOF RPR-equivalent parallel mechanisms. IEEE Trans Robot 30(6):1333–1343 12. Zhang DS, Xu YD, Yao JT, Zhao YS (2019) Analysis and optimization of a spatial parallel mechanism for a new 5-DOF hybrid serial-parallel manipulator. Chin J Mech Eng 31(1) 13. Xu YD, Zhang DS, Yao JT, Zhao Y (2017) Type synthesis of the 2R1T parallel mechanism with two continuous rotational axes and study on the principle of its motion decoupling. Mech Mach Theory 108:27–40 14. Xu Y, Yang F, Xu Z, Yao J, Zhou Y, Zhao Y (2020) TriRhino: a five-DOF hybrid serial-parallel manipulator with all rotating axes being continuous: stiffness analysis and experiments. ASME J Mech Robot 11:23 15. Yang Y, Wang LP (2005) Inverse dynamics analysis of a new type of parallel machine tool. Tool Technol 06:11–13

Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

379

16. Wu J, Wang J, Wang L, Li T (2009) Dynamic and control of a planar 3-DOF parallel manipulator with actuation redundancy. Mech Mach Theory 44:835–849 17. Wang L, Wu J, Wang J, You Z (2009) An experimental study of a redundantly actuated parallel manipulator for a 5-DOF hybrid machine tool. IEEE/ASME Trans Mechatron 14(1):72–81

Force Modulation Mode Harmonic Atomic Force Microscopy for Enhanced Image Resolution of Cell Ke Feng, Jiarui Gao, Benliang Zhu, Hongchuan Zhang, and Xianmin Zhang

Abstract This paper presents a force modulation (FM) mode harmonic atomic force microscope (AFM) for enhanced cell scanning quality. Compared with tapping mode AFM, FM mode AFM has less contact force and less damage to biological samples. Harmonic AFM has been widely applied to characterize the local mechanical properties of the specimens. In this paper, we integrate FM mode AFM and harmonic AFM to scan Hela cells. Experimental results show that the comprehensive scan mode can perform better image resolution than the conventional tapping mode AFM. Our comprehensive scanning strategy is a powerful and proper technique to study the complete mechanical properties of cells and improve the sensitivity of nanometer detection. Keywords Atomic force microscope · Harmonic imaging · Cantilever design

1 Introduction Atomic force microscope (AFM) is a high-precision detection instrument invented in 1986 [1], which has an extensive range of applications covering micro- and nanotechnology, life science, biology, material science, and semiconductor industries [2–4]. The working principle of AFM is straightforward and powerful: A micro-cantilever with a sharp tip moves over the surface of the sample while the interaction force between the tip and sample is measured. The topographic features of the sample are calculated through a closed-loop control system.

K. Feng · J. Gao · B. Zhu · H. Zhang · X. Zhang (B) Guangdong Province Key Laboratory of Precision Equipment and Manufacturing Technology, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_21

381

382

K. Feng et al.

AFM can work in many conditions. It can test mechanical and electrical properties [5] of samples, whether in liquid [6] or vacuum, especially for micro- and nanoscale materials [2, 7]. Basically, the working modes of AFM can be classified into two sub-modes: static force mode and dynamic force mode. In the dynamic force mode, changes in the dynamic behavior of the cantilever are detected by measuring changes in its vibration amplitude when it is excited with a sinusoidal signal with a frequency close to the free resonance frequency of the cantilever. The tip will contact the surface once in each vibration period. These periodic contacts produce pulse-like tip-sample forces, which generate higher harmonics of the excitation frequency [8]. Force modulation (FM) mode combines the advantages of static force mode and dynamic force mode. The static force acting on the cantilever is still used to produce a topography image of the sample. Simultaneously, the cantilever is excited, and the resulting vibration amplitude is measured. This technique can improve the nanometer spatial resolution and ultra-high force detection sensitivity of nanoscale imaging [9] and seek to estimate the material properties of the sample [10] using the response measured at harmonic of the excitation. The harmonic response signal of a traditional cantilever is small. The signal-tonoise ratio is too low to be detected [11] because the frequency response curve of a commercial cantilever only has peaks at each resonant frequency. The natural resonant frequency is not equal to any of the harmonics. Although some researchers have proposed a method to obtain cellular mechanics by multi-harmonic atomic force microscopy [12], this method is not suitable for all samples because of the difficulty of data processing. How to increase the harmonic response has been within the focus. Several approaches have been proposed to tune the frequency characteristics of a conventional AFM cantilever for better harmonic signals. Among them, the simplest way is to change the mass distribution of the cantilever by adding [13, 14] or subtracting mass [15, 16], and the high-order Eigen-resonance frequency can be adjusted to an integer multiple of the first-order natural frequency. When the excitation frequency is equal to the first-order natural frequency, the amplitude of the corresponding harmonic can be amplified. In addition, changing the cantilever’s overall width [17] or local width [18], deciding the opening position by binary coding of the cantilever [19], or adding an inner-paddled cantilever to a commercial cantilever [20] can change its frequency characteristics and improve the imaging quality. This paper proposed a comprehensive scanning strategy combing FM mode and harmonic AFM. We redesign the mass distribution of the cantilever, making the higher-order resonant frequencies of the cantilever be integer multiples of the excitation frequency. And the frequency response of the cantilever was calculated using finite element analysis (FEA) to verify the results and improve the design accuracy. Focused ion beam (FIB) etching was utilized to fabricate the final optimized structures. Under the same experimental conditions, the harmonic probe has higher detection sensitivity and higher scanning image quality than that of the commercial probe when scanning soft biological samples, which is reflected in the higher marginal definition of images.

Force Modulation Mode Harmonic Atomic Force Microscopy …

383

Fig. 1 The top-down view of an AFM cantilever with one inserted hole

2 Cantilever Design and Optimization 2.1 Cantilever Configuration For the conventional FM-AFM, the higher eigenmode can determine the physical properties of the material surface, and the basic eigenmode can obtain the surface topography of the material. However, the high-order flexural eigenfrequency generally does not align with any of the harmonics, making the vibration amplitudes at higher harmonics too small to detect. One possible way is to redesign the shape of the cantilever to make its higher eigenfrequency be an integer harmonic of the basic eigenfrequency. Here, we tune the basic eigenfrequency of the cantilever by inserting a rectangle hole. The schematic illustration of the harmonic cantilever is presented in Fig. 1, which is a beam with a uniform cross-section. The width a and length b of the slot and its position x are adjusted in the optimization processes. It should be pointed out that rectangular is not the only choice of the shape of the inserted holes. One may use other shapes, such as circles, and the number of the inserted holes can be two or more. However, the chosen size and number of holes will affect the sensitivity of frequency regulation. From our experience, a single rectangular hole is a good one since it ensures good accuracy and convenient fabrication.

2.2 Cantilever Design Theory According to the vibration theory of the Timoshenko beam [8], for a cantilever with uniform cross-section and without axial effects, the governing equations take the form   ρ  ∂ 4 y(x, t) ρ P ∂ 2 y(x, t) ρ 2 ∂ 4 y(x, t)  ρ + + 1 − = (1) EI ∂t 2 EεG ∂x4 E εG ∂ x 2 ∂t 2   ρ  ∂ 4 θ (x, t) ρ P ∂ 2 θ (x, t) ρ 2 ∂ 4 θ (x, t)  ρ + + 1 − = (2) 2 4 EI ∂t EεG ∂x E εG ∂ x 2 ∂t 2

384

K. Feng et al.

where y(x, t) and θ (x, t) are the time-dependent deflection displacement and bending rotation. E is Young’s modulus, I is the moment of inertia, ρ is the density, P is the cross-section area, G is the shear modulus, and ε is the Timoshenko shear coefficient. Its value depends on the shape of the cross-section. For a rectangular cross-section, ε = 5/6. For a conventional AFM cantilever with dimensional of L (length), W (width), and H (thickness), by writing y(x, t) = Y (x)T (t), and θ (x, t) = Θ(x)T (t), we have the mode shapes Y (x) = Aξ1 + Bξ2 + Cξ3 + Dξ4

(3)

(x) = −AMξ2 + B Mξ1 + C N ξ4 + D N ξ3

(4)

where M, N, and ξ i (i = 1, 2, 3, 4) are parameters determined by the shape of the cantilever and material properties. A, B, C, and D can be determined using the boundary conditions at both ends of the cantilever. Using the above equations, the resonant frequencies of the cantilever can be determined. Considering the fixed side and the free side of the cantilever, we have F(ω, x1 , . . . xn , a1 , . . . an ) = 0

(5)

which represents the relationship between resonant frequencies and the characteristics of the inserted holes. Let ωk (k = 1, 2, …) denote the consecutive roots of Eq. (5). We have (ωk /ω1 ) = f k (ω, x1 , . . . xn , a1 , . . . an ), which means that the holes’ sizes and locations determine the (ωk /ω1 ) ratio entirely. The harmonic characteristics of the cantilever can be enhanced when the value ωk /ω1 is an integer. One may tune the typical cantilever by using a comprehensive map provided by Eq. (5) while considering the effect of the inserted holes on the cantilever’s stiffness and resonant frequency characteristics.

2.3 Cantilever Optimization Above theoretical analysis can be roughly used for optimization, including the sizes and locations of the holes. However, for better approximations of the realistic AFM cantilever, such as the non-ideal rectangular shape near the free end, Finite Element Analysis (FEA) simulation will be much more accurate, in which the natural shape and dimension of the base cantilever can be modeled according to scanning electron microscopy (SEM) measurements. The scanning electron microscope (SEM) of the original rectangular cantilever is shown in Fig. 2. Its actual dimensions are Length L = 480 μm, width W = 55 μm, and thickness H = 2 μm. These parameters are averaged from multiple measurements.

Force Modulation Mode Harmonic Atomic Force Microscopy …

385

Fig. 2 SEM image of the original rectangular cantilever

The primary materials properties used in FEA calculations are E = 169 GPa, ρ = 2230 kg/m3 , and the Poisson’s ratio ν = 0.2872 for silicon. We listed the possible positions and sizes of the inserted holes and then calculated the first three natural frequencies corresponding to each group of values performed in finite element simulation software using a tetrahedral element and a higher-order 3D ten-node element. Figure 3 shows the relationship between the ratio of resonant frequencies and the characteristics of the inserted holes. The abscissa x and ordinate a represent the position and width of the hole in Fig. 1, respectively. The integer number n labeled on each contour curve indicates where a certain eigenfrequency matches the fundamental frequency at the nth harmonic. Each intersection point of the dotted line and the solid line yields a multi-harmonic cantilever in the (x, a) parameter space. The horizontal and vertical values of the significant intersection points and their corresponding harmonics are listed in Table 1.

3 Fabrication and Experimental Study 3.1 Experimental Setup and Calibration This section will demonstrate the proposed method by modifying a commercial cantilever using FIB milling. The designed harmonic cantilever prototype is fabricated starting from a commercial cantilever (ContAL-G, BudgetSensors). The experiments were carried out on a commercial AFM (Nanosurf, Switzerland). For the used rectangular cantilevers, the first three eigenfrequencies of the cantilever (ω1 , ω2 , and ω3 ) before and after FIB milling are listed in Table 2. Its second and third eigenfrequency for the rectangular cantilevers are about 6.19ω1 and 17.94ω1 . They are near the 6th and 18th harmonic, respectively. However, the corresponding parameters of the inserted hole will lead to a decrease in the stiffness of the cantilever. Therefore, it is more sensible to set the target eigenfrequency ratios to ω2 /ω1 = 6 and ω3 /ω1 = 17. After the structure optimization, FIB milling was applied to several new ContAlG cantilevers to fabricate the optimal one-hole cantilever. The SEM image of the

386

K. Feng et al.

Fig. 3 Relationships between ω3 /ω1 ω2 /ω1 and x and a when b = 30 μm

Table 1 Intersection points in the image and their corresponding harmonics

Table 2 The first three eigenfrequencies of the cantilever before and after FIB milling

ω2 /ω1

ω3 /ω1

x/μm

a/μm

6

17

173

23

6

18

251

41

7

19

415

32

8

20

370

41

8

21

410

42

ω1 /kHz

ω2 /kHz

ω3 /kHz

Before FIB milling

13.49

83.50

242.02

After FIB milling

13.51

81.70

234.26

resulted harmonic cantilevers is shown in Fig. 4. We can obtain the experimental frequency spectrum by mounting the cantilever onto our AFM. The normalized frequency response of the cantilever is shown in Fig. 5. Without the cutting clot, the second and third eigenfrequencies are not equal to any higher harmonics. Due to the fast decay at off-resonance, the harmonic amplitudes will be weak and unsuitable

Force Modulation Mode Harmonic Atomic Force Microscopy …

387

Fig. 4 SEM image of the cantilever after inserting hole using FIB

Fig. 5 Normalized frequency responses of the cantilevers

for stable signal detection. After cantilever modification, the second and third eigenfrequencies coincide with the 6th and 17th harmonic. Therefore, the corresponding harmonics are at the resonance, and their signals can be enhanced.

3.2 AFM Imaging This part will describe how to perform a harmonic AFM-imaging procedure. Firstly, there are three essential parameters that affect imaging: excitation frequency, excitation amplitude, and setpoint. The setpoint is equal to the maximum contact force between tip and sample. A significant excitation frequency and setpoint can ensure imaging accuracy but will cause irreversible damage to the tip and sample. In our experiment, the excitation frequency is the same as the free resonance frequency

388

K. Feng et al.

of the first eigenmode. This relatively low frequency can stimulate the harmonic cantilever’s high-frequency characteristics while reducing the negative effect of high frequency. Besides, The excitation amplitude is 200 mV, and the setpoint is 10 nN. The spring constant of the cantilever was calibrated to be 0.23 N/m. We seeded the Hela cells in a complete cell culture medium onto cell culture dishes to prepare cells for AFM experiments. Let the cells grow for an additional 1–2 d to final confluency of ~ 60–70%. To verify the performance of the harmonic signals in improving image edge acuity, we selected Hela cells fixed with a 4% Formalin solution for harmonic imaging. After that, we placed the samples in a 4-degree refrigerator overnight to keep the sample dry. Before the experiment, we took the samples out of the refrigerator until their temperature was equal to the environment to reduce the influence of temperature differences on imaging quality. The average Young’s modulus of Hela cells is Ecell = 1000 Pa. In our experiments, a commercial cantilever and a harmonic cantilever were driven at their fundamental imaging frequency. The PID gain was 10,000, 1500, and 0, respectively. It is vital to select the region of interest. The unmistakable outline and full shape of cell samples are the optimal choices. We choose the scan size of 40 μm × 40 μm to observe the entire or a large part of the cell, and the number of points (256 × 256), depending on the required resolution. There are four channels for recording during the scanning of the sample with a directly excited cantilever: Z-sensor signal (topography), deflection (A0 ), amplitude (A1 ), and phase (ψ). The topography image of a Hela cell is shown in Fig. 6. The amplitude and phase map scanned by commercial and harmonic cantilevers is shown in Fig. 7. Fig. 6 The topography image of Hela cell. Scale bar: 10 μm

Force Modulation Mode Harmonic Atomic Force Microscopy …

389

Fig. 7 The amplitude and phase image scanned by a, c harmonic and b, d commercial cantilever

3.3 Analysis of Amplitude and Phase Maps This part will analyze the good scanning performance of harmonic probes in amplitude map and phase map, respectively.

3.3.1

Amplitude Map Evaluation

To intuitively evaluate amplitude map resolution, we propose an evaluation method. The main procedures are briefly described as follows. (1) The amplitude maps were converted into grayscale images for the convenience of subsequent calculation. (2) The grayscale images were denoised by bilateral filtering. (3) Sobel edge detection operator was used to calculate the edge information in the horizontal and vertical directions. Then, TenenGrad function was used to calculate the convolution of Sobel edge detection operator in two directions.

390

K. Feng et al.

We defined the convolution kernels of the Sobel edge detection operator as Gx and Gy. The gradient of the image at pixel (x, y) is S(x, y) =

√

Gx · I (x, y) + Gy · I (x, y)

(6)

1   · x y S(x, y)2 n

(7)

TenenGrad function is Ten =

The TenenGrad function value of Fig. 7a, b are 164.19 and 152.11, respectively. We have repeated the experiment on other cells, and the TenenGrad function value is 172.71 and 168.47 for harmonic and commercial probes. We found that the larger the TenenGrad function value, the clearer the image. Besides, the TenenGrad function value of image resolution varies with the image type. The image type in this experiment is CV_16S.

3.3.2

Phase Map Evaluation

In FM mode, the vibration amplitude depends on the drive amplitude, the stiffness of the cantilever, and, most importantly, the stiffness of the tip-sample contact. Thus, the FM mode can produce material contrast when there is a significant difference in stiffness of the tip-sample connection of these materials. We define Contact Stiffness Sensitivity (CSS) as a parameter for evaluating the ability of probes to distinguish different materials. CSS is equal to the change in contact resonant frequency when the contact stiffness between the probe and the sample changes by one unit. We use the FEA method to calculate the CSS value. When scanning cells, the contact stiffness between the tip and the sample is within 0 ~ 3 N/m. We found that the ability of the probe to discriminate sample stiffness increases with the increase of CSS, and the CSS of the harmonic probe is greater than that of the commercial probe. The result is shown in Fig. 8. The phase map can reflect the change in sample stiffness. Clearly, the phase map scanned by the harmonic probe shows a higher contrast between cells and petri dish than the commercial probe.

4 Conclusions In summary, a comprehensive scanning strategy is proposed to improve the imaging quality of biological samples. We combine theoretical calculation and finite element simulation to design a harmonic cantilever structure. The optimized cantilever has higher contact stiffness sensitivity and better imaging quality. Our analysis of image

Force Modulation Mode Harmonic Atomic Force Microscopy …

391

Fig. 8 Comparison of contact stiffness sensitivity between the harmonic probe and commercial probe

resolution confirms this. Force modulation mode AFM can reduce damage to biological samples because of the small scanning frequency and contact force. Our comprehensive scanning strategy can give AFM a significant advantage in biological sample scanning. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 51820105007).

References 1. Binning G, Quate CF (1986) Atomic force microscope. Phys Rev Lett 56(9):930 2. Dufrene YF, Ando T, Garcia R et al (2017) Imaging modes of atomic force microscopy for application in molecular and cell biology. Nat Nanotechnol 12(4):295–307 3. Mrinalini RSM, Sriramshankar R, Jayanth GR (2015) Direct measurement of three-dimensional forces in atomic force microscopy. IEEE/ASME Trans Mechatron 20(5):2184–2193 4. Taffetani M, Raiteri R, Gottardi R et al (2015) A quantitative interpretation of the response of articular cartilage to atomic force microscopy-based dynamic nanoindentation tests. IEEE/ASME J Biomech Eng 137(7):071005 5. Zhang H, Gao H, Geng J et al (2021) Torsional harmonic Kelvin probe force microscopy for high-sensitivity mapping of surface potential. IEEE Trans Ind Electron 1 6. Sztilkovics M, Gerecsei T, Peter B et al (2020) Single-cell adhesion force kinetics of cell populations from combined label-free optical biosensor and robotic fluidic force microscopy. Sci Rep 10(1):61

392

K. Feng et al.

7. Li M, Xi N, Wang Y et al (2020) Atomic force microscopy as a powerful multifunctional tool for probing the behaviors of single proteins. IEEE Trans Nanobiosci 19(1):78–99 8. Benlinag Z, Zimmermann S, Xianmin Z et al (2017) A systematic method for developing harmonic cantilevers for atomic force microscopy. J Mech Des 139(1):012303 9. Wang W, Zhang W, Chen Y (2020) Enhancement of contact resonance atomic force microscopy subsurface imaging by mass-attached cantilevers. J Phys D Appl Phys 53(21):215301 10. Raman A, Trigueros S, Cartagena A et al (2011) Mapping nanomechanical properties of live cells using multi-harmonic atomic force microscopy. Nat Nanotechnol 6(12):809–814 11. Zhang WM, Meng G, Peng ZK (2011) Nonlinear dynamic analysis of atomic force microscopy under bounded noise parametric excitation. IEEE/ASME Trans Mechatron 16(6):1063–1072 12. Efremov YM, Cartagena-Rivera AX, Athamneh AIM et al (2018) Mapping heterogeneity of cellular mechanics by multi-harmonic atomic force microscopy. Nat Protoc 13(10):2200–2216 13. Xiang W, Tian Y, Yang Y et al (2019) Enhancing multiple harmonics in tapping mode atomic force microscopy by added mass with finite size. Appl Phys Express 12(12):126505 14. Li H, Chen Y, Dai L (2008) Concentrated-mass cantilever enhances multiple harmonics in tapping-mode atomic force microscopy. Appl Phys Lett 92(15):151903 15. Zhang W, Chen Y, Chu J (2017) Cantilever optimization for applications in enhanced harmonic atomic force microscopy. Sens Actuators A Phys 255:54–60 16. Sriramshankar R, Sri Muthu Mrinalini R, Jayanth GR (2017) Design and fabrication of a flexural harmonic AFM probe with an exchangeable tip. J Micro-Bio Robot 13(1–4):39–53 17. Cai J, Xia Q, Luo Y et al (2015) A variable-width harmonic probe for multifrequency atomic force microscopy. Appl Phys Lett 106(7):071901 18. Li Z, Shi T, Xia Q (2018) An optimized harmonic probe with tailored resonant mode for multifrequency atomic force microscopy. Adv Mech Eng 10(11):168781 19. Hou Y, Ma C, Wang W et al (2019) Binary coded cantilevers for enhancing multi-harmonic atomic force microscopy. Sens Actuators A Phys 300:111668 20. Dharmasena SM, Yang Z, Kim S et al (2018) Ultimate decoupling between surface topography and material functionality in atomic force microscopy using an inner-paddled cantilever. ACS Nano 12(6):5559–5569

Optimization Design of Buffering and Walking Foot for Planetary Legged Robots Chu Zhang, Liang Ding, Huaiguang Yang, Haibo Gao, Liyuan Ge, and Zongquan Deng

Abstract The legged robot has better adaptability to terrain in the process of moving and has been considered for future planetary exploration missions. As the part of direct contact with planet soil, the foot will directly affect movement performance and control effect of the legged robot. In this paper, the advantages and disadvantages of various foot configurations of ground legged robot are analyzed, and a coronal foot configuration is proposed. Based on the foot-terrain interaction mechanics model, the size of the coronal foot is optimized with the goal of anti-sinkage, anti-slip and light weight. Then, a high-performance coronal foot with two-stage buffering and touch sensing functions is designed. And the finite element analysis is carried out to verify the reliability of strength and stiffness of the foot in the ultimate working conditions. Finally, the anti-sinkage and tangential traction performance of the coronal foot is verified by quasi-static loading, loading with impact and tangential slip experiments. Keywords Planetary legged robots · Optimization design · Coronal foot · Performance verification

1 Introduction Until now, the machines used for planetary exploration are wheeled robots, because wheeled robot can apply the experimental experience on earth and avoid the high complexity of motion control and high energy consumption on flat terrain [1]. But wheeled robots’ limited mobility makes them difficult to operate in soft, unstructured and rugged terrain. Compared to wheeled robots, the legged robots have outstanding advantages in the complex planetary exploration missions. C. Zhang · L. Ding · H. Yang (B) · H. Gao · Z. Deng School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China e-mail: [email protected] L. Ge Goertek, Qingdao 266000, Shandong, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_22

393

394

C. Zhang et al.

Researchers at the National Aeronautics and Space Administration (NASA) developed the all-terrain hexapod robot ATHLELE [2–4], that uses cylindrical wheels with active degrees of freedom as the foot mechanisms. DFKI Robotics Innovation Center at the University of Bremen in Germany improved the design of SpaceClimber [5–7] and developed an energy-saving, adaptive multi-legged free-climbing robot. Inspired by mantis, researchers at the DFKI-Ric and University of Bremen developed the multi-legged robot MANTIS [1], whose foot is composed of two hemispheres and can be passively adjusted to adapt to uneven terrain. DFKI of the University of Bremen developed the CREX [8] (CRater EXplorer) for CRater and cave exploration, whose foot mechanism consists of a cylindrical rubber foot and cushioned leg. Arm et al. designed the quadruped robot SpaceBok [9], whose foot is made of carbon fiber material. And the contact surface between the terrain and the foot is an incomplete cylinder. Zhang et al. proposed and designed a two-degree-of-freedom flexible foot-end structure to solve the problems of large landing impact, vibration, and poor adaptability to complex ground surfaces in the motion of a foot-type robot [10]. Xu et al. presented a novel foot structure for the legged robots, design of the foot system with high-adaptability and large-adhesion is carried out to adapt extreme road conditions [11]. Izi et al. studied the influence of circular foot center position on robot step length, step speed, foot clearance and stability [12]. Catalano et al. presented the SoftFoot-Q, an articulated adaptive foot for quadrupeds [13]. Hakamada and Mikami proposed a passively driven gripping foot that can be attached to any mobile legged robot without intervening in the robot’s controller [14]. Through the investigation and research, facing the unknow environment of the planet, the planetary legged robot may face landing, walking and climbing, so the foot requires special properties: anti-sinkage, anti-slip and light weight. In this paper, antisinkage means that the sinkage is no more than 100% of the foot height during walking and no more than 150% of the foot height during landing. Anti-slip means that the legged robot can climb 30° slopes. Light weight means that keeping the weight of the foot as small as possible without compromising the above two requirements. However, at present, the above properties are not fully considered in the design of the legged robot. The organization of this paper is as follows: Sect. 2 analyzes and summarizes five kinds of foot configurations and proposes a coronal foot configuration. In Sect. 3, the size of the coronal foot is optimized based on the performance requirements of anti-sinkage, anti-slip and light weight. In Sect. 4, a coronal foot with double-stage buffering and touch sensing is designed and the stiffness and strength are analyzed by finite element method; Sect. 5 verifies the performances of the designed coronal foot through experiments; and Sect. 6 has concluding remarks.

2 Foot Configuration Design of Planetary Legged Robot At present, the foot structures commonly used by legged robot are mainly divided into five types, namely spherical foot, cylindrical foot, circular flat foot, rectangular

Optimization Design of Buffering and Walking Foot for Planetary …

395

Table 1 Summary of the characteristics of feet Features

Types Spherical

Cylindrical

Circular flat √

Rectangular flat √

Special shaped √

Ankle joint

×

×

Complexity

*

*

*

*

***

Weight

*

*

**

**

**

Adaptability

***

**

*

*

**

Anti-sinkage

*

**

***

***

–

Anti-slip

**

*

Anisotropy

× √

*** √

* √

– √

× represents no,

×

represents yes, the more * represents the more prominent the characteristics

flat foot and special shaped foot. The characteristics of the five kinds of feet are summarized in Table 1. The robots with circular flat foot or rectangular flat foot require additional joints to achieve ankle movement, so these two kinds of feet have complex structures, large mass and poor flexibility, which are suitable for heavy-load and low-speed robots. Special shaped foot is designed for robots with special functions, such as soft robots, which have different shapes and complex structures according to different design requirements. The planetary legged robot has no special requirements other than anti-sinkage, anti-slip and light weight, which there is no need for to use special shaped foot. Spherical and cylindrical feet have better adaptability to terrain without additional ankle joints, and have simple structure and small mass. However, cylindrical foot has anisotropy, which increases the complexity of the control system of the robot, so spherical foot is the first choice for planetary legged robot. According to the analysis, the top part of the spherical foot has little influence on anti-sinkage and tangential traction performance [15], and also increases foot’s mass. By removing the top part of the spherical foot, the coronal foot configuration is obtained. As shown in Fig. 1, the coronal foot has four parameters, namely, spherical radius R, coronal section radius r, coronal angle θ and coronal height h. The relationship between the four parameters is as follows: r = R sin θ

(1)

h = R(1 − cos θ )

(2)

396

C. Zhang et al. Spherical

Fig. 1 Coronal foot configuration

R h

r

Coronal

3 Optimization Design of Coronal Foot Size For the coronal foot proposed in Sect. 2, the influence of R and θ on anti-sinkage performance, anti-slip performance and mass is comprehensively considered in this section. And an optimization function for coronal foot size is proposed, which considers these three design requirements.

3.1 Analysis of Foot Scale Effect Based on Anti-sinkage Performance As shown in Eq. (3), the normal force model of spherical foot represents the relationship between normal force and sinkage. The soil mechanical parameters in Eq. (3) are shown in Table 2. When the normal force remains constant, the sinkage decreases with the increment of spherical radius R. Therefore, in order to ensure that the coronal foot has great anti-sinkage performance, it is necessary to choose a reasonable R.   FN = π kc + kϕ R   0.05 δ 0.55 (π/4 − 1/2) tan ϕ n+0.95 R δ + · √ 2 R

(3)

In order to express the degree of sinkage, the ratio of sinkage δ to coronal height h is defined as sinkage ratio ζ, as shown in Eq. (4). ζ =

δ(FN , R) h

(4)

Table 2 Soil mechanical parameters Parameters

k c (kPa/mn−1 )

k ϕ (kPa/mn )

c (kPa)

ϕ (°)

K (mm)

n

Value

− 20.7

1594.8

0.46

38.1

13.3

0.79

Optimization Design of Buffering and Walking Foot for Planetary …

397

70 8

50 40

6

(45,45)

4 ζ

δ (mm)

60

30 20 10 40

50 60 70 80 R (mm) (a) δ vs. R

90 100

2 0 60

74 68 76 82 84 92 100 90 R (mm) (b) ζ vs. R and θ

66

58

Fig. 2 The relationship between δ and R

After investigation, the maximum normal force of the planetary legged robot at work is about 1200 N. Under the maximum normal force, the relationship between δ and R is shown in Fig. 2a. When the spherical radius R is 45 mm, the sinkage δ is also 45 mm and the sinkage ratio ζ is 1. Therefore, the spherical radius R should be greater than 45 mm. The relationship between ζ and R, θ is shown in Fig. 2b. With the increment of R and θ, ζ gradually decreases and the anti-sinkage performance becomes better.

3.2 Analysis of Foot Scale Effect Based on Anti-slip Performance The tangential traction performance of the foot directly affects the walking and climbing ability of the planetary legged robot and must be taken into account in the design of the coronal foot. The sinkage and tangential force of the spherical foot in the sliding process can be calculated by Eqs. (5) and (6) [15], where γ is wrap angle coefficient, N is dynamic sinkage exponent, and j is shear displacement. The ratio of tangential force to normal force is defined as the equivalent friction coefficient μ, as shown in Eq. (7).   π kc + kϕ R FN = 2    1 (1 − γ )δ 0.55 tan ϕ π (1 + γ )R 0.05 − − δ N +0.95 · √ 4 π 2 R   π kc + kϕ R FT = 2     1 (1 + γ )R 0.05 tan ϕ N +0.95 (1 − γ )δ 0.55 π − + δ · √ 4 π 2 R

(5)

(6)

398

C. Zhang et al. 0.98

Fig. 3 The relationship between R and μ

0.96

μ

0.94 0.92 0.9 0.88 0.86 40

u=

50

60

70

80

R (mm)

FT FN

90 100

(7)

As shown in Fig. 3, when R increases from 40 to 100 mm, the equivalent friction coefficient μ increases from 0.865 to 0.977.

3.3 Analysis of Foot Scale Effect Based on Light Weight When the Legged robot is working, the inertia of the leg will increase if the mass of the foot is large, which will directly affect the mobility of the robot. And as a space machine, with the increment of the mass of robot, the cost is significantly higher. In order to reasonably design the geometric size of the coronal and realize light weight on the basis of ensuring the anti-sinkage and tangential traction performance, the relationship between the mass of the coronal foot and R, θ is analyzed. In order to establish the mass function, the structure in the foot rack that is not affected by R and θ is removed. The simplified foot rack structure is shown in Fig. 4. According to the principle of mechanical design, in order to ensure the strength and stiffness of the coronal foot rack, the following relationship should be satisfied: ⎧ R = R1 + 3 ⎪ ⎪ ⎪ ⎨ r = 0.52r 1 ⎪ h 1 = 0.2h ⎪ ⎪ ⎩ h 2 = 0.4h The volume of each part can be calculated by the following formula:

(8)

Optimization Design of Buffering and Walking Foot for Planetary …

R

θ

θ1

R1

399

V2

r V1 h

r1

h1

V3

h2

V4

Fig. 4 Simplified diagram of coronal foot rack. R—The radius of the outer coronal; r—the section radius of the outer coronal; θ—the coronal angle of the outer coronal; h—the height of the outer coronal; R1 —the radius of the inner coronal; θ 1 —the coronal angle of the inner coronal; r 1 —the radius of the inner cylinder; h1 —the height of the inner cylinder; h2 —the height of the dashed coronal

⎧ ⎪ ⎪ ⎪ V1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨V 2 ⎪ ⎪ ⎪ ⎪ V3 ⎪ ⎪ ⎪ ⎪ ⎩ V4

1 π R 3 (1 − cos θ )2 (2 + cos θ ) 3 1 = π R13 (1 − cos θ1 )2 (2 + cos θ1 ) 3 1 2 = π h 2 (3R1 − h 2 ) 3 = πr12 h 1 =

(9)

where, V 1 —The volume of the outer coronal V 2 —The volume of the inner coronal V 3 —The volume of the dashed coronal V 4 —The volume of the inner cylinder. The volume V and mass m of the coronal foot rack can be calculated by Eqs. (10) and (11): V = V1 − V2 + V3 − V4

(10)

m = ρV

(11)

where, ρ is the density of foot rack’s material (Al alloy).

400

C. Zhang et al.

3.4 Analysis of Foot Comprehensive Effect Based on Anti-sinkage Performance, Anti-slip Performance and Light Weight According to the above analysis, by increasing R and θ, the anti-sinkage and anti-slip performance of the coronal foot can be improved, but the mass will increase. This section comprehensively considers the requirements of anti-sinkage, anti-slip and light weight, and establishes the objective function of size optimization design of coronal foot, as shown in Eq. (12):    ζ (R, θ ) − ζobj 2 m(R, θ ) − m obj 2 f obj (R, θ ) = p1 + p2 m max − m min ζmax − ζmin   u(R, θ ) − u obj + p3 u max − u min 

(12)

where, obj is the expected value, p1 is the mass coefficient, p2 is the sinkage ratio coefficient and p3 is the coefficient of equivalent friction coefficient. The values of each parameter in Eq. (12) are shown in Table 3. The parameters R and θ can be obtained by satisfying the optimal values of Eq. (13). f = min f obj (R, θ )

(13)

(R,θ )

The design parameters of the coronal foot rack are finally obtained as shown in Table 4. The radius R is 100 mm, the coronal angle is 42°, the equivalent friction coefficient is 0.9740, and the sinkage ratio under the maximum normal force is 1.179. Table 3 Parameter values of the objective function

Parameters

Values

Parameters

Values

ζ obj

1.20

μobj

0.98

ζ max

1.82

μmax

0.98

ζ min

μmin

0.87

mobj (g)

595.50

p1

0.2

mmax (g)

4811.16

p2

0.4

mmin (g)

595.50

p3

0.4

0.29

Optimization Design of Buffering and Walking Foot for Planetary … Table 4 Optimization results of design parameters

Design parameters

Optimal values

R (mm)

100

θ (°)

42

h (mm)

26

ζ

1.179

u

0.9740

m (g)

391.75

401

4 Detailed Structural Design of Coronal Foot Including Analysis of Stiffness and Strength In this section, the structure of the proposed coronal foot is designed in detail, and the finite element analysis under extreme working conditions is carried out to verify the reliability of the strength and stiffness.

4.1 Detail Design of the Coronal Foot Considering Double-Stage Buffering The coronal foot designed in this paper not only ensures anti-sinkage, anti-slip performance and light weight, but also has the functions of touch sensing and double-stage buffering. The three-dimensional section view of the foot is shown in Fig. 5. The coronal foot rack is designed as stepped structure, which is convenient for machining, and can provide enough installation space for other components. The foot rack has raised block pattern at the bottom, which can improve the tangential traction performance. Because of the existence of normal force and block pattern, soil are fixed in the pattern grooves, which allows the friction between foot and soil to be converted into the friction between soil. Therefore, the tangential traction performance of foot can be enhanced by increasing the friction coefficient. A double-stage buffering structure with metal rubber and buffer spring is mounted inside the foot rack. When the planetary legged robot lands, there will be a big impact force between foot and terrain, and both metal rubber and buffer spring can play roles of buffer energy absorption. However, when metal rubber is pressed, its reset process has certain hysteresis, and the buffer spring has a certain pre-tightening force to ensure that the external guide pillar can quickly return to the initial position after the foot is detached from the terrain. The trigger metal and the inner guide sleeve are the two electrodes of the touch sensing circuit respectively. When the foot is in contact with the terrain, the external guide pillar will compress the buffer spring and the reset spring, and the circuit is connected realizing the touch sensing function when the trigger metal and the inner guide sleeve come into contact. The reset spring ensures that the trigger metal and

402

C. Zhang et al.

Screw

Clamp

Trigger metal

Reset spring

Precision screw

Inner guide sleeve Hose clip

Fixed disk Supporting ring

Metal rubber

Sealing cover

External guide pillar

External guide sleeve

Foot rack

Sealing clamp Screw Buffer spring

Fig. 5 Three-dimensional section view of coronal foot

inner guide sleeve are separated from each other when the foot is not in contact with the terrain.

4.2 Finite Element Analysis Under Extreme Working Conditions During landing and walking, the robot’s foot will be subjected to the impact force of the terrain, which will shock the internal components. In order to verify the reliability of stiffness and strength of the coronal foot, the finite element analysis on components with large force (foot rack, external guide pillar, external guide sleeve) is carried out. The material of the foot rack, external guide pillar and external guide sleeve is Al alloy. Through investigation, the working conditions of the planetary legged robot include three types: landing, climbing and steering. The simulation conditions are set respectively for the above three working conditions, as shown in Fig. 6. The finite element analysis results are shown in Fig. 7. Under the three working conditions, the maximum deformation of the three components is 4.79 × 10−3 mm, and the maximum stress is 13.76 MPa, which is far less than the yield strength of Al alloy. The results of finite element simulation show that the designed coronal foot has sufficient strength and stiffness.

Optimization Design of Buffering and Walking Foot for Planetary …

Fixed

403

Fixed

Fixed

Torque

The pressure (a) Landing

The pressure (b) Climbing

The pressure (c) Steering

Fig. 6 Simulation condition setting diagram

3.89e-4 Max 3.46e-4 3.03e-4 2.60e-4 2.16e-4 1.73e-4 1.30e-4 8.66e-5 4.33e-5 0 Min 0 Unit:mm

3.86 Max 3.43 3.00 2.57 2.14 1.71 1.28 0.86 0.43 4.31e-5 Min 0 Unit:MPa

4.79e-3 Max 4.26e-3 3.73e-3 3.19e-3 2.66e-3 2.13e-3 1.60e-3 1.06e-3 5.3e-4 80(mm) 0 Min 0 40 Unit:mm 20 60 (c) Climbing deformation 2.56e-3 Max 2.27e-3 1.99e-3 1.70e-3 1.42e-3 1.14e-3 8.52e-4 5.68e-4 2.84e-4 80(mm) 0 Min 0 40 Unit:mm 20 60 (e) Steering deformation

13.76 Max 12.23 10.70 9.17 7.65 6.12 4.59 3.06 1.53 3.53e-4 Min 80(mm) 40 0 Unit:MPa 20 60 (d) Climbing stress 9.86 Max 8.77 7.67 6.58 5.48 4.38 3.29 2.19 1.10 9.21e-5 Min 80(mm) 40 0 Unit:MPa 20 60 (f) Steering stress

80(mm) 40 20 60 (a) Landing deformation

Fig. 7 The finite element analysis results diagram

80(mm) 40 20 60 (b) Landing stress

404

C. Zhang et al.

5 Experimental Verification of Coronal Foot Performances In order to verify the anti-sinkage performance and tangential traction performance of the designed coronal foot, the prototype is processed, and the quasi-static loading, load with impact and tangential slip experiments are carried out by using the footterrain interaction testbed, as shown in Fig. 8. The experimental conditions are shown in Table 5. In order to reduce the influence of accidental factors on the experimental results, each experiment should be repeated at least 3 times.

5.1 Results of Quasi-static Loading Experiment In order to analyze the sinkage ratio of the coronal foot in the quasi-static process, the quasi-static loading experiment is carried out, and the experimental results are shown in Fig. 9. It can be seen from Fig. 9a that the interaction process between

(a) Quasi-static loading

(b) Loading with impact

(c) Tangential slip Fig. 8 Experimental diagram of coronal foot

Optimization Design of Buffering and Walking Foot for Planetary … Table 5 Experimental setup of coronal foot

Experiments

Normal load (N)

Initial height (mm)

Quasi-static loading

250

0

0

Load with impact

160

100

0

Data1 Data2 Data3

0

δ (mm)

FN (N)

Tangential slip 250

300 250 200 150 100 50 0 -50 -5

405

5 10 15 20 25 30 δ (mm) (a) FN vs. δ

30 25 20 15 10 5 0 -5 0

Static sinkage

Tangential velocity (mm/s)

10

Data1 Data2 Data3

50 100 150 200 250 300 t (s) (b) δ vs. t

Fig. 9 Results of quasi-static loading experiment

foot and soil can be divided into three stages. The first stage is soil elastic-plastic deformation stage, in which the interaction force between foot and terrain is less than the spring’s pre-tightening force, and the equivalent stiffness increases with the increment of sinkage. The second stage is the spring buffer stage, in which the interaction force between foot and terrain is greater than the spring’s pre-tightening force, the metal rubber is not compressed, and the equivalent stiffness is about equal to the linear stiffness of the spring. The third stage is the spring and metal rubber buffer stage, in which the spring and metal rubber are compressed at the same time, the equivalent stiffness of the foot is nonlinear and large. The area enclosed by the loading curve and the unloading curve can be expressed as the energy absorbed by the buffer element. As shown in Fig. 9b, under the normal load of 250 N, the sinkage of the foot is about 25 mm, and the sinkage ratio is about 0.96, which meets the design requirement.

5.2 Results of Loading with Impact Experiment In order to verify the anti-sinkage performance of the foot in the normal impact process, the loading with impact experiment is carried out, and the experimental results are shown in Fig. 10.

C. Zhang et al.

14 12 10 8 6 4 2 0 -2 0

50 Data1 Data2 Data3

δ (mm)

FN (×10 2N)

406

0 Data1 Data2 Data3

-50

1

4 3 t(s) (a) FN vs. t 2

5

6

-100 0

1

4 3 t(s) (a) δ vs. t

2

5

6

Fig. 10 Results of loading with impact experiment

When the foot touches the terrain, the normal force reaches the peak value immediately (about 1200 N) and then decreases to be equal to the normal load (250 N). Due to the rebound of the foot after contact with the terrain, the sinkage reaches the peak value first, then decreases to the minimum value, and finally increases to the stable value (about 36 mm). Thus, the sinkage ratio of the coronal foot under loading with impact is 1.38, which meets the design requirement.

5.3 Results of Tangential Slip Experiment In order to verify the tangential traction performance of the coronal foot, the tangential slip experiment is carried out, and the results are shown in Fig. 11. With the increment of shear displacement, both the sinkage and tangential force increase first and then reach a stable value. As shown in Fig. 11b, when the shear displacement is 120 mm, the tangential force is 145 N, and the corresponding equivalent friction coefficient is 0.58. The maximum equivalent friction coefficient is about 1, which can meet the requirement of climbing the 30° slopes.

6 Conclusions By comparing the characteristics of the foot configurations, the coronal foot suitable for planetary legged robot is proposed. According to the specific design requirements, the sinkage ratio and equivalent friction coefficient are defined based on the footterrain interaction mechanics model, and an optimization design method of coronal foot considering anti-sinkage, anti-slip and light weight is proposed. Then a coronal foot with the functions of double-stage cushioning and touch sensing is designed. The reliability of stiffness and strength of the designed coronal foot is analyzed by finite element simulation. Finally, through experiments, it is verified that the sinkage ratios

Optimization Design of Buffering and Walking Foot for Planetary … 14

F T (×10N)

δ (×10mm)

12 10 8 Data1 Data2 Data3

6 4 2 0

0.5

1 1.5 2 j (×10 2 mm) (a) δ vs. j

2.5

3

27 24 21 18 15 12 9 6 3 0 -3

407

(120mm,145N) Data1 Data2 Data3 0

0.5

1 1.5 2 j (×10 2 mm) (b) F T vs. j

2.5

3

Fig. 11 Results of tangential slip experiment

under quasi-static loading and loading with impact are 0.96 and 1.38, respectively, and the saturation value of the equivalent friction coefficient is greater than 0.58, meeting the requirements of anti-sinkage, anti-slip and light weight. Acknowledgements Supported by the National Key Research and Development Program of China (No. 2019YFB1309500), the National Natural Science Foundation of China (Grant No. 91948202), Heilongjiang Postdoctoral Fund (No. LBH-Z20136), Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (No. SKLRS202101A03), the Fundamental Research Funds for the Central Universities (No. FRFCU9803500621).

References 1. Bartsch S, Manz M, Kampmann P et al (2016) Development and control of the multi-legged robot MANTIS. In: Proceedings of ISR 2016: 47th international symposium on robotics, Munich, Germany, 21–22 June 2016 2. Wilcox BH, Litwin T, Biesiadecki J et al (2012) ATHLETE: a limbed vehicle for solar system exploration. In: IEEE aerospace conference, Big Sky, MT, 03–10 Mar 2012 3. Howe AS, Wilcox B (2016) Outpost assembly using the ATHLETE mobility system. In: IEEE aerospace conference, Big Sky, MT, 05–12 Mar 2016 4. Julie T (2011) ATHLETE mobility performance in long-range traverse. In: AIAA SPACE 2011 conference & exposition, Long Beach, California, 27–29 Sept 2011 5. Bartsch S, Birnschein T, Cordes F et al (2011) SpaceClimber: development of a six-legged climbing robot for space exploration. In: ISR 2010 (41st international symposium on robotics) and ROBOTIK 2010 (6th German conference on robotics), Munich, Germany, 01 June 2010 6. Bartsch S, Birnschein T, Rommermann M (2012) Development of the six-legged walking and climbing robot SpaceClimber. J Field Robot 29(3):506–532 7. Roehr TM, Cordes F, Kirchner F (2014) Reconfigurable integrated multirobot exploration system (RIMRES): heterogeneous modular reconfigurable robots for space exploration. J Field Robot 31(1):3–34 8. Machowinski J, Böckmann A, Arnold S et al (2017) Climbing steep inclines with a six-legged robot using locomotion planning. In: International conference on robotics and automation

408

C. Zhang et al.

9. Arm P, Zenkl R, Sun B et al (2019) SpaceBok: a dynamic legged robot for space exploration. In: 2019 IEEE international conference on robotics and automation (ICRA), Montreal, QC, Canada, 20–24 May 2019 10. Zhang L, Liu XZ, Ren P et al (2019) Design and research of a flexible foot for a multi-foot bionic robot. Appl Sci-Basel 9(17) 11. Xu ZY, Chen XD, Liu Y (2019) Design and implementation of a novel robot foot with high-adaptability and high-adhesion for heavy-load walking robots. In: 9th IEEE annual international conference on cyber technology in automation, control, and intelligent systems (IEEE-CYBER), Suzhou, China, 29 July–02 Aug 2019 12. Izi H, Naraghi M, Safa AT (2019) Semi-passive kneed walker: analysis of foot parameters for an effective gait balance. In: 7th international conference on robotics and mechatronics (ICRoM), Sharif University of Technology, Tehran, Iran, 20–21 Nov 2019 13. Catalano MG, Pollayil MJ, Grioli G et al (2021) IEEE Trans Robot 38(1):302–316 14. Hakamada S, Mikami S (2022) Passive gripping foot for a legged robot to move over rough terrain. In: Lecture notes in networks and systems, vol 324, pp 203–212 15. Ge LY (2021) Design of buffering walking foot of legged leaping robot in lunar exploration based on modeling of foot-terrain interaction. Harbin Institute of Technology

Uncertainty Distribution Estimation Based on Unified Uncertainty Analysis Under Probabilistic, Evidence, Fuzzy and Interval Uncertainties Xiangyun Long, Mengchen Yu, Donglin Mao, and Chao Jiang

Abstract A unified uncertainty analysis method for estimating uncertainty distribution is proposed to deal with the uncertain propagation problem involving probabilistic, evidence, fuzzy and interval uncertainties. First, the unified moment analysis based on dimensional reduction integral and efficient global optimization is performed to calculate the first four statistical moments. Then, based on these moments, a family of Johnson distributions fitting to the distribution function of response are obtained using the moment matching method. Afterwards, the uncertainty distribution bounds of the response function are obtained by using optimization techniques. Finally, three numerical examples are investigated to demonstrate the validity of the proposed method. The results show that the proposed method can not only be applied to the uncertain propagation analysis under probabilistic, evidence, fuzzy and interval uncertainties, but also be suitable for general application problems with strong nonlinearity and implicit response function. Keywords Unified uncertainty analysis · Evidence theory · Fuzzy sets · Interval analysis · Johnson p-box

X. Long · M. Yu · D. Mao · C. Jiang (B) State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, P. R. China e-mail: [email protected] X. Long e-mail: [email protected] M. Yu e-mail: [email protected] D. Mao e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_23

409

410

X. Long et al.

1 Introduction In uncertainty analysis, probabilistic model quantifies uncertain variables through uncertainty distribution, which can handle situations with sufficient information, and its research and development has been relatively mature [1–6]. The available information of uncertainties can appear as imprecise, incomplete, ambiguous, or linguistic. To quantify these uncertainties suitably, besides probabilistic model, many other uncertainty models have been developed, including interval method [7, 8], fuzzy set [9], and evidence theory [10]. Though there exist different uncertain models, whether we are using probability, evidence theory, interval, or fuzzy set for uncertainty quantification, we are drinking the same water of truth from different sides of the same well. When different kinds of uncertainties involved in a problem, it is difficult to perform uncertainty analysis without ignoring important information and without introducing unwarranted assumptions at the same time [11]. To solve this problem, recently, many unified uncertainty analysis (UUA) methods integrating different types of uncertainties into a unified framework have been proposed. Based on the first-order reliability method, Du [12] studied a method to quantify the influence of mixed random and interval variables on the structural reliability. Jiang et al. [13] developed a probability and interval hybrid model, in which the uncertainty distribution parameters of random variables are intervals rather than exact values. Du [14] considered the combination of probability and epistemic uncertainties by using evidence theory, and investigated the corresponding sensitivity analysis [15]. Xiao et al. [16] provided a unified uncertainty analysis based on mean first-order saddle point approximation. Kang and Luo [17] established a hybrid reliability model based on probability and convex model. Wang et al. [18] proposed an uncertainty analysis method based on fuzzy and random variables. Lü et al. [19] proposed a unified stability analysis method with two types of random fuzzy uncertainty. Wang et al. [20] constructed a novel method based on active learning Kriging model and importance sampling to solve rare-event hybrid reliability problems with random and interval variables. Yang et al. [21] developed a hybrid adaptive Kriging model for complex reliability-based design optimization. Based on quadratic response surface and polynomial chaos expansion, Zhang and Qiu [22] exploited a fatigue reliability analysis to predict the structural fatigue life under random and interval hybrid uncertainties. For structure of small failure probability, Hong et al. [23] combined the Kriging model with subset simulation importance sampling for hybrid reliability analysis under random and ellipsoidal variables. These unified uncertainty analysis methods are important and meaningful for uncertain quantification and propagation analysis. Although the above UUA methods were proposed, the study field of unified uncertainty analysis methods is still in the stage of urgent development on the whole. On the one hand, the existing unified uncertainty analysis was mostly limited to the analysis of two kinds of uncertainty, that is, considering both probabilistic uncertainty and a certain cognitive uncertainty, and rarely consider three or more kinds of uncertainty at the same time of uncertainty. On the other hand, although a small amount of work

Uncertainty Distribution Estimation Based on Unified Uncertainty …

411

has built a unified analytical framework to deal with the coexistence of probability, evidence, fuzzy and interval variables, only the mean and variance of the response can be calculated, but the uncertainty distribution function of the response cannot be obtained [24]. However, the reliability analysis and risk assessment of practical engineering structures or systems usually require the uncertainty distribution of the response. Therefore, it is necessary to propose a UUA method involving probability variables, evidence variables, interval variables and fuzzy variables to effectively obtain the uncertainty distribution of the response. This paper presents a unified uncertainty distribution analysis method under probabilistic, evidence, fuzzy and interval uncertainties. This method extends our previous work [24] where only the mean and variance can be calculated to estimate the uncertainty distribution of the structural response. The organization of this paper is as follows: Sect. 2 introduces the unified moment analysis method, Sect. 3 presents the unified uncertainty distribution analysis, Sect. 4 analyzes three numerical examples to verify the effectiveness of the method, and Sect. 5 has concluding remarks.

2 Unified Moment Analysis According to the definition of the statistical moment of probability theory, the nth origin moment of the response function g(X, Y, Z, P) can be expressed as:   m gn = E [g(X, Y, Z, P)]n ∞ ∞ ··· = [g(X, Y, Z, P)]n f x (X)dX −∞

−∞

n = 1, 2, . . .

(1)

where E is the expection operator, f x (X) is the joint probability density funcT  tion, X = X 1 , X 2 , . . . , X k1 represents a k 1 dimensional vector of random variT  ables, Y = Y1 , Y2 , . . . , Yk2 denotes a k 2 dimensional vector of evidence variT  able, Z = Z 1 , Z 2 , . . . , Z k3 indicates a k 3 dimensional fuzzy variable vector, T  P = P1 , P2 , . . . , Pk4 is a k 4 dimensional interval variable vector. All the uncertain variables are independent. Assumed that Y, Z, P are constant vectors. According to the univariate dimension reduction (UDR) approach [25], Eq. (1) can be expressed as:

m gn

⎧⎡ ⎤n ⎫ k1  ⎪   ⎪ ⎪ ⎪ ⎨⎢ g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P ⎥ ⎬ ⎥ , =E ⎢ ⎣ j=1 ⎦ ⎪ ⎪ ⎪ ⎪   ⎭ ⎩ −(k1 − 1)g μ1 , . . . , μk1 , Y, Z, P

412

X. Long et al.

n = 1, . . . , 4

(2)

  where g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P represents the structural response based on X j , μ j , j = 1, 2, . . . , k1 represents the mean of jth random variable. Equation (2) can be extended into the following form based on the binomial theorem: n    n

m gn =

i=0

i

⎤i k1    g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P ⎦ E⎣ ⎡

j=1

  (n−i ) −(k1 − 1)g μ1 , . . . , μk1 , Y, Z, P n     (n−i) n i  = Sk1 −(k1 − 1)g μ1 , . . . , μk1 , Y, Z, P i

(3)

i=0

    n n represents the combination operator, that is = Cni = i i can be solved by the following recursive equation: where

n! , i!(n−i)!

Sik1

  i  , i = 1, . . . , n Si1 = E g X 1 , μ2 , . . . , μk1 , Y, Z, P i      (i−k)  i i Sj = , Skj−1 E g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P k k=0

j = 2, . . . , k1 − 1; i = 1, . . . , n i      (i−k)  i Sik1 = , Skk1 −1 E g μ1 , μ2 , . . . , μk1 −1 , X k1 , Y, Z, P k k=0

i = 1, . . . , n

(4)

According to Eqs. (2)–(4), the k 1 -dimensional integral represented by Eq. (1) can be decomposed into a series of one-dimensional integrals, which can be solved by the following equation: Igi = E

  i  g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P

∞ =

  i   g μ1 , . . . , μ j−1 , X j , μ j+1 , . . . , μk1 , Y, Z, P f x j X j dX j ,

−∞

i = 1, 2, . . . , n; j = 1, . . . , k1 r i 1    wq g μ1 , . . . , μ j−1 , X jq , μ j+1 , . . . , μk1 =√ π q=1

(5)

Uncertainty Distribution Estimation Based on Unified Uncertainty …

413

  √ where X jq = FX−1j  2h q , wq and h q are the qth gaussian integral weight and gaussian integral point, respectively, and r is the number of integral points. Substituting Eqs. (4) and (5) into Eq. (1), the statistical moment of the response function g(X, Y, Z, P) can be expressed as an integral function containing Y, Z, and P. The mean value M1 , the variance M2 , the third order central moment M3 and the fourth order central moment M4 of the response function g(X, Y, Z, P) can be obtained from the relationship as below: M1 = m g1

2  M2 = m g2 − m g1

3  M3 = m g3 − 3m g1 m g2 + 2 m g1   2 4 M4 = m g4 − 4m g4 m g3 + 6 m g1 m g2 − 3 m g1

(6)

Considering the evidence vector Y and the fuzzy vector Z, the first four order central moment of the response at a specific membership level can be obtained by the evidence analysis and fuzzy discretization [24]: Mn (g(X, Y, Zα , P)) =

l 

     Mn g X, Ys j , Zα , P · m Y s j

(7)

j=1

  where, Mn is the nth order central moment of the response function, and m Y s j is the basic probability assignment of focal element s j under the framework of joint identification of evidence variables, l is the number of focal elements. Since the Ys j , Zα and P in Eq. (7) have the nature of interval, Mn also belongs to an interval. Based on the efficient global optimization (EGO) [26], the upper and lower bounds of the first four order central moments can be calculated by using: M n (g(X, Y, Zα , P)) =

l 

     M n g X, Ys j , Zα , P · m Y s j

(8)

     M n g X, Ys j , Zα , P · m Y s j

(9)

j=1

M n (g(X, Y, Zα , P)) =

l  j=1

where M n (g(X, Y, Zα , P)) and M n (g(X, Y, Zα , P)) are the lower and upper bounds of the central moment of the structural response.

414

X. Long et al.

3 Unified Uncertainty Distribution Analysis The cumulative distribution function (CDF) of response function needs to be obtained to perform subsequent reliability analysis and design for practical engineering problems. However, as descript in Sect. 2, the statistical moment of response function is an interval with different sources of inputs. And it is difficult to obtain the CDF boundaries of the response function by using traditional methods such as saddle point approximation [27] or maximum entropy principle [28]. In this paper, Johnson p-box modeling method [29] is introduced to calculate the boundaries of the CDF based on the boundaries of statistical moment. A Johnson distribution cluster is a generalized distribution cluster with four parameters γ , δ, λ, and ξ . The Johnson distribution cluster has four types of distributions, namely, the normal distribution S N , the lognormal distribution SL , the bounded distribution S B , and the unbounded distribution SU [29]. And the standard normal distribution can be transformed into the four types of Johnson distribution above by using the previous four parameters and transformation formula. The relationship between the four parameters of Johnson distribution and the standard normal distribution is as follows:   g−ξ (10) w = γ + δf λ where w ∼ N (0, 1) obeys standard normal distribution, g represents the response quantified by Johnson variable, and f represents the transformation function corresponding to the Johnson distribution type. The transformation function of the different distribution types in the Johnson cluster to the standard normal distribution is as follows:  ⎧  ψ ⎪ ln bounded distribution S B ⎪ ψ+1 ⎪  ⎪ √ ⎨  2 f (ψ) = ln ψ + ψ + 1 unbounded distribution SU (11) ⎪ ⎪ ln(ψ) lognormal distribution S ⎪ L ⎪ ⎩ ψ normal distribution S N where ψ = (g − ξ )/λ. Under the membership α, the upper and lower boundaries of statistical moment of the response function g(X, Y, Zα , P) is calculated based on Eqs. (8) and (9). And the response function can be transformed to unbounded Johnson distribution based on the first four central moments. All the Johnson distribution matching the boundaries envelope of Johnson distribution cluster are defined as Johnson probability box (Johnson p-box). Johnson p-box model is used to characterize the response function g(X, Y, Zα , P) under each membership α, and calculate the probability boundaries of response. According to Eqs. (10) and (11), g ω of unbounded Johnson distribution can be expressed as:

Uncertainty Distribution Estimation Based on Unified Uncertainty …

415

   g ω = ξ + λ sinh −1 (ω) − γ /δ

(12)

where ω = (w) and 0 < ω < 1, g ω is the response function g with a specific ω. To calculate the probability boundaries of the response function, multiple optimizations need to be performed under different values of ω. For a specific ω, the boundaries of gαω can be calculated by the following optimization [30]: min / max gαω (δ, γ , λ, ξ ) δ,γ ,λ,ξ

s.t. M 1 (g(X, Y, Zα , P)) ≤ Mα J (1) (δ, γ , λ, ξ ) ≤ M 1 (g(X, Y, Zα , P)) M 2 (g(X, Y, Zα , P)) ≤ Mα J (2) (δ, γ , λ, ξ ) ≤ M 2 (g(X, Y, Zα , P)) M 3 (g(X, Y, Zα , P)) ≤ Mα J (3) (δ, γ , λ, ξ ) ≤ M 3 (g(X, Y, Zα , P))

(13)

M 4 (g(X, Y, Zα , P)) ≤ Mα J (4) (δ, γ , λ, ξ ) ≤ M 4 (g(X, Y, Zα , P)) 0.2 ≤ δ −50 ≤ γ ≤ 50 where Mα J (i) (δ, γ , λ, ξ ), i = 1, . . . , 4 are the first four central moments under the membership α of Johnson distribution with parameters γ , δ, λ and ξ. If δ → 0, the Johnson distribution will be close to an invalid region and the divisor in Eq. (12) cannot be 0, thus the constraint condition of parameter δ above is δ ≥ 0.2. The constraint − 50 ≤ γ ≤ 50 is to ensure that the calculated result of Eq. (12) is a finite non-zero value. Through calculating the maximum and minimum values of gαω under different values of α and ω, the uncertainty distribution boundaries of the response function g(X, Y, Zα , P) can be finally obtained.

4 Example Analysis 4.1 Example 1 This example was a crank slider mechanism in construction machinery as shown in Fig. 1 [31]. Considering the maximum stress of the connecting rod BC under the action of external force P, the strength of the connecting rod is analyzed. When the crank overlaps with the connecting rod, the maximum stress of the connecting rod is expressed as follows: σmax =

π

√

4P(b − a)

  (b − a)2 − e2 − μe d22 − d12

(14)

where a is the length of the crank AB, b is the length of the connecting rod BC, d1 and d2 are the inner and outer diameters of the connecting rod, respectively. The variables

416

X. Long et al.

Fig. 1 The crank slider mechanism [31]

a, b, d1 and d2 are all random variables. P is the external force on the slider, according to the theory of evidence, P is given in the form of evidence variable and assigned to each focal element basic uncertainty distribution. e is the offset distance between the crank and the slider, and it is ambiguous since different construction sites have different requirements for the installation position of the slider. Thus, the offset e is set as a fuzzy variable. μ represents the friction coefficient between the slider and the ground. Due to the uncertainty of the working environment, the precise distribution information of μ can not be obtained, hence the uncertainty of μ is represented by an interval variable. Details of all uncertain variables are shown in Table 1. Table 1 Uncertainty variable information in the crank slider mechanism Uncertain variables Distribution

Parameter 1

Parameter 2

Parameter 3

Uncertainty (%)

Crank length a

Normal

100 mm

5 mm

–

5

Connecting rod length b

Normal

300 mm

15 mm

–

5

Inner diameters of connecting rod d 1

Normal

30 mm

1.5 mm

–

5

Outer diameters of connecting rod d 1

Normal

55 mm

2.75 mm

–

5

External force P

Evidence

48,500 N

49,250 N

30%

0.77

49,250 N

50,750 N

30%

1.5

50,750 N

51,500 N

40%

0.73

72 mm

80 mm

88 mm

10

0.2

0.004

Offset distance e

Fuzzy

Friction coefficient Interval μ

2

In random variables, parameters 1 and 2 are the mean and the standard deviation of variables, respectively In evidence variables, parameter 1 is the lower limit of the focal element, parameter 2 is the upper limit of the focal element, and parameter 3 is the basic probability assignment In interval variables, parameters 1 and 2 are the midpoint and radius of variables, respectively In fuzzy variables, parameters 1 and 3 are the lower limit and upper limit of the variable when α = 0, respectively, and parameter 2 is the value of the variable when α = 1

Uncertainty Distribution Estimation Based on Unified Uncertainty …

417

The uncertainty distribution of the maximum stress on the connecting rod was calculated by the proposed method. First, the boundaries of the first four order central moments of the response function were calculated based on the UDR and the EGO. In the process of the EGO, 10 samples were taken to establish the initial Kriging surrogate model of the first four order central moments, and the expected improvement (EI) criterion [26] was used to update the model to solve the boundaries of the first four order moments. ε is the maximum EI index in EGO algorithm, and take ε = 0.00001 as the convergence condition. Then, the response function was converted into the Johnson variable, and the Johnson distribution was calculated based on the moment matching method. Finally, the uncertainty distribution Johnson p-box of the response was solved. The calculation result is shown in Fig. 2. The Monte Carlo method was given to verify the validity of the proposed method. In the MCS, the outer layer was the uncertainty analysis of fuzzy variables, evidence variables and interval variables, and the inner layer was the Monte Carlo simulation of probability analysis for random variables as the verification method. Figure 2 shows the uncertainty distribution of responses for membership levels of 0, 0.2, 0.4, and 0.7. The blue line in Fig. 2 indicates the upper and lower bounds of the uncertainty distribution calculated by the MCS, and the red line indicates the upper and lower bounds of the uncertainty distribution obtained by the proposed method. It can be seen that the calculation results of the proposed method are close to those of the MCS. The uncertainty distribution boundaries of the proposed method envelop the boundaries of the MCS. However, the number of function calls for this method is 25,428, which is much less compared to MCS’s 3.3E9 function calls. Therefore, the proposed method can efficiently calculate the uncertainty distribution of the response under the mixed uncertainty problem.

4.2 Example 2 Automobile crashworthiness analysis is an essential and important part in automobile safety design. There are mainly three kinds of collisions including frontal, side and rear collisions. Frontal collision includes head-on collision and offset collision. The final form of car crash safety design not only affects the overall performance of the car, but also affects the safety of drivers and passengers directly. Therefore, the ultimate goal of the crash test is to comprehensively consider the test results under different collisions, and then design the optimal vehicle structure and determine the external dimensions of key energy-absorbing components. However, the unavoidable error in the actual manufacturing process will lead to the uncertainty of the actual size of the parts, and it is of great significance to analyze the influence of the size uncertainty on the collision result. This example was a crashworthiness analysis of a vehicle in a low-speed offset collision [32]. The test layout scheme and the finite element model of the collision result of a frontal low-speed offset collision are shown in Fig. 3a, b, respectively. V0L is the collision speed, and it is set as 15 km/h. The offset is 40% to the driver’s side. The uncertain variables are the thickness of front

X. Long et al. Cumulative distribution function

Cumulative distribution function

418 The M CS The proposed method

The M CS The proposed method

(b) α =0.2 Cumulative distribution function

Cumulative distribution function

(a) α =0 The M CS The proposed method

The M CS The proposed method

(c) α =0.4

(d) α =0.7

Fig. 2 Uncertainty distribution of the maximum stress of crank-slider mechanism

bumper d 1 , the inner and outer panels thickness of energy-absorbing box d 2 and d 3 , and the inner and outer panels thickness of front longitudinal beam d4 and d5 . Table 2 gives the specific information of uncertainty variables. Through the response surface approach, the quadratic response surface of the total energy absorbed by the energy-absorbing component under the vehicle low-speed Fig. 3 The car frontal low-speed offset collision [32]

V0L = 15km / h

Engine

Front bumper Absorbing box front rails

(a) Crash test layout

(b) Collision finite element model

Uncertainty Distribution Estimation Based on Unified Uncertainty …

419

Table 2 Uncertainty variable information in the vehicle frontal low-speed offset collision Uncertain variables

Distribution

Parameter 1 (mm)

Parameter 2 (mm)

Parameter 3

Uncertainty (%)

Front bumper thickness d 1

Normal

2

0.04

–

2

Inner panel thickness of energy absorbing box d2

Normal

1.5

0.015

–

1

Outer panel thickness of energy absorbing box d3

Interval

1

0.02

–

2

Inner panel thickness of front longitudinal d 4

Evidence

1.26

1.33

20%

2.7

1.33

1.47

40%

5

1.47

1.54

40%

2.3

Outer panel thickness of front longitudinal d 5

Fuzzy

1.425

1.5

1.575 mm

5

In random variables, parameters 1 and 2 are the mean and the standard deviation of variables, respectively In evidence variables, parameter 1 is the lower limit of the focal element, parameter 2 is the upper limit of the focal element, and parameter 3 is the basic probability assignment In interval variables, parameters 1 and 2 are the midpoint and radius of variables, respectively In fuzzy variables, parameters 1 and 3 are the lower limit and upper limit of the variable when α = 0 respectively, and parameter 2 is the value of the variable when α = 1

offset collision condition can be represented as following [32]: u = 1094.281d1 + 4468.162d2 + 2921.606d3 − 7831.190d4 − 14,550.227d5 − 789.12d1 d2 − 1798.221d1 d3 + 557.348d1 d4 + 689.273d2 d3 + 975.459d1 d5 − 990.46d2 d4 − 884.136d2 d5 − 354.607d3 d4 + 522.592d3 d5 + 1857.173d4 d5 + 148.5d22 + 1349.944d42 + 2753.084d52 + 12,579.336

(15)

As is shown in Fig. 4, the proposed method was applied to calculate the uncertainty distribution of the total absorbed energy E. Figure 4 presents the Johnson p-box uncertainty distribution results of total energy absorption E when the membership degrees were 0, 0.3, 0.6, and 1. It can be seen that the uncertainty distribution boundaries calculated by the proposed method completely envelopes the calculation results

X. Long et al. The M CS The Proposed method

Cumulative distribution function

Cumulative distribution function

420

The M CS The Proposed method

The M CS The Proposed method

b) α =0.3

Cumulative distribution function

Cumulative distribution function

a) α =0

The M CS The Proposed method

c) α =0.6

d) α =1

Fig. 4 Uncertainty distribution of vehicle low-speed offset collision

of the MCS. Furthermore, the computational cost of this method is significantly lower than that of MCS, with 15,015 and 3.3E9 function calls, respectively.

4.3 Example 3 This example is a trestle bridge, and its appearance is shown in Fig. 5a. The trestle is composed of piers, abutments and steel beams, of which the steel beams include support frames and Bailey beams. A Bailey beam consists of five Bailey pieces that are composed of chords and webs. The structure diagram is shown in Fig. 5b. The maximum stress σmax of Bailey beam web under the condition of wind load W, the pedestrian load M1 , temperature T and the vehicle load M2 was considered. The finite element model of trestle bridge was established by using Midas Civil software, as shown in Fig. 5c. The section of web member was an I-shaped section, as shown in Fig. 5d. Beam element was applied to finite element model of web member. The wind load was equivalent to a line load applied on the node of the outermost Bailey beam. In this example, the bridge deck element was not established in order to simplify

Uncertainty Distribution Estimation Based on Unified Uncertainty …

chord

Bailey beam

Support frame

421

web

(a) Schematic diagram of trestle bridge Vehicle load M2

Distribution beam

(b) Schematic diagram of Bailey piece Flange thickness tf

Pedestrian load M2

Waist thickness tw Wind load W

(c) Finite element model

(d) I-shaped section

Fig. 5 The trestle bridge and finite element analysis

the model. However, for the purpose of meeting the actual force transfer situation, a distribution beam was established above the Bailey beam, and the lane load was distributed to the Bailey beam through the distribution beam. The lane load includes the pedestrian load M 1 and the vehicle load M 2 . There are six uncertain variables in this problem. The web waist thickness t w , the flange thickness t f of the web I-shaped section and the wind load W were characterized by random variables. The temperature T was modeled according to the evidence theory. The evidence variable T contains three focal elements, and its BPA structure is shown in Table 3. The pedestrian load M 1 was expressed as an interval variable. The fuzziness of triangular fuzzy number was applied to represent the uncertainty of vehicle load M 2 , and the α-cut method was used to discretize M 2 . All variables are independent variables, and their specific information is shown in Table 3. The relationship between the maximum stress σmax borne by the web member of Bailey beam and the input variable is shown as follows:   σmax = g tw , t f , W, M1 , M2 , T

(16)

The proposed UUA method was used to calculate the uncertainty distribution of the maximum stress borne by the web member of trestle bridge. The calculated

422

X. Long et al.

Table 3 Uncertainty variable information in the trestle bridges Uncertain variables

Distribution

Parameter 1

Parameter 2

Parameter 3

Uncertainty (%)

Web waist thickness t w

Normal

6.5 mm

0.65 mm

–

10

Flange thickness t f

Normal

4.5 mm

0.45 mm

–

10

Wind load W

Normal

0.5 N/mm

0.1 N/mm

–

20

Pedestrian load M 1

Interval

0.6 N/mm

0.12 N/mm

–

20

Temperature T

Evidence

Vehicle load M2

Fuzzy

7.5 °C

11.25 °C

20%

20

11.25 °C

18.75 °C

30%

25

18.75 °C

22.5 °C

50%

9.1

64 kN

80 kN

96 kN

20

In random variables, parameters 1 and 2 are the mean and the standard deviation of variables, respectively In evidence variables, parameter 1 is the lower limit of the focal element, parameter 2 is the upper limit of the focal element, and parameter 3 is the basic probability assignment In interval variables, parameters 1 and 2 are the midpoint and radius of variables, respectively In fuzzy variables, parameters 1 and 3 are the lower limit and upper limit of the variable when α = 0 respectively, and parameter 2 is the value of the variable when α = 1

uncertainty distribution results of the response are shown in Fig. 6. The figure shows the uncertainty distribution of the response function when the membership degree α is 0, 0.3, 0.6 and 1, respectively. It can be seen from Fig. 6 that the response probability boundaries lines calculated by the two methods are very close. The convergence conditions of the EGO in calculating statistical moments are extremely harsh, resulting in as many as 28,940 function calls to calculate the above accurate results. However, the number of function calls of MCS is 3.3E9, and the computational efficiency of this method still has a great advantage over the MCS. This example again demonstrates the accuracy and efficiency of the proposed method.

5 Conclusions A unified uncertainty analysis method is proposed to calculate the probability boundaries of structural responses under probability, evidence, fuzzy and interval uncertainties. Three numerical examples are performed to verify the validity of the proposed method. The results show that the response uncertainty distribution calculated by the proposed method is extremely close to the calculation results of double loop sampling MCS. In terms of computational efficiency, the computational cost of the proposed

The MCS

The proposed method

Cumulative distribution function

Cumulative distribution function

Uncertainty Distribution Estimation Based on Unified Uncertainty …

The MCS

The proposed method

(b) α = 0.3

The proposed method

(c) α = 0.6

Cumulative distribution function

Cumulative distribution function

(a) α = 0 The MCS

423

The MCS

The proposed method

(d) α = 1

Fig. 6 Uncertainty distribution of the maximum stress of trestle bridge

method is much less than that of double loop sampling MCS method, which demonstrates the effectiveness of the proposed method. In the future, the proposed method is promised to solve the unified uncertainty analysis in nonlinear and black-box problems. Acknowledgements Supported by National Science Foundation (Grant No. 52175134), National Science Foundation for Distinguished Young Scholars (Grant No. 51725502), and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51621004).

References 1. Xiong F (2015) Engineering probabilistic uncertainty analysis method. Science Press 2. Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19(1):3–19

424

X. Long et al.

3. Xiong F, Xiong Y, Greene S, Chen W, Yang S (2009) A new sparse grid based method for uncertainty propagation. In: International design engineering technical conferences and computers and information in engineering conference, vol 49026, pp 1205–1215 4. Bucher CG, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7(1):57–66 5. Zhang L, Lu Z, Wang P (2015) Efficient structural reliability analysis method based on advanced Kriging model. Appl Math Model 39(2):781–793 6. Hong HP (1998) An efficient point estimate method for probabilistic analysis. Reliab Eng Syst Saf 59(3):261–267 7. Jiang C, Han X, Lu GY, Liu J, Zhang Z, Bai YC (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200(33):2528–2546 8. Mourelatos ZP, Zhou J (2005) Reliability estimation and design with insufficient data based on possibility theory. AIAA J 43(8):1696–1705 9. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28 10. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B Methodol 39(1):1–22 11. Beer M, Ferson S, Kreinovich V (2013) Imprecise probabilities in engineering analyses. Mech Syst Signal Process 37(1):4–29 12. Du X (2007) Interval reliability analysis. In: ASME 2007 international design engineering technical conferences and computers and information in engineering conference, volume 6: 33rd design automation conference, parts A and B, pp 1103–1109 13. Jiang C, Han X, Li WX, Liu J, Zhang Z (2012) A hybrid reliability approach based on probability and interval for uncertain structures. J Mech Des 134(3):031001 14. Du X (2008) Unified uncertainty analysis by the first order reliability method. J Mech Des 130(9):091401 15. Guo J, Du X (2007) Sensitivity analysis with mixture of epistemic and aleatory uncertainties. AIAA J 45(9):2337–2349 16. Xiao NC, Huang HZ, Wang Z, Liu Y, Zhang XL (2012) Unified uncertainty analysis by the mean value first order saddlepoint approximation. Struct Multidiscip Optim 46(6):803–812 17. Kang Z, Luo Y (2010) Reliability-based structural optimization with probability and convex set hybrid models. Struct Multidiscip Optim 42(1):89–102 18. Wang Z, Huang HZ, Li Y, Pang Y, Xiao NC (2012) An approach to system reliability analysis with fuzzy random variables. Mech Mach Theory 52:35–46 19. Lü H, Shangguan WB, Yu D (2017) A unified approach for squeal instability analysis of disc brakes with two types of random-fuzzy uncertainties. Mech Syst Signal Process 93:281–298 20. Wang T, Yang X, Mi C (2021) An efficient hybrid reliability analysis method based on active learning Kriging model and multimodal-optimization-based importance sampling. Int J Numer Methods Eng 122(24):7664–7682 21. Yang M, Zhang D, Jiang C, Han X, Li Q (2021) A hybrid adaptive Kriging-based single loop approach for complex reliability-based design optimization problems. Reliab Eng Syst Saf 215:107736 22. Zhang Z, Qiu Z (2021) Fatigue reliability analysis for structures with hybrid uncertainties combining quadratic response surface and polynomial chaos expansion. Int J Fatigue 144:106071 23. Hong L, Li H, Fu J (2022) Novel Kriging-based variance reduction sampling method for hybrid reliability analysis with small failure probability. ASCE-ASME J Risk Uncertainty Eng Syst Part A Civ Eng 8(2):04022017 24. Long XY, Mao DL, Jiang C, Wei FY, Li GJ (2019) Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties. Comput Methods Appl Mech Eng 355:1–26 25. Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408

Uncertainty Distribution Estimation Based on Unified Uncertainty …

425

26. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Global Optim 13(4):455–492 27. Rowe NC (1988) Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative transformations. SIAM J Sci Stat Comput 9(6):1098–1113 28. Siddall JN, Diab Y (1975) The use in probabilistic design of probability curves generated by maximizing the Shannon entropy function constrained by moments. J Eng Ind 97(3):843–852 29. Johnson NL (1949) Systems of frequency curves generated by methods of translation. Biometrika 36(1/2):149–176 30. Liu HB, Jiang C, Jia XY, Long XY, Zhang ZY, Guan FJ (2018) A new uncertainty propagation method for problems with parameterized probability-boxes. Reliab Eng Syst Saf 172:64–73 31. Du X (2006) Uncertainty analysis with probability and evidence theories. In: ASME 2006 international design engineering technical conferences and computers and information in engineering conference, volume 1: 32nd design automation conference, parts A and B, pp 1025–1038 32. Jiang C, Deng S (2014) Multi-objective optimization and design considering automotive highspeed and low-speed crashworthiness. Chin J Comput Mech 31(4):474–479

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel Mechanism Hui Wang, Jiale Han, Yulei Hou, Haitao Liu, and Ke Xu

Abstract This paper deals with the conceptual design and kinematic analysis of a novel six degrees of freedom (DOF) parallel mechanism. The newly invented mechanism is a modified version of the F-200iB industrial robot, achieved by integrating the universal joints of a pair of limbs into a two-in-one part. The mobility of the proposed parallel mechanism is analyzed with the aid of the theory of mechanism topology. The inverse and forward kinematic models of the proposed parallel mechanism are derived in brief and validated by numerical simulation. Keywords Parallel robot · Kinematics · Conceptual design

1 Introduction Parallel mechanisms have drawn interest in both academia and industry for over forty years due to their potential advantages of high load carrying capacity, high static H. Wang · Y. Hou School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China e-mail: [email protected] Y. Hou e-mail: [email protected] H. Wang Basic Experimental and Training Center, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China J. Han (B) · H. Liu · K. Xu Key Laboratory of Modern Mechanisms and Equipment Design of the State Ministry of Education, Tianjin University, Tianjin 300072, China e-mail: [email protected] H. Liu e-mail: [email protected] K. Xu e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_24

427

428

H. Wang et al.

and dynamic performance, and high accuracy [1]. From the perspective of mobility, parallel mechanisms can be classified into types with fewer degrees of freedom (DOF) and 6-DOF. Among such mechanisms with 6-DOF, the most celebrated one is the generalized Stewart platform, successfully employed for flight simulators, machine tools, and high-precision positioning systems [2]. The very typical design of a Stewart platform consists of six UPS (universalprismatic-spherical) limbs connecting the moving platform to the base. By driving the prismatic joints independently, six linear translations lead to three positional and three orientational degrees of freedom of the moving platform. According to the arrangement of the universal and spherical joints, there are 6–6, 6–3, and 3–3 architectures, respectively [2]. Pioneering work on this kind of parallel mechanisms can be traced back to the 1980s [3, 4]. Since then, extensive research activities have been carried out toward the design, analysis, and application of the Stewart platform and its variants [5–10]. However, it is frustrating to see that the topologies of most 6DOF parallel mechanisms proposed in literature are similar to that of the generalized Stewart platform using UPS limbs. One outstanding design worth mentioning is the F-200iB industrial robot developed by FANUC, as shown in Fig. 1. The particularity of the architecture is that a pair of limbs connect to the moving platform via a unique arrangement of four revolute joints instead of spherical joints. In this manner, six 1-DOF revolute joints are saved compared to the 6–6 type Stewart platform, resulting in a more rigid and compact mechanical design. Although little attention is paid to analyzing the mechanism itself [11], this robot has drawn great interest from users in robotic machining [12–14]. Inspired by the design of F-200iB, this paper presents a new 6-DOF parallel mechanism. The proposed mechanism simplifies the F-200iB by integrating the universal joints of a pair of limbs into a two-in-one part while keeping the motion unchanged. The conceptual design and mobility analysis of the new design are introduced, followed by deriving its inverse and forward kinematics. Numerical simulation is taken to validate the kinematic models before summarizing the conclusions.

2 Conceptual Design and Mobility Analysis Typically, a 6-DOF parallel mechanism is composed of six UPS limbs. Each individual limb connects to the base via a universal joint and to the moving platform via a spherical join or three revolute joints intersecting at a common center. To break this arrangement, we are keen to seek a new architecture with a topological structure simpler than the Stewart platform while having the same motion capability. By investigating the structure of the F-200iB robot, it can be observed that a pair of limbs share two revolute joints connecting to the moving platform while the limbs are still attached to the base through a universal joint. It naturally asks: is it possible to further simplify the structure by making universal joints share common rotational axes while maintaining their original functionality? In this way, more revolute joints can be saved. Bearing this notion in mind, we indeed find a feasible arrangement

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel …

429

Fig. 1 FANUC F-200iB industrial robot

of the universal joints, leading to the innovation of a 3-R(2RPR)U (R: revolute joint) parallel mechanism with 6-DOF. Figure 2 shows the three-dimensional (3D) model of the proposed mechanism. It can be seen that the mechanism is composed of three symmetrically distributed hybrid sub-chains. The schematic diagram of a sub-chain is illustrated in Fig. 3, where a composite joint (a planar six-bar linkage 2RPR) connects to the base and the moving platform through a revolute joint and a universal joint, respectively. The significance of the innovative design is that a pair of universal joints are elaborately designed by using a single (two-in-one) part, thereby allowing them to share a common rotational axis. In total, there are six prismatic joints and twenty-one revolute joints. Compared with a 6–6 type Stewart platform using three intersecting revolute joints as the spherical joint, the new design can save nine revolute joints. Theoretically, a fewer number of joints would have a beneficial effect on the static performance of the mechanism. To verify the DOF of the proposed parallel mechanism, the mobility analysis is carried out. Noting the existence of composite joints (overconstraint), the Chebychev–Grüble–Kutzbach’s criterion is not applicable. Therefore, the theory of mechanism topology [15] is employed in this paper. According to this method, the position and orientation characteristics (POC) sets of limbs, independent loops, and independent displacement equations of each loop need to be evaluated [16–18]. As shown in Fig. 3, the proposed parallel mechanism has three hybrid sub-chains with the same structure. In each sub-chain, the planar linkage 2RPR is taken as a

430

H. Wang et al.

Fig. 2 3D model of the proposed parallel mechanism

Fig. 3 Schematic diagram of the ith hybrid sub-chain

Ui

Moving platform

RPR limb

Base Two-in-one part R 2, i R 1,i

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel …

431

composite joint with two translations and one rotation. According to [15], the POC set of the ith (i = 1, 2, 3) sub-chain can be expressed as Mbi = MR1i ∪ M2RPR ∪ MUi  1   2   2  t (⊥ R1i ) t (⊥ R2i ) t (⊥ Ui ) = ∪ ∪ , i = 1, 2, 3 r 1 (|| R1i ) r 2 (|| Ui ) r 1 (|| R2i )  3 t = r3

(1)

where R1i , R2i , and Ui represent the rotational axes of the revolute and universal joints, respectively (see Fig. 3); t and r denote translation and rotation, respectively. According to the approach for decomposition of independent loops, the first independent loop can be defined by the sub-chains 1 and 2, while the second independent loop is obtained by regarding the first independent loop as a closed-loop mechanism and attaching the sub-chain 3. With these definitions available, the number of independent displacement equations of two loops can be achieved by  3  t   ξ1 = dim . Mb1 ∪ Mb2 = dim . =6 r3  3  t   ξ2 = dim . Mb1 ∩ Mb2 ∪ Mb3 = dim . =6 r3

(2)

(3)

where dim.{*} denotes the dimension of a POC set, i.e., the number of independent motions. Subsequently, the DOF of the parallel mechanism can be obtained by F=

3

3

i=1 j=1

f j,i −

2

ξk = 18 − 12 = 6

(4)

k=1

where f j,i is the DOF of the jth joint in the ith sub-chain. Here, f 1,i = 1, f 2,i = 3, and f 3,i = 2. To assess the complexity of the mechanism, the coupling degree can be considered as a metric. For the evaluation of coupling degree, the mechanism should be decomposed into single open chains (SOCs) and the constraint degree describing the constraint relationship of each individual SOC has to be evaluated [15]. For this mechanism, the first independent loop can be taken as the first SOC, and the sub-chain 3 is the second one. Hence, the constraint degrees of two SOCs are 1 =

2

3

i=1 j=1

f i, j − I1 − ξ1 = 12 − 4 − 6 = 2

(5)

432

H. Wang et al.

2 =

3

f 3, j − I2 − ξ2 = 6 − 2 − 6 = − 2

(6)

j=1

where I1 and I2 are the numbers of actuated prismatic joints in two SOCs, respectively. Finally, the coupling degree of the parallel mechanism can be obtained by κ=

1 (|1 | + |2 |) = 2 2

(7)

Equation (7) demonstrates that the constraints of two SOCs are highly coupled. According to [15], it can be concluded that there is no analytical solution for the forward kinematics. However, its coupling degree is lower than that of a 6–6 Stewart platform, whose coupling degree is 3 [19], meaning that the topology of the proposed mechanism is simpler.

3 Kinematic Analysis The inverse and forward kinematic modeling is the basis of design, analysis, and control of a parallel mechanism. The inverse kinematics solves the actuated joint variables (rotational angles and/or sliding lengths) given the position and orientation of the moving platform. In contrast, the forward kinematics takes the vice versa procedure. This section derives the inverse and forward kinematics of the proposed parallel mechanism and validates the models by numerical simulation.

3.1 Inverse Kinematics Figure 4 illustrates the definition of vectors for kinematic analysis in the ith hybrid sub-chain. For convenience, two reference frames K and K A are fixed on the base and platform at their center points B and A, respectively. All vectors are measured in frame K unless otherwise specified. For inverse kinematics, the position vector r of A measured in K and the rotation matrix R describing the orientation of K A with respect to K are known. Given r and R, the closed-loop constraint equations associated with the ith (i = 1, 2, 3) sub-chain can be formulated as r = b j,i + q j,i w j,i + (− 1) j+1 l1 cˆ i + l2 dˆ i − Rai ,

j = 1, 2

(8)

where ai and b j,i are the constant position vectors of Ai and B j,i measured in K A and K, respectively; l1 cˆ i = C1,i Di and l2 dˆ i = Di Ai with cˆ i and dˆ i denoting the unit vectors; q j,i and w j,i are the length and the unit vector of translational direction of

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel …

433

Fig. 4 Definition of vectors in the ith hybrid sub-chain

the RPR limb j ( j = 1, 2) in the ith sub-chain (see Fig. 3). In Eq. (8), q j,i , w j,i , cˆ i , and dˆ i are unknowns to be solved. Letting s1,i and s3,i be the unit vectors along the axial axes of the first and last revolute joints connecting to the base and moving platform, respectively, and s2,i the unit vector along axial axes of the rear revolute joint of the RPR limb, the following geometrical relationships can be found  s2,i ⊥ s1,i , s2,i ⊥ r − b1,i + Rai,0 dˆ i ⊥ s2,i , dˆ i ⊥ s3,i

(9)

cˆ i ⊥ s2,i , cˆ i ⊥ dˆ i Noting that s3,i = Rs3,i where s3,i is a constant vector measured in K A , we can derive  s1,i × r − b1,i + Rai,0   (10) s2,i =   s1,i × r − b1,i + Rai,0  s2,i × s3,i  dˆ i =   s2,i × s3,i 

(11)

cˆ i = s2,i × dˆ i

(12)

Hence, Eqs. (11) and (12) solve the unknown vectors cˆ i and dˆ i , respectively. Then, rearranging and taking the norm on both sides of Eq. (8) leads to the solutions of q j,i and w j,i , i.e.,

434

H. Wang et al.

    q j,i =  r − b j,i + Rai − (− 1) j+1 l1 cˆ i − l2 dˆ i , w j,i =

r − b j,i + Rai − (− 1) j+1l1 cˆ i − l2 dˆ i , q j,i

j = 1, 2

(13)

j = 1, 2

(14)

Consequently, the inverse kinematics problem of the proposed parallel mechanism can be solved by taking the same procedure for all sub-chains.

3.2 Forward Kinematics The forward kinematics deals with determining the position and orientation of the moving platform given six limb lengths q j,i ( j = 1, 2, i = 1, 2, 3). Since the mobility analysis confirms that there is no analytical solution for the forward kinematics of the proposed mechanism, the Newton–Raphson method is employed in this study. This method has been widely used by researchers for solving this problem of different parallel mechanisms [20, 21]. Due to the complexity of the mechanism, the position vectors of C1,i and C2,i measured in K are defined as intermediate variables to formulate the forward kinematics, i.e., T  r c, j,i = xc, j,i yc, j,i z c, j,i ,

j = 1, 2, i = 1, 2, 3

(15)

where the coordinates xc, j,i , yc, j,i , and z c, j,i are unknowns to be determined. Given r c, j,i , the rest vectors of the ith sub-chain involved in the forward kinematics can be expressed as  s1,i × r c,1,i − b1,i   =  s1,i × r c,1,i − b1,i 

(16)

r c,2,i − r c,1,i  cˆ i =   r c,2,i − r c,1,i 

(17)

dˆ i = s2,i × cˆ i

(18)

r a,i = r c, j,i + l1 cˆ i + l2 dˆ i

(19)

s2,i

where r a,i is the position vector of Ai measured in K. Then, the constraint equations in the ith sub-chain can be derived as    r c,2,i − r c,1,i  = 2l1

(20)

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel …

   r c, j,i − b j,i  = q j,i , 

r c,2,i − r c,1,i

T

435

j = 1, 2

(21)

s2,i = 0

(22)

In addition, the constraint equations among sub-chains can be formulated as 

T r a,2 − r a,3 dˆ 1 = 0

(23a)



T r a,1 − r a,3 dˆ 2 = 0

(23b)



T r a,1 − r a,2 dˆ 3 = 0

(23c)

√    r a,1 − r a,2  = a 3

(24a)

√    r a,1 − r a,3  = a 3

(24b)

√    r a,2 − r a,3  = a 3

(24c)

  where a =  ai  (i = 1, 2, 3) is the radius of the moving platform. Using Eqs. (20)– (24), the forward kinematics can be expressed as a system of 18 equations with 18 unknowns. Having these equations, we can define the optimization function as F(X) =

2 3 3



     r c,2,i − r c,1,i  − 2l1 2 +  r c, j,i − b j,i  − q j,i 2 i=1

+

i=1 j=1

3



r c,2,i − r c,1,i

T

2 s2,i

+



T 2 r a,2 − r a,3 dˆ 1

i=1

T 2  T 2 r a,1 − r a,3 dˆ 2 + r a,1 − r a,2 dˆ 3  √ 2  √ 2   +  r a,1 − r a,2  − a 3 +  r a,1 − r a,3  − a 3  √ 2  +  r a,2 − r a,3  − a 3  T X = X T1 X T2 X T3  T X i = xc,1,i yc,1,i z c,1,i xc,2,i yc,2,i z c,2,i +



(25)

The simplification of Eq. (25) will not be presented in this paper due to the length limits. By applying the Newton–Raphson method to solve Eq. (25) and with the aid of Eqs. (16)–(19), the position vector of Ai , i.e., r a,i , can be determined. Then, the position vector r of A measured in K can be obtained by

436 Table 1 Dimensional parameters for numerical simulation

H. Wang et al.    a  (mm)

   b j,i  (mm)

l 1 (mm)

l 2 (mm)

100

300

40

52

i

r=

r a,1 + r a,2 + r a,3 3

(26)

Meanwhile, the rotation matrix R of K A with respect to K can be determined using cross products of r a,1 − r a,2 , r a,1 − r a,3 , and r a,2 − r a,3 according to the specific definition of K A . Hence, the forward kinematics problem of the proposed parallel mechanism is solved.

3.3 Numerical Simulation This section aims to validate the algorithm for the forward kinematic analysis based on the Newton–Raphson method. The approach involves randomly selecting a pose of the mechanism within its workspace, evaluating the limb lengths using the derived inverse kinematics, and taking the limb lengths as inputs to the forward kinematic equations. For the numerical simulation, a set of dimensional parameters of the parallel mechanism is specified in Table 1. Then, given the ranges of the x, y, and z coordinates of point A measured in K and three rotational angles α, β, and γ consecutively rotating about the x-, y-, and z-axis (see Table 2), a number of 30 random poses across the workspace are taken into analysis. The position and angular values evaluated by the forward kinematics are compared with those given for the inverse kinematics. The results given in Table 2 shows that the forward kinematic analysis exhibits high accuracy, and the errors are mainly attributed to the round-off errors. It also finds that the algorithm converges to the true values with an average of seven iterations.

4 Conclusions This paper proposes a new 6-DOF parallel mechanism with a topology different from the Stewart platform. Due to the elaborate design of the universal joints, it can save up to 9 revolute joints, resulting in a very compact yet rigid design. The mobility analysis of the mechanism verifies its DOF and evaluates its coupling degree, demonstrating that the forward kinematics has no analytical solutions. The analytical expression of the inverse kinematics is derived, while the numerical approach for the forward kinematics is presented based on the Newton–Raphson method. By taking random poses of the parallel mechanism, the results of numerical simulation validate the forward kinematic analysis and show highly accurate solutions to the position and orientation of the moving platform.

Conceptual Design and Kinematic Analysis of a New 6-DOF Parallel … Table 2 Motion ranges and simulation results

Range

Minimum

437 Maximum

x (mm)

− 200

200

y (mm)

− 200

200

800

1200

z (mm) α (°)

− 20

20

β (°)

− 20

20

0

40

γ (°) Position error (x, y, z)

Value

Maximum (mm)

6 × 10−8

Minimum (mm)

0

Mean (mm)

3 × 10−9

Standard deviation (mm)

2 × 10−9

Angular error (α, β, γ )

Value

Maximum (°)

2 × 10−7

Minimum (°)

0

Mean (°)

5 × 10−8

Standard deviation (°)

2 × 10−8

Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 91948301 and 51721003) and Foundation of Tianjin Sino-German University of Applied Sciences (Grant No. zdkt2018-009), and EU H2020-RISE-ECSASDP (Grant No. 734272).

References 1. Merlet JP (2005) Parallel robots. Springer 2. Dasgupta B, Mruthyunjaya TS (2000) The Stewart platform manipulator: a review. Mech Mach Theory 35(1):15–40 3. Earl CF, Rooney J (1983) Some kinematic structures for robot manipulator designs. ASME J Mech Transm Autom Des 15(1):15–22 4. Hunt KH (1983) Structural kinematics of in-parallel-actuated robot-arms. ASME J Mech Transm Autom Des 15(4):705–712 5. Husty ML (1996) An algorithm for solving the direct kinematics of general Stewart–Gough platforms. Mech Mach Theory 31(4):365–379 6. Davliakos I, Papadopoulos E (2008) Model-based control of a 6-DOF electrohydraulic Stewart– Gough platform. Mech Mach Theory 43(11):1385–1400 7. Ophaswongse C, Murray RC, Agrawal SK (2018) Wrench capability of a Stewart platform with series elastic actuators. J Mech Robot 10(2):021002 8. Kim Y-S, Shi H, Dagalakis N et al (2019) Design of a six-DOF motion tracking system based on a Stewart platform and ball-and-socket joints. Mech Mach Theory 133:84–94 9. Markou AA, Elmas S, Filz GH (2021) Revisiting Stewart–Gough platform applications: a kinematic pavilion. Eng Struct 249:113304 10. Ono T, Eto R, Yamakawa J et al (2022) Analysis and control of a Stewart platform as base motion compensators—part I: kinematics using moving frames. Nonlinear Dyn 107(1):51–76

438

H. Wang et al.

11. Pham M-N, Champliaud H, Liu Z et al (2022) Parameterized finite element modeling and experimental modal testing for vibration analysis of an industrial hexapod for machining. Mech Mach Theory 167:104502 12. Tunc LT, Shaw J (2016) Experimental study on investigation of dynamics of hexapod robot for mobile machining. Int J Adv Manuf Technol 84(5–8):817–830 13. Tunc LT, Shaw J (2016) Investigation of the effects of Stewart platform-type industrial robot on stability of robotic milling. Int J Adv Manuf Technol 87(1–4):189–199 14. Tunc LT, Stoddart D (2017) Tool path pattern and feed direction selection in robotic milling for increased chatter-free material removal rate. Int J Adv Manuf Technol 89(9–12):2907–2918 15. Yang TL, Liu AX, Shen HP et al (2018) Topology design of robot mechanisms. Springer 16. Yang TL, Liu AX, Luo YF et al (2009) Position and orientation characteristic equation for topological design of robot mechanisms. J Mech Des 131(2):021001 17. Yang TL, Liu AX, Shen HP et al (2013) On the correctness and strictness of the POC equation for topological structure design of robot mechanisms. J Mech Robot 5(2):021009 18. Yang TL, Liu AX, Shen HP et al (2018) Composition principle based on single-open-chain unit for general spatial mechanisms and its application. J Mech Robot 10(5):051005 19. Shen HP, Yin HB, Wang Z et al (2013) Research on forward position solutions for 6-SPS parallel mechanisms based on topology structure analysis. J Mech Eng 49(21):70–80 20. Yang C-F, Zheng S-T, Jin J et al (2010) Forward kinematics analysis of parallel manipulator using modified global Newton–Raphson method. J Cent South Univ Technol (Engl Ed) 17(6):1264–1270 21. Abo-Shanab RF (2014) An efficient method for solving the direct kinematics of parallel manipulators following a trajectory. J Autom Control Eng 2(3):228–233

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit into Human Body Curve Weilin Lv, Wansui Nie, and Jianjun Zhang

Abstract A nursing bed is helpful for patients confined to the bed for a long period of time and rigid origami is a potential source for designing support mechanism of nursing beds. This paper presents a rigid origami nursing bed support mechanism that is able to fit into human body curve. Firstly, the relation between degree-4 origami vertex and spherical 4R linkages is introduced, together with the kinematics of spherical 4R linkages. Secondly, the origami pattern of the support mechanism is proposed, and the kinematics of the back board, which can be folded to fit into human body curve, is analyzed. Finally, the back-lifting, turning and leg-lifting mechanism of the support mechanism is introduced together with their kinematics. The mechanism proposed in this paper provides multiple choices for the design of nursing beds. Keywords Nursing bed support mechanisms · Rigid origami · Shape-morphing structure

1 Introduction A nursing bed is helpful for patients confined to the bed for a long period of time [1]. There have been considerable efforts to improve comfort of nursing beds, such as the Care Assist bed designed by Hill-Rom Company [2], the RotoFlex standup bed produced by PhysioNova in Germany [3], and the robotic bed designed by W. Lv (B) · W. Nie · J. Zhang School of Mechanical Engineering, Hebei University of Technology, Tianjin 300400, China e-mail: [email protected] W. Nie e-mail: [email protected] J. Zhang e-mail: [email protected] Hebei Provincial Key Laboratory of Robot Perception and Human-Machine Fusion, Hebei University of Technology, Tianjin 300400, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_25

439

440

W. Lv et al.

Panasonic Corporation in Japan [4]. However, the support mechanism of the nursing beds outlined above do not fit into human curve, which may cause local pressure concentrations on a bedridden body and give rise to bedsores. Rigid origami is referred to the case that each paper facet surrounded with creases is not stretching or bending during folding, is a continuous, one-to-one mapping of a crease pattern to create a three-dimensional object, which makes rigid origami a potential source for designing support mechanism of nursing beds that fit into human body curve. Efforts have been done to find out origami patterns that are able to fold into specific configuration. On the one hand, rigid origami has been investigated from the viewpoint of geometry. Miura [5] presented a proposition of intrinsic geometry of origami based on an arbitrary point on the surface of origami works. Watanabe and Kawaguchi [6] proposed two methods to judging rigid foldability of origami patterns from the compatibility matrix. Based on separating each crease of an origami pattern into two parallel creases, Hull and Tachi [7] presented the double line method to obtain new origami patterns. He and Guest [8] studied the configuration space of four-crease origami patterns and generated two families of rigid-foldable origami patterns with four-crease vertexes. And recently, Dang et al. develop a general, efficient and widely applicable inverse design framework to achieve a targeted surface by controlled deployment of a rigidly and flat-foldable quadrilateral mesh origami crease pattern [9]. On the other hand, rigid origami can be studied with a kinematic approach, where its facets and crease lines can be replaced by rigid panels and hinges [10]. In particular, a four-crease vertex can be represented by a spherical 4R linkage [11–13]. The typical origami crease patterns and their corresponding equivalent closed-loop linkage were investigated by Zhang and Dai [14]. Wei and Dai [15] analyzed an origami carton by representing it with one planar four-bar loop and two spherical 4R linkage loops. With the tessellation method for mobile assemblies of spatial linkages [16–18], Liu [19] used the assemblies of spherical 4R linkages to analyze the rigid origami patterns and presented several new patterns. The research on rigid origami prompted us to think whether it could be adopted to create novel support mechanism of nursing beds. Our endeavor leads to this paper. This paper presents a nursing bed support mechanism with rigid origami patterns formed by degree-4-vertex, which makes the nursing bed fit into human curve and is able to help patients turn over and lift their back and legs. In Sect. 2, the relation between a degree-4 vertex of rigid origami pattern and a spherical 4R linkage is introduced, and the kinematics of a single spherical 4R linkage is presented. In Sect. 3, the origami support mechanism, whose back board is designed to fit into human body curve, is introduced, and the kinematics of its back board is provided by regarding the back board as an assembly of spherical 4R linkages. The back-lifting, turning and leg-lifting mechanism are presented in Sect. 4, and their kinematics are analyzed according to the closing-vector-circle method. Section 5 gives conclusions and ends the paper.

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit …

441

2 Kinematics of Spherical 4R Linkages Rigid origami can be analyzed using a kinematic approach. Since the research of Cundy and Rollett [20], it has been widely acknowledged that for every rigid origami structure, there exists an equivalent linkage. The left part of Fig. 1 shows a degree-4 origami vertex containing four panels or sectors 1–4, and four creases AO, BO, CO and DO; the four creases intersect at a common point O. The four sector angles between the adjacent creases are α12 , α23 , α34 and α41 ; and the four dihedral angles between the adjacent sectors are ω1 , ω2 , ω3 and ω4 . From the mechanism viewpoint, by considering the sectors and creases as links and revolute joints, respectively, an equivalent spherical 4R linkage can be obtained, as shown in the right part of Fig. 1, and the frames are set up according to the D-H notation [21]. Wherein the zi -axis (i = 1, 2, 3 and 4) is along the joint axis of joint Ai ; xi -axis is normal to the plane formed by the zi and zi+1 axes such that xi = zi × zi+1 (it should be noted that when the subscript i + 1 = 5, it is replaced with 1); yi -axis can be found with right-hand rule and the origin Oi of each coordinate frame coincides with point O. Angle θ i+1 is defined as joint angle from the xi -axis to the xi+1 -axis, positively about zi+1 -axis; and angle αi (i+1) is the twist angle from zi to zi+1 positively about axis xi , and we have Q 21 Q 32 Q 43 Q 14 = I 3 ,

(1)

where ⎡

Q (i+1)i

⎤ cos θi − sin θi cos αi (i+1) sin θi sin αi(i+1) = ⎣ sin θi cos θi cos αi(i+1) − cos θi sin αi (i+1) ⎦. 0 sin αi(i+1) cos αi(i+1)

Merging Eqs. (1) and (2) gives sin αi (i+1) cos α(i+1)(i+2) sin α(i+3)(i+4) cos θi

Fig. 1 A four-crease origami vertex and its corresponding spherical 4R linkage

(2)

442

W. Lv et al.

+ sin αi (i+1) sin α(i+1)(i+2) cos α(i+3)(i+4) cos θi+1 + cos αi (i+1) sin α(i+1)(i+2) sin α(i+3)(i+4) cos θi cos θi+1 − sin α(i+1)(i+2) sin α(i+3)(i+4) sin θi sin θi+1 − cos αi (i+1) cos α(i+1)(i+2) cos α(i+3)(i+4) + cos α(i+2)(i+3) = 0,

(3a)

which leads to ⎞

⎛ θi+1 = ± cos−1 ⎝ √

k3 k12

+

k22

⎠ − tan−1 k2 k1

(3b)

With k 1 = sinα i(i+1) sinα (i+1)(i+2) cosα (i+3)(i+4) + cosα i(i+1) sinα (i+1)(i+2) sinα (i+3)(i+4) cosθ i , k 2 = sinα (i+2)(i+3) sinα (i+3)(i+4) sinθ i and k 3 = cosα i(i+1) cosα (i+1)(i+2) cosα (i+3)(i+4) − cosα (i+2)(i+3) − sinα i(i+1) cosα (i+1)(i+2) sinα (i+3)(i+4) cosθ i , all being the functions of θ i , and can be represented in implicit function form as θi+1 = f i(i+1) (θi )

(3c)

And multi-degree-4-vertex patterns are the assemblies of spherical linkages.

3 The Origami Support Mechanism and Kinematics of the Back Board The diagram of human spine (see Fig. 2a) involves three curvatures, the cervical curvature, the thoracic curvature and the lumbar curvature, according to which the support mechanism of the nursing bed is proposed. Figure 2b illustrates the flat state of the origami support mechanism, together with its mountain-valley crease lines distribution, in which the mountain crease lines are represented by solid lines while the valley crease lines are in dash lines. The pattern of the support mechanism is divided into four parts, the back board (red crease lines), the haunch board, the thigh board and the calf board with green crease lines. The origami support mechanism is symmetric with respect to the symmetry axis S, and the two blue crease lines T1 and T2 are the turn-over crease lines, which do not take effect in the back- and leg-lifting motion. In this paper, the summation of the twist angles for each spherical 4R linkage is 2π , and the support mechanism is flat-deployable. The back board can be folded into the configuration that fit into human body curve, as shown in Fig. 2c, where the back board curve is in purple and the turnover crease lines are in grey because they do not take effect under this condition. Since the back board is symmetric with respect to axis S, only the left part of it is studied. The connection between adjacent degree-4 vertexes B1 and B2 , which can

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit …

443

Fig. 2 The support mechanism. a The diagram of human spine; b the flat state of the support mechanism; c the folded state of the back board; d the connection between adjacent degree-4 vertexes of the back board; e the diagram of the back board curve

444

W. Lv et al.

be regarded as the assembly of spherical 4R linages, is shown in Fig. 2d, where the coordinate systems are set up and the parameters are provided according to Fig. 1. The superscripts B1 and B2 represent vertexes B1 and B2 . According to Eq. (3a), we have B1 B1 B1 sin α23 cos α34 sin α12 cos θ2B1 B1 B1 B1 + sin α23 sin α34 cos α12 cos θ3B1 B1 B1 B1 + cos α23 sin α34 sin α12 cos θ2B1 cos θ3B1 B1 B1 − sin α34 sin α12 sin θ2B1 sin θ3B1 B1 B1 B1 B1 − cos α23 cos α34 cos α12 + cos α41 = 0,

(4a)

B2 B2 B2 sin α34 cos α41 sin α23 cos θ3B2 B2 B2 B2 + sin α34 sin α41 cos α23 cos θ4B2 B2 B2 B2 + cos α34 sin α41 sin α23 cos θ3B2 cos θ4B2 B2 B2 − sin α41 sin α23 sin θ3B2 sin θ4B2 B2 B2 B2 B2 − cos α34 cos α41 cos α23 + cos α12 = 0,

θ3B1 = θ3B2 .

(4b) (4c)

According to Eq. (3c), Eqs. (4a)–(4c) can be represented in implicit function form as   θ4B2 = f 34 θ3B2 = f 34 f 23 (θ2B1 )

(5a)

Besides Eq. (5a) we can also obtain that:   θ2B3 = f 12 θ1B3 = f 12 f 41 (θ4B2 ) ,

(5b)

  θ4B4 = f 34 θ3B4 = f 34 f 23 (θ2B3 ) ,

(5c)

  θ2B5 = f 12 θ1B5 = f 12 f 41 (θ4B4 ) .

(5d)

If Eq. (5a) is represented as f 23

f 34

θ2B1 −→ θ3B1 = θ3B2 −→ θ4B2 , the connection among vertexes B1 and B5 can be represented as

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit … f 23

f 34

f 41

f 12

f 23

f 34

f 41

f 12

445

θ2B1 −→ θ3B1 = θ3B2 −→ θ4B2 −→ θ1B2 = θ1B3 −→ θ2B3 −→ θ3B3 = θ3B4 −→ θ4B4 −→ θ1B4 = θ1B5 −→ θ2B5 Figure 2e provides the diagram of boards B0 B1 to B5 B6 , which determine the shape of the back board curve. φ0 is the dihedral angle between the board B0 B1 and the haunch board and φi (i = 1–5) represents the dihedral angle between boards Bi−1 Bi and Bi Bi+1 , and we have φ1 = π − θ2B1 ,

(6a)

φ2 = π − θ4B2 ,

(6b)

φ3 = θ2B3 − π,

(6c)

φ4 = θ4B4 − π,

(6d)

φ5 = π − θ2B5 .

(6e)

Thus the back board curve can be obtained by giving φ0 and φ1 , i.e. the back board is 2-DOF (degree of freedom). The shape of the back board curve can be Bj and the lengths of boards Bi Bi+1 to make optimized by changing the value of αi(i+1) the nursing bed support mechanism fit into human body curve, which will be studied in the future.

4 The Back-Lifting, Turning and Leg-Lifting Mechanisms 4.1 The Back-Lifting Mechanism Figure 3a shows the configuration of the nursing bed when the back board is lifted, and the back-lifting mechanism is a planar five bar mechanism which is shown in Fig. 3b, and its diagram is illustrated in Fig. 3c, wherein ψi (i = 1–4, ψ1 = φ0 , ψ2 = π + ψ1 − φ1 ) refers to the angle between links and the horizontal plane. Links 3 and 4, which is connected by a translational joint, refer to the driving links, and links 1 and 2 represent boards B1 B2 and B0 B1 respectively. The diagram of the backlifting mechanism is illustrated in Fig. 3c, in which the origin is at point O, x-axis is the horizontal axis and y-axis is the vertical axis. The back-lifting mechanism is also 2-DOF, according to the closing-vector-circle method, we have

446

W. Lv et al.

lOA cos ψ3 − lB1 A cos ψ2 − lB0 B1 cos ψ1 − lOB0 cos ψ4 = 0, lOA sin ψ3 − lB1 A sin ψ2 − lB0 B1 sin ψ1 − lOB0 sin ψ4 = 0.

(7)

According to Eqs. (4), (6) and (7), the relation between φi (i = 0–5, in which positive angles represent mountain crease-lines and negative angles represent valley crease-lines) and lOA (the elongation of the translational joint between links 3 and 4) is shown in Fig. 3d with ψ3 = π/4, ψ4 = π/6, lB1 A = 10, lB0 B1 = 20 and lOB0 = 100 (the link lengths are non-dimensional). The sector angles around vertices B1 B1 B1 B1 = 2π/3, α23 = π/2, α34 = π/3, and α41 = π/2; B1 to B5 are as follows: α12 B2 B2 B2 B2 B3 B3 α12 = π/2, α23 = 4π/9, α34 = π/2, and α41 = 5π/9; α12 = 4π/9, α23 = π/2, B3 B3 B4 B4 B4 B4 α34 = 5π/9, and α41 = π/2; α12 = π/2, α23 = 5π/9, α34 = π/2, and α41 = B5 B5 B5 B5 4π/9; α12 = 5π/9, α23 = π/2, α34 = 4π/9, and α41 = π/2.

Fig. 3 Back-lifting mechanism. a The back-lifting configuration of the nursing bed; b the backlifting mechanism; c the diagram of the back-lifting mechanism and d the relation between ψ1 , ψ2 and lOA when ψ3 is given

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit …

447

4.2 The Turning Mechanism According to the superimposed method of rigid origami [22], the turn-over crease lines only take effect in the turning motion. The configuration of the nursing bed during turning is shown in Fig. 4a. The turning mechanism consists of two sets of planar four-bar mechanisms, as shown in Fig. 4b, and its diagram is illustrated in Fig. 4c. Links 3, 4 and 5 represent bed boards. The arrangement of the two sets of planar four-bar mechanisms is completely symmetrical, so only the right part of the turning mechanism, in which link 2 is the driving link, is studied. The coordinate frame and parameters of the turning mechanism are set up in Fig. 4d, and ψi (i = 5–7) refers to the angle between links and horizontal plane. In the turning mechanism, lOD and lDC are constants. We can obtain that:

ψ7 = arccos

2 2 2 − lOD − lDC lOC 2lODlDC

 − ψ6

(8)

Fig. 4 The turning mechanism. a The turning configuration of the nursing bed; b the turning mechanism; c the diagram of the turning mechanism and d the diagram of the right part of the turning mechanism

448

W. Lv et al.

4.3 The Leg-Lifting Mechanism In Fig. 5a, the configuration of the nursing bed in the leg-lifting motion is illustrated. Figure 5b shows the 1-DOF leg-lifting mechanism and its diagram is provided in Fig. 5c, in which link 2 is the driving link, links 3 and 4 represent the thigh board and calf board respectively, and points O, L and K are co-linear. In the close loop of LHOL and OIJKO, it can be obtained that 2 2 2 lOH + lLO − lLH − 2lOHlLO cos(ψ9 + ψ10 ) = 0,   1 2 2 2 lKJ − lOI − lIJ2 − lOK − lOIlIJ cos(ψ9 + ψ11 ) 2 + lOIlOK cos(ψ9 + ψ10 ) + lIJlOK cos(ψ11 − ψ10 ) = 0.

(9)

Fig. 5 Leg-lifting mechanism. a The leg-lifting configuration of the nursing bed; b the leg-lifting mechanism; c the diagram of the leg-lifting mechanism and d the diagram of the leg-lifting mechanism

A Rigid Origami Nursing Bed Support Mechanism That Is Able to Fit …

449

In which link lengths lLO , lOH , lOI , lIJ , lOK , lKJ are constants. The relation between ψi (i = 9 and 11) and lLH is shown in Fig. 5d. Parameter settings are as follows: lOH = 32, lOI = 50, lIJ = 30, lLO = 20, lOK = 30, lKJ = 70 and ψ10 = 7π/4, in which all link lengths are non-dimensional. This section introduces the back-lifting, turning and leg-lifting mechanisms of the origami nursing bed support mechanism, together with their kinematics. According to the study mentioned above, the nursing bed support mechanism is 5-DOF (2 for back-lifting, 2 for turning-over and 1 for leg-lifting).

5 Conclusions This paper has proposed a rigid origami nursing bed support mechanism that is able to fit into human body curve. The origami pattern of the support mechanism is firstly presented, and the kinematics of its back board is analyzed based on the kinematics of spherical 4R linkages. Subsequently, the back-lifting, turning and leglifting mechanisms of the support mechanism are introduced, whose kinematics are analyzed by the closing-vector-circle method. The mechanism proposed in this paper provides multiple choices for the design of nursing beds and may contribute to improving comfort of nursing beds. Acknowledgements Supported by the National Natural Science Foundation of China (Project 52075145), the Science and Technology Program of Hebei (Project 20281805Z) and the Central Government Guides Basic Research Projects of Local Science and Technology Development Funds (Project 206Z1801G).

References 1. Ning M, Ren M, Fan Q, Zhang L (2017) Mechanism design of a robotic chair/bed system for bedridden aged. Adv Mech Eng 9(3):1687814017695691 2. Seo K-H, Choi T-Y, Oh C (2010) Development of a robotic system for the bed-ridden. Mechatronics 21(1):227–238 3. Mercader A, Biersack M, Sun Y et al (2019) A mechanical bed for elderly care to assist while standing, sitting and lying. In: 2019 IEEE international conference on robotics and biomimetics (ROBIO), Dali, China, 06–08 Dec 2019, pp 965–970 4. Ye A, Zou W, Yuan K et al (2012) A reconfigurable wheelchair/bed system for the elderly and handicapped. In: 2012 IEEE international conference on mechatronics and automation, Chengdu, China, 05–08 Aug 2012, pp 1627–1632 5. Miura K (1989) A note on the intrinsic geometry of origami. KTK Scientific Publishers, Tokyo, Japan, pp 91–102 6. Watanabe N, Kawaguchi K (2009) The method for judging rigid foldability. Origami 4:165–174 7. Hull TC, Tachi T (2017) Double-line rigid origami. arXiv preprint arXiv:1709.03210 8. He Z, Guest SD (2020) On rigid origami II: quadrilateral creased papers. Proc R Soc A 476(2237):20200020

450

W. Lv et al.

9. Dang X, Feng F, Plucinsky P et al (2021) Inverse design of deployable origami structures that approximate a general surface. Int J Solids Struct 234:111224 10. Dai J, Ress JJ (1999) Mobility in metamorphic mechanisms of foldable/erectable kinds. J Mech Des 121(3):375–382 11. Stachel H (2010) A kinematic approach to Kokotsakis meshes. Comput Aided Geom Des 27(6):428–437 12. Medellín-Castillo HI, Cervantes-Sánchez JJ (2005) An improved mobility analysis for spherical 4R linkages. Mech Mach Theory 40(8):931–947 13. Chiang CH (1984) On the classification of spherical four-bar linkages. Mech Mach Theory 19(3):283–287 14. Zhang K, Dai J (2013) Classification of origami-enabled foldable linkages and emerging applications. In: ASME 2013 international design engineering technical conferences and computers and information in engineering conference, 4–7 Aug 2013 15. Wei G, Dai J (2014) Origami-inspired integrated planar-spherical overconstrained mechanisms. ASME J Mech Des 136(5):051003 16. Chen Y, You Z (2005) Mobile assemblies based on the Bennett linkage. Proc R Soc Lond A Math Phys Eng Sci 461:1229–1245 17. Chen Y, You Z (2008) On mobile assemblies of Bennett linkages. Proc R Soc Lond A Math Phys Eng Sci 464:1275–1293 18. Liu S, Chen Y (2009) Myard linkage and its mobile assemblies. Mech Mach Theory 44(10):1950–1963 19. Liu S (2014) Deployable structure associated with rigid origami and its mechanics. PhD thesis, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 20. Cundy HM, Rollett AP (1952) Mathematical models. Oxford University Press, New York 21. Denavit J, Hartenberg RS (1955) A kinematic notation for lower-pair mechanisms based on matrices. J Appl Mech 215–221 22. Liu X, Gattas JM, Chen Y (2016) One-DOF superimposed rigid origami with multiple states. Sci Rep 6(1):1–9

Interval Principal Component Analysis of Non-probabilistic Convex Model Shuofeng Hou, Bingyu Ni, Wanyi Tian, Jinwu Li, and Chao Jiang

Abstract The non-probabilistic convex model describes uncertainty of parameters in the form of a bounded set rather than the probability distribution and is generally applied to interval uncertainty analysis based on the fluctuation ranges of structural parameters. When performing the structural response bounds analysis or uncertain optimization design, too many interval parameters is not beneficial in most circumstances. It may lead to problems such as inefficient computation and non-convergence of the optimization process. This paper proposes an interval principal component analysis method for the non-probabilistic multidimensional ellipsoid convex model, which aims to reduce the dimensionality of the uncertainty domain of correlated intervals, while retaining as much as possible of the variation present in the bounded uncertainty. Firstly, the mathematical mapping between the coordinate axes and the marginal intervals of the multidimensional ellipsoid convex model on these axes is established for an arbitrary coordinate system. Secondly, the difference index is introduced to describe the differentiation degree of the group of marginal intervals, the maximizing of which yields the principal axes of the multidimensional ellipsoid convex model. With the magnitudes of the marginal intervals, the most critical axes and corresponding interval components in uncertainty quantification can then be identified. By neglecting those axes with small marginal intervals, the dimensionality can be thus reduced. An error index is also suggested to indicate the accuracy of uncertainty quantification under the dimensionality reduction representation. Finally, two numerical examples are presented to show the dimensionality reduction representation procedure by the proposed interval principal component analysis. Keywords Interval uncertainty · Non-probabilistic convex model · Principal component analysis · Dimensionality reduction representation S. Hou · B. Ni (B) · W. Tian · J. Li · C. Jiang College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, Hunan, China e-mail: [email protected] State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_26

451

452

S. Hou et al.

1 Introduction Uncertainties in parameters such as material properties, loads, and structural sizes are widely present in practical engineering and are important factors that cannot be ignored in structural uncertainty analysis and reliability design. Traditional approaches are mainly based on probability models to quantify the above uncertain parameters and describe the randomness of parameters with probability distribution functions or statistical moments. It has been successfully applied in areas such as structural uncertainty propagation analysis, reliability assessment, and optimization design. However, due to the limitations of test techniques and costs for many engineering structures, it is difficult to obtain accurate probability distributions or statistical information of the uncertain parameters. Thus it may not meet the assumptions or requirements in constructing a probability model [1]. For this reason, a series of non-probability or imprecise probability methods were proposed, such as evidence theory [2–4], probability-box model [5, 6], convex model [7, 8], fuzzy set theory [9, 10], and interval analysis methods [11–16], etc. Among them, the convex model method quantitatively describes the uncertainty domain of the parameters through a convex set. This method mainly requires the uncertainty boundary of the parameters instead of the precise probability distributions, which has gained more and more attention in theoretical research and practical applications [17–26]. The non-probabilistic convex models are also called non-probabilistic settheoretical models, uncertain-but-bounded models, etc. which were firstly applied in cybernetics. In 1968, Schweppe [27] studied the state estimation of a linear dynamic system. Considering that only the amplitude boundary or energy boundary of the uncertain parameters such as the system inputs or observation errors can be known, a convex model was introduced to describe the uncertainty domain of such parameters. Subsequently, the research investigation on the convex modeling method began to spread in various engineering areas, initially in the field of control and filtering [28–31]. In 1990, Ben-Haim and Elishakoff [7] published a monograph on convex models, which discussed in detail the applicability and significance of convex model methods in the field of uncertainty analysis. Since then, the convex models have been furtherly applied to the material uncertainty analysis of viscoelastic structures, the nonlinear buckling analysis of cylinders, the prediction of fatigue life and the S–N curves [32–36], etc. In 1998, Pantelides and Ganzerli [37] used three criteria to establish a non-probabilistic ellipsoid convex model that considered uncertainty. Zhu et al. [38] used the Gram–Schmidt orthogonalization method to determine the best ellipsoid model of limited experimental data through the general transformation matrix of the rotation of the multi-dimensional coordinate system. Jiang et al. [18] proposed an interval correlation analysis method and defined the correlation of interval variables based on the multi-dimensional ellipsoid convex model. It effectively avoids the optimization difficulties encountered in constructing high-dimensional ellipsoid convex models through the minimum volume method. Ni et al. [24] proposed a unified framework for establishing the mathematical expressions of convex models by correlation analysis of interval variables, and offered a measure for validation of the convex

Interval Principal Component Analysis of Non-probabilistic Convex Model

453

modeling methods. Kang and Zhang [20] solved the multidimensional ellipsoid convex modeling by the semi-definite programming, which significantly improved the efficiency. Qiu et al. [39] proposed a non-probabilistic convex set-based structural optimization method, in which the uncertain parameters in the objective function and constraints are quantified with convex sets. Guo et al. [40] described the uncertain structural parameters as interval variables and proposed a non-probabilistic reliability measurement. Wang et al. [41] proposed a non-probabilistic reliability index of the structure by using the ratio of the hypervolume of the safety region. Kang and Luo [42] used a performance measurement-based method to transform reliability constraints into related performance index constraints. Besides, the convex model method has also been extended to other areas such as interval finite element analysis [43–46], reliability analysis [47, 48], uncertain optimization [49–55], etc. The existing studies on convex models are mainly focused on the development of new convex models, their uncertainty propagation analysis methods and corresponding structural reliability analysis methods, etc. However, the current research and application in these areas, such as uncertainty propagation, are limited to modeling and analyzing non-probabilistic uncertainty with relatively low dimensionality. Theoretically, a high-dimensional convex model needs to be established for problems with many interval variables, which leads to high-dimensional uncertainty propagation problems in structural response uncertainty analysis and poses significant challenges to accuracy and efficiency. Therefore, it would be beneficial to reduce the dimensionality of the convex model of input interval uncertainties while ensuring accuracy. This paper proposes an interval principal component analysis method based on the non-probabilistic multidimensional ellipsoid convex model. In terms of uncertainty quantification, the principal axes on which the marginal intervals of the multidimensional ellipsoid convex model can be differentiated to the fullest extent are identified. The proposed method could provide a theoretical basis for the accurate and efficient dimensionality reduction representation of high-dimensional non-probabilistic convex models. The remainder of this paper is organized as follows: Sect. 2 briefly introduces the concept of non-probabilistic multidimensional ellipsoid convex model; Sect. 3 proposes the interval principal component analysis method of the non-probabilistic convex model; Sect. 4 is devoted to numerical examples; Sect. 5 gives the conclusions.

2 Non-probabilistic Multidimensional Ellipsoid Convex Model Among the existing convex models, the interval and ellipsoid models are the most widely used. The ellipsoid model is currently the most commonly used nonprobabilistic convex model for dealing with correlated interval variables because of its good mathematical properties. Traditionally, the ellipsoid with the minimum volume enveloping all samples is regarded as the best ellipsoid convex model. To

454

S. Hou et al.

find the minimum-volume convex model efficiently, several techniques such as the Gram–Schmidt orthogonal transformation method and semi-definite programming have already been developed. Another type of convex modeling method is based on correlation analysis [37, 38], which creates the multidimensional model by the correlation quantification for all of the bivariant interval variables, thus effectively avoiding the high-dimensional volume minimization problem. In this section, we only briefly introduce the multidimensional ellipsoid model and its construction method based on correlation analysis. Assume that there exist n uncertain parameters X i , i = 1, 2, . . . , n with Ns groups of experimental samples x(s) , s = 1, 2, . . . , Ns , which, for a practical structure, could be material properties, external loads, geometrical sizes, etc. These parameters constitute an n-dimensional parameter space, namely X-space. By using the convex model approach, the variation of each uncertain parameter X i , i = 1, 2, . . . , n is quantified by an interval, namely:   X i ∈ X iI = X iL , X iU   = X im − X ir , X im + X ir , i = 1, 2, . . . , n

(1)

where the superscripts I, L and U denote interval, lower bound and upper bound, respectively; X im and X ir are midpoint and radius, respectively. This kind of uncertain parameters can also be called as interval variables. The uncertainty domain of the multiple interval variables X = [X 1 , X 2 , . . . , X n ]T constitutes a bounded set, denoted as  X in this paper. The generalized marginal interval indicates the variation range of the bounded set along a specific coordinate direction, namely, the projection interval along this coordinate. For an arbitrary coordinate axis P, the marginal interval of the uncertainty domain  X can be defined as:   P I = P L , P U = {P|X ∈  X }

(2)

Specifically, the marginal interval of the uncertainty domain  X on the coordinate axis of X i is defined as:   X iI = X iL , X iU = {X i |X ∈  X }, i = 1, 2, . . . , n

(3)

which indicates the variation range of X i regardless of the values of other interval variables. Specifically, the marginal interval X iI = [− 1, 1] is also called as a standard marginal interval; and the convex set with standard marginal intervals is called as a regularized convex set. For the two-dimensional case of the ellipsoid convex model, the uncertainty domain of the interval variables X 1 and X 2 is the ellipse shown in Fig. 1, denoted as E.2 E.2 X or  , where the superscript E represents the ellipsoid model. The degree of correlation between the variables X 1 and X 2 is measured by the geometric characteristics of the ellipse.

Interval Principal Component Analysis of Non-probabilistic Convex Model

455

Fig. 1 Correlation description of interval variables

The ellipse uncertainty domain E.2 can be expressed as:  

E.2

= 

| T X 1 || X 1 − X 1m X 2 | X 2 − X 2m

Cov(X 1 , X 1 ) Cov(X 1 , X 2 ) Cov(X 2 , X 1 ) Cov(X 2 , X 2 )

−1 

X 1 − X 1m X 2 − X 2m



 ≤1

(4)

or 



E.2

| T X 1 || X 1 − X 1m / X 1r = X 2 | X 2 − X 2m / X 2r 

r  −1  m − X / X 1 ρ(X 1 , X 2 ) X 1 1 1

≤1 ρ(X 1 , X 2 ) X 2 − X 2m / X 2r 1

(5)

where Cov(X 1 , X 2 ) represents the covariance of interval variables X 1 and X 2 , and ρ(X 1 , X 2 ) represents the correlation coefficient of X 1 and X 2 . The covariance can be calculated by the following formula:

Cov(X 1 , X 2 ) = sin(θ ) cos(θ ) r12 − r22

(6)

in which r1 and r2 respectively represent lengths of the two semi-axises of the ellipse, θ represents the rotation angle of the ellipse from its normal state. When X 1r /= X 2r , the covariance Cov(X 1 , X 2 ) of X 1 and X 2 can also be expressed as: Cov(X 1 , X 2 ) =

tan(θ )  r 2 r 2 X1 − X2 1 − tan2 (θ )

The correlation coefficient ρ(X 1 , X 2 ) is:

(7)

456

S. Hou et al.

ρ(X 1 , X 2 ) =

Cov(X 1 , X 2 ) X 1r X 2r

(8)

In particular, the condition of Cov(X 1 , X 1 ) = D(X 1 ) is satisfied, that is, the covariance between X 1 and itself is the variance of X 1 . The interval variable X i ∈  X iI = X im − X ir , X im + X ir can be transformed into a standard interval variable by the following transformation: Ui =

X i − X im X ir

(9)

which is called the standardization of the interval variable X i . For the correlated standard interval variables U1 ∈ U1I = [− 1, 1] and U2 ∈ U2I = [− 1, 1], their joint E.2 uncertainty domain shown in Fig. 2 is denoted as  , where θ = π/4. From Eqs. (6) and (8), there is: ρ(U1 , U2 ) = Cov(U1 , U2 ) =

r 21 − r 22 2

(10) E.2

in which r 1 and r 2 are lengths the two semi-axial of the ellipse  , respectively. E.2 The joint uncertainty domain  of U1 and U2 can be then expressed as: E.2



 =

 | T  −1   U1 || U1 1 ρ(U1 , U2 ) U1 ≤1 U2 | U2 1 U2 ρ(U2 , U1 )

(11)

  For multiple interval variables X i ∈ X iI = X iL , X iU , i = 1, 2, . . . , n, if the variables X i are independent of each other, their joint uncertainty domains form an “n-dimensional box”, denoted as I.n X : Fig. 2 Joint uncertainty domain of standard interval variables U 1 and U 2

Interval Principal Component Analysis of Non-probabilistic Convex Model I I I I.n X = X1 × X2 × · · · × Xn    L U  L U = X 1 , X 1 × X 2 , X 2 × · · · × X nL , X nU

457

(12)

If the interval variables X i , i = 1, 2, . . . , n are correlated, its n-dimensional ellipsoid uncertainty domain E.n X can be expressed as:

|  m T m | E.n X = X (X − X ) GE (X − X ) ≤ 1

(13)

in which X = [X 1 X 2 . . . X n ]T is the vector of interval variables. Xm = T  m m X 1 X 2 . . . X nm represents the center point of the ellipsoid and is also the midpoint vector of the interval variables. G E is a symmetric positive definite matrix, reflecting the size and shape of the uncertainty domain of the multidimensional ellipsoid. In constructing the characteristic matrix based on correlation analysis technique, the characteristic matrix can be represented by the inverse of the covariance matrix, namely: GE = C−1

(14)

where C is the covariance matrix of the interval variable X i , i = 1, 2, . . . , n: ⎡

Cov(X 1I , X 1I ) Cov(X 1I , X 2I ) ⎢ Cov(X I , X I ) Cov(X I , X I ) 2 1 2 2 ⎢ C=⎢ .. .. ⎣ . . Cov(X nI , X 1I ) Cov(X nI , X 2I )

⎤ · · · Cov(X 1I , X nI ) · · · Cov(X 2I , X nI ) ⎥ ⎥ ⎥ .. .. ⎦ . . I I · · · Cov(X n , X n )

(15)

Furthermore, the uncertainty domain of the standard interval variables Ui , i = 1, 2, . . . , n can be obtained as:

|  E.n U = U|UT ρ−1 U ≤ 1

(16)

where ρ is the correlation coefficient matrix of interval variable X i , i = 1, 2, . . . , n: ⎡

ρ(X 1I , X 1I ) ρ(X 1I , X 2I ) · · · ⎢ ρ(X I , X I ) ρ(X I , X I ) · · · 2 1 2 2 ⎢ ρ=⎢ .. .. .. ⎣ . . . I I I I ρ(X n , X 1 ) ρ(X n , X 2 ) · · · ⎡ 1 ρ(X 1I , X 2I ) · · · ⎢ ρ(X I , X I ) 1 ··· 2 1 ⎢ =⎢ .. .. .. ⎣ . . . ρ(X nI , X 1I ) ρ(X nI , X 2I ) · · ·

⎤ ρ(X 1I , X nI ) ρ(X 2I , X nI ) ⎥ ⎥ ⎥ .. ⎦ .

ρ(X nI , X nI )

⎤ ρ(X 1I , X nI ) ρ(X 2I , X nI ) ⎥ ⎥ ⎥ .. ⎦ . 1

(17)

458

S. Hou et al.

With the above correlation analysis method or other minimum-volume methods, a multidimensional ellipsoid convex model can be then constructed for multiple interval variables. This multidimensional ellipsoid is then regarded as the uncertainty domain of these correlated intervals.

3 The Proposed Interval Principal Component Analysis With the increase of the number of uncertain parameters especially for complex engineering problems, the uncertainty analysis and reliability design for structures become more and more tough. For interval uncertainty analysis, reducing the number of interval variables while ensuring the accuracy of uncertainty quantification is meaningful for subsequent uncertainty propagation and reliability design. The basic idea of principal component analysis [56, 57] is to represent a group of variables by uncorrelated principal components through coordinate transformation. The coordinate axes of the principal components are the principal axes, and the variance of the variable on the principal axis represents the uncertainty along this coordinate. The greater variance indicates the stronger uncertainty; conversely, the smaller variance means closer to certainty. In extreme cases, if the variance along a coordinate axis equals to zero, the uncertainty in this direction disappears and this component can be regarded as a deterministic value. Therefore, the principal component analysis could identify the independent factors as much as possible, and retains the principal components with larger variances while ignoring the variables with smaller variances, so as to achieve the reduction of the number of variables, thus achieving dimensionality reduction representation. As shown in Fig. 3a is a two-dimensional description of the ellipsoid model and the marginal intervals. For an arbitrary coordinate system w1 − w2 , the marginal intervals of the ellipse uncertainty domain are respectively W1I and W2I . The interval principle analysis aims to find the principal axes, in which those marginal intervals of the convex uncertainty domain can have the most obvious difference. For the three-dimensional case, the schematic diagram is shown in Fig. 3b.

3.1 Marginal Intervals of the Multidimensional Ellipsoid Convex Model in an Arbitrary Coordinate System Taking a two-dimensional model as an example, the orthogonal coordinate system {w1 , w2 } is established with the center point of the ellipse as the origin. In this coordinate system, the bounded domain constituted by the marginal intervals WiI , i = 1, 2 is a rectangle circumscribing the ellipse, as shown in Fig. 3a. For multiple interval  variables X i ∈ X iI = X iL , X iU , i = 1, 2, . . . , n, they need to be transformed to standard interval variables Ui ∈ UiI = [− 1, 1], i = 1, 2, . . . , n firstly. Assume

Interval Principal Component Analysis of Non-probabilistic Convex Model

459

Fig. 3 The ellipsoid model and the marginal intervals in an arbitrary coordinate system

that an arbitrary orthogonal coordinate system {w1 , w2 , . . . , wn } is established with the center O of this multi-dimensional ellipsoid as the origin. In this coordinate system, an arbitrary point U = {U1 , U2 , . . . , Un } in the uncertainty domain of the multi-dimensional ellipsoid in this coordinate system can be denoted as: U = W1 · w1 + W2 · w2 + · · · + Wn · wn namely:

(18)

460

S. Hou et al.

U = Tw W

(19)

in which the transformation matrix Tw = [w1 , w2 , . . . , wn ]n×n is a unit orthogonal matrix. Substituting formula (19) into the multi-dimensional ellipsoid Eq. (16), the expression of the multi-dimensional ellipsoid model in the coordinate system {w1 , w2 , . . . , wn } can be obtained as: (Tw W)T ρ−1 (Tw W) ≤ 1

(20)

WT C−1 W W ≤1

(21)

or

T −1 where C−1 W = Tw ρ Tw . According to the relationship between the characteristic matrix of the multidimensional ellipsoid and the covariance matrix of the interval variables, it can be seen that it is equivalent to the inverse of the covariance matrix of the interval variables W1 , W2 , . . . , Wn , which yields:

[D(W1 ), D(W2 ), . . . , D(Wn )] = diag(CW )

(22)

where D(Wi ) represents the variance of the interval variable Wi , namely, the square of the radius of the interval WiI ; diag(CW ) represents the vector formed by the diagonal elements of the covariance matrix CW . Since Tw = [w1 , w2 , . . . , wn ]n×n T −1 is an identity orthogonal matrix and C−1 W = Tw ρ Tw , there is: CW = TTw ρTw

(23)

D(Wi ) = wiT ρwi

(24)

which yields:

Then the length of the interval WiI can be obtained: √ √ L Wi = 2 D(Wi ) = 2 wiT ρwi , i = 1, 2, . . . , n.

(25)

In order to determine the principal interval components in the ellipsoid convex uncertainty, the coordinate system {w1 , w2 , . . . , wn } requires to be determined, in which the marginal intervals of the n-dimensional ellipsoid along the principal axes present with the most significant difference. On this basis, to retain as much as possible of the variation present in the bounded uncertainty while reducing the dimensionality of the uncertainty domain, those principal interval components with larger ranges are suggested to be retained, thus realizing the dimensionality reduction representation for convex modeling of interval uncertainties.

Interval Principal Component Analysis of Non-probabilistic Convex Model

461

3.2 Identification of the Principal Axes for the Multi-dimensional Ellipsoid Convex Model According to the outer quasi circle theorem, the vertices of all the circumscribed rectangles of the ellipse √ locate on the same circle, as shown in Fig. 4. The radius

of the circle is R = r12 + r22 , where ri , i = 1, 2 represents the length of the ith semi-major axis of the ellipse. For the three-dimensional and multi-dimensional cases, the outer quasi circle theorem is also applicable, namely, the vertices of all the circumscribed cubes of the multidimensional ellipsoid √ locate on the same hypern 2 sphere, and the radius of this hyper-sphere is R = i=1 ri , which is a fixed value for a specific multidimensional ellipsoid. Meanwhile, the side width of the circumscribed cube is also of the n the length L 2Wi = R 2 = marginal interval L Wi , i = 1, 2, . . . , n. Therefore, there is i=1 n 2 i=1 ri , which is a fixed value. In this case, the maximization of the difference of the marginal intervals of the n-dimensional ellipsoid along the principal axes is equivalent to the minimization of the sum of the lengths, which leads to minimization of the following index: L=

n √  wiT ρwi

(26)

i=1

When L takes the minimum value, the difference of lengths of the marginal intervals is then maximized. Therefore, in order to find the best principal axes, the set of orthogonal vectors {w1 , w2 , . . . , wn } needs to be identified. Without loss of generality, we search for a set of identity orthogonal vectors {w1 , w2 , . . . , wn } with wi w j = δi j . Herein δi j is the Kronecker-delta function. Thus, it leads to the following optimization problem: Fig. 4 Schematic diagram of outer quasi circle

462

S. Hou et al.

min L =

n √  wiT ρwi

i=1

(27)

s.t. wi w j = δi j , i, j = 1, 2, . . . , n By using the Lagrange multiplier method to solve the above optimization model, there is:

f = wiT ρwi − λi wiT wi − 1 ∂f = 2ρwi − 2λi wi = 0 ∂wi ∂f = wiT wi − 1 = 0 ∂λi

(28)

which yields: ρwi = λi wi

(29)

in which f is Lagrangian Function; λi is the parameter of each additional condition ∂f ∂f wiT wi −1; ∂w means that f takes the partial derivative of wi ; ∂λ means that f takes the i i partial derivative of λi . It can be seen that the main coordinate axis {w1 , w2 , . . . , wn } of the multidimensional ellipsoid convex model can be obtained by eigen decomposition of the correlation coefficient matrix ρ of the interval variable X i , i = 1, 2, . . . , n, and the principal axes are right the eigenvectors of the correlation coefficient matrix ρ. Meanwhile, the transformation matrix Tw = [w1 , w2 , . . . , wn ]n×n in Eq. (19) is right the characteristic matrix of the correlation coefficient matrix ρ. Multiplying the left and right sides of Eq. (29) by wiT , the following equation can be obtained: wiT ρwi = wiT λi wi = λi wiT wi = λi

(30)

Furtherly, by combining with Eq. (25), it can be obtained that the marginal interval length L Wi of the multi-dimensional ellipsoid convex model along the direction of the coordinate axis wi is: √ (31) L Wi = 2 λi

3.3 Dimensionality Reduction Representation and Error Analysis According to the above-mentioned interval principal component analysis, the principal axes of the multi-dimensional ellipsoid convex model and the marginal intervals under this coordinate system can be established. In this coordinate system, on one

Interval Principal Component Analysis of Non-probabilistic Convex Model

463

hand, the principal component variables are relatively independent, and the reduction of some variables with tiny variations has little influence on the other variables. On the other hand, the marginal intervals of the multidimensional convex model along these axes differ with each other to the largest extent. Therefore, it is convenient and reasonable to implement the dimensionality reduction representation within the coordinate system of the above mentioned principal axes. It can be known that the sum of the eigenvalues of the matrix is equal to the trace of the matrix. For the n dimensional correlation coefficient matrix, there is: n 

λi = n

(32)

i=1

namely:  n   LW 2 i

i=1

2

=n

(33)

When performing the dimensionality reduction representation of interval uncertainty, if only the m principal components are retained, e.g., the principal interval variables W1 , W2 , . . . , Wm , then there is:  m   LW 2 i

i=1

2

≤n

(34)

Therefore, the following error index ε can be used to represent the error caused by the above-mentioned dimensionality reduction representation: m  L Wi 2 ε =1−

i=1

2

n

(35)

that is m ε =1− m

i=1

n

λ

λi

(36)

i We also take 1 − ε = i=1 as the cumulative contribution rate of the m principal n component interval variables in the uncertainty representation. According to the mapping relationship between the standard interval variables Ui , i = 1, 2, . . . , n and the principal component interval variables Wi , i = 1, 2, . . . , m in formula (19), there is:  ˜w (37) [U1 , U2 , . . . , Un ]T ≈ T [W1 , W2 , . . . , Wm ]T

n×m

464

S. Hou et al.

 where T˜ w

n×m

is the n × m-order matrix, which is composed of the eigenvectors

corresponding to Wi , i = 1, 2, . . . , m in the n × n-order eigen matrix Tw . Finally, according to formulas (9) and (13), the analytical expression of the m-dimensional ellipsoid convex model can be established. This dimensionality reduction model uses as few interval variables as possible to approximate the original n-dimensional ellipsoid uncertainty domain under the given precision.

4 Numerical Examples In this section, two examples are used to verify the feasibility and effectiveness of the proposed interval principal component analysis method. The first example is a 52-bar space truss, where the cross-sectional areas of some bars are treated as dependent interval variables. The second example discusses a heavy-duty conveyor chain device subjected to the dependent uncertain loads. To investigate the applicability of the proposed method, several kinds of degrees of correlation of the uncertainty are considered in each example.

4.1 A 52-Bar Space Truss As shown in Fig. 5, a hemispherical space 52-bar space truss (like a dome) [58] is considered in this example, the radius of which is 6 m. In this example, six external loads are applied on the space truss, which are P1 in the inner normal direction of point 1, P2 in the inner normal direction of points 2 and 4, P3 in the inner normal direction of points 3 and 5, P4 in the inner normal direction of points 6 and 10, P5 in the inner normal direction of points 8 and 12, P6 in the inner normal direction of points 7, 9, 11, 13. The elastic modulus of each bar is E = 2.5 × 104 N/m2 . The cross-sectional area of bars is denoted as Si , i = 1, 2, . . . , 52. The cross-sectional area of bars 1–16, namely Si , i = 1, 2, . . . , 16, are treated as the interval variables with upper and lower bounds and the others are treated as the constant. The details are given in Table 1. To investigate the applicability of the proposed method, four kinds of correlation of the interval variables are considered in this example, namely (1) the coexistence of strong correlation and weak correlation, (2) the strong correlation, (3) the weak correlation and (4) no correlation. The first kind of correlation of the cross-sectional area of each bar, namely the coexistence of strong correlation and weak correlation, is given in Table 2. The interval principal component analysis result is shown in Fig. 6a. Under the condition that the cumulative contribution rate is about 90%, the principal components of the 16-dimensional ellipsoid convex model are w13, w7, w8, w12, w10, w9, w14, w11, w5, w16, w3, w6 and the cumulative contribution rate of principal components increases as the number of principal components increases.

Interval Principal Component Analysis of Non-probabilistic Convex Model

465

(a) Top view of the 52-bar space truss

(b) Side view of the 52-bar space truss Fig. 5 A 52-bar space truss [58]

The second kind of correlation of the cross-sectional area of each bar, namely the strong correlation, is shown in Table 3. The result of the interval principal components is shown in Fig. 6b. From Fig. 6b, it can be found that only several individual interval components can represent the whole uncertainty to a great extent since there is a strong correlation between the original interval variables. When the correlation between interval variables is weak as shown in Table 4, the contribution rate of each

466 Table 1 The ranges or values of the variables in the 52-bar space truss

S. Hou et al. Variables

Value/value range

Type

Unit

Si , i = 1, 2, . . . , 8

[12, 14]

Interval variables

cm2

Si , i = 9, 10, . . . , 16

[7, 8.5]

Interval variables

cm2

Si , i = 29, 30, . . . , 36

13

Constant

cm2

Si , i = 17, . . . , 28

4

Constant

cm2

Si , i = 37, . . . , 52

4

Constant

cm2

Pi , i = 1, 2, . . . , 6

40

Constant

kN

variable is almost the same. The result of the interval principal component analysis is shown in Fig. 6c. From Fig. 6c, it can be found that the importance of the components differs slightly. When the interval variables of the cross-sectional areas of the truss bars are uncorrelated, the contribution rates of all the interval components are exactly the same. It means that every interval component is equally important for uncertainty quantification. The result of interval principal component analysis is given in Fig. 6d. The above analysis shows that principal components and their contribution to uncertainty are highly related with the correlation between the interval variables. When it coexists of the strong and weak correlation, the principal components are relatively easy to be identified. When the interval variables are strongly correlated, only a few principal component variables are needed to represent the original uncertainty considerably. When the interval variables are weakly correlated, the importance of most interval components are almost the same, thus it generally requires relatively more variables to accurately represent the convex uncertainty. When the variables are uncorrelated to each other, the importance of each component is exactly the same, and the principal components cannot be identified.

4.2 A Heavy-Duty Conveyor Chain Device As shown in Fig. 7, a heavy-duty conveyor chain device is considered in this example to illustrate the feasibility of the proposed interval principal component analysis method. The length of the conveyor chain is 10 m, and 8 weights are placed on the movable stage. The loads Pi , i = 1, 2, . . . , 8 applied by the 8 weights as well as their locations, quantified by the distances L i , i = 1, 2, . . . , 8 to the driving axis, are uncertain and described as interval variables. The details of are given in Table 5. Under different correlation cases, the principal interval components for the convex uncertainty quantification of the uncertain loads and distances can be quite different. In this numerical investigation, two kinds of correlation cases of the above interval

Interval Principal Component Analysis of Non-probabilistic Convex Model

467

Table 2 The correlation coefficients of interval variables in the 52-bar space truss example (coexistence of strong correlation and weak correlation) X1

X2

X3

X4

X5

X6

X7

X8

X1

1.000

0.858

0.858

0.885

0.803

0.889

0.841

0.804

X2

0.858

1.000

0.814

0.861

0.825

0.832

0.840

0.841

X3

0.858

0.814

1.000

0.819

0.809

0.832

0.877

0.823

X4

0.885

0.861

0.819

1.000

0.829

0.831

0.852

0.833

X5

0.803

0.825

0.809

0.829

1.000

0.823

0.846

0.838

X6

0.889

0.832

0.832

0.831

0.823

1.000

0.805

0.876

X7

0.841

0.840

0.877

0.852

0.846

0.805

1.000

0.821

X8

0.804

0.841

0.823

0.833

0.838

0.876

0.821

1.000

X9

0.155

0.196

0.189

0.136

0.155

0.135

0.162

0.180

X 10

0.175

0.113

0.182

0.103

0.141

0.173

0.178

0.137

X 11

0.174

0.189

0.124

0.113

0.123

0.135

0.129

0.193

X 12

0.105

0.159

0.116

0.184

0.117

0.150

0.200

0.136

X 13

0.105

0.121

0.140

0.133

0.123

0.194

0.168

0.196

X 14

0.144

0.194

0.101

0.161

0.180

0.123

0.193

0.176

X 15

0.183

0.157

0.179

0.133

0.122

0.131

0.158

0.183

X 16

0.129

0.140

0.186

0.161

0.199

0.120

0.183

0.168

X9

X 10

X 11

X 12

X 13

X 14

X 15

X 16

X1

0.155

0.175

0.174

0.105

0.105

0.144

0.183

0.129

X2

0.196

0.113

0.189

0.159

0.121

0.194

0.157

0.140

X3

0.189

0.182

0.124

0.116

0.140

0.101

0.179

0.186

X4

0.136

0.103

0.113

0.184

0.133

0.161

0.133

0.161

X5

0.155

0.141

0.123

0.117

0.123

0.180

0.122

0.199

X6

0.135

0.173

0.135

0.150

0.194

0.123

0.131

0.120

X7

0.162

0.178

0.129

0.200

0.168

0.193

0.158

0.183

X8

0.180

0.137

0.193

0.136

0.196

0.176

0.183

0.168

X9

1.000

0.735

0.796

0.750

0.762

0.780

0.764

0.748

X 10

0.735

1.000

0.757

0.752

0.726

0.799

0.786

0.764

X 11

0.796

0.757

1.000

0.709

0.745

0.716

0.740

0.789

X 12

0.750

0.752

0.709

1.000

0.784

0.724

0.763

0.720

X 13

0.762

0.726

0.745

0.784

1.000

0.770

0.799

0.740

X 14

0.780

0.799

0.716

0.724

0.770

1.000

0.756

0.799

X 15

0.764

0.786

0.740

0.763

0.799

0.756

1.000

0.740

X 16

0.748

0.764

0.789

0.720

0.740

0.799

0.740

1.000

468

S. Hou et al.

Fig. 6 Interval principal component analysis for the interval variables of the 52-bar space truss

variables are considered for interval principal component analysis, namely a strong correlation case and a weak correlation. The principal components and their contribution rates are given in Fig. 8. It can be observed that for the given strong correlation, only 8 principal interval components could achieve an uncertainty representation with over 89% contribution rate, and 10 components could reach an estimation with over 95% accuracy, as shown in Fig. 8a. While for the weak correlation case, even the most important component interval variable contributes only 10.32% to the original uncertainty. And totally 15 variables are required to ensure an accuracy of over 94%, which poses great challenge for dimension reduction. Furtherly, for the given strong correlation case, the mapping relationship between the original correlated interval variables and the set of principal intervals with reduced dimension can be given as follows:

Interval Principal Component Analysis of Non-probabilistic Convex Model

469

Table 3 The correlation coefficients of interval variables in the 52-bar space truss example (strong correlation) X1

X2

X3

X4

X5

X6

X7

X8

X1

1.000

0.990

0.980

0.970

0.961

0.951

0.942

0.932

X2

0.990

1.000

0.990

0.980

0.970

0.961

0.951

0.942

X3

0.980

0.990

1.000

0.990

0.980

0.970

0.961

0.951

X4

0.970

0.980

0.990

1.000

0.990

0.980

0.970

0.961

X5

0.961

0.970

0.980

0.990

1.000

0.990

0.980

0.970

X6

0.951

0.961

0.970

0.980

0.990

1.000

0.990

0.980

X7

0.942

0.951

0.961

0.970

0.980

0.990

1.000

0.990

X8

0.932

0.942

0.951

0.961

0.970

0.980

0.990

1.000

X9

0.923

0.932

0.942

0.951

0.961

0.970

0.980

0.990

X 10

0.914

0.923

0.932

0.942

0.951

0.961

0.970

0.980

X 11

0.905

0.914

0.923

0.932

0.942

0.951

0.961

0.970

X 12

0.896

0.905

0.914

0.923

0.932

0.942

0.951

0.961

X 13

0.887

0.896

0.905

0.914

0.923

0.932

0.942

0.951

X 14

0.878

0.887

0.896

0.905

0.914

0.923

0.932

0.942

X 15

0.869

0.878

0.887

0.896

0.905

0.914

0.923

0.932

X 16

0.861

0.869

0.878

0.887

0.896

0.905

0.914

0.923

X9

X 10

X 11

X 12

X 13

X 14

X 15

X 16

X1

0.923

0.914

0.905

0.896

0.887

0.878

0.869

0.861

X2

0.932

0.923

0.914

0.905

0.896

0.887

0.878

0.869

X3

0.942

0.932

0.923

0.914

0.905

0.896

0.887

0.878

X4

0.951

0.942

0.932

0.923

0.914

0.905

0.896

0.887

X5

0.961

0.951

0.942

0.932

0.923

0.914

0.905

0.896

X6

0.970

0.961

0.951

0.942

0.932

0.923

0.914

0.905

X7

0.980

0.970

0.961

0.951

0.942

0.932

0.923

0.914

X8

0.990

0.980

0.970

0.961

0.951

0.942

0.932

0.923

X9

1.000

0.990

0.980

0.970

0.961

0.951

0.942

0.932

X 10

0.990

1.000

0.990

0.980

0.970

0.961

0.951

0.942

X 11

0.980

0.990

1.000

0.990

0.980

0.970

0.961

0.951

X 12

0.970

0.980

0.990

1.000

0.990

0.980

0.970

0.961

X 13

0.961

0.970

0.980

0.990

1.000

0.990

0.980

0.970

X 14

0.951

0.961

0.970

0.980

0.990

1.000

0.990

0.980

X 15

0.942

0.951

0.961

0.970

0.980

0.990

1.000

0.990

X 16

0.932

0.942

0.951

0.961

0.970

0.980

0.990

1.000

470

S. Hou et al.

Table 4 The correlation coefficients of interval variables in the 52-bar space truss example (weak correlation) X1

X2

X3

X4

X5

X6

X7

X8

X1

1.000

0.210

0.140

0.160

0.150

0.160

0.150

0.320

X2

0.210

1.000

0.170

0.190

0.140

0.130

0.220

0.330

X3

0.140

0.170

1.000

0.190

0.150

0.190

0.160

0.140

X4

0.160

0.190

0.190

1.000

0.200

0.170

0.260

0.120

X5

0.150

0.140

0.150

0.200

1.000

0.110

0.190

0.150

X6

0.160

0.130

0.190

0.170

0.110

1.000

0.170

0.140

X7

0.150

0.220

0.160

0.260

0.190

0.170

1.000

0.190

X8

0.320

0.330

0.140

0.120

0.150

0.140

0.190

1.000

X9

0.120

0.110

0.140

0.210

0.120

0.110

0.130

0.170

X 10

0.200

0.180

0.150

0.120

0.180

0.140

0.150

0.110

X 11

0.230

0.120

0.210

0.240

0.160

0.180

0.170

0.140

X 12

0.250

0.140

0.260

0.250

0.240

0.170

0.160

0.120

X 13

0.240

0.230

0.220

0.210

0.250

0.220

0.140

0.130

X 14

0.220

0.210

0.140

0.230

0.220

0.110

0.120

0.140

X 15

0.210

0.180

0.220

0.200

0.270

0.240

0.110

0.190

X 16

0.180

0.240

0.230

0.210

0.160

0.150

0.130

0.120

X9

X 10

X 11

X 12

X 13

X 14

X 15

X 16

X1

0.120

0.200

0.230

0.250

0.240

0.220

0.210

0.180

X2

0.110

0.180

0.120

0.140

0.230

0.210

0.180

0.240

X3

0.140

0.150

0.210

0.260

0.220

0.140

0.220

0.230

X4

0.210

0.120

0.240

0.250

0.210

0.230

0.200

0.210

X5

0.120

0.180

0.160

0.240

0.250

0.220

0.270

0.160

X6

0.110

0.140

0.180

0.170

0.220

0.110

0.240

0.150

X7

0.130

0.150

0.170

0.160

0.140

0.120

0.110

0.130

X8

0.170

0.110

0.140

0.120

0.130

0.140

0.190

0.120

X9

1.000

0.210

0.140

0.220

0.130

0.140

0.190

0.120

X 10

0.210

1.000

0.110

0.190

0.170

0.160

0.140

0.130

X 11

0.140

0.110

1.000

0.120

0.150

0.110

0.180

0.140

X 12

0.220

0.190

0.120

1.000

0.180

0.220

0.140

0.170

X 13

0.130

0.170

0.150

0.180

1.000

0.170

0.160

0.220

X 14

0.140

0.160

0.110

0.220

0.170

1.000

0.220

0.140

X 15

0.190

0.140

0.180

0.140

0.160

0.220

1.000

0.160

X 16

0.120

0.130

0.140

0.170

0.220

0.140

0.160

1.000

Interval Principal Component Analysis of Non-probabilistic Convex Model

471

Fig. 7 A heavy-duty conveyor chain device Table 5 The ranges of the interval variables of loads and their distances

Variables

Ranges/kN

Variables

Ranges/kN

P1

[9.8, 10.2]

P5

[9.0, 11.0]

P2

[9.5, 10.5]

P6

[9.4, 10.6]

P3

[9.3, 10.7]

P7

[9.2, 10.8]

P4

[9.4, 10.6]

P8

[9.3, 10.7]

L1

[1.4, 1.6]

L5

[5.4, 6.0]

L2

[2.7, 3.3]

L6

[6.2, 6.9]

L3

[3.3, 4.0]

L7

[7.0, 7.6]

L4

[4.2, 5.2]

L8

[8.0, 8.5]

472

S. Hou et al.

⎡ m⎤ ⎤ P1 P1 ⎢ Pm ⎥ ⎢P ⎥ ⎡ ⎤ ⎢ 2 ⎥ ⎢ 2⎥ w16 ⎢ ⎢ . ⎥ ⎥ ⎢ .. ⎥ ⎢ w ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎥ ⎢ 1⎥ ⎢ ⎢ ⎥  ⎢ ⎥ ⎢ m⎥ w P ⎢ ⎢ P8 ⎥ ⎥ ⎢ ⎥ 10 ⎢ ⎥ ≈ T˜ w ⎢ ⎥ + ⎢ 8m ⎥ ⎢ L1 ⎥ ⎢ L1 ⎥ 6×16 ⎢ w12 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ L2 ⎥ ⎥ ⎣ w5 ⎦ ⎢ L m ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎢ .. ⎥ ⎥ . w7 ⎣ .. ⎦ ⎣ . ⎦ L8 Lm 8 ⎡

Fig. 8 Interval principal component analysis for the interval variables of the heavy-duty conveyor chain device

(38)

Interval Principal Component Analysis of Non-probabilistic Convex Model

 ˜w in which T

6×16

473

is given in Eq. (39), Pim , i = 1, 2, . . . , 8 and L im , i = 1, 2, . . . , 8

are midpoint values of the loads and distances, which can be obtained from Table 5. ⎡



˜w T

6×16

− 0.298 ⎢ 0.099 ⎢ ⎢ ⎢ 0.073 =⎢ ⎢ 0.318 ⎢ ⎣ − 0.105 0.367

− 0.308 0.133 0.037 0.291 − 0.125 0.391

− 0.350 0.153 0.014 0.130 0.011 0.028

− 0.344 0.148 0.023 0.037 0.050 0.071

− 0.345 0.144 0.000 − 0.070 0.028 0.044

− 0.336 0.146 − 0.034 − 0.202 0.044 − 0.125

− 0.324 0.135 − 0.068 − 0.320 0.051 − 0.300

− 0.297 0.125 − 0.102 − 0.185 0.014 − 0.417

− 0.417 − 0.328 0.417 − 0.392 0.130 0.124

− 0.147 − 0.372 0.342 − 0.032 − 0.273 0.053

− 0.126 − 0.304 − 0.511 − 0.200 − 0.105 0.300

− 0.137 − 0.335 0.245 − 0.204 − 0.360 0.017

− 0.135 − 0.379 − 0.364 0.035 0.380 0.075

− 0.118 − 0.308 − 0.444 0.093 − 0.310 − 0.119

− 0.138 − 0.261 0.098 0.560 − 0.173 − 0.546

⎤ − 0.141 − 0.308 ⎥ ⎥ ⎥ 0.159 ⎥ ⎥ 0.231 ⎥ ⎥ 0.681 ⎦ − 0.023

(39)

5 Conclusions This paper proposes an interval principal component analysis method for correlated interval variables based on the non-probabilistic ellipsoid convex model. The basic idea of this method is to identify the principal coordinate axes and corresponding interval components for uncertainty quantification. More specifically, for arbitrary coordinate axes, the marginal intervals of the multidimensional ellipsoid convex model are obtained by establishing a mathematical mapping. Then, the principal axes of the multi-dimensional ellipsoid convex model can be obtained by maximizing the

474

S. Hou et al.

differential index, which describes the degree of differentiation of the marginal intervals. Finally, the principal coordinate axes and corresponding interval components in uncertainty quantification can be identified according to the ranges of the marginal intervals. It is suggested to pay more attention to those principal component interval variables and the corresponding coordinate axes, which may significantly impact the structural uncertainty quantification and dimension reduction. Two numerical examples show that the principal component variables and their quantities are mainly determined by the correlation degree between the original interval variables. They further demonstrated the feasibility and applicability of the proposed method. The dimensionality reduction can be achieved through such an analysis technique, which might provide a more efficient way for the following reliability analysis and the reliability-based design optimization. Acknowledgements Supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51621004), the National Science Foundation of China (Grant No. 52175224, 52075156), the National Science Fund for Distinguished Young Scholars (Grant No. 51725502).

References 1. Elishakoff I (2000) Possible limitations of probabilistic methods in engineering. Appl Mech Rev 53(2):19–36 2. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc Ser B (Methodol) 39(1):1–22 3. Shafer G (1976) A mathematical theory of evidence. Princeton, Princeton University Press 4. Yager R, Fedrizzi M, Kacprzyk J (1994) Advances in the Dempster–Shafer theory of evidence. Wiley, New York 5. Zhang H, Mullen RL, Muhanna RL (2012) Structural analysis with probability-boxes. Int J Reliab Saf 6(1–3):110–129 6. Fetz T, Tonon F (2008) Probability bounds for series systems with variables constrained by sets of probability measures. Int J Reliab Saf 2(4):309–339 7. Ben-Haim Y, Elishakoff I (2013) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam 8. Kang Z, Luo Y (2006) On structural optimization for non-probabilistic reliability based on convex models. Chin J Theoret Appl Mech 38(6):807–815 9. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28 10. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353 11. Alefeld G, Claudio D (1998) The basic properties of interval arithmetic, its software realizations and some applications. Comput Struct 67(1–3):3–8 12. Hansen ER (1968) On solving systems of equations using interval arithmetic. Math Comput 22(102):374–384 13. Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs 14. Yang C, Lu Z (2017) An interval effective independence method for optimal sensor placement based on non-probabilistic approach. Sci China Technol Sci 60(2):186–198 15. Zhang J, Du H, Xue D, Gu P (2021) Robust design approach to the minimization of functional performance variations of products and systems. Front Mech Eng 16(2):379–392 16. Zhou C, Chang Q, Zhao H, Ji M, Shi Z (2020) Fault tree analysis with interval uncertainty: a case study of the aircraft flap mechanism. IEEE Trans Reliab 70(3):944–956

Interval Principal Component Analysis of Non-probabilistic Convex Model

475

17. Ben-Haim Y (1993) Convex models of uncertainty in radial pulse buckling of shells. J Appl Mech 60(3):683–688 18. Jiang C, Han X, Lu G, Liu J, Zhang Z, Bai Y (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200(33–36):2528–2546 19. Jiang C, Li J, Ni B, Fang T (2019) Some significant improvements for interval process model and non-random vibration analysis method. Comput Methods Appl Mech Eng 357:112565 20. Kang Z, Zhang W (2016) Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data. Comput Methods Appl Mech Eng 300:461–489 21. Kaya O, Oktem AS, Adali S (2021) Closed form buckling and sensitivity analysis of isotropic and orthotropic columns with uncertain material and geometric properties via convex modeling. Part I: theory and formulation. Eng Struct 247:112935 22. Meng Z, Hu H, Zhou H (2018) Super parametric convex model and its application for nonprobabilistic reliability-based design optimization. Appl Math Model 55:354–370 23. Ni B, Jiang C, Han X (2016) An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis. Appl Math Model 40(7–8):4727–4745 24. Ni B, Jiang C, Huang Z (2018) Discussions on non-probabilistic convex modelling for uncertain problems. Appl Math Model 59:54–85 25. Qiao X, Wang B, Fang X, Liu P (2021) Non-probabilistic reliability bounds for series structural systems. Int J Comput Methods 18(09):2150038 26. Wang C, Matthies HG (2020) A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis. Comput Methods Appl Mech Eng 369:113209 27. Schweppe F (1968) Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans Autom Control 13(1):22–28 28. Bertsekas D, Rhodes I (1971) Recursive state estimation for a set-membership description of uncertainty. IEEE Trans Autom Control 16(2):117–128 29. Chernousko F (1980) Optimal guaranteed estimates of uncertainties by means of ellipsoids. Parts I, II, III, Izv USSR Acad Sci Eng Cybern 3:3–11 30. Schlaepfer F, Schweppe F (1972) Continuous-time state estimation under disturbances bounded by convex sets. IEEE Trans Autom Control 17(2):197–205 31. Witsenhausen H (1968) Sets of possible states of linear systems given perturbed observations. IEEE Trans Autom Control 13(5):556–558 32. Ben-Haim Y (1994) Fatigue lifetime with load uncertainty represented by convex model. J Eng Mech 120(3):445–462 33. Ben-Haim Y (1995) A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Struct Saf 17(2):91–109 34. Elishakoff I, Cai G, Starnes J Jr (1994) Non-linear buckling of a column with initial imperfection via stochastic and non-stochastic convex models. Int J Non-Linear Mech 29(1):71–82 35. Elishakoff I, Colombi P (1993) Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Comput Methods Appl Mech Eng 104(2):187–209 36. Elishakoff I, Elisseeff P, Glegg SA (1994) Nonprobabilistic, convex-theoretic modeling of scatter in material properties. AIAA J 32(4):843–849 37. Pantelides CP, Ganzerli S (1998) Design of trusses under uncertain loads using convex models. J Struct Eng 124(3):318–329 38. Zhu LP, Elishakoff I, Starnes J Jr (1996) Derivation of multi-dimensional ellipsoidal convex model for experimental data. Math Comput Model 24(2):103–114 39. Qiu Z, Chen J, Wang X (2005) Robust optimization for structures using non-probabilistic convex method of set theory. Acta Mech Sin 295–300 40. Guo S, Lv Z, Feng Y (2001) A non-probabilistic model of structural reliability based on interval analysis. Jsuan Lixue Xuebao (Chin J Comput Mech) (China) 18(1):56–60 41. Wang X, Wang L, Jia X, Qiu Z (2012) Structural optimization design based on non-probabilistic convex modeling reliability. J Beijing Univ Aeronaut Astronaut 38(5):630–635

476

S. Hou et al.

42. Kang Z, Luo Y (2009) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198(41– 44):3228–3238 43. Faes M, Moens D (2019) Multivariate dependent interval finite element analysis via convex hull pair constructions and the extended transformation method. Comput Methods Appl Mech Eng 347:85–102 44. Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194(12–16):1527–1555 45. Rao MR, Mullen RL, Muhanna RL (2011) A new interval finite element formulation with the same accuracy in primary and derived variables. Int J Reliab Saf 5(3–4):336–357 46. Sofi A, Romeo E, Barrera O, Cocks A (2019) An interval finite element method for the analysis of structures with spatially varying uncertainties. Adv Eng Softw 128:1–19 47. Jiang C, Zhang Q, Han X, Qian Y (2014) A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model. Acta Mech 225(2):383–395 48. Wang L, Wang X, Wang R, Chen X (2015) Time-dependent reliability modeling and analysis method for mechanics based on convex process. Math Probl Eng 1–16 49. Du X (2012) Reliability-based design optimization with dependent interval variables. Int J Numer Meth Eng 91(2):218–228 50. Ganzerli S, Pantelides CP (2000) Optimum structural design via convex model superposition. Comput Struct 74(6):639–647 51. Jiang C, Zhang Z, Zhang Q, Han X, Xie H, Liu J (2014) A new nonlinear interval programming method for uncertain problems with dependent interval variables. Eur J Oper Res 238(1):245– 253 52. Lombardi M (1998) Optimization of uncertain structures using non-probabilistic models. Comput Struct 67(1–3):99–103 53. Wang L, Wang X, Wang R, Chen X (2016) Reliability-based design optimization under mixture of random, interval and convex uncertainties. Arch Appl Mech 86(7):1341–1367 54. Zhan J, Luo Y (2019) Robust topology optimization of hinge-free compliant mechanisms with material uncertainties based on a non-probabilistic field model. Front Mech Eng 14(2):201–212 55. Meng Z, Zhou H (2018) New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization. Comput Methods Appl Mech Eng 339:644–662 56. Jackson JE (2005) A user’s guide to principal components. Wiley-Interscience, Hoboken 57. Wold S, Esbensen K, Geladi P (1987) Principal component analysis. Chemom Intell Lab Syst 2(1–3):37–52 58. Gomes HM (2011) Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst Appl 38(1):957–968

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator with Scissor Legs Jianxun Fu, Rongfu Lin, Chengze Liu, Jiayi Zhou, and Feng Gao

Abstract In this article, a new six degrees-of-freedom (DOF) deployable parallel manipulator (DPM) with three straight scissors legs is briefly described. The configuration design method of the DPM is proposed by means of GF set theory, using this method the mechanism configuration can be determined. The decoupling of the proposed kinematic chains is discussed, a new deployable parallel manipulator is designed. Its position only depends on the elongation of three deployable legs, there such legs are employed to build 6-DOF DPM manipulator. And the effect factors of error were investigated, its accuracy and precision are verified by checkout location errors and clearance and driving errors, and the kinematic accuracy models are proposed for the proposed manipulator. Keywords DPM · Configuration principle · Semi-decoupling · Kinematics accuracy

1 Introduction Nowadays some parallel manipulators (PM) have been performed significant applications due to its advantages over serial mechanisms, such as high strength, large carrying capacity, perfect performance of force balance and high accuracy, etc. [1–3]. As a representative example of such PMs, Gough [4] developed a 6-DOF PM and is used as tire detecting device. A class of parallel manipulators termed Stewart are designed for flight simulation [5–7]. Since then, a vast scope of theoretical researches and PMs to be devised for a wide variety of applications, Gosselin [8], Huang and Li J. Fu (B) · C. Liu Shanghai Institute of Applied Physics, CAS, Shanghai 200240, China e-mail: [email protected] R. Lin · F. Gao School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 201800, China J. Zhou Yindu Kitchen Equipment Co., Ltd., Hangzhou 311199, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_27

477

478

J. Fu et al.

[9], Merlet [10], Gogu [11], Lee et al. [12, 13], Gao et al. [14], Dai and Jones [15], Kong et al. [16, 17], and Tsai and Joshi [18], e.g. Consequently, the screw theory as a productive way is used for the kinematic analysis and dimensional synthesis, and achieve good practical effects. The GF sets presented a relatively simple and practical approach in the precision kinematics synthesis of PMs. The design and analysis theory for PMs have long been a current topic in the international robotic research area. In recent years, the research for the 3-UP3R 6-DOF parallel robot gets great advance. Fu et al. [19, 20] studied optimum design of a 6-DOF PM manipulator with three legs based on its workspace. Lin et al. [21] proposed a family of SLBMtype mechanisms with three limbs, and type synthesis procedures are abstracted correspondingly. Hui et al. [22] proposed a novel 3-UP3R parallel robot, and the inverse kinematic, dexterity and kinematic performance have been studied. In order to enlarge the range of motion, many researchers have done a lot of effective work. Gonzalez and Asada [23] proposed a 6-DOF PM with a deployable scissor structure, which could be deployed and retracted respectively. Yang et al. [24] introduced a dual scissor-like mechanisms, and two kinds of deployable 3-DOF PMs are assembled, and discussed the kinematics performances of the proposed manipulators. Wang et al. [25] designed a kind of planar deployable mechanisms, its dimensions and diagrams are analyzed, and a petal-inspired unit is constructed. In recent years, many researchers have made valuable contributions in the area of PMs. However, there still have much design work for the novel PMs need to be done in order to meet potential applications in many fields. The purpose of this study is concentrates on design and analysis a novel 6-DOF parallel manipulator. This article is arranged as follows: The characteristics of the robot and configuration principle are introduced in Sect. 2. The kinematic equations and decoupling are addressed in Sect. 3. Kinematic accuracy is discussed in Sect. 4 and final part is the conclusion.

2 Configuration of the DPM Mechanisms This section presents a novel parallel robot named as DPM. The characteristics of this robot and its configuration will be introduced in this section.

2.1 Characteristics and Model of the DPM As shown in Fig. 1, the DPM manipulator has a parallel mechanism type with three deployable scissor structure legs, one side of the leg is fixed on the fixed platform and another is connected to moving platform. Each kinematic chain is assembled by two passive revolute (R) joints, an active R joint, a deployable structure limb with straight scissors and a universal (U) joint, as shown in Fig. 2a, the structure of the scissors is in contraction status, and the extension status is shown in Fig. 2b.

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator …

479

Fig. 1 Prototype of the manipulator

Three legs can be assembled into the proposed 6-DOF parallel manipulator. All rotational axes of the R joint are coincident at one point p called the rotation center of the mechanism, and the straight scissors legs generate the extendable translation motion at the rotation common center p also. Compared with other parallel manipulators, the three legs constructure allows the manipulator has large workspace, less possibility of interference. And the kinematic decoupling property of the DPM is a theoretical foundation for its kinematic analysis, control and its simplification of demarcation.

2.2 Configuration Principle for the DPM Manipulator As mentioned above, the proposed manipulator is the intersection of the characteristics of each leg, the configuration can be modeled based on GF sets:  GF =

3 

 G 3T3R Fileg

(1)

i=1

where GF represents the 6-DOF of the moving platform of the DPM manipulator, and GFileg refers to the ith leg (i = 1, 2, 3) motion pattern.

480

J. Fu et al.

Fig. 2 Deployable structure with straight scissors in contraction and extension status

As shown in Fig. 3, all R joints and all the extendable translation axes intersect at the common constraint center p. The moving platform has three rotational motions (Fig. 3a), and translational motions (Fig. 3b), which means that each leg has same freedoms too, that is G3T3R Fileg (T i1 ; T i2 ; T i3 ; Ri1 ; Ri2 ; Ri3 ). According to Eq. (1), the number of kinematic chains is determined, and each leg is driven by two actuators.

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator …

481

Fig. 3 DPM mechanisms and its configuration

3 Kinematics Analysis for the DPM Manipulator 3.1 Kinematics As illustrated in Fig. 4, the fixed coordinate frame (FCF) is attached to the top of the triangular pyramid, the origin of the FCF is placed at point p(px , py , pz ), which is a virtual common constraint center of three identical scissor structure limbs. And the rotary coordinate frame (MCF) is also constructed at the point p. The vectors vi and wi , i = 1, 2, 3, run along the axes of the ith leg. According to Fig. 2, a location vector of the U joints in fixed platform is defined as a(ax , ay , az ), and its vector quaternion q ' can be expressed as,

482

J. Fu et al.

 T q' = 0 a

Fig. 4 Kinematic diagram of the DPM

(2)

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator …

483

The location vector quaternions of the U joints in fixed platform qi can be expressed as,     q i = M i er M i er q '

(3)

where  ηi1 

 ⎞ 0  − sin η2i1 0ηi1  ⎟   ⎜ 0 0 cos 2  − sin η2i1 ⎟ M i er = ⎜ ηi1 ηi1 ⎠ ⎝ cos 2 0ηi1  0ηi1  sin 2 0 0 cos 2 sin 2  ηi1  ⎞   ⎛ cos η2i1 0ηi1  sin 2 0ηi1  ⎟   ⎜ 0 0 cos 2  sin  2  ⎟. M i er = ⎜ ηi1 ηi1 ⎝ 0 ⎠ 0  − sin 2 cos 2  0 0 cos η2i1 − sin η2i1 ⎛

cos

2

As shown in Fig. 5, L 0 is the length of the straight scissors at is initial status, and the expansion length of the ith (i = 1, 2, 3) deployable leg can be expressed as, li = 6 cos(ϕ − θi )s0

(4)

The expansion length of the ith (i = 1, 2, 3) deployable leg satisfies the following equations, 

P − qi



 P − q i = (L 0 + li )2

(5)

From the structural features of the PDM manipulator, the configuration angles between two components (wi -axis and the vi -axis, i = 1, 2, 3) are fixed, then building the equations according to angle constraints of orientation vectors wi and vi , α2

eu i 2 v i eu i

α2 2

= wi

(6)

And then from Eqs. (5) and (6), we can get the solutions of inverse and forward kinematics for the proposed DPM manipulator.

3.2 Decoupling Analysis According to Sect. 3.1, the forward kinematics of the DPM can be written as K = F(Q), where K(x, y, z, α, β, γ )T is the vector of kinematic output parameters, Q(θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 )T is the vector of kinematic input parameters. The differential kinematics equation with respect to time can be expressed as,

484

J. Fu et al.

Fig. 5 Scissor structure limbs of the DPM

˙ V = JQ

(7)

˙ where Q—First derivative of the input parameters, ˙Q = (θ˙1 , θ˙2 , θ˙3 , θ˙4 , θ˙5 , θ˙6 )T , V—Velocity of the moving platform, V = (vx , vy , vz , ωx , ωy , ωz )T , J—6 × 6 square matrix. According to Eq. (7), the forms of kinematic decoupling properties related to the Jacobian matrix can be seen in Table 1 [20]. Taking the DPM manipulator for example, according to Eq. (7), the output parameters of the moving platform of module a1 , a2 , a3 along three directions whose unit vectors are i1 , j1 , k 1 , and the orientation motion of module b1 , b2 , b3 , c1 , c2 , c3 along three directions whose unit vectors are i2 , j2 , k 2 , i3 , j3 , k 3 . According to Eq. (5), the matrix J has the following form:

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator … Table 1 Decoupling properties related to the Jacobian

Decoupling pattern

485

Jacobian



J = diag j11 j22 j33 j44 j55 j66

Complete decoupling

⎛

j11 0

⎜ ⎜ ⎜ j21 j22 J =⎜ ⎜. ⎜. ⎝.

Partial decoupling

··· 0 .. . .. . 0

j61 j62 · · · j66  J=

Semi-decoupling

⎛

a1 i 1x ⎜a i ⎜ 1 1y ⎜ ⎜a i J = ⎜ 1 1z ⎜ b1 i 2x ⎜ ⎝ b1 i 1y b1 i 1z

a2 j1x a2 j1y a2 j1z b2 j2x b2 j2y b2 j2z

a3 k1x a3 k1y a3 k1z b3 k2x b3 k2y b3 k2z

0 0 0 c1 i 3x c1 i 3y c1 i 3z

0 0 0 c2 j3x c2 j3y c2 j3z

JT 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 6×6



0 JR

6×6

⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟. c3 k3x ⎟ ⎟ c3 k3y ⎠ c3 k3z

(8)

And then we can get the solution of the forward position of the end-effector using Eq. (3), that means of ∂p/∂l i = 0. Therefore Eq. (8) can be simplified as, ⎛

 J=

JT 0 0 JR

 6×6

a1 i 1x ⎜a i ⎜ 1 1y ⎜ ⎜a i = ⎜ 1 1z ⎜0 ⎜ ⎝0 0

a2 j1x a2 j1y a2 j1z 0 0 0

a3 k1x a3 k1y a3 k1z 0 0 0

0 0 0 c1 i 3x c1 i 3y c1 i 3z

0 0 0 c2 j3x c2 j3y c2 j3z

⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟, c3 k3x ⎟ ⎟ c3 k3y ⎠ c3 k3z

(9)

where J T = R1 D1 = R1 diag(a1 , a2 , a3 ), J R = R2 D2 = R2 diag(c1 , c2 , c3 ), where Ri is a coefficient matrix, the manipulator is in a state of semi-decoupling.

4 Kinematics Accuracy Analysis As above mentioned, the DPM manipulator has three identical scissor structure legs, each leg is configured by ten R joints, and a U joint. There are two main categories about the error sources: (1) location errors of the U joints, (δpU , δΩU )i ; (2) Clearance originating at R joints (δpC , δΩC )i .

486

J. Fu et al.

Fig. 6 Location error of the U joint

As shown in Fig. 6, referring to the kinematic constraint of Eq. (3), the error models of the location of the U joints are derived as,   ˜ i + er q ' qi = M

(10)

Differentiating this equation yields,     δq i = δ M˜ i + e r q ' + M˜ i + e r δq '

(11)

For clearance error, assuming there is no deformation for the shaft and bearing. Referring to Fig. 7, there are two ways of shaft and hole mating for R joints, in practical work one of the clearances (Fig. 7b) is considered, in that case will cause the worst deviation by the amount moved in both directions from the nominal configuration. The shaft vi is rotatable about the axis ui , the differential rotation angle is δα i , taking shaft vi to v' I after rotation. The maximum value of its differential rotation angle value δα i takes as,

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator …

487

Fig. 7 Shaft and hole mating for R joints

 δαi = arctan

σi ai

 ≈

σi ai

(12)

Referring to Fig. 8, the differential distance of the rotation center from p to p' is ppi' = δαi rai =

σi rai ai

And the differential angle δζ i can be derived as, Fig. 8 Location error due to clearance

(13)

488

J. Fu et al.

 δζi = ζ − arctan

h − ppi' L

 (14)

By means of Eqs. (11) and (14) we can get the position and orientation accuracies for the proposed DPM manipulator.

5 Conclusions A new 6-DOF parallel robot DPM with three scissor-like legs is briefly described in this article, a deployable structure leg with straight scissors is introduced. The moving platform of the proposed manipulator has six freedom motion, and analysed the kinematic decoupling properties of the proposed deployable parallel manipulator related to the Jacobian matrix. Meanwhile, the kinematic equations are built with the method of quaternion. A method of accuracy which is based on the position of the U joints errors and clearance of the R joints is derived. And we hope the DPM manipulator will have some positive applications in industrial robots, and other fields. Acknowledgements Supported by Natural Science Foundation of China (Grant No. 51905338).

References 1. Monsarrat B, Gosselin CM (2003) Workspace analysis and optimal design of a 3-leg 6-DOF parallel platform mechanism. IEEE Trans Robot Autom 19(6):954–966 2. Fu J et al (2015) Forward kinematics solutions of a special six-degree-of-freedom parallel manipulator with three limbs. Adv Mech Eng 7(5):1687814015582118 3. Huang T, Li Z, Li M et al (2004) Conceptual design and dimensional synthesis of a novel 2-DOF translational parallel robot for pick-and-place operations. J Mech Des 126(3):449–455 4. Liu M-J, Li C-X, Li C-N (2000) Dynamics analysis of the Gough–Stewart platform manipulator. IEEE Trans Robot Autom 16(1):94–98 5. Dasgupta B, Mruthyunjaya TS (2000) The Stewart platform manipulator: a review. Mech Mach Theory 35(1):15–40 6. Fichter EF (1986) A Stewart platform-based manipulator: general theory and practical construction. Int J Robot Res 5(2):157–182 7. Raghavan M (1993) The Stewart platform of general geometry has 40 configurations. J Mech Des 277–282 8. Gosselin C (1990) Determination of the workspace of 6-DOF parallel manipulators. J Mech Des 112(3):331–336 9. Huang Z, Li QC (2002) General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. Int J Robot Res 21(2):131–145 10. Merlet JP (1997) Designing a parallel manipulator for a specific workspace. Int J Robot Res 16(4):545–556 11. Gogu G (ed) (2008) Structural synthesis of parallel robots. Springer Netherlands, Dordrecht 12. Lee K-M, Shah DK (1988) Kinematic analysis of a three-degrees-of-freedom in-parallel actuated manipulator. IEEE J Robot Autom 4(3):354–360

Design and Analysis of a New 6-DOF Deployable Parallel Manipulator …

489

13. Lee MK, Park KW (2000) Workspace and singularity analysis of a double parallel manipulator. IEEE/ASME Trans Mechatron 5(4):367–375 14. Gao F et al (2002) New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs. Mech Mach Theory 37(11):1395–1411 15. Dai JS, Jones JR (2001) Interrelationship between screw systems and corresponding reciprocal systems and applications. Mech Mach Theory 36(5):633–651 16. Kong X, Gosselin CM, Richard P-L (2007) Type synthesis of parallel mechanisms with multiple operation modes. J Mech Des 595–601 17. Kong X, Gosselin CM (2002) Kinematics and singularity analysis of a novel type of 3-CRR 3-DOF translational parallel manipulator. Int J Robot Res 21(9):791–798 18. Tsai L-W, Joshi S (2000) Kinematics and optimization of a spatial 3-UPU parallel manipulator. J Mech Des 122(4):439–446 19. Fu J et al (2016) Kinematic accuracy research of a novel six-degree-of-freedom parallel robot with three legs. Mech Mach Theory 102:86–102 20. Fu J, Gao F (2016) Optimal design of a 3-leg 6-DOF parallel manipulator for a specific workspace. Chin J Mech Eng 29(4):659–668 21. Lin R, Guo W, Gao F (2016) Type synthesis of a family of novel four, five, and six degreesof-freedom sea lion ball mechanisms with three limbs. J Mech Robot 8(2):021023 22. Hui DU, Feng GAO, Yang PAN (2015) Kinematic analysis and design of a novel 6-degree of freedom parallel robot. Proc Inst Mech Eng Part C J Mech Eng Sci 229(2):291–303 23. Gonzalez DJ, Asada HH (2017) Design and analysis of 6-DOF triple scissor extender robots with applications in aircraft assembly. IEEE Robot Autom Lett 2(3):1420–1427 24. Yang Y et al (2019) Deployable parallel lower-mobility manipulators with scissor-like elements. Mech Mach Theory 135:226–250 25. Wang R, Sun J, Dai JS (2019) Design analysis and type synthesis of a petal-inspired space deployable-foldable mechanism. Mech Mach Theory 141:151–170

An Optimized Cable Layout Method for Cable-Driven Continuum Robots Zheshuai Yang, Laihao Yang, Dong Yang, Yu Lan, Yu Sun, and Xuefeng Chen

Abstract Cable-driven continuum robots, inspired by natural biologies such as elephants and octopuses, have countless numbers of degrees of freedom, permitting them to carry out special tasks in confined space. However, the slender manipulator driven by cables also results in a significant number of driving motors integrated at the end of manipulator, which complicates the control of continuum robot. In this paper, to solve the problem, we optimize the cable layout based on the analysis of the cable length variation principle. First and foremost, a cable layout method of the shared motor is proposed, which can achieve the complete control of a continuum robot by employing half the number of motors. And then, the kinematics for the proposed cable layout is established based on the assumption of piecewise constant curvature. Finally, the experimental platform is constructed to evaluate the effectiveness of the proposed methods. The experimental results suggest that the optimized cable layout method can achieve the control of a double-section continuum robot by employing four motors. Moreover, it is worthy to be noted that, to some degree, the proposed cable layout method can effectively decrease the influence of gravity, which maximally improves 92.16% of the tip position accuracy. Keywords Continuum robots · Cable layout · Kinematics · Shape estimation

1 Introduction Cable-driven continuum robots, novel bionic robots inspired by natural biologies such as elephants and octopuses, have widely received attention in the last two decades due to their unique flexibility. Owing to their adaptability to narrow and confined spaces by compliantly changing their shape [1], continuum robots have been Z. Yang · L. Yang (B) · D. Yang · Y. Lan · Y. Sun · X. Chen School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China e-mail: [email protected] Z. Yang e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_28

491

492

Z. Yang et al.

extensively applied in minimally invasive surgery [2, 3], nuclear reactor maintenance [4, 5], as well as emerging in the field of aero-engine inspection recently [6–8]. Benefitting from the cable-driven mechanism of continuum robots, it is easier to minimize the structure of the manipulators than conventional rigid-link robots, which permits them to operate easily in long and narrow environments [9, 10]. However, the slender manipulator also results in a significant number of driving motors integrated at the end of manipulator, which greatly complicates the control of continuum robot. To solve the problem, Dong et al. [11] developed a spool mechanism that was employed in the continuum robot system. Yang et al. [12] investigated the solution of reducing the number of cables and proposed a hybrid modular cable routing method. Xu et al. [13] presented a passive-active linkage segment design to optimize the number of motors. Ji et al. [14] proposed a layered-drive design. Although the above optimizations of cable layout simplify the continuum robot driving system, it is tough to establish an accurate kinematic and dynamic model of the continuum robots because of the cable-driven mechanism. Considerable efforts have been made on this issue in previous reports [1, 15]. To improve the modeling accuracy of continuum robots, kinetostatic and dynamic modeling of flexure-based compliant mechanisms are gradually considered since the model mismatch problem of the widely used assumption of piecewise constant curvature (PCC) [16–19]. For example, Xu et al. [20] established a comprehensive static model. Chen et al. [21] proposed a dynamic modeling method for cable-driven rigid backbone continuum manipulator. Besides, Cosserat-rod theory [22], beam constraint model [23], and the principle of virtual power [24] are investigated in static or dynamic models. In contrast, the PCC-based method is still a reliable modeling method because of its simple principle of modeling and low computation cost with an acceptable error. This paper proposed an optimized cable layout method based on the analysis of the principle of cable length variation, which can achieve the control of continuum robot by employing half the number of motors. The kinematics for the proposed cable layout method is established. And the experimental platform is constructed to validate the effectiveness of the proposed methods. The remainder of this paper is organized as follows. Section 2 introduces the mechanism design of continuum robot and proposed an optimized cable layout method. Section 3 establishes the kinematics for the proposed cable layout method. In Sect. 4, the experimental platform is constructed to validate the proposed methods. And the last section summarizes the whole paper and gives the conclusion.

2 Mechanism Design and Optimization In this section, a continuum robot with twin-pivot compliant joint is constructed. And based on the analysis of the cable length variation principle, the cable layout is optimized.

An Optimized Cable Layout Method for Cable-Driven Continuum Robots

493

Fig. 1 The structure of continuum robot

2.1 Overview of Continuum Robot Structure In the previous work [25], the structure design of the continuum robot was reported in detail. A double-section structure of continuum robot is illustrated in Fig. 1. Each section is made up of several identical segments and driven by four cables. And each segment comprises three disks connected by NiTi rods. The disk is approximately a ring in shape with a diameter of 17 mm and a thickness of 5 mm.

2.2 Cable Layout Optimization In most cable layout method of continuum robots, each cable is actuated by one motor, which results in a complex and bulky driving system and increases the cost of system integration, especially as the number of sections increases. In this study, by analyzing the cable length variation principle, it is found that the length variations of two cables that are symmetrical to the bending plane are the same, as illustrated in Fig. 2. Inspired by this mechanism, a cable layout method of the shared motor is proposed, in which the cable layout can be optimized based on the fact that two cables with identical length variation can be driven by one motor. As a result, one can achieve the complete control of a continuum robot by employing half the number of motors. As illustrated in Fig. 2, four motors enable the control of a double-section continuum robot by employing the optimized cable layout method, whereas the original method requires eight. It should be noted that the optimized cable layout method leads to one degree of freedom (DOF) for each section of continuum robot. But in some scenarios, such as aero-engine combustors detection [26], the continuum robot utilizing the proposed cable layout method can still meet the task requirement. Compared with the original cable layout method, the proposed method can better overcome the torsion problem caused by gravity. Furthermore, the proposed method enables the reduction of control complexity, optimization of robot structure, and saving of the system integration cost.

494

Z. Yang et al.

Fig. 2 The cable layout principle

3 Kinematics Based on the analysis of previous work [25], we established the kinematic model of the proposed cable layout method based on the assumption of piecewise constant curvature in this section.

3.1 Forward Kinematics As illustrated in Fig. 3, the transformation matrix from coordinate {i} to {i + 1} can be expressed as i−1 i T

= Rot(x, αi−1 ) · Trans(ai−1 , 0, 0) · Rot(z, θi ) · Trans(0, 0, di )

(1)

where i = 1, 2, 3. And the forward kinematics of one segment can be written as 0 3T

= 01 T · 12 T · 23 T

(2)

Different from the kinematic model reported in previous work, one of the bending angles (β1 , β2 ) will turn to zero since one section only owns one bending DOF. Thus, the transformation matrix 01 T or 23 T can be written as

An Optimized Cable Layout Method for Cable-Driven Continuum Robots

495

Fig. 3 The joint coordinate system of one segment

⎡

⎤ 0 l20 0 0⎥ ⎥ or 1 0⎦ 0 1

⎡

0h+ 1 0 0 0 0 1

l0 2

cos β2 − sin β2 0 l20 + h + βl02 · tan ⎢ ⎢ 0 0 −1 0 1 2T = ⎢ ⎣ sin β2 cos β2 0 0 0 0 0 1 ⎡ ⎤ β 1 0 0 βl01 · tan 21 + h + l20 ⎢ ⎥ ⎢0 0 −1 ⎥ 0 or 12 T = ⎢ ⎥ ⎣0 1 0 ⎦ 0 00 0 1

β2 2

1 ⎢ 0 0 ⎢ 1T = ⎣ 0 0

0 1 0 0

1 ⎢ 0 2 ⎢ 3T = ⎣ 0 0

0 0 −1 0

⎤ ⎥ ⎥ ⎦

and 12 T can be expressed as ⎡

⎤ ⎥ ⎥ ⎥ ⎦

where l0 is the length of NiTi rod, h is the thickness of the disk.

3.2 Inverse Kinematics When it comes to the inverse kinematics for the proposed cable layout, the length of cables varies only in one plane, i.e., Bending Plane 1 or Bending Plane 2, as illustrated in Fig. 4. And we assume that Bending Plane 1 is bent. Thus, the inverse kinematics of section 1st can be given by

496

Z. Yang et al.

Fig. 4 a One segment; b top view of disk A

  ⎧ β1 l0 1 1 ⎪ ⎪ l · sin 2 = l = n − r · cos δ − l 1 1 0 ⎨ 1,1 1,2 β1 2   ⎪ l0 β1 ⎪ 1 1 ⎩ l1,3 − l0 = l1,4 = n1 2 + r · sin δ1 · sin β1 2

(3)

where n 1 is the number of segments in section 1st. Hence, the inverse kinematics of the continuum robot can be obtained li, j =

i 

li,k j

(4)

k=1

4 Experimental Validation A prototype of single-section continuum robot is fabricated to compare the influence of different cable layout method on the motion performance of the continuum robot, and to validate the effectiveness of the proposed cable layout method.

4.1 Experimental Setup As shown in Fig. 5, the experimental platform mainly consists of a continuum robot prototype, a vision system, pulleys, and motors. The shape of continuum robot is collected by the vision system with a 5472 × 3648-pixel camera and reshaped by detecting the positions of marker points on the disks.

An Optimized Cable Layout Method for Cable-Driven Continuum Robots

497

Fig. 5 Experimental platform

4.2 Cable Layout Experiment The cable layout experiment is performed to evaluate the motion performance of the continuum robot with different cable layout methods. Two cable layout methods are illustrated in Fig. 6 and the shape estimation results are presented in Fig. 7. The mass of the continuum robot without cables is 165.5 g. The mass of the continuum robot utilizing original cable layout method 178.5 g while the proposed cable layout method is 189.5 g. The experimental results suggest that the shape of the proposed cable layout method is in good agreement with the desired results obtained by the PCC-based kinematic model when the bending angle ranges from 0 to 20°. Compared to the original cable layout result, the proposed method can better resist the influence of gravity, which maximally improves 92.16% of the tip position accuracy (when the bending angle is 10°). However, with the enhancement of bending angle, the position accuracy of the two cable layout methods gradually decreases, which further indicates the significance of motion compensation for continuum robot.

Fig. 6 a Original cable layout; b proposed cable layout

498

Z. Yang et al.

Fig. 7 Shape estimation of two cable layout methods

5 Conclusion This paper presented an optimized cable layout method for continuum robot. An experimental platform is constructed to evaluate the effectiveness of the proposed cable layout method by comparing it with the original method. The main conclusions can be summarized as follows. First, a novel cable layout method is proposed through the analysis of cable length variation principle. The control of continuum robot is simplified by optimizing the cable layout. Second, the kinematics for the optimized cable layout is established based on the assumption of PCC. At last, the experimental results suggest that the optimized cable layout can effectively decrease the influence of gravity, which maximally improves 92.16% of the tip position accuracy. In the coming future, motion compensation for continuum robot will be further studied to improve motion accuracy. Acknowledgements Supported by National Natural Science Foundation of China (No. 52105117, No. 52105118, and No. 92060302).

An Optimized Cable Layout Method for Cable-Driven Continuum Robots

499

References 1. Webster RJ III, Jones BA (2010) Design and kinematic modeling of constant curvature continuum robots: a review. Int J Robot Res 29(13):1661–1683 2. Gao A, Liu N, Shen M, Abdelaziz MEMK, Temelkuran B, Yang G-Z (2020) Laser-profiled continuum robot with integrated tension sensing for simultaneous shape and tip force estimation. Soft Robot 7(4):421–443 3. Ping Z, Zhang T, Zhang C, Liu J, Zuo S (2021) Design of contact-aided compliant flexure hinge mechanism using superelastic nitinol. J Mech Des 143(11) 4. Qin G, Cheng Y, Pan H, Zhao W, Shi S, Ji A, Wu H (2022) Systematic design of snake arm maintainer in nuclear industry. Fus Eng Des 176 5. Buckingham R, Graham A (2012) Nuclear snake-arm robots. Ind Robot Int J 39(1):6–11 6. Dong X, Palmer D, Axinte D, Kell J (2019) In-situ repair/maintenance with a continuum robotic machine tool in confined space. J Manuf Process 38:313–318 7. Dong X, Axinte D, Palmer D, Cobos S, Raffles M, Rabani A, Kell J (2017) Development of a slender continuum robotic system for on-wing inspection/repair of gas turbine engines. Robot Comput Integr Manuf 44:218–229 8. Wang M, Dong X, Ba W, Mohammad A, Axinte D, Norton A (2021) Design, modelling and validation of a novel extra slender continuum robot for in-situ inspection and repair in aeroengine. Robot Comput Integr Manuf 67 9. Xu W, Liu T, Li Y (2018) Kinematics, dynamics, and control of a cable-driven hyper-redundant manipulator. IEEE-ASME Trans Mechatron 23(4):1693–1704 10. Tang L, Huang J, Zhu L-M, Zhu X, Gu G (2019) Path tracking of a cable-driven snake robot with a two-level motion planning method. IEEE-ASME Trans Mechatron 24(3):935–946 11. Dong X, Raffles M, Cobos-Guzman S, Axinte D, Kell J (2016) A novel continuum robot using twin-pivot compliant joints: design, modeling, and validation. J Mech Robot Trans ASME 8 12. Wang Y, Song C, Zheng T, Lau D, Yang K, Yang G (2019) Cable routing design and performance evaluation for multi-link cable-driven robots with minimal number of actuating cables. IEEE Access 7:135790–135800 13. Liu T, Xu W, Yang T, Li Y (2021) A hybrid active and passive cable-driven segmented redundant manipulator: design, kinematics, and planning. IEEE-ASME Trans Mechatron 26(2):930–942 14. Qin G, Ji A, Cheng Y, Zhao W, Pan H, Shi S, Song Y. A snake-inspired layer-driven continuum robot. Soft Robot 15. Garriga-Casanovas A, Rodriguez y Baena F (2019) Kinematics of continuum robots with constant curvature bending and extension capabilities. J Mech Robot Trans ASME 11(1) 16. Ling M, Howell LL, Cao J, Chen G (2020) Kinetostatic and dynamic modeling of flexure-based compliant mechanisms: a survey. Appl Mech Rev 72(3) 17. Peng J, Wu H, Liu T, Han Y (2022) Workspace, stiffness analysis and design optimization of coupled active-passive multilink cable-driven space robots for on-orbit services. Chin J Aeronaut 18. Xu K, Simaan N (2010) Analytic formulation for kinematics, statics, and shape restoration of multibackbone continuum robots via elliptic integrals. J Mech Robot Trans ASME 2(1) 19. Ba W, Dong X, Mohammad A, Wang M, Axinte D, Norton A (2021) Design and validation of a novel fuzzy-logic-based static feedback controller for tendon-driven continuum robots. IEEE-ASME Trans Mechatron 26(6):3010–3021 20. Yuan H, Zhou L, Xu W (2019) A comprehensive static model of cable-driven multi-section continuum robots considering friction effect. Mech Mach Theory 135:130–149 21. Zheng X, Yang T, Zhu X, Chen Z, Wang X, Liang B (2022) Dynamic modeling and experimental verification of a cable-driven continuum manipulator with cable-constrained synchronous rotating mechanisms. Nonlinear Dyn 107(1):153–172 22. Ghafoori M, Khalaji AK (2020) Modeling and experimental analysis of a multi-rod parallel continuum robot using the Cosserat theory. Robot Auton Syst 134 23. Awtar S, Sen S (2010) A generalized constraint model for two-dimensional beam flexures: nonlinear strain energy formulation. J Mech Des 132(8)

500

Z. Yang et al.

24. Rone WS, Ben-Tzvi P (2014) Continuum robot dynamics utilizing the principle of virtual power. IEEE Trans Robot 30(1):275–287 25. Yang Z, Yang L, Xu L, Chen X, Guo Y, Liu J, Sun Y (2021) A continuum robot with twin-pivot structure: the kinematics and shape estimation. In: Intelligent robotics and applications, ICIRA 2021, PT I, pp 466–475 26. Russo M, Raimondi L, Dong X, Axinte D, Kell J (2021) Task-oriented optimal dimensional synthesis of robotic manipulators with limited mobility. Robot Comput Integr Manuf 69

Design of a Miniature Three-Dimensional Force Sensor for Force Feedback in Minimally Invasive Surgery Qinjian Zhang, Wu Zhang, Haiyuan Li, and Lutao Yan

Abstract Force feedback in robot-assisted surgery is important to enhance the immersion of intraoperative operations. In this paper, a miniaturized small-range three-dimensional force sensor is designed for surgical robots. We presented the elastomer structure design and performed mechanical modeling analysis, finite element force and deformation verification as well as the analysis and comparison to choose a better elastomer structure. According to the optimized elastomer structure, the strain gage arrangement method is carried out to ensure higher sensitivity and better dynamic performance. Finally, the calibration experiment verifies that the designed miniature three-dimensional force sensor has high sensitivity, high resolution and low-dimensional coupling, resulting in potential application in surgical robotic instruments. Keywords Surgical robot · Force feedback · Robot-assisted surgery

1 Introduction With the rapid development of today’s medical technology, modern minimally invasive surgery has brought great changes to the traditional surgery, and surgical robots are more and more widely used in modern minimally invasive surgery [1–4]. As Q. Zhang · W. Zhang School of Mechanical and Electrical Engineering, Beijing Information Science and Technology University, Beijing 100096, China e-mail: [email protected] W. Zhang e-mail: [email protected] H. Li (B) · L. Yan School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China e-mail: [email protected] L. Yan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_29

501

502

Q. Zhang et al.

an important part of modern medical field, compared with the traditional surgery in the past, surgical robot can reduce the trauma of surgery to patients and shorten the recovery period of surgery [5], but also enable surgeons to better adapt to the complex surgical environment, better complete the operation [6, 7]. However, in the actual minimally invasive surgery, surgeons do not directly operate the surgical instrument arm to contact the pathogenic tissue, but indirectly operate the main hand through the doctor’s operating table to perform surgery, which is prone to cause tissue trauma and other problems. Therefore, there is still a lack of force perception feedback between the operated pathogenic tissue and the surgical instrument [8–10]. Force measurement in minimally invasive surgery robot is important to avoid misoperation or excessive clamping. Due to the narrow operation space and the maximum operation, force (cutting, holding, knotting, etc.) used is about 9 N [11–13], a new type of small-scale three-dimensional force sensor is designed and it can be integrated on the arm of surgical instrument. This force sensor has the characteristics of small range, high sensitivity, and small coupling between dimensions, and can better perform force-sensing feedback operations.

2 Design of Three-Dimensional Force Sensor 2.1 The Overall Design of the Three-Dimensional Force Sensor At present, there are many types of three-dimensional force sensor elastomer structures, such as tubular structure, cantilever beam, tripod, and cross structure [14, 15]. However, in practical applications, small size and simple structure are essential as force sensors for force feedback in minimally invasive surgery. Based on the consideration of the simplification of the structure, a miniature three-dimensional force sensor with an optimized cross structure was designed. The overall size of the sensor is 8 mm in outer diameter and 6 mm in height. The structure is divided into four parts: cylindrical shell, cross elastomer, cross fixing block and the top cover, as shown in Fig. 1. In the above-designed miniature three-dimensional force sensor, the cylindrical shell is mainly used to fix and encapsulate the cross elastomer structure. The cross elastomer structure consists of two beams and a square column, which is used to measure the forces of the X, Y and Z-axis. The total length of the beam is 7.8 mm, the height and thickness of the beam are 2 mm and 1.9 mm respectively. The height of the square column is 2.5 mm. The cross fixing block fixes the cross elastomer structure in the cylindrical shell by screws to ensure the stability of the cross elastomer when the force is measured. The top cover is used to encapsulate the overall three-dimensional force sensor. There are four hollow structures inside the sensor, and the remaining space is used for connecting strain gauges and subsequent circuit cables. There are 3 outputs according to the three-dimensional force to be measured, and the strain

Design of a Miniature Three-Dimensional Force Sensor for Force …

503 the top cover cross fixing block

cross elastomer cylindrical shell

Fig. 1 The overall structure of the three-dimensional force sensor

gauges are respectively arranged on the two beams of the elastic cross and on the square column.

2.2 Cross Elastomer Design of Three-Dimensional Force Sensor In the structure of the force sensor, the design of the cross elastomer structure is the most critical. Force-sensing feedback depends entirely on the structural properties of the elastomer. In order to ensure that the three-dimensional force sensor has a better strain effect and well performance, the structural optimization design of elastomer structure is needed. In the five schemes, the overall structure adopts the cross structure. The thickness of the elastomer is large, which ensures the stability, but the sensitivity measured by the sensor is small. In order to solve the influence of the thickness of the elastomer structure on the sensitivity of the sensor, a hole was dug between the cross structure of the X-axis and Y-axis. The elastomer was designed in the following 5 schemes, as shown in Fig. 2.

2.3 Finite Element Analysis of Cross Elastomer In order to study the deformation and stress of the five sensor elastic bodies, theoretically verify the optimality of the design, and obtain the best sticking position of the strain gauges, a finite element analysis was carried out. In order to ensure the environmental requirements during minimally invasive surgery, aluminum alloy 2024 is selected as the elastomer material [16, 17], and its relevant parameters are: E = 7.2 × 104 MPa, Poisson’s ratio ν = 0.33, and density ρ = 2.7 × 103 kg/m3 . In the finite element analysis, the solid element is divided into a network with a unit size of 0.3 mm, and the finite element model is obtained. Constraints are imposed according to the sensor’s assembly mode and force analysis. The maximum load

504

Q. Zhang et al.

Fig. 2 Design scheme of cross elastomer structure

required to apply force is 9 N. Since the elastomer has a highly symmetrical structure, for convenience and efficiency, the force analysis results on the X-axis and the Y-axis are consistent. The following analysis is based on the X-axis. When the constraint is applied to the force analysis of the X-axis, the constraint fixation is applied to both sides of the Y-axis of the elastomer model. A force reference point is selected and the action of the force reference point is coupled to the inner surface of the center hole, so that the force applied at the reference point along the X-axis is equivalent to the force applied to the coupling surface. Similarly, the force applied to the Y-axis is the same as the force applied to the X-axis. When analyzing the force on the Z-axis, constraint fixation is applied to the fixed structure on the base. The force surface is selected as the upper surface of the elastomer, and the maximum load force is applied in the Z-axis direction. Then the static analysis of the model was carried out in the three directions of X-axis, Y-axis and Z-axis respectively. For each of the above elastomer schemes, the corresponding strain nephogram, stress nephogram and strain diagram along the length path were made, and the results were as follows: For scheme (1), according to the finite element analysis of the above method, the strain cloud diagram, the stress cloud diagram and the stress diagram on the length path on the X-axis and the length path stress diagram of the elastomer force path ABCD are respectively obtained, as shown in Fig. 3. The maximum stress distribution can be analyzed to be generated at 0.6 and 2.3 mm.

Design of a Miniature Three-Dimensional Force Sensor for Force …

505

Fig. 3 Finite element analysis results scheme (1) a strain cloud diagram, b stress diagram on the length path; c length path stress diagram of the elastomer force path ABCD

For scheme (2), similarly, the strain nephogram and stress nephogram on the Xaxis, the strain diagram on the length path and the strain diagram on the length path of the elastomer stress path ABCD were obtained by the finite element analysis, as shown in Fig. 4. The maximum stress distribution can be analyzed to be uneven. For scheme (3), similarly, the strain nephogram and stress nephogram on the Xaxis, the strain diagram on the length path and the strain diagram on the length path of the elastomer stress path ABCD were obtained by finite element analysis, as shown in Fig. 5. The maximum stress distribution can be analyzed to be 1.1 and 1.7 mm. For scheme (4), similarly, the strain nephogram and stress nephogram on the X axis, the strain diagram on the length path and the strain diagram on the length path of the elastomer stress path ABCD were obtained by the finite element analysis, as shown in Fig. 6. The maximum stress distribution can be analyzed to be 1.1 and 1.7 mm. For scheme (5), similarly, the strain nephogram and stress nephogram on the X axis, the strain diagram on the length path and the strain diagram on the length path of the elastomer stress path ABCD were obtained by finite element analysis, as shown in Fig. 7. The maximum stress distribution can be analyzed to be 1.1 and 1.7 mm. For the five schemes, since the design on the Z-axis is the same, it can be analyzed in the direction of the Z-axis in accordance with a structure. As shown in Fig. 8, strain

506

Q. Zhang et al.

Fig. 4 Finite element analysis results scheme (2) a strain cloud diagram; b stress diagram on the length path; c length path stress diagram of elastomer force path ABCD

nephogram, stress nephogram, vertical side stress nephogram and transverse side stress nephogram are respectively shown on the Z-axis. Because two corresponding surfaces on the Z-axis are vertical patches, and the other two corresponding surfaces are horizontal patches, the strain pressure difference can be generated in this way. Based on the above analysis, in the scheme (1), the maximum stress distribution is 0.6 and 2.3 mm, with an interval of about 1.7 mm, while the size of the strain gauge is 1.5 × 1.3 mm, which is relatively far apart; In scheme (2), the maximum stress distribution of AC and BD is not uneven; The maximum strains in solutions (3), (4) and (5) are almost all distributed at 1.1 and 1.7 mm, with an interval of 0.6 mm, which is within the range of the strain gauge size. In order to make the elastomer get better effect, it is more symmetrical at the point of maximum stress and point of maximum deformation. For the path ABCD, the A and C paths should be subjected to the same maximum stress, and the B and D paths should be subjected to the same maximum stress. However, in scheme (3), (4) and (5), only scheme (5) has better maximum stress symmetry. Combined with the processing technology and strain effect of the above five schemes, the final choice of scheme (5) as the design of the elastomer structure.

Design of a Miniature Three-Dimensional Force Sensor for Force …

507

Fig. 5 Finite element analysis results scheme (3) a strain cloud diagram; b stress diagram on the length path; c length path stress diagram of elastomer force path ABCD

3 Three-Dimensional Force Sensor Bridge Arm Circuit Analysis The force signal acquisition of the sensor is designed based on the working principle of the strain gauge. When the external force acts on the end of the sensor, the elastomer of the sensor will produce the corresponding stress and strain, so that it expands with the strain of the elastomer, and the strain is measured by measuring the change of the resistance. The deformation of the resistance wire causes the resistance of the resistance wire to change, so that the strain gauge produces a corresponding voltage change, and the corresponding force signal can be measured. Next, the bridge circuit and material strain are analyzed. When the material is stressed, the relationship between the internal stress and strain is expressed according to Hooke’s law: σ = E ·ε

(1)

508

Q. Zhang et al.

Fig. 6 Finite element analysis results scheme (4) a strain cloud diagram; b stress diagram on the length path; c length path stress diagram of elastomer force path ABCD

When the aluminum alloy stretches or contracts during the application of force, its resistance will change. The strain is measured by measuring the change in resistance. Therefore, once it receives external tension (or pressure), that is, stretching (or shortening), the value of the resistance will also increase (or decrease). When the metal material is strained, assuming that the resistance R changes R under its influence, then: R = Ks · ε R

(2)

In the formula, R represents the original resistance value of the strain gauge Ω (ohm); R represents the resistance change Ω (ohm) caused by elongation or contraction; K s represents the proportional constant (called the strain rate); ε represents the strain. The strain rate K s varies with metal materials. In the case of aluminum alloy, the value of the strain rate is approximately 2.

Design of a Miniature Three-Dimensional Force Sensor for Force …

509

Fig. 7 Finite element analysis results scheme (5) a strain cloud diagram; b stress diagram on the length path; c length path stress diagram of elastomer force path ABCD

In practice, it is very difficult to accurately measure such a small change in resistance. In order to measure such small resistance changes, a circuit called a Wheatstone bridge is used, which is suitable for detecting small resistance changes, and the resistance changes of the strain gauge are also measured using this circuit. As shown in Fig. 9, a Wheatstone bridge consists of four resistors combined. The measuring circuit can be divided into three types according to the different ways of accessing the strain gauge, namely, single arm, half bridge and full bridge [18]. In order to improve the sensitivity of the bridge circuit and eliminate the instability of temperature to the output voltage, we adopt the full bridge circuit form among the three kinds of bridge arm selection. In addition, the full bridge circuit has the highest measurement accuracy, and according to the symmetry of the elastomer structure, the full bridge arm circuit is finally selected. If R1 = R2 = R3 = R4 , or R1 × R2 = R3 × R4 , no matter how much voltage is input, the output voltage U0 is always zero. The state of this bridge is called the equilibrium state. If the balance is unbalanced, an output voltage corresponding to the change in resistance will be generated. In the full-bridge circuit, all four sides are connected to strain gauges. The output voltage of the bridge is shown in Fig. 9. When the resistance of the strain gauges on the four sides changes respectively:

510

Q. Zhang et al.

Fig. 8 Finite element analysis results of Z-axis a Z-axis strain cloud diagram; b Z-axis stress cloud diagram; c vertical side stress cloud diagram; d horizontal side stress cloud diagram Fig. 9 Wheatstone bridge

Design of a Miniature Three-Dimensional Force Sensor for Force …

U0 =

  1 R1 R2 R3 R4 E − + − 4 R1 R2 R3 R4

511

(3)

Since the specifications of the four strain gauges are exactly the same, when the strain rates are all K s, the strains received by the four strain gauges are ε1 , ε2 , ε3 , ε4 , and the above Eq. (3) can be written as the following form: U0 =

1 K s(ε1 − ε2 + ε3 − ε4 )E 4

(4)

As for the force forms on X-axis and Y-axis, it can be concluded that when the external force F is applied, the elastomer is subjected to the deformation of tension and compression on both sides. Due to the symmetrical structure, the strain generated by tension and compression is the same in size and opposite in direction. It can be obtained: ε1 = −ε2 = ε3 = −ε4 = ε

(5)

Therefore, the output voltage is: U0 = K s · ε · E

(6)

For the direct arrangement method, when the material is subjected to the axial force F, it elongates in the axial direction and shortens in the rectangular direction. The elongation of the material along the axial direction is called the axial strain, the shortening of the rectangular direction is called the transverse strain, and the absolute value of the ratio of the longitudinal strain to the transverse strain is called the Poisson’s ratio. v = |ε2 /ε1 |

(7)

In the formula, ε1 means longitudinal strain; ε2 means transverse strain. Therefore, when external force is applied on the Z-axis, the resistance of strain gauges on the four sides changes respectively:   1 R1 R2 R3 R4 U0 = − + − 4 R1 R2 R3 R4 U0 =

(8)

1 (ε − vε + ε − vε) 4

(9)

(1 + v)E Ks · ε 2

(10)

U0 =

512

Q. Zhang et al.

Fig. 10 The patch position of the strain gauge on the cross elastomer

Through the patch position of the above finite element analysis, and combined with the size of the cross elastomer structure, the strain gauge of TSK-05R-120-1A model was selected, and the base size of the strain gauge was 1.3 * 1.5 mm. Attach three sets of full bridge strain gauges to the position where the strain is most obvious in the X, Y, and Z-axis of the cross elastomer, and measure the three-dimensional force by changing the strain. The attachment position of the strain gauges is shown in Fig. 10. When the external force is generated in the direction of X-axis, the beam on which strain gauges 5, 6, 7 and 8 are located produce bending deformation, and the corresponding strain gauges 5, 6, 7, and 8 measure the force FX in the direction of Xaxis. Similarly, when the external force is generated in the Y-axis direction, the beam on which strain gauges 1, 2, 3, and 4 are located will produce bending deformation, and the corresponding strain gauges 1, 2, 3, and 4 measure the force FY in the direction of Y-axis. When the external force is generated in the Z-axis direction, the square column where strain gauges 9, 10, 11 and 12 are located produces compression deformation, and the corresponding strain gauges 9, 10, 11 and 12 measure the force FZ in the direction of Z-axis.

4 Calibration Experiment In order to complete the loading calibration of the three-dimensional force sensor, a simulated force loading experiment was carried out to realize the loading of the positive and negative ranges of each dimension. The real object is shown in Fig. 11 and force sensor calibration experiment as shown in Fig. 12. According to the requirement of minimally invasive surgical force, the positive and negative directions of 0–10 N are applied on the X-axis, Y-axis and Z-axis respectively. Through the simulated force effect applied by simulation, the elastic strain distribution can be analyzed to be mainly concentrated in the position of the center hole of the elastic body. When the external force is exerted on the X-axis direction, the beam along the X-axis direction produces bending deformation and corresponding elastic strain. Similarly, when the

Design of a Miniature Three-Dimensional Force Sensor for Force …

513

external force is exerted on the Y-axis direction, the beam along the Y-axis direction generates bending deformation and corresponding elastic strain. When the external force is exerted on the Z-axis direction, the beam along the Z-axis direction produces bending deformation and corresponding elastic strain. Generally, under ideal conditions, when a force is applied in a single direction, only the corresponding elastic strain will be produced in this direction, and the other directions will not be affected. However in engineering practice, there are still relatively small corresponding elastic strains in other directions. Therefore, the

Fig. 11 Miniature three-dimensional force sensor a overall structure; b disassembly structure

Fig. 12 Calibration experiment of force sensor

514

Q. Zhang et al.

directions of the force sensor are not independent of each other, and it is inevitable for the elastic strain in each direction to appear inter-dimensional coupling [19, 20]. The relationship between the elastic strain on the sensor elastomer and the multidimensional force acting on the sensor is established. The relationship between the generated elastic strain and the force on the sensor is: ⎛

⎞ ⎛ ⎞⎛ ⎞ εX CX X CXY CX Z FX ⎝ εY ⎠ = ⎝ CY X CY Y CY Z ⎠⎝ FY ⎠ εZ CZ X CZY CZ Z FZ

(11)

By multiplying formula (11) by C −1 , the relationship between force and elastic strain of the sensor can be obtained as follows: ⎛

⎞ ⎛ ⎞−1 ⎛ ⎞ FX CX X CXY CX Z εX ⎝ FY ⎠ = ⎝ CY X CY Y CY Z ⎠ ⎝ εY ⎠ FZ CZ X CZY CZ Z εZ

(12)

where, ε X , εY , ε Z are the elastic strains generated in three directions, FX , FY , FZ are the forces applied, and C is the calibration coupling matrix. Linear least square method was used to fit the output function of the sensor, and static decoupling processing was carried out on the experimental data, as shown in Fig. 13, to obtain the relationship between the stress applied in each direction and the elastic strain. The parameters of the coupling matrix after fitting can be obtained according to the data obtained by applying the forces in three axial directions and the corresponding elastic strain respectively: C X X = 43.1591, CY X = 0.2223, C Z X = 0.5866, C X Y = 0.3108, CY Y = 42.9398, C Z Y = 0.8886, C X Z = 0.0440, CY Z = 0.0633, C Z Z = 8.9310. That is, the coupling matrix C can be obtained: ⎛

⎞ 43.1591 0.3108 0.0440 C = ⎝ 0.2223 42.9398 0.0633 ⎠ 0.5866 0.8886 8.9310

(13)

Invert the coupling matrix (13): ⎛

C −1

⎞ 0.0232 − 0.0002 − 0.0001 = ⎝ − 0.0001 0.0233 − 0.0002 ⎠ − 0.0015 − 0.0023 0.1120

(14)

The relationship between the generated elastic strain and the force on the sensor can be obtained by substituting the parameters of the coupling matrix C and C −1 solved above respectively:

Design of a Miniature Three-Dimensional Force Sensor for Force … Fig. 13 a Elastic strain corresponding to the X-axis under stress; b the corresponding elastic strain under stress on the Y-axis; c corresponding elastic strain under stress on the Z-axis

515

516

Q. Zhang et al.

⎛

⎞ ⎛ ⎞ ⎞⎛ εX 43.1591 0.3108 0.0440 FX ⎝ εY ⎠ = ⎝ 0.2223 42.9398 0.0633 ⎠⎝ FY ⎠ εZ FZ 0.5866 0.8886 8.9310 The stress and elastic strain of the sensor: ⎛ ⎞ ⎛ ⎞⎛ ⎞ FX 0.0232 − 0.0002 − 0.0001 εX ⎝ FY ⎠ = ⎝ − 0.0001 0.0233 − 0.0002 ⎠⎝ εY ⎠ FZ εZ − 0.0015 − 0.0023 0.1120

(15)

(16)

According to the formula, the force on the known elastomer can be obtained and the corresponding elastic strain of the elastomer can be obtained. Conversely, given the corresponding elastic strain of the elastomer, the force exerted on the elastomer can also be obtained. For the coupling matrix C and C −1 , both have non-diagonal coefficients, indicating that there is a certain coupling error between the dimensions of the sensor [14]. The inter-dimensional coupling error is shown in Fig. 14 and the coupling error analysis table is shown in Table 1 when the force is applied in the triaxial direction respectively. Through the above coupling error analysis, it can be obtained that when the force is measured in the whole process, the maximum coupling error between dimensions of the miniature three-dimensional force sensor in the design is 1.35% when the force is in the X-axis. When the force is exerted in the Y-axis, the maximum coupling error between dimensions is 2.07%. In the case of force in the Z-axis, the maximum interdimensional coupling error is 0.71%. It can be concluded that the output linearity of the miniature three-dimensional force sensor is well and the interdimensional coupling is small.

5 Conclusions In this paper, combined with the problem of force feedback technology in the actual operation of minimally invasive surgical robots, a small-range three-dimensional force sensor for surgical robots was designed. The force sensor can measure threedimensional force through the combination of full bridge arm circuit and the patch method of orthogonal configuration method. It also has the characteristics of miniaturization and structural integration. In order to ensure that the micro three-dimensional force sensor has a better strain effect and well performance, the structure of the elastomer structure is optimized. Five kinds of elastic body structure are designed, through the finite element simulation analysis of the structure corresponding to the elastic strain and the post-processing results on the length path, the elastic body structure with the best effect among the designed miniature sensors is obtained.

Design of a Miniature Three-Dimensional Force Sensor for Force … Fig. 14 a Variation of the coupling error corresponding to the X-axis under stress; b the change of the coupling error corresponding to the force on the Y-axis; c the variation of the corresponding coupling error when the Z-axis is subjected to force

517

518 Table 1 Sensor coupling error analysis table

Q. Zhang et al. The coupling error (%) FX

Applied force FX

FY

FZ

–

0.72

0.49

FY

0.51

–

0.71

FZ

1.35

2.07

–

According to the requirement of minimally invasive surgical force, load the positive and negative ranges of each dimension of the sensor by the simulated force, and complete the simulation coupling calibration of the three-dimensional force sensor by linear least square fitting, and establish the elastic strain and applied force. The multidimensional relationship between. Through the simulation experiment to calculate the corresponding coupling error between dimensions. The calibration and calculation results show that the sensor can realize the measurement of three-dimensional force, and has the performance of well output linearity and small coupling error, which initially meets the design requirements of micro-sensors in minimally invasive surgery. The next step will continue to optimize the structure of the sensor and complete the force loading experiment test of the sensor, and the next step will be studied. Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFC0119203 and 2019YFB1309802) and National Natural Science Foundation of China (Grant No. 62003048).

References 1. Yang GZ, Bellingham J, Dupont PE et al (2018) The grand challenges of science robotics. Sci Robot 3(14):eaar7650 2. Vaida C, Al Hajjar N, Lazar V et al (2019) Robotics in minimally invasive procedures: history, current trends and future challenges. IFMBE Proc 71:267–273 3. Ranev D, Teixeira J (2020) History of computer-assisted surgery. Surg Clin North Am 100:209– 218 4. Zhou J, Ma X, Zhang X et al (2018) Overview of medical robot technology development. Chin Control Conf 6:5169–5174 5. Feyza S (2019) Robotically assisted surgical devices raise caution. JAMA 321(15):1449 6. Peters BS, Armijo PR, Krause C et al (2018) Review of emerging surgical robotic technology. Surg Endosc 32(4):1636–1655 7. Hagn U, Konietschke R, Tobergte A et al (2010) DLR MiroSurge: a versatile system for research in endoscopic telesurgery. Int J Comput 5(2):183–193 8. Merrifield R, Taylor R (2017) The surgical robot challenge [from the guest editors]. IEEE Robot Autom Mag 24(2):22–23 9. Merrifield R (2017) A journey to the surgical robot challenge. IEEE Robot Autom Mag 24(2):90 10. Holsinger FC (2016) A flexible, single-arm robotic surgical system for transoral resection of the tonsil and lateral pharyngeal wall: next-generation robotic head and neck surgery. Laryngoscope 126(4):864–869

Design of a Miniature Three-Dimensional Force Sensor for Force …

519

11. Bos J, Doornebosch EWLJ, Engbers JG et al (2013) Methods for reducing peak pressure in laparoscopic grasping. Proc Inst Mech Eng Part H J Eng Med 227(12):1292–1300 12. Payandeh S, Li T (2003) Toward new designs of haptic devices for minimally invasive surgery. In: International congress series, vol 1256. Elsevier, pp 775–781 13. Thielmann S, Seibold U, Haslinger R et al (2010) MICA—a new generation of versatile instruments in robotic surgery. In: IROS 2010, IEEE international conference on intelligent robots and systems, pp 871–878 14. Yuan C, Luo LP, Yuan Q et al (2015) Development and evaluation of a compact 6-axis force/moment sensor with a serial structure for the humanoid robot foot. Measurement 70:110–122 15. Chen D, Song A, Li A (2015) Six-axis force torque sensor with large measurement range used for the space manipulator. Procedia Eng 99:1164–1170 16. Yu H, Jiang J, Xie L et al (2014) Design and static calibration of a six-dimensional force/torque sensor for minimally invasive surgery. Minim Invasive Ther 23(3):136–143 17. Shi Y, Zhou C, Xie L et al (2017) Research of the master–slave robot surgical system with the function of force feedback. Int J Med Robot Comput Assist Surg 13:e1826 18. Hu S, Wang H, Wang Y et al (2018) Design of a novel six-axis wrist force sensor. Sensors 18(9):3120 19. Li Y, Wang G, Zhao D et al (2013) Research on a novel parallel spoke piezoelectric 6-DOF heavy force/torque sensor. Mech Syst Signal Process 36:152–167 20. Wang Z, Yao J, Xu Y et al (2012) Hyperstatic analysis of a fully pre-stressed six-axis force/torque sensor. Mech Mach Theory 57:84–94

Configuration Design of Lower-Mobility Parallel Driving Mechanisms Jinzhu Zhang, Xinjun Liu, Hanqing Shi, Yangyang Huang, and Qingxue Huang

Abstract A type of parallel driving mechanism (PDM) is proposed in this paper. PDM is different from parallel mechanism and hybrid mechanism, which is composed of the operating mechanism, driving mechanism, and connecting joint. Both the operating mechanism and driving mechanism are connected to the frame which can reduce the inertia at the end of the mechanism and improve the dynamic response. Based on the PDM’s concept, configuration design process of lowermobility PDM is proposed. In this process, the matching procedure between the driving mechanism, operating mechanism and connecting joint in PDM during configuration design is presented. On this basis, the configuration design of 3– 5 degrees-of-freedom (DOFs) PDM is carried out. PDM can enrich the types of mechanisms and provide more configuration options for mechanical equipment. The configuration design process provides theoretical support for developing new PDMs and three PDMs has good potential to be used in different application. Keywords Parallel driving mechanism · Configuration design · Lower-mobility PDM

J. Zhang (B) · X. Liu The State Key Laboratory of Tribology and Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected] X. Liu e-mail: [email protected] J. Zhang · H. Shi · Y. Huang · Q. Huang College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China J. Zhang · Q. Huang Engineering Research Center of Advanced Metal Composites Forming Technology and Equipment, Ministry of Education, Taiyuan 030024, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_30

521

522

J. Zhang et al.

1 Introduction Equipment innovation and development is a general recognition which is gradually formed in the process of human continuous exploration of nature [1]. Mechanism is the skeleton of industrial equipment, which is the key to determine the performance of industrial equipment [2, 3]. From the single-DOF mechanism of drilling wood for fire, to the classical planar mechanism, such as crank slider mechanism and four-bar mechanism [4, 5], and then to the modern spatial mechanism, such as series mechanism [6], parallel mechanism [7], hybrid mechanism [8, 9], reconfigurable mechanism etc. [10]. The invention of each novel mechanism type will promote the improvement of equipment types and performance, which is beneficial to further expand the boundaries of human exploration of nature. PDM is a new kind of mechanism type which is different from parallel mechanism and hybrid mechanism. It is composed of driving mechanism and operating mechanism which are directly connected with the frame. PDM has the characteristics of high stiffness, good dynamic performance, easy protection and so on, which is a supplement to the existing mechanism types [11, 12]. The parallel mechanism with passive constrained branches is considered as a typical example of PDM because its passive constraint branches and driving branches are connected to the frame. Meanwhile, the driving mechanism of this kind of mechanism is connected to the end of the operating mechanism through a fixed pair. Its structure is the same as that of PDM, so that this kind of mechanism can be regarded as a subset of PDM. At the same time, compared with the lower-mobility parallel mechanism without constrained branches, it has good mechanism stiffness and eliminates the negative self-micromotion at the end of the mechanism. Gosselin [13], Huang [14], Lu [15], Cui [16], Guo [17] and others have done a lot of work for this kind of mechanism. Gao et al. [18] proposed a variety of 3-DOF PDM and used it to design a firefighting hexapod robot. After this, Jin et al. [19] designed a 3-DOF leg mechanism and used it to design a quadruped robot. Recently, Zhang et al. [20, 21] proposed a variety of 3-DOF leg mechanisms with protectable and 5-DOF PDM [22, 23] for special steel bar grinding. However, the existing configuration design based on PDM was mainly carried out for special limiting conditions and application scenarios, the theoretical basis is weak and scattered, and the configuration design theory is not systematic. The main contribution of this paper lies in two aspects. (i) The number and type of lower-mobility PDM are extended. (ii) The theoretical basis of PDM industrial application is established. The remainder of this paper is organized as follows. Section 2 presents the characteristics of mechanism composition of PDM. In Sect. 3, the configuration design process of lower-mobility PDM, and the matching process between driving mechanism, operating mechanism and connecting joint in PDM during configuration design are presented. The configuration design of 3–5 DOFs PDM are proposed in Sect. 4. Conclusions are made in Sect. 5.

Configuration Design of Lower-Mobility Parallel Driving Mechanisms

523

2 The Mechanism Composition of PDM PDM is composed of the driving mechanism, operating mechanism, and connecting joint, as shown in Fig. 1. The driving mechanism is active, and its possible compositions can be • DC1: an independent mechanism; • DC2: n discrete branches. The operating mechanism is passive, and its possible compositions can be • OC1: a single open chain mechanism; • OC2: a single loop chain mechanism; • OC3: a mechanism with closed-loop subchain. The connecting joint is passive, and its possible compositions can be • CC1: The number of connecting joint is 1, it can be a fixed pair, revolute joint (R), universal joint (U) or spherical joint (S); • CC2: The number of connecting joint are m, that may be a combination some or all of fixed pair, R, U or S. Based on the possible compositions of the above driving mechanism, operating mechanism and connecting joint, we can obtain the mapping relationship between them, as shown in Fig. 2. As can be seen from Fig. 2, through the combination of the driving mechanism, operating mechanism and connecting joint, 12 composition modes of PDM can be obtained. Meanwhile, there is n = m when the driving mechanism is DC2. Fig. 1 Composition scheme of PDM

524

J. Zhang et al.

Fig. 2 Matching diagram of driving mechanism, operating mechanism, and connecting joint

3 Configuration Design Process of PDM According to Fig. 2, if we want to obtain the specific configuration of a PDM, we need to determine the configuration of the driving mechanism, operating mechanism and connecting joint respectively. By comparing some type synthesis methods such as screw theory, GF set, Grassmann line geometry and so on, we can find that the existing type synthesis methods need to determine the DOF of the mechanism firstly, then solve the combinations of links and DOF and obtain mechanism configuration finally. Therefore, this paper adopts the same process to carry out mechanism configuration design of the lowermobility PDM. The flowchart of PDM mechanism configuration design is shown in Fig. 3. Step 1: Synthesize configuration of the operating mechanism according to the required DOF. The method of type synthesis of the operating mechanism is the same as that of existing open chain mechanism, closed chain mechanism or mechanism with closed loop subchain. Therefore, the detailed process of configuration design is not described in this paper. Step 2: Analyze the controllability of the acquired operating mechanism, determine the number and position of the motion input points of the operating mechanism and the motion characteristics at each motion input point. The selection of motion input points is simple. We only need to find the combination of one or several links related to all joints, and randomly select a point on the link as the motion input point. The motion characteristics at motion input point can be solved by screw theory or GF set. Step 3: Match the appropriate driving mechanism and connecting joint for the operating mechanism, and determine the type and position of the connecting joint. The design method of driving mechanism is the same as that of parallel mechanism or other rigid mechanisms. Step 4: Stiffening the PDM’s driving joints, then judge the effectiveness of the selected driving joints by comparing the connection point’s motion characteristics of PDM operating and driving mechanism and the DOF of the whole machine before and after rigidity.

Configuration Design of Lower-Mobility Parallel Driving Mechanisms

525

Fig. 3 Flowchart of PDM configuration design

In the process of configuration design of PDM, a crucial problem is to match the DOFs among driving mechanism, operating mechanism and connecting joint. The matching process needs to be classified and described according to the conditions of the connecting joint. And the specific process is shown in Fig. 4.

526

J. Zhang et al.

Fig. 4 Matching process between the driving mechanism, operating mechanism and connecting joint

• When the condition of connecting joint is CC1, the driving mechanism is a parallel mechanism. And then this motion characteristics are assigned to driving mechanism and connecting joint. The sum of the DOFs of the driving mechanism and the connecting joint is greater than or equal to the dimension of motion characteristics at the motion input point. Meanwhile, the DOFs of the driving mechanism are greater than operating mechanism. • When the condition of connecting joint is CC2, the driving mechanism is composed of some independent branches distributed in parallel. And then this motion characteristics are assigned to driving branch and connecting joint. The number of branches is the same as the number of connected joints. And each branch has only one driving joint. The sum of the DOFs of the driving branch and the connecting joint is greater than or equal to the dimension of motion characteristics at the motion input point. During the matching of motion characteristics of mechanism, in order to ensure that there are no local DOFs between driving mechanism (or driving branch) and connecting joint, there are two problems to be attended.

Configuration Design of Lower-Mobility Parallel Driving Mechanisms

527

• The sequence of movement and rotation in the motion characteristics of the driving branch or driving mechanism. • The geometric relationship between the motion characteristics of the driving mechanism (or driving branch) and the connecting joint.

4 Lower-Mobility PDM Configuration Design for Different Applications PDMs with lower-mobility have the advantages of lower cost of manufacturing, simple control compared with its 6-DOF counterpart. Meanwhile, it is suitable for many tasks requiring less than 6-DOF. This section focuses on three different applications requiring lower-mobility: leg mechanism of multi-legged robot, riding simulation platform and special steel bar grinding manipulator. Based on the PDM configuration design process given above, 3–5 DOFs PDM configuration design is carried out, respectively.

4.1 The 3-DOF Parallel Driving Leg Mechanism The multi-legged robot has the characteristics of flexible movement and strong ground adaptability. The mechanical leg mechanism is the main operating mechanism of multi legged robot to realize motion and load. In order to reduce the manufacturing cost, it is a general solution to add passive joints at the end of the mechanical leg with lower-mobility, then the mechanical leg has 3R3T full mobility. For application situations such as high temperature, high humidity and strong radiation, all or part of the actuators and their accessories of leg mechanisms always move with the mechanical leg in most legged robot. So that the protection of the driver and its accessories is difficult, which makes the failure rate of the foot robot high, and affects the application of the multi-legged robot in the harsh environment such as firefighting and rescue. Based on the above mechanism configuration design method, we design a 3-DOF parallel driving protectable leg mechanism, as shown in Fig. 5.

4.2 The 4-DOF Parallel Driving Riding Motion Simulation Mechanism Recently, motion simulation platform can simulate all kinds of space motion, so that it is widely used in automobile manufacturing, aerospace, shipbuilding industry, civil entertainment and other fields. According to the real experience of the bicycle riding process, a 4-DOF PDM is designed based on the above mechanism configuration design method, as shown in Fig. 6. This mechanism can simulate main

528

J. Zhang et al.

Fig. 5 The proposed parallel driving leg mechanism

(a) the kinematic scheme

(b) the joint-and-loop graph of mechanism principle.

cycling motion for riding processes such as uphill and downhill, acceleration and deceleration, cornering, vibration, etc.

4.3 The 5-DOF Parallel Driving Grinding Mechanism The steel industry is one of the process industries with a high level of automation, and the robotization of steel production lines is the mainstream direction in the future. Special steel rod finishing operation has the characteristics of high transfer frequency, and high safety risk. As a key process of special steel rod finishing, grinding can eliminate product defects and improve the added value of products. The hardness of

Configuration Design of Lower-Mobility Parallel Driving Mechanisms

529

Fig. 6 The proposed parallel driving riding simulation mechanism

(a) the kinematic scheme

(b) the joint-and-loop graph of mechanism principle. special steel rod is high (HV ≤ 260), which requires high precision, bearing capacity and rigidity of grinding mechanism. In addition, the length of special steel rod is very long (10 ~ 12 m). Therefore, there is an urgent need for high stiffness mechanism with the ability to adjust orientation flexibly and large orientational workspace. We design a 5-DOF parallel driving grinding mechanism based on the above mechanism configuration design method, as shown in Fig. 7.

5 Conclusions In this paper, the PDM composed of the operating mechanism, driving mechanism, and connecting joint, is presented. The possible form and the characteristics of PDM is summarized. Configuration design process of the PDM is proposed by comparing type synthesis methods. To carry out the design of the driving mechanism, operating mechanism, and connecting joint in PDM, matching process is presented by considering relationship between translational and rotational characteristics of the driving mechanism, and geometric relationship between driving mechanism and connecting joint. Afterward, three kinds of lower-mobility PDM configurations are designed

530

J. Zhang et al.

Fig. 7 The proposed parallel driving grinding mechanism

(a) the kinematic scheme

(b) the joint-and-loop graph of mechanism principle.

including 3- DOF parallel driving leg mechanism, 4-DOF parallel driving riding simulation mechanism, and 5-DOF parallel driving grinding mechanism. This paper provides theoretical support for PDM configuration design and is helpful for potential application of PDM. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 51905367), The National Key Research and Development Program of China (No. 2018YFB1308702), The Open Foundation of the State key Laboratory (No. SKLRS-2020KF-17), The Foundation of Applied Basic Research General Youth Program of Shanxi (No. 201901D211011).

Configuration Design of Lower-Mobility Parallel Driving Mechanisms

531

References 1. Gao F, Guo W (2016) Thinking of the development strategy of robots in China. J Mech Eng 52(7):1–5 2. Yang W, Ding H, Andrés K (2019) Automatic synthesis of plane kinematic chains with prismatic pairs and up to 14 links. Mech Mach Theory 132:236–247 3. Zhang W, Lu S, Ding X (2019) Recent development on innovation design of reconfigurable mechanisms in China. Front Mech Eng 14(1):15–20 4. Chen Z (1963) Synthesis of plane articulated four-member mechanism according to the law of given transmission ratio. J Xi’an Jiaotong Univ 3:16–25 5. Westneat MW (1990) Feeding mechanics of teleost fishes: a test of four-bar linkage models. J Morphol 205(3):269–295 6. Wang X, Zhang D, Zhao C, Zhang P, Zhang Y, Cai Y (2019) Optimal design of lightweight serial robots by integrating topology optimization and parametric system optimization. Mech Mach Theory 132:48–65 7. Li Y, Wang Z, Chen C, Xu T, Chen B (2020) Dynamic accuracy analysis of a 5PSS/UPU parallel mechanism based on rigid-flexible coupled modeling. Chin J Mech Eng 35(1):1–14 8. Liu Q, Huang T (2019) Inverse kinematics of a 5-axis hybrid robot with non-singular tool path generation. Robot Comput Integr Manuf 56:140–148 9. Xi D, Gao F (2018) Type synthesis of walking robot legs. Chin J Mech Eng 31(1):44–56 10. Wang T, Olivoni E, Dai J (2022) Novel design of a rotation center auto-matched ankle rehabilitation exoskeleton with decoupled control capacity. J Mech Des 144(5):1–12 11. Liu XJ, Xie F, Yang D, Xie Z, Meng Q (2022) Discussion on research mode of advanced scientific and technological innovation. J Mech Eng 58:1–10 12. Wang T, Tao Y (2014) Research status and industrialization development strategy of Chinese industrial robot. J Mech Eng 50(9):1–13 13. Zhang D, Gosselin C (2001) Kinetostatic modeling of N-DOF parallel mechanisms with a passive constraining leg and prismatic actuators. J Mech Des 123(3):375–381 14. Wu Z, Wang Y, Huang T (2003) Optimal dimensional synthesis of tricept robot. J Mech Eng 39(6):22–25, 30 15. Lu Y (2010) Research on the type synthesis and properties of 3 and 4 DOF parallel manipulator with passive branches. J Mech Eng 46(20):194 16. Cui G, Zhang H, Xu F, Sun C (2014) Kinematic dexterity and stiffness performance of spatial 3-PUS-UP parallel manipulator. Trans Chin Soc Agric Mach 45(12):348–354 17. Lin R, Guo W, Chen X, Li M (2018) Type synthesis of legged mobile landers with one passive limb using the singularity property. Robotica 36(12):1836–1856 18. He J, Gao F (2020) Mechanism, actuation, perception, and control of highly dynamic multilegged robots: a review. Chin J Mech Eng 33(1):1–30 19. Jin Z, Zhang J, Gao F (2016) A firefighting six-legged robot and its kinematics analysis of leg mechanisms. China Mech Eng 27(7):865–871 20. Zhang J, Jin Z, Feng H (2018) Type synthesis of a 3-mixed-DOF protectable leg mechanism of a firefighting multi-legged robot based on GF set theory. Mech Mach Theory 130:567–584 21. Zhang J, Jin Z, Zhao Y (2018) Dynamics analysis of leg mechanism of six-legged firefighting robot. J Mech Sci Technol 32(1):351–361 22. Zhang J, Shi H, Wang T, Huang Q, Jiang L (2021) Analysis of 5-DOF parallel driving mechanism and its position resolution. China Mech Eng 32(12):1414–1422 23. Zhang J, Shi H, Wang T, Liu XJ, Jiang L, Huang Q (2021) Configuration design and evaluation of bionic grinding manipulator based on human upper limb. In: ICIRA 2021: intelligent robotics and applications, vol 13015, pp 642–652

Dimensional Analysis of Transmission Mechanism of Novel Simulated “Soft” Mechanical Adaptive Grasper Haibo Huang, Xinpeng Li, and Rugui Wang

Abstract In this paper, a kind of simulated “soft” mechanical adaptive grasper (GXU-Grasper) with the characteristics of adaptive object shape, uniform grasping force, single-degree-of-freedom drive and full rigid structure is taken as the research object, and the dimensional of its transmission mechanism is analyzed. The correlation model between the motion parameters and dimensional parameters of the transmission mechanism of each knuckle element under different constraints is established, and the linkage correlation model of the transmission mechanism between adjacent knuckle elements is established. The relationship diagram of all feasible solution of the transmission mechanism dimensional of each knuckle unit of the grasper is obtained through the analysis of the example. A prototype was made and the grasping experiment was carried out. The experiment showed that the GXU-Grasper can relatively smoothly and softly grasp the objects that are variable, fragile, brittle and uneven in shape, which verifies the effectiveness of the dimensional analysis of the grasper transmission mechanism. Keywords Grasper · Transmission mechanism · Dimensional analysis · Correlation model

1 Introduction As the end effector of industrial robots, the robot gripper is one of the important research plans of my country’s “14th Five-Year Plan” for the development of the robot industry [1], and it is also one of the frontiers and main hot topics of scientific research and discussion in the world. In the past decades, many researchers have

H. Huang · X. Li · R. Wang (B) School of Mechanical Engineering, Guangxi University, Nanning 530004, Guangxi, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_31

533

534

H. Huang et al.

devoted to designing a simpler and more effective finger mechanism [2]. Zhang et al. [3–5] proposed a new type of linkage for an indirect adaptive underactuated hand, which can utilize the inverse forces from object grasping to realise the action of the next joint without any motor. Choi et al. [6, 7] designed an underactuated finger mechanism that can conduct both self-adaptive grasping and natural motions and can realise a three-degrees-of-freedom mechanism composed of stackable four-bar linkages and contractible slider cranks with a linear spring in each mechanism layer. Stavenuiter et al. [8] introduces the concept design of an underactuated grasper with the ability to adjust its level of self-adaptability by changing the rotational stiffness of its differential mechanism. Hua et al. [9] present a 1-Dof four-bar finger mechanism which could produce bidirectional symmetric grasping motion. Wang et al. [10] proposed a passive adaptive five-finger underactuated dexterous hand, which has the ability to powerfully grasp objects of different shapes and sizes by using linkageslider and rack-pinion mechanisms. Zhang et al. [11] developed a one-DOF six-bar gripper with multiple operation modes and good force adaptability. However, there are few reports of mechanical graspers with fully rigid structure and single degree of freedom drive, which can use multi-level transmission mechanism to realize adaptive envelope grasping of irregular objects and uniform grasping force. Using the link mechanism as the transmission mechanism of the robot grasper is one of the schemes that most scholars often consider when designing the robot grasper [12–15]. The link mechanism has the advantages of high transmission precision and large transmission torque. Wang et al. [16] proposed a new simulated “soft” mechanical self-adaptive grasper, which has the characteristics of adaptive object shape, uniform grasping force, single-degree-of-freedom drive, and full rigid structure. When the grasper works, its transmission mechanism realizes active drive through the common cooperation of link mechanism and slider mechanism, passively adapts to any irregular contour shape of the object, and there are differences between the grasper transmission mechanism and the transmission mechanism of the traditional manipulator. The transmission mechanism dimensional of the grasper has an important influence on its grasping sequence, grasping workspace, and the mapping relationship between the grasper and the contour of the object. Therefore, a kind of simulated “soft” mechanical adaptive grasper with the characteristics of adaptive object shape, uniform grasping force, single-degree-of-freedom drive and full rigid structure will be taken as the research object in this paper, and the dimensional analysis of its transmission mechanism will be carried out. The correlation model between the kinematic parameters and dimensional parameters of the transmission mechanism of each knuckle element under different constraints is established, which provides a certain theoretical basis for the engineering application of the grasper.

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

535

2 Mechanism Model of Grasper In order to achieve adaptive envelope grasping of irregularly shaped objects, a type of new simulated “soft” mechanical adaptive grasper was proposed [16]. No matter how many segments each finger contains, only one degree of freedom is needed to make the transmission mechanism actively drive, the knuckle link passively adapts to the contour shape of the object, envelope and grasp any object with irregular shape, which greatly reduces the control difficulty. We define the simulated “soft” mechanical adaptive grasper as GXU-Grasper in this paper. The fingers of GXU-Grasper are composed of multiple knuckle units in series. Each knuckle unit includes a knuckle link, four-bar mechanism, rocker slider, contact plate and multiple springs. The four-bar mechanism and rocker slider mechanism are the transmission mechanism of knuckle unit. The overall mechanism model of grasper and knuckle unit model are shown in Fig. 1. It is assumed that the grasped object is fixed on the plane. Taking a knuckle unit as an example, the grasping principle of the GXU-Grasper knuckle unit is analyzed, as shown in Fig. 2. First of all, slider S1 is initially confined into the card slot of the contact plate of the object that can slide laterally along knuckle link O1 O2 , and the Fig. 1 GXU-Grasper mechanism model. a Overall structure of the grasper, b knuckle unit

536

H. Huang et al.

O1 C1 S1 forms a fixed triangle structure, the driver drives active rocker link O1 A1 to rotate, and the knuckle unit as a whole begins to rotate around point O1 , as shown in Fig. 2a. Then, with the continuous driving of the driver, the contact plate begins to contact the surface of the object, overcoming the elastic pressure of the spring and sliding along the link O1 O2 laterally, as shown in Fig. 2b. In the end, as the object contact plate slides for a certain distance, the mechanical restriction of the card slot on slider S1 is released, and slider S1 becomes free, as shown in Fig. 2c. At this time, knuckle link O1 O2 stops rotating, and the reversal of link O1 O2 is prevented under the action of the ratchet wheel self-locking mechanism at kinematic pair O1 , as shown in Fig. 3. Such a knuckle unit is a novel design because even if the GXUGrasper finger has three or more knuckle units, it can always be controlled by one driving source, as shown in Fig. 2d. Fig. 2 Knuckle unit grasping principle. a Initial state, b intermediate state, c final state, d motion of the next knuckle unit

Fig. 3 Schematic diagram of self-locking mechanism of the knuckles

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

537

3 Dimensional Analysis of Transmission Mechanism The transmission mechanism dimensional of GXU-Grasper has an important influence on its grasping sequence, grasping workspace, and the mapping relationship between the grasper and the contour of the object. Therefore, it is necessary to carry out a dimensional analysis of the grasper transmission mechanism. Define the dimensional of link Oi Oi+1 as li4 , the dimensional of link Oi Ai as li1 , the dimensional of link Ai Bi as li2 , the dimensional of link Oi+1 Bi as li3 , the dimensional of link Oi Ci as li5 , and the dimensional of link Ci Si as li6 , as shown in Fig. 4. First, according to the final grasping state of the knuckle unit transmission mechanism, as shown in Fig. 4c, the dimensional model of the transmission mechanism is established. Transmission mechanism dimensional parameters li1 , li2 , li3 , li4 and motion parameters θi min , ϕi max are related as follows Fig. 4 Schematic diagram of transmission mechanism dimensional parameters li1 , li2 , li3 , li4 and motion parameters θi min , θi max , ϕi max , ϕi min under different grasping states of the knuckle unit. a Initial state, b intermediate state, c final state

538

H. Huang et al. 2 2 li2 = ri1 + li3 − 2ri1li3 cos(ϕi max 2 )

(1)

ϕi max 2 = ϕi max − ϕi max 1

(2)

where 

ϕi max 1 ri1 =

l 2 sin θi min = arcsin i1 ri1



/ 2 2 li1 + li4 − 2li1li4 cos(θi max )

In Eqs. (1), (2), ϕi max and ϕi min respectively represent the maximum and minimum angle between links Oi Oi+1 and Oi+1 Bi , where ϕi min ≤ ϕi ≤ ϕi max ; θi max and θi min respectively represent the maximum and minimum angle between links Oi Oi+1 and Oi+1 Ai , where θi min ≤ θi ≤ θi max . Then, according to Fig. 3a, the dimensional of the initial grasping state of the transmission mechanism of a single knuckle unit is established. Due to the limitation of the motion of the transmission mechanism, in order to enable the grasper fingers to grasp various irregular objects, the knuckle unit needs to meet 90 < θi max < 180◦ and ϕ1 min < 90◦ in the initial state. Assuming that the rotation range of the knuckle unit is δi , when the first knuckle unit is in the initial grasping state, there are ϕi min = ϕi max − δi

(3)

θi max = θi max 1 + θi max 2

(4)

where ri2 =

/ 2 2 li4 + li3 − 2li4 li3 cos(ϕi min ) 

θi max 1 = arcsin θi max = arccos

2 sin(ϕi min ) li3 ri2



2 2 2 + ri2 − li2 li1 2li1ri2

Since each knuckle unit of GXU-Grasper is installed 20° counterclockwise, for the dimensional of the transmission mechanism of the second and above knuckle units, under the constraint of the dimensional of the transmission mechanism of the previous knuckle unit, the adjacent knuckle unit the linkage constraint of the transmission mechanism between the two should also meet the initial grasping state. θi max = 180◦ − 20◦ − ϕi−1 min ,

i ≥2

(5)

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

ϕi min = ϕi min 1 + ϕi min 2

539

(6)

where ri2 =

/ 2 2 li1 + li4 − 2li1li4 cos(θi max ) 

ϕi min 1

l 2 sin(θi max ) = arcsin i1 ri2

ϕi min 2 = arccos



2 2 2 + ri2 − li2 li3 2li3ri2

It can be seen from the structural design of the second section that in order to make the grasper have a clear grasping sequence of the knuckle unit and a single degree of freedom, the slider mechanism and the contact plate are used to cooperate together to achieve. Due to the limited length of the knuckle link Oi Oi+1 during the movement of the slider, the moving distance si of the slider must move within the angular range of the rotation of the transmission mechanism, that is si ≤ li4 . According to the geometric relationship of the slider mechanism, the dimensional parameters li4 , li5 , li6 , ei and the motion parameters θi , si of the link are established θi = arcsin

2 2 ei2 + li5 + s 2 − li6 si / i − arctan ei 2li5 ei2 + si2

(7)

Among them, ei is the offset distance from the slider Si to Oi Oi+1 . In Eq. (7), the numerical solutions of link dimensional parameters li5 , li6 and ei can be obtained when any three sets of motion parameters θi and si are given.

4 Example Analysis In order to enable the grasper to successfully adaptive envelope grasp objects of any shape and size, the rotation range of the transmission mechanism and the size of each link should not be too large or too small. Assuming that the value range of link Oi Ai is 20 mm ≤ li1 ≤ 40 mm, and the value range of link Oi+1 Bi is 10 mm ≤ li3 ≤ 40 mm, the dimensional analysis is carried out on the transmission mechanism of each knuckle unit. Firstly, the dimensional analysis of the first knuckle unit transmission mechanism is carried out, taking θ1 min = 50◦ and ϕ1 max = 180◦ , and then under the constraint of the final grasping state, according to Eq. (1) to find all solutions that meet the first knuckle unit transmission dimensional l12 , and the dimensional relationship of the first knuckle unit transmission mechanism l11 ,

540

H. Huang et al.

l13 and l12 is given, as shown in Fig. 5a. At this time, the transmission mechanism dimensional of the first knuckle unit calculated by Eq. (1) do not all meet the finger motion requirements. Therefore, it is also necessary to find out the transmission mechanism dimensional of the first knuckle unit in both the final grasping state and the initial grasping state in all the solutions calculated by Eq. (1). Therefore, when the first knuckle unit returns to the initial grasping state, taking ϕ1 min = 50◦ , and calculate the θ1 max of the first knuckle unit transmission mechanism in the initial grasping state according to Eq. (4), as shown in Fig. 5b, and the judgment is given as following: In all the solutions obtained by Eq. (4), if θ1 max ≤ 160◦ , then the value obtained according to Eq. (4) is the dimensional that conforms to the motion of the first knuckle unit transmission mechanism, and the area covered by the red dot is the feasible solution dimensional conforming to l11 , l13 , l12 and θ1 max . According to the dimensional constraint of the transmission mechanism of the first knuckle unit, it can be known that ϕ1 min = 50◦ . Substitute it into Eq. (5) to obtain θ2 max = 110◦ . Taking θ2 min = 40◦ and ϕ2 max = 180◦ . According to Eqs. (1), (2), and (5), (6), all solutions and feasible solutions for the dimensional of the transmission mechanism of the second knuckle unit are obtained, as shown in Fig. 5c, d. Similarly, according to the dimensional calculation method of the transmission mechanism of the second knuckle unit, take ϕ2 min = 50◦ , where ϕ2 min and ϕ3 min are one of the feasible solutions in Figs. 5d and 6b, then obtain θ3 max = 109.21◦ and θ4 max = 80.50◦ according to Eqs. (5) and (6). Taking θ3 min = 50◦ , ϕ3 max = 180◦ and θ4 min = 50◦ , ϕ3 max = 167◦ , and calculate all solutions and feasible solutions of the transmission mechanism dimensional of the third and fourth knuckle units according to Eqs. (1)–(6), as shown in Fig. 6. According to the rotation range of the above link mechanism, take the second knuckle unit as an example, θ2 max = 110◦ , θ2 min = 40◦ to illustrate the influence of the slider moving distance si (si min ≤ si ≤ si max ≤ li4 ) on the change of the link Oi Ci and the link Ci Si , the relationship between the slider moving distance si and the link Oi Ci and the link Ci Si is calculated according to Eq. (7), as shown in Fig. 7. In Figs. 5 and 6, the area covered by the red dot is all the dimensional of the knuckle unit transmission mechanism under the constraints of the initial state and the final state of grasping. By comparing the eight diagrams in Figs. 5 and 6 can be found that the area covered by the red dot (feasible solution dimensional) calculated by the transmission mechanism dimensional of each knuckle unit shows a decreasing trend. This is because the dimensional of the transmission mechanism of the first knuckle unit only needs to meet the constraints in the final grasping state and the initial grasping state, so the area covered by the red dot (feasible solution dimensional) is the largest. While the dimensional of the second knuckle unit transmission mechanism needs to meet the constraints of the final grasping state and the

Dimensional Analysis of Transmission Mechanism of Novel Simulated … Fig. 5 Dimensional analysis of transmission mechanism of the first knuckle unit and the second knuckle unit. a Dimensional of links li1 , li3 and li2 , b dimensional of links li1 , li3 and θ1 max , c dimensional of links l21 , l23 and l22 , d dimensional of links l21 , l23 and ϕ2 min . The area covered by the red dot is the feasible solution dimensional in accordance with li1 , li3 , li2 and θ1 max , ϕi min

541

542 Fig. 6 Dimensional analysis of transmission mechanism of the third knuckle unit and the fourth knuckle unit. a Dimensional of links l31 , l33 and l32 , b dimensional of links l31 , l33 and ϕ3 min , c dimensional of links l41 , l43 and l42 , d dimensional of links l41 , l43 and ϕ4 min

H. Huang et al.

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

543

Fig. 7 Dimensional analysis of slider mechanism. a Dimensional parameters li5 is affected by the moving distance si of the slider and the offset distance ei of the slider, b dimensional parameters li6 is affected by the moving distance si of the slider and the offset distance ei of the slider

initial grasping state, it also needs to meet the constraints of the first knuckle unit transmission mechanism. Therefore, the area covered by the dimensional red dot of the second knuckle unit transmission mechanism (feasible solution dimensional) is smaller than that covered by the dimensional red dot of the first knuckle unit transmission mechanism (feasible solution dimensional). Similarly, the area covered by the transmission mechanism dimensional red dot of the third and fourth knuckle units (feasible solution dimensional) is smaller than that covered by the transmission mechanism dimensional red dot of the previous knuckle unit (feasible solution dimensional). Analysis of Fig. 7 shows that the length li5 of the link Oi Ci increases with the increase of the slider moving distance si and the offset distance ei , while the length li6 of the Ci Si decreases with the increase of the offset distance.

5 Experiment According to the dimensional analysis of the transmission mechanism in Sect. 3, a GXU-Grasper physical prototype was made by selecting a value in the area covered by the red dot (feasible solution dimensional) in Figs. 5 and 6 and the appropriate parameters in Fig. 7, as shown in Table 1. The relevant experimental platforms were

544

H. Huang et al.

Table 1 Dimensional of the transmission mechanism of the knuckle unit (mm) i

li1

li2

li3

li4

li5

li6

1

33.10

32.68

11.90

30

16.67

10.61

2

32.10

32.25

11.50

30

16.67

10.61

3

30.83

28.72

11.16

30

16.67

10.61

4

23.01

23.35

11.39

30

16.67

10.61

Fig. 8 GXU-Grasper prototype

Finger1 Finger2

Motor1 Motor2 Motor4

Finger3

Motor3 Motor driver

Dc Power supply(12V)

established, as shown in Fig. 8. The experimental platform consists of one GXUGrasper, four motors, one driving source and driver. Motor 1, motor 2 and motor 3 drive finger 1, finger 2 and finger 3 respectively, and motor 4 adjusts the azimuth angle of finger 1 and finger 2. In order to verify the dimensional analysis of the GXU-Grasper transmission mechanism, grasping experiments were carried out on balloons, sponges, light bulbs, pears, tomatoes and various irregularly shaped objects, as shown in Fig. 9. The experimental results show that GXU-Grasper can grasp objects that are volatile, rotten, brittle and uneven in shape in a smooth and gentle way.

6 Conclusion According to the proposed GXU-Grasper mechanism model, the transmission mechanism is modeled and analyzed. The constraint relationship between the transmission mechanism of each knuckle element in the final grasping state and the initial grasping

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

545

Fig. 9 GXU-Grasper envelope grasping experiment. a Grasp balloon, b grasp sponge, c grasp the bulb, d grasp objects with uneven shape, e grasp pear, f grasp tomatoes

state was analyzed, and the correlation model between the kinematic parameters and dimensional parameters of the transmission mechanism of each knuckle element under different constraints was established. The linkage association model of the transmission mechanism between adjacent phalangeal units is established. Through the calculation example, the dimensional parameters li1 , li2 , li3 and motion parameters θi min , θi max , ϕi max , ϕi min of the transmission mechanism of each knuckle element of the grasper under the constraint conditions and all solutions and feasible solutions are given, as well as the variation relationship between the link dimensional parameters li4 , li5 , li6 , ei and the motion parameters θi , si . The results of each knuckle unit transmission mechanism are different, and the dimensional range of the feasible

546

H. Huang et al.

solution shows a decreasing trend. The GXU-Grasper prototype was made and the corresponding experimental platform was built to carry out grasping experiments on balloons, sponges, light bulbs, pears, tomatoes and various irregular-shaped objects. The experiment showed that the GXU-Grasper can relatively smoothly and softly grasp the objects that are variable, fragile, brittle and uneven in shape, which verifies the effectiveness of the dimensional analysis of the grasper transmission mechanism. Acknowledgements Supported by National Natural Science Foundation of China (Grant No. 51865001).

References 1. Notice of the 15th Department on printing and distributing the “14th five-year plan for the development of robot industry”, no 206. Ministry of Industry and Information Technology (2021). http://www.gov.cn/zhengce/zhengceku/2021-12/28/content_5664988.htm,2021-12-21 2. Kashef SR, Amini S, Akbarzadeh A (2020) Robotic hand: a review on linkage-driven finger mechanisms of prosthetic hands and evaluation of the performance criteria. Mech Mach Theory 145:103677 3. Tan S, Zhang W, Chen Q, Du D (2009) Design and analysis of underactuated humanoid robotic hand based on slip block-cam mechanism. In: IEEE international conference on robotics and biomimetics, Guilin, China, Dec 19–23, 2009, pp 2356–2361 4. Jin J, Zhang W, Sun Z, Chen Q (2012) LISA hand: indirect self-adaptive robotic hand for robust grasping and simplicity. In: IEEE international conference on robotics and biomimetics, Guangzhou, China, pp 2393–2398 5. Liu S, Zhang W, Sun J (2019) A coupled and indirectly self-adaptive under-actuated hand with double-linkage-slider mechanism. Ind Robot Int J Robot Res Appl 46(5):660–671 6. Jang G, Lee C, Lee H, Choi Y (2013) Robotic index finger prosthesis using stackable double 4-BAR mechanisms. Mechatronics 23(3):318–325 7. Yoon D, Choi Y (2017) Underactuated finger mechanism using contractible slider-cranks and stackable four-bar linkages. IEEE/ASME Trans Mechatron 22(5):2046–2057 8. Stavenuiter RAJ, Birglen L, Herder JL (2017) A planar underactuated grasper with adjustable compliance. Mech Mach Theory 112:295–306 9. Hua H, Liao Z, Chen YJ (2020) A 1-Dof bidirectional graspable finger mechanism for robotic gripper. J Mech Sci Technol 34(11):4735–4741 10. Wang D, Xiong Y, Zi B, Qian S, Wang Z, Zhu W (2021) Design, analysis and experiment of a passively adaptive underactuated robotic hand with linkage-slider and rack-pinion mechanisms. Mech Mach Theory 155:104092 11. Zhang Z, Zhang Y, Ning M, Zhou Z, Wu Z (2022) One-DOF six-bar space gripper with multiple operation modes and force adaptability. Aerosp Sci Technol 123:107485 12. Difonzo E, Zappatore G, Mantriota G, Reina G (2020) Advances in finger and partial hand prosthetic mechanisms. Robotics 9(4):80 13. Birglen L, Gosselin CM (2004) Kinetostatic analysis of underactuated fingers. IEEE Trans Robot Autom 20(2):211–221

Dimensional Analysis of Transmission Mechanism of Novel Simulated …

547

14. Birglen L, Gosselin CM (2006) Geometric design of three-phalanx underactuated fingers. ASME J Mech Des 128(2):356–364 15. Wu L, Kong Y, Li X (2016) Fully rotational joint underactuated finger mechanism and its kinematics analysis. J Mech Eng 5(13):47–54 16. Wang R, Huang H, Xu R, Li K, Dai J (2021) Design of a novel simulated “soft” mechanical grasper. Mech Mach Theory 158:104240

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces Zezheng Qi, Shibing Hao, Qianqian Zhang, Ran Shi, and Yunjiang Lou

Abstract Compared to other types of robots, snake robots have many advantages and are able to work in special environments such as narrow spaces. In this paper, we design a remote-controlled snake robot with orthogonal connections by investigating the structural design and motion planning algorithms according to performance requirements. Experiments have shown that the snake robot designed in this paper can move quickly on flat surfaces and can also enter narrow environments for detection. In post-disaster rescue, the snake robot can be used to enter through narrow passages, detect trapped people and improve rescue efficiency. Keywords Snake robot · System design · Control algorithm

1 Introduction Existing conventional robots face various constraints in terms of size and structure in the complex environment of a disaster scene, and can only move on the surface of debris, making it difficult to pass through narrow spaces in order to search for and detect casualties. To fulfil the task of detecting casualties in such situations, this paper investigates a snake robot with a modular structure and a slender shape that can locate and detect casualties through narrow spaces, which is important for improving the rescue efficiency and survival rate of casualties after disasters such as earthquakes and mudslides. The world’s first snake robot, developed by Professor Hirose Shigeo’s team in Japan, was named ACM-III [1]. Subsequently, they developed the ACM-R2 and ACM-R3 terrestrial snake robots [2, 3]. ACM-R3 uses orthogonal connections and Z. Qi · S. Hao · Q. Zhang · Y. Lou (B) School of Mechanical Engineering and Automation, Harbin Institute of Technology Shenzhen, Shenzhen 518000, Guangdong, China e-mail: [email protected] R. Shi School of Mechanical and Electrical Engineering, Shenzhen Polytechnic, Shenzhen 518000, Guangdong, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_32

549

550

Z. Qi et al.

has the ability to move in three dimensions [2]. Compared to the ACM-R3, the ACM-R4 has a better seal performance and is more reliable. The ACM-R4.1 has not changed significantly from the ACM-R4 in terms of shape and has an active wheel structure [3]. The ACM-R5 [4] is an amphibious robot. The ACM-R7 differs from previous generations of robots in the way it moves and how it is controlled. This form of its motion, known as loop gait [5], enhances obstacle-crossing capabilities. The ACM-R8 is the latest generation of snake robots developed by Professor Hirose Shigeo’s research team, using a unidirectional rotating active wheel structure and consisting of four jointed modules [6]. The snake robots GMD-Snake [7] and GMD-Snake2 [8], developed by the German National Technology Centre. The GMD-Snake head is fitted with four photoresistors and a haptic sensor to detect changes in the brightness of the environment and whether the head is touching an obstacle, respectively [9]. The GMDSnake2 is based on the GMD-Snake and has some autonomous motion capabilities [10]. The snake robot OmniTread [11], developed by the University of Michigan, lack the ability to move in a three-dimensional environment. Besides, Carnegie Mellon University focuses on the climbing motion of snake robots and their Unified Snake Robot is extremely capable of climbing [12]. The Unified Snake Robot can also perform side-to-side motions, tumbling motions on flat ground and has the ability to roll over obstacles [13]. In addition, the Japanese company NEC has developed Orochi, a snake robot with a gimbaled joint design that allows for more flexible joint motions and can be used for detection and rescue work in hazardous environments [14]. Besides, Anna Konda [15], a snake robot that can be used in fire rescue, was developed at the Norwegian University of Science and Technology (NTNU). Furthermore, Aiko [16] and Kullo [17], snake robots, were developed by NTNU to study obstacle-aided locomotion in unstructured environments. In summary, although research on snake robot has been relatively fruitful, most snake robot are still in the laboratory research stage. There are no snake robots specifically designed to enter narrow spaces to detect casualties and improve rescue efficiency after a disaster such as an earthquake. In this case, snake robots have unique advantages that necessitate our research. In this paper, we first complete the structural design. Next, the motion planning algorithm and simulation are completed. Then, the control system is designed. Finally, experiments are conducted to test the performance of the snake robot.

2 Snake Robot Structure Design The snake robot designed in this paper is primarily used for detection through narrow spaces in complex disaster scenarios. In this application scenario, we propose the following functional requirements and performance indicators. The snake robot needs to have basic motion capabilities, image acquisition and wireless transmission capabilities as well as remote control capabilities. The performance indicators

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

551

for snake robot are as follows: maximum motion speed of 0.3 m/s on a flat surface and the ability to pass through narrow passages of 0.15 × 0.15 m2 . The spinal structure of the snake is very special, consisting of many similar joints connected in sequence. Each joint has a limited angle of rotation, but the cumulative effect of these small rotations eventually takes the form of the flexible motion of the biological snake. This inspired us to come up with a modular design solution. The snake robot consists of several identical or similar joints connected in series, and by controlling the rotation of each joint, it can mimic the motion of a biological snake. The snake robot is a multi-degree-of-freedom chain robot, consisting of several similar joint modules connected in sequence. In order to achieve a three-dimensional motion capability, it is necessary to select the connection method in such a way that each joint has a complete three-dimensional working space. The snake robot designed in this paper uses an orthogonal connection as the joint connection method. The joints of the snake robot are connected by rotating pairs whose axes are orthogonal to each other and perpendicular to the longitudinal axis of the robot, giving the robot the ability to move in three dimensions. The robot is made up of joint modules with two orthogonal degrees of freedom of rotation connected in sequence. The joint module consists mainly of the servo, the snake body, the servo connector and the passive wheel, as shown in Fig. 1. We mount the two servos to the servo connectors in an orthogonal posture of the rotor shafts, and use the metal servo mounts to fix each rotation axis to the snake body. This installation allows each module to have two mutually perpendicular degrees of rotational freedom for pitch and yaw rotation. The robot designed in this paper is fitted with passive wheels which are installed all around the robot to improve robustness. The passive wheels act similarly to the scales of a biological snake, providing different ground friction characteristics, which is necessary for the robot to be able to move smoothly. In this case, the normal friction coefficient perpendicular to the robot’s forward direction is greater than the tangential friction coefficient along the robot’s forward direction, preventing the robot from slipping sideways during motion. Besides, the head and tail of robot are designed, as shown in Fig. 1. The robot should be able to acquire images in real time, so the head structure is designed for

Fig. 1 The snake robot structure

552

Z. Qi et al.

camera mounting. The tail of the robot carries mainly the controller, communication module and battery. Because they are relatively long and heavy, two sets of passive wheels are used to improve reliability.

3 Motion Planning Algorithm and Simulation For the motion control problem of the snake robot, we use a method based on motion control functions to plan the pitch and yaw angles of the snake robot respectively to achieve different motions.

3.1 Morphological Models of Biological Snakes Hirose Shigeo, one of the first to investigate snake robots, developed the serpenoid curve, after years of recording and studying the characteristics of biological snakes. The shape of the curve mimics the motion of a biological snake very well. The definition of the serpenoid curve is given by describing the curvature over a period [1], as shown in Eq. (1). ρ(s) = −

  2K n π α0 2K n π sin s L L

(1)

where ρ represents the serpenoid curve curvature, L is the whole length of snake body, K n is the number of the wave shapes, α0 is the initial winding angle of the curve and s is the length between a point and the starting point along the curve. In the Cartesian coordinate system, when the curve passes through the origin of the coordinates, Eq. (1) can be expressed as follows: ⎧ s ⎪ ⎪ ⎨ x(s) = cos(ζσ )dσ 0

s ⎪ ⎪ ⎩ y(s) = sin(ζσ )dσ

(2)

0

where ζσ = a cos(bσ )+cσ , a, b and c are the three parameters that control the shape of the serpenoid curve. By taking different values for a, b and c respectively, the shapes of the serpenoid curve, as shown in Fig. 2. As can be seen from Fig. 2, the smaller the value of a, the smaller the initial phase and amplitude of the curve. The larger the value of b, the smaller the period and amplitude of the curve, and the more waveforms per unit length. The parameter c affects the direction of curve deflection. When c = 0, the direction is constant, when c > 0, the direction is left, when c < 0, the direction is right.

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces Fig. 2 The serpentine curve shape

553

554

Z. Qi et al.

3.2 Two-Dimensional Motion Control Functions To facilitate obtaining the motion control functions for the snake robot, we change Eqs. (1) to (4) based on Eq. (3). /  ρ(s) =

d2x ds 2

2

 +

d2 y ds 2

2 = | − ab sin(bs) + c|

ρ(s) = −ab sin(bs) + c

(3) (4)

where s is the arc length between a point on the snake curve and the start point, a affects the amplitude of the curve and the initial bend angle, b affects the number of waveforms contained in the snake curve as well as the amplitude of the curve, c affects the direction of curve deflection. Integrating the curvature equation shown in Eq. (4) over the arc length s gives the absolute angle θ (s) at arc length s on the curve, as shown in Eq. (5). θ (s) = a cos(bs) + cs

(5)

A sketch of the serpentine curve is shown in Fig. 3. Because each joint module of the snake robot can be considered as a rod of length 2l, the curve can be discretized as a collection of n rods of length 2l connected end to end, as shown in Fig. 4. We assume that each rod is tangent to the serpenoid curve and the tangent point is at the centre of the rod. Then from Eq. (5), the relative rotation angle of the two adjacent rods can be calculated as shown in Eq. (6). ϕ(s) = θ (s − l) − θ (s + l) = 2a sin(bl) sin(bs) − 2cl

(6)

Since a change in the relative angle of rotation ϕ(s) of two adjacent rods can cause a change in the shape of the snake, the locomotion of the snake can be controlled by controlling the relative angle of rotation of each joint of the snake. Fig. 3 Serpentine curve sketch

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

555

Fig. 4 Serpentine curve fitting diagram

The length of each joint module of the snake robot is 2l. According to Eq. (6), the relative rotation angles ϕ(s) at the i-th joint when s = 0, 2l, 4l, 6l . . . can be written in the following form. ϕi = α sin((i − 1)β) + γ

(7)

where α = 2a sin(bl), β = 2bl, γ = −2cl. According to Eq. (7), the snake robot waveform is stationary and cannot be transmitted with time. Therefore, in order to make the waveform be transmitted dynamically in the snake robot, a time variable t should be introduced into the value of the arc length s, i.e. s = ht, ht + 2bl, ht + 4bl, ht + 6bl . . ., where h is a constant. Then the relative rotation angle function of the snake robot at the i-th joint over time is ϕi (t) = α sin(ωt + (i − 1)β) + γ

(8)

where ω represents joint angular frequency.

3.3 Meandering Motion Control Functions The two-dimensional motion control function for the snake robot is shown in Eq. (8). where γ = −2cl and l is half of the length of each module of the snake robot. When c = 0 i.e. γ = 0 the meandering motion proceeds along a straight line. In this case, the corresponding control function is shown in Eq. (9).

556

Z. Qi et al.

Fig. 5 Yaw joint angle



θi (t) = α sin(ωt + (i − 1)β) ψi (t) = 0

(9)

where θi (t) represents the angle of the i-th yaw joint at time t, ψi (t) represents the angle of the i-th pitch joint at time t. We let α = 0.5 rad, ω = 2 rad/s and β = 1 rad and use MATLAB to analyse Eq. (9), the angular change of each joint is shown in Fig. 5. It can be seen from the figure that there is an abrupt angle change at t = 0. This abrupt change can cause the snake robot to jitter, which can have a detrimental effect on our control and the stability of the snake robot’s operation. As shown in Eq. (10), an optimization function f(t) is added by us to Eq. (9) to suppress sudden changes in joint angles and thus improve the reliability of the snake robot. The optimization function curve is shown in Fig. 6. The optimization function f(t) is non-negative, with an initial value of 0, increasing monotonically and a limit value of 1. The rate of convergence of f(t), accelerates as η increases. Using this function, the meandering motion control function is optimised as shown in Eq. (11). f (t) =

1 − e−ηt 1 + e−ηt

(10)

where η represents optimization parameter. 

θi (t) =



1−e−ηt 1+e−ηt

ψi (t) = 0

α sin(ωt + (i − 1)β)

(11)

The changes in joint angle after optimization of the yaw joint for the optimization parameters η = 2 and η = 5 are shown in Fig. 7a, b. From the graphs, it can be seen that the angle increases gradually from 0 to the desired angle, avoiding the sudden

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

557

Fig. 6 Optimization function f(t)

change that occurred before the optimization, eliminating the jitter at start-up of the snake robot and making its motion smoother. We perform kinematic simulation in Webots using the optimised meandering motion control function. The control parameters were selected as α = 0.5 rad, ω = 2 rad/s, β = 1 rad, η = 2 and the control function is shown in Eq. (12). 

−2t θi (t) = 0.5 1−e sin(2t + (i − 1)) 1+e−2t ψi (t) = 0

(12)

The simulation results for a motion period T = π s are shown in Fig. 8. From the simulation results, it can be seen that the optimised meandering motion control functions are effective for the snake robot.

Fig. 7 Yaw joint angle after optimization

558

Z. Qi et al.

Fig. 8 Simulation of meandering motion

3.4 Turning Motion Control Functions During the motion of the snake robot, unpredictable influences may cause it to deviate from its intended route. The snake robot requires a turning motion to adjust its direction of motion. Therefore, it is necessary to investigate turning motion. After the previous analysis, the snake robot turns when γ /= 0 in Eq. (8). The turning motion control function is shown in Eq. (13). 

θi (t) =



1−e−ηt 1+e−ηt

(α sin(ωt + (i − 1)β) + γ )

ψi (t) = 0

(13)

The turning radius of a snake robot in a turning motion is shown in Eq. (14), the larger the absolute value of γ, the smaller the turning radius. Besides, the positive or negative γ affects the direction of the turn. r=

2l |γ |

(14)

We simulate the turning motion of the snake robot in Webots by setting α = 0.5 rad, ω = 2 rad/s, β = 1 rad and η = 2. Then, we let γ = ±0.5 rad. The turning motion control function is shown in Eq. (15). The angle of each yaw joint is shown in Fig. 9. The simulation results are shown in Fig. 10. 

θi (t) =



1−e−2t 1+e−2t

ψi (t) = 0

(0.5 sin(2t + (i − 1)) ± 0.5)

(15)

4 Control System Design Servos are a class of position servo motors with the advantages of compactness, ease of control and high accuracy. They are commonly used in scenarios where angle

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

559

Fig. 9 Yaw joint angle curve for turning motion

Fig. 10 Simulation of turning motion

changes are controlled in real time and need to be maintained. The snake robot designed in this paper simulates biological snake motion by controlling the rotation and speed of each joint, in line with the function of the servo. Therefore, we chose digital servos as the actuator. The snake robot controller is the Arduino Nano3.0, which is very small and has a main chip, the ATmega328. It has 14 digital I/O ports, six of which can be used as PWM outputs, just as required for the control of the snake robot’s six servos. The image acquisition and transmission system consists of a CCD high-definition camera, a wireless module and a display. The CCD HD camera has a focal length of 2.1 mm and a 1/3,, SONY SUPER HAD II light sensor. The wireless module has

560

Z. Qi et al.

a power of 200 mW and a reference transmission range of 100–500 m in an ideal environment with no interference and no obstruction, which meets the functional requirements of the snake robot. The entire control process is based on the operator issuing a command, which is received by a signal receiver mounted on the snake robot and transmitted to the controller. The controller makes decision based on the type of command and executes the corresponding motion control function, completing the calculation of the servo angle. Once the calculation is complete, the PWM signal corresponding to that rotation angle is obtained. The controller communicates with the servos and sends PWM signal to change the rotation angle of each servo in real time to achieve the motion corresponding to the command. In order to provide the snake robot with the ability to detect its surroundings, an all-in-one graphics camera is installed in its head with image acquisition and transmission capabilities. The operator can see the real time picture on the accompanying display and control the snake robot to move straight or turn to reach the casualty through narrow passages to complete the detection. The snake robot system structure is shown in Fig. 11. The electronic components installed on the snake robot all require a power supply and a good power supply system is extremely important for the stable operation of the whole system. We chose the 2S lithium battery as the power source for the snake robot, as shown in Fig. 12.

Fig. 11 System structure

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

561

Fig. 12 Power supply system

5 Experiment The assembled snake robot, shown in Fig. 13, consists of three joint modules, approximately 75 cm long, 10 cm wide and 13 cm high.

Fig. 13 Snake robot experimental platform

562

Z. Qi et al.

The experimental platform mainly consists of a snake robot and a remote control. The operator issues commands to the snake robot via the remote control and the snake robot performs the corresponding motions.

5.1 Meandering Motion Experiment The control function that moves the snake robot forward is shown in Eq. (9), where η is the optimization parameter that does not change the motion waveform. According to Eq. (8), we analytically obtain that ω does not affect the shape of the meandering motion waveform. The effect of the control parameters α and β on the waveform will be investigated experimentally in the following. Keeping ω = 1.5 rad/s, β = 1.2 rad, η = 2, the parameters α is taken as 0.3 rad and 1 rad respectively. The motion waveforms when the control parameter α at different values are obtained, as shown in Fig. 14. Keeping α = 0.3 rad, ω = 1.5 rad/s, η = 2, the parameters β is taken to be 0.5 rad and 2 rad respectively. The motion waveforms when the control parameter β at different values are obtained, as shown in Fig. 15. The graphs show that adjusting parameter α changes the amplitude of swing during the meandering motion of the snake robot. While setting the other parameters to fixed values, the higher the value of α, the greater the amplitude of swing. The parameter β affects the number of waveforms contained in the meandering motion of the snake robot, and the larger β, the more waveforms the robot contains per unit length. The parameters α influences the amplitude of swing so that affects the snake robot’s ability to pass through narrow environments. When α = 0.15 rad, ω = 1.5 rad/s, β = 1.2 rad and η = 2, the snake robot is able to pass through a narrow passage with a width of 0.15 m at a speed of 0.05 m/s, as shown in Fig. 16, satisfying performance indicator of passing capability. Next, the effect of the control parameters on the speed of motion is investigated. Fig. 14 Motion waveforms with different α

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

563

Fig. 15 Motion waveforms with different β

Fig. 16 Snake robot through narrow passages

Keeping parameters ω = 1.5 rad/s, β = 1.5 rad, η = 2, we adjust the parameter α and record the speed of the snake robot. The relationship between α and velocity is shown in Fig. 17. We obtain that when the other variables are fixed and the value of α is increased, the meander speed increases and then decreases. The maximum speed, when α is taken between 1 rad and 1.3 rad, is approximately 0.6 m/s which meets the performance indicator of speed. Keeping α = 0.3 rad, β = 1.2 rad and η = 2, we change the value of ω. The relationship between ω and velocity is shown in Fig. 18. The analysis shows that as ω increases, the speed of the snake robot’s meandering motion also increases. Keeping α = 0.3 rad, ω = 1.5 rad/s and η = 2, we change the value of β. The relationship between β and velocity is shown in Fig. 19. The analysis shows that as β increases, the speed of the snake robot’s meandering motion increases and then decreases.

564

Z. Qi et al.

Fig. 17 The relationship between α and velocity

Fig. 18 The relationship between ω and velocity

5.2 Turning Motion Experiment According to the turning motion control function proposed in Eq. (13), we let α = 0.3 rad, ω = 1.2 rad/s, β = 1.2 rad, η = 2 and γ = ± 0.5 rad. The experimental results are shown in Fig. 20. The positive or negative of γ affects the direction of the turn. When γ > 0, the snake robot turns to the right, when γ < 0, it turns to the left.

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

Fig. 19 The relationship between β and velocity

Fig. 20 Turning motion experiment

565

566

Z. Qi et al.

Fig. 21 Image capture

5.3 Image Acquisition and Transmission Experiments With a camera and image wireless transmission module mounted on the head of the snake robot, the operator can use a portable display within a certain distance to view the captured images and obtain information about the snake robot’s surroundings. The operator can use this image to control the snake robot to move to the appropriate position and detect trapped people. The results of the experiment are shown in Fig. 21.

6 Conclusion In this paper, we focus on the practical application of snake robots for post-disaster search and rescue, and design a snake robot that can be operated by remote control to detect through narrow spaces. Starting from functional requirements and performance indicators, we design the snake robot structure. Then, the motion planning algorithm of the snake robot is investigated. We use Webots to perform kinematic simulations of meandering and turning motion, verifying that the algorithms are effective. Finally, experiments are carried out to verify the reliability of the snake robot which has the ability to pass through narrow spaces and can move quickly on flat surfaces. Besides, the image acquisition and transmission system is tested and shows clear images. The snake robot designed in this paper can meet the functional requirements and performance indicators, and can operate stably. It can be used in post-disaster rescue to detect casualties through narrow spaces and improve rescue efficiency.

Design and Control of a Detecting Snake Robot by Passing Narrow Spaces

567

Acknowledgements Supported by the National Key R&D Program of China (Grant No. 2020YFB1313900).

References 1. Hirose S (1993) Biologically inspired robots: snake-like locomotors and manipulators. Oxford University Press, Oxford 2. Mori M, Hirose S (2002) Three-dimensional serpentine motion and lateral rolling by active cord mechanism ACM-R3. In: IEEE/RSJ international conference on intelligent robots and systems, vol 1, pp 829–834 3. Takaoka S, Yamada H, Hirose S (2011) Snake-like active wheel robot ACM-R4.1 with joint torque sensor and limiter. In: 2011 IEEE/RSJ international conference on intelligent robots and systems, pp 1081–1086 4. Yamada H, Chigisaki S, Mori M, Takita K, Ogami K, Hirose S (2005) Development of amphibious snake-like robot ACM-R5. In: The 36th international symposium on robotics (ISR 2005), Tokyo 5. Ohashi T, Yamada H, Hirose S (2010) Loop forming snake-like robot ACM-R7 and its serpenoid oval control. In: 2010 IEEE/RSJ international conference on intelligent robots and systems, pp 413–418 6. Komura H, Yamada H, Hirose S (2015) Development of snake-like robot ACM-R8 with large and mono-tread wheel. Adv Robot 29(17):1081–1094 7. Paap KL, Dehlwisch M, Klaassen B (1996) GMD-snake: a semi-autonomous snake-like robot. In: Distributed autonomous robotic systems, vol 2. Springer, Tokyo, pp 71–77 8. Klaassen B, Paap KL (1999) GMD-SNAKE2: a snake-like robot driven by wheels and a method for motion control. In: Proceedings 1999 IEEE international conference on robotics and automation (Cat. No. 99CH36288C), vol 4, pp 3014–3019 9. Worst R, Linnemann R (1996) Construction and operation of a snake-like robot. In: Proceedings IEEE international joint symposia on intelligence and systems, pp 164–169 10. Linnemann R, Paap KL, Klaassen B, Vollmer J (1999) Motion control of a snakelike robot. In: 1999 third European workshop on advanced mobile robots (Eurobot’99). Proceedings (Cat. No. 99EX355), pp 1–8 11. Granosik G, Hansen MG, Borenstein J (2005) The OmniTread serpentine robot for industrial inspection and surveillance. Ind Robot Int J 32(2):139–148 12. Wright C, Buchan A, Brown B, Geist J, Schwerin M, Rollinson D, Tesch M, Choset H (2012) Design and architecture of the unified modular snake robot. In: 2012 IEEE international conference on robotics and automation, pp 4347–4354 13. Tesch M, Schneider J, Choset H (2011) Using response surfaces and expected improvement to optimize snake robot gait parameters. In: 2011 IEEE/RSJ international conference on intelligent robots and systems, pp 1069–1074 14. Takanashi N (1996) A gait control for the hyper-redundant robot O-RO-CHI. In: Proceedings of 8th JSME annual conference on robotics and mechatronics, pp 78–80 15. Liljeback P, Stavdahl O, Beitnes A (2006) SnakeFighter-development of a water hydraulic fire fighting snake robot. In: 2006 9th international conference on control, automation, robotics and vision, pp 1–6 16. Transeth AA, Liljeback P, Pettersen KY (2007) Snake robot obstacle aided locomotion: an experimental validation of a non-smooth modeling approach. In: 2007 IEEE/RSJ international conference on intelligent robots and systems, pp 2582–2589 17. Liljebäck P, Pettersen KY, Stavdahl Ø (2010) A snake robot with a contact force measurement system for obstacle-aided locomotion. In: 2010 IEEE international conference on robotics and automation, pp 683–690

Design of a 3DOF XYZ Precision Positioning Platform Using Novel Z-Shaped Flexure Hinges Lejin Wan, Jiarong Long, Juncang Zhang, and Jinqiang Gan

Abstract This paper presents a design of a 3DOF XYZ precision positioning platform using novel Z-shaped flexure hinges. In the platform, bridge-type mechanism and leverage mechanism are adopted to amplify its output displacement. Besides, a new type of Z-shaped flexure hinge is proposed to achieve the amplification of the displacement in Z-axis. The bi-directional motion in the X-axis of the platform is realized using the differential moving principle. Kinematics and statics analysis of the proposed platform are carried out by pseudo-rigid-body-modeling method and matrix-based compliance modeling method, respectively. In addition, finite element method is used to verify the validity of theoretical analysis. The simulation results show that the maximum displacement of the platform in the XYZ-axis are ± 535.25 μm, 439.98 μm and 1578.7 μm, respectively. Keywords Precision positioning platform · Z-Shaped flexure hinges · Compliant mechanisms

1 Introduction With the development of science and technology, the performance of traditional rigid mechanism can no longer meet the requirements of modern precision machinery due to their disadvantages such as clearance, friction and wear [1]. The compliant mechanism with the advantages of no clearance, no friction and high precision has gradually L. Wan · J. Long · J. Zhang · J. Gan (B) School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China e-mail: [email protected] L. Wan e-mail: [email protected] J. Long e-mail: [email protected] J. Zhang e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Liu (ed.), Advances in Mechanism, Machine Science and Engineering in China, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9398-5_33

569

570

L. Wan et al.

become a new development direction of modern mechanism and mechanical equipment [2]. Due to the excellent characteristics of compliant mechanism, the precision positioning platform based on compliant mechanisms has the advantages of high precision, good stability and high response speed [3]. In recent years, precision positioning platforms based on compliant mechanism have been widely used in many micromanipulation tasks, such as atomic force microscopes, bio-cell manipulation, high-resolution manufacturing and assembly [4–8]. In order to achieve better performance of the platform, different types of actuators are applied to precision positioning platforms, such as electromagnetic actuator (EMA), electrostatic actuator (ESA), electrothermal actuator (ETA) and piezoelectric actuator (PEA) [9]. Among them, the PEA is widely used in precision positioning platform due to its advantages of high response speed, large generated force, subnanometer-scale resolution and the ease of use [10, 11]. However, the disadvantage of PEA is its limited displacement [12]. It is usually necessary to design mechanical displacement amplifiers such as bridge-type mechanism and leverage mechanism to amplify the output displacement of PEA [13]. Since the amplification ratio of one single amplifier is usually limited, the multi-level amplifications can be realized by the serial connections of amplifiers to obtain larger moving ranges of the platform [14]. The selection, structural design and configuration of amplification mechanisms are very important considering that they will affect the total size, conduction of motion and stress distribution of the platform. The XYZ platform is one type of the precision positioning platforms, which has stronger functionality and higher applicability compared with 1-DOF and 2-DOF platforms. The XYZ-platform can be classified as serial platform or parallel platform, depending on the connection form of the mechanism [15]. The serial platform is generally composed of multi-level one-direction mechanism nested. Its output platform is only connected with the last level of one-direction mechanism. The serial platform is simple in structure and easy to control, but it has the limitation of large inertia, low natural frequency and low repetitive positioning accuracy [16–18]. The output platform of parallel platform is directly connected with the mechanism in each direction. As a consequence, its structure is usually complicated. And the motion in different directions is easy to be coupled. However, the parallel platform has the advantages of low inertia, high natural frequency, and high load capacity [19–21]. Therefore, the design and research surrounding the parallel XYZ platform has been extensively carried out. For example, Zhu et al. [22] developed a triaxial compliant mechanism for nano-cutting to obtain tri-axial translational motions with decoupled features. A novel 3-DOF parallel platform proposed by Li et al. [23] can perform high dexterous manipulation within the desired workspace. Gao et al. [24] used three pairs of modified differential lever displacement amplifiers which are arranged orthogonally to enhance the mechanism workspace. By using the bridge-type and lever-type compound amplifier, the platform presented by Zhang et al. [25] provides a large output displacement, which is over 30 times the input displacement. However, most XYZ-platforms have three orthogonal working axes, which results in their large overall sizes. This not only makes the platform unusable in some workplaces where size is required, but also adversely affects the natural frequency of the platform.

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

571

Z-Shaped flexure hinge (ZFH) is a recently proposed flexure hinge composed of several flexible beams. With a simple structure, ZFH can change the direction of input displacement and amplify the displacement in the output direction. In order to obtain structures that may be even more optimal, some researchers used ZFHs in the structural design of XYZ-platforms. For example, by using ZFHs, Xie et al. [26] proposed a platform with more compact size in the height comparing with existing XYZ-platforms. Gan et al. [27] presented a XYZ bi-directional motion platform by means of four PEAs and the reverse arrangement of the ZFHs along the X-axis and Y-axis. Although the traditional ZFH can amplify the input displacement, its amplification ratio is inadequate. Besides, if ZFHs are symmetrically arranged at both ends of the moving platform, the inevitable coupling error will occur when the moving platform moves horizontally along the ZFH. Therefore, the structural optimization of ZFH is meaningful in order to achieve a larger amplification ratio and reduce the coupling error. In this paper, a novel ZFH with larger amplification ratio than the traditional ZFH is proposed. Novel ZFHs, bridge-type mechanisms and leverage mechanisms are introduced into the proposed XYZ-platform to amplify the displacement in the XYZaxis. In consequence, the platform has a large stroke in the Z-axis and bidirectional motion in the X-axis. And the platform achieves a large accessible workspace with a nearly flat structure. The rest of the paper is organized as follows. Section 2 introduces the structure design and working principle of the XYZ-platform; In Sect. 3, statics analysis of the platform is carried out, and the amplification ratio of the novel ZFH is calculated; In Sect. 4, the finite element simulation of the platform is carried out to verify the rationality of theoretical analysis; In Sect. 5, the whole paper is summarized.

2 Design of the XYZ Precision Positioning Platform In this paper, the structure of ZFH is optimized. Figure 1 shows the structure of the traditional ZFH and the novel ZFH. The traditional ZFH consists of two horizontal beams and a vertical beam. The novel ZFH with two steps is obtained by adding a horizontal beam and a vertical beam on the basis of the traditional ZFH. Compared with the traditional ZFH, the novel ZFH has a higher amplification ratio, which varies with the length of the horizontal beam in the middle. In addition, the novel ZFH operates with less coupling error due to its greater stiffness along X-axis. Figure 2 illustrates the working principle of the novel ZFH. The principle of the novel ZFH and the traditional ZFH is the same. To make them play the role of changing the direction of motion and amplifying the displacement, ZFHs need to be symmetrically arranged on both sides of the moving stage. When the opposite ends of a pair of ZFHs are subjected to the force along X-axis, the elastic deformation of ZFHs will drive the middle stage to move along Z-axis. It is noteworthy that the displacement along the Z-axis is larger than the displacement on the X-axis, which shows the displacement amplification effect of ZFH.

572

L. Wan et al.

Fig. 1 The structure of the traditional ZFH and the novel ZFH: a the structure of the traditional ZFH; b the structure of the novel ZFH

Fig. 2 The working principle of the novel ZFH

The structural design of the XYZ precision positioning platform is shown in Fig. 3. All the mechanisms of the platform are arranged symmetrically. There are four parallel branched chains connected to the moving stage in the middle. The branched chain above the moving stage consists of two special flexure hinges and guiding mechanisms. The special flexure hinge is a kind of hollow hinge with two degrees of freedom in X-axis and Z-axis. The hollowing is done to reduce the stress on the hinge. The branched chain below the moving stage consists of two special flexure hinges, leverage mechanism and bridge-type mechanism. The branched chains on the left and right sides of the moving stage are the same, which are composed of a pair of novel ZFHs arranged up and down, leverage mechanism and bridge-type mechanism. The platform is driven by three PEAs placed in bridge-type mechanism. When driving PEA 1, piezoelectric material’s elongation causes the bridge-type mechanism and leverage mechanism to move, thereby driving the moving stage to move along the Y-axis. PEA 2 and PEA 3 are identical. According to the differential moving principle, the difference of driving force between the two coaxial drivers can cause bidirectional motion of the platform along the X-axis [27]. The displacement of X-axis is amplified by bridge-type mechanism and leverage mechanism. If PEA 2 and PEA 3 output the same driving force, the levers on both sides of the platform

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

573

Fig. 3 Structural design of the platform: a top view of the platform; b spatial view of the platform

move towards the middle together, squeezing the novel ZFHs and deforming them to produce the motion along Z-axis. The multi-level amplification of the displacement along Z-axis is achieved through bridge-type mechanism, leverage mechanism and novel ZFHs.

3 Modeling and Analysis of the Platform In this section, the XYZ precision positioning platform is modeled and analyzed, including kinematics modeling and statics modeling. Kinematics modeling mainly analyzes the motion characteristics of the whole platform. Statics modeling includes the analysis of the novel ZFH’s amplification ratio and the output compliance of the platform.

574

L. Wan et al.

3.1 Kinematics Modeling In kinematic analysis, according to the pseudo-rigid-body-modeling method, leafshaped flexure hinges are regarded as revolute joints with torsion springs. And the connecting rod is regarded as rigid body without deformation [2]. Figure 4 shows the partial structure of the left half platform, including the amplifiers of the platform’s two branched chains. Figure 5 shows the quarter kinematics diagram of the bridge-type mechanism. C and D denote the centers of flexure hinges marked in Fig. 4. H and L represent the distances of point C and D along X-axis and Y-axis, respectively. According to the geometric relations between input displacement and output displacement [28], the even displacement amplification ratio of bridge-type mechanism 2 can be obtained:         H ln √ HH2 +L 2 − ln sin arctan HL − 2XHin A B2 = (1) 2X in where X in denotes half of the input displacement of PEA 2. In the same way, the amplification ratio of bridge-type mechanism 1 can be obtained. The kinematics diagrams of two branched chains of the platform are shown in Figs. 6 and 7, respectively, where A-G and a-j represent the centers of the flexure hinges marked in Fig. 4. In order to obtain the relationship between the output and input displacements of the platform, the displacements of revolute joints are set and Fig. 4 Partial structure of the platform

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

575

Fig. 5 Quarter of kinematics diagram of bridge-type mechanism

they can be obtained: X F = X D = A B2 X in XG =

lEG XF lE F

(3) y

X e = X d = A B1 X in Xi = Xg = Xj =

l fg Xe le f

lh j Xi lhi

(2)

(4) (5) (6)

where A B1 denotes the amplification ratio of the bridge−type mechanism 1. X G and X j are equal to the output displacement of the platform along the X-axis and Y-axis respectively, while the displacement along the Z-axis is further amplified by novel Fig. 6 Kinematics diagram of the branched chain on the left of the moving stage

576

L. Wan et al.

Fig. 7 Kinematics diagram of the branched chain below the moving stage

ZFHs. Then the final output displacement of the platform can be deduced as: x X out = XG y

(7)

X out = X j

(8)

z X out = 2 Az X G

(9)

where A z denotes the amplification ratio of the novel ZFH. Since a novel ZFH is composed of five flexible beams, its amplification ratio cannot be calculated by kinematic method. The amplification ratio of the novel ZFH will be calculated in Sect. 3.2 by the energy method.

3.2 Statics Modeling In this section, the output compliance of the platform is calculated by matrixbased compliance modeling method, and the amplification ratio of the novel ZFH is analyzed by the energy method. The matrix method has been well developed and is suitable for the analysis of compliant mechanism. The details of matrix-based compliance modeling method can refer to Refs. [29, 30].

3.2.1

Compliance Model of the Platform

In the XYZ precision positioning platform, leaf-shaped flexure hinges and the right circular flexure hinges are utilized. As shown in Fig. 8, local coordinate systems are established on two different flexible hinges. The compliance matrix of the flexible hinges in local coordinate system can be derived:

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

⎡

c1 ⎢ 0 ⎢ ⎢ ⎢ 0 Cb = ⎢ ⎢ 0 ⎢ ⎣ −c3 0

0 c2 0 c4 0 0

0 0 c5 0 0 0

0 c4 0 c6 0 0

−c3 0 0 0 c7 0

577

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ c8

(10)

where the flexibility matrix parameters ci (i = 1 … 8) of two different flexible hinges is shown in Table 1. Where E and G denote the Young’s modulus and shear modulus of the material, respectively, and other notations are shown in Fig. 8. In the compliance model, the compliance of flexure hinge is usually expressed by the compliance matrix in global coordinate system. The transformation from the local coordinate matrix Cb to the global coordinate system matrix C can be obtained by: C = AdCb Ad T

(11)

where Ad denotes the adjoint matrix for coordinate transformation, which can be expressed as: Fig. 8 Schematic diagram of leaf-shaped flexure hinge and right circular flexure hinge

Table 1 The flexibility matrix parameters of two different flexible hinges Parameter Leaf-shaped Right circular   1 12l 12 c1 π rt 2 − Eab3 Eh 3 c2 c3 c4

12l Ea 3 b 6l 2 Eab3 6l 2 Ea 3 b

 2+π 2

Parameter Leaf-shaped Right circular c5

12l G(a 3 b+ab3 )

c6

4l 3 Ea 3 b

1

9πr 2 2Eht

5 2



π

12r Eh 3

r 1 t

2

 −

2+π 2

c7

5

2Eht 2

5

2Ght 2 5

+

9πr 2 2Eht

5 2

12πr 2 Eh 3





3

9πr 2

4l 3 Eab3

1

9πr 2

c8

l Eab

1 Eh

π

3

3πr 2

3

2Eht 2

r 1 2

t

r 1 t

2

− −

 1 4

π 2



578

L. Wan et al.

 Ad =

R 03×3 PR R

 (12)

where R denotes the rotation matrix from the local coordinate system to the global coordinate system, and P describes the translational coordinate transformation. They can be defined in Eq. (13) and Eq. (14), respectively. ⎧ ⎡ ⎤ 1 0 0 ⎪ ⎪ ⎪ ⎪ Rx = ⎣ 0 cos α − sin α ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ 0 sin α cos α ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ cos β 0 sin β ⎨ Ry = ⎣ 0 1 0 ⎦ ⎪ ⎪ ⎪ − sin β 0 cos β ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ cos γ − sin γ 0 ⎪ ⎪ ⎪ ⎪ R = ⎣ sin γ cos γ 0 ⎦ ⎪ ⎪ ⎩ z 0 0 1 ⎡ ⎤ 0 −z y P = ⎣ z 0 −x ⎦ −y x 0

(13)

(14)

Note that Rx represents the rotation matrix around the x-axis of the global coordinate, and α is the rotation angle. The meanings of other notations in Eq. (13) can be obtained similarly. Let C j and Ci be the compliance matrices of flexure hinge j and flexure hinge i in global coordinate system, respectively. If the local coordinate system of flexure hinge i can be obtained by rotating the local coordinate system of flexure hinge j around an axis of the global coordinate system, then: Ci = RdC j Rd T

(15)

where Rd can be expressed as:  Rd =

R 03×3 03×3 R

 (16)

As shown in Fig. 9, the proposed platform consists of four parallel branched chains marked by 1, 2, 3 and 4. Chain 1 consists of bridge-type mechanism, leverage mechanism and novel ZFHs. The bridge-type mechanism is first connected in parallel with the leaf-shaped flexure hinge in the lever mechanism. And then they are in series with two novel ZFHs arranged up and down. Finally, the parallel novel ZFHs are connected with the moving stage. The compliance of bridge-type mechanism in Chain 1 can be obtained by:

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

579

Fig. 9 Output compliance model of the platform A B C D F C B1 = C B1 + C B1 + C B1 + C B1 + C B1

(17)

A where C B1 denote the compliance of flexure hinges shown in Fig. 4 marked by A − D and F. As shown in Fig. 10, right circular flexure hinges are added to both ends of the novel ZFH to provide guidance function. Therefore, a novel ZFH can be conceived as a series of seven flexure hinges, including two right circular flexure hinges and five leaf-shaped flexure hinges. The compliance of the novel ZFHs in Chain 1 can be obtained by:

CZ1

⎡ −1  7 −1 ⎤−1 7   ⎦ =⎣ C Zi 1up + C Zi 1down i=1

(18)

i=1

where C Zi 1up and C Zi 1down respectively denote the compliance of the flexure hinges in the upper and lower novel ZFHs. According to the series and parallel relationship, the compliance of Chain 1 can be expressed as: −1  C1 = (C B1 )−1 + (C L1 )−1 + CZ1

(19)

580

L. Wan et al.

Fig. 10 Schematic diagram of the novel ZFH

where C L1 denotes the compliance of the leaf-shaped flexure hinge in leverage mechanism. Since Chain 1 and Chain 2 are identical and symmetric, the compliance of Chain 2 can be derived: C2 = Rdzπ C1 Rdzπ T

(20)

where Rdzπ denotes the rotation change matrix 180° along the Z-axis in global coordinate system. Chain 3 is composed of two special flexure hinges in parallel and leaf-shaped guide beams. Four parallel leaf-shaped flexure hinges are connected in series with two special flexure hinges. The special flexure hinges are connected to the moving stage. The compliance of guide beams can be obtained by: C G3 =

4 

i C G3

(21)

i=1 i where C G3 denotes the compliance of the leaf-shaped flexure in Chain 3. As shown in Fig. 11, The special flexure hinge can be regarded as a combination of six flexure hinges, including two right circular flexure hinges and four leaf-shaped flexure hinges. According to the series and parallel connections of these flexure hinges, the compliance of a special flexure hinge in Chain 3 can be derived: l 1 2 C G2 = C G2 + C G2 +

   4 −1 −1 3 −1 5 6 C G2 + C G2 + C G2 + C G2

(22)

l where C G2 denote the compliance of compliant elements in special flexure hinge. Since the two special flexure hinges in Chain 3 are arranged symmetrically, the compliance of special flexure hinges in Chain 3 can be expressed as:

C G2 =

 −1  π l −1 −1 l C G2 + Rdz C G2 Rdzπ T

As a result, the compliance of Chain 3 can be derived:

(23)

Design of a 3DOF XYZ Precision Positioning Platform Using Novel …

581

Fig. 11 Schematic diagram of the special flexure hinge

C3 = C G2 + C G3

(24)

Chain 4 consists of bridge-type mechanism, leverage mechanism and two parallel special flexure hinges. The bridge-type mechanism and leverage mechanism can be divided into two symmetrical branched chains in parallel. They are connected in series with special flexure hinges. Finally, the special flexure hinges are connected to the moving stage. The compliance of half bridge-type mechanism and leverage mechanism in Chain 4 can be obtained by: C A1 =

 −1 C aA1 + C bA1 + C cA1 + C dA1 + C eA1 ⎫−1 −1 −1 −1  −1 ⎬  f + C A1 + C iA1 + C hA1 ⎭

(25)

where C aA1 denote the compliance of flexure hinges shown in Fig. 4 marked by a − i. The compliance of the other half of the bridge-type mechanism and leverage mechanism can be obtained from the following equation because they are symmetrically arranged: C A2 = Rdzπ C A1 Rdzπ T

(26)

And the compliance of special flexure hinges in Chain 4 can be obtained by: C G1 = Rdxπ C G2 Rdxπ T

(27)

where Rdxπ denotes the rotation change matrix 180° along the X-axis in global coordinate system. As a result, the compliance of Chain 4 can be derived: −1  C4 = (C A1 )−1 + (C A2 )−1 + C G1

(28)

582

L. Wan et al.

Fig. 12 Schematic diagram of a Z-shaped beam

Since the four branched chains connected to the moving stage are in parallel, the output compliance of the platform can be expressed as: −1  Cout = (C1 )−1 + (C2 )−1 + (C3 )−1 + (C4 )−1

3.2.2

(29)

Amplification Ratio of the Novel ZFH

The amplification ratio of the novel ZFH is calculated by the energy method proposed by Guan et al. [31]. In this method, the relationship between the input displacement and the output displacement is obtained by analyzing the force and moment on each beam of ZFH. Since the novel ZFHs are arranged symmetrically, only one Z-shaped beam as shown in Fig. 12 need to be analyzed in order to obtain the amplification ratio of the novel ZFH. The Z-shaped beam is composed of two horizontal beams with length L 1 , two vertical beams with length L 2 and a central horizontal beam with length L 3 . The thickness of the beams is w. In the analysis, a reaction moment M, a reaction force Fx along X-axis, and a virtual force P are present. The moments at any point on the five beams can be obtained separately by: ⎧ ⎪ M1 (x1 ) = ⎪ ⎪ ⎪ ⎪ ⎨ M2 (x2 ) = M3 (x3 ) = ⎪ ⎪ ⎪ M4 (x4 ) = ⎪ ⎪ ⎩ M (x ) = 5 5

P x1 − M P L 1 − M + x2 Fx P(L 1 + x3 ) − M + L 2 Fx P(L 1 + L 3 ) − M + (L 2 + x4 )Fx P(L 1 + L 3 + x5 ) − M + 2L 2 F x

0< 0< 0< 0< 0