Parallel PnP Robots: Parametric Modeling, Performance Evaluation and Design Optimization [1st ed.] 9789811566707, 9789811566714

Table of contents :
Front Matter ....Pages i-xvii
Introduction (Guanglei Wu, Huiping Shen)....Pages 1-15
Kinematic Geometry of the PnP Robots (Guanglei Wu, Huiping Shen)....Pages 17-56
Differential Kinematics of the Robots (Guanglei Wu, Huiping Shen)....Pages 57-85
Kinematic Performance Criteria (Guanglei Wu, Huiping Shen)....Pages 87-120
Elastostatics and Stiffness (Guanglei Wu, Huiping Shen)....Pages 121-148
Rigid Multibody Dynamics (Guanglei Wu, Huiping Shen)....Pages 149-170
Robot Elastodynamics (Guanglei Wu, Huiping Shen)....Pages 171-190
Design Optimization of Parallel PnP Robots (Guanglei Wu, Huiping Shen)....Pages 191-220
Back Matter ....Pages 221-251

Citation preview

Research on Intelligent Manufacturing

Guanglei Wu Huiping Shen

Parallel PnP Robots Parametric Modeling, Performance Evaluation and Design Optimization

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

Research on Intelligent Manufacturing (RIM) publishes the latest developments and applications of research in intelligent manufacturing—rapidly, informally and in high quality. It combines theory and practice to analyse related cases in fields including but not limited to: Intelligent Intelligent Intelligent Intelligent

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This book series aims to address hot technological spots and solve challenging problems in the field of intelligent manufacturing. It brings together scientists and engineers working in all related branches from both East and West, under the support of national strategies like Industry 4.0 and Made in China 2025. With its wide coverage in all related branches, such as Industrial Internet of Things (IoT), Cloud Computing, 3D Printing and Virtual Reality Technology, we hope this book series can provide the researchers with a scientific platform to exchange and share the latest findings, ideas, and advances, and to chart the frontiers of intelligent manufacturing. The series’ scope includes monographs, professional books and graduate textbooks, edited volumes, and reference works intended to support education in related areas at the graduate and post-graduate levels. If you are interested in publishing with the series, please contact Dr. Mengchu Huang, Senior Editor, Applied Sciences Email: [email protected] Tel: +86-21-2422 5094.

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Guanglei Wu Huiping Shen •

Parallel PnP Robots Parametric Modeling, Performance Evaluation and Design Optimization

123

Guanglei Wu School of Mechanical Engineering Dalian University of Technology Dalian, Liaoning, China

Huiping Shen Mechanical Engineering Changzhou University Changzhou, Jiangsu, China

ISSN 2523-3386 ISSN 2523-3394 (electronic) Research on Intelligent Manufacturing ISBN 978-981-15-6670-7 ISBN 978-981-15-6671-4 (eBook) https://doi.org/10.1007/978-981-15-6671-4 Jointly published with Huazhong University of Science and Technology Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Huazhong University of Science and Technology Press. © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To our families

Preface

Over the past several decades, parallel robots have attracted lots of attention in both academia and industries, which have been widely used as mechanical architectures. Amongst the various types of parallel robots, pick-and-place (PnP) robot is a family of the most popular robots for material handling, leading to its successful industrial application. As the multi-arm articulated mechanical systems with closed-loop structure, the design of parallel PnP robots is a challengeable task due to their highly nonlinear and nonintuitive behaviors. This book aims to provide some material in the context of the most important aspects of parallel robots for the performance analysis, evaluation and design: • As the fundamentals for the modeling and programming of a robot, robot kinematics reveal the functions and mappings between the actuated inputs and end-effector outputs, subject to the robot topological geometry and link parameters. Most of the important performance indices are related to the kinematic Jacobian matrix that is an issue of utmost importance in the field of robotic manipulators. • Robot elastostatics and dynamics pertains to the dynamic characteristics of robotic manipulators. Elastostatics is measured by the manipulator stiffness that reflects the capability of a robot resisting to the payload. Robot dynamics is concerned with the relationship between the forces acting on a robot mechanism and the accelerations they produce. Typically, robot dynamics is classified into rigid-body dynamics and flexible robot dynamics. • Performance indices play an important role in the evaluation and design of parallel robots, which are to formulate the design criteria and guidelines. An index is usually a norm derived from the parametric model of the robot mechanism. The criteria of performance evaluation for parallel mechanisms can be roughly classified into two groups: one relates to the kinematic performance while the other relates to the kinetostatic/dynamic performance. • The design process of the parallel robots, in general, simultaneously considers a number of performance indices that essentially vary throughout the operational workspace but remain within the prescribed bounds. An efficient and systematic

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Preface

way to cope with this problem is to apply optimization technique in the design process. Design optimization can take all or most of the evaluation criteria into account towards the comprehensive design. The present book, in context of the published research output and technical reports by the two authors in the past 10 years, aims to provide the material for engineers, master and Ph.D. students both in academia and industrials dealing with the parametric modeling, performance evaluation and design optimization of parallel PnP robots. Most of the presented methods and results can be generalized to other robotic manipulators. The authors are genuinely grateful to Ms. Li Shen and Mr. Arul Venkatasalam from Springer Nature as well as Mr. Yajun Wan from Huazhong University of Science and Technology Press Co., Ltd, who have made this publication possible. The authors are also grateful to the master students under our supervision for their contributions to this book. Special thanks go to Professor Shaoping Bai of Aalborg Universitet and Former CEO Preben Hjørnet of Blue WorkForce A/S, for the collaboration with the first author in the industrial Post Doc project of developing novel high-speed parallel robots funded by InnovationsFonden Denmark. In addition, the authors gratefully appreciate the publishers, namely, Springer Nature, Elsevier, Cambridge University Press, American Society of Mechanical Engineers (ASME), Institute of Electrical and Electronics Engineers (IEEE) and SAGE Publishing for the permission of reuse of the published materials. Last but not least, the first author would like to acknowledge Associator Professor Xuping Zhang from Department of Engineering, Aarhus Universitet, Denmark, for hosting a visiting scholar to accomplish the documentation of this book. The authors will be also grateful to the readers for any critical comment on the contents of this book and for any suggestion and insightful discussion for further improvement. Aarhus, Denmark Changzhou, China May 2020

Guanglei Wu Huiping Shen

Contents

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1 1 4 5 14

2 Kinematic Geometry of the PnP Robots . . . . . . . . . . . . . . . . . 2.1 Position and Orientation Representation . . . . . . . . . . . . . . . 2.1.1 Position and Translation . . . . . . . . . . . . . . . . . . . . . 2.1.2 Orientation and Rotation . . . . . . . . . . . . . . . . . . . . 2.2 Homogeneous Transformations . . . . . . . . . . . . . . . . . . . . . 2.2.1 Elementary Transformation Matrices . . . . . . . . . . . . 2.2.2 Properties of Transformation Matrices . . . . . . . . . . . 2.3 Planar Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Kinematic Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Geometry of Kinematic Linkages . . . . . . . . . . . . . . . . . . . 2.5.1 Forward Geometry: Denavit–Hartenberg Convention 2.5.2 Inverse Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Planar Five-Bar Mechanism . . . . . . . . . . . . . . . . . . 2.6.2 Redundantly Actuated Translational Robot . . . . . . . 2.6.3 Four-Limb SMG with a Single Platform . . . . . . . . . 2.6.4 Two-Limb SMG with Articulated Double Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 3T3R Double-Delta Robot . . . . . . . . . . . . . . . . . . .

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17 17 17 19 25 26 26 28 28 29 29 30 31 31 37 42

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3 Differential Kinematics of the Robots . . . . . 3.1 Instantaneous Kinematics . . . . . . . . . . . 3.1.1 Velocity and Jacobian Matrix . . . 3.1.2 Acceleration and Hessian Matrix

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57 57 58 59

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Parallel Manipulators . . . . . . . . . . . . . . . . . 1.2 General Definitions on Parallel Mechanisms . 1.3 Parallel PnP Robots . . . . . . . . . . . . . . . . . . 1.4 Objective and Organization of This Book . .

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3.2 Twist and Kinematic Screw . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Planar Five-Bar Mechanism . . . . . . . . . . . . . . . . . . . 3.3.2 SMG with Planetary Gear Train Based End-Effector . 3.3.3 Kinematic Sensitivity of SMG with a Single Platform 3.3.4 Isotropic Configuration of the Two-Limb SMG . . . . .

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60 62 62 64 70 83

4 Kinematic Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . 4.1 Workspace Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kinematic Dexterity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Condition Index of Jacobian Matrix . . . . . . . . . . . . 4.2.2 Homogenization of Jacobian Matrix . . . . . . . . . . . . 4.3 Motion/Force Transmission . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Transmission Wrench and Twist Screw . . . . . . . . . . 4.3.2 Transmission and Pressure Angle . . . . . . . . . . . . . . 4.4 Singularity Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Examples: Four-Limb Parallel SMGs . . . . . . . . . . . . . . . . . 4.5.1 Asymmetrical Robot with a Single Moving Platform 4.5.2 Symmetric Robot with PGT Based Platform . . . . . . 4.5.3 SMGs with Articulated Double Sub-platforms . . . . .

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5 Elastostatics and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Stiffness Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stiffness Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Eigenscrew Decomposition of the Stiffness Matrix . 5.2.2 Decoupling of the Stiffness Matrix . . . . . . . . . . . . 5.2.3 Nondimensionalization of the Motion Variables . . . 5.2.4 Vertex Velocity Transformation Based Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Stiffness Characterization of PnP Robots . . . . . . . . . . . . . 5.3.1 Example I: Planar Five-Bar Mechanism . . . . . . . . 5.3.2 Example II: Translational Delta Robots . . . . . . . . . 5.3.3 Example III: Four-Limb PnP Robots . . . . . . . . . . .

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6 Rigid Multibody Dynamics . . . . . . . . . . . . . . . . . . . . . . . 6.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . 6.1.2 Recursive Newton–Euler Equation . . . . . . . . . 6.1.3 Principle of Virtual Work . . . . . . . . . . . . . . . . 6.2 Dynamic Simulation of Four-Limb SMGs . . . . . . . . . 6.2.1 SMG with a Single Mobile Platform . . . . . . . . 6.2.2 SMG with PGT-Based Mobile Platform . . . . . 6.2.3 SMG with Screw-Pair Based Mobile Platform .

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7 Robot Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Elastodynamics of Robotic Manipulators . . . . . . . . . . . . . . . 7.1.1 Bernoulli–Euler Beam Stiffness and Mass Matrices . . 7.1.2 Flexible Dynamic Model . . . . . . . . . . . . . . . . . . . . . 7.1.3 Natural Frequencies and Displacement Response . . . . 7.2 Elastodynamic Analysis of a Redundant Translational Robot 7.2.1 Elastodynamic Model in High-Dimensional Generalized Coordinate Space . . . . . . . . . . . . . . . . . 7.2.2 Jacobian-Based Elastodynamic Model in Cartesian Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Numerical Simulation Results . . . . . . . . . . . . . . . . . . 7.3 Elastodynamic Analysis of the SMG with a Single Platform .

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8 Design Optimization of Parallel PnP Robots . . . . . . . . . . . . . . 8.1 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . 8.3 Examples of Design Optimization . . . . . . . . . . . . . . . . . . . 8.3.1 Planar Five-Bar Linkage . . . . . . . . . . . . . . . . . . . . 8.3.2 Kinematic Design of SMG with Two Objectives . . . 8.3.3 SMG Comprehensive Design with Three Objectives

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Appendix A: PnP Trajectory Under Working Cycle 120 cpm . . . . . . . . . 221 Appendix B: Matlab Codes: Eigenscrew Decomposition of a Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Appendix C: Matlab Codes: Decoupling of a Stiffness Matrix . . . . . . . . 227 Appendix D: Matlab Codes: Finding Maximum Square Workspace of the Five-bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

List of Symbols and Abbreviations

Throughout this book, bold lowercase and uppercase symbols stand for vectors and matrices, respectively, and IN represents a N-dimensional identity matrix, and ON represents a N-dimensional null matrix, while qn stands for a n-dimensional column vector.

Acronyms a.k.a. cpm mTnR pdf w.r.t. CAD CCT CPM DE DOF EA EE EOM FEA FKP GA GCI GTI GUI IKP ITI KC

also known as cycles per minute m Translations and n Rotations probability density function with respect to Computer-Aided Design Conservative Congruence Transformation Cross-Product matrix Differential evolution Degree-Of-Freedom Evolutionary algorithms End-Effector Equations of motion Finite Element Analysis Forward kinematic problem Genetic Algorithm Global Conditioning/Condition Index Global transmission index Graphical user interface Inverse kinematic problem Input transmission index Kinematic Chain xiii

xiv

LCI LTI MOEA MOOP MP MSA NSGA OTI PCB PGT PKM PM PnP RPY angle RWS SAD SCARA SMG SMT T&T TI TWS VJM WM WS

List of Symbols and Abbreviations

Local Condition Index Local Transmission Index Multi-Objective Evolutionary Algorithm Multi-Objective Optimization Problem Mobile/Moving Platform Matrix Structural Analysis Non-dominated Sorting Genetic Algorithm Output transmission index Printed Circuit Board Planetary gear train Parallel Kinematic Machine Parallel Manipulator/Mechanism Pick-and-place Roll, Pitch and Yaw Regular workspace Small-amplitude displacement Selective Compliance Assembly/Articulated Robot Arm Schönflies Motion Generator Surface mount technology Tilt-and-Torsion Transmission index Transmission wrench screw Virtual Joint Method Working Mode Workspace

Nomenclature

½u = CPMðuÞ $T ; ^$T $T  $W $W ; ^$W v C k x Uq si u ¼ ½ a b c T €v dW dhi dqi v_ _ €q q; ^$P ^xi ; ^yi ; ^zi i; j; k

cross product matrix by vector u twist screw and unit twist screw reciprocal product between twist and wrench screws wrench screw and unit wrench screw end-effector pose of a parallel robot 6  6 permutation matrix vector of Lagrange multipliers vector of angular velocity constraint Jacobian matrix vector of the internal force/moment in the ith linkage of a parallel robot orientation angles of the mobile platform of parallel robot in its reference coordinate frame linear and angular accelerations of a parallel robot’s end-effector virtual work of a mechanical system deflections of the actuated joint and flexible links in the ith linkage of a parallel robot small-amplitude displacements of passive joints in the ith linkage of a parallel robot linear and angular velocities of a parallel robot’s end-effector first-, second-order differentiation of motion variable q w.r.t time elastic deformation of the reference point on the mobile platform of a parallel robot unit vectors of the xi -, yi - and zi -axis of Cartesian coordinate Fi unit vectors of the x-, y- and z-axis of the reference coordinate frame

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Nomenclature

J ¼ J1v Jq ð¼ A1 ; BÞ Jv ; Jq ðA; BÞ Jh; i Jq; i KC ðKÞ 2 R6 Ki KJ Kh; i Krr ; Ktt ; Krt M M j ; Jj p ¼ ½x y

z T

Qex Rot Ru ðhÞ Rx ðÞ; Ry ðÞ; Rz ðÞ Trans v w x; xlb ; xub Fi ðFp Þ Fj Fðb Þ cðÞ ; sðÞ mod rankðÞ sgnðÞ trðÞ vectð½Þ  /–h–r

kinematic Jacobian matrix forward and inverse Jacobian matrices Jacobian matrix mapping the elastic deformations in the ith linkage of a parallel robot Jacobian matrix related to the small-amplitude displacements of passive joints in the ith linkage of a parallel robot Cartesian stiffness matrix Cartesian stiffness matrix of the ith linkage of a parallel robot stiffness matrix in joint space stiffness matrix of the ith linkage in the joint space of a parallel robot linear, angular stiffness entries and their coupling entry mass/inertia matrix mass and inertia matrices of the ith rigid body in its inertial frame position vector of the mobile platform of parallel robot in its reference coordinate frame vector of external forces homogeneous transformation of a pure rotation about an axis rotation matrix about an arbitrary axis u by angle h basic or elemental rotation matrix about x-, y-, z-axis of a coordinate system homogeneous transformation of a pure translation along an axis vector of linear velocity static external wrench applied to the mobile platform of a parallel robot vector of design variables and their lower and upper bounds moving coordinate system/frame relative to the reference one reference coordinate system/frame cosine and sine functions modulo operation rank of a matrix ½ sign function of a term ðÞ trace of a matrix ðÞ operation of matrix vector tensor product of two vectors or Kronecker product of two matrices Azimuth–Tilt–Torsion angles

Nomenclature

tan1 hj j

pi ¼

j

Ri Ti

j

j1

j

xi

Aj dj E; G; m L qj  T ¼ Tðq; qÞ V ¼ VðqÞ ½1 ; ½T

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j

yi

j

zi

T

inverse tangent function variable of the jth R joint position vector of the origin of coordinate frame Fi relative to Fj in the Cartesian coordinate system rotation matrix of coordinate frame Fi relative to Fj 4  4 homogeneous transformation matrix from frame Fi to Fj homogeneous transformation of DH convention variable of the jth P joint Young’s modulus, shear modulus and Possion’s ratio Lagrangian of a robotic system if it is not redefined variable of the jth joint kinetic energy potential energy inverse/transpose of a matrix ½

Chapter 1

Introduction

Abstract Over the past several decades, the research and development of parallel manipulators that have been widely used as mechanical architectures have been extensively carried out. This introductory chapter presents the definitions and advantages of the parallel manipulators, compared to their serial counterparts, together with some typical examples of industrial applications of the parallel mechanisms. After the description of some important terminologies commonly used in the field of parallel robots, a brief historical development of the parallel pick-and-place robots is presented, serving in material handling. At the end of this chapter, the main objectives and organization of this book are depicted.

1.1 Parallel Manipulators Parallel manipulators (PMs), also called parallel robots or parallel kinematic machines (PKMs), of which the definition of the generalized parallel manipulators was introduced by Merlet [186]: a generalized parallel manipulator is defined as a closed-loop kinematic chain mechanism whose end-effector is linked to the base by several independent kinematic chains, in comparison to the serial manipulators, as shown in Fig. 1.1. From the previous definition, it is seen that the parallel mechanisms has the following characteristics: • a manipulator is composed of a fixed base and a moving platform • at least two kinematic linkages that contain at least one simple actuator in each chain to support the end-effector • a linkage is usually a kinematic chain of serial or tree-structure type • the mobility of the manipulator degenerates to null when the actuators are locked Sequentially, parallel robots can therefore be defined as follows: A parallel robot is made up of an end-effector with n degrees of freedom, and of a fixed base, linked together by at least two independent kinematic chains [160, 186]. Their unique structures allow the parallel manipulators to perform better over the serial manipulators: © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2021 G. Wu and H. Shen, Parallel PnP Robots, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-6671-4_1

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1 Introduction

Fig. 1.1 Comparison of serial and parallel manipulators: a serial 6-axis robotic arm, b planar five-bar mechanism

• The actuators of a parallel manipulator are usually arranged on or close to its fixed platform that enables the manipulator perform higher performances such as low mass in motion, high speed and better dynamic behaviors. • The non-accumulated errors of the manipulator end-effector provide high accuracy and repeatability. • The multiple closed-loop kinematic chains lead to high specific structural stiffness and payload-weight ratio. • The symmetric structure of a parallel manipulator ensures better kinematic isotropy, i.e., approximately equivalent performance in all directions. • PMs can readily achieve real-time control due to their simple inverse kinematics.

1.1 Parallel Manipulators

3

Parallel manipulators were originally proposed to cope with the previous problems encountered with their serial counterparts [13, 186]. The milestone of the development of parallel manipulators is the appearance of the well-known “Gough–Stewart” platform (see Fig. 1.2) introduced by Gough as a tire testing machine [113] and then by Stewart as an aircraft simulator [249]. Afterwards, parallel mechanisms have

Fig. 1.2 Gough–Stewart platform: a Gough platform as a tire testing device (1956) [113], b schematic of Stewart platform (1965) [249]

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1 Introduction

been extensively studied, particularly the past several decades, focusing on different aspects, i.e., type synthesis, mobility analysis, dimensional synthesis, kinematic and dynamic modeling, static/elastostatic analysis, design optimization, etc. Up to date, parallel manipulators have found their applications in many fields, such as: • • • • • • • • • •

machining [19, 28, 127, 181, 218] orientating device [25, 112, 284] aircraft/vehicle simulators [3, 17, 86, 129, 130, 149, 197, 204, 273] medical/surgical devices [47, 53, 67, 142, 165, 178, 188, 230, 242, 268] adjustable articulated trusses [70, 115, 117, 192, 253, 303] accessing offshore structure [62, 101] force/torque sensor [89, 90, 97, 138, 166, 220, 229, 245] micro-positioning devices [77, 78, 95, 118, 137, 216, 257, 304, 309] pick-and-place operations [1, 26, 34, 200, 201, 287] ...

Parallel manipulators have become an indispensable part of general robotic manipulators both in industry and in academia, and their research and development have been a key technology of robotic applications in engineering [275].

1.2 General Definitions on Parallel Mechanisms From the definition, the end-effector of one parallel robot will generate motions to accomplish the task, which can be decomposed into kinematic linkages that are consisted of links and joints. Throughout this book, some terms regards to the parallel mechanism are defined as: • degree of freedom: the number of independent motions of a body or the number of possible independent relative motions between the links of a mechanism, abbreviated as DOF. • mobility: the number and the types (rotation/translation) of independent motions of a body. • lower-mobility: the mobility of a body or a robot is less than six. • mTnR: the motion of a body includes m translational and n rotational movements. • R: a revolute joint, also called pin joint or hinge joint, allows a single-axis rotation, shown in Fig. 1.3a, and an underlined (R) letter denotes an active R joint, otherwise passive. • P: a prismatic joint, allows a linear translational movement between two bodies along a prescribed axis, shown in Fig. 1.3b, and an underlined (P) letter denotes an active P joint, otherwise passive. • C: a cylindrical joint (Fig. 1.3c), is a 2-DOF kinematic pair providing a single axis rotation as well as a translation along the same axis.

1.2 General Definitions on Parallel Mechanisms

5

Fig. 1.3 Commonly used kinematic pairs in PMs: a revolute; b prismatic; c cylinderical; d helical; e universal; e spherical

• H: a screw joint (Fig. 1.3d), is a 1-DOF kinematic pair providing single-axis translation by transforming the rotation around the same axis. • U: a universal joint (Fig. 1.3e), also called universal coupling, Cardan joint or Hooke’s joint, allowing 2-DOF independent rotations around two perpendicular axes, is equivalent to two R joints, of which the two axes of rotation are perpendicular that intersects at one point. • S: a spherical joint (Fig. 1.3f), also called ball joint, allowing 3-DOF independent rotations, is equivalent to three R joints, of which the three axes of rotation are orthogonal that intersects at one point. • kinematic linkage: also called limb, leg or kinematic chain (KC), is depicted by the kinematic joints in serial, for instance, PURS standing for prismatic-universalrevolute-spherical linkage with actuated prismatic joint.

1.3 Parallel PnP Robots Amongst the various types of parallel robots, pick-and-place (PnP) robots are one class of popular robots dedicating to material handling, such as auto, electronics, semiconductors and food industries. The pick-and-place robots are well suited for a

6

1 Introduction

Fig. 1.4 Kinematic diagram of serial SCARA configuration

static environment where the task is repeated and demands precise tolerances. Initially used in the packaging, case filling and palletizing, which are normally referred as downstream operations, such the robots are nowadays increasingly deployed in upstream operations. The types of PnP robots are diverse, from the general sixdegree-of-freedom (6-DOF) robots, to 2-DOF planar robots, and now more with high-speed parallel robots, to meet a variety of requirements. The first appeared robot deployed in PnP applications was seen in the early 1980s in the electronic assembly sector, such as printed circuit board (PCB) and electronic parts. This robot is known as SCARA robot,1,2 developed under the guidance of Prof. Hiroshi Makino at the University of Yamanashi, which can generate three independent translations and one rotation around the vertical axis (3T1R), as shown in Fig. 1.4. For this type of serial robots, the driving units are placed within the joints, resulting in the high load inertia, which is a disadvantage to the improvement of high operating speed. The robot was able to pick and place at a reasonable rate of 55–80 cycles,3 which is much lower compared with today’s parallel robots. To design the PnP robots with high-speed operation to improve the productivity, the parallel mechanisms received the attentions, thanks to their advantages of low inertia and high speed. Based on the concept of PKMs, Clavel proposed the well-known Delta robot [71], as shown in Fig. 1.5. The Delta robot can generate three translations and one rotation around a fixed vertical axis (SCARA motion or 1 The

SCARA acronym stands for Selective Compliance Assembly Robot Arm or Selective Compliance Articulated Robot Arm. 2 A robot with SCARA motion, a.k.a. Schönflies motion, namely, three independent translations and one rotation around a fixed direction of an axis, are also named as Schönflies-motion robot or Schönflies motion generator (SMG). 3 Here, a cycle is defined as the movement along a trajectory of 25 mm × 300 mm × 25 mm with a 1 kg payload per minute (cpm) in the definition of Adept test cycle [100], which can also be a trajectory of 25 mm × 305 mm × 25 mm instead in some definitions.

1.3 Parallel PnP Robots

7

Fig. 1.5 Schematic of the Delta robot [71]

Schönflies motion), where a RUPUR chain called as the fourth telescopic limb is added to the robot architecture in-parallel for the realization of the rotary output of the end-effector. The RUPUR chain is composed of an actuated revolute joint in series with a passive universal joint, a passive prismatic joint, an additional passive universal joint and an additional passive revolute joint, transferring the independent rotation from the actuator fixed to the mounting base to the end-effector. The Delta robot was commercialized by a number of companies, which was capable to significantly improve robot performance. In the late 1990s, ABB Flexible Automation launched its Delta robot under the name IRB 340 FlexPicker [1], served in three industry sectors, i.e., the food, pharmaceutical, and electronics industries. The FlexPicker is equipped with an integrated vacuum system capable of rapid PnP operation weighing up to 1 kg, with a machine vision system by Cognex and an ABB S4C controller. The maximum allowable velocities are 10 m/s and 3.6 ◦ /s (about 150 cpm), and the accelerations are up to 100 m/s2 and 1.2 rad/s2 , almost double the performance provided by SCARA or Cartesian robot configurations. Delta robot has been successfully used in many industrial sectors. In a sense, Delta robot starts the era of the high-speed PnP robots constructed with the parallel mechanisms.

8

1 Introduction

Fig. 1.6 CAD model of the H4 robot for PnP operation [72]

The performance improvement is made available with new design to the Delta robot, where the mass and inertia are reduced, allowing for high-speed movement. For instance, FANUC Robots developed the Delta-architecture robots to achieve better performance with high payload and high level of flexibility, and full mobilities as well [93]. Differing from the structure displayed in Fig. 1.5, FANUC M-2iA/3S robot is with the fourth telescopic limb attached to the end of one of three actuated arms, with the axis of rotation parallel to the parallelogram. This design ensures a constant link length with reduced kinematic joints to simplify the topological structure. In addition, the FANUC M-2iA/3A robot adopts three additional RU chains mounted on the parallelogram to transfer actuator rotation to rotation relative to the manipulated platform, where the three rotations are achieved by the combined rotations or independent rotation of the RU chains through a complex gearbox as the wrist mechanism. This three-axis wrist is mounted at the mobile platform to have additional degrees of freedom for improved flexibility, which allows the robot to pick up objects and place at a simple or compound angle for the efficient utilization of available workspace. On the other hand, despite the end-effector rotations by the telescopic limbs, it is difficult to design a lightweight passive prismatic joint with a long service life, as noted by Pierrot et al. [211]. Moreover, as the telescopic limb is long, it raises the stability problem when this long chain undergoes high-speed rotations around the axes of the U joint [109, 279, 298]. To overcome the previous problems, the research group “Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM)” devoted to develop the fully parallel Schönflies-motion robots with four limbs. The LIRMM group firstly developed the H4 [210] robot in two versions, with four identical limbs and an articulated traveling plate [72], as shown in Fig. 1.6. Most of the following inventions or proposals regarding the four-limb parallel PnP robots inherit the H4 architecture. Later on, the similar versions of H4 robot, i.e., the I4 robot [156] and the symmetrical Par4 [195] were developed. The main difference among those robots lies in the different articulated platforms [213]. One successful application is the commercialization of the Par4. The commercial version of Par4, namely, the Quattro robot [200] by Adept Technologies Inc. in Fig. 1.7, hit the market in 2007, is the fastest industrial PnP robot

1.3 Parallel PnP Robots

9

Fig. 1.7 CAD model of the Adept Quattro robot and the mobile platform without amplification mechanism

Fig. 1.8 An articulated traveling plate with double-platform structure [200]

available. Besides the three independent translations of end-effector, the rotation is realized by the relative movement in parallel between the two sub-platforms in this double-platform architecture (see Fig. 1.7), and the pulley-belt mechanism is adopted to enhance the rotational capability, for which the articulated system is depicted in Fig. 1.8. The Adept Quattro robot can accelerate at 200 m/s2 with a 2 kg payload, comparing other robots moving at 100 m/s2 , thus is capable of moving 240 cycles per minute. The same group proposed and patented the Heli4 robot with a compact articulated traveling plate that is connected by a screw pair for enhanced rotational capability [157]. The license of this ingenious design was released to the industry, and its commercial version “Veloce. robot” [209] was developed by the Penta Robotics as displayed in Fig. 1.9. The design of this screw-pair-based platform allows the full-circle and even 720◦ rotation of the tool attached to the end-effector.

10

1 Introduction

Fig. 1.9 CAD model of the Veloce. robot with high rotational capability

It is noteworthy that the parallel Schönflies-motion PnP robots are the most popular ones, despite the presence of the various counterparts. Therefore, the design and development of the Schönflies-motion generators (SMGs) attracted the interests from both the industry and the academia. A number of different types of parallel SMGs should be noticeable for their high performance [42, 73, 102, 152, 164, 224, 239, 306], such as the Delta based robots [214] and the two-limb SMGs [15, 43, 119, 139, 146, 159, 244, 302], etc. For instance, Figs. 1.10a, b show two designs of the four-limb SMGs with reduced mounting space to utilize the shop-floor space more efficiently, which admits a compact installation in the production line, allowing more robots to be deployed side-byside in shop-floors to improve the productivity [167, 277, 283]. Figure 1.10c displays a Quattro-architecture robot with two RUU linkages instead of the conventional parallelogram structure to simplify the topological architecture [310]. Besides, a linearly actuated robot with enhanced stability, is shown in Fig. 1.10d, used for surface mount technology (SMT) in industrial electronics. With the considerations of manufacturing costs, structural complexity and working environments, parallel PnP robots with different mobilities were developed and found their industrial applications. Lexium Delta 2 robot from Schneider Electrics [235], a 2-DOF planar parallel robot, allows the maximum payload 40 kg and maximum speed 5.8 m/s, compared to the net weight 56 kg of the robot, i.e., high payload ratio. The main structure of the Micro working robot of Mitsubishi Electric RP-ADH series is based on planar five-bar linkage [191], leading to the structural compactness of space-saving, high-speed operation and higher positioning accuracy. These merits makes that the robot corresponds to the implementation, transport, assembly and inspection of micro-electronic components. The 3-DOF parallel PnP robots were also developed, such as the classical Delta robot and the ABB SCARA-Tau robot [135]. Compared to the ABB FlexPicker, the translational Delta robot does not include the RUPUR chain in the center. The SCARA-Tau robot is built on the fact that differing from the 2/2/2 structure of the DELTA robot, of which the 3/2/1 structure allows a common axis of rotation for the three actuated arms with constant platform pitch and roll angles. Therefore the 3/2/1

1.3 Parallel PnP Robots

11

Fig. 1.10 Four-limb parallel SMGs: a asymmetrical robot with screw-pair-based mobile platform; b a wall-mounted “V4”robot; c a Quattro-architecture robot with two RUU linkages; d linear actuated robot

distal link configuration opens up for a parallel robot with similar workspace as a traditional serial SCARA robot. Modification to the architectures of some existing robots may result in performance enhancement. Figure 1.11a shows a semi-symmetric Delta-architecture robot with three translations, named Delta-CU robot [163], which adopts the design concept of the RUU linkage instead of one parallelogram structure for structural simplification, still having similar static stiffness [238]. Actuation redundancy is another approach to improve the robot performance, particularly, on the aspect of

12

1 Introduction

Fig. 1.11 a Delta-CU robot and b redundantly actuated Ragnar robot

high dynamic response [74]. For instance, the translational Veloce. robot [209] is redundantly actuated with four limbs. Figure 1.11b displays the same design concept although the robot has an asymmetrical architecture. Parallel PnP robots with five degrees of freedom have not been seen on the market available. Technische Universiteit Delft developed a 5-DOF parallel PnP robot for electric board assembly [158], with two translational end-effectors. This robot is consisted of five limbs with parallelogram structures, of which the connecting bars of the mobile platforms constitute a six-bar linkage for the purpose of reconfiguration [158]. The two end-effectors can have independent translations in a same horizontal plane, allowing the improvement of the productivity. Inheriting the same concept, architectural modifications, as shown in Fig. 1.12, is applied to the previous reconfigurable platforms [158] with six RUU limbs to generate an additional rotation around an axis of vertical direction for each end-effector. This HEXA [212] base 6-RUU parallel robot has two end-effectors Schönflies motion by means of the reconfigurability. A major advantage of the full-mobility parallel robot is to overcome the limitation of “plane to plane” PnP operation, thanks to the three-axis rotations. For instance, the HEXA robot [212] proposed by Pierrot et al. can realize the unidirectional rotations with range of ±30◦ . Experimental results showed that the prototyped robot can achieve the Adept motion in 0.465 s/cycle and proved that the HEXA robot is suitable for very fast motion. From the perspective of mobilities, the FUNAC M-2iA/3A robot is a 6-DOF manipulator. On the other hand, the floating actuators mounted in the robot linkages will decrease the dynamic performance due to the increased inertia in the highspeed motion. In light of the stability problem cause by the telescopic limb and high dynamic inertia, a 6-DOF robot, consisting of two Delta robots was proposed with

1.3 Parallel PnP Robots

13

Fig. 1.12 The HEXA robot with reconfigurable platforms producing double Schönflies motions Fig. 1.13 6-DOF double Delta PnP robot

a similar wrist mechanism [280], where the three rotations are realized through the differential motions of the two robots’ mobile platforms, as displayed in Fig. 1.13. Besides the foregoing parallel PnP robots, a number of robot architectures towards PnP operations were proposed in the industry and academia. Therefore, it is impos-

14

1 Introduction

sible to show in this book all possible architectures of parallel PnP robots. As no uniform classification methods have been proposed, parallel PnP robots can be grouped as a function of the number and/or types of the DOF of the robot to carry the endeffector.4 Some examples are given below: • Classification by Robot DOFs 1. 2-DOF: Lexium Delta 2, IRSBot-2 [43] 2. 3-DOF: ABB SCARA-Tau, Delta-CU (Fig. 1.11a) 3. 4-DOF: ABB FlexPicker, FANUC M-2iA/3S, H4 (Fig. 1.6), Omron Quattro (Fig. 1.7), Penta Veloce. (Fig. 1.9), four-limb SMGs (Fig. 1.10), translational Ragnar robot (Fig. 1.11b) 4. 5-DOF: 5-DOF parallel robot with configurable platforms [158] 5. 6-DOF: FUNAC M-2iA/3A, HEXA [212], double Delta robot (Fig. 1.13), HEXA based robot with reconfigurable platforms (Fig. 1.12) • Classification by End-effector Mobilities 1. Planar Motions of the End-effector: Lexium Delta 2, IRSBot-2 2. Spatial Motions of the End-effector: the previous 3-, 4-, 5- and 6-DOF robots • Redundant Robots: translational Ragnar robot (Fig. 1.11b) and among others [18, 74, 236] It should be noted that not all the proposed parallel PnP robots can be used for the practical applications, despite the numerous types of robots presented through type synthesis of parallel mechanisms [103, 151, 306]. In the present work, the focus will be the parallel PnP robots of potential in industrial applications.

1.4 Objective and Organization of This Book As a major objective, this book aims at presenting the architectural and structural design of parallel PnP robots that have the potential of industrial applications, from parametric modeling to design optimization. In addition, some performance evaluation indices will be revisited and comparative studies among the robot counterparts will be carried out. This book includes seven aspects of importance for the parallel PnP robots. Chapter 2 revisits the fundamentals of mechanism kinematics, for instance, the homogeneous transformation, rotation matrix, and the representations of the points, vectors and transformation between them. The kinematic problems of some representative robots, including the inverse/forward geometry and workspace, are illustrated symbolically and numerically, where the kinematic constraint equations, forward/inverse geometry problems, the workspace searching methods are described. 4 Throughout

this book, the end-effector is referred to the moving platform, without additional devices attached to the former.

1.4 Objective and Organization of This Book

15

Chapter 3 presents the differential kinematics of parallel manipulators, referring to the mapping of the velocities/accelerations between the motion variables in the joint space and those in the Cartesian space. An issue of importance in the field of robotic manipulators, known as Jacobian matrix, is introduced from the first-order differential kinematics that relates the joint rates and end-effector velocities. The Hessian matrix, together with Jacobian matrix, to link the accelerations and velocities of the endeffector and joint rates, is derived from the second-order differential kinematics. Chapter 4 presents the kinematic performances and the corresponding measurement indices, such as Jacobian matrices, dexterity, singularity and motion/force transmission. The first/second-order kinematic models of the parallel robots are investigated, to derive the kinematic Jacobian matrix. Examples on the dexterous workspace of representative robots are numerically illustrated as well as the identified singular configurations. Motion/force transmission performance is analyzed and comparative study among the robot counterparts is carried out. Chapter 5 deals with the analytical elastostatic modeling and stiffness characterization. Detailed stiffness modeling method of virtual joint method for parallel robot is depicted. Stiffness indices based on the stiffness matrix to evaluate the robot performance are presented in local and global manners. Illustrative examples of parallel PnP robots are studied to show the stiffness modeling procedure in detail and the application of the stiffness indices for the evaluation of robots. Chapter 6 details the robot dynamics for the rigid parallel robots to calculate the motor torque/power/speed and robot acceleration. Three main dynamic modeling approaches, i.e., the Lagrangian formulation, the Newton–Euler equations and the principle of virtual work, are introduced. Different approaches are applied to several four-limb parallel PnP robots as illustrative examples for the illustration of motor selection and robot evaluation, where the computer-aided modeling approach was adopted for comparison to show their effectiveness Chapter 7 presents elastodynamics of parallel manipulators. Mass/inertia matrix of flexible body is briefly introduced from the structural dynamics. Lagrangian based approach for elastodynamic modeling is presented either in high-dimensional generalized coordinate space or Cartesian space of robot end-effector. Inertia and stiffness matrices of the robots are calculated to find the natural frequencies, and the performances of the same robot subject to different models in different dimensions are compared. Chapter 8 presents the design optimization of parallel PnP robots, where the design criteria, optimization algorithm and mathematic formulation are introduced. Design criteria, including the kinematic and kinetostatic/dynamic aspects are introduced. Numerical example on the multiobjective design optimization of parallel robot is illustrated, through which the Pareto-fronts for the proposed optimization problems are obtained as well as the scatter matrices for both objective functions and variables.

Chapter 2

Kinematic Geometry of the PnP Robots

Abstract Kinematic geometry is the fundamentals in the modeling and programming of a robot. In this chapter, rigid body kinematics, purely geometrical aspects of individual positions and of continuous motions of rigid bodies, are studied. We firstly introduce some basic material, e.g., the homogeneous transformation, rotation matrix, and the representations of the points, vectors and transformation between them. Several parallel PnP robots with different mobilities are illustrated as examples. The kinematic workspace envelope, namely, the set of positions that a robot can reach, is studied with either numerical searching or geometric approaches.

2.1 Position and Orientation Representation Spatially, combination of position and/or orientation is considered as the representation of a body’s pose [198]. Translations and rotations, referred to in combination as rigid-body displacements in robotics, are also expressed with position and orientation representation. Currently, these notations has been extensively used in robotics.

2.1.1 Position and Translation Figure 2.1 depicts two coordinates systems (F j can be viewed as reference frame). The position of the origin of coordinate frame Fi relative to coordinate frame F j can be represented by the 3 × 1 vector as follow ⎡j ⎤ xi j pi = ⎣ j yi ⎦ j zi

(2.1)

The three components in the vector (2.1) stand for the Cartesian coordinates of point Oi in frame F j , which are the projects of the vector j pi onto the three © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2021 G. Wu and H. Shen, Parallel PnP Robots, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-6671-4_2

17

18

2 Kinematic Geometry of the PnP Robots

Fig. 2.1 The translation between a reference and a moving coordinate frames

independent coordinate axes. The vector components can also be represented in cylindrical coordinates of Oi in F j below ⎡ ⎤ r cos θ j pi = ⎣ r sin θ ⎦ j zi

(2.2)

with the radial distance r from the z j -axis to the point Oi and the azimuth angle θ between the projection of segment O j Oi in the x y plane and x-axis, respectively, expressed by r=



( j xi )2 + ( j yi )2

θ = tan−1

j

yi

jx

(2.3a) (2.3b)

i

or in spherical coordinates ⎡

⎤ ρ sin ϕ cos θ j pi = ⎣ ρ sin ϕ sin θ ⎦ ρ cos ϕ

(2.4)

2.1 Position and Orientation Representation

19

with ρ=

 ( j xi )2 + ( j yi )2 + ( j z i )2

ϕ = cos−1 θ = tan−1

(2.5a)

j

zi ρ j yi jx i

(2.5b) (2.5c)

where ρ is the radius from the origin O j to the point Oi , ϕ is the polar angle between z-axis and O j Oi , and θ is the azimuth angle between the projection of segment O j Oi in the x y plane and x-axis, respectively. A translation is a displacement that a rigid body shifts from one position to another one with all the straight lines in the rigid body parallel to their initial orientations. The translation of a body in space can be represented by the combination of its positions prior to and following the translation [243]. Conversely, the position of a body can be represented as a translation that takes the body from an initial position1 to the current position.2

2.1.2 Orientation and Rotation A rotation is an angular displacement in which at least one line in the rigid body is not parallel to their initial orientations without linear translation. For instance, the moving coordinate frame Fi shares the same origin with the reference frame F j , but with nonparallel coordinate axes (Fig. 2.2).

2.1.2.1

Rotation Matrix

The rotation matrix maps the orientation of the moving coordinate frame relative to the reference one [256]. For instance, the orientation of coordinate frame Fi relative to frame F j can be denoted by expressing the unit vectors (ˆxi , yˆ i , zˆ i ) in terms of the unit vectors (ˆx j , yˆ j , zˆ j ). Therefore, the components of the rotation matrix j Ri related frames Fi and F j are the dot products of the unit vectors of the two coordinate frames, namely, ⎡ ⎤ xˆ i · xˆ j yˆ i · xˆ j zˆ i · xˆ j j Ri = ⎣xˆ i · yˆ j yˆ i · yˆ j zˆ i · yˆ j ⎦ (2.6) xˆ i · zˆ j yˆ i · zˆ j zˆ i · zˆ j

1 The inherent coordinate frame attached to the body coincides with the reference coordinate frame. 2 The

two frames are not coincident but with all the coordinate axes parallel.

20

2 Kinematic Geometry of the PnP Robots

Fig. 2.2 The rotation between a reference and a moving coordinate frames

which means that each component is the cosine of the angle between any two axes [20], referring to as directional cosines.

2.1.2.2

Basic Rotations

A basic rotation, also called elemental rotation, is a rotation about one of the axes of a coordinate system [193], e.g., the three basic rotation matrices of frame Fi about the x-, y-, or z-axis: ⎡

⎤ 1 0 0 Rx (θ ) = ⎣0 cos θ − sin θ ⎦ 0 sin θ cos θ ⎡ ⎤ cos θ 0 sin θ R y (θ ) = ⎣ 0 1 0 ⎦ − sin θ 0 cos θ ⎡ ⎤ cos θ − sin θ 0 Rz (θ ) = ⎣ sin θ cos θ 0⎦ 0 0 1

2.1.2.3

(2.7a)

(2.7b)

(2.7c)

Euler Angles

The general orientation of coordinate frame Fi relative to coordinate frame F j can be denoted as a vector of three angles (α, β, γ ), which are known as Euler angles when each represents a rotation about an axis of a moving coordinate frame [114].

2.1 Position and Orientation Representation

21

Fig. 2.3 The successive intrinsic rotations that define the z–y–x Euler angles

The corresponding rotation matrix can be derived using matrix multiplication [85], such as ⎡ ⎤ cβ cγ cγ sα sβ − cα sγ sα sγ + cα cγ sβ j Ri = Rz (α)R y (β)Rx (γ ) = ⎣cβ sγ cα cγ + sα sβ sγ cα sβ sγ − cγ sα ⎦ (2.8) −sβ cβ sα cα cβ which represents an intrinsic rotation whose Tait–Bryan angles [94] are α, β, γ , about axes z, y, x, respectively, as displayed in Fig. 2.3. Moreover, c(·) := cos(·) and s(·) := sin(·). Euler angles can also be defined by extrinsic rotations, in which the elemental rotations occur about the axes of the reference coordinate system F j . Figure 2.4 depicts an extrinsic rotation whose Euler angles are α, β, γ , about axes x, y, z, respectively, leading to the rotation matrix j

Ri = Rz (γ )R y (β)Rx (α)

(2.9)

Without consideration of using two different conventions for the definition of the intrinsic or extrinsic rotation axes, there exist twelve possible sequences of rotation axes, grouped by: • Proper Euler angles: z–x–z, x–y–x, y–z–y, z–y–z, x–z–x, y–x–y • Tait-Bryan angles: x–y–z, y–z–x, z–x–y, x–z–y, z–y–x, y–x–z The sequences of the first group are called proper or classic Euler angles. The latter ones are also called Cardan angles, which can be used to describe the orientation of a

22

2 Kinematic Geometry of the PnP Robots

Fig. 2.4 The successive extrinsic rotations that define the z–y–x Euler angles

ship or aircraft, therefore, they are sometimes called yaw, pitch and roll (RPY angle) in aerospace. It is noteworthy that this will not work if the rotations are applied in any other order if the airplane axes start in any position without the initial alignment to the reference frame.

2.1.2.4

Azimuth–Tilt–Torsion Angles

Apart from the orientation representations such as Euler angles, the Tilt-and-Torsion (T&T) angles take full advantage of a mechanism’s symmetry [35]. As shown in Fig. 2.5, the rotation matrix is kinematically equivalent to the Euler convention z–y– z, namely, ⎤ cφ cθ cφ−σ + sφ sφ−σ cφ cθ sφ−σ − cφ cφ−σ cφ sθ j Ri = Rz (φ)R y (θ )Rz (σ − φ) = ⎣sφ cθ cφ−σ − sφ sφ−σ sφ cθ sφ−σ + cφ cφ−σ sφ sθ ⎦ −sθ cφ−σ −sθ sφ−σ cθ (2.10) where φ ∈ (−π, π ], θ ∈ [0, π ) and σ ∈ (−π, π ]. ⎡

2.1.2.5

Axis–Angle Representation

The pure rotation of angle θ around an arbitrary axis u parameterized by the unit T  vector u = u x u y u z can be represented by [262]:

2.1 Position and Orientation Representation

23

Fig. 2.5 The Azimuth–Tilt–Torsion Angles

⎡

⎤ cθ + u 2x (1 − cθ ) u x u y (1 − cθ ) − u z sθ u x u z (1 − cθ ) + u y sθ Ru (θ ) = ⎣u x u y (1 − cθ ) + u z sθ cθ + u 2y (1 − cθ ) u y u z (1 − cθ ) − u x sθ ⎦ u x u z (1 − cθ ) − u y sθ u y u z (1 − cθ ) + u x sθ cθ + u 2z (1 − cθ ) (2.11) which can be written more concisely [13] as Ru (θ ) = cos θ I3 + sin θ [u]× + (1 − cos θ )u ⊗ u

(2.12)

where I3 3 is the 3-dimensional identity matrix, and [u]× is the cross-product matrix (CPM)4 of u, and the expression u ⊗ u is the tensor product, namely, ⎡

⎤ 0 −u z u y [u]× = ⎣ u z 0 −u x ⎦ −u y u x 0 ⎡ 2 ⎤ ux ux u y ux uz u ⊗ u = ⎣ ... u 2y u y u z ⎦ sym. ... u 2z

3 Throughout

(2.13a)

(2.13b)

this book, I N stands for a N -dimensional identity matrix, and O N stands for a N dimensional null matrix, while qn stands for a n-dimensional column vector. 4 Capital CPM stands for cross-product matrix, differing from cpm (cycles per minute) in lowercase.

24

2 Kinematic Geometry of the PnP Robots

2.1.2.6

Euler Parameters and Quaternions

The quaternions [116, 240] are referred to Euler parameters or Olinde-Rodrigues parameters [246], which do not suffer from singularities as Euler angles. This is a different way to parameterize the rotation of an angle θ about an axis u. The orientation is represented by four parameters defined as: θ e0 = cos ⎡ ⎤2 ⎡ ⎤ e1 ux θ θ e = ⎣e2 ⎦ = sin ⎣u y ⎦ = u sin 2 u 2 e3 z

(2.14a) (2.14b)

where vector e is along the axis of rotation u having a magnitude of sin θ2 , as depicted in Fig. 2.6. The four Euler parameters are not independent constrained by the following equation: (2.15) e02 + eT e = e02 + e12 + e22 + e32 = 1 The rotation matrix is expressed as ⎡

⎤ e02 + e12 − 21 e1 e2 − e0 e3 e1 e3 − e0 e2 j Ri = Ru (θ ) = 2 ⎣ e1 e2 − e0 e3 e02 + e22 − 21 e2 e3 − e0 e1 ⎦ e1 e3 − e0 e2 e2 e3 − e0 e1 e02 + e32 − 21 which can be written in a compact form below

Fig. 2.6 The rotation matrix from axis-angle representation

(2.16)

2.1 Position and Orientation Representation

25

Ru (θ ) = (2e02 − 1)I3 + 2(eeT + e0 e) 2.1.2.7

(2.17)

Properties of Rotation Matrices

In the study of robot kinematics, the following properties of a rotation matrix [60] are commonly used: • Property 1 The rotation matrix is orthogonal and its determinant is equal to one. • Property 2 The inverse of the rotation matrix R is equal to its transpose: R−1 = R T

(2.18)

where the superscript “T ” indicates the transpose of a matrix. • Property 3 The dot product of any row or column with itself is equal to one. • Property 4 The dot product of any row (column) with any other row (column) is equal to zero.

2.2 Homogeneous Transformations Homogeneous transformations [194] couples the preceding position vectors and rotation matrices in a compact notation. An arbitrary vector ri in the coordinate frame Fi can be expressed in the coordinate frame F j if the position and orientation of the frame Fi relative to F j are known, as depicted in Fig. 2.7. The position of the origin of coordinate frame Fi relative to coordinate frame F j can be denoted by the vector j pi , and the orientation of frame Fi relative to frame F j can be denoted by the rotation matrix j Ri . Sequentially, one obtains r j = j Ri ri + j pi

(2.19)

which can be cast in a matrix form, namely, j



Ri j pi ri rj = 1 03T 1 1 Let j

Ti =

j

Ri j pi 03T 1

(2.20)

(2.21)

here, j Ti is the 4 × 4 homogeneous transformation matrix, transforming vectors T   T from frame Fi to frame F j . Moreover, r Tj 1 and riT 1 are the homogeneous representations of the position vectors r j and ri , respectively.

26

2 Kinematic Geometry of the PnP Robots

Fig. 2.7 Transformation of a vector

2.2.1 Elementary Transformation Matrices The homogeneous transformation of a pure translation along an axis is sometimes denoted by Trans, say a translation of d along an axis zˆ [76]: ⎡

1 ⎢0 Trans(ˆz, d) = ⎢ ⎣0 0

0 1 0 0

0 0 1 0

⎤ 0 0⎥ ⎥ d⎦ 1

(2.22)

Similarly, the homogeneous transformation of a pure rotation about an axis is sometimes denoted by Rot, such as a rotation of θ about an axis xˆ : ⎡

1 0 0 ⎢0 cos θ − sin θ Rot(ˆx, θ ) = ⎢ ⎣0 sin θ cos θ 0 0 0

⎤ 0 0⎥ ⎥ 0⎦ 1

(2.23)

2.2.2 Properties of Transformation Matrices Before going further, the following properties of the homogeneous transformation matrices [45] are defined:

2.2 Homogeneous Transformations

27

• Property 1 A transformation matrix can be written as

j

Ti =

j

Ri j pi 03T 1

⎡

ux ⎢u y =⎢ ⎣u z 0



vx vy vz 0

wx wy wz 0

⎤ px   py ⎥ ⎥ = u v w j pi pz ⎦ 1

(2.24)

  where the matrix j Ri = u v w and vector j pi represent the rotation and translation, respectively. For a transformation of pure translation, j Ri = I3 , and j pi = 0 for a transformation of pure rotation. • Property 2 The inverse of matrix j Ti , denoted by j Ti−1 , transforms vectors from frame F j to frame Fi , namely, j

Ti−1

= Tj = i

j

RiT − j RiT j pi 03T 1

(2.25)

• Property 3 The inverse of the elementary transformation matrices along or about an arbitrary axis u is written as Trans−1 (u, d) = Trans(u, −d) = Trans(−u, d) −1

Rot (u, θ ) = Rot(u, −θ ) = Rot(−u, θ )

(2.26a) (2.26b)

• Property 4 The multiplication of two transformation matrices results in a transformation matrix:



k j k j

k R j k p j j Ri j pi R j Ri R j pi + k p j k k j = (2.27) Ti = T j Ti = 03T 1 03T 1 03T 1 wherein the sequence is important, since matrix multiplications do not commute, thus, k T j j Ti = j Ti k T j in most cases. • Property 5 If a frame F j is achieved after j consecutive transformations from the initial frame F0 , and each transformation i (i = 1, . . . , j) is defined with respect to the current frame Fi−1 , the homogeneous transformation matrix is calculated through the sequential multiplications: 0

T j = T1 T2 · · · 0

1

j−2

T j−1

j−1

Tj =

j

i=1

i−1

Ti

(2.28)

28

2 Kinematic Geometry of the PnP Robots

Fig. 2.8 Planar displacements

2.3 Planar Displacements The set of planar displacements is a special case of the rigid-body motion, where the rotation is normal to the plane that the linear displacements are confined in. Figure 2.8 depicts the coordinate systems of planar displacements, from which the affine map is defined as: j



xi xj cos θ − sin θ xi = + (2.29) j yi yj yi sin θ cos θ    jR i

which can be rewritten as homogeneous (linear) map: ⎡

⎤ ⎡ ⎤⎡ ⎤ 1 1 0 0 1 ⎣ j xi ⎦ = ⎣x j cos θ − sin θ ⎦ ⎣xi ⎦ j yi yi y j sin θ cos θ   

(2.30)

jT i

2.4 Kinematic Linkages A robotic manipulator is composed of a set of links connected by joints. The joints can simply be either a revolute (R) or a prismatic (P) joint, or they can be more complex, such as a universal or spherical joint. Throughout this book, all the joints between any two links are supposed to have one degree of freedom. This assumption does not lose any generality, as joints such as a universal (two degrees-of-freedom)

2.4 Kinematic Linkages

29

or a spherical (three degrees-of-freedom) joint can viewed as a succession of singledegree-of-freedom revolute joints with zero-length links in between. A kinematic linkage with n joints has n + 1 links, since each joint connects two links. In this linkage, the joints can be numbered from 1 to n, while the links will be numbered from 0 to n, the link 0 is fixed to the base. With this convention, joint j connects link j − 1 to link j, and joint j is fixed to link j − 1. The actuated joint j drives link j. In the case that the robotic manipulator is amounted on a mobile platform, the mobile articulated robots are not considered. To depict the robot kinematics [223], the jth joint is associated with a variable q j . In the case of R joint, q j is parameterized by θ j , and in the case of P joint, q j is parameterized by d j :  qj =

θ j , if joint j is a R joint d j , if joint j is a P joint

(2.31)

With the position/orientation representations of the rigid bodies and the concept of the homogeneous transformations as well as the joint variables, geometric models can be built to depict the robot (or mechanism, or linkage) configurations or kinematic architectures with respect to the link dimensions and architectural parameters, which will be introduced in the following section.

2.5 Geometry of Kinematic Linkages Throughout this book, the term “geometry” is used to depict the models defining the configurations of the robot or mechanism or linkage, whereas the term “kinematics” extends to the velocities/accelerations together with the geometric models (closedform solutions). This issue involves two aspects, i.e., the forward (or direct) and inverse geometry.

2.5.1 Forward Geometry: Denavit–Hartenberg Convention The forward geometry problem is concerned with the mapping between the known individual joint variables of the robotic manipulator and the unknown position and orientation of the tool or end-effector. Unified and systematic convention for selecting frames of reference in robotic applications is required to simplify the analysis procedure. In robot engineering, a commonly used convention is the Denavit–Hartenberg, or DH convention [120]. In this convention, each homogeneous transformation j−1 A j is represented as a product of four basic transformations:

30

2 Kinematic Geometry of the PnP Robots

Fig. 2.9 The DH convention and their parameters

j−1

A j = Rot(ˆz j−1 , θ j )Trans(ˆz j−1 , d j )Trans(ˆx j , a j )Rot(ˆx j , α j ) ⎡ ⎤⎡ ⎤ ⎤⎡ ⎤⎡ cos θ j − sin θ j 0 0 1 0 0 aj 100 0 1 0 0 0 ⎢ sin θ j cos θ j 0 0⎥ ⎢0 1 0 0 ⎥ ⎢0 1 0 0 ⎥ ⎢0 cos α j − sin α j 0⎥ ⎥⎢ ⎥ ⎥⎢ ⎥⎢ =⎢ ⎣ 0 0 1 0⎦ ⎣0 0 1 d j ⎦ ⎣0 0 1 0 ⎦ ⎣0 sin α j cos α j 0⎦ 000 1 0 0 0 1 0 0 01 000 1 ⎡ ⎤ cθ j −sθ j cα j sθ j sα j ai cθ j ⎢sθ j cθ j cα j −cθ j sα j a j sθ j ⎥ ⎥ =⎢ (2.32) ⎣0 sα j cα j dj ⎦ 0 0 0 1

where the four parameters a j , α j , d j , and θ j are generally named as link length, link twist, link offset, and joint angle, respectively, as shown in Fig. 2.9, depicted as: • • • •

θ j : the rotation angle between the axes of x j−1 and x j about the z j−1 -axis d j : the translation along the z j−1 -axis a j : the translation along the x j -axis α j : the rotation angle between the axes of z j−1 and z j about the x j -axis

2.5.2 Inverse Geometry The forward geometry refers to the determination of the end-effector position and orientation in terms of the joint variables. Conversely, the inverse geometry is con-

2.5 Geometry of Kinematic Linkages

31

cerned with the findings of the joint variables in terms of the end-effector position and orientation. The most parallel mechanisms are composed of simple linkages that do not require the use of complex methods for solving their inverse geometric models. Suppose that a parallel mechanism has n identical kinematic linkages consisting of j joints in-parallel, with known end-effector pose χ (a vector of combination of position and orientation) with respect to the fixed frame F0 , the loop-closure equation for the ith linkage can be established: f i (χ, qi , vi ) = 0, i = 1, 2, . . . , n

(2.33)

in which there exists only one unknown qi generally, vi being a vector of the geometric parameters of the robot linkage. The inverse geometry problem in most cases, namely, the joint variables, qi , can be solved from the foregoing equation (also called kinematic constraint equation):  qj =

2tan−1 ( f i−1 (χ, vi )), for an actuated revolute joint f i−1 (χ, vi ), for an actuated prismatic joint

(2.34)

where the Trigonometric half angle are usually employed when solving the joint variables in revolute joints, namely, sin θ =

1 − t2 θ 2t , cos θ = , t = tan 2 2 1+t 1+t 2

(2.35)

Finding the solution of the inverse geometric model of an open kinematic linkage, i.e., the leg configuration, is of importance for the study of parallel mechanism whose linkages connecting the end-effector (mobile platform) to the base.

2.6 Illustrative Examples In this section, the robot geometry problems, i.e., closed-from solution and workspace, are illustrated with three examples, e.g., a planar five-bar mechanism, a redundantly actuated translational robot, an asymmetric four-limb SMG, and a double Delta based robot producing three translations and three rotations (3T3R).

2.6.1 Planar Five-Bar Mechanism Figure 2.10 depicts the parametrization of a general five-bar mechanism [263]. The reference coordinate frame is built on the fixed frame with the origin located at the center of the segment AC = 2l0 , and the X -axis is parallel to the latter. The Cartesian

32

2 Kinematic Geometry of the PnP Robots

Fig. 2.10 Parameterization of a general five-bar mechanism

 T coordinates of the end-effector P in reference frame is denoted by p = x y . Moreover, AB = l1 , B P = l2 , C D = l3 and P D = l4 .

2.6.1.1

Inverse Geometry

The Cartesian coordinates of point P can be derived from the following loop-closure equation: p = l0 ui + l2i−1 vi + l2i wi , i = 1, 2 (2.36)

with ui =





cos(iπ ) cos θi cos(θi + ψi ) , wi = , vi = sin θi sin(θi + ψi ) sin(iπ )

(2.37)

Equation (2.36) establishes a system of four equations, from which the inverse geometry problem can be solved, namely,

θi = 2 tan

−1

−Ii ±



Ii2 + Ji2 − K i2

K i − Ji

, i = 1, 2

(2.38)

2.6 Illustrative Examples

33 Solution 2

Solution 1 100

100

50

50

0 −50

0 0

50

−50

100

0

Solution 3

50

100

Solution 4

120 100

100

80 60

50

40 20 0

−50

0

0

50

−50

0

50

Fig. 2.11 The four solutions of the inverse geometry of the five-bar mechanism

with Ii = 2yl2i−1

(2.39a)

Ji = 2l2i−1 [x − (−1) l0 ] i

Ki =

l2i2

2 − l2i−1

(2.39b)

− y − [x − (−1) l0 ] 2

i

2

(2.39c)

Equation (2.38) can have up to four solutions, corresponding to the four working modes (W M s) of the mechanism in a prescribed configuration, which are characterized by the sign +/−, i.e., working mode “++”, “+−”, “−+”, and “−−”. The four working modes associates with the inverse geometry, with the geometric parameters li = 100 mm, i = 1, 2, 3, 4, are illustrated in Fig. 2.11.

2.6.1.2

Forward Geometry

The Cartesian coordinates of points B and D can be deduced with the known input angles θ1 and θ2 : b=



l + l3 cos θ2 −l0 + l1 cos θ1 ,d= 0 l1 sin θ1 l1 sin θ2

(2.40)

34

2 Kinematic Geometry of the PnP Robots

Sequentially, the position vector of segment B D is expressed as

−→ 2l0 + l3 cos θ2 − l1 cos θ1 BD = d − b = l3 sin θ2 − l1 sin θ1

(2.41)

and the corresponding unit vector e is written as

−→ BD ex = ey |B D|

(2.42)

(2l0 + l3 cos θ2 − l1 cos θ1 )2 + (l3 sin θ2 − l1 sin θ1 )2

(2.43)

e= with |B D| =



Consequently, the angle ∠B P D is expressed as ∠B P D = cos−1

l22 + l42 − |B D|2 2l2 l4

(2.44)

As a result, the two solutions of the forward geometry problem are  x = −l0 + l1 cos θ1 + h 1 ex + l42 − h 22 e y  y = l1 sin θ1 + h 1 e y + l42 − h 22 ex

(2.45a) (2.45b)

or  x = −l0 + l1 cos θ1 + h 1 ex − l42 − h 22 e y  y = l1 sin θ1 + h 1 e y − l42 − h 22 ex with h1 =

|B D|2 + l22 − l42 |B D|2 + l42 − l22 , h2 = 2|B D| 2|B D|

(2.46a) (2.46b)

(2.47)

The previous two solutions of the forward geometry problem, as displayed in Fig. 2.12, correspond to the two assembly modes, with the same link lengths in foregoing section.

2.6.1.3

Reachable Workspace

In general, the workspace of a robotic manipulator is the entire spatial volume that the robot end-effector can reach to in all the possible configurations. The workspace

2.6 Illustrative Examples Fig. 2.12 The two solutions of the forward geometry problem of the five-bar mechanism

35 solution 1 160 140 120 100 80 60 40 20 0 −50

0

50

solution 2 100

shape and size depend on the geometric architecture of the manipulator and the limits of the joint motions. As a matter of fact, in the kinematic considerations of design of parallel robots, the quality of the workspace that reflects the shape, size, dexterity and presence of singularities is of primary importance. The workspace with desired kinematic performances will be discussed in the section of kinematic performance evaluation. Here, the reachable workspace is the focus. Workspace of the robot with mixed mobilities of end-effector, i.e., combination of translations and rotations, is usually uneasy to intuitively depict graphically. An alternative approach is to find the workspace envelope with constant orientations. The common method to find the workspace of the parallel robots includes the discrete numerical searching method and geometric approach. • Discrete numerical searching method This method is usually applied to roughly assess the robot by qualifying the workspace size compared to the robot space. The general procedure is to check if the inverse solution corresponding to a workspace point can meet the kinematic constraints (inverse solution should be a real number and is subject to the motion range of joints) or not one by one, where the workspace points are discretized in a reasonable space according to the link dimensions. The general algorithm is depicted below: In this approach, the accurate workspace boundaries depends on the interval step size. The smaller the step size is, the more accurate the calculation is. On the other hand, it will introduce the higher computational cost when the robotic manipulator has a large workspace. Figure 2.13 shows the workspace of the five-bar mechanism in a working mode, where the discrete searching method is deployed. It can be seen that it is difficult to get a solid workspace area with the consideration of the less

36

2 Kinematic Geometry of the PnP Robots

Algorithm 1 Algorithm of discrete numerical searching method 1: procedure SEARCH(q, x, y, z)  q = (q1 , q2 , . . . , qi , . . .) the actuated joints 2: for 3: x = xmin : Δx : xmax  discrete x-, y-, z-coordinate 4: y = ymin : Δy : ymax 5: z = z min : Δz : z max do 6: qi = f −1 (x, y, z, v)  solving the inverse geometry problem 7: while isreal(qi ) & qmin ≤ qi ≤ qmax do  check the inverse solution 8: plot (x, y, z) 9: end while 10: continue  calculate and check the next workspace point 11: end for 12: end procedure Fig. 2.13 The reachable workspace of the planar five-bar mechanism with links P B and P D up in unit length of li = 1, i = 1, 2, 3, 4

2 1.8 1.6 1.4

y

1.2 1 0.8 0.6 0.4 0.2 0 −1

−0.5

0 x

0.5

1

time consumption. This numerical approach is similar to the Monte Carlo method in finding the workspace. • Geometric approach The geometric approach to find the workspace is based on the Boolean operation by making use of the maximum reachable workspace volume of the single kinematic linkage. The procedure of the geometric approach, as displayed in Fig. 2.14, to find the workspace of the five-bar mechanism is described as follow, which can be generalized to other parallel mechanisms: 1. Find the circles centered at points A and C with the radius of rmax = l1 + l2 , note their areas as S A and SC , respectively; 2. Use the Boolean’s Intersection to find the common area S AC (blue area) between S A and SC ; 3. Find the circles centered at points A and C with the radius of rmin = |l2 − l1 |, note their areas as S A and SC , respectively;

2.6 Illustrative Examples

37

Fig. 2.14 The geometric approach to find the reachable workspace of the five-bar mechanism

4. Use the Boolean’s Subtraction to find the remaining area S by removing S A and SC from S AC . To this end, the region enveloped by the solid black curves is the reachable workspace of the five-bar mechanism. The geometric approach can be used to find the solid and continuous workspace volume of the robotic manipulator. Moreover, the reachable workspace surfaces can also be depicted with the mathematic models in terms of regular geometry, after the operations of the kinematic constraint equation.

2.6.2 Redundantly Actuated Translational Robot The redundantly actuated translational robot as displayed in Fig. 1.11b is with four limbs composed of four identical RR RR linkages to connect the base and the mobile platform. Different from its symmetrical counterparts, such as Quattro and Penta Veloce. robots, the motors of this robot are mounted at different orientations on the base frame, of which the four axes of rotation of the actuated revolute joints are not coplanar. The robot parameterization is depicted in Fig. 2.15. The global coordinate frame Fb is built with the origin located at the geometric center of the base frame, the x-axis being parallel to segment A2 A1 (A3 A4 ). The moving coordinate frame F p is attached to the mobile platform and the origin is at the geometric center, with the frame axes parallel to those of frame Fb . Throughout this book, vectors i, j and k represent the unit vectors of x-, y- and z-axis, respectively. The orientation of the ith

38

2 Kinematic Geometry of the PnP Robots

Fig. 2.15 Parameterization of the redundant translational robot with four limbs

motor is described by its yaw angle αi and pitch angle βi , which means ei = Ei k, Ei = R(z, αi )R(y, βi ), and −α1(3) = α2(4) = α, −β1(4) = β2(3) = β. Under the coordinate systems of Fig. 2.15, the position vectors of point Ai in frame Fb are denoted by T T   a1 = −a3 = ax a y 0 ; a2 = −a4 = −ax a y 0

(2.48)

and the position vector of point Bi is derived as bi = bhi + ai ; hi = Ri i, i = 1, . . . , 4

(2.49)

where Ri = Ei R(z, θi ), and θi is the angle of rotation of motor from a reference position.  T Let the mobile platform position be denoted p = x y z , the Cartesian coordinates of point Ci is expressed asci = ci + p, where ci is the position vector of Ci in the frame F p : T  ci = r cos γi sin γi 0 ; γ1 = −γ4 = γ , γ2 = −γ3 = π/2 + γ 2.6.2.1

(2.50)

Inverse Geometry Problem

The kinematic constraint equation of the ith limb of the robot is expressed as ci − bi 2 = l 2 , i = 1, . . . , 4

(2.51)

2.6 Illustrative Examples

39

Expanding and simplifying the above equation leads to Ii sin θi + Ji cos θi + K i = 0

(2.52)

with Ii = 2b(ai − ci )T Ei j, Ji = 2b(ai − ci )T Ei i, K i = (ai − ci ) 2 + b2 − l 2 (2.53) Finally, the inverse geometry problem is solved as θi = 2 tan−1

−Ii ±



Ii2 + Ji2 − K i2

K i − Ji

(2.54)

From Eq. (2.54), it is seen that each limb has two solutions corresponding to two working modes, which is characterized by the sign “−/+” in the equation. Here, the “− + −+” mode is the best selection from the perspective of application.

2.6.2.2

Forward Geometry Problem

Expanding Eq. (2.51), four implicit functions can be derived below: Fi = (x + u i )2 + (y + vi )2 + (z + wi )2 − l 2 = 0

(2.55)

with u i = −ai x + r cos γi + bi (sin αi sin θi − cos αi cos βi cos θi ) vi = −ai y + r sin γi − bi (cos αi sin θi + cos αi cos βi cos θi )

(2.56a) (2.56b)

wi = bi sin βi cos θi

(2.56c)

As this translational robot is redundantly actuated, the first three equations, i.e., F1 , F2 , F3 , are adopted to solve the forward kinematic problem. Making use of F1 − F2 and F1 − F3 , one obtains M1 x + N1 y + P1 z + Q 1 = 0 M2 x + N2 y + P2 z + Q 2 = 0

(2.57a) (2.57b)

with M1 = 2u 1 − 2u 2 ; N1 = 2v1 − 2v2 ; P1 = 2w1 − 2w2 ; Q 1 = u 21 − u 22 + v12 − v22 + w12 − w22

(2.58a)

M2 = 2u 1 − 2u 3 ; N2 = 2v1 − 2v3 ; P2 = 2w1 − 2w3 ; Q 2 = u 21 − u 23 + v12 − v32 + w12 − w32

(2.58b)

40

2 Kinematic Geometry of the PnP Robots

As this robot has asymmetric structure and x, y cannot reach to zeros simultaneously, hence, the FKP can be solve by using x, y to represent z first. Equations (2.57a) and (2.57b) can be cast in matrix form, namely,

M1 N 1 M2 N 2







x x P1 z + Q 1 =D =− P2 z + Q 2 y y

(2.59)

The determinant value of matrix D is not equal to zero, thus, using Cramer’s rule, the x- and y-coordinate can be solved as the functions of z, namely,

|Dx | −P1 z − Q 1 N1 ; Dx = −P2 z − Q 2 N2 |D|

|D y | M1 −P1 z − Q 1 ; Dy = y= M2 −P2 z − Q 2 |D|

x=

(2.60) (2.61)

subsequently, substituting x and y in to F1 in Eq. (3.45) leads to (|Dx | + |D|u 1 )2 + (|D y | + |D|v1 )2 + |D|2 (z + w1 )2 − |D|2 l 2 = 0

(2.62)

Expanding the above equation, a quadratic equation with respect to z is expressed as I z2 + J z + K = 0

(2.63)

with I = |S|2 + |T|2 + |D|2

(2.64a)

J = 2|T|(|D|u 1 − Q 1 N2 + Q 2 N1 ) − 2|S|(|D|v1 + Q 1 M2 − Q 2 M1 ) + 2|D|2 w1 (2.64b) K = (|D|u 1 − Q 1 N2 + Q 2 N1 )2 + (|D|v1 + Q 1 M2 − Q 2 M1 )2 − l 2 |D|2 + |D|2 w12 (2.64c)

where S=



N1 P1 M1 P1 ,T= M2 P2 N2 P2

(2.65)

To this end, the z-coordinate is solved as z=

−J ±

√

J 2 − 4I K 2I

consequently, x and y can be solved from Eqs. (2.60) and (2.61).

(2.66)

2.6 Illustrative Examples

41

Table 2.1 Geometrical parameters of the four-limb translational robot [ax , a y ] (mm) α β b (mm) l (mm) r (mm) [280, 114]

2.6.2.3

15o

45o

260

550

γ 30o

70

Evaluation of the FKP and IKP

The parameters of the robot are designated as listed in Table 2.1. To evaluate the forward and inverse geometry models, the IKP is solved firstly with given EE positions, subsequently, the corresponding joint angles of the IKP solutions are substituted into the FKP model to calculate the errors between them. For example,  T with position p = 120 450 −320 , the IKP solution is calculated as θ1 = 0.8865, θ2 = 1.5427, θ3 = 0.1950 and θ4 = 2.9368, then, substituting θ1 , θ2 and θ3 into the  T FKP model, the EE position is solved as p = 120.0000 450.0000 −320.0000 ,  T with the differences δp = 10−13 · 0.4263 −0.5684 0.5684 . To evaluate the FKP model effectively, 105 random workspace points, are used to compute forward kinematics. Figure 2.16 displays the differences between the forward and inverse kinematic problems, from which it is seen that the differences are extremely tiny to validate the FKP model. −10

δx

5

x 10

0 −5

0

2

4

6

8

10 4

x 10

−10

δy

5

x 10

0 −5

0

2

4

6

8

10 4

x 10

−8

δz

1

x 10

0 −1

0

2

4

6 point index

8

10 4

x 10

Fig. 2.16 Differences between the FKP and IKP models of the redundant translational robot

42

2 Kinematic Geometry of the PnP Robots

2.6.2.4

Reachable Workspace Display

The reachable workspace of the translational robot can be found by either discrete searching or geometric approach, while the latter is an efficient way to fully characterize the workspace. The actuated angles θi of the robot from the inverse geometry problem have to be real values, which requires to meet the following conditions from Eq. (2.54) [287]: (2.67) Ii2 + Ji2 − K i2 ≥ 0, i = 1, . . . , 4 Expanding the above equation yields 

(x − X i )2 + (y − Yi )2 + z 2 + b2 − l 2 )

2

≤ Δi , i = 1, . . . , 4

(2.68)

with Δi =4b2 [(x − X i ) cos αi cos βi + (y − Yi ) sin αi cos βi − z sin βi ]2

(2.69)

+ 4b2 [(x − X i ) sin αi − (y − Yi ) cos αi ]2 and X i = ai x − r cos γ , Yi = ai y − r sin γ

(2.70)

Equation (2.68) expressed the Cartesian coordinates of the robot end-effector in the equation of the four tori [2] with center located at point (X i , Yi , 0), axis  T parallel to the unit vector cos αi cos βi sin αi cos βi − sin βi , radius l of the tube, and distance b from the center of the tube to the center of each torus [7]. As a consequence, the reachable workspace will be the common volume of the four tori. Finally, the reachable workspace can be graphically visualized, e.g., the workspace, determined by the set of parameters in Table 2.1, displayed in Fig. 2.17. This figure depicts a graphical user interface (GUI), where the workspace with different sets of geometric parameters can be plotted.

2.6.3 Four-Limb SMG with a Single Platform The parallel Schönflies-motion robot [289] displayed in Fig. 2.18a has similar structure as the translational one in Fig. 1.11b. The difference lies that the SMG MP has four revolute joints with the axes of rotation about the vertical directions to generate an additional rotation. Figure 2.18 illustrates the kinematic structure of the ith limb with the global coordinate frame Fb same to Fig. 2.15. The moving coordinate frame F p is attached to the mobile platform and the origin is at the geometric center, where X -axis is parallel to segment C2 C1 (C3 C4 ).

2.6 Illustrative Examples

43

Fig. 2.17 Graphical user interface of the translational robot with its reachable workspace

The procedure of solving the robot’s inverse geometry refers to Sect. 2.6.2, but T  with different coordinates of point Ci . Here, the MP pose is denoted by χ = pT φ ,  T p = x y z , thus, the Cartesian coordinates of point Ci is expressed as ci = Qci + p

(2.71)

where Q = Rz (φ) is the rotation matrix of the MP and ci is the position vector of Ci in the frame F p : T  ci = r cos γi sin γi 0 ; γ1 = −γ4 = γ , γ2(3) = π ∓ γ

(2.72)

After a tedious operation, the inverse geometry of the robot has the same expression of Eq. (2.54).

2.6.3.1

Reachable and Regular Workspace

The workspace of this robot can be found by referring to Sect. 2.6.2.4. Here, CADbased geometric approach can also be adopted to find the workspace intuitively with the aid of some CAD software, for instance, SolidWorksTM . In practice, a workspace in the shape of regular geometry can better evaluate the robot performance, and is also beneficial to the applications and programming. With geometric parameters listed in Table 2.2, the reachable and regular workspace is visualized in Fig. 2.19a. By changing the motor position and orientation, the robot can have workspace of different shapes, such as cylinder/trapezoid displayed in Figs. 2.19b, c.

44

2 Kinematic Geometry of the PnP Robots

Fig. 2.18 An asymmetric four-limb parallel Schönflies-motion robot: a CAD model, b parameterizations of the ith limb Table 2.2 Geometrical parameters of the SMG with a single mobile platform [ax , a y ] (mm) α β b (mm) l (mm) r (mm) [200, 55]

15o

45o

315

596

112

γ 30o

2.6 Illustrative Examples

45

Fig. 2.19 Robot topologies of different shaped workspace [287]: a cuboid; b cylinder; c trapezoid

For an efficient usage of the workspace, a superellipsoid is adopted for fitting to approximate the workspace, as the top view of the reachable workspace has the cross-section close to superellipse, taking the form:  x 3  y 3  z 3         +  +  =1 a b c

(2.73)

As shown in Fig. 2.20, a super-ellipsoidal workspace is fitted, which is located inside the reachable workspace through the discrete searching method [281].

2.6.4 Two-Limb SMG with Articulated Double Platform Figure 2.21 depicts the conceptual design of a two-limb parallel Schönflies-motion manipulator. As displayed in Fig. 2.21a, each limb of the robot is actuated by two perpendicular actuators on the base. A pair of spatial modules [43, 300], instead of the conventional joints, are adopted to the connect the mobile platform and the proximal links. The parameterized kinematic structure of the robot is illustrated in Fig. 2.21b, where the global coordinate frame Fb is built with the origin located at the geometric center of the base platform, the x-axis being collinear to segment A1 A2 . The moving coordinate frame F p is attached to the mobile platform and the origin is at the geometric center, where X -axis is parallel to segment C1 C2 . Here and after, vectors i, j and k represent the unit vectors of x-, y- and z-axis, respectively. The axis of rotation of the i jth actuated joint of angular displacement θi j ∈ (−π/2, π/2), i = 1, 2, j = 1, 2, is parallel to ui j = Rz (αi j )j, αi j = (−1)i+1 π/2 + (−1) j π/4. Moreover, as the offset distance e on the elbow bracket does not affect the output variable, but is added as an offset to z-coordinate, henceforth, the offset distance e is assumed to be zero for convenience.

46

2 Kinematic Geometry of the PnP Robots

Fig. 2.20 The superellipsoid workspace with constant orientation φ = 0: a 3D view, b top view, c front view, b side view

Under the coordinate systems in Fig. 2.21b, the position vectors of point Ai in frame Fb are denoted by ai = (−1)i ai, then the Cartesian coordinates of point Bi is derived as (2.74) bi = bvi + ai ; vi = Ri k = Ri1 Ri2 k where Ri j = Rz (αi j )R y (θi j ) and θi j is the input angle from the reference plane. T  T  Let the mobile platform pose be denoted by χ = pT φ , p = x y z , the Cartesian coordinates of point Ci is expressed as ci = Qci + p

(2.75)

where Q = R(z, φ) is the rotation matrix of the MP and ci = (−1)i r i is the position vector of point Ci in the frame F p .

2.6.4.1

Forward Geometric Problem

Due to translational motion of the distal spatial modules, the y-coordinate of point Bi is equal to that of Ci , thus, the y-coordinate of point P and the orientation of the mobile platform, respectively, are derived as below:

2.6 Illustrative Examples

47

Fig. 2.21 The conceptual design of the two-limb parallel Schönflies-motion robot: a CAD model; b kinematic structure

48

2 Kinematic Geometry of the PnP Robots

Fig. 2.22 The planar four-bar linkage B1 C1 C2 B2 in the forward geometry

(b2 − b1 )j 1 T bi j; φ = sin−1 2 i=1 2r 2

y=

(2.76)

On the other hand, the x- and z-coordinates of point P of the mobile platform can be solved from the four-bar linkage as shown in Fig. 2.22, namely, l(sin q1 − sin q2 ) + g1 = 0; l(cos q1 + cos q2 ) + g2 = 0

(2.77)

where g1 = (b2 − b1 )T k and g2 = 2r cos φ + (b1 − b2 )T i. Consequently, the angles q1 , q2 ∈ [0, π/2] are obtained as q1 = 2 tan−1

2g1 ±

 4g12 + 4g22 − (g12 + g22 )2 g12 + g22 − 2g2

; q2 = sin−1 (sin q1 + g1 ) (2.78)

To this end, one obtains the x- and z-coordinates of point P in the frame Fb : 1 T 1 T ci i; z = c k 2 i=1 2 i=1 i 2

x=

2

(2.79)

T  where ci = bi + lwi , wi = (−1)i+1 cos qi 0 sin qi , stands for the position vector of point Ci .

2.6.4.2

Inverse Geometric Problem

The inverse geometric problem is solved from the following kinematic constraints: (ci − bi ) · j = 0, i = 1, 2

(2.80)

(ci − bi )T (ci − bi ) = l 2

(2.81)

2.6 Illustrative Examples

49

Expanding the above equations leads to i i sin θi1 + D12 cos θi1 + d1i = 0 D11 i D21

sin θi1 +

i D22

cos θi1 +

d2i

(2.82a)

=0

(2.82b)

which can be cast in a matrix form as below:





i

i i sin θi1 sin θi1 d1 D11 D12 = D = − i i i cos θi1 cos θi1 D21 D22 d2i

(2.83)

with the coefficients expressed as i i i i D11 = h i1 cθi2 ; D12 = h i2 sθi2 ; D21 = h i3 sθi2 + h i4 cθi2 ; D22 = h i5 sθi2 + h i6 cθi2 (2.84a)

d1i = di1 sθi2 + di2 ; d2i = di3 sθi2 + di4

(2.84b)

where s and c stand for the sine and cosine functions, respectively, and h i1 = −bsαi1 , h i2 = −bcαi2 sαi1 , h i3 = 2bzcαi2 h i4 = −2b((x + a)cαi1 + ysαi1 − r c(αi1 − φ)) h i5 = −2bcαi2 ((x + a)cαi1 + ysαi1 − r c(αi1 − φ)), h i6 = −2bz

(2.85)

di1 = −bcαi1 sαi2 , di2 = y − r sφ di3 = 2bsαi2 ((x + a)sαi1 − ycαi1 − r s(αi1 − φ)), di4 = ci − ai 2 + b2 − l 2 The determinant of matrix Di is calculated as: |Di | = 2b2 z sin αi1 (1 − sin2 αi2 sin2 θi2 ) = 0

(2.86)

Solving Eq. (2.83) using Cramer’s rule, sin θi1 and cos θi1 can be obtained as the functions of θi2 . Expanding and simplifying equation sin2 θi1 + cos2 θi1 = 1 yields S1 sin4 θi2 + S2 sin3 θi2 + S3 cos2 θi2 sin2 θi2 + S4 sin2 θi2

(2.87)

+ S5 cos θi2 sin θi2 + S6 cos θi2 + S7 cos θi2 = 0 2

4

2

with 2 2 2 2 2 2 S1 = di1 (h i3 + h i5 ) − 2di1 di3 h i2 di5 + h i2 (di3 − h i3 )

S3 =

2 2 2 2di1 di2 (h i3 + h i5 ) + 2di3 di4 h i2 − 2h i2 di5 (di1 di4 + di2 di3 ) 2 2 2 2 2 2 di1 (h i4 + h i6 ) − 2di1 di3 h i1 di4 + h i1 (di3 − h i5 ) + 2h i1 h i2 (h i4 h i5

S2 =

S4 =

2 2 di2 (h i3

S5 =

2 2 2 + h i5 ) − 2di2 di4 h i2 di5 + h i2 di4 2 2 2 2di1 di2 (h i4 + h i6 ) + 2di3 di4 h i1 − 2h i1 di4 (di1 di4

+ di2 di3 )

(2.88a) (2.88b) 2 2 + h i3 h i6 ) − h i2 h i4

(2.88c) (2.88d) (2.88e)

50

2 Kinematic Geometry of the PnP Robots 2 2 S6 = −h i1 h i6

S7 =

2 2 di2 (h i4

2 + h i6 ) − 2di2 di4 h i1 di4

(2.88f) (2.88g)

2 2 + di4 h i1

Equation (2.87) can be written in the following equivalent form: sin4 θi2 + L i sin3 θi2 + Mi sin2 θi2 + Ni sin θi2 + Pi = 0

(2.89)

with Li =

S2 − S5 S3 + S4 − 2S6 − S7 S5 S6 + S7 ; Mi = ; Ni = ; Pi = S1 − S3 + S6 S1 − S3 + S6 S1 − S3 + S6 S1 − S3 + S6

(2.90)

Finally, angle θi2 can be obtained by solving the Quartic equation (2.89): ⎛



1 Li θi2 = sin−1 ⎝− − 4 2

 L i2 2Mi 1 − + Ii + Ji ± 4 3 2

⎞ L i2 4Mi − − Ii − Ji ± K i ⎠ 2 3 (2.91)

with  √ 3 1 −L i3 + 4L i Mi − 8Ni 2Ui 3 Wi Ii = √ ; d J = ; Ui = Mi2 − 3L i Ni + 12Pi ; Ki =  i 3 54 4 L 2 /4 − 2M /3 + I + J 3 Wi i i i i  3 2 2 Vi = 2Mi − 9L i Mi Ni + 27Ni + 27L i Pi − 72Mi Pi ; Wi = Vi + −4Ui3 + Vi2 (2.92)

Substituting θi2 into Eq. (2.80), angle θi1 is then solved as θi1 = 2 tan−1

−X i ±



X i2 + Yi2 − Z i2

Z i − Yi

(2.93)

with X i = −b sin αi1 cos θi2 ; Yi = −b cos αi2 sin αi1 sin θi2

(2.94)

Z i = y − r sin φ − b sin αi2 cos αi1 sin θi2 It should be noted that the forward geometry of this two-limb SMG can be solved readily while the inverse one leads to a complex solution, compared to the foregoing four-limb SMG. This is due to that each limb involves two actuated variables as a serial linkage with a simpler forward kinematics rather than the parallel-mechanism architecture.

2.6 Illustrative Examples

51

2.6.5 3T3R Double-Delta Robot This robot is a combination of two Delta robots with two mobile platforms coupled by a complex gearbox [280]. Figure 2.23 illustrates kinematic structure of the robot as well as the gearbox. The global coordinate frame Fb is built with the origin located at the geometric center of the base frame, namely, point O, where the x-axis is perpendicular to the axes of rotation of the first actuator in the first Delta robot and the second actuator in the second one. The moving coordinate frame F p in Fig. 2.23b is attached to the end-effector and the origin is at the end, namely, point P, where X -axis is parallel to x-axis in the home configuration. Here and after, vectors i, j and k represent the unit vectors of x-, y- and z-axis, respectively. The orientation of the ith actuator in the first and second Delta robots, respectively, are described by unit vectors ui and ri : T T   ui = − sin ηi cos ηi 0 ; ri = − sin ζi cos ζi 0

(2.95)

where ηi = 2(i − 1)π/3, ζi = ηi + π/3, i = 1, 2, 3. Moreover, unit vectors vi and wi are parallel to the segments Ai Bi and Bi Ci , respectively, namely, the unit vectors along the active link and parallelogram in the ith limb of the first Delta robot. Likewise, the same interpretation of unit vectors si and ti are applicable to the second Delta robot. The kinematic parameters of the gearbox are depicted in Fig. 2.23b. The orientation of the end-effector follows the Euler convention Z –X –Z , where the three successive rotations are denoted by Euler angles α ∈ [0, ∞), β ∈ [0, 2π ] and γ ∈ [0, 2π ]. The gear transmission ratios for the last two rotations are both equal to 1, and the leads of the two screw pairs 1 and 2 are represented by h 1 and h 2 , respectively.

2.6.5.1

Inverse Geometry

Under the coordinate systems in Fig. 2.23a, the position vectors of points Ai and Di in frame Fb are denoted by T T   ai = R1 cos ηi sin ηi 0 ; di = R2 cos ζi sin ζi a

(2.96)

and the position vectors of points Bi and Ci are derived as bi = b1 vi + ai ; vi = Rz (ηi )R y (θi )i

(2.97)

ei = b2 si + di ; si = Rz (ζi )R y (φi )i

(2.98)

where θi and φi are the angles of rotation of motors from a reference position for the first and second Delta robots, respectively. According to Fig. 2.23b, the homogeneous transformation from the frame (x2 , y2 , z 2 ) on the sub-platform 2 to the end-effector frame F p is:

52

2 Kinematic Geometry of the PnP Robots

Fig. 2.23 The kinematic structure of the robot: a double Delta robot; b home configuration of the mobile platform

2.6 Illustrative Examples

53

T = T1 T2 T3

(2.99)

with







Rz (−α) d3 k I3 d5 i Rx (−β) 0 I3 d4 k Rz (−γ ) 0 ; T2 = ; T3 = 0 1 0 1 0 1 0 1 0 1 (2.100) where I3 is the 3 × 3 identity matrix. Thus, the rotation matrix of the end-effector will be described by (2.101) Q = Rz (−α)Rx (−β)Rz (−γ ) T1 =

 T T  Let the robot end-effector pose be denoted by x = pT ξ T , p = x y z , ξ =  T α β γ , subsequently, the Cartesian coordinates of point P2 of sub-platform 2 can be derived as ⎡ ⎤ ⎡ ⎤ x2 x − d5 cos α − d4 sin α sin β p2 = ⎣ y2 ⎦ = ⎣ y + d5 sin α − d4 cos α sin β ⎦ (2.102) z2 z − d3 − d4 cos β and the Cartesian coordinates of point P1 of sub-platform 1 are expressed as ⎡ ⎤ ⎡ ⎤ x1 x − d5 cos α − cos α(d1 + h 1 β/(2π )) − d4 sin α sin β p1 = ⎣ y1 ⎦ = ⎣ y + d5 sin α + sin α(d1 + h 1 β/(2π )) − d4 cos α sin β ⎦ (2.103) z1 z − d2 − d3 − d4 cos β − h 2 γ /(2π ) The inverse geometry of the robot can be solved from the following kinematic constraints: (2.104) ci − bi − l1 = 0; fi − ei − l2 = 0 where ci and fi , respectively, are the Cartesian coordinates of points Ci and Fi in the global frame Fb : T T   ci = p1 + r1 cos ηi sin ηi 0 ; fi = p2 + r2 cos ζi sin ζi 0

(2.105)

T  Let mi = m i x m i y m i z = ci − ai , the angular displacement of the ith actuated joint of the first robot is solved as

θi = 2 tan

−1

−Ii ±



Ii2 + Ji2 − K i2

K i − Ji

(2.106)

with Ii = 2b1 m i z , Ji = −2b1 (m i x cos ηi − m i y sin ηi ), K i = mi 2 + b12 − l12 (2.107)

54

2 Kinematic Geometry of the PnP Robots

By the same token, the inverse geometry of the second robot is: φi = 2 tan−1

−Ui ±

 Ui2 + Vi2 − Wi2

(2.108)

Wi − Vi

with Ui = 2b2 n i z , Vi = −2b2 (n i x cos ζi − n i y sin ζi ), Wi = ni 2 + b22 − l22 (2.109) T  where ni = n i x n i y n i z = fi − di .

Table 2.3 Geometric parameters (unit: mm) of the 6-dof robot R r1 r2 l h 1 / h 2 d1 d2 265

70

150

800

20

104

210

d3

d4

d5

75

90

90

Fig. 2.24 The reachable workspace of the two Delta robots and their common cylindrical volume: (a) b = 350, c = 300; (b) b = 300, c = 350

2.6 Illustrative Examples

2.6.5.2

55

Workspace Analysis

This robot is composed of two Delta robots, thus, the operational workspace of the 6-dof robot is included in their common volume, where the workspace of a Delta robot is determined by the common envelope of three tori that are the functions of the link lengths, actuator position/orientations and manipulator pose. For the workspace maximization and interchangeability, here, the influence of the link lengths of the inner arms on the workspace volume is to be analyzed. With the geometric parameters listed in Table 2.3, the workspace of the robot with different link lengths of the inner arms are shown in Fig. 2.24. It is seen that the combination of the robot Δb with longer inner arm and smaller platform radius and the robot Δc with

Fig. 2.25 The workspace volume of the robot with the different rotational capabilities: (a) b = 350, c = 300; (b) b = 400, c = 300

56

2 Kinematic Geometry of the PnP Robots

shorter inner arm and larger platform radius leads to a relatively larger workspace volume of the 6-dof robot. It should be noted that the robot’s operational workspace is not equal to the common volume of the two Delta robots, due to the presence of the dimensions of the gearbox. Alternatively, by integrating the vertical and horizontal offsets determined by the gearbox dimensions, the boundaries of the operational workspace can be found, such as the regular workspace with different rotational capabilities as depicted in Fig. 2.25. With the identical parameters except the inner arm length b of robot Δb , the longer the inner arm, the larger the operational workspace. Usually, the link length ratio of the outer and inner arms is bounded between 1.8 and 2.7, henceforth, the link length of the inner arm is set to the range [300, 400] mm. By limiting the rotational capabilities α ∈ [0, ∞], β = ±300◦ , γ = 360◦ , the robot admits a cylindrical workspace with diameter 1130 mm with the maximum range of the vertical distance 300 mm, which corresponds to the vertical workspace of the FlexPicker manipulator from ABB Robotics. From the perspective view of the workspace, the set of geometric parameters in Fig. 2.25a and Table 2.3 can be a candidate for the robot design.

Chapter 3

Differential Kinematics of the Robots

Abstract Differential kinematics of a robotic manipulator pertains to the mapping between the motion variables in the joint space and those in the Cartesian (or operational, or task) space for the velocity/acceleration analysis, upon the differentiation of the geometric constraint equation with respect to time. The first-order differential kinematics relates the joint rates and end-effector velocities by means of the Jacobian matrix. Jacobian matrix is a concept of importance in the robotic manipulators, from which a number of crucial performance indices, such as dexterity, singularity, even force capability, etc., are derived. Besides the kinematic Jacobians, the second-order differential kinematics involves the Hessian matrix, to link the accelerations and velocities of the end-effector and joint rates in the joint space.

3.1 Instantaneous Kinematics The instantaneous kinematics problem is important when deriving the Jacobian matrices for dexterity evaluation or singularity identification or doing acceleration analysis for the purpose of studying dynamics. This problem can be handled by means of the differentiation of the geometric constraint equations w.r.t. time. Figure 3.1 depicts the kinematic structure of a general parallel mechanism, consisting of n independent limbs. Suppose that the pose of the reference point on  T  T T  the mobile platform is χ = pT ϕ T , p = x y z , ϕ = α β γ , the geometric constraint equation of the ith limb is a function of the variables and parameters: Φi (g, χ , qi ) = 0, i = 1, 2, . . . , n

(3.1)

where g is a vector of the known architectural and geometric parameters, and qi is the joint variables.

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2021 G. Wu and H. Shen, Parallel PnP Robots, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-6671-4_3

57

58

3 Differential Kinematics of the Robots

Fig. 3.1 The kinematic structure of a general parallel mechanism

3.1.1 Velocity and Jacobian Matrix Differentiating Eqs. (3.1) w.r.t. time yields a set of equations of the form ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi x˙ + y˙ + z˙ + α˙ + β˙ + γ˙ + q˙i = 0, i = 1, 2, . . . , n ∂x ∂y ∂z ∂α ∂β ∂γ ∂qi (3.2) which is cast in a matrix form ⎡ ∂Φ1 ⎤⎡ ⎤ ⎡ ∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ⎤ ⎡ ⎤ q˙1 ∂q1 ∂Φ2 x˙ ∂x ∂y ∂z ∂α ∂β ∂γ ⎢ ⎥ ⎢q˙ ⎥ ⎥ ⎢ 2⎥ ⎢ ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ⎥ ⎢ ⎥ ⎢ ∂q2 ⎥⎢ . ⎥ ⎢ ∂ x ∂ y ∂z ∂α ∂β ∂γ ⎥ ⎢ y˙ ⎥ ⎢ .. ⎥⎢ . ⎥ ⎥⎢ ⎥ ⎢ ⎢ . ⎥⎢ . ⎥ ⎢ · · · · · · · · · · · · · · · · · · ⎥ ⎢ z˙ ⎥ ⎢ ⎥⎢ ⎥ = 0 ⎢ ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ⎥ ⎢ ⎥ + ⎢ ∂Φi ⎥ ⎢ q˙i ⎥ ⎢ ∂ x ∂ y ∂z ∂α ∂β ∂γ ⎥ ⎢α˙ ⎥ ⎢ ∂q i ⎢ ⎥⎢ . ⎥ ⎥ ⎢ ⎥⎣ . ⎦ ⎣ · · · · · · · · · · · · · · · · · · ⎦ ⎣β˙ ⎦ ⎢ .. ⎣ ⎦ . . ∂Φn ∂Φn ∂Φn ∂Φn ∂Φn ∂Φn γ˙ ∂Φn ∂x ∂y ∂z ∂α ∂β ∂γ q˙n ∂qn

(3.3)

Written in a more compact form leads to Jχ χ˙ + Jq q˙ = 0 with

(3.4)

3.1 Instantaneous Kinematics

59

∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ⎤ ∂x ∂y ∂z ∂α ∂β ∂γ ⎢ ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ∂Φ2 ⎥ ⎢ ∂ x ∂ y ∂z ∂α ∂β ∂γ ⎥

⎡ ∂Φ1

⎡ ⎤ x˙ ⎢ y˙ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ z˙ ⎥ ⎢··· ··· ··· ··· ··· ···⎥ ⎥ Jχ = ⎢ ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ∂Φi ⎥ , χ˙ = ⎢ ⎢α˙ ⎥ ⎢ ∂ x ∂ y ∂z ∂α ∂β ∂γ ⎥ ⎢ ⎥ ⎥ ⎢ ⎣β˙ ⎦ ⎣··· ··· ··· ··· ··· ···⎦ ∂Φn ∂Φn ∂Φn ∂Φn ∂Φn ∂Φn γ˙ ∂x ∂y ∂z ∂α ∂β ∂γ 

 T i n 1 ∂Φ2 · · · ∂Φ · · · ∂Φ ˙ = q˙1 q˙2 · · · q˙i · · · q˙n Jq = diag ∂Φ ∂q1 ∂q2 ∂qi ∂qn , q

(3.5a)

(3.5b)

where Jχ ∈ Rn×6 and Jq ∈ Rn×n are named the forward and inverse Jacobian matrices, respectively. For a fully parallel mechanism, both kinematic Jacobians are square matrices, with the dimensions n ≤ 6. The kinematic Jacobian matrix of a parallel robot can be obtained as long as the forward Jacobian is nonsingular: J = −Jχ−1 Jq

(3.6)

For a parallel robot, its kinematic Jacobian matrix plays an important role, since the dexterity/manipulability of the robot can be evaluated through the conditioning number of the kinematic Jacobian matrix. Jacobian matrix is an utmost important concept in robotics, as it is a mapping between the joint space and the operational space, relates the joint rates to the end˙ In the design and application of a robotic manipeffector velocities, i.e., χ˙ = Jq. ulator, the kinematic dexterity and manipulability indices are proposed from the Jacobian matrix for the performance evaluation, such as the condition number of the Jacobian matrix. This quantity can reflects the accuracy/wrench capability [185] between the actuators and end-effector. Moreover, it is applicable for singularity identification, and it is related to the kinetostatic and dynamic performance indices due to its presence in the derivation of the stiffness and inertia matrices.

3.1.2 Acceleration and Hessian Matrix The end-effector accelerations of the mechanism can be derived by differentiating Eq. (3.4) w.r.t. time again, namely, Jχ χ¨ + J˙ χ χ˙ + Jq q¨ + J˙ q q˙ = 0 with

(3.7)

60

3 Differential Kinematics of the Robots

 T ∂Jχ ∂Jχ ∂Jχ ∂Jχ ∂Jχ ∂Jχ x˙ + y˙ + z˙ + α˙ + β˙ + γ˙ , χ¨ = x¨ y¨ z¨ α¨ β¨ γ¨ J˙ χ = ∂x ∂y ∂z ∂α ∂β ∂γ (3.8a) T  ∂J ∂J ∂J ∂J q q q q q˙1 + q˙2 + · · · + q˙i + · · · q˙n , q¨ = q¨1 q¨2 · · · q¨i · · · q¨n J˙ q = ∂q1 ∂q2 ∂qi ∂qn (3.8b) For the fully parallel mechanism, each independent limb is actuated by an independent actuator, thus, ∂Jq /∂qi is a matrix full of zero elements except the ith diagonal T  one. Let Φ = Φ1 Φ2 · · · Φi · · · Φn , Eq. (3.7) can be rewritten as Jχ χ¨ + χ˙ T ⊗ Hχ χ˙ + Jq q¨ + q˙ T ⊗ Hq q˙ = 0

(3.9)

with ⎡

∂ 2 Φ ∂Φ 2 x∂ y ⎢ ∂ x ∂∂Φ ⎢ ∂ y2 ⎢

⎢ Hχ = ⎢ ⎢ ⎢ ⎢ ⎣

Hq = diag

∂Φ ∂ x∂z ∂Φ ∂ y∂z ∂Φ ∂z 2

∂Φ ∂ x∂α ∂Φ ∂ y∂α ∂Φ ∂z∂α ∂Φ ∂α 2

sym.

∂Φ ∂Φ ∂q12 ∂q22

···

∂Φ ∂qi2

∂Φ ∂ x∂β ∂Φ ∂ y∂β ∂Φ ∂z∂β ∂Φ ∂α∂β ∂Φ ∂β 2

···

∂Φ ∂ x∂γ ∂Φ ∂ y∂γ ∂Φ ∂z∂γ ∂Φ ∂α∂γ ∂Φ ∂β∂γ ∂Φ ∂γ 2

∂Φ ∂qn2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.10a)



(3.10b)

where Hχ and Hq are the Hessian matrices, and the operator ⊗ stands for the Kronecker product. Moreover, the ith diagonal element in matrix ∂Φ/∂qi2 is ∂Φi /∂qi2 while the remaining elements are equal to zero. The end-effector accelerations is now solved as

(3.11) χ¨ = Jq¨ − Jχ−1 χ˙ T Hχ χ˙ + q˙ T Hq q˙

3.2 Twist and Kinematic Screw The vector χ˙ consists of six elements, which can be classified into two sets of vectors,  T  T say, as shown in Fig. 3.2, vi = p˙ = x˙ y˙ z˙ , ωi = ϕ˙ = α˙ β˙ γ˙ : • vi : the linear velocity at point Oi with respect to the reference frame F j ; • ωi : the angular velocity of the frame Fi with respect to the frame F j . The velocity v j of point P (fixed in frame Fi ) is calculated in terms of the linear velocity of the point Oi and the angular velocity of the frame F j : v j = vi + ωi × j ri = vi + ωi × (r j − j pi )

(3.12)

3.2 Twist and Kinematic Screw

61

Fig. 3.2 The linear and angular velocity of a frame Fi w.r.t. frame F j

thus, vector χ˙ is called the twist, written as T  ti = viT ωiT

(3.13)

For a rigid body, the spatial motion can be viewed as a rotation about one axis together with the translation along the same axis, similar to the motion of a pair of nut and screw. For instance, the motion of point P is viewed as the screw motion along axis s, as displayed in Fig. 3.2. If the screw pitch is equal to h, the velocity at point P can be decomposed into the translation hω and the velocity perpendicular to s: (3.14) v = v p = hω + ω × (r j − r0 ) after the deduction, the velocity screw is defined as       ω ω s = =ω S= v r × ω + hω r × s + hs

(3.15)

where r is the distance of point P pointing to screw axis s. The spatial motion of a rigid body can be depicted by a velocity screw with zero pitch if it is actuated by a revolute joint: S=

    ω s = θ˙ v r ×s

(3.16)

62

3 Differential Kinematics of the Robots

Fig. 3.3 The velocity screw of a reference point in an open chain

and the motion is depicted by a velocity twist with infinite pitch if actuated by a prismatic pair:   0 S=v (3.17) s Thus, in the case that an open chain is composed of p = m + n joints (m revolute joints and n prismatic joints) and p + 1 links, as depicted in Fig. 3.3, the velocity screw of the end-effector is calculated as         m n si 0 ω ˙ ˙ + θi dj S= = ri × si sj v i=1

(3.18)

i=1

where θ˙i and d˙ j stands for the joint rates of the revolute and prismatic joints, respectively. Moreover, si( j) stands for the unit vector of the joint axis, and ri = pi − p is the position vector from the ith revolute joint to the reference point on the end-effector.

3.3 Examples 3.3.1 Planar Five-Bar Mechanism The first example to illustrate the differential kinematics is the planar five-bar mechanism, with the kinematic structure shown in Fig. 2.10 of Sect. 2, and the link lengths are set to AB = C D = l1 , B P = P D = l2 ,

3.3 Examples

3.3.1.1

63

Velocity and Acceleration Analysis

The kinematic constraint equations are written as (x + l0 − l1 cos θ1 )2 + (y − l1 sin θ1 )2 = l22

(3.19a)

(x − l0 − l1 cos θ2 ) + (y − l1 sin θ2 ) =

(3.19b)

2

2

l22

Differentiation of the previous equations w.r.t. time leads to Jχ p˙ = Jθ θ˙

(3.20)

with  Jχ =

x + l0 − l1 cos θ1 y − l1 sin θ1 x − l0 − l1 cos θ2 y − l1 sin θ2



(3.21a)   0 l1 (y cos θ1 − x sin θ1 − l0 sin θ1 ) Jθ = 0 l1 (y cos θ2 − x sin θ2 + l0 sin θ2 ) (3.21b) T   T (3.21c) p˙ = x˙ y˙ , θ˙ = θ˙1 θ˙2 where Jχ and Jθ are the forward and inverse Jacobian matrices of the planar five-bar mechanisms, respectively. If Jχ is not rank-deficiency, the kinematic Jacobian matrix is derived as   1 j11 j12 (3.22) J = Jχ−1 Jθ = Δ j21 j22 with Δ = 2yl0 − l12 sθ1 −θ2 − yl1 (cθ1 − cθ2 ) − l0 l1 (sθ1 + sθ2 ) + xl1 (sθ1 − sθ2 )

(3.23a)

j11 = l1 (y − l1 sin θ2 )(l0 sin θ1 − y cos θ1 + x sin θ1 ) j12 = l1 (y − l1 sin θ1 )(y cos θ2 + l0 sin θ2 − x sin θ2 )

(3.23b) (3.23c)

j21 = l1 (l0 − x + l1 cos θ2 )(l0 sin θ1 − y cos θ1 + x sin θ1 ) j22 = −l1 (l0 + x − l1 cos θ1 )(y cos θ2 + l0 sin θ2 − x sin θ2 )

(3.23d) (3.23e)

Differentiating Eq. (3.20) w.r.t. time yields Jχ p¨ + q˙ T ⊗ Hqq q˙ = Jq θ¨ with

(3.24)

64

3 Differential Kinematics of the Robots

   ⎤  10 l 1 s θ1 0 ⎥ ⎢ ⎥ ⎢ 1 0  0 l 1 s θ2  ⎥ ⎢ 01 −l1 cθ1 0 ⎥ ⎢ ⎥ ⎢ 0 −l1 cθ2 01    ⎥ Hqq = ⎢ ⎢ l1 sθ1 −l1 cθ1 l1 (l0 cθ1 + xcθ1 + ysθ1 ) 0 ⎥ ⎥ ⎢ ⎢ 0 0  0 0 ⎥ ⎥ ⎢ ⎦ ⎣ 0 0 0 0 l1 sθ2 −l1 cθ2 0 l1 (xcθ2 − l0 cθ2 + +ysθ2 )   T  T T q˙ = x˙ y˙ θ˙1 θ˙2 , p¨ = x¨ y¨ , θ¨ = θ¨1 θ¨2 ⎡

3.3.1.2

(3.25a)

(3.25b)

Numerical Simulation

In this simulation, the link lengths of the five-bar mechanism are set to l0 = 50 mm and li = 100 mm, i = 1, 2, 3, 4. When the joint angular displacements follow the motion law below: π π π π (3.26) θ1 = + p(τ ), θ2 = + p(τ ) 4 3 3 4 with p(τ ) = −20τ 7 + 70τ 6 − 84τ 5 + 35τ 4 , τ ∈ [0, 1]

(3.27)

the corresponding angular displacements, velocities and accelerations are calculated as shown in Fig. 3.4a. By making use of the velocity and acceleration Eqs. (3.20) and (3.24), the end-effector motions are solved in Fig. 3.4b.

3.3.2 SMG with Planetary Gear Train Based End-Effector Figure 3.5a depicts the kinematic structure of the robot with a planetary gear train (PGT) as the amplification mechanism to construct the end-effector, aiming to realize the full-circle rotation. The geometric centers of the four actuated joints are located at the vertices of a rectangle on the base platform to avoid the singularities when the robot reaches some symmetric configurations. The mobile platform is consisted of a constant-orientation sub-platform as the gear ring and a rotational sub-platform as the carrier in the PGT. Both sub-platforms connect a pair of opposite limbs, where the rotational one has two additional revolute pairs at the connecting positions rather than the remaining limbs. The end-effector orientation is amplified by the PGT, which can be simplified as a revolute pair. The orientation of the rotational sub-platform is hence can be computed from the required end-effector orientation and speed ratio of the PGT below ωe zr =1+ (3.28) i eH = ωH ze

3.3 Examples

65

angle [rad]

joint angular displacement 2 1.5

θ1

1

θ2

0.5

0

0.2

0.4

0.6

0.8

1

time [sec]

[rad/s]

joint angular velocity 3

dθ1

2

dθ

2

1 0

0

0.2

0.4

0.6

0.8

1

time [sec] joint angular acceleration

ddθ

1

[rad/s2 ]

10

ddθ

2

0 −10

0

0.2

0.4

0.6

0.8

1

time [sec]

(a) end−effector displacement

y [mm]

170 160 150 140 −30

−20

−10

0

10

20

30

40

50

x [mm] vx

end−effector velocity

[mm/s]

100

v

y

0 −100 −200

0

0.2

0.4

0.6

0.8

1

time [sec] 4

end−effector acceleration

[mm/s2 ]

x 10 0

a

−2

x

ay

−4 0

0.2

0.4

0.6

0.8

time [sec]

(b) Fig. 3.4 Motions of the five-bar mechanism: a joint space, b end-effector

1

66

3 Differential Kinematics of the Robots

where ωe and ω H stand for the angular velocities of the output end-effector and the rotational sub-platform, respectively. Moreover, zr and z e are the number of the teeth of the gear ring and the output sun gear, respectively. One example is that i eH = 8 with zr = 140 and z e = 20 to realize the full-circle end-effector rotation with the sub-platform rotation 45◦ . The parametrization of the robot is shown in Fig. 3.5b. The reference coordinate frame Fb is built with the origin located at the geometric center O of the base platform, with x-axis being parallel to the segment A2 A1 (A3 A4 ). The moving coordinate frame F p is attached to the rotational sub-platform with the origin located at the geometric center P, and its x-axis is parallel to segment C3 C1 . The base platform dimension is depicted by the radius R with two arc centers shifted by distance 2a along y-axis, and the mobile platform dimension is depicted by radius r that is equal to the length of segment PCi . The segments Ai Bi = b and Bi Ci = l stand for the lengths of the proximal and distal links, respectively. In the ith limb, the axis of rotation of the actuated joint is parallel to unit vector ui , and unit vectors vi and wi are parallel to segments Ai Bi and Bi Ci , respectively.

3.3.2.1

Kinematic Analysis

With the unit vector previously defined, the Cartesian coordinates of point Ai in the frame Fb are denoted as  ai = with

Rei + aj, i = 1, 2 Rei − aj, i = 3, 4

T  2i − 1 π ei = cos αi sin αi 0 , αi = 4

(3.29)

(3.30)

and the Cartesian coordinates of point Bi in the frame Fb are given by bi = bvi + ai , vi = Rz (αi )R y (θi )i

(3.31)

with the corresponding axis of rotation of the ith actuated joint is described by T  ui = − sin αi cos αi 0 . T   T Let the end-effector pose be χ = pT φ , p = x y z , the Cartesian coordinates of point Ci in frame Fb is expressed as  ci =

r Qi + p, i = 1, 3 r Rz (αi )i + p, i = 2, 4

(3.32)

where Q = Rz (φ + φ0 + (i − 1)π/2) is the rotation matrix of the rotational subplatform, and φ0 is the neutral orientation angle of the rotational sub-platform. By making use of the following kinematic constraints:

3.3 Examples

67

Fig. 3.5 Kinematic structure of (a) robot architecture and (b) PGT based mobile platform

(ci − bi )T (ci − bi ) = l 2 , i = 1, . . . , 4

(3.33)

the inverse geometry problem of the robot can be solved as

θi = 2 tan with

−1

−Ii ±



Ii2 + Ji2 − K i2

K i − Ji

(3.34)

68

3 Differential Kinematics of the Robots

Fig. 3.6 The two postures of one robot linkage

Ii = −2b(ci − ai )T k, Ji = 2b(ci − ai )T ei , K i = l 2 − b2 − ci − ai 2

(3.35)

For a given pose of the robot end-effector, each limb can have two postures that are characterized by the sign “−/+” in Eq. (3.34), as shown in Fig. 3.6, which means that the robot can have up to 16 working modes (W M s). Here, the signs “+” and “−” stand for two different postures for each limb, respectively, as depicted in Fig. 3.6. To avoid the mechanical collision, the working mode corresponding to the four limbs in posture “+” is selected from the 16 working modes for this type of robot. Differentiating the four Eqs. (3.33) with respect to time yields Aχ˙ = Bθ˙

(3.36)

with ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ θ˙1 x˙ h1 w1T r w1T s ⎢ h2 ⎥ ⎢θ˙2 ⎥ ⎥ ⎢ y˙ ⎥ ⎢w T 0 2 ⎢ ⎥ , θ˙ = ⎢ ⎥ ⎥ ⎢ ⎥ A=⎢ ⎣θ˙3 ⎦ ⎣w3T −r w3T s⎦ , χ˙ = ⎣ z˙ ⎦ ; B = ⎣ h3 ⎦ 0 h4 w4T φ˙ θ˙4 ⎡

where

(3.37)

3.3 Examples

69

T  s = − sin(φ + φ0 ) cos(φ + φ0 ) 0 ; h i = b(ui × vi ) · wi , wi = (ci − bi )/l (3.38) In Eq. (3.36), A and B are named as the forward and inverse Jacobian matrices, respectively. The kinematic Jacobian matrix is hence obtained as J = A−1 B as long as matrix A is not rank deficient. Differentiating Eq. (3.36) with respect to time again leads to ˙ χ˙ = Bθ¨ + B˙ θ˙ Aχ¨ + A

(3.39)

with ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ h˙ 1 θ¨1 x¨ ˙ 1T s + r w1T s˙ ˙ 1T r w w T ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢w ˙ ¨⎥ 0 ⎥ , θ¨ = ⎢θ2 ⎥ ⎥ ¨ = ⎢ y¨ ⎥ ; B˙ = ⎢ h 2 ˙ = ⎢ ˙ 2T A T T ⎦, χ ⎦ ⎣ ⎦ ⎣ ⎣ ⎣w ˙ z¨ ˙ 3 s − r w3 s˙ ˙ 3 −r w θ¨3 ⎦ h3 T ¨ ˙ ˙4 w 0 φ θ¨4 h4 (3.40) where ⎡

T  s˙ = φ˙ − cos(φ + φ0 ) − sin(φ + φ0 ) 0 ˙i h˙ i = θ˙i b(ui × v˙ i ) · wi + b(ui × vi ) · w

(3.41a) (3.41b)

and c˙ i − b˙ i , c˙ i = l



˙ Qj + p, ˙ i = 1, 3 ˙ φr , bi = bv˙ i ˙ i = 2, 4 p, (3.42) The forward and inverse acceleration equations, respectively, are derived as

˙i = v˙ i = θ˙i Rz (αi )R y (θi )j; w



˙ χ˙ − B˙ θ˙ ˙ χ˙ + B ˙ θ˙ , θ¨ = J−1 χ¨ + A χ¨ = Jθ¨ − A

3.3.2.2

(3.43)

Numerical Simulation

The kinematic simulation is studied along an energy-minimized PnP trajectory lied in the plane x = 0 as show in Fig. 3.7a, for which the test trajectory1 is mathematically defined in Appendix A. Based upon the design analysis in previous work [297], the geometric parameters in the unit of mm are listed in Table 3.1. With the inverse kinematic model in Eq. (3.43), the velocities and accelerations of the actuated joints are calculated and visualized in Fig. 3.7b. In this test, the maximum velocities and accelerations appear in the period when the robot endeffector accelerates or decelerates.

1 Throughout

this book, the SMG working cycle is set to 120 cpm in all the illustrative examples, with the identical PnP test trajectory for numerical analysis.

70

3 Differential Kinematics of the Robots

Table 3.1 Geometric design parameters of the SMG with PGT based End-effectors a R b l r 100

150

300

600

100

z [mm]

−370 −380 −390 −400 0

50

100

150

200

250

300

y [mm]

(a)

velocity [rad/s]

8 6 4 2 0 −2 −4 −6 0

0.05

0.1

0.15

time [sec]

2

acceletation [rad /s]

600 400

0.2

0.25

joint 1 joint 2 joint 3 joint 4

200 0 −200 −400 0

0.05

0.1

0.15

0.2

0.25

time [sec]

(b) Fig. 3.7 Robot motion: (a) end-effector displacement; (b) velocities and accelerations of the actuated joints

3.3.3 Kinematic Sensitivity of SMG with a Single Platform The foregoing Jacobian analysis deals with the mapping between the joint rates and end-effector velocities, which is confined to the manipulator mobilities. Besides, full Jacobian matrix is usually derived to analyze the sensitivity of the design variables or the robot accuracy/errors. This section presents the kinematic sensitivity analysis based on the full Jacobian matrix. In the design of parallel robots, a fundamental problem is that their performance heavily depends on their geometry. The variations in the geometric parameters of parallel robots can be either compensated or amplified,

3.3 Examples

71

Fig. 3.8 CAD models of the (a) asymmetrical and (b) symmetrical SMGs with a single platform

therefore, it is important to analyze the sensitivity of their end-effectors to variations in its geometric parameters [282]. Two sensitivity models of the asymmetrical robot in Sect. 2.6.3, of which the simplified CAD model is depicted in Fig. 3.8a, are developed by means of linkage kinematic analysis method and variational method, respectively, which are applicable to the Delta robot and Quattro architecture based robots, such as the symmetrical robot shown in Fig. 3.8b. Taking the asymmetrical robot as an example, the sensitivity model is derived with the parameterized structure in Fig. 2.18.

3.3.3.1

Sensitivity Model by Linkage Kinematic Analysis

As derived in Sect. 2.6.3, the kinematic constraint equation for each independent leg is expressed as (3.44) (ci − bi )T (ci − bi ) = li2 , i = 1, . . . , 4 Expanding the above equation, thus, four implicit functions can be derived below:  2 Fi = x − ai x + ri c(γi + φ) + bi (sαi sθi − cαi cβi cθi )  2 + y − ai y + ri s(γi + φ) − bi (cαi sθi + cαi cβi cθi )

(3.45)

+ (z + bi sβi cθi )2 − li2 = 0, i = 1, . . . , 4 where s and c stand for the sine and cosine functions, respectively. Differentiating Eq. (3.45) with respect to the design parameters, manipulator pose and input angle, the pose error δx of the manipulator and the variations of the actuated joints and design parameters δqi is related by δ Fi = AiT δx + BiT δqi = 0; Ai =

∂ Fi ∂ Fi , Bi = ∂x ∂qi

(3.46)

72

3 Differential Kinematics of the Robots

 T where qi = θi ai x ai y αi βi bi li ri γi . Equations (3.46) can be cast in a matrix form, namely, Aδx + Bδq = 0

(3.47)

with T  A = A1 A2 A3 A4 ∈ R4×4   B = diag B1T B2T B3T B4T ∈ R4×36 T  δq = δq1T δq2T δq3T δq4T ∈ R36×1

(3.48a) (3.48b) (3.48c)

Assuming that A is not singular, the sensitivity Jacobian matrix C of the EE pose to variations in geometric parameters is denoted below: δx = Cδq; C = −A−1 B ∈ R4×36

(3.49)

In order to investigate the influence of the geometric parameters errors on the position and the orientation of the end-effector, sensitivity indices need to be defined. Assuming that variations in the design parameters are independent, as variations in the design parameters of the same type from one leg to the other have the same influence on the end-effector pose [58], the sensitivity of the EE pose to the variations in the kth design parameter responsible for its pose error, i.e., δq(1,2,3,4)k , is defined by μλ,k below:   4 4   2 Cink ; k = 1, . . . , 9, λ ∈ {x, y, z, φ} (3.50) μλ,k =  i=1 n=1

Hence, the sensitivity index of positioning error δp to the kth parameter is defined as:  μk = μ2x,k + μ2y,k + μ2z,k (3.51) Based upon the foregoing defined indices, the mean sensitivity indices of the Ragnar robot throughout the workspace are shown in Fig. 3.9. It is seen that all the motion parameters are very sensitive to the angular geometric parameters, particularly, the actuated joint θ , due to the large dimensional link lengths. Among influence of the linear geometric parameters to the EE linear motion, the proximal link is the most sensitive one as it has the longest link length. Moreover, the EE orientational error is free of the linear geometric parameters except the slight influence of the distal links, while, the EE position is sensitive to all the geometric parameters.

3.3 Examples

73

Mean sensitivity index of y−coordinates to parameters

Mean sensitivity index of x−coordinates to parameters

45

60

∂y/∂q [mm/mm, mm/deg]

∂x/∂q [mm/mm, mm/deg] 40

50 35 30

40

25

30

20 15

20

10

10

0

5

θ

ax

ay

α

b

β

L

r

0

γ

θ

ax

ay

α

β

b

L

r

γ

(b)

(a) Mean sensitivity index of z−coordinates to parameters

Mean sensitivity index of φ to parameters

25

50

∂z/∂q [mm/mm, mm/deg]

∂φ/∂q [deg/mm, deg/deg] 45

20

40 35

15

30 25

10

20 15

5

10 5

0 θ

a

x

a

y

α

b

β

L

r

0

γ

a

θ

x

ay

α

(c)

β

b

L

(d) Mean sensitivity index of p to parameters

80 ∂p/∂q [mm/deg, mm/deg] 70 60 50 40 30 20 10 0

θ

ax

ay

α

β

b

L

r

γ

(e) Fig. 3.9 Mean of Sensitivity indices of x, y, z, φ throughout the regular workspace

r

γ

74

3.3.3.2

3 Differential Kinematics of the Robots

Variational Method

With the coordinate system set in Fig. 2.18, the ith limb of the robot can be split to depict the variations in the design parameters in a vectorial form. The closed-loop kinematic chains is described by O–Ai –Bi –Bi j –Ci j –Ci –P, i = 1, . . . , 4, j = 1, 2. Vectors o, ai , bi , bi j , ci j , ci , p, are the Cartesian coordinates of points O, Ai , Bi , Bi j , Ci j , Ci , P in the frame Fb , respectively. According to Fig. 3.10, (3.52) ai − o = a0i + δai where a0i is the nominal position vector of Ai with respect to O in the frame Fb and δai is the positioning error of Ai . According to Fig. 3.11,

with

bi − ai = (bi + δbi )Q Ai i

(3.53)

    Q Ai = (I + δϕ i )Ei (I + δψ Ai )Rz (θi )(I + δθi [k])

(3.54)

where [·] = CPM(·) denotes the cross-product matrix and I is the 3 × 3 identity  T matrix. Term δbi is the variation in bi , and δϕ i = 0 δβi δαi , δαi and δβi being the variations in αi and βi , respectively. Moreover, δθi is the error of input angle θi T  of the ith actuator and δψ Ai = δψ Ai x δψ Ai y δψ Ai z being the angular variation of manufacturing tolerance [58].

Fig. 3.10 Variations in O–Ai chain

Fig. 3.11 Variations in Ai –Bi chain

3.3 Examples

75

Fig. 3.12 Variations in Bi –Bi j chain Fig. 3.13 Variations in Bi j –Ci j chain

According to Fig. 3.12,  bi j − bi = ε( j)(di /2 + δdi j )(I + [δψ Bi ])Q Ai k; ε( j) =

1, j = 1 −1, j = 2

(3.55)

T  where δψ Bi j = δψ Bi x δψ Bi y δψ Bi z is the orientation error of link Bi Bi j with respect to ei , and δdi j is the variation of di in the jth direction. According to Fig. 3.13, ci j − bi j = li wi + δli j wi + li δwi j

(3.56)

where δli is the variation in the length of link Bi j Ci j along the direction wi , and δwi j is the variation in this direction, orthogonal to wi . Let assume all four axes of the four holes of the EE plate parallel to Z -axis. According to Fig. 3.14, ci j − ci = (I + δφCi [k])ε( j)(di /2 + δci j )(I + [δψ Ci ])ei

(3.57)

T  where δψ Ci j = δψCi x δψCi y δψCi z is the orientation error of link Ci Ci j with respect to ei , and δφCi is the angular displacement of the ith passive revolute joint on the end-effector. Moreover, from Fig. 3.14,

76

3 Differential Kinematics of the Robots

Fig. 3.14 Variations in Ci j –Ci –P chain

ci − p = (ri + δri )Rz (φ + γi )(I + δφ[k] + δγi [k])i

(3.58)

Equations (3.52), (3.53), (3.55), (3.56), (3.57) and (3.58) can be linearized by ignoring the high-order small terms. Afterwards, removing the nominal kinematic constraint equation p0 = a0i + bi hi + li wi − c0i leads to δp = p − p0 = δθi bi Rj + δai + δαi bi [k]Ri + δbi hi + δβi bi cos θi ei

(3.59)

− bi Ei [Rz (θi )i]δψ Ai + δβi ε( j)di /2Ei i − ε( j)di /2Ei [k]δψ Ai + ε( j)δsi j ei − ε( j)di /2[ei ]δψ di + δli j wi + li j δwi j − δφCi ε( j)d/2[k]ei − Rz (φ + γi )(δri i + δφri j + δγi ri j) where δsi j = δdi j − δci j and δψ di = δψ Bi − δψ Ci are the relative length and orientation errors of links Bi j Bi and Ci j Ci with respect to ei , respectively. By addition of Eq. (3.59) written for j = 1 and j = 2, the relationship between the pose errors and geometric/joint variations of the ith limb is expressed as δp = δθi bi Rj + δai + δαi bi [k]Ri + δbi hi + δβi bi cθi ei − bi Ei [Rz (θi )i]δψ Ai + δdi ei + li /2(δwi1 + δwi2 )

(3.60)

+ δli wi − Rz (φ + γi )(δri i + δφri j + δγi ri j) where δli = (δli1 + δli2 )/2 is the mean of relative length error of links Bi1 Ci1 and Bi2 Ci2 , δdi = (δsi1 − δsi2 )/2 is the mean of the relative length error of the connecting bars Bi1 Bi2 and Ci1 Ci2 in the parallelogram. Dot-multiplying to Eq. (3.60) on both sides with wi yields wiT δp = jθi δθi + jai δai + jαi δαi + jβi δβi + jbi δbi + j Ai δψ Ai + jdi δdi + δli + jri δri + jγ i δγi −

δφri wiT Rz (φ

+ γi )j

(3.61)

3.3 Examples

77

with jθi = bi wiT Rj; jai = wiT ; jαi = wiT [k]Ri jβi = jdi =

(3.62)

bi cθi wiT ei ; jbi = wiT hi ; j Ai = −bi wiT Ei [Rz (θi )i] wiT ei ; jri = −wiT Rz (φ + γi )i; jγ i = −ri wiT Rz (φ

+ γi )j

Equation (3.61) can be rewritten in a vector form, namely, Jxi δx = Jqi  qi

(3.63)

  Jxi = wiT ri wiT Rz (φ + γi )j   Jqi = jθi jai jαi jβi jbi j Ai jdi 1 jCi

(3.64)

with

(3.65)

T    where  qi = δθi , δaiT , δαi , δβi , δbi , δψ TAi , δdi , δli , δciT , and jCi = jri jγ i −1 RCi , RCi and δci being defined as below        δci X cos γi −ri sin γi δri δri = or δci = RCi δciY sin γi ri cos γi δγi δγi

(3.66)

It is noted that δci X and δciY are the positioning errors of point Ci along X - and Y axis, respectively, namely, the variations in the Cartesian coordinates of Ci in frame Fp. Equations (3.63) can be written in a matrix form, namely, Jx δx = Jq  q

(3.67)

with  T T T T T Jx2 Jx3 Jx4 ∈ R4×4 Jx = Jx1   Jq = diag Jq1 Jq1 Jq3 Jq4 ∈ R4×56 T  δq = δq1T δq2T δq3T δq4T ∈ R56×1

(3.68a) (3.68b) (3.68c)

By the same token, the sensitivity matrix J is obtained, as long as Jx is not singular: δx = J q ; J = J−1 x Jq =

  Jp jφ

(3.69)

Likewise, the sensitivity indices of the mth parameter can be defined similar to Eqs. (3.50) and (3.51), m = 1, . . . , 14. In order to evaluate the sensitivity globally, two aggregate sensitivity indices [58] of the position and orientation of the manipulator

78

3 Differential Kinematics of the Robots

end-effector to variations in its geometric parameters and actuated joints are defined from Eq. (3.69), expressed as: νp =

J p 2 jφ 2 ; νφ = nq nq

(3.70)

where  ·  denotes the 2-norm, and n q is the number of variations that are considered. The lower ν p and νφ , the lower the aggregate sensitivity of the EE pose of the manipulators to variations in its geometric parameters.

3.3.3.3

Sensitivity Comparison of the SMG Counterparts

The sensitivity analysis is numerically illustrated with the asymmetrical robot in comparison with the symmetrical counterpart [301], for which the design parameters are given in Table 3.2, the length and angle being in the units of mm and deg, respectively. In order to approximate the maximum regular workspace, a superellipsoid is adopted to describe the workspace (WS), which can contain different shaped cuboid WS. The corresponding workspace with the rotation range of ±45o are shown in Fig. 3.15, from which it is seen that the asymmetrical robot has a long and narrow workspace coincident with the PnP trajectories, as revealed in Sect. 2.6.3. Figure 3.16 depicts the mean of the sensitivity coefficients of p and φ with respect to the nominal geometric parameters throughout the workspace, defined by Eqs. (3.50) and (3.51), respectively. Due to the large link lengths, the position and orien-

Table 3.2 Geometric design parameters of the asymmetrical and symmetrical robots Robots ax ay α β b l r γ 109 162.3

15 45

45 90

295 292.5

536 526.5

103 153

d

30 60

100 100

−300

−400

z [mm]

z [mm]

Asymmetrical 310 Symmetrical 162.3

−500 −600

−400 −500 −600

400 200

y [mm]

0

200 0

−200 −400

(a)

−200

x [mm]

300 200 100

y [mm]

0 −100 −200 −300

200 0 −200

(b)

Fig. 3.15 Super-ellipsoidal workspace of the robots: a asymmetrical; b symmetrical

x [mm]

∂p/∂q [mm/mm, mm/deg]

3.3 Examples

79 10 9 8 7 6 5 4 3 2 1 0

asymmetrical symmetrical

θ

a

x

a

y

α

β

b

l

r

γ

∂φ/∂q [deg/mm, deg/deg]

(a) 5 asymmetrical symmetrical

4 3 2 1 0

θ

ax

ay

α

β

b

l

r

γ

(b) Fig. 3.16 Mean of sensitivity indices of p and φ with respect to the nominal geometric parameters throughout the super-ellipsoidal workspace: (a) μ p ; (b) μφ

tation of the end-effector of the asymmetrical robot are very sensitive to the angular variations, particularly the actuated joints. Besides, contrary to orientation, the EE position is sensitive to the position of points Ai , variations in the lengths of the inner and outer arms b and l, and variation in the radius r of the mobile platform. Due to the similar linkages, the sensitivity of EE pose of the symmetrical robot to the actuated joints and geometric parameters is similar to that of the asymmetrical one. Compared to the former, the pose of the symmetrical robot is less sensitive to the actuated joints and angle α. By contrast, the orientation of the asymmetrical robot is more sensitive to the variations, while the position of the symmetrical is more sensitive to the variations. Aiming to carry out a comparative study between the asymmetrical and symmetrical robots, the sensitivity indices in Eq. (3.70) and pose errors are compared within a cuboid WS of 400 × 600 × 220 mm3 . In order to compare the sensitivity with respect to the workspace, the geometric parameters ax , a y , b, l, r and d are normalized by a normalizing factor [174] defined as Nf =

  ax2 + a 2y + r /2

(3.71)

80

3 Differential Kinematics of the Robots

asymmetrical symmetrical

0.25

0.3

0.2 φ

ν

ν

p

0.25

0.15

0.2 0.15

0.1 300

asymmetrical symmetrical

0.35

200

0.1 300 100

0

y [mm]

200 −100

−200

0 −300 −200

x [mm]

200

100

y [mm]

0

200 −100

(a)

−200

0 −300 −200

x [mm]

(b)

Fig. 3.17 Isocontours of the global sensitivity indices: (a) ν p ; (b) νφ

The isocontours of mean of the indices ν p and νφ defined by Eq. (3.70) over the WS bottom cross-section of asymmetrical and symmetrical robots are displayed in Fig. 3.17. We can notice that the closer EE location to the central region of the workspace, the smaller the sensitivity of the MP pose to variations in the geometric parameters and active joints. From Fig. 3.17a, it is seen that the indices corresponding to position of the asymmetrical robot is bounded between 0.1 and 0.15, which means that the asymmetrical robot admits a rectangular area with lower sensitivity indices evenly distributed. Figure 3.17b shows that the sensitivity indices associated with the orientation of the asymmetrical robot has close values along y-axis, particulary in the region of |x| < 100 mm. Both the maximum values of ν p and νφ occur at the WS corners. Contrary to asymmetrical, the symmetrical robot has larger ν p but smaller νφ at most workspace points, i.e., the position of the symmetrical robot is much more sensitive to the variations while its orientation is not so sensitive, which agrees with the results given in Figs. 3.16, 3.18 and Table 3.3. Moreover, Eq. (3.69) can be used to compute the pose errors of the end-effector with known variations in design parameters. For instance, let the linear and angular tolerances be 0.02 mm and 0.03 deg, respectively, the maximum pose errors of the asymmetrical and symmetrical robots throughout the workspace are visualized in Fig. 3.19. The error distributions are consistent with the isocontours of the sensitivity indices in Fig. 3.17. It is noted that the asymmetrical robot has relatively higher positioning accuracy and lower orientation accuracy but acceptable for food handling industry that does not require very high accuracy. Moreover, the maximum pose errors of the two robots appear at the four corners on the bottom surface of the WS, where PnP motions rarely take place in these regions. The probability density function (pdf) of the pose errors of the asymmetrical and symmetrical robots at the center and corner on the bottom cross-section are shown in Figs. 3.20 and 3.21, respectively. Based on the previous analysis, the architecture of the asymmetrical robot still allows relatively better operational accuracy, particularly on the positioning accuracy, and

3.3 Examples

81

100

[%]

80 60 40 20 0