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Kinematic Analysis of Parallel Manipulatorsby Algebraic Screw Theory

Jaime Gallardo-Alvarado

Kinematic Analysisof Parallel Manipulatorsby Algebraic Screw Theory

123

Jaime Gallardo-AlvaradoDepartment of Mechanical EngineeringInstituto Tecnologico de CelayaCelaya, Mexico

ISBN 978-3-319-31124-1 ISBN 978-3-319-31126-5 (eBook)DOI 10.1007/978-3-319-31126-5

Library of Congress Control Number: 2016939238

© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe 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 informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.

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This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AG Switzerland

A la entrañable memoria de mis padres,Josefina y José.A mi esposa Alma Delia e hijos DeliaJosefina,Juan Pablo y José.To my family and friends!

Preface

Robotics is a prolific and intensive research field whose applications have hada highly positive impact on both academia and industry. While old and moderntheories are applied to solve challenging problems and improve existing solutions,mathematical models are currently used in the design, simulation, and controlof industrial robots. Certainly, the possibilities offered by robot manipulators areimmense, causing the variety of robot designs to continue to grow, which demandsefficient and confident mathematical methods of analysis. It is noticeable thatdespite the diversity of journals and books concerned with the study of robotmanipulators, there are particular issues that deserve to be deeply investigatedor in some cases elucidated in the light of modern kinematics; these includesingularities and higher-order kinematic analyses of robot manipulators. In this area,contributions approaching the jerk and hyper-jerk analyses of robot manipulators viascrew theory are rather scarce, perhaps because of the lack of practical applicationsor the lack of credibility about the validity of the equations in screw form of theacceleration analysis of kinematic chains introduced almost two decades ago. Inthis book, relevant fundamentals of screw theory concerning the so-called first-orderkinematic analysis of a rigid body are carefully reviewed through the introductionof propositions that are confirmed through detailed and explicit proofs. In addition,the acceleration, jerk, and hyper-jerk analyses of a rigid body are exhaustivelyinvestigated by taking advantage of the method outlined for the first-order kinematicanalysis. Because of the recent successful introduction of parallel manipulators inthe industry, those mechanical devices are selected as the means to exemplify theapplication of screw theory exposed in this book. With the material provided inthis book, readers can extend screw theory into the kinematics of optional orderof a rigid body and apply it to the analysis of parallel manipulators. Illustrativeexamples and exercises to reinforce learning are provided. Of particular note, thekinematics of emblematic parallel manipulators, such as the Delta robot as wellas the original Gough and Stewart platforms, are revisited. In addition to screwtheory, we apply new methods devoted to simplifying the corresponding forwarddisplacement analysis, a challenging task for most parallel manipulators.

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viii Preface

The material presented in this book is based on research published by the authorover the last 20 years, including his doctoral dissertation developed at InstitutoTecnológico de la Laguna, México, under the supervision of Prof. José María Rico-Martínez. To the author’s best knowledge, this is one of the first books devotedto the higher-order kinematic analyses of a rigid body by screw theory and itsapplication to the study of mechanisms possessing parallel kinematic structures.The author is confident about the usefulness of the material provided in this book,which represents an excellent argument to ensure that screw theory is not confinedto the so-called first-order kinematic analysis. The material presented here providestools for engineers, researchers, and students dealing with the kinematics of parallelmanipulators. As a consideration for readers unfamiliar with screw theory, the bookcontains a reasonable quantity of examples. Furthermore, as an unusual outcome,full answers to selected exercises are provided at the end of the volume.

Celaya, GTO, Mexico Jaime Gallardo-AlvaradoMay 2016

Acknowledgments

The author wishes to thank the support of two Mexican institutions: the ConsejoNacional de Ciencia y Tecnología (CONACYT) and the Tecnológico Nacionalde México (TecNM). The author is especially indebted to Michael Luby, Editor,Mechanical Engineering, Springer, for his advice and guidance in the preparationof the final text. Also, Nicole Lowary and Brian Halm are to be thanked for theassistance they provided during the publication of this book.

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Contents

Part I General Introduction

1 An Overview of the Theory of Screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 The Theory of Screws: Historical Contributions . . . . . . . . . . . . . . . . . . . 31.2 Notable Scientists in Screw Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 An Overview of Parallel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Typical Parallel Manipulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Part II Fundamentals of the Theory of Screws

3 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Velocity State or Twist About a Screw of aRigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 The Velocity State as an Equivalence Relationship . . . . . . . 353.2.3 The Velocity State and the Associated Screw

of Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.4 Plücker Coordinates of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.5 Lower Kinematic Pairs and Their Equivalent

Infinitesimal Screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 The Lie Algebra se.3/ of the Euclidean Group SE.3/ . . . . . . . . . . . . . . 45

3.3.1 The Lie Product and the Euclidean Motion . . . . . . . . . . . . . . . 533.3.2 Geometric Interpretation of the Lie Product . . . . . . . . . . . . . . 55

3.4 Helicoidal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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4 Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Fundamental Equations of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Equations of Velocity in Screw Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Part III Higher-Order Kinematic Analyses of Rigid Body

5 Acceleration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Fundamental Equations of Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 Equations of Acceleration in Screw Form . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Jerk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 Fundamental Jerk Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Jerk Equations in Screw Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Hyper-Jerk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.2 Fundamental Hyper-Jerk Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.3 Hyper-Jerk Equations in Screw Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Part IV Kinematics of Parallel Manipulators by Means ofScrew Theory Exemplified

8 3R2T Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.2 Description of the 3R2T Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . 1908.3 Finite Kinematics of the 3R2T Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.3.1 Forward Displacement Analysis of the 3R2T Robot . . . . . 1918.3.2 Inverse Displacement Analysis of the 3R2T Robot . . . . . . . 193

8.4 Infinitesimal Kinematics of the 3R2T Robot . . . . . . . . . . . . . . . . . . . . . . . 1938.4.1 Velocity Analysis of the 3R2T Parallel Manipulator . . . . . 1938.4.2 Acceleration Analysis of the 3R2T Parallel Manipulator. 196

8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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9 Two-Degree-of-Freedom Parallel Wrist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.2 Description of the Parallel Wrist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.3 Finite Kinematics of the Parallel Wrist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.3.1 Forward Displacement Analysis of the Parallel Wrist . . . . 2079.3.2 Inverse Displacement Analysis of the Parallel Wrist . . . . . 209

9.4 Infinitesimal Kinematics of the Parallel Wrist . . . . . . . . . . . . . . . . . . . . . . 2099.4.1 Velocity Analysis of the Parallel Wrist . . . . . . . . . . . . . . . . . . . . 2109.4.2 Acceleration Analysis of the Parallel Wrist . . . . . . . . . . . . . . . 2139.4.3 Jerk Analysis of the Parallel Wrist . . . . . . . . . . . . . . . . . . . . . . . . . 2149.4.4 Singularity Analysis of the Parallel Wrist . . . . . . . . . . . . . . . . . 215


10 3-RRPS Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21910.2 Description of the 3-RRPS Parallel Manipulator . . . . . . . . . . . . . . . . . . . 22010.3 Finite Kinematics of the 3-RRPS Manipulator . . . . . . . . . . . . . . . . . . . . . 221

10.3.1 Forward Displacement Analysis of the3-RRPS Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

10.3.2 Inverse Displacement Analysis of the 3-RRPSManipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.4 Infinitesimal Kinematics of the 3-RRPS Manipulator . . . . . . . . . . . . . . 22710.4.1 Velocity Analysis of the 3-RRPS Manipulator . . . . . . . . . . . . 22710.4.2 Acceleration Analysis of the 3-RRPS Manipulator . . . . . . . 22810.4.3 Jerk Analysis of the 3-RRPS Manipulator . . . . . . . . . . . . . . . . 230


11 3RRRSC3RRPS Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.2 Description of the 3RRRSC3RRPS Parallel Manipulator . . . . . . . . . 23711.3 Finite Kinematics of the 3RRRSC3RRPS Parallel Manipulator . . 239

11.3.1 Forward Displacement Analysis of the3RRRSC3RRPS Parallel Manipulator . . . . . . . . . . . . . . . . . . . . 239

11.3.2 Closed-Form Solution of the Forward PositionAnalysis Using Three Rotary Sensors . . . . . . . . . . . . . . . . . . . . . 241

11.3.3 Inverse Displacement Analysis of the3RRRSC3RRPS Parallel Manipulator . . . . . . . . . . . . . . . . . . . . 242

11.4 Infinitesimal Kinematics of the 3RRRSC3RRPSParallel Manipulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24211.4.1 Velocity Analysis of the 3RRRS+3RRPS

Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24311.4.2 Acceleration Analysis of the 3RRRSC3RRPS

Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

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11.4.3 Jerk Analysis of the 3RRRSC3RRPS ParallelManipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

11.4.4 Hyper-Jerk Analysis of the 3RRRSC3RRPSParallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248


Part V Emblematic Parallel Manipulators

12 Gough’s Tyre Testing Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25512.2 Description of the General Parallel Manipulator . . . . . . . . . . . . . . . . . . . 25712.3 Finite Kinematics of the General Hexapod . . . . . . . . . . . . . . . . . . . . . . . . . 258

12.3.1 Forward Displacement Analysis of the Hexapod . . . . . . . . . 25812.3.2 Inverse Displacement Analysis of the Hexapod. . . . . . . . . . . 273

12.4 Infinitesimal Kinematics of the Six-Legged Parallel Manipulator . 27612.4.1 Velocity Analysis of the Six-Legged Parallel Manipulator 27712.4.2 Acceleration Analysis of the Six-Legged

Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27812.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

13 The Original Stewart Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28113.2 Description of the Stewart Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28313.3 Finite Kinematics of the Stewart Platform . . . . . . . . . . . . . . . . . . . . . . . . . . 285

13.3.1 Forward Displacement Analysis of the Stewart Platform. 28513.3.2 Inverse Displacement Analysis of the Stewart Platform . . 289

13.4 Infinitesimal Kinematics of the Stewart Platform. . . . . . . . . . . . . . . . . . . 29013.4.1 Velocity Analysis of the Stewart Platform. . . . . . . . . . . . . . . . . 29013.4.2 Acceleration Analysis of the Stewart Platform. . . . . . . . . . . . 29213.4.3 Singularity Analysis of the Stewart Platform . . . . . . . . . . . . . 295


14 Delta Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30114.2 Description of the Delta Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30214.3 Finite Kinematics of the Delta Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

14.3.1 Forward Displacement Analysis of the Delta Robot . . . . . . 30414.3.2 Inverse Displacement Analysis of the Delta Robot . . . . . . . 308

14.4 Infinitesimal Kinematics of the Delta Robot . . . . . . . . . . . . . . . . . . . . . . . . 31014.4.1 Velocity Analysis of the Delta Robot . . . . . . . . . . . . . . . . . . . . . . 31114.4.2 Acceleration Analysis of the Delta Robot . . . . . . . . . . . . . . . . . 312


Contents xv

Part VI Solved Exercises

15 Full Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31915.1 Chapter 3: Mathematical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31915.2 Chapter 4: Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32415.3 Chapter 5: Acceleration Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33215.4 Chapter 6: Jerk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33915.5 Chapter 7: Hyper-Jerk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34515.6 Chapter 12: Gough’s Tyre Testing Machine . . . . . . . . . . . . . . . . . . . . . . . . 352References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

16 Appendix 1: A Simple Method to Compute the Rotation Matrix . . . . . 355

17 Appendix 2: Computer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35717.1 Maple Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

List of Tables

Table 8.1 Example 8.1. Reachable values for X1, Y2, and Z3 . . . . . . . . . . . . . . . . 200

Table 10.1 Example 10.1. Solutions of the unknowns wi.i D 1; 2; : : : ; 9/ . . . 224Table 10.2 Example 10.1. Available coordinates for the center of

the moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Table 12.1 Example 12.1. Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Table 12.2 Example 12.1. The 40 positions of point O2 . . . . . . . . . . . . . . . . . . . . . . . 264Table 12.3 Example 12.2. Solutions of the forward displacement analysis . . 270Table 12.4 Example 12.2. Real solutions of the forward

displacement analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Table 12.5 Example 12.3. The 40 reachable positions of point B1 . . . . . . . . . . . . 274Table 12.6 Example 12.3. Coordinates of points Bi, solution 6 of

the forward displacement analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Table 13.1 Example 13.1. Solutions of the characteristic equations . . . . . . . . . . 287Table 13.2 Example 13.1. Imaginary solutions of the forward

displacement analysis of the Stewart platform. . . . . . . . . . . . . . . . . . . . . 288

Table 14.1 Example 14.3. Inverse displacement analysis . . . . . . . . . . . . . . . . . . . . . 309

xvii

List of Figures

Fig. 1.1 The portrait of the work Discorso matematico sopra ilrotamento momentaneo dei corpi (Mozzi 1763) . . . . . . . . . . . . . . . . . . . 4

Fig. 1.2 Book cover of the work The Screw Calculus and ItsApplications in Mechanics (Dimentberg 1965) . . . . . . . . . . . . . . . . . . . . 5

Fig. 2.1 Possibly the first parallel robot, an invention creditedto Gwinnett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Fig. 2.2 Pollard’s spray painting machine (U.S. Patent No. 2,286,571) . . . 21Fig. 2.3 Three parallel manipulators and a polemic discussion

about their origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Fig. 2.4 The Delta robot (U.S. Patent No. 4,976,582) . . . . . . . . . . . . . . . . . . . . . . 24

Fig. 3.1 A body B in motion with respect to another body Acomprising angular and linear velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Fig. 3.2 Typical lower kinematic pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Fig. 3.3 Geometric interpretation of the Lie product. . . . . . . . . . . . . . . . . . . . . . . . 55

Fig. 4.1 Bodies m and j in relative motion; angular and linear velocities . . 66Fig. 4.2 Bodies k and l connected to each other through a

helical pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Fig. 4.3 A right-hand orthonormal basis fb1; b2; b3g attached

to a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Fig. 4.4 A free vector ˇ in a body l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Fig. 4.5 Bodies j, k, and m in relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Fig. 4.6 Multibody mechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Fig. 4.7 Example 4.3. Actuating mechanism for a telescoping

antenna on a spacecraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Fig. 4.8 Example 4.3. Decomposition of the velocity analysis . . . . . . . . . . . . . 79Fig. 4.9 Serial kinematic chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Fig. 4.10 Example 4.4. Actuating mechanism for a telescoping

antenna on a spacecraft, infinitesimal screws . . . . . . . . . . . . . . . . . . . . . . 83Fig. 4.11 Four-bar linkage mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xix

xx List of Figures

Fig. 4.12 Crank-slider linkage mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Fig. 4.13 Geneva wheel and its geometry scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Fig. 4.14 Screws associated to the kinematic pairs of the Geneva wheel . . . 91Fig. 4.15 Singularities in loci form of the Geneva wheel . . . . . . . . . . . . . . . . . . . . 94Fig. 4.16 Nonredundant planar parallel manipulators . . . . . . . . . . . . . . . . . . . . . . . . 95Fig. 4.17 Six-bar dwell linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Fig. 4.18 High-torque mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Fig. 5.1 Example 5.1. Two trucks in relative motion . . . . . . . . . . . . . . . . . . . . . . . 102Fig. 5.2 Example 5.2. Planar two-degree-of-freedom serial manipulator . . 109Fig. 5.3 Example 5.3. Acceleration analysis of the Geneva wheel . . . . . . . . . 112Fig. 5.4 Example 5.4. 2RRR+PPR planar parallel manipulator

and its geometric scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Fig. 5.5 The gradual improvement of a planar parallel manipulator . . . . . . . 115Fig. 5.6 2RRR+PPR planar parallel manipulator. Infinitesimal screws . . . . 118Fig. 5.7 Example 5.5. Trucks in relative motion and its

kinematically equivalent mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Fig. 5.8 6-PSU parallel manipulator. A Hexaglide-type robot . . . . . . . . . . . . . 127Fig. 5.9 Planar timing mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Fig. 6.1 Example 6.1. Rocket fired vertically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Fig. 6.2 Example 6.2. Two trucks in relative motion . . . . . . . . . . . . . . . . . . . . . . . 140Fig. 6.3 Example 6.3. Three-degree-of-freedom robot arm . . . . . . . . . . . . . . . . 144Fig. 6.4 Planar two-degree-of-freedom parallel manipulator . . . . . . . . . . . . . . . 150Fig. 6.5 Jerk: problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Fig. 6.6 Jerk: problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Fig. 6.7 Jerk: problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Fig. 6.8 Jerk: problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Fig. 6.9 Jerk: problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Fig. 7.1 Example 7.1. Rotating disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Fig. 7.2 Example 7.1. Decomposition of the velocity analysis . . . . . . . . . . . . . 165Fig. 7.3 Example 7.1. Decomposition of the acceleration analysis . . . . . . . . 166Fig. 7.4 Example 7.2. Paint-spraying robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Fig. 7.5 Example 7.2. Decomposition of the velocity analysis . . . . . . . . . . . . . 176Fig. 7.6 Example 7.2. Decomposition of the acceleration analysis . . . . . . . . 177Fig. 7.7 Example 7.2. Model of the screws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Fig. 7.8 Problem 1. Hyper-jerk analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Fig. 7.9 Problem 2. Hyper-jerk analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Fig. 7.10 Problem 3. Hyper-jerk analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Fig. 8.1 3R2T parallel manipulator and its geometric scheme . . . . . . . . . . . . . 190Fig. 8.2 Infinitesimal screws of the 3R2T parallel manipulator . . . . . . . . . . . . 194Fig. 8.3 Example 8.1. Time history of the angular and linear

velocities of the center of the moving platform . . . . . . . . . . . . . . . . . . . . 201

List of Figures xxi

Fig. 8.4 Example 8.1. Time history of the angular and linearaccelerations of the center of the moving platform . . . . . . . . . . . . . . . . 201

Fig. 9.1 The RR + RRR spherical parallel manipulator . . . . . . . . . . . . . . . . . . . . 206Fig. 9.2 Geometric scheme of the RR + RRR parallel wrist. . . . . . . . . . . . . . . . 207Fig. 9.3 Infinitesimal screws of the RR + RRR parallel wrist . . . . . . . . . . . . . . 211Fig. 9.4 Example 9.3. Time history of the angular quantities of

the knob as measured from the base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Fig. 10.1 The 3-RRPS parallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Fig. 10.2 3-RRPS parallel manipulator. Active revolute joint . . . . . . . . . . . . . . . 222Fig. 10.3 Example 10.2. Time history of the infinitesimal

kinematics of the center of the moving platform . . . . . . . . . . . . . . . . . . 233

Fig. 11.1 Six-legged parallel manipulator provided with linearand rotary actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Fig. 11.2 Hexagonal moving platform of the 3RRRS+3RRPSparallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Fig. 11.3 Infinitesimal screws of two distinct limbs of the3RRRS+3RRPS parallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Fig. 11.4 Example 11.1. Time history of the velocity of thecenter of the moving platform of the 3RRRS+3RRPSparallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Fig. 11.5 Example 11.1. Time history of the acceleration of thecenter of the moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Fig. 11.6 Example 11.1. Time history of the jerk of the centerof the moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Fig. 11.7 Example 11.1. Time history of the hyper-jerk of thecenter of the moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Fig. 12.1 The original Gough platform, perhaps the firstoctahedral hexapod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Fig. 12.2 Geometric scheme of the general six-legged in-parallel platform 257Fig. 12.3 Moving platform with three control points . . . . . . . . . . . . . . . . . . . . . . . . . 260Fig. 12.4 Example 12.1. Hexapod with planar platforms . . . . . . . . . . . . . . . . . . . . 262Fig. 12.5 Three-dimensional platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Fig. 12.6 Example 12.2. Parallel manipulator provided with a

triangular-prism moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Fig. 12.7 Example 12.3. Parallel manipulator provided with

three-dimensional platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Fig. 12.8 Example 12.3. Real solutions of the forward

displacement analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275Fig. 12.9 Infinitesimal screws of the 6-SPS parallel manipulator . . . . . . . . . . . 276Fig. 12.10 Six-legged parallel manipulator with triangular

moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Fig. 13.1 The original Stewart platform proposed as a flight simulator . . . . . 282

xxii List of Figures

Fig. 13.2 The flight simulator of Klaus Cappel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Fig. 13.3 Geometric scheme of the Stewart platform . . . . . . . . . . . . . . . . . . . . . . . . 284Fig. 13.4 Infinitesimal screws of the Stewart platform . . . . . . . . . . . . . . . . . . . . . . . 290Fig. 13.5 Example 13.2. Temporal behavior of the velocity and

acceleration of the center of the moving platform of aStewart-type platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Fig. 14.1 Geometric scheme of the Delta robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Fig. 14.2 Example 14.2. Workspace 1 (volume) of the Delta robot . . . . . . . . . 307Fig. 14.3 Example 14.2. Workspace 2 (volume) of the Delta robot . . . . . . . . . 308Fig. 14.4 Infinitesimal screws of a kinematically equivalent

Delta robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Fig. 14.5 Example 14.4. Time history of the displacement of

the center of the moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Fig. 14.6 Example 14.4. Time history of the velocity of the

moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Fig. 14.7 Example 14.4. Time history of the acceleration of the

moving platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Fig. 16.1 Parameters of the rotation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

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