Robot modeling and kinematics

R o b o t M o d e lin g

AND K in em a t ic s

LIM ITED W A R R A N TY AN D D ISC LA IM ER OF LIA BILITY

THE CD-RO M TH AT A C C O M PA N IES THE B O O K M A Y BE U SED O N A SIN G LE

PC ONLY. THE LICEN SE DOES N O T PERM IT THE USE ON A N E T W O R K (OF A N Y

K IN D). YOU FU RTH ER A G REE TH A T THIS LIC EN SE G R A N TS PE R M ISSIO N TO

U SE THE PRO DUCTS C O N TA IN ED HEREIN , B U T DO ES N O T G IV E Y O U W G H T

O F O W N ERSH IP TO AN Y O F TH E CO N TEN T O R PR O D U C T C O N TA IN E D O N

THIS CD-ROM . USE OF TH IR D -PA R TY SO FTW A RE C O N TA IN ED O N T H IS CD￾ROM IS LIM ITED TO AN D SU BJECT TO LIC EN SIN G TERM S FO R TH E R E SP E C ­

TIVE PRODUCTS.

CH A RLES R IV ER M ED IA , INC. (“C R M ”) A N D /O R A N Y O N E W H O H A S B EEN

INV OLV ED IN THE W RITIN G , C R EA TIO N , O R PR O D U C TIO N OF TH E A C C O M ­

PA N YIN G COD E (“THE SO FTW A R E”) O R THE TH IR D -PA R TY PR O D U C TS C O N ­

TAINED ON THE CD-RO M OR TEX TU A L M A TER IA L IN TH E BOO K, C A N N O T

A N D DO N O T W A R R A N T THE PE R FO R M A N C E O R R ESU LTS TH A T M A Y BE

O B TA IN ED BY U SIN G THE SO FTW A R E O R C O N TEN TS O F TH E B O O K . THE

A U TH O R A N D PU B LISH ER H A V E U SED T H E IR B E ST E FFO R TS TO EN SU R E

THE A CCU RA CY A N D FU N C TIO N A LITY OF THE TEX TU A L M A TE R IA L A N D

PRO G RA M S CON TA IN ED H EREIN. W E H O W EV ER, M A K E N O W A R R A N TY OF

A NY KIN D, EX PRESS O R IM PLIED , R EG A R D IN G TH E PE R FO R M A N C E OF

THESE PRO GRAM S O R CONTENTS. THE SO FTW A RE IS SOLD “AS IS” W ITH O U T

W ARRAN TY (EX CEPT FOR D EFECTIV E M A TER IA LS U SED IN M A N U FA C T U R ­

ING THE D ISK OR DUE TO FA U LTY W O RK M A N SHIP).

THE AU TH OR, THE PU BLISHER, D EV ELO PER S OF TH IR D -PA R TY SO FTW A R E,

A ND AN YONE IN V O LVED IN THE PR O D U C TIO N A N D M A N U FA C TU R IN G OF

THIS W O RK SH ALL NO T BE LIABLE FO R D A M A G ES OF A N Y FOND A R ISIN G

OU T OF THE USE OF (OR THE INA BILITY TO USE) TH E PR O G R A M S, SO U RCE

CO D E O R TEX TU AL M A TERIA L C O N TA IN ED IN TH IS PU B LIC A TIO N . TH IS

IN CLU D ES, BUT IS N O T LIM ITED TO, LOSS OF R EV EN U E O R PR O FIT OR

OTH ER INCID EN TA L OR CO N SEQ U EN TIA L D A M AG ES A RISIN G O U T OF THE

USE OF THE PRODUCT.

THE SOLE REM EDY IN THE EVENT OF A CLA IM OF AN Y KIN D IS E X PR ESSLY

LIM ITED TO R EPLA CEM EN T OF THE BOO K A N D/O R CD -RO M , A N D O N LY AT

THE D ISCRETION OF CRM.

THE USE OF “IM PLIED W A R R A N TY ” AND CERTA IN “E X C LU SIO N S” V A RIES

FROM STATE TO STATE. AND M AY N O T APPLY TO THE PU R C H A SER OF THIS

PRODUCT.

R o b o t M o d elin g

AND K in em a t ic s

ĩ RưỮN6B,y:!i::!< ĩ r:,MiiNHH!PP

TH Í-' \ '! Ễ N

P M U N G Đ Ộ O

R a c h i d M a n s e u r

ĐẠIHỌCTHẤI NGUYÊN

' TRUNGTẦMHỌCLIỆU

D

D a V inci

DA VINCI ENGINEERING PRESS

Boston. Massachusetts

Copyright 2006 Career & Professional Group, a division of Thomson Learning Inc.

Published by Charles River Media, an imprint of Thomson Learning Inc.

All rights reserved.

No part of this publication may be reproduced in any way, stored in a retrieval system of any type, or

transmitted by any means or media, electronic or mechanical, including, but not limited to, photocopy,

recording, or scanning, without prior permission in writing from the publisher.

Cover Design: Tyler Creative

Charles River Media

25 Thomson Place

Boston, Massachusetts 02210

617-757-7900

617-757-7969 (FAX)

[email protected]

www.charlesriver.com

This book is printed on acid-free paper.

Rachid Manseur. Robot Modeling and Kinematics.

ISBN: 1-58450-851-5

All brand names and product names mentioned in this book are trademarks or service marks of their

respective companies. Any omission or misuse (of any kind) of service marks or trademarks should not

be regarded as intent to infringe on the property of others. The publisher recognizes and respects all

marks used by companies, manufacturers, and developers as a means to distinguish their products.

Library of Congress Cataloging-in-Publication Data

Manseur, Rachid.

Robot modeling & kinematics / Rachid Manseur.

p, cm.

Includes bibliographical references and index.

ISBN 1-58450-851-5 (hardcover wiứi cd-rom ; alk. paper)

1. Robots-Kinematics. I. Title.

TJ211.412.M36 2006

629.8'92-dc22

2006002975

Printed in the United States of America

06 7 6 5 4 3 2 First Edition

Charles River Media titles are available for site license or bulk purchase by institutions, user groups

corporations, etc. For additional information, please contact the Special Sales Department

at 800-347-7707.

Requests for replacement of a defective CD-ROM must be accompanied by the original disc your

mailing address, telephone number, date of purchase and purchase price. Please state the nature of

the problem, and send the information to Charles River Media, 25 Thomson Place, Boston

Massachusetts 02210. CRM’s sole obligation to the purchaser is to replace the disc, based on defective

materials or faulty workmanship, but not on the operation or functionality of the product.

Contents

Preface ix

1 Introduction 1

1.1 Viewing VRML Files 2

1.2 Textbook Organization 2

1.3 Robotics and Mechaưonics 7

1.4 Why Robots? 9

1.5 Types of Robots: General Overview 10

1.6 W hafsinaR obot? 21

1.7 Links and Joints 23

1.8 Representation and Visualization of Robot Joints 23

1.9 Robot Arm Sưuctures 25

1.10 Modeling the Human Arm 26

1.11 Joints and Degrees of Freedom 27

1.12 Robot End-Effectors 29

1.13 Indusnial Applications of Robot Manipulators 30

1.14 Robot Specifications 30

1.15 Robot Manufacturers 33

1.16 Flexible Manufacturing 33

1.17 The Future of Robotics 34

1.18 Summar>' 35

Exercises 35

2 Object Location 37

2.1 Object Frames 38

2.2 Position and Orientation in 3D 44

2.3 The Frame Transfomi 47

v i Contents

2.4 3D R otations and T ranslations 50

2.5 RPY A ngles A lgorithm 62

2.6 V isualizing RPY A ngles 65

2.7 V isualizing Euler A ngles 67

2.8 Hom ogeneous Fram e Transform s 68

2.9 Sum m ary 72

Exercises 72

Robot Modeling 77

3.1 M odeling Robotic A rm s 78

3.2 The D enavit-H artenberg Param eters 79

3.3 Link Fram e A ssignm ents 81

3.4 R obot M anipulator M odeling 81

3.5 Exam ples o f R obot Link Sequences 84

3.6 Hom ogeneous Fram e Transform s 87

3.7 Virtual D H Param eters 89

3.8 The R obot Hom e Position 93

3.9 C onverse Robots: Exchanging Base and End-Effector Fram es 96

3.10 M odeling Exam ples 100

3.11 RM odelo: A R obot V irtual M odeling Program 103

3.12 The PU M A 560 V irtual M odel 109

3.13 Sum m ary 109

Exercises 110

Forward Kinematics 117

4.1 Forw ard Kinem atics o f R obot M anipulators 118

4.2 Inverse o f a H om ogeneous Fram e Transform M atrix 122

4.3 Inverse o f a Pose M atrix 123

4.4 Com puting Separate Pose M atrix Elem ents 125

4.5 End-Effector Fram e Param eters 127

4.6 Forw ard K inem atics o f a Cartesian R obot 129

4.7 W rist-Partitioned Robots 131

4.8 Forward Kinematics of the PUMA 560 132

4.9 Forward Kinematics of the GP66 134

4.10 Symbolic Kinematic Equations Generator Program (SKEG) 138

4.11 Summary 145

Exercises 146

Inverse Kinem atics 151

5.1 General Inverse Kinematics Considerations 152

5.2 Solving a Three-Joint Robot 156

5.3 General Inverse Kinematics Procedure 163

5.4 Other Kinematics Equations 169

5.5 Solving Inverse Kinematics Equations 169

5.6 Joint, Cartesian, and Task Space 171

5.7 Inverse Kinematics of the PUMA 560 Robot 172

5.8 Number of Solutions 178

5.9 Summary 180

Exercises 181

Kinem atics of 4-Joint Robot Arms 185

6.1 The Four-DOF Reduced Set of Equations 186

6.2 Inverse Kinematics of the General Four-Revolute Joint Arm 191

6.3 Special Four-Joint Structures 193

6.4 Special Structures Examples 199

6.5 Summary 203

Exercises 204

Kinem atics of Five-Joint Robots 209

7.1 Kinematic Analysis of Five-DOF Arms 210

7.2 Numerical Inverse Kinematics for Five-Revolute-Joint Arms 213

7.3 Closed-Fomi Solution for Five-Joint Arms 221

7.4 The KAP5 Program 224

7.5 Summary 226

Exercises 226

Contents v ii

8 Kinematics of Six-Joint Robot Arms 231

8.1 The Six-Joint K inem atics Problem 232

8.2 Classification o f Six-Joint A rm s 233

8.3 Solving the General 6R Inverse K inem atics Problem 234

8.4 2D Iterative M ethod 236

8.5 The K A P6R Program 245

8.6 ID Iterative M ethod 248

8.7 Kinem atics o f the OM 25 R obot A rm 249

8.8 C om puting Joint V ariables in C losed-Fonn 253

8.9 Sum m ary 257

Exercises 258

9 The Jacobian and Velocity Kinematics 265

9.1 C artesian and Joint V elocities 265

9.2 The M anipulator Jacobian M atrix 266

9.3 The End-Effector L inear V elocity 268

9.4 The End-Effector A ngular V elocity 272

9.5 Com puting the Jacobian 274

9.6 Jacobian o f Six-Joint Robots 278

9.7 D ifferential Inverse Kinem atics 284

9.8 O rientation and A ngular V elocity 285

9.9 Link V elocities 289

9.10 Sum m ary 293

Exercises 294

10 Robot Singularities and Redundant Arms 299

10.1 R obot Singularities 300

10.2 K inem atics o f Redundant M anipulators 306

10.3 D ifferential K inem atics o f R edundant M anipulators 310

10.4 Sum m ary 314

Exercises 314

v i i i Contents

Appendix A About the CD-ROM 317

General Layout 317

System Requirements 320

Appendix B: VRML and 3D Visualization 321

\-RML Viewers 321

RModelo Displays 325

Appendix C: Mathematics Review 329

Points and X'ectors 329

Maữices 335

Appendix D: Trigonometric Equations 339

Triaonomettic Functions and the Unit Circle 339

Triansular Relations of Trisonometric Functions 340

Trigonomettic Idennties and Relations 341

The Atan2 Function 343

Common Robotics Equations 344

Bibliography 347

Index 351

Contents IX

Preface

T

ih is book is written to provide an introduction to robotics. It is intended for

a first course in robotics and covers modeling and kinematics of robot ma￾nipulators. It is suitable for a senior level undergraduate course or for a first

year graduate course. It differs from most other robotics textbooks in two major

ways:

■ Its content is limited to robot modeling and kinematics, whereas most other

textbooks cover dynamics, control, trajectory generation, and programming as

well as modeling and kinematics.

■ It offers extensive software support in terms of visualization, symbolic equa￾tion generation, and computation.

These differences make this textbook suitable to use either as a main textbook

for a first course in robotics or as a supplement to another textbook if the course

coverage is more extensive. Modeling and kinematics are foundation topics in ro￾botics. and a robust understanding of this material is necessary for further studies

of robotics. The supporting software offers great assistance in learning the material

and alleviates much of the computational burden associated with robotics.

The ability to generate and animate a virtual model of any robot manipulator,

change its structure, and view the effects of the changes on the operation of the

robot offers a unique and great learning experience and advantage over traditional

methods.

The need for this textbook arose from teaching robotics for many years and

facing the difficulty of explaining three-dimensional motions and object represen￾tations based on drawings and diagrams that can be confusing. With the rise of the

Internet, software for easy development of computer graphics and visualization

tools has become available and accessible. It is now possible for instructors and

xii Preface

textbook authors to show students realistic renditions o f three-dim ensional system s

in m otion to illustrate textbook discussions. This textbook is one o f the first at￾tem pts to use these m odem visualization tools in support o f a textbook in robotics.

ACKNOWLEDGMENTS__________________________________________________________

This textbook is the result o f m any years o f w ork, discussions, softw are develop￾ment, teaching, and learning. M any people have influenced its outcom e in various

ways and deserve som e recognition. First, I m ust acknow ledge m y m entor and

friend, Keith L. Doty, emeritus professor. University o f Florida, for teaching m e ro￾botics, for his guidance, and for m any fruitful exchanges. A few o f m y students

have also offered assistance sim ply by engaging in projects and courses that have

allowed me to leam with them and from them. A m ong these bright students, I m ust

recognize M elinda Robinette, who w orked on the first version o f my Internet￾based software, Robot-Draw, Richard M eserve, and M aynard Factor, who chose to

work on VRM L projects as well, and all those who assisted by reporting errors and

typos from m y notes while taking m y robotics class. M y colleague and friend,

Douglas Jordan, U niversity o f Florida, w ho shared with m e his extensive know l￾edge o f software and com puters w henever needed, deserves som e heartfelt thanks

as well. The review ers who have read my work and offered so m any w ords o f en￾couragem ent and suggestions for a better textbook are gratefully recognized here.

Finally, a w ord o f recognition goes to my wife, colleague, and friend, Z ohra

Manseur, and m y fantastic children, Mehdi and M aya, for m aking it all worthw hile.

Rachid M anseur

■

■

1 i Introduction

In This Chapter

■ Viewina VRAIL Files

■ Textbook Organization

■ Robotics Oven iew

■ Why Robots?

■ Types of Robots

■ W hat's in a Robot?

■ Links and Joints

■ Representation and Visualization of Robot Joints

■ Robot Arm Sữuctures

■ Modeling the Human Arm

■ Joints and Degrees of Freedom

■ Robot End-Effectors

■ Indusưial Applications of Robot Manipulators

■ Robot Specifications

■ Robot Manufacturers

■ Flexible Manufacturing

■ The PuUưe of Robotics

■ Summan'

T

his text presents an inưoduction to robotics, and this chapter provides a gen￾eral oveniew of the field of robotics and a description of the book's content

and organization. The visualization software is one of the main features of

ứiis textbook. It allows readers to view 3D renderings of several robotic systems.

Robot models and animations displayed on computer screens are generated using

i 'RML. the \'irm al Reality Modeling Language. Many of the fiaures in the early

chapters refer ứie reader to a i'R-XlL file with a file extension o f ,\\t1 on the accom￾panx ina CD-ROM for visualization. In later chapters, the reader is pro\ ided with

sofhvare that generates these \ ’RML files and is given the ability to generate anv