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Fundamentals of Robotic Mechanical Systems:Theory, Methods, and Algorithms, Second Edition Jorge Angeles... Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Alg

Fundamentals of Robotic Mechanical Systems:

Theory, Methods, and

Algorithms, Second Edition

Jorge Angeles

Mechanical Engineering Series

Mechanical Engineering Series

J Angeles, Fundamentals of Robotic Mechanical Systems:

Theory, Methods, and Algorithms, 2nd ed.

P Basu, C Kefa, and L Jestin, Boilers and Burners: Design and Theory

J.M Berthelot, Composite Materials:

Mechanical Behavior and Structural Analysis

I.J Busch-Vishniac, Electromechanical Sensors and Actuators

J Chakrabarty, Applied Plasticity

G Chryssolouris, Laser Machining: Theory and Practice

V.N Constantinescu, Laminar Viscous Flow

G.A Costello, Theory of Wire Rope, 2nd ed.

K Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems M.S Darlow, Balancing of High-Speed Machinery

J.F Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics,

Dynamics, and Stability

J.F Doyle, Wave Propagation in Structures:

Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed.

P.A Engel, Structural Analysis of Printed Circuit Board Systems

A.C Fischer-Cripps, Introduction to Contact Mechanics

A.C Fischer-Cripps, Nanoindentation

J García de Jalón and E Bayo, Kinematic and Dynamic Simulation of

Multibody Systems: The Real-Time Challenge

W.K Gawronski, Dynamics and Control of Structures: A Modal Approach K.C Gupta, Mechanics and Control of Robots

J Ida and J.P.A Bastos, Electromagnetics and Calculations of Fields

M Kaviany, Principles of Convective Heat Transfer, 2nd ed.

M Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed.

E.N Kuznetsov, Underconstrained Structural Systems

(continued after index)

Mechanical Engineering Series (continued from page ii)

P Ladevèze, Nonlinear Computational Structural Mechanics:

New Approaches and Non-Incremental Methods of Calculation

A Lawrence, Modern Inertial Technology: Navigation, Guidance, and

Control, 2nd ed.

R.A Layton, Principles of Analytical System Dynamics

F.F Ling, W.M Lai, D.A Lucca, Fundamentals of Surface Mechanics With

Applications, 2nd ed.

C.V Madhusudana, Thermal Contact Conductance

D.P Miannay, Fracture Mechanics

D.P Miannay, Time-Dependent Fracture Mechanics

D.K Miu, Mechatronics: Electromechanics and Contromechanics

D Post, B Han, and P Ifju, High Sensitivity Moiré:

Experimental Analysis for Mechanics and Materials

F.P Rimrott, Introductory Attitude Dynamics

S.S Sadhal, P.S Ayyaswamy, and J.N Chung, Transport Phenomena

with Drops and Bubbles

A.A Shabana, Theory of Vibration: An Introduction, 2nd ed.

A.A Shabana, Theory of Vibration: Discrete and Continuous Systems,

2nd ed.

Jorge Angeles

Department of Mechanical Engineering

and Centre for Intelligent Machines

Ernest F Gloyna Regents Chair in Engineering

Department of Mechanical Engineering

The University of Texas at Austin

Austin, TX 78712-1063, USA

and

William Howard Hart Professor Emeritus

Department of Mechanical Engineering,

Aeronautical Engineering and Mechanics

Rensselaer Polytechnic Institute

p cm.—(Mechanical engineering series)

Includes bibliographical references and index.

ISBN 0-387-95368-X (alk paper)

1 Robotics I Title II Mechanical engineering series (Berlin, Germany)

TJ211 A545 2002

629.8'92—dc21 2001054911

ISBN 0-387-95368-X Printed on acid-free paper.

© 2003 Springer-Verlag New York, Inc.

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY

10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in nection with any form of information storage and retrieval, electronic adaptation, computer software,

con-or by similar con-or dissimilar methodology now known con-or hereafter developed is fcon-orbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America

To Anne-Marie, who has given me not only her love, but also her precious time, without which this book would not have been possible.

Mechanical Engineering Series

Biomechanics V.C Mow

Columbia University

Computational Mechanics H.T Yang

University of California,Santa Barbara

Dynamical Systems and Control K.M Marshek

University of Texas, Austin

Energetics J.R Welty

University of Oregon, Eugene

Mechanics of Materials I Finnie

University of California, Berkeley

Processing K.K Wang

Cornell University

Production Systems G.-A Klutke

Texas A&M University

Thermal Science A.E Bergles

Rensselaer Polyechnic Institute

Tribology W.O Winer

Georgia Institute of Technology

Series Preface

Mechanical engineering, an engineering discipline borne of the needs of theindustrial revolution, is once again asked to do its substantial share in thecall for industrial renewal The general call is urgent as we face profound is-sues of productivity and competitiveness that require engineering solutions,among others The Mechanical Engineering Series features graduate textsand research monographs intended to address the need for information incontemporary areas of mechanical engineering

The series is conceived as a comprehensive one that covers a broad range

of concentrations important to mechanical engineering graduate educationand research We are fortunate to have a distinguished roster of consult-ing editors on the advisory board, each an expert in one of the areas ofconcentration The names of the consulting editors are listed on the facingpage of this volume The areas of concentration are: applied mechanics;biomechanics; computational mechanics; dynamic systems and control; en-ergetics; mechanics of materials; processing; production systems; thermalscience; and tribology

Preface to the Second Edition

The theory, methods and algorithms behind the development of roboticmechanical systems continue developing at a rate faster than they can be

recorded The second edition of Fundamentals of Robotic Mechanical

Sys-tems does not claim a comprehensive account of developments up-to-date.

Nevertheless, an attempt has been made to update the most impactingdevelopments in these activities Since the appearance of the first edition,many milestones can be cited Advances in a host of applications areas can

be mentioned, e.g., laparoscopy, haptics, and manufacturing, to mention arepresentative sample

Perhaps the most impressive achievements to be cited lie in the realm ofspace exploration Indeed, in the period of interest we have seen the suc-

cessful landing of the Sojourner on Mars, with the wheeled robot Pathfinder

roaming on the Martian landscape in 1997 Along the same lines, the frastructure of the International Space Station was set in orbit in 2000,

in-with the installation of Canadarm2, the successor of Canadarm, following

suit in 2001 Not less impressive are the achievements recorded on the oretical side of the areas of interest, although these have received much less

the-media attention To cite just one such accomplishment, one open question

mentioned in the first edition was definitely closed in 1998 with a paper

pre-sented at the International Workshop on Advances in Robot Kinematics.

This question pertains to the 40th-degree polynomial derived by Husty—

as reported in 1996 in a paper in Mechanism and Machine Theory—and

allowing the computation of all forward-kinematics solutions of a generalStewart-Gough platform Dietmaier reported an algorithm in that work-shop that is capable of generating a set of geometric parameters of the

x Preface to the Second Edition

platform that indeed lead to 40 real solutions The conclusion then is thatHusty’s polynomial is indeed minimal

In producing the Second Edition, we took the opportunity to clear themanuscript of errors and inaccuracies An in-depth revision was conductedin-between Special thanks go to Dr Kourosh Etemadi Zanganeh, Can-met (Nepean, Ontario, Canada), for his invaluable help in the rewrit-ing of Chapter 8 Profs Carlos L´opez-Caj´un, Universidad Aut´onoma deQuer´etaro (Mexico), and J Jes´us Cervantes-S´anchez, Universidad de Gua-najuato (Mexico) pointed out many inconsistencies in the first edition.Moreover, Dr Zheng Liu, Canadian Space Agency, St.-Hubert (Quebec,Canada), who is teaching a course based on the first six chapters of thebook at McGill University, pointed out mistakes and gave valuable sugges-tions for improving the readability of the book All these suggestions wereincorporated in the Second Edition as suggested, except for one: While

Dr Liu suggested to expand on the use of Euler angles in Chapter 2, cause of their appeal to robotics engineers in industry, we decided to add,instead, a couple of exercises to the list corresponding to this chapter Thereason is that, in the author’s personal opinion, Euler angles are a neces-sary evil Not being frame-invariant, their manipulation tends to becomeextremely cumbersome, as illustrated with those examples Euler anglesmay be good for visualizing rigid-body rotations, but they are very bad

be-at solving problems associbe-ated with these rotbe-ations using a computer orsimple longhand calculations Needless to say, the feedback received fromstudents throughout over 15 years of using this material in the classroom,

is highly acknowledged

One word of caution is in order: RVS, the software system used to sualize robot motions and highlighted in the first edition, has not receivedeither maintenance or updating It still runs on SGI machines, but we have

vi-no plans for its porting into Windows

Since there is always room for improvement, we welcome suggestions fromour readership Please address these to the author, to the e-mail addressincluded below Updates on the book will be posted at

www.cim.mcgill.ca/~rmslThe Solutions Manual has been expanded, to include more solutions ofsampled problems By the same token, the number of exercises at the end ofthe book has been expanded The manual is typeset in LATEX with Autocaddrawings; it is available upon request from the publisher

Last, but by no means least, thanks are due to Dr Svetlana Ostrovskaya,

a Postdoctoral Fellow at McGill University, for her help with Chapter 10and the editing of the Second Edition

Montreal, January 2002 Jorge Angeles

angeles@cim.mcgill.ca

Preface to the First Edition

No todos los pensamientos son algor´ıtmicos.

—Mario Bunge1

The beginnings of modern robotics can be traced back to the late sixtieswith the advent of the microprocessor, which made possible the computercontrol of a multiaxial manipulator Since those days, robotics has evolvedfrom a technology developed around this class of manipulators for the re-playing of a preprogrammed task to a multidiscipline encompassing manybranches of science and engineering Research areas such as computer vi-sion, artificial intelligence, and speech recognition play key roles in thedevelopment and implementation of robotics; these are, in turn, multidis-ciplines supported by computer science, electronics, and control, at theirvery foundations Thus we see that robotics covers a rather broad spec-trum of knowledge, the scope of this book being only a narrow band of thisspectrum, as outlined below

Contemporary robotics aims at the design, control, and implementation

1Not all thinking processes are algorithmic—translation of the author—

personal communication during the Symposium on the Brain-Mind Problem A

Tribute to Professor Mario Bunge on His 75th Birthday, Montreal, September

30, 1994

xii Preface to the First Edition

of systems capable of performing a task defined at a high level, in a guage resembling those used by humans to communicate among themselves.Moreover, robotic systems can take on forms of all kinds, ranging from themost intangible, such as interpreting images collected by a space sound, tothe most concrete, such as cutting tissue in a surgical operation We can,therefore, notice that motion is not essential to a robotic system, for thissystem is meant to replace humans in many of their activities, moving beingbut one of them However, since robots evolved from early programmablemanipulators, one tends to identify robots with motion and manipulation.Certainly, robots may rely on a mechanical system to perform their in-

lan-tended tasks When this is the case, we can speak of robotic mechanical

systems, which are the subject of this book These tasks, in turn, can be

of a most varied nature, mainly involving motions such as manipulation,but they can also involve locomotion Moreover, manipulation can be assimple as displacing objects from a belt conveyor to a magazine On theother hand, manipulation can also be as complex as displacing these objectswhile observing constraints on both motion and force, e.g., when cuttinglive tissue of vital organs We can, thus, distinguish between plain manipu-lation and dextrous manipulation Furthermore, manipulation can involvelocomotion as well

The task of a robotic mechanical system is, hence, intimately related

to motion control, which warrants a detailed study of mechanical systems

as elements of a robotic system The aim of this book can, therefore, be

stated as establishing the foundations on which the design, control, and

implementation of robotic mechanical systems are based.

The book evolved from sets of lecture notes developed at McGill versity over the last twelve years, while I was teaching a two-semester se-quence of courses on robotic mechanical systems For this reason, the bookcomprises two parts—an introductory and an intermediate part on roboticmechanical systems Advanced topics, such as redundant manipulators, ma-nipulators with flexible links and joints, and force control, are omitted Thefeedback control of robotic mechanical systems is also omitted, althoughthe book refers the reader, when appropriate, to the specialized literature

Uni-An aim of the book is to serve as a textbook in a one-year robotics course;another aim is to serve as a reference to the practicing engineer

The book assumes some familiarity with the mathematics taught in anyengineering or science curriculum in the first two years of college Familiar-ity with elementary mechanics is helpful, but not essential, for the elements

of this science needed to understand the mechanics of robotic systems arecovered in the first three chapters, thereby making the book self-contained.These three chapters, moreover, are meant to introduce the reader to thenotation and the basics of mathematics and rigid-body mechanics needed

in the study of the systems at hand The material covered in the samechapters can thus serve as reading material for a course on the mathemat-ics of robotics, intended for sophomore students of science and engineering,

Preface to the First Edition xiii

prior to a more formal course on robotics

The first chapter is intended to give the reader an overview of the subjectmatter and to highlight the major issues in the realm of robotic mechanicalsystems Chapter 2 is devoted to notation, nomenclature, and the basics oflinear transformations to understand best the essence of rigid-body kine-matics, an area that is covered in great detail throughout the book Aunique feature of this chapter is the discussion of the hand-eye calibrationproblem: Many a paper has been written in an attempt to solve this fun-damental problem, always leading to a cumbersome solution that invokesnonlinear-equation solving, a task that invariably calls for an iterative pro-cedure; moreover, within each iteration, a singular-value decomposition,itself iterative as well, is required In Chapter 2, a novel approach is in-troduced, which resorts to invariant properties of rotations and leads to a

direct solution, involving straightforward matrix and vector multiplications.

Chapter 3 reviews, in turn, the basic theorems of rigid-body kinetostaticsand dynamics The viewpoint here represents a major departure from mostexisting books on robotic manipulators: proper orthogonal matrices can beregarded as coordinate transformations indeed, but they can also be re-garded as representations, once a coordinate frame has been selected, ofrigid-body rotations I adopt the latter viewpoint, and hence, fundamentalconcepts are explained in terms of their invariant properties, i.e., proper-ties that are independent of the coordinate frame adopted Hence, matricesare used first and foremost to represent the physical motions undergone byrigid bodies and systems thereof; they are to be interpreted as such whenstudying the basics of rigid-body mechanics in this chapter Chapter 4 isthe first chapter entirely devoted to robotic mechanical systems, properlyspeaking This chapter covers extensively the kinematics of robotic ma-nipulators of the serial type However, as far as displacement analysis isconcerned, the chapter limits itself to the simplest robotic manipulators,

namely, those with a decoupled architecture, i.e., those that can be posed into a regional architecture for the positioning of one point of their end-effector (EE), and a local architecture for the orientation of their EE.

decom-In this chapter, the notation of Denavit and Hartenberg is introduced andapplied consistently throughout the book Jacobian matrices, workspaces,singularities, and kinetostatic performance indices are concepts studied inthis chapter A novel algorithm is included for the determination of theworkspace boundary of positioning manipulators Furthermore, Chapter 5

is devoted to the topic of trajectory planning, while limiting its scope toproblems suitable to a first course on robotics; this chapter thus focuses onpick-and-place operations Chapter 6, moreover, introduces the dynamics

of robotic manipulators of the serial type, while discussing extensively therecursive Newton-Euler algorithm and laying the foundations of multibodydynamics, with an introduction to the Euler-Lagrange formulation Thelatter is used to derive the general algebraic structure of the mathematicalmodels of the systems under study, thus completing the introductory part

xiv Preface to the First Edition

of the book

The intermediate part comprises four chapters Chapter 7 is devoted tothe increasingly important problem of determining the angular velocity andthe angular acceleration of a rigid body, when the velocity and acceleration

of a set of its points are known Moreover, given the intermediate level ofthe chapter, only the theoretical aspects of the problem are studied, andhence, perfect measurements of point position, velocity, and accelerationare assumed, thereby laying the foundations for the study of the sameproblems in the presence of noisy measurements This problem is findingapplications in the control of parallel manipulators, which is the reasonwhy it is included here If time constraints so dictate, this chapter can beomitted, for it is not needed in the balance of the book

The formulation of the inverse kinematics of the most general robotic nipulator of the serial type, leading to a univariate polynomial of the 16thdegree, not discussed in previous books on robotics, is included in Chap-ter 8 Likewise, the direct kinematics of the platform manipulator popularly

ma-known as the Stewart platform, a.k.a the Stewart-Gough platform, leading

to a 16th-degree monovariate polynomial, is also given due attention in thischapter Moreover, an alternative approach to the monovariate-polynomial

solution of the two foregoing problems, that is aimed at solving them

semi-graphically, is introduced in this chapter With this approach, the

under-lying multivariate algebraic system of equations is reduced to a system oftwo nonlinear bivariate equations that are trigonometric rather than poly-nomial Each of these two equations, then, leads to a contour in the plane

of the two variables, the desired solutions being found as the coordinates

of the intersections of the two contours

Discussed in Chapter 9 is the problem of trajectory planning as taining to continuous paths, which calls for some concepts of differentialgeometry, namely, the Frenet-Serret equations relating the tangent, nor-mal, and binormal vectors of a smooth curve to their rates of change withrespect to the arc length The chapter relies on cubic parametric splinesfor the synthesis of the generated trajectories in joint space, starting fromtheir descriptions in Cartesian space Finally, Chapter 10 completes thediscussion initiated in Chapter 6, with an outline of the dynamics of paral-lel manipulators and rolling robots Here, a multibody dynamics approach

per-is introduced, as in the foregoing chapter, that eases the formulation of theunderlying mathematical models

Two appendices are included: Appendix A summarizes a series of factsfrom the kinematics of rotations, that are available elsewhere, with thepurpose of rendering the book self-contained; Appendix B is devoted to thenumerical solution of over- and underdetermined linear algebraic systems,its purpose being to guide the reader to the existing robust techniques forthe computation of least-square and minimum-norm solutions The bookconcludes with a set of problems, along with a list of references, for all tenchapters