机器人运动学、动力学、动量、能量等基础理论Monographs in computer ScienceEditorsDavid griesFred b. schneiderMonographs in Computer ScienceAbadi and Cardelli, a Theory of objectsBenosman and Kang [editors ] Panoramic Vision: Sensors, Theory, and ApplicationsBroy and Stolen, Specification and Development of Interactive Systems: FOCUS onStreams. Interfaces, and RefinementBrzozowski and Seger, Asynchronous CircuitsBurgin, Super-Recursive AlgorithmsCantone, Omodeo, and Policriti, Set Theory for Computing: From DecisionProcedures to Declarative Programming with SetsCastillo, Gutierrez, and Hadi, expert systems and probabilistic Network ModelsDowney and Fellows, Parameterized ComplexityFeijen and van Gasteren, On a Method of MultiprogrammingHerbert and Sparck Jones [editors, Computer Systems: Theory, Technology, andApplicationseiss, Language equationsevin, Heydon, and Mann, Software Configuration Management with VESTAMclver and Morgan [editors ], Programming MethodologyMclver and Morgan [editors), Abstraction, Refinement and Proof for ProbabilisticSystemsMisra, A Discipline of Multiprogramming: Programming Theory for DistributedApplicationsNielson [editor], ML with ConcurrencyPaton [editor], Active Rules in Database SystemsSelig, Geometrical Methods in RoboticsSelig, Geometric Fundamentals of Robotics, Second EditionShasha and Zhu, High Performance Discovery in Time Series: Techniques and CaseStudiesTonella and Potrich, Reverse Engineering of Object Oriented CodeJ M seliGeometricFundamentalsof roboticsSecond edition②SpringerJ.M. SeligLondon South Bank UniversityFaculty of Business, Computing and Information ManagementLondon SE1 OAAU.Kseligim@ lsbu ac ukSeries EditorsDavid griesFred B. SchneideCornell UniversityCornell UniversityDepartment of Computer ScienceDepartment of Computer ScienceIthaca. NY 14853Ithaca, NY 14853U.S.AU.S.AMathematics Subject Classification (2000): 70B15, 70E60, 53A17, 22E99SBN0-387-208747Printed on acid-free paper.@2005 Springer Science+Business Media IncBased on Geometrical Methods in robotics, Springer New york @1996All rights reserved. This work may not be translated or copied in whoie or in part withoutthe written permission of the publisher(Springer Science+Business Media Inc, Rights andPermissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts inconnection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimi-lar methodology now known or hereafter developed is forbiddenThe 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 towhether or not they are subject to proprietary rightsPrinted in the United States of America. (TXQ/HP)987654321SPN10983911springeronline corTo KathyPrefaceThis book is an extended and corrected version of an earlier work GeometricalMethods in Robotics"published by Springer-Verlag in 1996. I am extremelyglad of the opportunity to publish this work which contains many correctionsand additions. The extra material, two new chapters and several new sectionsrefects some of the advances in the field over the past few years as well as somematerial that was missed in the original workas before this book aims to introduce Lie groups and allied algebraic andgeometric concepts to a robotics audience. I hope that the power and eleganceof these methods as they apply to problems in robotics is still clear. By nowthe pioneering work of Ball is well known. However, the work of Study and hiscolleagues is not so widely appreciated, at least not in the English speakingworld. This book is also an attempt to bring at least some of their work to theattention of a wider audienceIn the first four chapters, a careful exposition of the theory of lie groups andtheir Lie algebras is given. All examples used to illustrate these ideas, except forthe simplest ones, are taken from robotics. So, unlike most standard texts onLie groups, emphasis is placed on a group that is not semi-simple the groupof proper Euclidean motions in three dimensions. In particular, the continuoussubgroups of this group are found, and the elements of its Lie algebra areidentified with the surfaces of the lower Reuleaux pairs. These surfaces werefirst identified by reuleaux in the latter half of the 19th century. They allow usto associate a lie algebra, element to every basic mechanical joint. The motionsallowed by the joint are then just the one-parameter subgroups generated by theviii PrefaceLic algebra element. a detailed study of the exponential map and its derivativeis given for the rotation and rigid body motion groupsChapter 5 looks at soine geometrical problems that are basic to robotics andthe theory of mechanisms. Having developed in thc previous chapter the description of robot kinematics using exponentials of Lic algebra elements. theseideas arc used to generalise and simplify some standard results in kinematics. The chapter looks at, the kinematics of 3-joint wrists and 3-joint regionalmanipulatorsSome of the classical thcory of ruled surfaces and line complexes is introducedin Chapter 6. This naterial also benefits from the Lic algebra point of viewFor robotics, the most important ruled surfaces are the cylindrical hyperboloidand the cylindroid. A full description of these surfaces is givenIn Chapter 7, the thcory of group representations is introduced. Once againthe emphasis is on the group of proper Euclidean motions. Many representa-tions of this group are used in robotics. a benefit of this is that it allows aconcise statement and proof of the Principle of transference, a result thatuntil recently had the status of a 'folk theorem,in the mechanism theory communityBall's thcory of screws underlies much of the work in this book. Ball's treatisewas written at the turn of the twentieth century, just before Lics and Cartan'swork on continuous groups. The infinitesimal screws of Ball can now be scenas elements of the Lie algebra of the group of proper Euclidean motions. InChapter 8, on screw systems, the linear subspaces of this Lie algebra arc ex-plored The Gibson Hunt classification of these systems is derived using a grouptheoretic approach.Clifford algebra is introduced in Chapter 9. Again, attention is quickly spe-cialised to the case of the Clifford algebra for the group of proper Euclideanmotions. This is something of an esoteric case in the standard mathematicalliterature. since it is the clifford algebra of a degenerate bilinear form. This algebra is a very cfficient vehicle for carrying out comput ation both in the groupand in some of its geometrical representations. Moreover, it allows us to definethe Study quadric, an algebraic variety that contains the elements of the groupof proper euclidean motionsChapter 10 explores this Clifford algebra in more detail. It is shown howpoints, lines and planes can be represented in this algebra, and how geometricoperations can be modelled by algebraic operations in the algebra. The resultsare used to look at the kinematics of six-joint industrial robots and prove animportant, theorem concerning designs of robots that have solvable inverse kinematIcsThe Study quadric is more fully explored in Chapter 11 where its subspacesand quotients are examined in some depth. The intersection theory of the varietyis introduced and used to solve some simple enumerative problems like themumber of postures of the general 6-R robotPreChapters 12, 13 and 14 cover the statics and dynamics of robots. The dualspace to the Lie algebra is identified with the space of wrenches, that is, forcetorque vectors. This facilitates a simple description of some standard problemsin robotics. in particular, the problern of gripping solid objects. The group theory helps to isolate the surfaces that cannot be completely immobilised withoutfriction. They turn out to be exactly the surfaces of the lower Reuleaux pairsIn ordcr to deal with the dynamics of robots, the inertia properties of rigidbodies must be studied. In standard dynamics texts, the motion of the centreof mass and the rotation about the centre of mass are treated separately. Forrobots. it is more convenient to use a six-dimensional notation. which docsnot separate thc rotational and translational motion This leads to a six-by-sixinertia matrix for a rigid body and also allows a modern exposition of somedeas due to Ball, namely conjugate screws and principal screws of inertiaThe standard theory of robot dynamics is presented in two ways, first as asimple Newtonian-style approach, and then using Lagrangian dynamics. TheLagrangian approach leads to a simple study of small oscillations of the endeffector of a robot and reintroduces what ball termed harmonic screws Theneat formalism used means that the equations of motion for a simple robot canbe studied quite easily. This advantage is used to look at the design of robotswith a view to simplifying their dynamics. Several approaches to this problemare consideredThe dynamics of robots with end-effector constraints and the dynamics ofrobots with star structures is also investigated. This allows the description ofthe dynamics of parallel manipulators and some simple examples of these arepresentedIn Chapter 15 some deeper applications of differential geometry are exploredThree applications are studied the mobility of overconstrained mechanisms, thecontrol of robots along geodesic paths, and hybrid controlThe original book was never intended as an encyclopedic account of "robotgeometry", but over the last few years this field has expanded so much that itis no longer even feasible to catalogue the omissions. The criterion for selectingmaterial for this book is still a reliance on the methods outlined in the first fewchapters of the book, essentially elenentary differential geometryHowever, one omission that i would like to mention is the field of robot visionA central problem in robot vision is to find the rigid motion undergone bthe camera using information derived from the images. There are many otherinteresting geometric problems in this area, see Kanatani [61 for example. Ifeel that this area is so large and with very specific problems that it deservesseparate treatnentI would like to thank the many people who pointed out errors in the originalbook, in particular Charles Wampler, Andreas Ruf and Ross Mcaree. I metPertti Lounesto shortly before his untimely death in 2002. Naturally he foundan error in the chapter on Clifford algebra in the original book, but this is