IEE ntE erdisciplinary Applied Mathematics Editors S.S.Antman J.E.Marsden L.Sirovich GeophysicsandPlanetarySciences Imaging,Vision,andGraphics D.Geman MathematicalBiology L.Glass,J.D.Murray MechanicsandMaterials R.V.Kohn SystemsandControl S.S.Sastry,P.S.Krishnaprasad Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspond- ingly increased dialog between the disciplines has led to the establishment of the series: InterdisciplinaryAppliedMathematics. Thepurposeofthisseriesistomeetthecurrentandfutureneedsfortheinteractionbetween various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouraging the ways that mathematics may be applied in traditional areas,aswellaspoint towardsnew andinnovative areasofapplications;and, secondly, by encouragingotherscientificdisciplinestoengageinadialogwithmathematiciansoutlining their problems to both access new methods and suggest innovative developments within mathematicsitself. The series will consist of monographs and high-level texts from researchers working on theinterplaybetweenmathematicsandotherfieldsofscienceandtechnology. Interdisciplinary Applied Mathematics Forothertitlespublishedinthisseries,goto www.springer.com/series/139095590 J. Michael McCarthy • Gim Song Soh Geometric Design of Linkages Second Edition J. Michael McCarthy Gim Song Soh Department of Mechanical and Singapore Institute of Manufacturing Aerospace Engineering Technology University of California, Irvine 71 Nanyang Drive Irvine, CA 92697 Singapore 638075 USA [email protected] [email protected] Series Editors S.S.Antman J.E.Marsden DepartmentofMathematics ControlandDynamicalSystems and MailCode107-81 InstituteforPhysicalScience CaliforniaInstituteofTechnology andTechnology Pasadena,CA91125,USA UniversityofMaryland [email protected] CollegePark,MD20742,USA [email protected] L.Sirovich DivisionofAppliedMathematics BrownUniversity Providence,RI02912,USA [email protected] ISSN0939-6047 ISBN978-1-4419-7891-2 e-ISBN978-1-4419-7892-9 DOI10.1007/978-1-4419-7892-9 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): 53A17, 13P15, 15A66 ©SpringerScience + BusinessMedia,LLC2011 Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) ForKendallandWyatt Preface 2ndEdition This second edition of Geometric Design of Linkages revises and updates our for- mulation of the kinematic theory of linkages. Four new chapters have been added that present the analysis and synthesis of multiloop planar and spherical linkages, andthesynthesistheoryforspatialserialchains. An introduction to linkage graphs and linkage enumeration has been added to Chapter1toprovidebackgroundforthenewchaptersonthesynthesisofmultiloop linkages,andtheuseoftheDixondeterminanttoanalyzeplanarmultilooplinkages hasbeenaddedtoChaptertwo. Chapters three, four, and five are the same as before with corrections of minor errors. Chapter six is new and presents a methodology for the synthesis of pla- narsix-barandeight-barlinkagesbyconstrainingthree-degree-of-freedom3Ropen chains or 6R closed chains, respectively. Examples are provided that demonstrate thetechnique. ChaptersevenisthesameasChaptersixinthepreviouseditionbutnowincludes asectionontheanalysisofmultiloopsphericallinkages.Chapterseightandnineare thesameasChapterssevenandeightinthepreviousedition,againwithcorrections ofminorerrors,.ThenewChapter10parallelsthenewChaptersixandpresentsa waytodesignsphericalsix-barandeight-barlinkagesbyconstraining3Ropenand 6Rclosedsphericalchains. Chapters 11, 12, and 13 are the same as Chapters 9, 10, and 11 in the previ- ous edition. Chapters 14 and 15 on the synthesis of spatial serial chains are new andweretheresultofresearchwithHaijunSuandAlbaPerez-Gracia.Chapter14 formulates and solves using numerical homotopy the synthesis equations for five- degree-of-freedomserialchainsthatcanpositionawristcenteronspecificalgebraic surfaces, termed reachable surfaces. Chapter 15 introduces the Clifford algebra of dualquaternionsanditsuseinformulatingthesynthesisequationsforgeneralspa- tialserialchains.Chapter16isthesameasChapter12inthepreviousedition. vii viii Preface Icontinuetobenefitfromthecontributionsofteachers,colleagues,andstudents toward a geometric synthesis theory for linkage systems, recently from Jeff Ge, Mohan Bodduluri, John Dooley, Pierre Larochelle, Andrew Murray, Fangli Hao, Curtis Collins, Alba Perez-Gracia, Haijun Su, Nina Robson, Duanling Li, and my coauthorGimSongSoh.Inaddition,Iamgratefulforthecontinuedinspirationof QizhengLiaoandBernardRoth. Finally,IgratefullyacknowledgethesupportoftheEngineeringDesignProgram oftheNationalScienceFoundationthathasmadethisbookpossible. Irvine,CA, J.MichaelMcCarthy September2010 FirstEdition This book is an introduction to the mathematical theory of design for articulated devicesthatrelyonsimplemechanicalconstraintstoprovideacomplexworkspace for a workpiece or end-effector. Devices ranging from windshield wipers to robot manipulators and mechanical hands are examples of these systems each of which hasaskeletonoflinksconnectedbyjointscalledalinkage.Thefunctionortaskfor thedeviceisdefinedasasetofpositionstobereachedbytheend-effector.Thegoal istodeterminethedimensionsofallofthedevicesthatcanachieveaspecifictask. Formulatedinthiswaythedesignproblemispurelygeometricincharacter. This text blends two approaches to this design problem in order to develop the intuitionneededtomovefromplanartospatiallinkagedesign.Oneapproachcon- sidersthegeometricconfigurationsofpointsandlinesgeneratedasamovingbody is displaced through a finite set of positions. This is the foundation for graphical methodsforplanarlinkagesynthesisandcanbegeneralizedtosphericalandspatial linkagedesign.Aseparateapproachfocusesdirectlyonsolvingthenonlinearcon- straint equations that characterize a mechanical connection. This provides conve- nientequationsforplanarandsphericallinkagedesign,andiscrucialtoaddressing thegeometricchallengeofspatiallinkagedesign. This unified formulation requires a range of mathematical tools. The basic lan- guage is vector algebra and matrix theory, which should be familiar to junior and senioruniversitystudents.However,somethingamongthetechniquesrangingfrom graphicalconstructions,sphericaltrigonometry,complexvectors,andquaternionsto linegeometryanddualvectoralgebraiscertaintobeunfamiliar.Forthisreason,the presentation is designed to introduce these techniques, and additional background isprovidedinappendices. The first chapter presents an overview of the articulated systems that we will be considering in this book. The generic mobility of a linkage is defined, and we separatethemintotheprimaryclassesofplanar,spherical,andspatialchains. Thesecondchapterpresentstheanalysisofplanarchainsanddetailstheirmove- mentandclassification.Chapterthreedevelopsthegraphicaldesigntheoryforpla- Preface ix nar linkages and introduces many of the geometric principles that appear in the remainderofthebook.Inparticular,geometricderivationsofthepoletriangleand the center-point theorem anticipate analytical results for the spherical and spatial cases. Chapter four presents the theory of planar displacements, and Chapter five presentsthealgebraicdesigntheory.Thebilinearstructureofthedesignequations providesasolutionstrategythatemphasizesthegeometryunderlyinglinearalgebra. Thefive-positionsolutionincludesaneliminationstepthatisprobablynewtomost students,thoughitisunderstoodandwellreceivedintheclassroom. Chapters six and seven introduce the properties of spherical linkages and de- tail the geometric theory of spatial rotations (now Chapters seven and eight, 2nd edition). Chapter eight presents the design theory for these linkages (now Chapter nine),whichisanalogoustotheplanartheory.Thismaterialexercisesthestudent’s use of vector methods to represent geometry in three dimensions. Perpendicular bisectorsintheplanardesigntheorybecomeperpendicularbisectingplanesthatin- tersecttodefineaxes.Theanalogueprovidesstudentswithageometricperspective ofthelinearequationsthattheyaresolving. Chapter nine introduces the analysis of spatial linkages including open chains thatarecloselyrelatedtorobotmanipulators(nowChapter11).Thecomplexityof spatiallinkagesrequirestheintroductionofnewtechniques.However,wemaintain apointofviewthatemphasizesthesimilaritytotheplanarandsphericaltheories. Forexample,theconstraintequationsofplanarandsphericallinkagesareshownto bespecialcasesofthoseforspatiallinkages. Chapter 10 develops the geometry of spatial displacements (now Chapter 12). Here,wefindthatthescrewtriangleandthecenter-axistheoremmustbeformulated using lines rather than points. Dual vector algebra is introduced to provide vector operationsforcalculationswiththePlu¨ckercoordinatesoflines.Theresultisthat geometriccalculationswithlinecoordinatesareidenticaltothemorefamiliarvector calculationswithpointcoordinates. Chapter 11 presents the design theory for spatial chains, and Chapter 12 intro- ducesthegeometryoflinearcombinationsoflinesthatariseintheconstructionof spatiallinkagesystems(nowChapters13and16).Whilethedesigntechniquesfor planarlinkagesarewelldeveloped,thereisroomformuchmoreworkinthedesign anduseofspatiallinkages. I am pleasedto express my gratitude for thecontribution of many teachers and colleagues whose work over the years has developed and clarified linkage design theory. This book includes results of the insight, commitment, and hard work by JeffGe,MohanBodduluri,JohnDooley,PierreLarochelle,AndrewMurray,Fangli Hao,CurtisCollins,AlanRuth,ShawnAhlers,andAlbaPerez.Ihavealsobenefit- tedfromtheinsightofQizhengLiaoandtheinspirationofBernardRoth.Finally, the support by the Division of Design, Manufacturing, and Industrial Innovation of the National Science Foundation that has made this book possible is gratefully acknowledged. Irvine,CA, J.MichaelMcCarthy