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Mathematical Engineering Jorge Angeles Shaoping Bai Kinematics of Mechanical Systems Fundamentals, Analysis and Synthesis Mathematical Engineering SeriesEditors JörgSchröder,InstituteofMechanics,UniversityofDuisburg-Essen,Essen, Germany BernhardWeigand,InstituteofAerospaceThermodynamics, UniversityofStuttgart,Stuttgart,Germany Jan-PhilipSchmidt,UniversitätofHeidelberg,Heidelberg,Germany AdvisoryEditors GünterBrenn,InstitutfürStrömungslehreundWärmeübertragung,TUGraz,Graz, Austria DavidKatoshevski,Ben-GurionUniversityoftheNegev,Beer-Sheva,Israel JeanLevine,CAS-MathematiquesetSystemes,MINES-ParsTech,Fontainebleau, France GabrielWittum,Goethe-UniversityFrankfurtamMain,FrankfurtamMain, Germany BassamYounis,CivilandEnvironmentalEngineering,UniversityofCalifornia, Davis,Davis,CA,USA Today,thedevelopmentofhigh-techsystemsisunthinkablewithoutmathematical modeling and analysis of system behavior. As such, many fields in the modern engineering sciences (e.g. control engineering, communications engineering, mechanicalengineering,androbotics)callforsophisticatedmathematicalmethods inordertosolvethetasksathand. The series Mathematical Engineering presents new or heretofore little-known methods to support engineers in finding suitable answers to their questions, presentingthosemethodsinsuchmannerastomakethemideallycomprehensible andapplicableinpractice. Therefore,theprimaryfocusis—withoutneglectingmathematicalaccuracy—on comprehensibilityandreal-worldapplicability. Tosubmitaproposalorrequestfurtherinformation,pleaseusethePDFProposal Formorcontactdirectly:Dr.ThomasDitzinger([email protected]) IndexedbySCOPUS,zbMATH,SCImago. · Jorge Angeles Shaoping Bai Kinematics of Mechanical Systems Fundamentals, Analysis and Synthesis JorgeAngeles ShaopingBai DepartmentofMechanicalEngineering, DepartmentofMaterialsandProduction CentreforIntelligentMachines AalborgUniversity McGillUniversity Aalborg,Denmark Montreal,QC,Canada ISSN 2192-4732 ISSN 2192-4740 (electronic) MathematicalEngineering ISBN 978-3-031-09543-6 ISBN 978-3-031-09544-3 (eBook) https://doi.org/10.1007/978-3-031-09544-3 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The kinematics of mechanisms is a classical discipline in the broader domain of mechanical systems. The latter include systems of rigid and deformable solids, while the former has mainly focused on systems of rigid bodies. Given that the mechanics of deformable solids depends on the forces applied to the body under study, their motion is not determined solely by the motion undergone by some of thesystembodies.Indeed,theinterplayofforcesandmoments(wrenches)withthe displacements of the system bodies must be taken into account via the principles of continuum mechanics, which is a significantly broad multidiscipline, involving fluidsanddeformablesolids.Inthismultidiscipline,therelationsbetweenwrenches anddisplacementsareincludedviatheconstitutiveequationsrelatingthem. On the contrary, the motion of a system of coupled rigid bodies depends only onthemotionundergonebyagivensubsetofthebodies.Inthiscase,knowingthe motion of that subset, the motion of all the bodies of the system is determined— modulo the possible finite number of postures adopted by a subset of the system bodies,aconcepttermedassemblymodes.Themechanicalsystemsofinteresttothe bookinvolveonlyrigidbodies. Broadly speaking, the systems of rigid bodies studied here can form open and closed chains.Theparadigmmechanicalsystemstudiedintherealmofkinematics is the planar four-bar linkage, composed of four rigid bodies (the links) coupled bymeansofwhatisknownaslowerkinematicpairs (LKP).Thebookfocuseson systems of rigid bodies forming closed loops, mainly intended for one-degree-of- freedom (one-dof) tasks. The four-bar linkage is given due attention, in its three domains: planar, spherical, and spatial. Multi-degree-of-freedom mechanisms of eitheropenorclosedkinematicchainsareofutmostimportanceinrobotics.These systemsareextensivelyandintensivelystudiedintherealmofroboticmechanical systems,wherebyarichliteratureisavailable. Thebookcontainssixchapters.Chapter1providesthefundamentalsforkinematic synthesis, covering two basic concepts, synthesis-equation solving, and numerical methods.Chapter2isdevotedtothequalitativesynthesisoflinkages,inwhichthe layoutofkinematicchainsandpairsisconsidered,todeterminethelinkagedegreeof freedom.Chapters3–5coverinfulldetailsingle-doflinkagesofthreemajortypes, v vi Preface namelyplanar,spherical,andspatialfour-barlinkages.Theirfunctional,motionand pathsynthesesaredescribedinsequenceinthethreechapters.Inthelastchapter,the synthesisofsingle-dofcomplexlinkages,includingsix-barandten-barlinkages,is introducedascasesofapplicationofthemethodologystudiedinthebook. Our aim is to provide comprehensive and systematic knowledge of kinematic- synthesistheoryandmethods.Thebookincludesnotonlytheclassicalfoundationsof kinematicsynthesis,suchastheBurmestertheorem,theRoberts–Chebyshevtheorem, andtheprincipleoftransference,butalsothelatestadvancesinthisareathatwere made by the authors, particularly on coupler-curve synthesis. Moreover, synthesis examples are included in the book to illustrate the synthesis theory and methods. Thebookissuitableforgraduatestudentsofmechanicalengineering,researchersof mechanismandrobotdesign,andmachinedesignengineers.Wetrustthatthebook will guide and inspire engineers, researchers, and students in the design, analysis, anddevelopmentofinnovativeandhigh-performancemachinesinallrelevantfields ofapplications. JorgeAngelesacknowledgesthefinancialsupportreceivedfromCanada’sNatural Sciences and Engineering Research Council (NSERC) throughout his academic careerinCanada.Indeed,NSERC’sDiscoveryGrant programallowedhimtohire toptalentattheRoboticMechanicalSystemsLaboratory(RMSLab),McGillUniver- sity,sincethemid-1980s.Variousgraduatestudentsandvisitingfellowscontributed, withtheirwork,tothedevelopmentofideaswhich,eventually,foundtheirwayinto thebook.AcompletelistofthesetraineesisavailableintheRMSLabwebpage.1 Shaoping Baiacknowledges Danishnational andEUfunding bodies,including Innovationsfonden,IndependentResearchFundDenmark,andEUAALProgramme, amongothers,forprovidingfinancialsupportforresearchininnovativemechanisms andexoskeletonsatAalborgUniversity,Denmark.Heacknowledgesalsocollabora- tionwithlocalindustryandinternationaluniversities,suchasPNBeslagFabricand DalianUniversityofTechnology,amongothers,whichmotivatedresearchtopicson linkagesynthesisandapplications. JorgeAngeles,Ph.D. ProfessorEmeritus,McGillUniversity Montreal,QC,Canada ShaopingBai,Ph.D. Professor,AalborgUniversity Aalborg,Denmark 1http://www.cim.mcgill.ca/:rmsl/Index/index.htm. Contents 1 IntroductiontoKinematicSynthesis ............................. 1 1.1 TheRoleofKinematicSynthesisinMechanicalDesign .......... 1 1.2 Glossary .................................................. 5 1.3 KinematicAnalysisVersusKinematicSynthesis ................ 8 1.3.1 ASummaryofSystemsofAlgebraicEquations .......... 10 1.4 AlgebraicandComputationalTools ........................... 10 1.4.1 TheTwo-DimensionalRepresentationoftheCross Product ............................................. 11 1.4.2 Algebraof2×2Matrices ............................. 13 1.4.3 Algebraof3×3Matrices ............................. 14 1.4.4 Linear-EquationSolving:DeterminedSystems ........... 14 1.4.5 Linear-EquationSolving:OverdeterminedSystems ....... 18 1.5 Nonlinear-EquationSolving:TheDeterminedCase .............. 29 1.5.1 TheNewton–RaphsonMethod ......................... 31 1.6 OverdeterminedNonlinearSystemsofEquations ............... 32 1.6.1 TheNewton–GaussMethod ........................... 33 1.7 PackagesRelevanttoLinkageSynthesis ....................... 36 References ..................................................... 38 2 TheQualitativeSynthesisofKinematicChains .................... 41 2.1 Notation .................................................. 41 2.2 Background ............................................... 42 2.3 KinematicPairs ............................................ 47 2.3.1 The(cid:2)KinematicPair ................................ 49 2.4 GraphRepresentationofKinematicChains ..................... 50 2.5 GroupsofDisplacements .................................... 53 2.5.1 DisplacementSubgroups .............................. 56 2.6 KinematicBonds ........................................... 60 vii viii Contents 2.7 TheChebyshev–Grübler–Kutzbach–HervéFormula ............. 63 2.7.1 TrivialChains ....................................... 63 2.7.2 ExceptionalChains ................................... 66 2.7.3 ParadoxicalChains ................................... 69 2.8 ApplicationstoRobotics .................................... 69 2.8.1 TheSynthesisofRoboticArchitecturesandTheir Drives .............................................. 69 References ..................................................... 73 3 LinkageSynthesisforFunctionGeneration ....................... 75 3.1 Introduction ............................................... 75 3.2 Input-Output(IO)Functions ................................. 76 3.2.1 PlanarFour-BarLinkages ............................. 76 3.2.2 TheDenavit-HartenbergNotation ...................... 79 3.2.3 SphericalFour-Bar-Linkages .......................... 80 3.2.4 SpatialFour-Bar-Linkages ............................. 86 3.3 ExactSynthesis ............................................ 91 3.3.1 PlanarLinkages ...................................... 91 3.3.2 SphericalLinkages ................................... 96 3.3.3 SpatialLinkages ..................................... 98 3.4 AnalysisoftheSynthesizedLinkage .......................... 99 3.4.1 PlanarLinkages ...................................... 99 3.4.2 SphericalFour-BarLinkages .......................... 110 3.4.3 SpatialFour-BarLinkages ............................. 113 3.5 ApproximateSynthesis ...................................... 120 3.5.1 The Approximate Synthesis of Planar Four-Bar Linkages ............................................ 123 3.5.2 TheApproximateSynthesisofSphericalLinkages ........ 125 3.5.3 TheApproximateSynthesisofSpatialLinkages .......... 127 3.6 LinkagePerformanceEvaluation ............................. 131 3.6.1 Planar Linkages: Transmission Angle andTransmissionQuality ............................. 131 3.6.2 Spherical Linkages: Transmission Angle andTransmissionQuality ............................. 136 3.6.3 Spatial Linkages: Transmission Angle andTransmissionQuality ............................. 137 3.7 DesignErrorVersusStructuralError .......................... 139 3.7.1 MinimizingtheStructuralError ........................ 142 3.7.2 Branch-SwitchingDetection ........................... 144 3.7.3 IntroducingaMassiveNumberofDataPoints ............ 145 3.8 SynthesisUnderMobilityConstraints ......................... 145 References ..................................................... 148 Contents ix 4 MotionGeneration ............................................. 151 4.1 Introduction ............................................... 151 4.2 PlanarFour-BarLinkages .................................... 151 4.2.1 DyadSynthesisforThreePoses ........................ 153 4.2.2 DyadSynthesisforFourPoses ......................... 154 4.2.3 DyadSynthesisforFivePoses ......................... 156 4.2.4 CaseStudy:SynthesisofaLandingGearMechanism ..... 158 4.2.5 ThePresenceofaPJointindyadSynthesis .............. 165 4.2.6 ApproximateSynthesis ............................... 170 4.3 SphericalFour-BarLinkages ................................. 174 4.3.1 DyadSynthesisforThreeAttitudes ..................... 177 4.3.2 DyadSynthesisforFourAttitudes ...................... 178 4.3.3 DyadSynthesisforFiveAttitudes ...................... 179 4.3.4 SphericaldyadswithaPJoint ......................... 181 4.3.5 ApproximatedyadSynthesis ........................... 181 4.3.6 Examples ........................................... 189 4.4 SpatialFour-BarLinkages ................................... 195 4.4.1 GeometricConstraintsofCCandRCdyads .............. 196 4.4.2 TheSynthesisoftheCCdyad .......................... 198 4.4.3 TheSynthesisoftheRCdyad .......................... 200 4.4.4 SynthesisofFour-BarLinkages ........................ 202 4.4.5 ASemigraphicalSolutionoftheDirectionEquations ...... 202 4.4.6 SolvingtheLinearEquationsoftheMomentVariables .... 204 4.4.7 CongruencesoftheFixedandtheMovingAxes .......... 204 4.4.8 Examples ........................................... 205 4.4.9 Summary ........................................... 208 References ..................................................... 210 5 TrajectoryGeneration .......................................... 213 5.1 PlanarLinkages ............................................ 214 5.1.1 PlanarPathGenerationwithPrescribedTiming ........... 214 5.1.2 Coupler-CurveSynthesisofPlanarFour-BarLinkages .... 221 5.2 TrajectoryGenerationforSphericalFour-BarLinkages .......... 234 5.2.1 PathSynthesiswithDiscretePositions .................. 235 5.2.2 PathSynthesiswithPrescribedTiming .................. 236 5.2.3 Examples ........................................... 237 5.2.4 Summary ........................................... 240 5.3 PathGenerationforRCCCLinkages .......................... 240 5.3.1 AGenericRCCCLinkage ............................. 240 5.3.2 AnAlternativeCoordinateFrame ...................... 242 5.3.3 ConstraintEquationsoftheCCandRCdyads ............ 243 5.3.4 N-pointPathSynthesis ................................ 245 5.3.5 Example ............................................ 247 5.3.6 Summary ........................................... 249 References ..................................................... 249