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DYNAMICMODELINGOF PARALLELMANIPULATORS -'- By IVANJ.BAIGES-VALENTIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1996 Copyright1995 by IvanJ.Baiges-Valentin TheauthorwouldliketodedicatethisworktohisparentsSalvadorandMonsitaBaiges andtohisbrothersandsisterswhosupportedandencouragedhimalltheway. Thiswork isalsodedicatedtoZitaSumazawhosefriendshipwasagreatsourceofinspirationduring themostdifficultmoments. Finallyandmostimportant,thisworkisdedicatedtothe author'sdaughterGiannaMariewithallhislove. ThisworkisdedicatedtothememoryofProf.RichardH.Zimmerman. ACKNOWLEDGEMENTS TheauthorwouldliketothanktheUniversityofFloridaandtheCenterforRobotics andIntelligentMachinesfortheopportunityofcompletingthePh.D.degreeandforthe financialsupport. TheauthorwouldliketothanktheUniverityofPuertoRicoat MayagiiezandtheEconomicDevelopmentAgencyofPuertoRicofortheirsupport. TheauthorwouldliketorecognizethegreatcontributionstothisworkbyDr.Joseph DuffyandDr.CarlCrane. TheauthorisgreatlyindebtedtothestudentsandstaffoftheCenterforRoboticsand IntelligentMachinesfortheirhelpandmostimportantlyfortheirgreatfriendshipwhich madeallthisworkpossible. iv TABLEOFCONTENTS page ACKNOWLEDGMENTS iv ABSTRACT vii CHAPTERS INTRODUCTION 1 1 1.1 Previous Work 2 1.2 Objectives 4 1.3 CAE Tool for Parallel Manipulators 5 1.4 Research Outline 8 2 SYSTEM LUMPED PARAMETER MODEL 10 2.1 Energy Coupling 10 2.2 Basic Connector Model 13 2.3 Implicit Force Control 16 2.4 Modified Implicit Force Control Mode 18 2.5 CombinedHighFrequency/ Low Frequency Connector Model 20 2.6 Connector Model for Dynamic Modeling 22 3 PARALLEL MANIPULATOR KINEMATICS 24 3.1 Forward and Inverse Kinematics 24 3.2 Kinematic Relationships 25 3.3 Position Analysis 27 3.4 Velocity Analysis 31 3.5 Acceleration Analysis 41 3.6 Motion Planning 46 4 MANIPULATORFORCEANDTORQUEANALYSIS 54 4.1 Platform External Forces and Torques 54 4.2 Gravitational Forces and Torques 61 4.3 Connector Forces and Torques 62 4.4 Platform Resultant Forces and Torques 63 4.5 System Singularities 64 4.6 Connector Force Analysis 67 V 5 PARALLELMANIPULATORDYNAMICMODELING 75 5.1 SelectionofaDynamicFormulationMethod 75 5.2 Mobility Analysis 79 5.3 GeneralizedSpeeds,VelocityandAccelerationAnalysis 82 5.4 VelocityandAngularVelocityPartialDerivatives 88 5.5 Setting up the Dynamic Model 94 5.6 Deriving the Equations ofMotion 100 5.7 Dynamic Model Validation 117 5.8 Summary 126 6 DYNAMIC SIMULATION ALGORITHM 128 6.1 Motion and Task Planning 128 6.2 Inverse Kinematic Simulation Algorithm 129 6.3 Equations of Motion 133 6.4 DevelopmentoftheDynamicSimulationSoftware 137 6.5 Dynamic Simulation Software 143 7 TESTING AND RESULTS 146 7.1 System Geometrical Description 147 7.2 System Parameters 151 7.3 Motion and Task Planning for Testing 154 7.4 Test Cases, Results and Discussion 156 7.5 Summary 186 8 CONCLUSIONS AND RECOMMENDATIONS 192 8.1 Developing the Dynamic Model 192 8.2 Dynamic BehaviorofParallelManipulators 194 8.3 Recommendations 195 8.4 Summary 198 APPENDIX 200 REFERENCES 223 BIOGRAPHICAL SKETCH 226 vi , >:. AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DYNAMICMODELINGOF PARALLELMANIPULATORS By IVANJ.BAIGES-VALENTIN 1996 Chairman:Dr.JosephDuffy MajorDepartment:MechanicalEngineering Parallelmanipulators,whichconsistoftworigidbodies(abaseandaplatform) connectedbysixserialkinematicchains(connectors),offerdistinctadvantageswhen comparedtotheirserialcounterparts. Themotionoftheplatformwithrespecttothebaseis controlledbythedisplacementofthesueparallelconnectorswhichgenerateaverystiffand accuratemanipulatorcapableofhandlinghighpayloadswithminimalpositioningerrors. Thesecharacteristicshavegeneratedgreatinterestintheuseofthistypeofmanipulatorin applicationssuchasmachiningprocessesandautomatedassemblyoperations. Itisclearlydesirabletodevelopacomprehensivedynamicmodelforeffectively designingandcontrollingtheparallelmanipulators. Themajorobjectiveofthisresearchistoderiveexplicitequationsofmotionforparallel manipulators.Thiswillprovidethemeanstounderstandtheirdynamicbehaviorandenable thedesignofmoreefficientdevicescapableoffastandaccuratemotionswhilehandling heavypayloads. Asfarastheauthorisaware,thisisthefirsttimetheexplicitequationsof motionhavebeenderived(asopposedtonumericalsolutions). Theywerederivedusing Kane'sMethodwhichprovedtobewellsuitedtotheintricatekinematicsofparallel manipulators,andverifiedwiththeNewton-Eulerformulation. Subsequently,the equationswereusedforainversedynamicsimulation,whichcalculatedtheactuatorforces vii requiredforproducingadesiredmotionofagivendesignwhilstsupportingaworkpiecein amachiningoperation. Variousgeometricparameters,inertialpropertiesandmotion profileswereusedfortestinginordertounderstandsomeoftheeffectsonthedynamic behavior. Theequationsofmotionindicateahighdegreeofcouplingbetweentheconnectors causedbythegravitational,tangential,Coriohsandcentrifugalaccelerationsactingupon thesystem. Amostimportantresultoftheinversedynamicsimulationsisthatevenfor feedratesinexcessofthelimitsofexistingtechnology(1200inches/min),theonly significantcouplingwasduetogravity;theothercouplingeffectswerenegUgible.This suggeststhepossibiUtyofusingmassbalancingtoreducethecouplingeffectsbetweenthe connectorsandintheprocesscreatingfasterandmoreaccurateparallelmanipulators. viii CHAPTER1 INTRODUCTION Manipulatorsorrobotscanbeclassifiedaccordingtothetypeofkinematicchainused fortheirimplementation. Serialmanipulatorsarebasedonopenkinematicchainssuchas the3degree-of-freedommanipulatorshownonthelefthandsideoffigure1.1. Parallel manipulatorsarebasedonclosedkinematicchainssuchasthe3degree-of-freedomsystem shownontherighthandsideofFigure1.1. Saialmanipulatorsgenerallyhavelongreach,agreatdegreeofdexterity,coveralarge workspaceandcanentersmallspacesamongotheradvantages. However,serial manipulatorshavelowstiffnessandhighlycouplednonlineardynamicbehavior,specially forhighspeedandhighpayloadapplications. Themainreasonfortheselimitationsisthe thattheserialmanipulatorisachainofcantilevertypestructures,wherethecomplianceand positioningerrorsareaccumulativeeffectoftheindividualjoints. Figure1.1- SerialandParallelManipulators Platform Figure1.2- SpatialParallelManipulator Inparallelmanipulators (seeFigure1.2)thelinksactinparallel, generatingaverystiff andaccuratemanipulatorwhichcancarryahighpayloadwithminimalpositioningerrors. Ontheotherhand,parallelmanipulatorscoversmallerworkspacesandarelessdexterous thantheserialmanipulators.Theparticularnatureoftheparallelmanipulatorhasgenerated greatinterestinroboticapplicationswherehighloadsandprecisionarerequiredsuchasin machiningprocessesandautomatedassemblyoperations. Themosttypicalparallel manipulatoristheStewartPlatform[1]whichconsistsoftworigidbodiesconnectedbysix linearparallelactuators.Themotionofoneoftherigidbodies(namelytheplatform)with respecttotheother(thebase)iscontrolledbythedisplacementofthesixlinearactuators. 1.1PreviousWork Theparallelmanipulatorpresentsadifferentsetofproblemsthantheserial manipulators. Asanexampletheinversekinematicanalysisissimpletodetermine comparedtotheforwardkinematicanalysisproblem,whichistheoppositewithserial