机器人学导论chapter4.docx

机器人学导论chapter4Chapter4PlanarKinematicsKinematicsisGeometryofMotion.Itisoneofthemostfundamentaldisciplinesinrobotics,providingtoolsfordescribingthestructureandbehaviorofrobotmechanisms.Inthischapter,wewilldiscusshowthemotionofarobotmechanismisdescribed,howitrespondstoactuatormovements,andhowtheindividualactuatorsshouldbecoordinatedtoobtaindesiredmotionattherobotend-effecter.Thesearequestionscentraltothedesignandcontrolofrobotmechanisms.Tobeginwith,wewillrestrictourselvestoaclassofrobotmechanismsthatworkwithinaplane,i.e.PlanarKinematics.Planarkinematicsismuchmoretractablemathematically,comparedtogeneralthree-dimensionalkinematics.Nonetheless,mostoftherobotmechanismsofpracticalimportancecanbetreatedasplanarmechanisms,orcanbereducedtoplanarproblems.Generalthree-dimensionalkinematics,ontheotherhand,needsspecialmathematicaltools,whichwillbediscussedinlaterchapters.4.1PlanarKinematicsofSerialLinkMechanismsExample4.1Considerthethreedegree-of-freedomplanarrobotarmshowninFigure4.1.1.Thearmconsistsofonefixedlinkandthreemovablelinksthatmovewithintheplane.Allthelinksareconnectedbyrevolutejointswhosejointaxesareallperpendiculartotheplaneofthelinks.Thereisnoclosed-loopkinematicchain;hence,itisaseriallinkmechanism.Figure4.1.1ThreedofplanarrobotwiththreerevolutejointsTodescribethisrobotarm,afewgeometricparametersareneeded.First,thelengthofeachlinkisdefinedtobethedistancebetweenadjacentjointaxes.LetpointsO,A,andBbethelocationsofthethreejointaxes,respectively,andpointEbeapointfixedtotheend-effecter.ThenthelinklengthsareEBBAAO=321,AAA.LetusassumethatActuator1drivinglink1isfixedtothebaselink(link0),generatingangle1,whileActuator2drivinglink2isfixedtothetipofLink1,creatingangle2betweenthetwolinks,andActuator3drivingLink3isfixedtothetipofLink2,creatingangle3,asshowninthefigure.Sincethisrobotarmperformstasksbymovingitsend-effecteratpointE,weareconcernedwiththelocationoftheend-effecter.Todescribeitslocation,weuseacoordinatesystem,O-xy,fixedtothebaselinkwiththeoriginatthefirstjoint,anddescribetheend-effecterpositionwithcoordinateseande.Wecanrelatetheend-effectercoordinatestothejointanglesdeterminedbythethreeactuatorsbyusingthelinklengthsandjointanglesdefinedabove:xy)cos()cos(cos321321211+=AAAex(4.1.1)sin()sin(sin321321211+=AAAey(4.1.2)Thisthreedofrobotarmcanlocateitsend-effecteratadesiredorientationaswellasatadesiredposition.Theorientationoftheend-effectercanbedescribedastheanglethecenterlineoftheend-effectermeasuredfromthepositivexcoordinateaxis.Thisend-effecterorientationeisrelatedtotheactuatordisplacementsas321+=e(4.1.3)viewedfromthefixedcoordinatesysteminrelationtotheactuatordisplacements.Ingeneral,asetofalgebraicequationsrelatingthepositionandorientationofarobotend-effecter,oranysignificantpartoftherobot,toactuatororactivejointdisplacements,iscalledKinematicEquations,ormorespecifically,ForwardKinematicEquationsintheroboticsliterature.Exercise4.1ShownbelowinFigure4.1.2isaplanarrobotarmwithtworevolutejointsandoneprismaticjoint.Usingthegeometricparametersandjointdisplacements,obtainthekinematicequationsrelatingtheend-effecterpositionandorientationtothejointdisplacements.Figure4.1.2ThreedofrobotwithtworevolutejointsandoneprismaticjointNowthattheaboveExampleandExerciseproblemshaveillustratedkinematicequations,letusobtainaformalexpressionforkinematicequations.Asmentionedinthepreviouschapter,twotypesofjoints,prismaticandrevolutejoints,constituterobotmechanismsinmostcases.Thedisplacementofthei-thjointisdescribedbydistancediifitisaprismaticjoint,andbyangleiforarevolutejoint.Forformalexpression,letususeagenericnotation:qi.Namely,jointdisplacementqirepresentseitherdistancediorangleidependingonthetypeofjoint.iiidq=(4.1.4)PrismaticjointRevolutejointWecollectivelyrepresentallthejointdisplacementsinvolvedinarobotmechanismwithacolumnvector:,wherenisthenumberofjoints.Kinematicequationsrelatethesejointdisplacementstothepositionandorientationoftheend-effecter.Letuscollectivelydenotetheend-effecterpositionandorientationbyvectorp.Forplanarmechanisms,theend-effecterlocationisdescribedbythreevariables:Tnqqqq"21=?=eeeyxp(4.1.5)Usingthesenotations,werepresentkinematicequationsasavectorfunctionrelatingptoq:113,),(nxxqpqfp?=(4.1.6)Foraseriallinkmechanism,allthejointsareusuallyactivejointsdrivenbyindividualactuators.Exceptforsomespecialcases,theseactuatorsuniquelydeterminetheend-effecterpositionandorientationaswellastheconfigurationoftheentirerobotmechanism.Ifthereisalinkwhoselocationisnotfullydeterminedbytheactuatordisplacements,sucharobotmechanismissaidtobeunder-actuated.Unlessarobotmechanismisunder-actuated,thecollectionofthejointdisplacements,i.e.thevectorq,uniquelydeterminestheentirerobotconfiguration.Foraseriallinkmechanism,thesejointsareindependent,havingnogeometricconstraintotherthantheirstrokelimits.Therefore,thesejointdisplacementsaregeneralizedcoordinatesthatlocatetherobotmechanismuniquelyandcompletely.Formally,thenumberofgeneralizedcoordinatesiscalleddegreesoffreedom.Vectorqiscalledjointcoordinates,whentheyformacompleteandindependentsetofgeneralizedcoordinates.4.2InverseKinematicsofPlanarMechanismsThevectorkinematicequationderivedintheprevioussectionprovidesthefunctionalrelationshipbetweenthejointdisplacementsandtheresultantend-effecterpositionandorientation.Bysubstitutingvaluesofjointdisplacementsintotheright-handsideofthekinematicequation,onecanimmediatelyfindthecorrespondingend-effecterpositionandorientation.Theproblemoffindingtheend-effecterpositionandorientationforagivensetofjointdisplacementsisreferredtoasthedirectkinematicsproblem.Thisissimplytoevaluatetheright-handsideofthekinematicequationforknownjointdisplacements.Inthissection,wediscusstheproblemofmovingtheend-effecterofamanipulatorarmtoaspecifiedpositionandorientation.Weneedtofindthejointdisplacementsthatleadtheend-effectertothespecifiedpositionandorientation.Thisistheinverseofthepreviousproblem,andisthusreferredtoastheinversekinematicsproblem.Thekinematicequationmustbesolvedforjointdisplacements,giventheend-effecterpositionandorientation.Oncethekinematicequationissolved,thedesiredend-effectermotioncanbeachievedbymovingeachjointtothedeterminedvalue.Inthedirectkinematicsproblem,theend-effecterlocationisdetermineduniquelyforanygivensetofjointdisplacements.Ontheotherhand,theinversekinematicsismorecomplexinthesensethatmultiplesolutionsmayexistforthesameend-effecterlocation.Also,solutionsmaynotalwaysexistforaparticularrangeofend-effecterlocationsandarmstructures.Furthermore,sincethekinematicequationiscomprisedofnonlinearsimultaneousequationswithmanytrigonometricfunctions,itisnotalwayspossibletoderiveaclosed-formsolution,whichistheexplicitinversefunctionofthekinematicequation.Whenthekinematicequationcannotbesolvedanalytically,numericalmethodsareusedinordertoderivethedesiredjointdisplacements.Example4.2ConsiderthethreedofplanararmshowninFigure4.1.1again.Tosolveitsinversekinematicsproblem,thekinematicstructureisredrawninFigure4.2.1.Theproblemistofindthreejointangles321,thatleadtheendeffectertoadesiredpositionandorientation,eeeyx,.Wetakeatwo-stepapproach.First,wefindthepositionofthewrist,pointB,fromeeeyx,.Thenwefind21,fromthewristposition.Angle3canbedeterminedimmediatelyfromthewristposition.Figure4.2.1SkeletonstructureoftherobotarmofExample4.1Letwandwbethecoordinatesofthewrist.AsshowninFigure4.2.1,pointBisatdistance3fromthegivenend-effecterpositionE.MovingintheoppositedirectiontotheendeffecterorientationxyAe,thewristcoordinatesaregivenbyeeweewyyxxsincos33AA?=?=(4.2.1)Notethattherighthandsidesoftheaboveequationsarefunctionsofeeeyx,alone.Fromthesewristcoordinates,wecandeterminetheangleshowninthefigure.1wwxy1tan?=(4.2.2)Next,letusconsiderthetriangleOABanddefineangles,asshowninthefigure.ThistriangleisformedbythewristB,theelbowA,andtheshoulderO.Applyingthelawofcosinestotheelbowangleyields2212221cos2r=?+AAAA(4.2.3)where,thesquareddistancebetweenOandB.Solvingthisforangle222wwyxr+=yields21222221122cosAAAAwwyx?+?=?=?(4.2.4)Similarly,221212cos2AAA=?+rr(4.2.5)Solvingthisforyields2212221221112costanwwwwwwyxyxxy+?+?=?=?AAA(4.2.6)Fromtheabove21,wecanobtain213?=e(4.2.7)Eqs.(4),(6),and(7)provideasetofjointanglesthatlocatestheend-effecteratthedesiredpositionandorientation.Itisinterestingtonotethatthereisanotherwayofreachingthesameend-effecterpositionandorientation,i.e.anothersolutiontotheinversekinematicsproblem.Figure4.2.2showstwoconfigurationsofthearmleadingtothesameend-effecterlocation:theelbowdownconfigurationandtheelbowupconfiguration.Theformercorrespondstothesolutionobtainedabove.Thelatter,havingtheelbowpositionatpointA,issymmetrictotheformerconfigurationwithrespecttolineOB,asshowninthefigure.Therefore,thetwosolutionsarerelatedas22''''2'232132211?+=?=?=+=e(4.2.8)Inversekinematicsproblemsoftenpossessmultiplesolutions,liketheaboveexample,sincetheyarenonlinear.Specifyingend-effecterpositionandorientationdoesnotuniquelydeterminethewholeconfigurationofthesystem.Thisimpliesthatvectorp,thecollectivepositionandorientationoftheend-effecter,cannotbeusedasgeneralizedcoordinates.Theexistenceofmultiplesolutions,however,providestherobotwithanextradegreeofflexibility.Considerarobotworkinginacrowdedenvironment.Ifmultipleconfigurationsexistforthesameend-effecterlocation,therobotcantakeaconfigurationhavingnointerferencewith1Unlessnotedspecificallyweassumethatthearctangentfunctiontakesanangleinaproperquadrantconsistentwiththesignsofthetwooperands.theenvironment.Duetophysicallimitations,however,thesolutionstotheinversekinematicsproblemdonotnecessarilyprovidefeasibleconfigurations.Wemustcheckwhethereachsolutionsatisfiestheconstraintofmovablerange,i.e.strokelimitofeachjoint.11Elbow-UpConfigurationFigure4.2.2MultiplesolutionstotheinversekinematicsproblemofExample4.24.3KinematicsofParallelLinkMechanismsExample4.3Considerthefive-bar-linkplanarrobotarmshowninFigure4.3.1.22112211sinsincoscosAAAA+=+=eeyx(4.3.1)NotethatJoint2isapassivejoint.Hence,angle2isadependentvariable.Using2,however,wecanobtainthecoordinatesofpointA:25112511sinsincoscosAAAA+=+=AAyx(4.3.2)PointAmustbereachedviathebranchcomprisingLinks3and4.Therefore,44334433sinsincoscosAAAA+=+=AAyx (4.3.3)Equatingthesetwosetsofequationsyieldstwoconstraintequations:4433251144332511sinsinsinsincoscoscoscosAAAAAAAA+=+=+(4.3.4)Notethattherearefourvariablesandtwoconstraintequations.Therefore,twoofthevariables,suchas31,areindependent.Itshouldalsobenotedthatmultiplesolutionsexistfortheseconstraintequations.xLink0Figure4.3.1Five-bar-linkmechanismAlthoughtheforwardkinematicequationsaredifficulttowriteoutexplicitly,theinversekinematicequationscanbeobtainedforthisparallellinkmechanism.Theproblemistofind31,thatleadtheendpointtoadesiredposition:.Wewilltakethefollowingprocedure:eeyx,Step1Given,findeeyx,21,bysolvingthetwo-linkinversekinematicsproblem.Step2Given21,obtain.Thisisaforwardkinematicsproblem.AAyx,Step3Given,findAAyx,43,bysolvinganothertwo-linkinversekinematicsproblem.Example4.4Obtainthejointanglesofthedogslegs,giventhebodypositionandorientation.Figure4.3.2AdoggyrobotwithtwolegsonthegroundTheinversekinematicsproblem:Step1GivenBBByx,findandAAyx,CCyx,Step2Given,findAAyx,21,Step3Given,findCCyx,43,4.4RedundantmechanismsAmanipulatorarmmusthaveatleastsixdegreesoffreedominordertolocateitsend-effecteratanarbitrarypointwithanarbitraryorientationinspace.Manipulatorarmswithlessthan6degreesoffreedomarenotabletoperformsucharbitrarypositioning.Ontheotherhand,ifamanipulatorarmhasmorethan6degreesoffreedom,thereexistaninfinitenumberofsolutionstothekinematicequation.Considerforexamplethehumanarm,whichhassevendegreesoffreedom,excludingthejointsatthefingers.Evenifthehandisfixedonatable,onecanchangetheelbowpositioncontinuouslywithoutchangingthehandlocation.Thisimpliesthatthereexistaninfinitesetofjointdisplacementsthatleadthehandtothesamelocation.Manipulatorarmswithmorethansixdegreesoffreedomarereferredtoasredundantmanipulators.Wewilldiscussredundantmanipulatorsindetailinthefollowingchapter.