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1、机器人学导论chapter4Chapter4PlanarKinematicsKinematicsisGeometryofMotion.Itisoneofthemostfundamentaldisciplinesinrobotics,providingtoolsfordescribingthestructureandbehaviorofrobotmechanisms.Inthischapter,wewilldiscusshowthemotionofarobotmechanismisdescribed,howitrespondstoactuatormovements,andhowtheindivi
2、dualactuatorsshouldbecoordinatedtoobtaindesiredmotionattherobotend-effecter.Thesearequestionscentraltothedesignandcontrolofrobotmechanisms.Tobeginwith,wewillrestrictourselvestoaclassofrobotmechanismsthatworkwithinaplane,i.e.PlanarKinematics.Planarkinematicsismuchmoretractablemathematically,comparedt
3、ogeneralthree-dimensionalkinematics.Nonetheless,mostoftherobotmechanismsofpracticalimportancecanbetreatedasplanarmechanisms,orcanbereducedtoplanarproblems.Generalthree-dimensionalkinematics,ontheotherhand,needsspecialmathematicaltools,whichwillbediscussedinlaterchapters.4.1PlanarKinematicsofSerialLi
4、nkMechanismsExample4.1Considerthethreedegree-of-freedomplanarrobotarmshowninFigure4.1.1.Thearmconsistsofonefixedlinkandthreemovablelinksthatmovewithintheplane.Allthelinksareconnectedbyrevolutejointswhosejointaxesareallperpendiculartotheplaneofthelinks.Thereisnoclosed-loopkinematicchain;hence,itisase
5、riallinkmechanism.Figure4.1.1ThreedofplanarrobotwiththreerevolutejointsTodescribethisrobotarm,afewgeometricparametersareneeded.First,thelengthofeachlinkisdefinedtobethedistancebetweenadjacentjointaxes.LetpointsO,A,andBbethelocationsofthethreejointaxes,respectively,andpointEbeapointfixedtotheend-effe
6、cter.ThenthelinklengthsareEBBAAO=321,AAA.LetusassumethatActuator1drivinglink1isfixedtothebaselink(link0),generatingangle1,whileActuator2drivinglink2isfixedtothetipofLink1,creatingangle2betweenthetwolinks,andActuator3drivingLink3isfixedtothetipofLink2,creatingangle3,asshowninthefigure.Sincethisrobota
7、rmperformstasksbymovingitsend-effecteratpointE,weareconcernedwiththelocationoftheend-effecter.Todescribeitslocation,weuseacoordinatesystem,O-xy,fixedtothebaselinkwiththeoriginatthefirstjoint,anddescribetheend-effecterpositionwithcoordinateseande.Wecanrelatetheend-effectercoordinatestothejointanglesd
8、eterminedbythethreeactuatorsbyusingthelinklengthsandjointanglesdefinedabove:xy)cos()cos(cos321321211+=AAAex(4.1.1)sin()sin(sin321321211+=AAAey(4.1.2)Thisthreedofrobotarmcanlocateitsend-effecteratadesiredorientationaswellasatadesiredposition.Theorientationoftheend-effectercanbedescribedastheanglethec
9、enterlineoftheend-effectermeasuredfromthepositivexcoordinateaxis.Thisend-effecterorientationeisrelatedtotheactuatordisplacementsas321+=e(4.1.3)viewedfromthefixedcoordinatesysteminrelationtotheactuatordisplacements.Ingeneral,asetofalgebraicequationsrelatingthepositionandorientationofarobotend-effecte
10、r,oranysignificantpartoftherobot,toactuatororactivejointdisplacements,iscalledKinematicEquations,ormorespecifically,ForwardKinematicEquationsintheroboticsliterature.Exercise4.1ShownbelowinFigure4.1.2isaplanarrobotarmwithtworevolutejointsandoneprismaticjoint.Usingthegeometricparametersandjointdisplac
11、ements,obtainthekinematicequationsrelatingtheend-effecterpositionandorientationtothejointdisplacements.Figure4.1.2ThreedofrobotwithtworevolutejointsandoneprismaticjointNowthattheaboveExampleandExerciseproblemshaveillustratedkinematicequations,letusobtainaformalexpressionforkinematicequations.Asmenti
12、onedinthepreviouschapter,twotypesofjoints,prismaticandrevolutejoints,constituterobotmechanismsinmostcases.Thedisplacementofthei-thjointisdescribedbydistancediifitisaprismaticjoint,andbyangleiforarevolutejoint.Forformalexpression,letususeagenericnotation:qi.Namely,jointdisplacementqirepresentseitherd
13、istancediorangleidependingonthetypeofjoint.iiidq=(4.1.4)PrismaticjointRevolutejointWecollectivelyrepresentallthejointdisplacementsinvolvedinarobotmechanismwithacolumnvector:,wherenisthenumberofjoints.Kinematicequationsrelatethesejointdisplacementstothepositionandorientationoftheend-effecter.Letuscol
14、lectivelydenotetheend-effecterpositionandorientationbyvectorp.Forplanarmechanisms,theend-effecterlocationisdescribedbythreevariables:Tnqqqq21=?=eeeyxp(4.1.5)Usingthesenotations,werepresentkinematicequationsasavectorfunctionrelatingptoq:113,),(nxxqpqfp?=(4.1.6)Foraseriallinkmechanism,allthejointsareu
15、suallyactivejointsdrivenbyindividualactuators.Exceptforsomespecialcases,theseactuatorsuniquelydeterminetheend-effecterpositionandorientationaswellastheconfigurationoftheentirerobotmechanism.Ifthereisalinkwhoselocationisnotfullydeterminedbytheactuatordisplacements,sucharobotmechanismissaidtobeunder-a
16、ctuated.Unlessarobotmechanismisunder-actuated,thecollectionofthejointdisplacements,i.e.thevectorq,uniquelydeterminestheentirerobotconfiguration.Foraseriallinkmechanism,thesejointsareindependent,havingnogeometricconstraintotherthantheirstrokelimits.Therefore,thesejointdisplacementsaregeneralizedcoord
17、inatesthatlocatetherobotmechanismuniquelyandcompletely.Formally,thenumberofgeneralizedcoordinatesiscalleddegreesoffreedom.Vectorqiscalledjointcoordinates,whentheyformacompleteandindependentsetofgeneralizedcoordinates.4.2InverseKinematicsofPlanarMechanismsThevectorkinematicequationderivedinthepreviou
18、ssectionprovidesthefunctionalrelationshipbetweenthejointdisplacementsandtheresultantend-effecterpositionandorientation.Bysubstitutingvaluesofjointdisplacementsintotheright-handsideofthekinematicequation,onecanimmediatelyfindthecorrespondingend-effecterpositionandorientation.Theproblemoffindingtheend
19、-effecterpositionandorientationforagivensetofjointdisplacementsisreferredtoasthedirectkinematicsproblem.Thisissimplytoevaluatetheright-handsideofthekinematicequationforknownjointdisplacements.Inthissection,wediscusstheproblemofmovingtheend-effecterofamanipulatorarmtoaspecifiedpositionandorientation.
20、Weneedtofindthejointdisplacementsthatleadtheend-effectertothespecifiedpositionandorientation.Thisistheinverseofthepreviousproblem,andisthusreferredtoastheinversekinematicsproblem.Thekinematicequationmustbesolvedforjointdisplacements,giventheend-effecterpositionandorientation.Oncethekinematicequation
21、issolved,thedesiredend-effectermotioncanbeachievedbymovingeachjointtothedeterminedvalue.Inthedirectkinematicsproblem,theend-effecterlocationisdetermineduniquelyforanygivensetofjointdisplacements.Ontheotherhand,theinversekinematicsismorecomplexinthesensethatmultiplesolutionsmayexistforthesameend-effe
22、cterlocation.Also,solutionsmaynotalwaysexistforaparticularrangeofend-effecterlocationsandarmstructures.Furthermore,sincethekinematicequationiscomprisedofnonlinearsimultaneousequationswithmanytrigonometricfunctions,itisnotalwayspossibletoderiveaclosed-formsolution,whichistheexplicitinversefunctionoft
23、hekinematicequation.Whenthekinematicequationcannotbesolvedanalytically,numericalmethodsareusedinordertoderivethedesiredjointdisplacements.Example4.2ConsiderthethreedofplanararmshowninFigure4.1.1again.Tosolveitsinversekinematicsproblem,thekinematicstructureisredrawninFigure4.2.1.Theproblemistofindthr
24、eejointangles321,thatleadtheendeffectertoadesiredpositionandorientation,eeeyx,.Wetakeatwo-stepapproach.First,wefindthepositionofthewrist,pointB,fromeeeyx,.Thenwefind21,fromthewristposition.Angle3canbedeterminedimmediatelyfromthewristposition.Figure4.2.1SkeletonstructureoftherobotarmofExample4.1Letwa
25、ndwbethecoordinatesofthewrist.AsshowninFigure4.2.1,pointBisatdistance3fromthegivenend-effecterpositionE.MovingintheoppositedirectiontotheendeffecterorientationxyAe,thewristcoordinatesaregivenbyeeweewyyxxsincos33AA?=?=(4.2.1)Notethattherighthandsidesoftheaboveequationsarefunctionsofeeeyx,alone.Fromth
26、esewristcoordinates,wecandeterminetheangleshowninthefigure.1wwxy1tan?=(4.2.2)Next,letusconsiderthetriangleOABanddefineangles,asshowninthefigure.ThistriangleisformedbythewristB,theelbowA,andtheshoulderO.Applyingthelawofcosinestotheelbowangleyields2212221cos2r=?+AAAA(4.2.3)where,thesquareddistancebetw
27、eenOandB.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-effecteratthedesi
28、redpositionandorientation.Itisinterestingtonotethatthereisanotherwayofreachingthesameend-effecterpositionandorientation,i.e.anothersolutiontotheinversekinematicsproblem.Figure4.2.2showstwoconfigurationsofthearmleadingtothesameend-effecterlocation:theelbowdownconfigurationandtheelbowupconfiguration.T
29、heformercorrespondstothesolutionobtainedabove.Thelatter,havingtheelbowpositionatpointA,issymmetrictotheformerconfigurationwithrespecttolineOB,asshowninthefigure.Therefore,thetwosolutionsarerelatedas222232132211?+=?=?=+=e(4.2.8)Inversekinematicsproblemsoftenpossessmultiplesolutions,liketheaboveexampl
30、e,sincetheyarenonlinear.Specifyingend-effecterpositionandorientationdoesnotuniquelydeterminethewholeconfigurationofthesystem.Thisimpliesthatvectorp,thecollectivepositionandorientationoftheend-effecter,cannotbeusedasgeneralizedcoordinates.Theexistenceofmultiplesolutions,however,providestherobotwithan
31、extradegreeofflexibility.Considerarobotworkinginacrowdedenvironment.Ifmultipleconfigurationsexistforthesameend-effecterlocation,therobotcantakeaconfigurationhavingnointerferencewith1Unlessnotedspecificallyweassumethatthearctangentfunctiontakesanangleinaproperquadrantconsistentwiththesignsofthetwoope
32、rands.theenvironment.Duetophysicallimitations,however,thesolutionstotheinversekinematicsproblemdonotnecessarilyprovidefeasibleconfigurations.Wemustcheckwhethereachsolutionsatisfiestheconstraintofmovablerange,i.e.strokelimitofeachjoint.11Elbow-UpConfigurationFigure4.2.2Multiplesolutionstotheinverseki
33、nematicsproblemofExample4.24.3KinematicsofParallelLinkMechanismsExample4.3Considerthefive-bar-linkplanarrobotarmshowninFigure4.3.1.22112211sinsincoscosAAAA+=+=eeyx(4.3.1)NotethatJoint2isapassivejoint.Hence,angle2isadependentvariable.Using2,however,wecanobtainthecoordinatesofpointA:25112511sinsincosc
34、osAAAA+=+=AAyx(4.3.2)PointAmustbereachedviathebranchcomprisingLinks3and4.Therefore,44334433sinsincoscosAAAA+=+=AAyx (4.3.3)Equatingthesetwosetsofequationsyieldstwoconstraintequations:4433251144332511sinsinsinsincoscoscoscosAAAAAAAA+=+=+(4.3.4)Notethattherearefourvariablesandtwoconstraintequations.Th
35、erefore,twoofthevariables,suchas31,areindependent.Itshouldalsobenotedthatmultiplesolutionsexistfortheseconstraintequations.xLink0Figure4.3.1Five-bar-linkmechanismAlthoughtheforwardkinematicequationsaredifficulttowriteoutexplicitly,theinversekinematicequationscanbeobtainedforthisparallellinkmechanism
36、.Theproblemistofind31,thatleadtheendpointtoadesiredposition:.Wewilltakethefollowingprocedure:eeyx,Step1Given,findeeyx,21,bysolvingthetwo-linkinversekinematicsproblem.Step2Given21,obtain.Thisisaforwardkinematicsproblem.AAyx,Step3Given,findAAyx,43,bysolvinganothertwo-linkinversekinematicsproblem.Examp
37、le4.4Obtainthejointanglesofthedogslegs,giventhebodypositionandorientation.Figure4.3.2AdoggyrobotwithtwolegsonthegroundTheinversekinematicsproblem:Step1GivenBBByx,findandAAyx,CCyx,Step2Given,findAAyx,21,Step3Given,findCCyx,43,4.4RedundantmechanismsAmanipulatorarmmusthaveatleastsixdegreesoffreedominor
38、dertolocateitsend-effecteratanarbitrarypointwithanarbitraryorientationinspace.Manipulatorarmswithlessthan6degreesoffreedomarenotabletoperformsucharbitrarypositioning.Ontheotherhand,ifamanipulatorarmhasmorethan6degreesoffreedom,thereexistaninfinitenumberofsolutionstothekinematicequation.Considerforex
39、amplethehumanarm,whichhassevendegreesoffreedom,excludingthejointsatthefingers.Evenifthehandisfixedonatable,onecanchangetheelbowpositioncontinuouslywithoutchangingthehandlocation.Thisimpliesthatthereexistaninfinitesetofjointdisplacementsthatleadthehandtothesamelocation.Manipulatorarmswithmorethansixdegreesoffreedomarereferredtoasredundantmanipulators.Wewilldiscussredundantmanipulatorsindetailinthefollowingchapter.