RobotArm机器人手臂图文.docx
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RobotArm机器人手臂图文.docx
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RobotArm机器人手臂图文
v=d*2sdt2+x(51+2weXd*s+weX(woeXS+edtXJr='De+~SsandWs=WexWsSdt+dt(52(53d*WSUsingequations49through53,theNewton-Eulerequationsofmotionformanipulatorshavingallrotaryjointscanbederivedandarelistedbelow.Luhetal.giveamoredetailedderivationoftheequationsofmotion.5Forwardequations:
fori=1,2,...,ncxi=Ri-l(i-I+tizi-lGeneralizedd'Alembertequationsofmotion.TheL-Eapproachissimplebutcomputationallyinefficient;theN-Eapproachiscompactandfastincomputationbutbecauseofitsrecursivecomputation,itdoesnotprovidemuchinsightforthedesignofcontrollers.Toobtainanefficientsetofclosed-formmotionequations,wecanusethepositionvectorandrotationmatrixrepresentationtodescribeeachlink'skinematicsinformation,obtainthekineticandpotentialenergiesoftherobotarmtoformtheLagrangianfunction,andapplytheL-Eformulationtoobtaintheequationsofmotion.Detailedderivationispresentedelsewhere.6aij=ix(wjxrj+jixrj+Riai-Iai='aiX(cixr,+cx,Xri+aiBackwardequations:
fori=n,n-1,...Ifi=mji1+Rlfi+j=Fi+Ri+lfi+l+mi(r-i+riXa-i+rini=Ijaj+wjx(licoixRi+lfi+I+Ri+Inj+Iz.IDynamicssummary.Ihavebrieflypresentedthreedifferentformulationsforrobotarmdynamics,whicharecomparedinTable1.TheL-Eapproachiswellstructured(54andcanbeexpressedinmatrixnotationbutiscomputationallyimpossibletouseforreal-timecontrolunlesstheequationsofmotionaresimplified.TheN-Emethodresultsinaefficientsetofrecursiveequations,buttheyaredifficulttouseforderivingadvancedcontrollaws.TheG-Dequationsofmotiongivefairly"well-structured"equationsattheexpenseofhighercomputations.Inshort,ausercanchooseaformulationthatishighlystructuredbutcomputationallyinefficient(L-E,aformulationthathasefficientcomputationsattheexpenseofthemotionequationstructure(N-E,oraformulationthatretainsthestructureoftheproblemwithonlyamoderatecomputingpenalty(G-D.Ri-,wherefnj(60RobotarmcontrolGiventhemotionequationsofamanipulator,thepurposeofrobotarmcontrolistomaintainaprescribedmotionforthearmalongadesiredarmtrajectorybyapplyingcorrectivecompensationtorquestotheactuatorstoadjustforanydeviationsofthearmfromthetrajectory.Currentindustrialpracticeistouseconventionalservomechanismstocontrolmanipulators.However,themotiondynamicsofann-degree-of-freedommanipulatorareinherentlynonlinearandcanbedescribedonlybyasetofnhighlycoupled,nonlinear,second-orderordinarydifferentialequations.Thenonlinearitiesarisefromtheinertialloading,couplingbetweenneighboringjoints,andthegravitationalloadingofthelinks.Furthermore,thedynamicparametersofamanipulatorvarywiththepositionofthejointvariables,whicharethemselvesrelatedbycomplextrigonometrictransformations.Useofaservomechanismmodelsthevaryingdynamicsofamanipulatorinadequatelyandneglectsthecouplingeffectsofthejoints.Asaresult,manipulatorsmoveatslowspeedswithunnecessaryvibrations,makingthemappropriateonlyforlimited-precisiontasks.Severalrobotarmcontrolmethodsareavailable?
'4'8"1-22Themostnotableofthesearetheresolvedmotionratecontrol,orRMRC,IIthecerebellarmodelarticulationcontroller,orCMAC,13near-minimum-timecontrol,12thecomputedtorquetechnique,4'8andadaptivecontrolWi=Ci=ri=T=ai=Ai=RlIi=Fi=Ni=f=ni==ritheangularvelocityoflinkiwithrespecttotheithcoordinatesystemtheangularaccelerationoflinkiwithrespecttotheithcoordinatesystemtheoriginoftheithframewithrespecttothe(i-thframethecenterofmassoflinkiwithrespecttotheithframethelinearaccelerationoflinkiwithrespecttotheithcoordinatesystemthelinearaccelerationofthecenterofmassoflinkiwithrespecttotheithcoordinatesystem=therotationmatrixthatmapspositionvectorsfromtheithcoordinatesystemtothe(i-Ithcoordinatesystemtheinertiaaboutcenterofmassoflinkiwithrespecttotheithcoordinatesystemthetotalexternalforceexertedonlinkiwithrespecttotheithcoordinatesystemthetotalmomentexertedonlinkiwithrespecttotheithcoordinatesystemtheforceexertedonlinkibylinki-Iwithrespecttotheithcoordinatesystemthemomentexertedonlinkibylinki-1withrespecttotheithcoordinatesystemthetorqueexertedonlinkistrategies.18,20,21,22TheRMRCisatechniquefordeterminingthejointangleratesrequiredtocauseamanipulatorendpointtoDecember198277
moveincertaindirectionsthatareexpressedintheworldcoordinatesystem.Tofindtherequired7X,theinverseJacobianmatrixJ(tQ-Iisrequired.OneofthedrawbacksofthismethodistheaddedcomputationneededtofindtheinverseJacobianmatrixandthesingularityproblemassociatedwiththematrixinversion.TheCMACisatablelookupcontrolmethodthatisbasedonneurophysiologicaltheory.Itcomputescontrolfunctionsbyreferringtoatablestoredincomputermemoryratherthanbysolvinganalyticequations.Beforeusefulapplicationsarepossible,however,severalproblemssuchasmemorysizemanagementandaccuracyneedtobesolved.Becauseofthenonlinearityandcomplexityofthedynamicmanipulatormodel,aclosed-formsolutionoftheoptimalcontrolisverydifficult,ifnotimpossible.Near-minimum-timecontrolisbasedonthelinearizationoftheequationsofmotionaboutthenominaltrajectory,andlinearfeedbackand/orsuboptimalcontrollawsareobtainedanalytically.Thiscontrolmethodisstilltoocomplextobeusefulformanipulatorswithfourormoredegreesoffreedomandneglectstheeffectofunknownexternalloads.OneofthebasiccontrolschemesisthecomputedtorquetechniquebasedontheL-EortheN-Eequationsofmotion.Paulconcludedthatclosed-loopdigitalcontrolisimpossibleifthecompleteL-Eequationsofmotionareused,4since2000floatingpointmultiplictionsand1500floatingpointadditionsarerequiredtocomputeallthejointtorquespersetpointforaStanfordarm.IandmycolleagueChungappliedthecomputedtorquetechniquetotheN-Eequationsofmotionandderivedanefficientcontrollawinthejoint-variablespacetoservoaPUMArobotarm.8ThecontrollawiscomputedrecursivelyusingtheN-Eequationsofmotion.UsingaPDP-11/45computer,thefeedbackcontrolequationscanbecomputedwithinthreemillisecondsifallcomplextrigonometricfunctionsareimplementedasatablelookup.ThecomputedtorquetechniqueisbasicallyafeedforwardcontrolandassumesthatwecanaccuratelycomputethecounterpartsofD(a,H(9,7X,andG(6inequa-tion45tominimizetheirnonlineareffectsandcanuseapositionplusderivativecontroltoservothejoints.Thus,thestructureofthecontrollawhastheformT=Da(a[d+Kv+Ha(tJ,t5+Ga(t9(d-d0+KP(fd-](61whereKvisa6x6velocityfeedbackgainmatrix;Kpisa6x6positionfeedbackgainmatrix;andDa(O,Ha(it,?
andGa(JarethecounterpartsofD(a,H(O,0andG(0,respectively,inequation45.Theanalogouscontrollawderivedfromthecomputedtorquetechniquebasedonequations54through60canbeobtainedbysubstituting0iintheseequationswithnns=ornAl+d(0!
-QS+s=idOs(62ad+s=lFK5S+s=l1KAesnwhereKVandKsarethederivativeandpositionfeedbackgainsforjointi,respectively.Thephysicalinterpretationofputtingequation62intotheN-Erecursiveequationscanbeviewedasfollows:
(1Thefirsttermwillgeneratethedesiredtorqueforeachjointifnomodelingerrorexistsandsystemparametersareknown.However,errorsduetobacklash,gearfriction,uncertaintyabouttheinertiaparameters,andtimedelayintheservoloopexist,makingdeviationfromthedesiredjointtrajectoryinevitable.(2TheremainingtermsintheN-Eequationsofmotionwillgeneratethecorrectiontorquetocompensateforsmalldeviationsfromthedesiredjointtrajectory.(3Thecontrollawisaposition-plus-derivativecontrolandcompensatesforinertialloading,couplingeffects,andthegravityloadingofthelinks.(n=numberofdegreesoffreedomoftherobotarm.Table1.ComparisonofrobotarmdynamicsformulationsNEWTON-EULER132nAPPROACHMULTIPLICATIONSLAGRANGE-EULER128n4512n333739n2160n33++GENERALIZEDD'ALEMBERTn(312n2+44n+63298n4ADDITIONS3+781n36559n23+245n611n-4n(9n2+69n+53227878KINEMATICSREPRESENTATIONSEQUATIONSOFMOTION4x4HOMOGENEOUSMATRICESCLOSED-FORMDIFFERENTIALEQUATIONSROTATIONMATRICESANDPOSITIONVECTORSRECURSIVEEQUATIONSROTATIONMATRICESANDPOSITIONVECTORSCLOSED-FORMDIFFERENTIALEQUATIONSCOMPUTER~~~~~~~~~~~~~~~~~~~~~~COMPUTER
Toachievea"criticallydamped"systemforeachjointsubsystem(whichinturnlooselyimpliesthatthewholesystembehavesasacriticallydampedsystem,PauldiscusseshowthefeedbackgainmatricesKpandK,canbechosen.9Adaptivecontrolmethodscanbeusedtomaintaingoodperformanceoverawiderangeofmotionsandpayloads.Amongvariousadaptivemethods,modelreferencedadaptivecontrolisthemostwidelyusedandrelativelyeasytoimplement.IntheMRACmethodproposedbyDubowskyandDesForges,18alinear,secondorder,time-invariantdifferentialequationwasusedasthereferencemodelforeachdegreeoffreedom.Themanipulatoriscontrolledbyadjustingpositionandvelocityfeedbackgainstofollowthereferencemodel.Toadjustpositionandvelocitygains,thesteepestdescentmethodwasusedastheadaptivealgorithm.Inthiscase,designingastableadaptivecontrollawisnoteasy.Consequentlystabilityanalysisiscritica
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