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HybridPosition/ForceControloftheSCORBOTER4pcManipulatorwithNeuralCompensationofNonlinearitiesPiotrGierlakRzeszowUniversityofTechnology,DepartmentofAppliedMechanicsandRobotics8Powsta´nc´owWarszawySt.,35959Rzesz´ow,Polandpgierlakprz.edu.plAbstract.Theproblemofthemanipulatorhybridposition/forcecontrolisnottrivialbecausethemanipulatorisanonlinearobject,whoseparametersmaybeunknown,variableandtheworkingconditionsarechangeable.Theneuralcontrolsystemenablesthemanipulatortobehavecorrectly,evenifthemathematicalmodelofthecontrolobjectisunknown.Inthispaper,thehybridposition/forcecontrollerwithaneuralcompensationofnonlinearitiesfortheSCORBOTER4pcroboticmanipulatorispresented.ThepresentedcontrollawandadaptivelawguaranteepracticalstabilityoftheclosedloopsysteminthesenseofLyapunov.Theresultsofanumericalsimulationarepresented.KeywordsNeuralNetworks,RoboticManipulator,TrackingControl,ForceControl.1IntroductionRoboticmanipulatorsaredeviceswhichfinddifferentapplicationsinmanydomainsoftheeconomy.Therequirementsinrelationtoprecisionofmotionandautonomyofmanipulatorsareincreasingaswellasthetasksperformedbythemaremoreandmorecomplex.Incontemporaryindustrialapplicationsitisdesiredforthemanipulatortoexertspecifiedforcesandmovealongaprescribedpath.Manipulatorsareobjectswithnonlinearanduncertaindynamics,withunknownandvariableparametersmasses,massmomentsofinertia,frictioncoefficients,whichoperateinchangeableconditions.Controlofsuchcomplexsystemsisveryproblematic.Thecontrolsystemhastogeneratesuchcontrolsignalsthatwillguaranteetheexecutionofmovementalongapathwithasuitableforceandwithdesiredprecisioninspiteofthechangeableoperatingconditions.Inthecontrolsystemsofindustrialmanipulators,thecomputedtorquemethod1,2fornonlinearitycompensationisused.However,theseapproachesrequirepreciseknowledgeaboutthemathematicalmodelthestructureofmotionequationswithcoefficientsofthecontrolobject.Moreover,insuchanapproach,parametersinthecompensatorhavenominalvaluessothecontrolL.Rutkowskietal.Eds.ICAISC2012,PartII,LNCS7268,pp.433–441,2012.c©SpringerVerlagBerlinHeidelberg2012434P.Gierlaksystemactswithouttakingintoaccountthechangeableoperatingconditions.Intheliteratureexistsmanyvariationofalgorithms,inwhichparametersofthemathematicalmodelofmanipulatorareadapted1,2.Howevertheseapproachesdonoteliminatetheproblemwithstructuraluncertaintyofthemodel.Inconnectionwiththepresentdifficulties,neuralcontroltechniquesweredeveloped3,4,5,6.Inthesemethodsthemathematicalmodelisunnecessary.Thesetechniquesareusedinhybridposition/forcecontroller.Inworks7,8suchcontrollershavebeenpresented.Butinthefirstoftheworksonlyforcenormaltothecontactsurfaceistakingintoaccount,andinthesecondworksomeassumptionishardtosatisfyinpracticalapplications,namelysomestiffnessmatrixwhichcharacterizesfeaturesofenvironmentandallowstocalculatecontactforces,mustbeknown.Inpreviousauthorspaperonlypositioncontrollershavebeenconsidered.Inpresentpaperhybridposition/forceneuralcontrollerisshown.Thisapproachtakesintoaccountallforces/momentswhichactsontheendeffector.Theseforces/momentsaremeasuredbysensorlocatedintheendeffector.2DescriptionoftheSCORBOTER4pcRoboticManipulatorTheSCORBOTER4pcroboticmanipulatorispresentedinFig.1.Itisdrivenbydirectcurrentmotorswithgearsandopticalencoders.Themanipulatorhas5rotationalkinematicpairsthearmofthemanipulatorhas3degreesoffreedomwhereasthegripperhas2degrees.aA1q3yzxOBCOOOd1OAlABlBClCDd1235q1q223u2u1u3q4D4u4q5u5bcontactsurfacec108FEFig.1.aSCORBOTER4pcroboticmanipulator,bschemeThetransformationfromjointspacetoCartesianspaceisgivenbythefollowingequationykq,1HybridPosition/ForceControloftheSCORBOTER4pcManipulator435whereq∈Rnisavectorofgeneralizedcoordinatesanglesofrotationoflinks,kqisakinematicsfunction,y∈Rmisavectorofaposition/orientationoftheendeffectorpointD.Dynamicalequationsofmotionoftheanalysedmodelareinthefollowingform7,9Mq¨qCq,˙q˙qF˙qGqτdtuJThqλτF,2whereMq∈Rnxnisaninertiamatrix,Cq,˙q∈RnisavectorofcentrifugalandCoriolisforces/moments,F˙q∈Rnisafrictionvector,Gq∈Rnisagravityvector,τdt∈Rnisavectorofdisturbancesboundedby||τd||0,u∈Rnisacontrolinputvector,Jhq∈Rm1xnisaJacobianmatrixassociatedwiththecontactsurfacegeometry,λ∈Rm1isavectorofconstrainingforcesexertednormallyonthecontactsurfaceLagrangemultiplier,τF∈Rnisavectorofforces/momentsinjoints,whichcomefromforces/momentsFE∈Rmappliedtotheendeffectorexcepttheconstrainingforces.ThevectorτFisgivenbyτFJbTqFE,3whereJbq∈RmxnisageometricJacobianinbody2.TheJacobianmatrixJhqcanbecalculatedinthefollowingwayJhq∂hq∂q,4wherehq0isanequationoftheholonomicconstraint,whichdescribesthecontactsurface.Thisequationreducesthenumberofdegreesoffreedomton1n−m1,sotheanalysedsystemcanbedescribedbythereducedpositionvariableθ1∈Rn17.Theremainderofvariablesdependonθ1inthefollowingwayθ2γθ1,5whereθ2∈Rm1,andγarisefromtheholonomicconstraint.ThevectorofgeneralizedcoordinatesmaybewrittenasqθT1θT2T.LetdefinetheextendedJacobian7Lθ1bracketleftbiggIn1∂γ∂θ1bracketrightbigg,6whereIn1∈Rn1xn1isanidentitymatrix.Thisallowstowritetherelations˙qLθ1˙θ1,7¨qLθ1¨θ1˙Lθ1˙θ1,8andwriteareducedorderdynamicsintermsofθ1,asMθ1Lθ1¨θ1V1θ1,˙θ1˙θ1F˙θ1Gθ1τdtuJThθ1λJbTθ1FE,9whereV1θ1,˙θ1Mθ1˙Lθ1Cθ1,˙θ1Lθ1.Premultiplyingeq.9byLTθ1andtakingintoaccountthatJhθ1Lθ10,thereducedorderdynamicsisgivenbyM¨θ1V1˙θ1FGτdLTu,10whereMLTML,V1LTV1,FLTF,GLTG,τdLTbracketleftbigτd−JbTFEbracketrightbig.436P.Gierlak3NeuralNetworkHybridControlTheaimofahybridposition/forcecontrolistofollowadesiredtrajectoryofmotionθ1d∈Rn1,andexertdesiredcontactforceλd∈Rm1normallytothesurface.Bydefiningamotionerroreθ,afilteredmotionerrors,aforceerror˜λandanauxiliarysignalυ1aseθθ1d−θ1,11s˙eθΛeθ,12˜λλd−λ,13υ1˙θ1dΛeθ,14whereΛisapositivediagonaldesignmatrix,thedynamicequation10maybewrittenintermsofthefilteredmotionerrorasM˙s−V1sLTfxLTbracketleftbigτd−JbTFEbracketrightbig−LTu,15withanonlinearfunctionfxML˙υ1V1υ1FG,16wherexbracketleftBigeTθ˙eTθθT1dv˙θT1d¨θT1dbracketrightBigT.Themathematicalstructureofhybridposition/forcecontrollerhasaformof7uˆfxKDLs−JThbracketleftBigλdKF˜λbracketrightBig−ν,17whereKDandKFarepositivedefinitematrixesofpositionandforcegain,νisarobustifyingterm,ˆfxapproximatesthefunction16.Thisfunctionmaybeapproximatedbytheneuralnetwork.InthisworkatypicalfeedforwardneuralnetworkFig.2bwithonehiddenlayerisassumed.Thehiddenlayerwithsigmoidalneurons,isconnectedwithaninputlayerbyweightscollectedinamatrixD,andwithanoutputlayerbyweightscollectedinamatrixW.Theinputweightsarerandomlychosenandconstant,buttheoutputweightsinitiallyareequalzero,andwillbetunedduringadaptationprocess.Suchneuralnetworkislinearintheweights,andhasthefollowingdescription3,4fxWTϕxε,18withoutputfromhiddenlayerϕxSDTx,wherexisaninputvector,S.isavectorofneuronactivationfunctions,εisanestimationerrorboundedby||ε||0.ThematrixWisunknown,soanestimationˆWisused,andamathematicaldescriptionofarealneuralnetwork,whichapproximatesfunctionfxisgivenbyˆfxˆWTϕx.19
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