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外文资料-Sliding Mode PID Controller Design for Robot Manipulators.pdf

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外文资料-Sliding Mode PID Controller Design for Robot Manipulators.pdf

170Proceedingsofthe27thChineseControlConferenceJuly1618,2008,Kunming,Yunnan,ChinaSlidingModePIDControllerDesignforRobotManipulatorsbyUsingFuzzyTuningApproachAtaeiMohammad1,ShafieiS.Ehsan21.ElectronicDepartmentEngineeringFaculty,UniversityofIsfahan,Isfahan,IranEmailmataei1971yahoo.com2.ElectricalandRoboticEngineeringFaculty,ShahroodUniversityofTechnology,Shahrood,IranEmailsehshfyahoo.comAbstractInthispaper,achatteringfreeslidingmodecontrolSMCforarobotmanipulatorincludingPIDpartwithafuzzytunablegainisdesigned.ThemainideaisthattherobustnesspropertyofSMCandgoodresponsecharacteristicsofPIDarecombinedwithfuzzytuninggainapproachtoachievemoreacceptableperformance.Forthispurpose,inthefirststage,aPIDslidingsurfaceisconsideredsuchthattherobotdynamicequationscanberewrittenintermsofslidingsurfaceanditsderivativeandtherelatedcontrollawoftheSMCdesignwillcontainaPIDpart.ThestabilityguaranteeofthisslidingmodePIDcontrollerisprovedbyalemmausingdirectLyapunovmethod.Then,inthesecondstage,inordertodecreasethereachingtimetotheslidingsurfaceanddeletingtheoscillationsoftheresponse,afuzzytuningsystemisusedforadjustingbothcontrollergainsincludingslidingcontrollergainparameterandPIDcoefficient.Finally,theproposedmethodologyisappliedtoatwolinkrobotmanipulatorincludingmodeluncertaintyandexternaldisturbancesasacasestudy.ThesimulationresultsshowtheimprovementsoftheresultsinthecaseofusingtheproposedmethodincomparisonwiththeconventionalSMC.KeyWordsSlidingmodecontrol,Robotmanipulator,PIDcontrol,Fuzzycontrol,Lyapunovtheory1INTRODUCTIONArobotmanipulatorisanonlinearsystemwithhighcouplingtermwhosedynamicsconsistsofuncertaintyandencounteredwithpayloadchanges,frictionanddisturbance1.Ontheotherhand,slidingmodecontrolSMCasanonlineartechniquewiththecapabilitiesofrobustnessagainstthemodeluncertaintiesandabilityofthedisturbancerejectionhasbeenconsideredinmanyresearches24.AlthoughtherobustnessoftheSMCisoneofitsmaincharacteristics,thisisachievedonlyintheslidingphaseandthesystemissensitivetothestructureduncertaintiesandexternaldisturbancesinthereachingphasetotheslidingsurface.Therefore,differentapproachesforimprovingtheperformanceoftheSMChasbeenproposedwhichoneofthemisintelligentcontrolmethodsuchasfuzzycontrolsystem57.BecauseoftherelationsbetweenSMCandfuzzycontrol,8,thecombinationofthesetwoapproacheshasbeenconsideredasaresearchtopicinlastyears913suchthattheadvantagesofbothapproachescanbeused.Onesimplewaytodecreasethesensitivityofslidingmodecontrollertotheparametricuncertaintiesandexternaldisturbancesisusingofhighcontrolgainwhichdecreasealsothereachingtimeandtrackingerror.However,highcontrolgainincreasestheoscillationsinthecontrolsignalthatmayleadtotheexcitationofhighfrequencyunmodeleddynamicswhichisanundesiredphenomenon.Toovercomethisdrawback,thefuzzylogiccanbeusedfortuningofthisgain.Inthisregard,in14,anonlinearslidingsurfaceandfuzzylogichavebeenusedinthedesignofafuzzyterminalSMCforarobotmanipulator.Also,afuzzyvariableslidingsurfacebasedmethodhasbeenproposedin15inordertoimprovethetrackingperformance.In16,inadditiontousingvariableslidingsurface,theideaoffuzzygaintuningandboundarylayerhasbeenpresentedtoachievemoreimprovements.Inthispaper,inadditiontousingtheintegraltermintheslidingsurface,17,18,atfirst,theSMCwitchisincludingPIDpartisdesignedanditsstabilityguaranteeisprovedinalemma.Then,inordertoimprovethecontrollerperformance,afuzzysystemisusedtotunethegainofreachingphaseandalsoPIDpartgain.Thus,achatteringfreeSMCisachievedinwhichthetrackingerrorandreachingtimetoslidingsurfacehasbeenreducedwithoutneedtovariableslidingsurface.Thereminderofthepaperisorganizedasfollows.Inthesection2,themathematicalmodeloftherobotmanipulatorisgiven.TheSMCincludingthePIDlooptowhichisdenotedasSMCPIDispresentedinsection3.ThedesignoffuzzySMCPIDisdescribedinsection4.Insection5,thesimulationresultsareprovidedandfinally,summaryandsomeconclusionsarepresented.2THESYSTEMMATHEMATICALMODELThedynamicalequationofannlinkrobotmanipulatorinthestandardformisasfollows1ττdqGqqqCqqM,1wherennMq∈Risasymmetryandboundedpositivedefinitematrixwhichiscalledinertialmatrix.Moreover,,,nqqq∈Raretheposition,velocity,andangularaccelerationoftherobotjoint,respectively.Thematrix,nnCqq∈RisthematrixofCoriolisandcentrifugalforcessuchthatthematrix2,HqCqq−isasymmetry,i.e.,foranonzero1nvectorxwewillhaveT2,0xHqCqqx−.Also,nGq∈Risthegravityvector,ndτ∈RistheboundeddisturbancevectorsuchthatdDTτ0andnτ∈Ris171thecontrolinputvector.Inthefollowing,qH,,qqCandqGareshownbyH,C,andG,respectively.3SLIDINGMODECONTROLWITHPIDTheobjectiveoftrackingcontrolisdesignacontrollawforobtainingthesuitableinputtorqueτsuchthepositionvectorqcantrackthedesiredtrajectorydq.Inthisregard,thetrackingerrorvectorisdefinedasfollowsqqed−2InordertoapplytheSMC,theslidingsurfaceisconsideredastherelation3whichcontainstheintegralpartinadditiontothederivativeterm120dtseeetλλ³3whereiλisdiagonalpositivedefinitematrix.Therefore,0sisastableslidingsurfaceand0→eas∞→t.Therobotdynamicequationscanberewrittenbasedontheslidingsurfaceintermoffilterederrorasfollowsττ−−dfCssM4Where12120dtddfMqeeCqeetGλλλλ³5Now,thecontrolinputcanbeconsideredasfollowssgnˆsKsKfvτ6Where12120ˆˆˆˆdtddfMqeeCqeetGλλλλ³7isanestimationoffand0dtvvvvKsKeKeKetλλ³istheouterPIDtrackingloop,andKKv,arediagonalpositivedefinitematricesandaredefinedsuchthatthestabilityconditionsareguaranteed.Thesgnsisalsothesignfunction.Wehavealso12120dtddfMqeeCqeetGFλλλλ³08whereˆfff−,ˆMMM−,ˆCCC−,andˆGGG−.Fcanalsobeselectedasthefollowingrelation12120dtddFMqeeCqeetGλλλλ³9Inordertoreachthesystemstates,eetotheslidingsurface0sinalimitedtimeandremainthere,thecontrollawshouldbedesignedsuchthatthefollowingslidingconditionissatisfied2TT1/21d2dsMssstηªºVandbydifferentiatingoftherelation12andregardingthesymmetricpropertyofM,itcanbewrittenTT12VsMssMs13Bysubstituting4in13andconsideringthatTsM−20Cs,wehaveTTTT12ddVsMssCssfsfττττ−−−14Byreplacingtherelation6in14,VcanberewrittenasTT1ˆsgndvndviiiVsffKsKssfKsKsττ−−−−−¦15Sincethefollowinginequality16isvalidandregardingtherelation11,wehavevDdvFKsTfKsτ−116iidviiKfKsτη−117Finally,itcanbeconcludedthat1niiiVsη−¦018ThisindicatesthatVisaLyapunovfunctionandtheslidingcondition10hasbeensatisfied.Theuseofsignfunctioninthecontrollawleadstohighoscillationsincontroltorquewhichisundesiredphenomenonandiscalledchattering.Toovercomethisdrawback,therearesomesolutionswhichoneofthemisusingthefollowingsaturationfunctioninsteadofsignfunctioninthediscontinuouspartofthecontrollaw1,,1,ssssatssϕϕϕϕϕϕ­°§°−®¨¸©¹°°−¯1019Bythis,thereisaboundarylayerϕaroundtheslidingsurfacesuchthatwhenthestatetrajectoryreachtothislayerwillberemainingthere.4THEDESIGNOFFUZZYSMCPIDAsitwasmentionedbefore,byusingahighgaininSMCK,thesensitivityofthecontrollertothemodeluncertaintiesandexternaldisturbancescanbereduced.Moreover,ahighgaininPIDpartofthecontrolsystemvKcanreducethereachingtimetoslidingsurfaceandtrackingerror.However,increasingthegaincausestheincrementoftheoscillationsintheinputtorquearoundtheslidingsurface.Therefore,ifthisgaincanbetunedbasedonthedistanceofthestatestotheslidingsurface,amoreacceptableperformancecanbeachieved.Inotherwords,thevalueofgainshouldbeselectedhighwhenthestatetrajectory172isfarfromtheslidingsurfaceandwhenthedistanceisdecreasing,itsvalueshouldbedecreased.ThisideacanbeaccomplishedbyusingfuzzylogicincombinationwithSMCtotunethegainadaptively.Forthispurpose,twoinputoneoutputfuzzysystemisdesignedwhoseinputsaresandswhicharethedistancesofthestatetrajectoriestotheslidingsurfaceanditsderivative,respectively.ThemembershipfunctionsofthesetwoinputsareshowninFig.1.TheoutputofthefuzzysystemisdenotedbyfuzzKandhasbeenshowninFig.2.Forapplyingthesegainstothecontrolinput,thenormalizationfactorsNandvNasthefollowingrelationsareusedfuzzKNK⋅20fuzzvvKNK⋅21Thesefactorscanbeselectedbytrialanderrorsuchthatthestabilitycondition17issatisfied.10.500.5100.20.40.60.81inputvariablesDegreeofmembershipNBNSZEPSPBainputs10.500.5100.20.40.60.81inpuvariablesdDegreeofmembershipNZPbinputsFig.1Themembershipfunctions00.20.40.60.8100.20.40.60.81outputvariableKDegreeofmembershipSMBFig.2ThemembershipfunctionsoftheoutputfuzzKTab.1ThefuzzyrulebasefortuningfuzzKPBPSZNSNBssBSMBBNBMSMBZBBMSBPThemaximumvaluesofKandKvarelimitedaccordingtothesystemactuatorspower,andtheminimumvalueofKshouldnotbelessthantheprovidedamountinrelation17.Thefuzzybaserulehasbeengivenintable1inwhichthefollowingabbreviationshavebeenusedNBNegativeBigNSNegativeSmallZZeroPSPositiveSmallPBPositiveBigMMedium.Forexample,whensisnegativesmallNSandsispositiveP,thenfuzzKissmallS.5THECASESTUDYANDSIMULATIONRESULTSInordertoshowtheeffectivenessoftheproposedcontrollaw,itisappliedtotwolinksrobotwiththefollowingparameters2222coscoscosqqMqqαβγβγβγβªº«»¬¼222212212sinsin,sin0qqqqqCqqqqγγγ−−2311112112coscoscosqqqGqqqαδγδγ릧«»¬¼24where2121mmaα,222maβ,212maaγ,1agδ,and1m,2m,7.1a,5.2aarethemassesandlengthsofthefirstandsecondlinks,respectively.Themassesareassumedtobeintheendofthearmsandthegravityaccelerationisconsidered8.9g.Moreover,themassesareconsideredwith10uncertaintyasfollow0011112222,.4,.2mmmmmmmmΔΔΔΔ0025where014mand022m,andMˆ,Cˆ,andGˆareestimated.Thedesiredstatetrajectoryis1cosŒ2cosŒdtqt−ªº«»¬¼26andthedisturbancetorqueisconsideredasfollows0.5sin2Œ0.5sin2Œdttτªº«»¬¼27whichleadsto»¼º«¬ª5.05.0DT.Thedesignparametersaredeterminedasfollow»¼º«¬ª1500151λ,»¼º«¬ª4000402λ28173Thevaluesoftheϕandηareselectedas0.167ϕandT0.10.1η.Moreover,thefactorsNandvNareselectedasfollow»¼º«¬ª50050N,»¼º«¬ª10005vN29InordertoshowtheimprovementduetotheproposedmethodofthispaperFuzzySMCPID,thesimulationresultsofapplyingthismethodarecomparedwiththerelatedresultsoftheconventionalSMC.ThetrackingerrorandcontrollawinthecaseofconventionalSMChavebeenshowninfigures3and4,respectively.ThecorrespondinggraphsforthecaseofapplyingfuzzySMCPIDarealsoprovidedinfigures5,and6.02468100.0500.050.10.15timesecError1rad02468100.500.511.52timesecError2radFig.3ThetrackingerrorsinthecaseofusingconventionalSMC024681050050100150timesecinput1N.m024681050050100timesecinput2N.mFig.4ThecontrolinputsinthecaseofusingconventionalSMC02468100.0500.050.10.15timesecError1rad02468100.500.511.52timesecError2radFig.5ThetrackingerrorsinthecaseofusingFuzzySMCPID02468101000100200timesecinput1N.m024681010050050100timesecinput2N.mFig.6ThecontrolinputsinthecaseofusingFuzzySMCPIDAsitisseeninthesefigures,theproposedfuzzySMCPIDhasfasterresponseandlesstrackingerrorincomparisonwithconventionalSMC.Inordertoshowmoreclearlythedifferencebetweenthetrackingerrorsintwocases,theenlargedgraphshavebeenprovidedinFig.7andFig.8.02468100.010.00500.0050.01timesecError1rad0246810505x103timesecError2radFig.7TheenlargementofthetrackingerrorsinthecaseofusingconventionalSMC0246810505x104timesecError1rad024681010.500.51x103timesecError2radFig.8TheenlargementofthetrackingerrorsinthecaseofusingFuzzySMCPID6CONCLUSIONInthispaper,designofaslidingmodecontrolwithaPIDloopforrobotmanipulatorwaspresentedinwhichthegainof

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