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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,SHAFIEISEHSAN21ELECTRONICDEPARTMENTENGINEERINGFACULTY,UNIVERSITYOFISFAHAN,ISFAHAN,IRANEMAILMATAEI1971YAHOOCOM2ELECTRICALANDROBOTICENGINEERINGFACULTY,SHAHROODUNIVERSITYOFTECHNOLOGY,SHAHROOD,IRANEMAILSEHSHFYAHOOCOMABSTRACTINTHISPAPER,ACHATTERINGFREESLIDINGMODECONTROLSMCFORAROBOTMANIPULATORINCLUDINGPIDPARTWITHAFUZZYTUNABLEGAINISDESIGNEDTHEMAINIDEAISTHATTHEROBUSTNESSPROPERTYOFSMCANDGOODRESPONSECHARACTERISTICSOFPIDARECOMBINEDWITHFUZZYTUNINGGAINAPPROACHTOACHIEVEMOREACCEPTABLEPERFORMANCEFORTHISPURPOSE,INTHEFIRSTSTAGE,APIDSLIDINGSURFACEISCONSIDEREDSUCHTHATTHEROBOTDYNAMICEQUATIONSCANBEREWRITTENINTERMSOFSLIDINGSURFACEANDITSDERIVATIVEANDTHERELATEDCONTROLLAWOFTHESMCDESIGNWILLCONTAINAPIDPARTTHESTABILITYGUARANTEEOFTHISSLIDINGMODEPIDCONTROLLERISPROVEDBYALEMMAUSINGDIRECTLYAPUNOVMETHODTHEN,INTHESECONDSTAGE,INORDERTODECREASETHEREACHINGTIMETOTHESLIDINGSURFACEANDDELETINGTHEOSCILLATIONSOFTHERESPONSE,AFUZZYTUNINGSYSTEMISUSEDFORADJUSTINGBOTHCONTROLLERGAINSINCLUDINGSLIDINGCONTROLLERGAINPARAMETERANDPIDCOEFFICIENTFINALLY,THEPROPOSEDMETHODOLOGYISAPPLIEDTOATWOLINKROBOTMANIPULATORINCLUDINGMODELUNCERTAINTYANDEXTERNALDISTURBANCESASACASESTUDYTHESIMULATIONRESULTSSHOWTHEIMPROVEMENTSOFTHERESULTSINTHECASEOFUSINGTHEPROPOSEDMETHODINCOMPARISONWITHTHECONVENTIONALSMCKEYWORDSSLIDINGMODECONTROL,ROBOTMANIPULATOR,PIDCONTROL,FUZZYCONTROL,LYAPUNOVTHEORY1INTRODUCTIONAROBOTMANIPULATORISANONLINEARSYSTEMWITHHIGHCOUPLINGTERMWHOSEDYNAMICSCONSISTSOFUNCERTAINTYANDENCOUNTEREDWITHPAYLOADCHANGES,FRICTIONANDDISTURBANCE1ONTHEOTHERHAND,SLIDINGMODECONTROLSMCASANONLINEARTECHNIQUEWITHTHECAPABILITIESOFROBUSTNESSAGAINSTTHEMODELUNCERTAINTIESANDABILITYOFTHEDISTURBANCEREJECTIONHASBEENCONSIDEREDINMANYRESEARCHES24ALTHOUGHTHEROBUSTNESSOFTHESMCISONEOFITSMAINCHARACTERISTICS,THISISACHIEVEDONLYINTHESLIDINGPHASEANDTHESYSTEMISSENSITIVETOTHESTRUCTUREDUNCERTAINTIESANDEXTERNALDISTURBANCESINTHEREACHINGPHASETOTHESLIDINGSURFACETHEREFORE,DIFFERENTAPPROACHESFORIMPROVINGTHEPERFORMANCEOFTHESMCHASBEENPROPOSEDWHICHONEOFTHEMISINTELLIGENTCONTROLMETHODSUCHASFUZZYCONTROLSYSTEM57BECAUSEOFTHERELATIONSBETWEENSMCANDFUZZYCONTROL,8,THECOMBINATIONOFTHESETWOAPPROACHESHASBEENCONSIDEREDASARESEARCHTOPICINLASTYEARS913SUCHTHATTHEADVANTAGESOFBOTHAPPROACHESCANBEUSEDONESIMPLEWAYTODECREASETHESENSITIVITYOFSLIDINGMODECONTROLLERTOTHEPARAMETRICUNCERTAINTIESANDEXTERNALDISTURBANCESISUSINGOFHIGHCONTROLGAINWHICHDECREASEALSOTHEREACHINGTIMEANDTRACKINGERRORHOWEVER,HIGHCONTROLGAININCREASESTHEOSCILLATIONSINTHECONTROLSIGNALTHATMAYLEADTOTHEEXCITATIONOFHIGHFREQUENCYUNMODELEDDYNAMICSWHICHISANUNDESIREDPHENOMENONTOOVERCOMETHISDRAWBACK,THEFUZZYLOGICCANBEUSEDFORTUNINGOFTHISGAININTHISREGARD,IN14,ANONLINEARSLIDINGSURFACEANDFUZZYLOGICHAVEBEENUSEDINTHEDESIGNOFAFUZZYTERMINALSMCFORAROBOTMANIPULATORALSO,AFUZZYVARIABLESLIDINGSURFACEBASEDMETHODHASBEENPROPOSEDIN15INORDERTOIMPROVETHETRACKINGPERFORMANCEIN16,INADDITIONTOUSINGVARIABLESLIDINGSURFACE,THEIDEAOFFUZZYGAINTUNINGANDBOUNDARYLAYERHASBEENPRESENTEDTOACHIEVEMOREIMPROVEMENTSINTHISPAPER,INADDITIONTOUSINGTHEINTEGRALTERMINTHESLIDINGSURFACE,17,18,ATFIRST,THESMCWITCHISINCLUDINGPIDPARTISDESIGNEDANDITSSTABILITYGUARANTEEISPROVEDINALEMMATHEN,INORDERTOIMPROVETHECONTROLLERPERFORMANCE,AFUZZYSYSTEMISUSEDTOTUNETHEGAINOFREACHINGPHASEANDALSOPIDPARTGAINTHUS,ACHATTERINGFREESMCISACHIEVEDINWHICHTHETRACKINGERRORANDREACHINGTIMETOSLIDINGSURFACEHASBEENREDUCEDWITHOUTNEEDTOVARIABLESLIDINGSURFACETHEREMINDEROFTHEPAPERISORGANIZEDASFOLLOWSINTHESECTION2,THEMATHEMATICALMODELOFTHEROBOTMANIPULATORISGIVENTHESMCINCLUDINGTHEPIDLOOPTOWHICHISDENOTEDASSMCPIDISPRESENTEDINSECTION3THEDESIGNOFFUZZYSMCPIDISDESCRIBEDINSECTION4INSECTION5,THESIMULATIONRESULTSAREPROVIDEDANDFINALLY,SUMMARYANDSOMECONCLUSIONSAREPRESENTED2THESYSTEMMATHEMATICALMODELTHEDYNAMICALEQUATIONOFANNLINKROBOTMANIPULATORINTHESTANDARDFORMISASFOLLOWS1ΤΤDQGQQQCQQM,1WHERENNMQ∈RISASYMMETRYANDBOUNDEDPOSITIVEDEFINITEMATRIXWHICHISCALLEDINERTIALMATRIXMOREOVER,,,NQQQ∈RARETHEPOSITION,VELOCITY,ANDANGULARACCELERATIONOFTHEROBOTJOINT,RESPECTIVELYTHEMATRIX,NNCQQ∈RISTHEMATRIXOFCORIOLISANDCENTRIFUGALFORCESSUCHTHATTHEMATRIX2,HQCQQ−ISASYMMETRY,IE,FORANONZERO1NVECTORXWEWILLHAVET2,0XHQCQQX−ALSO,NGQ∈RISTHEGRAVITYVECTOR,NDΤ∈RISTHEBOUNDEDDISTURBANCEVECTORSUCHTHATDDTΤ0ANDNΤ∈RIS171THECONTROLINPUTVECTORINTHEFOLLOWING,QH,,QQCANDQGARESHOWNBYH,C,ANDG,RESPECTIVELY3SLIDINGMODECONTROLWITHPIDTHEOBJECTIVEOFTRACKINGCONTROLISDESIGNACONTROLLAWFOROBTAININGTHESUITABLEINPUTTORQUEΤSUCHTHEPOSITIONVECTORQCANTRACKTHEDESIREDTRAJECTORYDQINTHISREGARD,THETRACKINGERRORVECTORISDEFINEDASFOLLOWSQQED−2INORDERTOAPPLYTHESMC,THESLIDINGSURFACEISCONSIDEREDASTHERELATION3WHICHCONTAINSTHEINTEGRALPARTINADDITIONTOTHEDERIVATIVETERM120DTSEEETΛΛ3WHEREIΛISDIAGONALPOSITIVEDEFINITEMATRIXTHEREFORE,0SISASTABLESLIDINGSURFACEAND0→EAS∞→TTHEROBOTDYNAMICEQUATIONSCANBEREWRITTENBASEDONTHESLIDINGSURFACEINTERMOFFILTEREDERRORASFOLLOWSΤΤ−−DFCSSM4WHERE12120DTDDFMQEECQEETGΛΛΛΛ5NOW,THECONTROLINPUTCANBECONSIDEREDASFOLLOWSSGNˆSKSKFVΤ6WHERE12120ˆˆˆˆDTDDFMQEECQEETGΛΛΛΛ7ISANESTIMATIONOFFAND0DTVVVVKSKEKEKETΛΛISTHEOUTERPIDTRACKINGLOOP,ANDKKV,AREDIAGONALPOSITIVEDEFINITEMATRICESANDAREDEFINEDSUCHTHATTHESTABILITYCONDITIONSAREGUARANTEEDTHESGNSISALSOTHESIGNFUNCTIONWEHAVEALSO12120DTDDFMQEECQEETGFΛΛΛΛ08WHEREˆFFF−,ˆMMM−,ˆCCC−,ANDˆGGG−FCANALSOBESELECTEDASTHEFOLLOWINGRELATION12120DTDDFMQEECQEETGΛΛΛΛ9INORDERTOREACHTHESYSTEMSTATES,EETOTHESLIDINGSURFACE0SINALIMITEDTIMEANDREMAINTHERE,THECONTROLLAWSHOULDBEDESIGNEDSUCHTHATTHEFOLLOWINGSLIDINGCONDITIONISSATISFIED2TT1/21D2DSMSSSTΗ−FOR0≠S10THISSUBJECTISPROVEDINTHEFOLLOWINGLEMMALEMMAINTHESMCDESIGNOFASYSTEMWITHDYNAMICEQUATION1ANDSLIDINGSURFACE3,IFTHECONTROLINPUTΤISSELECTEDAS6,BYCONSIDERINGFAS9ANDK1122DIAG,,,NNKKKWITHTHEFOLLOWINGCOMPONENTSNITSKFKIDVII,,2,1,Η11THEN,THESLIDINGCONDITION10ISSATISFIEDBYEQUATION4PROOFCONSIDERTHEFOLLOWINGLYAPUNOVFUNCTIONCANDIDATET12VSMS12SINCEMISPOSITIVEDEFINITE,FOR0≠SWEHAVE0VANDBYDIFFERENTIATINGOFTHERELATION12ANDREGARDINGTHESYMMETRICPROPERTYOFM,ITCANBEWRITTENTT12VSMSSMS13BYSUBSTITUTING4IN13ANDCONSIDERINGTHATTSM−20CS,WEHAVETTTT12DDVSMSSCSSFSFΤΤΤΤ−−−14BYREPLACINGTHERELATION6IN14,VCANBEREWRITTENASTT1ˆSGNDVNDVIIIVSFFKSKSSFKSKSΤΤ−−−−−15SINCETHEFOLLOWINGINEQUALITY16ISVALIDANDREGARDINGTHERELATION11,WEHAVEVDDVFKSTFKSΤ−116IIDVIIKFKSΤΗ−117FINALLY,ITCANBECONCLUDEDTHAT1NIIIVSΗ−018THISINDICATESTHATVISALYAPUNOVFUNCTIONANDTHESLIDINGCONDITION10HASBEENSATISFIEDTHEUSEOFSIGNFUNCTIONINTHECONTROLLAWLEADSTOHIGHOSCILLATIONSINCONTROLTORQUEWHICHISUNDESIREDPHENOMENONANDISCALLEDCHATTERINGTOOVERCOMETHISDRAWBACK,THEREARESOMESOLUTIONSWHICHONEOFTHEMISUSINGTHEFOLLOWINGSATURATIONFUNCTIONINSTEADOFSIGNFUNCTIONINTHEDISCONTINUOUSPARTOFTHECONTROLLAW1,,1,SSSSATSSΦΦΦΦΦΦ−−1019BYTHIS,THEREISABOUNDARYLAYERΦAROUNDTHESLIDINGSURFACESUCHTHATWHENTHESTATETRAJECTORYREACHTOTHISLAYERWILLBEREMAININGTHERE4THEDESIGNOFFUZZYSMCPIDASITWASMENTIONEDBEFORE,BYUSINGAHIGHGAININSMCK,THESENSITIVITYOFTHECONTROLLERTOTHEMODELUNCERTAINTIESANDEXTERNALDISTURBANCESCANBEREDUCEDMOREOVER,AHIGHGAININPIDPARTOFTHECONTROLSYSTEMVKCANREDUCETHEREACHINGTIMETOSLIDINGSURFACEANDTRACKINGERRORHOWEVER,INCREASINGTHEGAINCAUSESTHEINCREMENTOFTHEOSCILLATIONSINTHEINPUTTORQUEAROUNDTHESLIDINGSURFACETHEREFORE,IFTHISGAINCANBETUNEDBASEDONTHEDISTANCEOFTHESTATESTOTHESLIDINGSURFACE,AMOREACCEPTABLEPERFORMANCECANBEACHIEVEDINOTHERWORDS,THEVALUEOFGAINSHOULDBESELECTEDHIGHWHENTHESTATETRAJECTORY172ISFARFROMTHESLIDINGSURFACEANDWHENTHEDISTANCEISDECREASING,ITSVALUESHOULDBEDECREASEDTHISIDEACANBEACCOMPLISHEDBYUSINGFUZZYLOGICINCOMBINATIONWITHSMCTOTUNETHEGAINADAPTIVELYFORTHISPURPOSE,TWOINPUTONEOUTPUTFUZZYSYSTEMISDESIGNEDWHOSEINPUTSARESANDSWHICHARETHEDISTANCESOFTHESTATETRAJECTORIESTOTHESLIDINGSURFACEANDITSDERIVATIVE,RESPECTIVELYTHEMEMBERSHIPFUNCTIONSOFTHESETWOINPUTSARESHOWNINFIG1THEOUTPUTOFTHEFUZZYSYSTEMISDENOTEDBYFUZZKANDHASBEENSHOWNINFIG2FORAPPLYINGTHESEGAINSTOTHECONTROLINPUT,THENORMALIZATIONFACTORSNANDVNASTHEFOLLOWINGRELATIONSAREUSEDFUZZKNK⋅20FUZZVVKNK⋅21THESEFACTORSCANBESELECTEDBYTRIALANDERRORSUCHTHATTHESTABILITYCONDITION17ISSATISFIED10500510020406081INPUTVARIABLESDEGREEOFMEMBERSHIPNBNSZEPSPBAINPUTS10500510020406081INPUVARIABLESDDEGREEOFMEMBERSHIPNZPBINPUTSFIG1THEMEMBERSHIPFUNCTIONS00204060810020406081OUTPUTVARIABLEKDEGREEOFMEMBERSHIPSMBFIG2THEMEMBERSHIPFUNCTIONSOFTHEOUTPUTFUZZKTAB1THEFUZZYRULEBASEFORTUNINGFUZZKPBPSZNSNBSSBSMBBNBMSMBZBBMSBPTHEMAXIMUMVALUESOFKANDKVARELIMITEDACCORDINGTOTHESYSTEMACTUATORSPOWER,ANDTHEMINIMUMVALUEOFKSHOULDNOTBELESSTHANTHEPROVIDEDAMOUNTINRELATION17THEFUZZYBASERULEHASBEENGIVENINTABLE1INWHICHTHEFOLLOWINGABBREVIATIONSHAVEBEENUSEDNBNEGATIVEBIG;NSNEGATIVESMALL;ZZERO;PSPOSITIVESMALL;PBPOSITIVEBIG;MMEDIUMFOREXAMPLE,WHENSISNEGATIVESMALLNSANDSISPOSITIVEP,THENFUZZKISSMALLS5THECASESTUDYANDSIMULATIONRESULTSINORDERTOSHOWTHEEFFECTIVENESSOFTHEPROPOSEDCONTROLLAW,ITISAPPLIEDTOTWOLINKSROBOTWITHTHEFOLLOWINGPARAMETERS2222COSCOSCOSQQMQQΑΒΓΒΓΒΓΒ222212212SINSIN,SIN0QQQQQCQQQQΓΓΓ−−2311112112COSCOSCOSQQQGQQQΑΔΓΔΓΔ24WHERE2121MMAΑ,222MAΒ,212MAAΓ,1AGΔ,AND1M,2M,71A,52AARETHEMASSESANDLENGTHSOFTHEFIRSTANDSECONDLINKS,RESPECTIVELYTHEMASSESAREASSUMEDTOBEINTHEENDOFTHEARMSANDTHEGRAVITYACCELERATIONISCONSIDERED89GMOREOVER,THEMASSESARECONSIDEREDWITH10UNCERTAINTYASFOLLOW0011112222,4,2MMMMMMMMΔΔΔΔ0025WHERE014MAND022M,ANDMˆ,Cˆ,ANDGˆAREESTIMATEDTHEDESIREDSTATETRAJECTORYIS1COSŒ2COSŒDTQT−26ANDTHEDISTURBANCETORQUEISCONSIDEREDASFOLLOWS05SIN2Œ05SIN2ŒDTTΤ27WHICHLEADSTO5050DTTHEDESIGNPARAMETERSAREDETERMINEDASFOLLOW1500151Λ,4000402Λ28173THEVALUESOFTHEΦANDΗARESELECTEDAS0167ΦANDT0101ΗMOREOVER,THEFACTORSNANDVNARESELECTEDASFOLLOW50050N,10005VN29INORDERTOSHOWTHEIMPROVEMENTDUETOTHEPROPOSEDMETHODOFTHISPAPERFUZZYSMCPID,THESIMULATIONRESULTSOFAPPLYINGTHISMETHODARECOMPAREDWITHTHERELATEDRESULTSOFTHECONVENTIONALSMCTHETRACKINGERRORANDCONTROLLAWINTHECASEOFCONVENTIONALSMCHAVEBEE

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