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IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009459ControlofaClassofUnderactuatedMechanicalSystemsUsingSlidingModesV.Sankaranarayanan,Member,IEEE,andArunD.Mahindrakar,Member,IEEEAbstractInthispaper,wepresentaslidingmodecontrolalgorithmtorobustlystabilizeaclassofunderactuatedmechanicalsystemsthatarenotlinearlycontrollableandviolateBrockettsnecessaryconditionforsmoothasymptoticstabilizationoftheequilibrium,withparametricuncertainties.Indefiningtheclassofsystems,afewsimplifyingassumptionsaremadeonthestructureofthedynamicsinparticular,thedampingforcesareassumedtobelinearinvelocities.Wefirstproposeaswitchingsurfacedesignforthisclassofsystems,andsubsequently,aswitchedalgorithmtoreachthissurfaceinfinitetimeusingconventionalandhigherorderslidingmodecontrollers.Thestabilityoftheclosedloopsystemisinvestigatedwithanundefinedrelativedegreeoftheslidingfunctions.Thecontrollergainsaredesignedsuchthatthecontrollerstabilizestheactualsystemwithparametricuncertainty.Theproposedcontrolalgorithmisappliedtotwobenchmarkproblemsamobilerobotandanunderactuatedunderwatervehicle.Simulationresultsarepresentedtovalidatetheproposedscheme.IndexTermsMobilerobot,nonholonomic,slidingmode,underactuatedunderwatervehicleUUV.I.INTRODUCTIONControlofunderactuatedmechanicalsystems,systemswithfewernumberofcontrolinputsthanthedegreesoffreedom,hasreceivedmuchattentionoverthepastfewyears.Thisisbecauseofthetheoreticalchallengesaswellaspracticalapplicability.Amongthehostofcontrolmethodsproposedfortheindividualapplications,somegeneralmethodologieshavealsobeenproposedtostabilizethesesystems1–3.Thesemethodologiesusepassivitytechniques,energyshaping,anddiscontinuouscontrolmethods.AchallengingproblemistostabilizeaparticularclassofsystemsthatarenotlinearlycontrollableanddisobeyBrockettsnecessarycondition4.Whilethereexistspiecewiseanalyticortimeperiodiccontinuousfeedbacklawsthatcanasymptoticallystabilizetheequilibrium,theproblemofrobustandboundedcontrollerdesignaretwoimportantissuesthatarestillopeninthisfield.Inthispaper,weconsidertherobuststabilizationofthisparticularclassofsystems.Inparticular,wepresentthedesignofaslidingmodecontrollerfortheaforementionedclassofsystems.Slidingmodeisoneoftherobustcontrollerdesignmethodsandhasbeensuccessfullyappliedtounderactuatedandnonholonomicsystems5–8.Stabilizationofseveralunderactuatedsystemsunderthisparticularclasshavebeensolvedusingslidingmodecontroltechniquessee,forexample,thenonholonomicintegratoranditsextendedversion9.However,toourknowledge,thereisnoconstructiveproManuscriptreceivedMarch11,2008revisedJuly1,2008andSeptember13,2008.FirstpublishedFebruary6,2009currentversionpublishedApril3,2009.ThispaperwasrecommendedforpublicationbyAssociateEditorE.PapadopoulosandEditorH.Araiuponevaluationofthereviewerscomments.TheworkofA.D.MahindrakarwassupportedbytheDepartmentofScienceandTechnology,India,undertheresearchGrantELE/0708/153/DSTX/ARUD.V.SankaranarayananiswiththeDepartmentofElectricalandElectronicsEngineering,NationalInstituteofTechnology,Tiruchirapalli620015,Indiaemailvsankarnitt.edu.A.D.MahindrakariswiththeDepartmentofElectricalEngineering,IndianInstituteofTechnologyMadras,Chennai600036,Indiaemailarun_dmiitm.ac.in.Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp//ieeexplore.ieee.org.DigitalObjectIdentifier10.1109/TRO.2008.2012338ceduretodesignaswitchingsurfaceandaslidingmodecontrollerforthisclassofsystems.First,wepresentanoutlinetodesigntheswitchingsurfacebyestablishingaconnectionbetweentheregularform10andthedefinitionoftheswitchingsurface.Threecasesareconsideredtodesigntheswitchingsurface.Subsequently,theslidingmodecontrollerdesignispresentedtoreachtheswitchingsurfaceinfinitetimeusingbothconventionalandhigherorderslidingmodecontrollers.Inaddition,aswitchedalgorithmisalsoproposedtoreachtheswitchingsurfaceiftherelativedegreeoftheslidingfunctionsarenotwelldefined.Finally,wepresentthedesignofthecontrollergainstorenderthesystemstablewithparametricuncertainties.Therestofthepaperisorganizedasfollows.AclassificationofunderactuatedmechanicalsystemsinthecontextofslidingmodeispresentedinSectionII.InSectionIII,adesignoutlineispresentedforthegeneration,finitetimereachabilitywithwelldefinedandundefinedrelativedegreesoftheslidingfunctions.InSectionsIVandV,theproposedmethodologyisappliedtotwobenchmarkexamplesamobilerobotandtheunderactuatedunderwatervehicleUUV,respectively.SimulationresultsarepresentedinSectionVI,followedbyconclusionsinSectionVII.Webeginwiththedefinitionoftheclassofunderactuatedmechanicalsystemsconsideredinthispaper.II.CLASSOFUNDERACTUATEDMECHANICALSYSTEMSThedynamicsofmanymechanicalsystemscanbeexpressedasDq¨qCq,˙qFu1whereqq1,...,qpistheparametrizationoftheconfigurationspaceQ,Dq∈IRppistheinertiamatrix,Cq,˙q∈IRpconsistsofdamping,Coriolis,stiffness,etc.,u∈IRmisthevectorofexternalinputs,andF∈IRpmisthecorrespondinginputmatrix.Weconsideraclassofunderactuatedmechanicalsystemsthatischaracterizedasfollows.1TheinertiamatrixDisdiagonalandconstant.2ThevectorCin1dependsonlyon˙q,andfurther,thedampingforcesarelinearinvelocities.3m0.Thisleadstoanequivalentdynamics˙x1−Kx1,whichisexponentiallystable.Butthischoiceoflx1resultsinaswitchingsurfacethatistheintersectionofn−mslidingsurfaces,andmoreover,ifthecorrespondingslidingfunctionsarelinearlyindependent,thentheycannotbereachedinfinitetimeusingmcontrolinputs.Case2Toovercometheconservativeprocedureoutlinedincase1,wepresentaLyapunovfunctionbaseddesignoflx1.ConsiderthefollowingcandidateLyapunovfunctionVIRn−m−→IRdefinedasVx11/2xlatticetop1x1,andfurtherconstructlx1suchthat˙Vx1≤0onO,andfurther,Oistheintersectionofmslidingsurfaces.Case3Thiscasepertainstothedesignoflx1whereintheequivalentdynamicsisnotLyapunovstablewithrespecttoVx1.Insuchacase,constructlx1suchthattheequivalentdynamicspossessesaweaknotionofstabilitysuchasconvergencetotheorigin.B.FiniteTimeReachabilityoftheSwitchingSurfaceOncetheswitchingsurfaceisdesigned,thecontroltaskisreducedtoreachingtheswitchingsurfaceinfinitetime.Wenextdefinefinitetimereachability13oftheswitchingsurface.Definition3.2TheswitchingsurfaceOissaidtobefinitetimereachableifforanyx0∈N⊆IRn,thereexistsT∈0,∞andanadmissiblecontrolu0,T−→IRmwiththepropertythatxT∈O.WeassumethatthedesignprocedureoutlinedintheearliersectionyieldsmfunctionsSjxsuchthatOintersectiontextmj1Sjx0.ThedynamicsobtainedafterdifferentiatingeachSjforsimplicity,wesuppressthedependenceonx,rjnumberoftimes,alongthetrajectoriesof5yieldsSr11Sr22...SrmmRxQxu7whereRxlatticetopLr1fS1Lr2fS2LrmfSmlatticetopandQxLg1Lr1−1fS1...LgmLr1−1fS1Lg1Lr2−1fS2...LgmLr2−1fS2.........Lg1Lrm−1fSm...LgmLrm−1fSm.Bythedefinitionofwelldefinedvectorrelativedegree14,Qxisinvertibleatxe,andthus,forallxintheneighbourhoodofxe,wehaveuQ−1x{Px−Rx}.8ThechoiceofPx∈IRmdependsontherelativedegreeoftheslidingfunctions.Forexample,ifthevectorrelativedegreeis{1,2},thenafewpossiblechoicesofPxP1xP2xlatticetopareP1xbraceleftbigg−K1signS1−Sλ1P2x−signS2|S2|a−sign˙S2|˙S2|b−sign˙S2|˙S2|1/3−signparenleftBigχS2,˙S2parenrightBigvextendsinglevextendsinglevextendsingleχS2,˙S2vextendsinglevextendsinglevextendsingle1/5−γ1sign˙S2γ2|S2|1/2signS2whereK10,λ,b∈0,1,ab2−b,γ1,γ20,andχS2,˙S2triangleS235˙S5/32.ThefinitetimeconvergencepropertiesofP1x,P2xcanbefoundin15–17.Ifrj1,thentheslidingsurfacescanbereachedinfinitetimeusinghigherorderslidingmodecontrollers.From7,itisclearthattheslidingsurfacesSj0,j1,...,m,correspondingtothesystem5arefinitetimereachable.ItisthenstraightforwardtodesignthecontrollersujsuchthatthetrajectorycanreachtheindividualslidingsurfacesSj0infinitetime.Oncetheindividualsurfaceisreached,thetrajectoriesareconfinedtotheswitchingsurfaceO.Sincetheswitchingsurfaceisconnectedandcontainstheorigin,andfurtherthecontrollerissuchthatitrenderseachoftheslidingsurfacesfinitetimestable,thetrajectoriesstaywithintheswitchingsurfaceOpositivelyinvariant.C.StabilizationWithUndefinedRelativeDegreeWenotethatfortheclassofsystemsconsideredinthispaper,thereexistsatleastonerjthatisundefinedatxe,i.e.,LgjLrj−1fSjxe0forsomej.ThisisreflectedinthelossofinvertibilitypropertyofthematrixQonasubsetcontainingtheequilibriumpointxe.LetΦj{x∈MLgjLrj−1fSjx0}anddefineΦ∪mj1Φj,andAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2012fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009461Fig.1.Illustrationofrelativestability.furtherletBδbeasmallopenballofradiusδ0aroundxe.Now,ifforanyδarbitrarilysmallΦ∩Bδnegationslash∅9then,theconventionalnotionofstability,suchasLyapunovstability,doesnothold.Thismotivatesustoaddressthestabilizationproblemusingthenotionofrelativestability18.Definition3.3ConsideranopensetDasshowninFig.1,andfurthertheequilibriumpointxe0issaidtoberelativelystablewithrespecttothesetO,ifgivenanyepsilon10,thereexistsδ0suchthatx0∈O∩Bδ⇒xt∈Bepsilon1∀t∈0,∞,whereBδandBepsilon1areopenballsaroundtheoriginofradiiδandepsilon1,respectively.Further,ifxt−→0ast−→∞,thenxe0issaidtoberelativelyasymptoticallystablewithrespecttothesetO.Fortheundefinedrelativedegreeoftheslidingfunctionswithcondition9,theclosedloopsystempossessesonlyrelativestability.Forexample,wehaveOtriangleM\Φ.Thus,inviewoftheundefinedrelativedegreeoncertainsets,thecontrolstrategyhasthefollowingswitchingmechanismuibraceleftbigPi−Ri−summationtextmj1inegationslashjLgjLri−1fSiujbracerightbigLgiLri−1fSi,ifxt/∈Φi0,otherwise.10D.StabilizationWithParametricUncertaintyNext,weanalyzetherobustnessoftheproposedcontrollaw8withparametricuncertainties.Forsimplicity,werestrictourattentiontosystemswithrelativedegree1foreachoftheslidingfunctions,andfurtherwithoutlossofgenerality,weassumethatQxislowertriangular.Wedenotebyˆfandˆgjthevectorfieldscontainingthenominalvaluesoftheparameters.Thecontrolobjectiveistostabilize5aroundtheequilibriumpointxewiththeknowledgeofthenominalsystemˆfx,ˆgx.Inotherwords,weseekfeedbackcontrollawoftheformuρˆf,ˆgi,Sj,i,j1,...,m,suchthattheclosedloopsystem˙xfxgxρˆf,ˆgi,Sjhasthedesiredstabilityproperty.Considertherobustcontrollawobtainedbyexpressing8intermsofthenominalvaluesofthesystemparametersuˆQ−1xbraceleftbigPx−ˆRxbracerightbig11Fig.2.Mobilerobot.whereP−K1signS1−KmsignSmlatticetopandKj0arefree.ThedynamicsofStriangleS1S2Smlatticetop∈IRmsubjectedtothecontrollaw11isgivenby˙SRxQxˆQ−1xPx−QxˆQ−1xˆRxandthederivativeof1/2S2j,j1,...,m,withrespecttotime,alongtheclosedloopsystemtrajectories5isgivenbySj˙Sjηjx−KjsignSjpjxSj12whereηjtriangleLgjSjLˆgjSjiswelldefinedprovidedLˆgjSjxnegationslash0.Thisfurtherimpliesthatthepjsp1triangleLˆg1S1Lg1S1LfS1−LˆfS1...pmtriangleLˆgmSmLgmSmLfSmLg1Smu1LgmSmum−1−LˆfSmLˆg1Smu1LˆgmSmum−1arewelldefined.Henceforth,weassumethefollowing.Assumption3.1Thefunctionsηjsatisfyηjx0,j1,...,m.Thisassumption,ingeneral,quantifiestheperturbationoftheparametersinthecontrolvectorfieldsabouttheirnominalvalues.Assumption3.2ThereexistsclosedsetsU0pjandconstantsµj0,suchthatifx0∈U0pj⇒xt∈Upjtriangle{x|pjx|≤µj},j1,...,m.Further,letU0ptriangle∩mj1U0pj.Ifx0∈U0p,then12reducesto˙SjSjηj−Kj|Sj|pjxSj,andusingtheboundsonpj,wehave˙SjSj≤ηj−Kjµj|Sj|13andwithgainsKjµjsoselected,theclosedlooptrajectoriesreachtheslidingsurfacesinfinitetime.Inmanysituations,itisnoteasytoanalyticallycharacterizethesetUpj.Insuchasituation,wefixKj0tobelarge,andinviewofassumption3.1,thereexistsasmalldomainUpjsuchthat13holds.Inthefollowingsection,weapplytheproposedslidingmodecontrolstrategytotwobenchmarkexamplesthemobilerobotandtheUUVwithparametricuncertainty.IV.EXAMPLE1MOBILEROBOTInthissection,weapplytheproposedmethodtorobustlystabilizethemobilerobot.WeconsideramobilerobotasshowninFig.2.TheAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2012fromIEEEXplore.Restrictionsapply.462IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009kinematicmodelofthismobilerobotcanbeexpressedas˙xvcosθ˙yvsinθ˙θω14wherethetriplex,y,θdenotesthepositionandtheorientationofthevehiclewithrespecttotheinertialframe,andvandωarethelinearandangularvelocitiesofthemobilerobot.ThedynamicmodelisobtainedbyusingthefollowingrelationsM˙vFI˙ωτ15whereMisthemassofthevehicle,Iisthemomentofinertia,τL/rτ1−τ2,andF1/rτ1τ2,withLbeingthedistancebetweenthecenterofmassandthewheel,τ1andτ2,theleftandrightwheelmotortorques,andristheradiusoftherearwheel.Twoassumptionsaremadetosimplifythemodelthecenterofmassandtherearaxiscentercoincide,andthewheelsdonotslide.Thesecondassumptionleadstoavelocitylevelnonholonomicconstraint˙xsinθ−˙ycosθ0.Withthefollowingchoiceofstatevectorx1θ,x2xcosθysinθ,x3θy−2xsinθ2yθxcosθ,x4ω,x5v−ωxsinθ−ycosθ,14and15canbecombinedtoyieldaregularformonthemanifoldS1IR4as˙x1f1x˙x2f2x¯g1x1τ¯g2F16wherex1x1,x2,x3,x2x4,x5,f1xx4,x5,x1x5−x2x4,f2x0,−x24x2,¯g1x1k1,−k1x1x2−x32,¯g20,k2,k1triangle1/I,k2triangle1/M.Notethatthechangeofcoordinatesfromθ,x,y,v,ωmapsto→x1,x2,x3,x4,x5isaglobaldiffeomorphism.Thestabilizationofthemobilerobothasbeensolvedusingadiscontinuouscontroller19–22usingthesigmaprocesswhiletimevaryingcontrollershavebeenusedin23–25,buttherobustnessissuesarenotdiscussedintheseworks.In26,theauthorshaveaddressedthetrajectoryandforcetrackingproblemforthemobilerobotusingthedynamicmodel.A.ControllerDesignThecontrolobjectiveistomovethemobilerobotfromanyinitialpositionandorientationalsowithinitialvelocitiestoadesiredpositionandorientationwithzerofinalvelocities.Withoutlossofgenerality,thepointtopointcontrolobjectiveisreducedtothestabilizationof16totheorigin.WenowdesigntheswitchingsurfaceusingthedesignoutlineproposedinSectionIII.WeonceagainwritetheswitchingsurfaceOtriangle{x∈Mf1xlx1}.Itcanbeverifiedthat,aspercase1inSectionIIIA,ifweassignalineardynamics−Kx1tolx1in16,theresultingswitchingsurfaceistheintersectionofthreeslidingsurfaces.Further,thecorrespondingslidingfunctionsarelinearlyindependent,andhence,theswitchingfunctionscannotbereachedinfinitetimeusingthetwocontrolinputs.Then,wetrytheLyapunovfunctionbaseddesignaspresentedincase2,whichagainfailsforthisparticularexample.Weareleftwiththeprocedureoutlinedincase3,andconstructlx1aslx1parenleftBigg−ax1−ax2−bx3x1−bx3parenrightBigg,x1negationslash017wherea,b0arefree.TheresultingswitchingsurfaceisOtriangle{x∈M⊂S1IR4S1S20},whereS1triangleax1x4,S2trianglex1x5−x2x4bx3,whichsatisfiesthebasicpropertiesoftheswitchingsurfacewithexponentialconvergenceasthestabilitypropertyoftheequivalentdynamics.Thistypeofstabilityismostcommonfornonholonomicsystems.Interestingly,theequivalentdynamicsofthissystemissimilartotheclosedloopsystemofthenonholonomicintegratorpresentedin27.Next,weshowthattheclosedlooptrajectoriesonOconvergeexponentiallytotheorigininO.Lemma4.1Thetrajectoriesof16originatinginOreachitsoriginx1x2x30exponentiallyifx1Tnegationslash0,T≥0,andba.ProofThereducedorderdynamicsonOcanberewrittenasztAzwt18whereztrianglex1x2x3T,Atrianglediag−a,−a,−b,wttriangle0vt0T,andvttrianglev0e−b−atwithv0triangle−bx30/x10.Withoutlossofgenerality,assumethatT0andx10negationslash0.Thesolutionof18isgivenbyzteAtz0integraldisplayt0eAt−τwτdτandsatisfiesbardblztbardbl≤parenleftbigbardblz0bardbl−v0aparenrightbige−btv0ae−b−at.squaresolidThecontrolobjectiveisnowreducedtoreachingthesetOinfinitetimeandrenderitpositivelyinvariant.Todoso,wefollowthecontrollerdesignprocedureoutlinedinSectionIIIB,andfrom8,wehavethetorqueandforcecontrollawfortheunperturbedsystemasτ−K1signS1−ax4k119FbraceleftBiggparenleftbig−k1x2−k12x21x2−x1x3parenrightbigK1signS1ax4k1k2x1−K2signS2bx1x5−x2x4−x1x2x24k2x1bracerightbigg.20B.RobustnessAnalysisToachievetherobuststability,wemodifythecontrollers19–20asafunctionofthenominalvalues.Wedefineˆk1andˆk2asthenominalvaluesoftheparametersk1andk2,respectively,anddenotethecontrollers19–20intermsofthesenominalvaluesasτˆk1andFˆk1,ˆk2.Thedynamicsof˙SjSjwiththecontrollawsτˆk1andFˆk1,ˆk2are˙SjSjηj−Kj|Sj|pjxSj,j1,221whereηiki/ˆki0,i1,2.Thus,therequirementofassumption3.1issatisfied.Equation21,withassumption3.2,canbeexpressedas˙SjSj≤ηj−Kj|pj||Sj|≤ηj−Kjµj|Sj|,wherethefunctionsp1andp2aregivenbyp1xax4parenleftbiggˆk1k1−1parenrightbigg22p2x,τparenleftbiggˆk2k2−1parenrightbigg{−x1x2x24bx1x5−x2x4}τ2parenleftbiggˆk1−ˆk2k1k2parenrightbiggx21x2−x1x32x2.23AuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2012fromIEEEXplore.Restrictionsapply.
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