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1、1556IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.46,NO.10,OCTOBER2001ControlledLagrangiansandtheStabilizationofMechanicalSystemsII:PotentialShapingAnthonyM.Bloch,Member,IEEE,DongEuiChang,NaomiEhrichLeonard,Member,IEEE,andJerroldE.MarsdenAbstract—Weextendthemethodof
2、controlledLagrangiansThesematchingconditionsensurethattheEuler–Lagrange(CL)toincludepotentialshaping,whichachievescompleteequationsderivedfromthecontrolledLagrangianareconsistentstate-spaceasymptoticstabilizationofmechanicalsystems.Thewithavailablecontrolinpu
3、ts,i.e.,theymatchthecontrolledCLmethoddealswithmechanicalsystemswithsymmetryandEuler–Lagrangeequationsforthegivenmechanicalsystem.providessymmetry-preservingkineticshapingandfeedback-con-trolleddissipationforstate-spacestabilizationinallbuttheTheCLhasareshape
4、dkineticenergythatretainstheoriginalsymmetryvariables.Potentialshapingcomplementsthekineticsymmetry.In[13],feedback-controlleddissipationwasaddedshapingbybreakingsymmetryandstabilizingtheremainingstatetoproveasymptoticstabilizationinallstatevariablesmodulovar
5、iables.Theapproachalsoextendsthemethodofcontrolledthesymmetrygroupvariables.FortheinvertedpendulumontheLagrangianstoincludeaclassofmechanicalsystemswithoutcart,wedrivethependulumtotheuprightpositionandthecartsymmetrysuchastheinvertedpendulumonacartthattravels
6、alonganincline.torestbutnotnecessarilypositionedattheorigin.Thislimita-tionwillbeovercomeinthepresentwork.IndexTerms—Lyapunovmethods,mechanicalsystems,non-linearcontrol,stabilization,tracking.B.HistoryandRelatedLiteratureTheCLmethodhasitsoriginsin[8]and[16].O
7、urpotentialI.INTRODUCTIONshapingapproachisinspiredby[12]and[28].OtherrelevantHISpapercontinuesthedevelopmentin[13]ofthemethodworkinvolvingenergymethodsincontrolandstabilizationin-TofcontrolledLagrangians(CL),aconstructivemethodforcludes[1],[3],[18],[26],[33],
8、[34],[37],[38],and[41].In[6],stabilizingmechanicalsystems.Varioussupplementaryandad-werelatethepotentialshapingapproachheretothatof[24],ditionalresultshaveappearedin[9]–[12]and[14].Ourmai
