OURNAL OF UIDANCE ONTROL AND YNAMICS ol - PDF document

JOURNALOFGUIDANCE,CONTROL,ANDDYNAMICSVol.25,No.1,January–February2002AdaptiveControlofDouble-GimbalControl-MomentGyrowithUnbalancedRotorJasimAhmed¤RobertBoschCorporation,PaloAlto,California94303andDennisS.Bernstein†UniversityofMichigan,AnnArbor,Michigan48109-2140Adouble-gimbalcontrol-momentgyro(CMG)ismodeledusingLagrange’sequations,andanadaptivefeedbackcontrollawisdevelopedtofollowacommandedCMGmotion.ThecontrollawdoesnotrequireknowledgeofthemasspropertiesoftheCMG.ALyapunovargumentisusedtoprovethatcommandfollowingisachievedgloballywithasymptoticconvergence.Numericalsimulationsareperformedtoillustratethecommandfollowingalgorithm.ACMGtestbedwasdesignedandconstructedtoimplementanddemonstratetheadaptivealgorithm.Thistestbedisdescribed,andexperimentalresultsaregiven.I.IntroductionACTUATORSforspacecraftfallintotwodistinctcategories,namely,inertialandnoninertial.Inertialactuatorsprovidetorquestoaspacecraftbyreactingagainstinertialspaceand,thus,changetheangularmomentumofthespacecraft.Thrusters,whichrequirefuel,aretheprincipaltypeofinertialactuators.Magneticactuators,whichreacttotheEarth’smagneticŽeldandwhichdonotrequirefuel,alsoserveasinertialactuators.Noninertialactuatorsincludereactionwheels,momentumwheels,andsingle-gimbalanddouble-gimbalcontrolmomentgy-ros(CMGs).Whereastheseactuatorsrequireelectricpower,theydonotrequirefuel,andtheydonotchangethetotalangularmomen-tumofthespacecraft.1Amongnoninertialactuators,thedual-axisCMGisthemostsophisticatedbecauseithastheabilitytoapplycontroltorquesaroundarbitraryaxeswiththeexceptionofsingularorientationscorrespondingtogimballock.Inapplications,multipleCMGsareusedforreliability,toavoidgimballock,andtoavoidlargeanglemotion.BecauseaCMGisamultibodysystem,itsdynamicsarecomplex,especiallyiflargeanglemotionisallowed.TheoreticalanalysisofCMGdynamicsisgiveninRef.2,whereasengineeringaspectsofCMGcontrolareconsideredinRefs.3–5.InthepresentpaperweareconcernedwiththeproblemofwheelimbalanceinCMGs.Athighrotationalspeeds(5,000–10,000rpmistypicalforCMGs),smallmassimbalancesintheCMGwheelorduetoasymmetriclubricantbuildupcanproducesigniŽcantdisturbanceforcesatthefrequencyofthewheelangularvelocity.6Massimbalanceeffectsalsoariseinmomentumwheelswhereactiveisolationstagesareusedfordistur-bancesuppression.7Thesestudiessuggestthatnoninertialactuatorsareoftenthesourceofthelargestcomponentofdisturbanceforcesonspacecraft.Inthepresentpaperwemodeladouble-gimbalCMGwithun-knownmassimbalance,mountedonasupportstructureŽxedtotheEarth.UsingLagrange’sequations(seeRef.8),weaccountforlargeanglemotionandthepresenceofimbalance,whichtothebestofourknowledgehasnotbeendone.Adouble-gimbalCMGtestbedwasdesignedandconstructedtopermitvariouscontrolexperimentsto Received26June2000;revisionreceived10April2001;acceptedforpublication13April2001.Copyrightc°2001bytheAmericanInstituteofAeronauticsandAstronautics,Inc.Allrightsreserved.Copiesofthispapermaybemadeforpersonalorinternaluse,onconditionthatthecopierpaythe$10.00per-copyfeetotheCopyrightClearanceCenter,Inc.,222Rose-woodDrive,Danvers,MA01923;includethecode0731-5090/02$10.00incorrespondencewiththeCCC.¤SystemsEngineer,ResearchandTechnologyCenter,4009MirandaAvenue;jasim.ahmed@rtc.bosch.com.†Professor,DepartmentofAerospaceEngineering;dsbaero@engin.umich.edu.beperformed.ThecontrolobjectiveistodevelopacontrollawthatcounteractstheeffectsofmassimbalancewithoutknowledgeofthemassdistributionoftheCMGwhileallowingtheCMGtofollowacommandedmotionthatincludesunboundedrotationalmotionoftheCMGgimbalsandwheel.Adaptivecontrollawsformechanicalsystemswithlineardepen-denceonparametersthatareabletofollowacommandedmotionhavebeendeveloped.9¡11However,inRef.9and10,theclassofcommandedmotionsdonotincludethosethatpermitunboundedrotationalmotionofthemechanicalsystemsbecausetherotationalmotionofthesystemisdescribedintermsofanglesandtheirderiva-tives.InRef.11,differentialgeometrictechniquesareusedtoguar-anteeconvergencetoasetconsistingoffourstates,oneofwhichisthedesiredstate,butrequiresknowledgeofboundsoftheunknownparameters.Inthispaper,therotationalmotionoftheCMGisdescribedbyusingtrigonometricfunctionsofthehalf-anglesofthegimbalsandwheel,whichtransformunboundedCMGrotationalmotiontomo-tiononacompactset.Thisformulationpermitsthedevelopmentofacontrolalgorithm,whichispresentedinSec.IV,thatallowsunboundedrotationalmotionoftheCMGgimbalsandwheel.AproofofthecontrollawisbasedonavariantofstandardLyapunovargumentsfoundinRef.12toaccountforthemotiononclosedsets.Thecontrollawisadirectadaptivecontrollaw13¡16andhastheformofadynamiccompensatorwhoseorderdependsonthenumberofuncertainparametersandwhosestatesprovideestimatesofthewheelandgimbalmomentsofinertiaandcentersofmass.OurcontrollercanbeviewedasanextensionofthecontrollawderivedinRef.17forspacecrafttrackingwithunknownmassdistribution.Thecontentsofthepaperareasfollows.InSec.II,wedescribetheequationsthatgoverntheCMGmotion;inSec.III,wedeŽnethecommandfollowingproblem;andinSec.IV,weformulateacontrolalgorithmthatpermitstheCMGtotrackadesiredtrajectory.InSec.V,weillustratetheadaptivecontrollawusinganumericalexample,inSec.VI,wedescribetheexperimentalsetup,andinSec.VII,wepresentexperimentalresults.II.EquationsofMotionInthissection,wemodeltheCMGasasystemofinterconnectedrigidbodiesandapplyLagrange’sequationsforasystemofrigidbodiestoobtaintheequationsthatgoverntheCMG’smotion.Forasystemofrigidbodies,Lagrange’sequationsaregivenbyd dt³@L @Pqi´¡@L @qiDQ0i;iD1;:::;n(1)wherenisthenumberofdegreesoffreedom,q1;:::;qn2aretheindependentgeneralizedcoordinates,Pq1;:::;Pqn2arethederiva-tivesofq1;:::;qn,andLDL.q;Pq/2istheLagrangianofthe105 106AHMEDANDBERNSTEIN Fig.1CMGnomenclature.system,whereqD[q1¢¢¢qn]T2nandPqD[Pq1¢¢¢Pqn]T2n,andLisgivenbyLDT¡V(2)whereTDT.q;Pq/2isthekineticenergyofthesystem,VDV.q/2isthepotentialenergyofthesystem,Q01;:::;Q0n2arethegeneralizedforcesnotderivablefromapotentialfunctionandaregivenbyQ0iDmXjD1Fj¢@½j @qi;iD1;:::;n(3)where,forjD1;:::;m;Fjisaforcenotderivablefromapotentialfunctionandactingatposition½jD½j.q/andmisapositiveintegerrepresentingthetotalnumberofforcesnotderivablefromapoten-tial.Iftherearenoforcesnotderivablefromapotentialfunction,thenQ01D¢¢¢DQ0nD0.ThekineticenergyofasinglerigidbodyrisTD1 2!r¢Irx¢!rC1 2mrvx¢vxCmr!r¢½xy£vx(4)where1 2!r¢Irx¢!rrepresentstherotationalkineticenergyand1 2mrvx¢vxCmr!r¢½xy£vxrepresentsthetranslationalkineticen-ergy,wheremr�0isthemassofr,xisapointonr,yisthecenterofmassofr;½xyD½xy.q/isthevectorfromxtoy,vxDvx.q;Pq/isthevelocityofx,!rD!r.q;Pq/istheangularvelocityofr,andIrxDIrx.q/isthepositivedeŽniteinertiatensorofraboutx.ThepotentialenergyofrinthepresenceofauniformgravitationalŽeldisVD¡mrg¢½y(5)where½yD½y.q/denotesthepositionofyandgisthegravityvector.TheCMGshowninFig.1consistsofarectangularoutergimbal,whichrotatesthroughanangleÃaboutanaxis³3,aninnergimbal,whichrotateswithintheoutergimbalthroughanangleµaboutan Git.q/1D2400000mi.f2£½bu/¢.³3£½cb/0mi.f2£½bu/¢.³3£½cb/mi[.³3£½cb/¢.³3£½cb/C2.³3£½bu/¢.³3£½cb/]35(12)Gwt.q/1D240mw.e1£½at/¢.f2£½ba/mw.e1£½at/¢.³3£½ca/mw.e1£½at/¢.f2£½ba/Z1Z2mw.e1£½at/¢.³3£½ca/Z2Z335(13) axisf2perpendiculartotheoutergimbalaxis,andawheelŽxedtotheinnergimbal,whichspinsthroughanangleÁaboutanaxise1perpendiculartotheinnergimbalaxis.WeassumethattheCMGisconstructedsothat³3isperpendiculartoe1initially.TheCMGhasthreeactuators,speciŽcally,anoutergimbalmotor,whichisŽxedtothesupportstructure,aninnergimbalmotor,whichisŽxedtotheoutergimbal,andawheelmotor,whichisattachedtotheinnergimbal.Wemodeltheoutergimbal,theinnergimbal,andthewheelasrigidbodiesandassumethatthesupportstructureonwhichtheCMGismountedisinertiallyŽxed.Weemploythefollowingbody-ŽxedframestodeterminethekineticenergyoftheCMG.Let(³1;³2;³3)beaframeŽxedtotheoutergimbal,where³3istheoutergimbalaxisand³2istheinnergimbalaxis.Let(f1;f2;f3)beaframeŽxedtotheinnergimbal,wheref1isthewheelaxisandisobtainedbyrotating(³1;³2;³3)throughanangleµabout³2sothat24f1f2f335D24cosµ0¡sinµ010sinµ0cosµ3524³1³2³335(6)Finally,let(e1;e2;e3)beaframeŽxedtothewheelobtainedbyrotating.f1;f2;f3/throughanangleÁaboutf1sothat24e1e2e335D241000cosÁsinÁ0¡sinÁcosÁ3524f1f2f335(7)Now,thekineticenergyoftheCMGisthesumofthekineticenergyoftheoutergimbal,theinnergimbal,andthewheel.WhenEq.(4)isappliedtotheoutergimbal,innergimbal,andwheel,thekineticenergyoftheCMGisgivenbyTD1 2PqTM.q/Pq(8)whereqD[ÁµÃ]T23,PqD[PÁPµPÃ]T23,andM:3!3£3isdeŽnedbyM.q/DGTwr.q/IwaGwr.q/CGTwt.q/Gwt.q/CGTir.q/IibGir.q/CGTit.q/Git.q/CGTo.q/IocGo.q/(9)wherea,b,andcarearbitrarypointsontheaxisofro-tationofthewheel,innergimbal,andoutergimbal,respec-tively;Iocistheinertiamatrixoftheoutergimbalaboutcex-pressedinframe(³1;³2;³3);Iibistheinertiamatrixofthein-nergimbalaboutbexpressedinframe(f1;f2;f3);andIwaistheinertiamatrixofthewheelaboutthepointaexpressedinframe(e1;e2;e3),whereGo:3!3£3,Gir:3!3£3,Git:3!3£3,Gwr:3!3£3,andGwt:3!3£3arede-ŽnedbyGo.q/1D2400000000135;Gir.q/1D2400¡sinµ01000cosµ35(10)Gwr.q/1D2410¡sinµ0cosÁcosµsinÁ0¡sinÁcosµcosÁ35(11)whereZ1Dmw[.f2£½ba/¢.f2£½ba/C2.f2£½at/£.f2£½ba/]Z2Dmw[.³3£½ca/¢.f2£½ba/C.³3£½at/¢.f2£½ba/C.f2£½at/¢.³3£½ca/]Z3Dmw[.³3£½ca/¢.³3£½ca/C2.³3£½at/¢.³3£½ca/] AHMEDANDBERNSTEIN107wheretanduarethecentersofmassofthewheelandinnergimbal,respectively.ThepotentialenergyoftheCMGisthesumofthepotentialenergyoftheoutergimbal,theinnergimbal,andthewheel.WeassumethatthegravitationalŽeldisuniformandwhenEq.(5)isapplied,thepotentialenergyoftheCMGisgivenbyV.q/1D¡g¢.mw½tCmi½uCmo½v/(14)wherevisthecenterofmassoftheoutergimbal.Thegeneralizedforcesnotderivablefromapotential,obtainedbyapplyingEq.(3),areQ0ÁD¿wCfwCsw(15)Q0µD¿iCfiCsi(16)Q0ÃD¿oCfoCso(17)where¿w;¿i,and¿oarethetorquesappliedbythewheel,theinnergimbal,andtheoutergimbalmotor,respectively;fw;fi,andfoarethetorquesduetofriction;andsw;si,andsoarethetorquesduetostiffnessactingonthewheel,theinnergimbal,andtheoutergimbal,respectively.FortheCMGdescribedinSec.VI,thestiffnesstorquesmodeltheeffectofthecablesontheCMG.ApplyingEq.(1),weobtainM.q/RqC[C.q;Pq/¡F.q;Pq/]PqCG.q/¡S.q/Du(18)whereC:3£3!3£3isdeŽnedbyC.q;Pq/1D1 2[PM.q;Pq/CBT.Pq;q/¡B.Pq;q/](19)PM:3£3!3£3isdeŽnedbyPM.q;Pq/1D@MT @q.Pq­I3/(20)Inisthenbynidentitymatrix,­istheKroneckerproduct,.@MT=@q/:3£3!3£9isdeŽnedby@MT @q1Dµ@M @Á@M @µ@M @ö(21)B:3£3!3£3isdeŽnedbyB.q;Pq/1D¡I3­PqT¢@M @q(22)G:3!3isdeŽnedby G.q/1D24¡mwg¢.e1£½at/¡g¢[mwf2£.½baC½at/Cmif2£½bu]¡g¢[mw³3£.½cbC½baC½at/Cmi³3£.½cbC½bu/Cmo³3£½cv]35(23) andu1D[¿w¿i¿o]T.Inaddition,weassumethefrictionandstiff-nesstorquesareoftheform24fwfifo35DF.q;Pq/Pq;24swsiso35DS.q/(24)whereF:3£3!3£3andS:3!3.III.ErrorEquationsCommandFollowingProblemInthissection,weemployasuitablechangeofcoordinatessothatunboundedcommandedrotationalmotionoftheCMGistrans-formedtomotiononacompactset.WethendeŽnesuitableerrorstatesandexpressthecommandfollowingproblemintermsoftheseerrorcoordinates.Considerthetransformationh:3!6givenbyh.p/1D266666664sinp1=2sinp2=2sinp3=2cosp1=2cosp2=2cosp3=2377777775(25)wherepD[p1p2p3]T.WeobservefromEq.(25)thath.3/DU(26)whereUisthecompactsetgivenbyU1D©.w1;w2;w3;w4;w5;w6/26:w2iCw2iC3D1;iD1;2;3ª(27)LetzdDh.qd/(28)whereqdD[ÁdµdÃd]T:[0;1/!3isthecommandedmotion.UsingEqs.(26)and(28),weobservethatzdisboundedforeveryqd,includingthoseqdthatareunbounded.Thus,unboundedrotationalcommandedmotionoftheCMGistransformedtomotiononthecompactsetU.Next,weshowthatEq.(18)canberewrittenintermsofz,wherezDh.q/(29)ThedependenceofMonqisonlyintheformoftrigonometricfunctionsofÁ;µ;andÃ.BecausesinÁD2sinÁ=2cosÁ=2andcosÁDsin2Á=2¡cos2Á=2,withsimilarexpressionsforµandÃ,itfollowsthatM.q/canberewrittenintermsofztoobtainthefunctionOM.z/.Similarly,becausethedependenceofCandGonqisonlyintheformofthetrigonometricfunctions,wecanexpressC.q;Pq/andG.q/intermsofzandPqtoobtainthefunctionsOC.z;Pq/andOG.z/.AssumingtheargumentsofFandSdependonlyontrigonometricfunctions,werewriteFandStoobtainOF.z;Pq/andOS.z/.RewritingEq.(18)intermsofzandPq,weobtainOM.z/RqC[OC.z;Pq/¡OF.z;Pq/]PqCOG.z/¡OS.z/Du(30)PzDµ0O.Pq/¡O.Pq/0¶z(31)whereO:3!3£3isdeŽnedbyO.p/D1 224p1000p2000p335(32)wherepD[p1p2p3]T.Next,theerrorstateEzisdeŽnedbyEz1DµEOE¶1DH.zd/z(33) 108AHMEDANDBERNSTEINwhereE23,OE23,andH:6!6£6isgivenbyH.w/1D266666664w400¡w1000w500¡w2000w600¡w3w100w4000w200w5000w300w6377777775(34)wherewD[w1w2w3w4w5w6]T.UsingEq.(34),weobservethatthatH.w/HT.w/DHT.w/H.w/DI6forallw2U,whereUisgivenbyEq.(27).Hence,Eq.(33)implieszDHT.zd/Ez(35)ED264sin.Á=2¡Ád=2/sin.µ=2¡µd=2/sin.Ã=2¡Ãd=2/375;OED264cos.Á=2¡Ád=2/cos.µ=2¡µd=2/cos.Ã=2¡Ãd=2/375(36)NotethatED0ifandonlyifÁ¡ÁdD0mod2¼,µ¡µdD0mod2¼;andÁ¡ÁdD0mod2¼.Furthermore,deŽnetheerrorstateePq1DPq¡Pqd(37)Forthecommandfollowingproblem,assumeqd:[0;1/!3isC2.FindadynamicfeedbackcontrollawoftheformPO®Df.zd;Pqd;Rqd;O®;z;Pq/(38)uDg.zd;Pqd;Rqd;O®;z;Pq/(39)forEqs.(30)and(31),whereO®.t/2º,t2[0;1/,suchthatE!0andePq!0ast!1forallinitialconditionsz.0/2U;Pq.0/23;andO®.0/2º.CommandfollowingproblemasstatedrequiresE!0andePq!0,whichusingEqs.(36)and(37)impliesthattheCMGfollowacommandedmotion,andpermitsallsufŽcientlysmoothqd,includingthosethatareunbounded.Notethatthecontrolalgo-rithmasstatedinEqs.(38)and(39)doesnothavetobeindependentofthemassdistributionoftheCMG.However,inSec.IVweshalldevelopacontrolalgorithmthatrequiresnoknowledgeofthemassorinertiapropertiesoftheCMG.Next,werecastEqs.(30)and(31)intermsoftheerrorstatesEzandePqandrestatethecommandfollowingproblemintermsofEzandePq.Todothis,deŽneQM:[0;1/£6!3£3byQM.t;Ez/1DOM©HT[zd.t/]Ezª(40)QC:[0;1/£6£3!3£3byQC.t;Ez;ePq/1DOC©HT[zd.t/]Ez;ePqCPqd.t/ª(41)andQG:[0;1/£6!3byG.t;Ez/1DOG©HT[zd.t/]Ezª(42)WhenEqs.(35)and(37)areused,Eqs.(30)and(31)becomeQM.t;Ez/.PePqCRqd/D[¡QC.t;Ez;ePq/CQF.t;Ez;ePq/].ePqCPqd/CQS.t;Ez/¡QG.t;Ez/Cu(43)PEzDµ0O.ePq/¡O.ePq/0¶Ez(44)Then,Eqs.(33)and(34)implythatE2iCOE2iDz2iCz2iC3;iD1;2;3(45)ItfollowsfromEq.(45)thatthecommandfollowingproblemissolvedifandonlyifE!0andePq!0inEqs.(43)and(44)forallinitialconditionsEz2UandePq.0/23.WeassumethatmeasurementsofqandPqareavailable.ItcanbeseenfromEqs.(25),(34),and(37)thatthequantitiesz,zd,E,OE,andePqcanbecalculated.InSec.IV,thecontrollawEqs.(38)and(39)isexpressedintermsofO®,Ez,ePq,zd,Pqd,andRqd.IV.AdaptiveControlLawInthissection,wepresentafeedbackcontrollawthatasymp-toticallyfollowsacommandedtrajectory.ThecontrollawdoesnotrequireknowledgeofthemassdistributionoftheCMG.UsingEqs.(9),(12),(13),(34),and(40),weobservethatQM.t;Ez/dependsontheinertia,mass,andcenterofmasslocationparameters,namely,Iwa11,Iwa22,Iwa33,Iwa23,Iwa13,Iwa12,Iib11,Iib22,Iib33,Iib23,Iib13,Iib12,Ioc33,mw,mi,mo,½at1,½at2,½at3,½bu1,½bu2,½bu3,½ba1,½ba2,½ba3,½cb1,½cb2,and½cb3,where½atisexpressedin.e1;e2;e3/,½buand½baareexpressedin.f1;f2;f3/,and½cbisexpressedin.³1;³2;³3/,andwherexiistheithcomponentofx2nandAijisthe.i;j/entryofA2m£n.Next,notefromEqs.(9),(12),(13),(34),and(40)thatQM.t;Ez/dependslinearlyon®m,where®mconsistsofinertia,massandcenterofmasslocationparameters,andproductsofthese.Inpractice,someoftheseparametersmaybeknown.Inthiscase,weassumethat®mconsistsonlyofuncertainparametersandproductsofparameters,atleastoneofwhichisuncertain.Itcanbeshownthatthedimensionof®misbetween0(nouncertainparameters)and50(alluncertainparameters).Similarly,usingEqs.(23),(34),and(42),weobservethatQG.t;Ez/dependsonthegravitationalparameters,namely,g1,g2,g3,mw,mi,mo,½at1,½at2,½at3,½bu1,½bu2,½bu3,½ba1,½ba2,½ba3,½cb1,½cb2,½cb3,½cv1;and½cv2wheregisexpressedinanarbitraryinertiallyŽxedframeand½cvisexpressedin(³1;³2;³3).Next,notefromEqs.(23),(34),and(42)thatQG.t;Ez/dependslinearlyon®g,where®gconsistsofthecenterofgravitylocationparametersandproductsofthese.Itcanbeshownthatthedimensionof®gisbetween0(nouncertainparameters)and15(alluncertainparameters).Forthefrictionandstiffnesstorques,weassumethatthereexistparameters®fand®s,sothatQF.t;Ez;ePq/andQS.t;Ez/dependlinearlyon®fand®s.ThenumberofuncertainparametersºdependsonassumptionsmadeabouttheCMGconŽguration,aswellasonQFandQS.Inthespecialcaseinwhichtherearenofrictionandstiffnesstorques,thereexistsacommonpointthatliesontheaxisofrotationofallofthesemotors,thatis,aDbDcsothatGwtDGitD0,andg2Dg3D0,wheregisexpressedinaninertiallyŽxedframe.²1;²2;²3/suchthat,attD0;.²1;²2;²3/coincideswith.³1;³2;³3/;thenitcanbeseenthatºD21.ThisisthecaseconsideredinSec.V.Thefollowinglemmaswillbeneeded.Lemma1:OM.z/andQM.t;Ez/arepositivedeŽniteforallz26,t2[0;1/,andEz26.Proof:RecallthatOM.z/isformedbyreplacingthetrigonometricfunctionsoftheanglesÃ;µ,andÁbyzusingEq.(25).WhenOGo.z/;OGir.z/;OGit.z/;OGwr.z/,andOGwt.z/aredeŽnedinasimilarmanner,itfollowsfromEqs.(9)and(29)thatOM.z/DOGTo.z/IocOGo.z/COGTir.z/IibOGir.z/COGTit.z/OGit.z/COGTwr.z/IwaOGwr.z/COGTwt.z/OGwt.z/(46)FromEq.(46),itfollowsthatOM.z/isthesumofpositivesemideŽ-nitetermsandis,thus,positivesemideŽniteforallz26.Letz26andletp23satisfypTOM.z/pD0.Thus,itfollowsthatOGo.z/pDOGir.z/pDOGwr.z/pD0(47)WhenEq.(10)isused,OGo.z/pD0impliesthatp3D0.Similarly,p3D0andOGir.z/pD0implyp2D0.Finally,whenEq.(11)isusedwithp2Dp3D0andOGwr.z/pD0,itfollowsthatp1D0;hence,pD0,whichimpliesthatOM.z/ispositivedeŽniteforallz26.FinallyitfollowsfromEq.(40)thatQM.t;Ez/ispositivedeŽniteforallt2[0;1/andEz26. AHMEDANDBERNSTEIN109Lemma2:Thereexist¹1�0and¹2�0suchthat¹1I3·OM.z/·¹2I3;z2U(48)¹1I3·QM.t;Ez/·¹2I3;t2[0;1/;Ez2U(49)whereUisgivenbyEq.(27).Proof:BecauseUisacompactsubsetof6;OMisacontinuousfunction,andOM.z/ispositivedeŽniteforallz26,itfollowsthatthereexistpositive¹1and¹2satisfyingEq.(48).Equation(49)isimmediate.Finally,weisolatetheparametersthatcharacterizetheinertia,mass,centerofmasslocations,centerofgravitylocations,andthefrictionandstiffnesstorquesbydeŽningY:[0;1/£6£3£3£3£3!3£ºbyY.t;Ez;ePq;·;Q·;O·/®1D¡QM.t;Ez/·¡QC.t;Ez;ePq/Q·CQF.t;Ez;ePq/O·¡QG.t;Ez/CQS.t;Ez/(50)where·23;Q·23;O·23,and®2ºisthevectorofparameters.Next,wepresentacontrollawthatsolvesthecommandfollowingproblemwithaproofbasedonLyapunovtheory.NotefromthedeŽnitionofthecommandfollowingproblemasstatedinSec.IIIthatweareonlyinterestedininitialconditionsthatbelongtotheclosedset3£U£º.ThestandardLyapunovtheoremasfoundinRef.18isonlyvalidforopensets,andsoweuseavariantofthestandardLyapunovargumentfoundinRef.12.Theorem:AssumethatPqdandRqdareboundedforallt2[0;1/.Let3:[0;1/!3£3becontinuous,K:[0;1/!3£3becon-tinuous,3123£3,3223£3,K123£3,andK223£3besuchthat31,32,and3.t/arediagonalforallt2[0;1/,0313.t/32;t2[0;1/(51)0K1K.t/K2;t2[0;1/(52)LetP23£3bediagonalandpositivedeŽnite,andletQ2º£ºbepositivedeŽnite.ThenthecontrollawPO®DQ¡1YT[t;Ez;ePq;Rqd¡3O.ePq/OE;¡3ECPqd;ePqCPqd].ePqC3E/(53)uD¡Y[t;Ez;ePq;Rqd¡3O.ePq/OE;¡3ECPqd;ePqCPqd]O®¡PE¡K.ePqC3E/(54)solvesthecommandfollowingproblem.Furthermore,O®isboundedforallt¸0,andPO®!0ast!1.Proof:DeŽne¾;Oe;ez;and¯by¾1DePqC3E(55)Oe1DOE¡´1(56)ez1D[ETOeT]T(57)¯1D®¡O®(58)where´1D[111]T.Forconciseness,wewriteYDY[t;ezC´2;¾¡3E;Rqd¡3O.¾¡3E/.OeC´1/;¡3ECPqd;¾¡3ECPqd](59)where´2D[000111]T.UsingEqs.(43),(44),and(53–59),weobtainQM.t;ezC´2/P¾DY¯¡QC.t;ezC´2;1E/¾¡PE¡K¾(60)PezDµ0O.¾¡3E/¡O.¾¡3E/0¶.ezC´2/(61)P¯D¡Q¡1YT¾(62)LetÂD[¾TeTz¯T]T.ThentheoriginÂD0isanequilibriumsolu-tionofthesystem(60–62).Next,weshowthat¾!0andE!0ast!1forinitialconditionsÂ.0/2D,whereD1D3£OU£º,whereOU1Dfw26:wC´22Ug.TodothisweuseTheorem3.2ofRef.12,whichentailsconstructingaLyapunovfunction,showingthatDisaninvariantsetandallsolutionsarebounded.ConsiderthecandidateLyapunovfunctionV:[0;1/£3£6£º!deŽnedbyV.t;Â/D1 2£¾TQM.t;ezC´2/¾C¯TQ¯¤CETPECOeTPOe(63)ThecandidateLyapunovfunctionisthesumofapseudokineticenergyterm1 2¾TQM.t;ezC´2/¾,apseudopotentialenergytermETPECOeTPOe,and1 2¯TQ¯,apositivedeŽnitefunctioninthepa-rametererror.ThetotaltimederivativeofValongthetrajectoriesofthesystemisgivenbyPV.t;Â/D¡¾TK.t/¾¡ETP3.t/E·¡¾TK1¾¡ETP31E(64)BecauseDisclosed,VispositivesemideŽniteon[0;1/£D,andPVsatisŽesEq.(64),itfollowsthatVisavalidLyapunovfunctiononD.NextweshowthatDisaninvariantsetandthatallsolutionsarebounded.LetÂ.t/beasolutionofEqs.(60–62)deŽnedonanintervalI,suchthatÂ.0/2D.NowÂ.t/remainsinDbecausee2z;i.t/C[ez;iC3.t/C1]2D1,t2I.WhenEq.(49)andEzDezC´2areused,itfollowsthat¹1I3·QM.t;ezC´2/·¹2I3;t2[0;1/;ez2OU(65)Now,itfollowsfromEq.(65)thatW1.Â/·V.t;Â/·W2.Â/;t2[0;1/;Â2D(66)whereW1:3£6£º!andW2:3£6£º!aretheradiallyunboundedpositivedeŽnitefunctionsW1.Â/D1 2£¹1¾T¾C¯TQ¯¤CETPECOeTPOe(67)W2.Â/D1 2£¹2¾T¾C¯TQ¯¤CETPECOeTPOe(68)WhenthesetsAi;±DfÂ2D:Wi.Â/·±garedeŽned,whereiD1;2;±�0,andÄt;±DfÂ2D:V.t;Â/·±g,wheret¸0and±�0,itfollowsthatA2;±½Ät‚±½A1‚±;t¸0;±�0(69)BecauseW1andW2areradiallyunbounded,itfollowsthatthesetsA1;±andA2;±areboundedforall±�0,and,furthermore,thereexistsO±�0largeenoughsuchthatÂ.0/2A2;O±.NowfromEq.(64)itfollowsthatV[t;Â.t/]isnotincreasingandwiththeuseofEq.(69)thatthesolutionÂ.t/remainsinthecompactsetA1;O±.ItnowfollowsfromTheorem2.4ofRef.18thatÂ.t/existsforallt¸0.NextnotethatthetotaltimederivativeofW3:3£6£º!deŽnedbyW3.Â/D¡¾TK1¾¡ETP31E(70)alongthetrajectoriesofthesystemisgivenbyPW3.t;Â/D¡2¾TK1[QM.t;ezC´2/]¡1[Y¯¡E¡K¾¡QC.t;ezC´2;¾¡3E/]¡2ET31F.¾¡3E/.ezC´2/BecauseQM¡1,QC,Y,F,Â,Pqd,andRqdarecontinuousfunctions,Â.t/isbounded,andbyassumption3,K,Pqd,andRqdarebounded,itfollowsthatPW3[t;Â.t/]isbounded.UsingTheorem3.2ofRef.12,weconcludethat¾!0andE!0.Furthermore,becausePqdandRqdareboundedand¾!0andE!0,itfollowsfromEq.(62)thatP¯!0and,thus,PO®!0.Because¾!0,E!0,and3isbounded,itfollowsfromEq.(55)thatePq!0.Hence,weconcludethatEqs.(53)and(54)solvethecommandfollowingproblem. 110AHMEDANDBERNSTEIN Fig.2ErrorstatesE1;E2,andE3. Fig.3ErrorstateseÇq=[eÇÁeǵeÇÃ]T.UsingEqs.(50),(53),and(54),weobservethatthecontrolalgo-rithmdoesnotrequireanyknowledgeofthemassdistributionoftheCMGandonlyrequiresknowledgeoftheCMGstatesz;Pq;zd;Pqd,andRqd.Furthermore,weobservethattheright-handsideofEq.(53)isindependentofO®andthattheright-handsideofEq.(54)isde-pendentonlyontheCMGstatesandO®.Hence,thecontrollaw(53)and(54)isaproportional–integralcompensator.TheparameterQrepresentsthegainoftheadaptationlaw,and3;K,andPrepresentthegainsoftheproportional–integralcon-troller.InSec.VII,wedescribehowwechosethesegainsforourexperimentalsetup.ThestateO®representsadjustableparameters,whereasEq.(53)representsthemechanismforadjustingthesepa-rameters.AlthoughthetimederivativeoftheadaptiveparameterO®convergestozeroast!1;O®doesnotnecessarilyconverge.SeeRef.17foradditionaldetailsconcerningtheuseofO®forparameteridentiŽcation.V.NumericalExampleInthissectionweillustratecommandfollowingforthedesiredtrajectory:Ád.t/D2000¼=60trad(71)µd.t/D.30¼=180/sin.15¼=180t/rad(72)Ãd.t/D.40¼=180/sin.10¼=180t/rad(73)ThiscommandrepresentsaCMGmotioninwhichthewheelspinsataconstantrateof1000rpm,theinnergimbaloscillatessinusoidallywithanamplitudeof30degandfrequencyof15deg/s,andtheoutergimbaloscillatessinusoidallywithanamplitudeof40degandafrequencyof10deg/s.ThenumericalsimulationsareperformedforamodeloftheplantgivenbyEqs.(30)and(31)basedontheCMGdescribedinSec.VI.Thenominalvaluesforthevariousmass,inertia,centerofmasslocation,andgravitationalparametersaregivenbyEqs.(76–81).NotefromEqs.(80)and(81)thatwehaveconsideredthecaseinwhichthereexistsacommonpointthatliesontheaxisofrotationofeachmotor,thatis,aDbDc,sothatGitDGwtD0andg2Dg3D0,wheregisexpressedintheinertiallyŽxedframe.²1;²2;²3/suchthat,attD0;.²1;²2;²3/coincideswith(³1;³2;³3).Inthissection AHMEDANDBERNSTEIN111 Fig.4Motortorques.weassumethattherearenofrictionandstiffnesstorquessothatQFDQSD0.Withthisassumption,thenumberofparametersisre-ducedfrom65to21.Inthiscase,®D[®Tm®Tg]T,where®mD264®Tw®Ti®To375;®wD26666666664Iwa11Iwa22Iwa33Iwa23Iwa13Iwa1237777777775;®iD26666666664Iib11Iib22Iib33Iib23Iib13IIb1237777777775®oDIoc33;®gD2666666666664¡mwg1½ar1¡mwg1½at2¡mwg1½at3¡mig1½bu1¡mig1½bu2¡mig1½bu3¡mog1½cv1¡mog1½cv23777777777775(74)TheinitialorientationisqD[000]Trad,theinitialrateisPqD[000]Trad/s,andtheinitialvalueoftheadaptiveparameterisO®D[O®TwO®TiO®ToO®Tg]T,whereO®wD1:0e¡3[1:47:51:21:0e¡33:0e¡34:0e¡4]Tkg¢m2,O®iD1:0e¡4[1:33:12:91:0e¡22:0e¡23:0e¡2]Tkg¢m2,O®oD7:2e¡3kg¢m2andO®gD[3:2e¡201:2e¡81:2e¡97:2e¡12:0e¡61:0e¡58:1e¡6]Tkg¢m2/s3.Thegainsarechosentobe3.t/D240:20004000435;K.t/D240:10002:40002:435t2[0;1/(75)PDI3andQD100;000I21.FortheCMGdescribedinSec.VI,thewheelmotorcangenerateamaximumtorqueof0.01332N¢m,theinnergimbalmotorcangenerateamaximumtorqueof0.113N¢m,andtheoutergimbalmotorcangenerateamaximumtorqueof1.769N¢m.WeapplythecontrollawgivenbyEqs.(53)and(54)toEqs.(30)and(31),butsaturatethecontrollersothatitdoesnotexceedthemaximumallowabletorque.WeobservefromFigs.2and3thatcommandfollowingisachieved.Figure4indicatesthecontroleffortrequired.Figures5and6indicatetheestimatesof®w,whicharegivenbyO®1;:::;O®6.NotefromFig.5thatO®1;O®2,andO®3donotconvergetotheapproximatevaluesasgiveninSec.VIandfromFig.6thatO®5andO®6areoscillatoryanddonotseemtoconverge.VI.ExperimentDescriptionTheCMGtestbedwasdesignedtoallowlargeanglerotationalmotionofthewheel.Eachofthegimbalsisabletorotatenearly180deginbothdirections,providingarangeofmotionsufŽcientforhighprecessionanglesandlargeangleslewingmaneuvers.Thegimbalscannotcompleteafullrevolutionbecausetheelectricalconnectionsaremadeusingwiresratherthansliprings.Theoutergimbalandinnergimbalsaremachinedfromasingleblockofalu-minumtoprovideprecisealignment.FordetailsseeRef.19.Figure7showsaphotographoftheactualCMGtestbedwithconnectorsandwiring.ThewheelshowninFig.7containsslotstowhichwecanaddmassestounbalancethewheel.Tofurthervarythecenterofmasslocation,thewheelcanbemovedtranslationallyalongitsrotationaxis.Theapproximatevaluesforthevariousmass,inertia,centerofmasslocation,andcenterofgravitylocationparametersbasedonanominalconŽgurationoftheCMGareIwaD244:79e¡41:0e¡62:0e¡61:0e¡66:1005e¡41:0e¡62:0e¡61:0e¡66:1005e¡435kg¢m2(76)IibD241:287e¡31:0e¡61:0e¡61:0e¡65:0185e¡40:01:0e¡60:01:605e¡335kg¢m2Ioc33D8:231e¡3kg¢m2(77)mwD0:3kg;miD0:35kg;moD1:23kg(78)g1D¡9:81m/s2;g2D0:0m/s2;g3D0:0m/s2(79)½atD[0:020:00:0]Tm;½buD[¡0:070:00:0]Tm½cvD[0:00:00:02]Tm(80)½cbD[0:00:00:0]Tm;½baD[0:00:00:0]Tm(81) 112AHMEDANDBERNSTEIN Fig.5Adaptiveparametersî1;î2,andî3(estimatesofIwa11,Iwa22,andIwa33,respectively). Fig.6Adaptiveparametersî4,î5,andî6(estimatesofIwa23;Iwa13,andIwa12,respectively). Fig.7CMGtestbed.whereweexpressginaninertiallyŽxedframe("1;"2;"3)suchthat,attD0,("1;"2;"3)coincideswith(³1;³2;³3).NotefromEqs.(80)and(81)thatinthenominalconŽgurationwehaveas-sumedthataDbDc.Allofthemotorsareequippedwithopticalincrementalencoders,providingmeasurementsoftheanglesofthegimbalsandwheel.WedifferentiateandŽltertheencodersignalstoobtainPÁ;Pµ,andPÃ.Theinnergimbalandwheelmotors,manufacturedbyMaxon,Inc.,werechosenfortheirhightorque-to-weightratio,lowinertia,andlowtorqueripple.ThecontrolprocessoristheDS1103boardmanufacturedbydSPACE,Inc.Thecodeforsimulationandcon-trollerimplementationiswritteninCusingtheS-functionblocksofSimulink®.Thesamplingrateis1000Hz.VII.ExperimentalResultsInthissectionwepresentexperimentalresultstoillustratecom-mandfollowingforthedesiredtrajectory AHMEDANDBERNSTEIN113 Fig.8Wheelrate. Fig.9Innergimbalangle.Ád.t/D2000¼=60trad(82)µd.t/D120¼=180rad(83)Ãd.t/D¡40¼=180rad(84)ThiscommandrepresentsaCMGmotioninwhichthewheelspinsataconstantrateof1000rpm,theinnergimbalisorientedtoanangleof120deg,andtheoutergimbalisorientedtoanangleof¡40deg.Onceconvergencehasbeenattained,thecommandisabruptlychangedsothattheinnergimbalisreorientedtoanangleof¡60degandtheoutergimbalisreorientedtoanangleof60deg.ThecontrollawgivenbyEqs.(53)and(54)isappliedtotheCMGdescribedinSec.VI.WeassumethefrictionandstiffnesstorquesareoftheformF.p;Op/D24F1000F2000F335(85)whereF1;F2,andF3arerealnumbersindependentofpandOp,S.p/D24s1sin.p1/000s2sin.p2/000s3sin.p3/35;s1;s2;s32(86)sothat®D[®Tm®Tg®Tf®Ts]T,where®mand®garegivenbyEq.(74),and®fD24f1f2f335;®sD24s1s2s335(87)Thetuningparametersarechosentobe3.t/D2450:00000:20000:235C.1¡e¡0:001t/24¡40:00001:800019:835(88) 114AHMEDANDBERNSTEIN Fig.10Outergimbalangle. Fig.11Wheelmotortorque.K.t/D242:0e¡40004:0e¡50004:0e¡435C.1¡e¡0:001t/240:00000:00000:039635(89)PD1:0e¡3I3QDdiag.Q1;Q2;Q3;Q4;Q5;Q6;Q7;Q8/(90)whereQ1Ddiag.1:0e¡11;1:0e¡11;1:0e¡11;1:0e¡14;1:0e¡14;1:0e¡14/(91)Q2Ddiag.1:0e¡6;1:0e¡6;1:0e¡6;1:0e¡9;1:0e¡9;1:0e¡9/;Q3D1:0e¡2(92)Q4D1:0e¡5I3;Q5D1:0e¡2I3;Q6D1:0e¡3I2(93)Q7Ddiag.1:0e¡9;1:0e¡5;1:0e¡1/Q8Ddiag.1:0e¡4;100:0;10:0/(94)Thegainswerechosentopreventsaturationofthemotorsforanyappreciableperiodoftime.Inourtestsonoursetup,wefoundthatsaturationofthemotorsforsigniŽcantperiodsoftimeresultedinthebuildupoflargeamplitudeoscillations.Also,saturationofthemotorsmightcausedamagetothemotorsifcontinuedforlongperiodsoftime.Withthisviewinmind,mostofthetime-varyinggainsareinitiallychosensmallbecauseparametricuncertainityisinitallylarge.Astheadaptationproceeds,thesegainsareincreased.However,notethat311.t/actuallydecreasesinmagnitude.ThetorquesthataretransmittedtothegimbalsandwheelcanbeturnedonoroffusingamasterswitchthatcanbecontrolledusingsoftwaredevelopedbydSPACE,Inc.Weapplythecontrollaw(53)and(54)attD0s,butthemasterswitchisturnedon AHMEDANDBERNSTEIN115 Fig.12Torquegeneratedbyinnerandoutergimbalmotors.onlyatapproximatelytD20s,andthus,themotorsareeffectivelyturnedofffort2[0;20/.ThespikesinFigs.8–11atapproximatelytD20sareduetothemasterswitchbeingturnedon.Figures11and12showthecontroleffortsasrequiredbythecontrollaw.BeforetD20s,thesetorquesarenottransmittedtotheCMGbecausethemasterswitchhasnotyetbeenturnedon.WeobservefromFig.8thatthewheelattainsaspeedof1000rpmatapproximatelytD30s.Figure9showsthattheinnergimbalat-tainsanangleof120degatapproximatelytD350s,andFig.10showsthattheoutergimbalattainsanangleof¡40degatap-proximatelytD200s.AtapproximatelytD350s,wemodifythecommandasdescribedearlier.WeobservefromFig.8thatthewheelattainsthespeedof1000rpmatapproximatelytD350s,fromFig.9thattheinnergimbalattainsanangleof¡60degatapproximatelytD600s,andfromFig.10thattheoutergimbalattainsanangleof60degatapproximatelytD400s.InFigs.9and10,itcanbeseenthatthegimbalsundergotransientsduetostartupaswellasatransientattD350sduetotheabruptchangeinsetpoint.VIII.ConclusionsInthispaper,weareinterestedindevelopingacontrolalgorithmthatfollowsacommandedCMGrotationalmotion,includingcom-mandedrotationalmotionsthatareunbounded.Todothis,wede-scribetherotationalmotionoftheCMGintermsofthetrigono-metricfunctionsofthehalf-anglesofthegimbalsandwheel.ThisformulationtransformsunboundedrotationalmotionoftheCMGontomotiononacompactsetandisthekeyingredientinthedevel-opmentofthecontrolalgorithm(53)and(54).Inasimilarvein,itistheuseoftime-varyinggainsthatpermitsthesuccessfuluseofEqs.(53)and(54)toachievecommandfollowinginourexperimentalsetup.TheuseofconstantgainsresultedineithersaturationofthecontrollerforsigniŽcantperiodsoftime,whichledtothebuildupoflargeamplitudeoscillations,ortoextremelyslowtimeresponses.InfutureresearchthecontrollawwillbemodiŽedtosuppressre-actiontorquestransmittedtothesupportstructureduetoimbalance.References1Bryson,A.E.,Jr.,ControlofSpacecraftandAircraft,PrincetonUniv.Press,Princeton,NJ,1994,pp.74–92.2Marguiles,G.,andAubrun,J.N.,“GeometricTheoryofSingle-GimbalControlMomentGyroSystems,”AIAAGuidanceandControlConference,AIAA,NewYork,1976,pp.255–267.3Liden,S.P.,“PrecisionCMGControlforHigh-AccuracyPointing,”AIAAGuidanceandControlConference,AIAA,NewYork,1973,pp.236–240.4Chubb,W.B.,Kennel,H.F.,Rupp,C.C.,andSeltzer,S.M.,“FlightPerformanceofSkylabAttitudeandPointingControlSystem,”AIAAMe-chanicsandControlofFlightConference,AIAA,NewYork,1974,pp.220–227.5Kurokawa,H.,Yajima,N.,andUsui,S.,“ANewSteeringLawofaSingleGimbalCMGSystemofPyramidConŽguration,”IFACAutomaticControlinSpace,IEEEPublications,Piscataway,NJ,1985,pp.251–257.6Bodora,J.A.,andBamlde,H.,“ExperimentalandSystemStudyofReactionWheels,”ESAContractReport,1982.7Neat,G.W.,Melody,J.W.,andLurie,B.J.,“VibrationAttenuationApproachforSpaceborneOpticalInterferometers,”IEEETransactionsonControlSystemsTechnology,Vol.6,No.6,1998,pp.687–700.8Greenwood,D.T.,PrinciplesofDynamics,Prentice–Hall,EnglewoodCliffs,NJ,1988,pp.239–299.9Bayard,D.S.,andWen,T.J.,“ANewClassofControlLawsforRoboticManipulators—PartII:AdaptiveCase,” InternationalJournalofControl , Vol.47,No.5,1988,pp.1387 – 1406. 10Arimoto,S.,ControlTheoryofNon-LinearMechanicalSystems:APassivity-BasedandCircuit-TheoreticApproach,Vol.49,OxfordUniv.Press,Oxford,England,U.K.,1996.11Koditschek,D.E.,“AppplicationofaNewLyapunovFunctiontoGlobalAdaptiveAttitudeTracking,”IEEEConferenceonDecisionandControl,Inst.ofElectricalandElectronicsEngineers,NewYork,1988,pp.63–68.12Hale,J.K.,OrdinaryDifferentialEquations,Wiley,NewYork,1969,p.305.13Slotine,J.J.E.,andLi,W.,AppliedNonlinearControl,Prentice–Hall,UpperSaddleRiver,NJ,1989.14Sastry,S.,andBodson,M.,AdaptiveControl:Stability,Convergence,andRobustness,Prentice–Hall,UpperSaddleRiver,NJ,1989.15Astrom,K.J.,andWittenmark,B.,AdaptiveControl,2nded.,Addison–Wesley,Reading,MA,1995.16Narendra,K.S.,andAnnaswamy,A.M.,StableAdaptiveSystems,PrenticeHall,EnglewoodCliffs,NJ,1988,Chap.1.17Ahmed,J.,Coppola,V.T.,andBernstein,D.S.,“AsymptoticTrackingofSpacecraftAttitudeMotionwithInertiaIdentiŽcation,” JournalofGuidance, Control,andDynamics ,Vol.21,No.5,1998,pp.684 – 691. 18Khalil,H.K.,NonlinearSystems,Prentice–Hall,UpperSaddleRiver,NJ,1996,p.77,138.19Ahmed,J.,Miller,R.H.,Hoopman,E.H.,Coppola,V.T.,Andrusiak,T.,Acton,D.,andBernstein,D.S.,“AnActivelyControlledControlMo-mentGyro/GyroPendulumTestbed,”ProceedingsofConferenceonControlApplications,IEEEPublications,Piscataway,NJ,1997,pp.250–252.