Control of Spacecraft Subject to Actuator Failures

CONTROLOFSPACECRAFTSUBJECTTOACTUATORFAILURES:STATE-OF-THE-ARTANDOPENPROBLEMSPanagiotisTsiotrasandViktoriaDoumtchenko†GeorgiaInstituteofTechnologyAtlanta,GA30332-0150,USAInthispaperwereviewthemainresultsintheareaofactivecontrolofspacecraftwithoneactuatorfailure.Weemphasizethequalitativechar-acteristicsthatmakethisachallengingcontrolproblem.Wepresentase-riesofnewresultsthatsolvetheproblemofdetumblingwithsimultaneousattitudestabilizationabouttheunactuatedaxis,thecompleteattitudesta-bilizationproblem,andthefeasibletrajectorygenerationproblemforaspacecraftwithoneactuatorfailure.Wepresentseveralnumericalexam-plesthatdemonstratetheefficacyoftheproposedcontrolalgorithms.Weconcludewithabriefreviewofsomeopenproblemsinthegeneralareaofspacecraftcontrolsubjecttoactuatorand/orsensorfailures.INTRODUCTIONRecentadvancesinspacecraftandsatellitecontrolsystemshavesucceededinsolvingseveralchal-lengingproblemsdealingwiththeattitudetracking,robustcontrolofrigidandflexiblespace-craft,optimalslewmaneuvers,precisionpointing,formationflying,etc.Techniquesfromnon-linear[1,2,3,4,5,6],adaptive[7,8,9,10,11,12],optimal[13,14,15,16,17,18,19,20,21]androbustcontrol[9,22,23,24,25]havebeenusedtothisendwithalotofsuccess.Most(ifnotall)oftheseresultsassumethatthespacecraftisactivelycontrolledwithasufficientnumberofactuatorsequalto,orlargerthan,thedegreesoffreedomofthesystem.AlthoughthisiscertainlythecaseAssociateProfessor,SchoolofAerospaceEngineering.Email:[e-mail].†GraduateStudent,SchoolofAerospaceEngineering.Email:[e-mail].1withmostcurrentspacecraft,thesecontrollaws–byandlarge–donotaccountforunexpectedactuatorand/orsensorfailures.Itappearsthattheissueofspacecraftcontrolincaseofactuatorandsensorfailureshasnotreceiveditsdueattentionintheliterature.Thescopeofthispaperistoemphasizethechallengesassociatedwiththeattitudestabilizationproblemincaseofactuatorfailures.Asitturnsout,thisproblem–apartfromitsobviousprac-ticalimportance–requiresansweringseveralinterestingcontrol-theoreticquestionsaswell.Wefirstpresentabriefsurveyofthemainstabilizationresultsforthecaseofspacecraftactuatorfail-ures.Subsequently,weintroduceaseriesofnewresultsincludingdetumblingmaneuvers,completeattitudestabilization,andfeasibletrajectorygeneration.ThefirsttheoreticalinvestigationoftheequationsoftherotationalmotionofarigidbodyisprobablyduetoCrouch[26],whereheprovidednecessaryandsufficientconditionsforthecon-trollabilityofarigidbodyinthecaseofone,twoandthreeindependentcontroltorques.Thispaperalsoshowedthatinthecaseofmomentumexchangedevices,controllabilityisimpossiblewithfewerthanthreedevices.Thisarticlesparkedarenewedinterestintheareaofcontrolofrigidspacecraftwithlessthanthreecontroltorques.Additionalresultsonthesmall-timecontrollabilityoftherigidbodyequationsweregivenbyKera¨ı[27],whereitwasshownthat,withonecontrol,thesystemisneversmall-timelocallycontrollable.Stabilizationoftheangularvelocityequationshasbeenaddressed,forexample,inRefs.[28,29,30,31,32,33].InRef.[28]itwasshownthattheangularvelocityequationscanbemadeasymptoticallystableabouttheoriginbymeansoftwotorques,eachappliedalongaprincipalaxis.InRef.[29]theauthoraddressestheproblemoffeed-backstabilizationofthezerosolutionofEuler’sangularvelocityequationsusingonetorquealignedwithaprincipalaxis.Itisshownthatthereexistsasmoothstabilizingfeedbackcontrollawifthemomentofinertiaoftherigidbodyalongthatprincipalaxisiseitherthelargerorsmallerthantheremainingtwo.Moreover,thiscontrollawisrobustrelativetochangesintheparametersdefiningthecontrollaw.InRef.[30]theauthorsprovideamethodologytoasymptoticallystabilizeEuler’sequationswithasingle(infact,linear)controllaw.Itisalsoshownthatasinglecontrolalignedwithaprincipalaxiscannotasymptoticallystabilizethesystem.ThecontrollawconstructioninRef.[30]requiresthatthebodyhasnosymmetryaxes.SontagandSussman[31]extendedtheresultsofRef.[30]byshowingthattheangularvelocityequationscanbesmoothlystabilizedwithasingle(nonlinear)torqueforarigidbodywithanaxisofsymmetry.ThisresultwasrederivedinRef.[32]asanapplicationoftheJurdjevic-Quinnmethod[34].Andriano[33]showedthattheangularvelocityequationsofanaxi-symmetricrigidbodycanbegloballyasymptoticallystabilizedbymeansofalinerfeedbackwhentwocontroltorquesactonthebody.References[35]and[36]deriveacontrollawthatstabilizesauniformrotationofarigidbodyaboutitsintermediateaxisusingasingletorqueaboutitsmajororminoraxis.ThecompleteattitudeequationswereaddressedinRef.[37]and[38].Inparticular,ByrnesandIsidori[38]provedthatthereisnosmoothstatevariablefeedbacklawthatlocallyasymptotically2stabilizesarigidspacecraftaboutadesiredreferenceattitudewithtwocontroltorques.Stabiliza-tionaboutanequilibriummanifoldis,however,possible[37,38].InRefs.[39,40]theauthorsinvestigatedtheattitudestabilizationofanaxiallysymmetricspacecraft.Thecontroltorqueswereassumedtobesuppliedbytwopairsofgasjetactuatorsaboutaxesspanningthetwo-dimensionalplaneorthogonaltotheaxisofsymmetry.AccordingtotheresultsofRefs.[39,40],thecom-pletedynamicsofthespacecraftsystemfailstobec