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Attitude control of spacecraft during propulsion of swing thruster

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Attitude control of spacecraft during propulsion of swing thruster

    Attitude control of spacecraft during

    propulsion of swing thruster

    JournalofHarbinInstituteofTechnolo(NewSeries),Vo1.19,No.1,2012

    Attitudecontrolofspacecraftduringpropulsionofswingthruster

    X/AXiwang,riNGWuxing,GAOChangsheng,WEIWenshu

    夏喜旺,荆武兴,高长生,韦文书

    (Dept.ofAerospaceEngineering,HarbinInstituteofTechnology,Harbin150001,China,xxiwang@yahoo.con.an)

    Abstract:Asfororbittransfervehicle(0TV)withmultiplesatellites/payloadscarried,thereleaseofeach

    payloadwillbringseriouschangetothemasscenterofOTVandthethrustproducedbytheswingthrusterwill

    formaratherlargedisturbancetotheattitudeofOTV.Steeringthenozzletotracktheestimatedcenterofmass

    (ECM)ofOTVcanreducebutnotremovethedisturbanceduetothedifferencebetweentheECMandtheprac

    ticalmasscenter(PCM)ofOTV.Thepracticalpropellingdirectionwillchangewiththeinternalmotionduring

    thepropulsionprocessandattitudecontrolsystemshouldbeenabledtoguaranteethatthepropellingdirectionis

    collinearwiththecommand.Sincethestructuralparametershavechanged.whichisduetointernalmotionand

    fuelconsumption.thedynamicmodelhavetobeformulatedtodeterminethesetimevaryingparametersandthe

    requiredattitudeof0TVshouldbedeterminedaswel1ModulatingattitudequateruionresultsinquasiEuleran

    gles.BasedontheresultingquasiEulerangles,anovelattitudeswitchingcontrollawisintrodu

cedtocontrol

    thevariable

    massOTV.Simulationresultsshowthat.eveninthecaseofstrueturalasymmetry.controltorq

    ue

    matrixasymmetry,attitudedisturbanceandstrongcouplingbetweenthechannels,theattitud

    eofOTVcanbe

    controlledperfectly,andtheproposedattitudecontrollawiseffectiveforthevariable

    massOTVwithswing

    thruster.

    Keywords:variable

    massspacecraft;relativemotiondynamicmodeling;quasiEulerangle;attitudeswitching

    controllaw

    CLCnumber:W48.22Documentcode:AArticleID:10059113(2012)01-0094-07 Orbittransfervehicles(OTV)maybeusedto

    sendmultiplesatellitestotheirorbits.Thereleaseof eachpayloadwillcauseratherlargechangeofthemass centerof0TV,whichleadstothefactthatthethrustof thethrusterwillheavilyinfluencetheOTV.Duetothe limitedattitudecontrolcapabilityoftheattitudecontrol system,steeringthenozzleoftheswingthrusterto trackthemasscenterof0TVisabetterchoicetore. movethisdisturbanee.Sincethementionedinternal motionwillchangethepracticalpropellingdirection, attitudecontrolsystemhastobeenabledtoguarantee thatitcantrackthecommandeddirection.Duetofuel consumptionorothercauses,theestimatedmasscenter (ECM)of0TVdoesnotcoincidewithitspractical masscenter(PCM),andthen,thenozzle'sbeing steeredwillnotremovethedisturbancecompletely. Obviously.aneffectiveattitudecontrollawhastobe

    developedtocontrolthevariable.massspacecraft. Theproblemofmodelingspacecraftwithswinging partsislikethecasethatspacemulti.bodysystem reorientationmaneuverusinginternalcontrol,which hasbeenstudiedextensivelyinRefs.25]andrefer.

    encestherein,inwhichEulerandLagrangemethods areemployedtodevelopthedynamicmode1.Pla

    nar-3jandnonplanarlrelativemotionbetweenindi. vidualrigidlinks(nolessthanthreelinksareneeded) Received201l0608

    ?

    94?

    ofmultibodysystemhavebeenconcerned.Inthe

    aforementionedcases.however.thelinksareintercon

    nectedbyfrictionlessiointswithonedegreeof~eedom. Inpracticalcase,whenbeingsteeredtotrackthemass centerof0TV,thenozzleneedstorotaterelativelyto theplatformwithtwodegreesoffreedom.Inthispa

    per,aclearandspecificrelativemotiondynamicmodel betweenthevariablemass0TVplatformandtherigid

    nozzle(oftheswingthruster)willbepresentedbyway oftheMomentofMomentumThe0reml6jandthe Roynald'sTransportTheorem.

    Attitudecontrolforspacecrafthasbeenstudiedex

    tensivelyandmanyattitudecontrollawsl"havebeen

    proposedtotreatvariousscenarios.Theabovemen

    tionedcontrollawsarebasedontheEulerangleoratti

    tudequaternionrepresentationl.However.these

    tworepresentationscoverthespecialorthogonalgroup multipletimes,introducingambiguities,whichwill

    leadstounwindingbehaviorsasnotedinRef.j141. Modulatingtheattitudequaternionyieldsanewrepre

    sentationofspacecraftattitude:quasiEulerangles15J whichinheritsthefeatureofsingularityfreeandre

    movesthesignambiguityofquaternion.Sincethere

    sultantthreequasiEuleranglescorrespondtothethree componentsofthecontroltorquerespectively,thequasi Euleranglesbasedattitudecontrollawscanbeusedto JournalofHarbinInstituteofTechnology(NewSeries),Vo1.19,No.1,2012

    decouplethecontrolchannelsaswellastoavoidthe unwindingphenomena.Inthispaper,aquasiEuler anglebasedphaseplaneswitchingattitudecontrollaw isproposedtocontroltheOTV.Whenasetofsuitable regulatingcoefficientsarespecified,thecontrollaw canensurethatthephasetraj.ectoriesslidealongthe switchingsurfacetothecorrespondingorigin(thede

    siredstate),respectively.Comparedwiththecontrol lawpresentedinRef.[15],thislawcantreatthe specificcaseofstructuralasymmetryandcoupling ternlsbetweenthehannels.

    Thispaperisarrangedasfollows.InSection1, thedynamicmodelbetweenthevariable-massOTV platformandtherigidnozzleispresentedandthenoz

    zle'srequiredgimbalanglesrelativetotheplatformare solvedaftersomemathematicdeductions.InSection2, therequiredattitudeofOTVisdeterminedfirstly,and thenanovelswitchingattitudecontrollawbasedon quasiEuleranglesisproposed.Finally,somesimula

    tionresultsarepresentedtoverifytheproposedcontrol 1aw.

    1DynamicModelforVariable-massSpacecraft Duringthepropellingprocess,theMomentofMo

    mentumTheoremisemployedtodevelopthedynamic modelofvariable.massOTVandtheRoynold'sTrans. portTheoremshouldbeadoptedtotreatthevariable. masssystem.

    1.1DynamicModel

    Theexhaustthatflowsoutoftheplatformorinto thenozzleisconsideredaslow.velocityfluidwhilethat flowsoutofthenozzleisconsideredashigh.velocity fluid.Asforthevariable.mass0TV(includingthe platform,thenozzleandtheexhaust),accordingto theMomentofMomentumTheorem.wehave

    ^=,;=r×i:dm=f,×/Mm+J,×/Mm

    (1)

    wherethesubscriptsMandmrepresenttheplatform (includingthelowvelocityfue1)andthenozzle(in

    cludingthelow?velocityinputexhaustandthehigh--ve?

    locityoutputexhaust),respectively.

    InFig.1,wehave

    d=C2Cl,k=M/m,

    :

    1/(1+k)d,12=k/(1+k)d

    whereC1andC2arethevectorsfromtherootofthe nozzle(pointD)toC1(centerofmassoftheplatform) andC2(centerofmassofthenozzle),respectively; 11,12representthevectorsfromC(centerofmassof theOTV)toC1andC2,respectively,andkisthemass ratiooftheplatformandthenozzle.

    Asfortheplatform,consideringr=11+P,we

have

    Lr×m:M(zp)×(-+ii)dm=l?×

    ildm+JP×71dm+Jl1×d,n+IP×//dm.(2)JMJMJM

    wherePdenotestheVect0rflr0mC1t0dm(anarhitrary unitmassintheplatform).

    Fig.1Sketchfortheplatformandthenozzle TakeJ2000flameastherefefenceframeandthe platform'sbodycoordinateflame(PBCF)astheno

    tionflame.AsforthevectorP,accordingtotherela

    tionshipoftheabsoluteandrelativederivatives,we

    have

    dZp/dt=~2p/St+2m×(t~p/St)+

    (dto/dt)×P+×(o.J×P)

    whereistheplatform'sangularvelocityvectorwith respecttoinertialspace.Hence,wehave l1l×dm=,l×f[62p/6t+2to×(f)+

    (dto/dt)×P+×(×P)]dm=11×

    f(gp/~t)+z1×J[20o×(4o/~t)3dmJP×:JMJMJM

    fP×[62p/6t+2to×(/6)+(dto/dt)×P+JM

    (p)]dm=fP×(6p/6t)dm+Ip×JMJM

    [2tO×(6p/6t)]dm+IP×[(dto/dt)xp]dm+JM

    ×[×((-O×P)]dm.(3)

    Accordingtothedefinitionoftheplatform'sno

    mentofinertia,wehave

    Jp×[×(xp)3dm×Jp×(toMJM×p)dm:J

    ×?l

    ×[dto/dt×p]dm=H1(4)

    whereHlistheplatformsinertiatensor. Accordingtothedefinitionsoftheplatform's massandcenterofmass,wehave

×l1dm=Z1xzfdm=JM

    MI1×11

    fP×dm=一×ModinJMM=0?(5)?

    ?

    95?

nalofHarbinInstituteofTechnology(NewSeries),Vo1.19,No.1,2012

    ApplyingtheReynold'sTransportTheoremtothe variablemassplatformandneglectingSomenegli

    giblequantities,wehave

    J(8p/at)dm:,dM

    j(32plat)dm=rnuM

    J[P×(32p/at)]dm=rap×H.M

    2J{P×[×(/6)]}dm=".×

    (×P)+(all1/6?)c?,.(6)

    wherePisthevectorfromCltodm;P.

    isthevector

    fromC1tothecenterofthefluid'sejectionsection(as showninFig.2);uistheexhaust'sjetvelocityvec- tor;istheangularvelocityvectorofthespacecraft: ,Jisthemomentofinertia.andrn:Idm/dtI. Fig.2Sketchforvariable-massspacecraft SubstitutingEqs.(3)(6)intoEq.(2)yields

    J,×/:dm=MIl×1+H1+×?1+JM

    1l×J2go×()dm+z1×I(32p/at.)dm+

    fMp×(32p/3)dm+fMp×[2go×(8p/31]dm=

    Ml1×l1+"1+×?1+nil1xu+

    Z1x(2goxP)+,xu+,,×(xP.)+

    311I?(7)

    Thevectorsappearedintheprecedingequation

aredescribedinPBCF.

    Asf0rthenozzle.consideringtheinputexhaustas thenegativeoutputexhaust,inlikemanner,wehave Jr×/'dm=ml2×2+(H2+×,2)+

    ×f21-..0×(i~o/at)dm+12×f(fp/&)dm+mJm

    JP×(32p/at)dm+JP×[2×(?)]dm=

    ml2x12+(H2+xH2)+(/6)L/2+

    {12x[2?2×(P+pe2)]+12×(u.1+uP2)+

    (Plx"j+p.2×u.2)+[Pl×(xpI)+

    P2x(xPe2)]}.(8)

    whereH2denotestheinertialtensorofthenozzle;Pthe vectorfromC2todm(anarbitraryunitmassintheplat

    form);LthetransformmatrixfromNBCF(the ?

    96?

    nozzle'sbodycoordinateframe)toPBCF:thenoz

    zle'sangularvelocityvectorrelativetoinertialspace (inNBCF),anditsrate.Moreover,Pl,P2denote thevectorsfromC2toC21(centerofthenozzle'supper section)andCa2(centerofthenozzle'slowersec

    tion),u1,ue2thevelocitiesoftheinputandoutput exhaust,respectively.InEq.(8),thevectorsP P2,u.1,ue2,and12aredescribedinPBCF.Since P.1//.p2//u.1//u2,wehaveP1xu1=0andP2x un=0.Inaddition,itiseasytoget8lI2/6t=0, Iue2lIuI=IuI0.

    SubstitutingEqs.(7)and(8)intoEq.(1)and simplifyingtheresultedequationyield(inPBCF) M={1xZ1+ml2×Z2+H1+x,,l+

[+-f2×,,2]}+rh{Z2xue2+l1x(2goXp)+

    P×(xP)+2x[2(L/2)x(Pe2P1)]

    P.1x[(L/2)xP1]+Pe2×[(L/2)×P2]}+

    ().(9)

    ThevectorsC1PandC2GarethedescriptionsofCl andC2inPBCFandNBCF,respectively.Accordingto thegeometricrelationship,wehavep.=ClP,Pl=

    1C2GandPe2=k2C2c(kl,k2areconstants). Inaddition,theexpressionin{?}canbyre

    placedbythefollowingexpression, {'}=MI1×{L[S×C2G+x(×C2G)]

    ×(C2G)}+"1+×H1+MI】×

    [C2Gx.C1P×]+[,,2+()+

    ×H2-f2].(10)

    whereS=?(+0)+L+,andC,isdescribed

    inNBCF.ThederivationforEq.(10)canbefoundin Ref.[1].

    RewriteEq.(9)yieldsthedynamicmodelofthe variablemassspacecraftas

    =

    [?1+?1L+MIl×(LC2G×L

    ClPx)]_.?{MI1x[L(S×C2G+×xC2G)

    x×CPJxH1L(S+x,J1)

    (3111~at)go+Me+([(C1Px)(2/1×)]x

    C.P+[(kk)(GGX)+2(k2k1)(12×)](2)x

    C2.)}.

    InEq.(11),wehave

    Md=Z2xu2(12)

    Whentheestimatedcenterofmass(ECM)ofthe 0TVplatfoFindoescoincidewithitspractica1centerof

mass(PCM),wehaveMd=0,thatis,thepropul

    sionoftheswingthrusterwillbringnoinfluencetothe attitudeoftheOTV.Otherwise,isnotazerovec. tor,namely,thepropulsionwillformasignificantin

    fluencetotheOTVduetotheratherlargeietvelocity oftheexhaust.

    1.2TheNozzle'sGimbaiAngels

    Taketherootofthenozzle(pointD)astheori

    gin,constructrighthandrectangularcoordinatesystem DXYZ.whereDxisalongtheplatform'slongitudinal

    JournalofHarbinInstituteofTechnolo

    (N

    ew

    

    Se

    ries)'VoZ.19,No.1,2012

    axisandDYliesinthelongitudinalsymmetryplaneand pointsupwards.ThevectorDH,theprojectionofthe vectorDCintheplaneDXY,canbeobtainedas DH=(IzxDC1)x1z(13)

    Fromgeometryrelationship,theanglebetweenthe vectorsDHand,randtheanglebetweenthevector DC1andtheplaneDXYcanbederivedas

    :c.s

    oH

    

    .

    ix

    .

    ),

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