Dynamics and Nonlinear Control of Electromagnetic Docking/Assembly and Proximity Operations

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DYNAMICSANDNONLINEARCONTROLOFELECTROMAGNETICDOCKING/ASSEMBLYANDPROXIMITYOPERATIONSByKEHUOATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2012

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c2012KeHuo 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldliketakethetimetopubliclyacknowledgeandthank:committeemembers,Dr.Wiens,CarlD.CraneandPrabirBarooah,forallthefeedbacksonmythesis,professorsfromclass,WarrenDixonandNormanG.Fitz-Coy,forlecturinggreatonnonlinearcontrolanddynamicsrespectively.IalsowanttothankYimingXu,KarlBrandt,YilunLiu,PaulMoore,forthegreatsupports.Ofcourse,anotherpersonIshouldthankismygirlfriendwhohasn'tbeenappearingforlasttwoyears.AdditionallythanksgoouttoallthehelpIhavereceivedinwriting. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 10 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1Motivation .................................... 15 1.2PreviousRelativeWork ............................ 16 1.3OverviewoftheResearchandThesis .................... 18 2MODELLINGTHEELECTROMAGNETICFORCESADCTORQUES ..... 20 2.1Overview .................................... 20 2.2DerivationoftheExactModel ......................... 20 2.2.1MagneticField ............................. 20 2.2.2ForceandTorque ............................ 23 2.3Far-FieldModelofElectromagneticForceandTorque ........... 24 2.3.1DerivationofFar-FieldModel ..................... 24 2.3.22Dimensional(2-D)CoplanarCase ................. 28 2.3.3Co-AxialTwistCase .......................... 30 2.3.43Dimensional(3-D)Representation ................. 32 2.3.5DipoleLinearSuperposition ...................... 35 2.4ModelEvaluation ................................ 36 3DYNAMICMODELS ................................. 38 3.1Overview .................................... 38 3.2SystemDescription .............................. 38 3.2.1GeometryofDifferentCoordinateSystem .............. 38 3.2.2DockingStrategy ............................ 42 3.3BasicDynamicFundamentalEquations ................... 43 3.4DynamicsforSpecicSteps .......................... 43 3.4.1Step1.A ................................. 43 3.4.1.1Translationaldynamics ................... 44 3.4.1.2Rotationaldynamics ..................... 47 3.4.2Step1.B ................................. 49 3.4.2.1Translationaldynamics ................... 50 5

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3.4.2.2Rotationaldynamics ..................... 51 3.4.3Step1.C ................................. 52 3.4.3.1Translationaldynamics ................... 53 3.4.3.2Rotationaldynamics ..................... 53 3.4.4Step2 .................................. 54 4CONTROLLAWSANDSIMULATIONRESULTS ................. 56 4.1Overview .................................... 56 4.2ControlStrategy ................................ 56 4.3ControllerDesignforEachStep ........................ 58 4.3.1ControllerforStep1.A ......................... 61 4.3.1.1Step1.A.0controlofdand ................ 62 4.3.1.2Step1.A.1controldand ................. 65 4.3.2ControllerforStep1.B ......................... 65 4.3.2.1Step1.B.0controldand ................. 66 4.3.2.2Step1.B.1controldand ................. 66 4.3.3ControllerforStep1.C ......................... 67 4.3.4ControllerforStep2 .......................... 68 4.4AssumptionsforSimulationParameters ................... 68 4.5SimulationandResult ............................. 70 4.5.1SeparateSimulationResultforEachController ........... 70 4.5.1.1SimulationforStep1.A.0 .................. 70 4.5.1.2SimulationforStep1.A.1 .................. 73 4.5.1.3SimulationforStep1.C ................... 74 4.5.1.4SimulationforStep2 .................... 76 4.5.2CompleteSimulation .......................... 76 5THELOWEARTHORBITCHALLENGES ..................... 82 5.1Overview .................................... 82 5.2GravitationalField ............................... 82 5.3GeomagneticField ............................... 85 6CONCLUSION .................................... 88 6.1SummaryoftheThesis ............................ 88 6.2FutureWork ................................... 89 APPENDIX AMATRIXNORM .................................... 90 REFERENCES ....................................... 91 BIOGRAPHICALSKETCH ................................ 93 6

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LISTOFTABLES Table page 4-1InitialconditionforsimulationofStep1.A.0 .................... 72 4-2InitialconditionsforsimulationofStep1.A.1 .................... 73 4-3InitialconditionsforsimulationofStep1.C ..................... 75 4-4Initialconditionsforcompletesimulation ...................... 78 4-5Completesimulationresult ............................. 79 7

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LISTOFFIGURES Figure page 1-1MagneticactuatorcongurationinaCubeSat. .................. 16 1-2Electromagneticformationightvehicle. ...................... 17 1-3Electromagneticdockingstrategy. ......................... 18 2-1Aloopofcurrent. ................................... 22 2-2Twoloopsofcurrent. ................................. 23 2-32dimensional(2-D)coplanarcoils. ......................... 29 2-4Alignedconguration. ................................ 30 2-5Torquesvarywithangles. .............................. 31 2-6Intersectinglinesforcontrollostfortorques. .................... 31 2-7Co-axialtwistcase. ................................. 32 2-8Euleranglerepresentation. ............................. 33 2-9Linearsuperposition. ................................. 35 2-10Comparingthefar-eldforcemodelagainsttheexactmodel. .......... 36 2-11Comparingthefar-eldtorquemodelagainsttheexactmodel. ......... 36 3-1Geometryofdifferentcoordinatesystem. ..................... 39 3-2Denitionofrotatedreferencecoordinatesystem. ................. 40 3-3Relationshipbetweenbodyxedframeandrotatedreferenceframe. ...... 41 3-4SubdivisionforStep1. ................................ 42 3-53dimensional(3-D)illustrationforStep1.A. .................... 44 3-6FrontviewofStep1.A. ................................ 45 3-73-DillustrationforStep1.B. ............................. 50 3-8FrontviewofStep1.B. ................................ 50 3-93-DillustrationforStep1.C. ............................. 52 3-10FrontviewofStep1.C. ................................ 53 3-113-DillustrationforStep2. .............................. 54 8

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4-1Repeatelectromagneticdockingstrategy. ..................... 57 4-2Roughdockingmechanism:foldabledockingMechanism ............ 58 4-3Controlowchartpart1. .............................. 59 4-4Controlowchartpart2. .............................. 60 4-5Controldiagram. ................................... 61 4-6Simulationresultforinitialcondition1. ....................... 71 4-7Simulationresultforinitialcondition2. ....................... 71 4-8Simulationresultforinitialcondition3. ....................... 72 4-9SimulationresultforStep1.A.1ininitialcondition1. ............... 74 4-10SimulationresultforStep1.A.1ininitialcondition2. ............... 75 4-11SimulationforStep1.Cininitialcondition1withsaturation. ........... 76 4-12SimulationforStep1.Cininitialcondition2withsaturation. ........... 77 4-13SimulationresultforStep2. ............................. 77 4-14Completesimulationforinitialcondition1:signal. ................. 79 4-15Completesimulationforinitialcondition1:magneticmoment. .......... 80 4-16Completesimulationforinitialcondition2:signal. ................. 80 4-17Completesimulationforinitialcondition2:magneticmoment. .......... 81 5-1A2-Dcaseconcernsorbitaldynamics. ....................... 83 5-2Asimple2-Dcaseconcernsgeomagneticeld. .................. 86 9

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LISTOFSYMBOLS ATheareaenclosedbyacoilloopA VectorpotentialB Magneticeldwithunitoftesla(symbolT)E ElectriceldwithunitofV/mF ForceF exExternalforceFAReferenceframexedonfoilAF AReactionforceoncurrentloopAFITheinterialreferenceframeFoTheorbitreferenceframeFRRotatedreferenceframeH AoAngularmomentumofbodyAaboutpointoiCurrentthroughacoilJ CurrentdensityvectorwithunitofA/mJ dDisplacementcurrentdensityvectorl LinesegmentofthecoilL1Thespaceofboundedsequences MagneticdipolemomentmMassmAMassofbodyA 10

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0Permeabilityoffreespaceconstant,410)]TJ /F8 7.97 Tf 6.58 0 Td[(6N/A)]TJ /F8 7.97 Tf 6.58 0 Td[(2NNumberofturnsinacoil^n TheunitvectoralongaxisofcoillooprGradientoperatorr Divergencer Curl! AngularvelocityC! A=BAngularvelocityofbodyAwithrespecttoFrameBexpressedinFrameCP ALinearmomentumofbodyATheratioofacircle'scircumferencetoitsdiameterR(x,)PrincipalrotationmatrixofangleaboutxaxisR(y,)PrincipalrotationmatrixofangleaboutyaxisR(z, )Principalrotationmatrixofangle aboutzaxisRA=BRotationmatrixfromFrameAtoFrameB Torqueaboutthecenterofaloop ATorqueaboutthecenterofaloopA exExternaltorqueUPotentialenergyuControlinputsignalu Controlinputsignalvector 11

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VLLyapunovcandidatefunction^x Unitvectoralongxaxis^y Unitvectoralongyaxis^z Unitvectoralongzaxis 12

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceDYNAMICSANDNONLINEARCONTROLOFELECTROMAGNETICDOCKING/ASSEMBLYANDPROXIMITYOPERATIONSByKeHuoAugust2012Chair:GloriaJ.WiensMajor:AerospaceEngineeringTheuseofelectromagneticactuatorsinattitudecontrolsystemhasbeenconsideredasaneffectiveandreliableapproachforlowEarthorbit(LEO)satellites.OnerecentapplicationisElectromagneticFormationFlight(EMFF)whichcontrolstherelativetranslationaldegreesoffreedombetweensatellites.Comparedtotheuseoftraditionalthrusters,usinganelectromagneticforceandtorqueinmulti-spacecraftsmissionshassomedistinctadvantages,suchasnopropellantconsumptionandplumecontamination,aswellascontinuouscontrollability.Theadvantagesofelectromagnetshowevercomeatthecostofhighlynonlinearandcoupleddynamics.ExtendingtheEMFFapproachtorendezvousanddocking,thispaperfocusesonprovidingsmallsatellitesadockingcapabilityinbothaxialandcircumferentialdirectionsthroughtheuseoftwosetsofelectromagneticcoils.Forimplementation,anovelcontrolstrategyisalsopresented.Therstproblemwhichneedstobesolvedisthemodelofelectromagneticeldandgeneratedforceandtorquebetweentwoelectromagneticcoils.Theforceandtorquearenotonlyfunctionallyrelatedtothecharacteristicsofthecoilsbutarealsodependentontherelativepositionandattitudeofthetwosatellitesinproximityofoneanother.Bothanumericallyexactmodelandananalyticfar-eldmodelhavebeenbuiltandcomparedinthispaper.Thisthesisprovidesacapableelectromagneticdockingstrategyfortwosatellites.Eachofthesatellitesinelectromagneticdockingsystemisequippedwithreaction 13

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wheelsandasetofthreeorthogonalcurrentdrivencoils.Withtheassistanceofreactionwheels,decouplingthe3-dimensionaldockingproblemtoseveralstepsofprincipalbasiccasesprovidesfulldockingcapabilitiesofthesmallsatellites.Dynamicsanalysisandnonlinearcontrollerdesignhavebeendeveloped.Aswell,anoverallcontrolstrategyandacompletesimulationhavebeendemonstrated.Powerconsumptionanddisturbancesduetobothgravityandtheearthmagneticeldareconsideredinanon-orbitscenario. 14

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CHAPTER1INTRODUCTION 1.1MotivationIngeneral,autonomousrendezvousanddockingtechnologyincludestwospacecraftsstartingataremotedistance,comingtogetherintoacommonorbit,rendezvous,dockingandcontrolofthenewcombinedspacecraft[ 19 ].Traditionaluseofthistechnologycontainsspaceexplorationandsupplyandrepairofvehicles[ 11 ].Moreover,therehasbeenatendencyinrecentyearstodevelopaspacecraftmodulararchitecturedesignconcept[ 14 ].Thisconceptexpandsapplicationandneedsofthedockingtechnologymorewidely.Proximityoperationsanddockingarecriticalphasesofarendezvousanddockingmissionduetobothtranslationalandrotationalmaneuversarerequired.Somecriticalissuesduetouseoftraditionalmaneuveringtechniques,suchasathrusterbasedpropellantsystem,arepronetoplumeimpingementandthepossibilityofcollisioncausedbythediscontinuouspropulsion[ 19 ].Inspiredbyacommondailyphenomenonthattwomagnetscanadjusttherelativepositionandattitude,andthenachieveself-docking,theideaofintroducingelectromagneticforceandtorqueintodocking/assemblyandproximityoperationsistheapproachofinterestinthisthesis.Comparedtotheuseofthrusters,usinganelectromagneticforceandtorquehassomedistinctadvantages,suchasnopropellantconsumptionandplumecontamination,aswellascontinuouscontrollability.Theadvantagesofelectromagnetshowevercomeatthecostofhighlynonlinearandcoupleddynamics.Thisthesisstartsfromtheideaofelectromagneticdocking,providesacapablewaytoimplementthisideainascenariowithfairassumptions,demonstratesacompletedockingmissionsimulationinthisscenario. 15

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1.2PreviousRelativeWorkTheexploitationoftheuseofmagneticforceandtorqueinspacemissionscanberoughlycategorizedintothefollowingthreeareas:usingelectromagneticactuatorsproventobeaneffectiveandreliableattitudecontrolsystemforlowEarthorbit(LEO)satellites[ 16 ];theelectromagneticformationight(EMFF)controlstherelativetranslationaldegreesoffreedombetweensatellites[ 15 ];andelectromagneticdocking/assemblyconsideringbothtranslationalandrotationaldegreesoffreedom[ 22 23 ].Inthefollowingparagraphs,previousworkrelatedtothesethreetypesofapplicationsisintroduced.Theelectromagneticactuatorsinanattitudecontrolsystemoperateonthebasisofinteractionbetweenasetofthreeorthogonalcurrent-drivenmagneticcoilsandthegeomagneticeld.Thesecoilscanthereforegeneratecorrespondingtorques.Thesetorquescaneitherbeusedtodumpangularmomentum[ 2 ]ortoactivelycontrolattitude.First,themagneticcontrolsystemisdesignedforusewithsomestabilizationmethodsuchasspinningandbiasmomentum[ 17 ].Then,inmorerecentdecades,purelymagneticattitudecontrolhasbeenstudied.Duetothelowcost,exibleshaping,lowenergyconsuming,simplehardwarerequirementformoderateattitudecontrolofmagneticattitudecontrol,magneticactuatorsarepreferredinsmallsatellitesandmicro-satellites,asshowninFigure 1-1 [ 5 ]. Figure1-1. MagneticactuatorcongurationinaCubeSat. Theconceptofusingelectromagneticforcetoprovidetherelativepositioningcontrolforsatellitesformationighthavebeenresearchedbytwomaingroups:MIT 16

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SpaceSystemsLab[ 15 ],andtheUniversityofTokyo/ISAS[ 9 ].TheEMFFuseshightemperaturesuperconductingwiretechnologytogeneratetheelectromagneticforcetomaintainandrecongurethesatellitesformation[ 1 ].TheEMFFconcentratesontranslationaldegreesoffreedomcontrol.Itisassumedthatthetorqueandangularmomentumgeneratedduringtheoperationwillbeabsorbedbyreactionwheels[ 15 ]. Figure1-2. Electromagneticformationightvehicle. SpacecraftelectromagneticdockingtechnologyissimilartotheEMFFconception(Figure 1-2 ),yetthemaindifferencebetweenthesetwoinclude,bothtranslationalandrotationalcontrolhavetobeconcerned.Analogyofthedailymagnetsattractionphenomenontodockingproblemisstraightforward.Aconceptofself-dockingcapabilityofelectromagnetspresentsthatundersomespecicconstraintsforinitialconditions,therelativeposition/attitudeautomaticallydecreasingtozero,notconsideringthedockingvelocity[ 22 23 ].Then,reference[ 22 23 ]hasconstrainedthedockingproblemtosmallrelativeattitudeassumption.Aswell,therelativevelocityistheonlycontrolobjective.However,thelinearizationbasedonsmallrelativeattitudeandcoplanarassumptionmakesthispaperinsubstantialinsomedegree.Basedontheabove,thisthesisprovidesafeasiblewaytouseelectromagneticindocking/assemblyandproximityoperationforsmallsatellitesundersomefairassumptions.Notonlythetranslationalcontrolbutalsotherotationalcontrolduringthedockingprocesshasbeeninvestigated.DifferentfromEMFFconception,thetorquesgeneratedduetothemisalignmentofdipoleshavebeenconsideredasthetorquesourceforattitudeadjustment,insteadofbeencanceledbyreactionwheels. 17

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1.3OverviewoftheResearchandThesisElectromagneticdockinginthisthesisistheideaofusingsetsoforthogonalcurrentdrivencoils,coupledwithreactionwheels,toprovidetherelativepositionandattitudecontrolindockingmission.Noticethat,thereactionwheelsinthisthesisareusedonlyforstabilizingthesatellitewhenthedockingstrategyneeds,thetorquegeneratedbyelectromagneticcoilsprovidethecontrolofrelativeattitude.Thedockingstrategyisdesignedtoadjusttheattitudeandtrackadesireddistancetrajectorysimultaneously,thenmaintaintheadjustedrelativeattitudeandtrackadesiredapproachingtrajectory.Theattitudeadjustmenthasbeendecoupledtotwosteps,alignmentofthedominantringsandtwistadjustmentofcoilsperpendiculartothedominantcoilaboutthealignedco-axial.TheFigure 1-3 demonstratesthedockingstrategy. Figure1-3. Electromagneticdockingstrategy. Theelectromagneticdockinghaswideprospectsofapplicationsinthefuture: 18

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Orbitservicecouldbenetfromtheimprovementofcontinuouscontrollabilityandzeropropellantconsumption. Thinkingaboutaseriesofmodulesequippedwithelectromagneticdockingsystem,eachofthemhasapartialfunction.Bydockingtogethertheymighthaveabilitytocooperateasacompletefunctionalsatellite. Ifaconceptualyingspaceroboticsisdesignedtobemanipulatedbyasetofcoilsequippedonthemotherstation,thenthepropulsionsystemandattitudecontrolsystemofroboticscouldbediminished. Also,consideringself-assemblyarchitectureinthespace,byusingautonomouselectromagneticdockingtechnology,onlythemainbodyneedsfullattitudecontrolsystemandpropulsionsystem.Thestructureofthisthesisisdevelopedassuch:InChapter 2 ,exactmodelandfareldmodelofelectromagneticforceandtorquehasbeenpresentedandcompared.Thischapteristhebasisofderivationofdynamicsequations.InChapter 3 ,detailsofthesystemdescriptionhavebeendemonstrated,followedbytheoveralldockingstrategyintroduction.Dockingmissionhasbeendividedtoseveralstepsofprincipalbasiccases.Thedevelopmentofdynamicmodelsofprincipalbasiccasesconstitutesthemainbodyofthischapter.InChapter 4 ,controlstrategyandcontrollerdesignforeachstephasbeeninvestigated.Aswell,simulationresultsofeachcontrolleraredemonstratedanddiscussed.Acompletesimulationcombinedthesecontrollersguidedbythecontrolstrategyispresented.InChapter 5 ,challengeswhenthissystemisoperatinginLEOscenariohavebeenshown.Difcultiesandpossiblesolutionshavebeendiscussed.InChapter 6 ,summaryandmaincontributesofthisthesishavebeenconcluded.Somesuggestionsonfutureworkalsoareincluded. 19

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CHAPTER2MODELLINGTHEELECTROMAGNETICFORCESADCTORQUES 2.1OverviewInthisthesis,theelectromagneticdockingsystemconsistsoftwosatellites.Eachsatelliteisequippedwithoneormorecoils.Tostudythedynamicsandderivethedynamicsequationsofthissystem,onemustrstdevelopthetheoreticalmodelsofthemagneticeld,forcesandtorquesgenerated.Usingtheprincipleofmagneticeldtheory[ 13 ],thefollowingsectionsdescribetheapplicabletheory.Basedonthistheory,thecorrespondingforceandtorqueequationsneededforgeneratingthefulldynamicmodelarepresented.Chapter 3 willdetailthecorrespondingdynamicsequations. 2.2DerivationoftheExactModel 2.2.1MagneticFieldThedifferencebetweenelectriceldE andthemagneticeldB :electricchargeisapointsourceofE ,whilemotionofchargedparticles,(i.e.current),isthesourceofB .Theliteraturesshowthattherearenoexperimentsindicatingexistenceofmagneticmonopolesormagneticcharge,thoughsearchingforthemcontinuestobeaninterestingchallenge[ 13 ].sinceamagneticelddoesnothaveapointsource,thedivergenceofthemagneticeldmustbezero,andcanbeexpressedas: r B =0 ,(2)whereristhegradientoperator,r B =0 isthedivergenceofB .ThecurlofmagneticeldisgivenbyAmpere'sequationformagnetostatics,i.e.fortheeldfromcurrentdistributionwhichisconstantintime r B =0J ,(2) 20

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whereJ isthecurrentdensityvector,and0isthepermeabilityoffreespace.Equation( 2 )canalsobemodiedasfollowingfortimevariantproblems. r B =0(J +J d),(2)whereJ disdisplacementcurrentdensityvector.Inalaboratorysystemofchargesandcurrents,thedisplacementcurrentisnormallyverysmallcomparedtotypicalchargecurrents.ToobserveJ dexperimentallyrequireshighfrequencies,largeelectriceld,orboth.Furthermore,thepresumptionthatthemagneticeldsandcurrentswillvaryslowlyovertimecouldbemade,thenmagnetostaticscanbeassumed.ByintroducingavectorpotentialA ,theelectromagneticeldcanbeexpressedas: B =r A .(2)Fromreference[ 13 ],thegeneralsolutionforA couldbewrittenas: A (s)=0 4ZZZVolJ ( ) js )]TJ /F4 11.955 Tf 11.96 0 Td[( jd3 ,(2)where isthepositionvectoroftheelementvolume(d3 =dxdydz)withinelectricalconductingmaterial,s isthepositionvectorfromtheelementvolumetopotentialvector'slocation.Foracurrentloop,asshowninFigure 2-1 ,thewireisassumedtohavenegligiblethickness,andthecurrentdensityiszeroeverywherebutwithinthewire.TheequationforthevectorpotentialatlocationPcanthenbereducedtoapathintegralaroundtheloopofcurrent. A (s )=0Ni 4I1 ks )]TJ /F3 11.955 Tf 11.96 0 Td[(a kdl ,(2)whereNisthenumberofturns;iisthecurrent,s isthevectorfromasmallsegmentofthecoil,dl ,topointP,a istheradialpositionvectorofsegmentdl relativetothecenterofcoil. 21

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Figure2-1. Aloopofcurrent. Substitutinginto( 2 )to( 2 ),themagneticeldcanbeexpressedas: B =r 0Ni 4I1 ks )]TJ /F3 11.955 Tf 11.96 0 Td[(a kdl .(2)SincetheoperationHisbasedons ,itcancommutewithintegrand.Also,byusing r 1 ks )]TJ /F3 11.955 Tf 11.95 0 Td[(a k=)]TJ /F3 11.955 Tf 18.94 8.09 Td[(s )]TJ /F3 11.955 Tf 11.95 0 Td[(a ks )]TJ /F3 11.955 Tf 11.96 0 Td[(a k3,(2)( 2 )gives B =0Ni 4Is )]TJ /F3 11.955 Tf 11.96 0 Td[(a ks )]TJ /F3 11.955 Tf 11.95 0 Td[(a k3dl .(2)Duetothedifcultyofintegratingthereciprocalofthesquareofmagnitudeoftheks )]TJ /F3 11.955 Tf 12.71 0 Td[(a k3,solvingforthemagneticeldisnotstraightforward.Assuch,themagneticeldB canonlybewrittenanalyticallyintermsofellipticalintegralswhenitisoffaxis.Fortheoffaxiscasethemagneticeldcanbeexpressedintermsofradialandaxialcomponentsas[ 4 ]:B =0Ni 2a1 p QE(m)1)]TJ /F4 11.955 Tf 11.95 0 Td[(2B)]TJ /F4 11.955 Tf 11.96 0 Td[(2B Q)]TJ /F9 11.955 Tf 11.96 0 Td[(4B+K(m)^z +0Ni 2a1 p QE(m)1+2B+2B Q)]TJ /F9 11.955 Tf 11.95 0 Td[(4B)]TJ /F3 11.955 Tf 11.95 0 Td[(K(m)^r (2) 22

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whereE(m)istherstkindofellipticalintegral,andK(m)isthesecondkindofellipticalintegral,and B=r a,B=z a,B=z r,QB=(1+B)2+2B,mB=r 4B QB.(2) 2.2.2ForceandTorqueInthissection,theforceandtorquearederivedsothatonemaymodeltheinteractionbehaviorbetweenofthetwosatellites.Figure 2-2 illustratesthefundamentalcongurationofthecoilseachrepresentsasatellite. Figure2-2. Twoloopsofcurrent. Whenawirecarryinganelectricalcurrentisplacedinamagneticeld,eachmovingcharges,experiencesheLorentzforce.ByusingLorentzforcelaw,foreachsmallsegmentofwiredl carryingcurrenti,theforceactingonthissegmentisgivenby: dF =idl B .(2)Integrating( 2 )overthelengthofthewire,totalelectromagneticforceonthewireisgivenby: F =iZdl B ,(2) 23

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ApplyingthisequationtothecaseillustratedinFigure 2-2 ,theforceactingonthesecondloopofcurrentis: F B=iBZdlB B A,(2)inwhichB AistheelectromagneticeldatpositionofdlB createdbyloopA.ThereactionforceactsoncurrentloopAisofcourse: F A=)]TJ /F3 11.955 Tf 9.29 0 Td[(F B=)]TJ /F3 11.955 Tf 9.3 0 Td[(iBZdl BB A.(2)ThetorqueonloopBaboutthecenteroftheloopduetoincrementalforcedF Bisgivenby: B=Za BdF B.(2)CombiningtheaboveequationswiththeeldresultsinSection 2.2.1 ,theequationsforbothforceandtorquehavedoubleintegrals,whichlimitstheabilitytosolvethemanalytically.Consequently,insightsabouttheforceandtorquemodelarehardtodetermine.Therefore,toobtainapproximationsthatcanbederivedanalytically,simplicationandlinearisationbyusingTaylorserieswillbeusedinthefollowingsection. 2.3Far-FieldModelofElectromagneticForceandTorque 2.3.1DerivationofFar-FieldModelAtasufcientlylargedistance,themagneticeldgeneratedbyacurrentloopbehavesasamagneticdipolewhenitcomestoasufcientlargedistance.Hence,acurrentloopcouldbevisualizedasabarmagnetalignedwiththeaxisofthelooppointinginthedirectiongivenbyrighthandrule.Toanalyticallyexpressthisbehavior,theforceandtorqueexpressioncontainsks )]TJ /F3 11.955 Tf 11.96 0 Td[(a kterminthedominator.Whenlargedistanceassumptionismade,wecansimplifythemodeldueto ka kks k.(2) 24

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Expand1=ks )]TJ /F3 11.955 Tf 11.95 0 Td[(a kabouta=s0usingTaylorseriesyields 1 ks )]TJ /F3 11.955 Tf 11.96 0 Td[(a k=1 s+s a s3+H.O.T,(2)wheres a isdotproductofvectorss anda ,andH.O.Tarethehigherordertermsthatareapproximatedtobezero.Substituting( 2 )into( 2 )and( 2 )forthemagneticvectorpotential,andobservingthersttermwith1=sintegratestozero,then:A (s )=0 4ZZZ1 s+s a s3J ( )d3 =0 4ZZZs a s3J ( )d3 (2)andA (s )=0Ni 4I1 s+s a s3dl =0Ni 4s3I(s a )dl (2)Usingvectorcalculus,ithasbeenproveninreference[ 13 ],theintegralin( 2 )and( 2 )canberewrittenas:ZZZs s3J ( )d3 = s (2)NiIs a dl = s (2)inwhich iscalledmagneticdipolemoment: =1 2ZZZ J ( )d3 .(2)Substitutebackto( 2 ),thevectorpotentialofamagneticdipolecanbeexpressedas: A (s )=0 4 s s3.(2)Foracurrentloop,themagneticdipolemomentis: =NiI1 2a dl =NiA^n ,(2)whereNisthenumberofturns,iisthecurrent,Aistheareaenclosedbytheloop,and^n isthevectoralongtheaxisoftheloop.Thecorrespondingmagneticeld 25

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mathematicalexpressionthentakestheform: B =r A =0 4r ( s s3),(2)Usingthepropertiesofthegradientoperator(r ) r (u v )=u (r v ))]TJ /F3 11.955 Tf 11.96 0 Td[(v (r u )+(v r )u )]TJ /F9 11.955 Tf 11.96 0 Td[((u r )v ,(2)yields r ( s s3)= (r s s3))]TJ /F3 11.955 Tf 15.61 8.09 Td[(s s3(r )+(s s3r ) )]TJ /F9 11.955 Tf 11.96 0 Td[(( r )s s3.(2)Since isnotafunctionofs,thenr 0and(s s3r ) 0 .Therefore,r s s3=@ @xx s3+@ @yy s3+@ @zz s3=1 s3)]TJ /F9 11.955 Tf 15.8 8.09 Td[(3 x5x2+1 s3)]TJ /F9 11.955 Tf 15.8 8.09 Td[(3 x5y2+1 s3)]TJ /F9 11.955 Tf 15.8 8.09 Td[(3 x5z2=0, (2)( r)s s3=x@ @xx s3+y@ @yy s3+z@ @zz s3=x^e x s3)]TJ /F9 11.955 Tf 15.38 8.08 Td[(3 s5xs +y^e y s3)]TJ /F9 11.955 Tf 15.38 8.08 Td[(3 s5ys +z^e z s3)]TJ /F9 11.955 Tf 15.37 8.08 Td[(3 s5zs =1 s3 )]TJ /F9 11.955 Tf 13.15 8.09 Td[(3s s5(s ). (2)Finally,wehave: B =0 43s (s ) s5)]TJ /F4 11.955 Tf 15 9.38 Td[( s3=B (s ),(2)whichvariesins and ).Forthefar-eldapproximations a .Hence,thepotentialenergyofamagneticdipoleofcoilBinamagneticeldofcoilA(B A(d )isgivenby: U(d )=)]TJ /F4 11.955 Tf 9.3 0 Td[( BB A(d ),(2)whered isthevectorconnectingthecenterofdipoleAtodipoleB,B A(d )istheapproximationofB A(s ),and BismagneticmomentduetocoilB.Theforceondipole 26

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Bissimplyderivedfromthegradientofthepotentialenergy. F B=rU=r( BB A)=0 4r3( Bd )(d A) d5)]TJ /F4 11.955 Tf 13.15 9.88 Td[( B A d3.(2)Again,noting Aand Barenotfunctionsofd,thereforer A=0,r B=0.Similartotheabovederivation,( 2 )resultsin F B=)]TJ /F9 11.955 Tf 10.5 8.09 Td[(30 4)]TJ /F9 11.955 Tf 9.3 0 Td[(( A B)d d5)]TJ /F9 11.955 Tf 11.96 0 Td[(( Ad ) B d5)]TJ /F9 11.955 Tf 11.96 0 Td[((d B) A d5+5( Ad )(d B)d d7.(2)Atorqueappliedoverarotationisequaltothechangeinpotentialenergy. dU=)]TJ /F4 11.955 Tf 9.3 0 Td[( d =)]TJ /F9 11.955 Tf 9.3 0 Td[(d B ,(2)where d =d .(2)Then, )]TJ /F4 11.955 Tf 11.96 0 Td[( d =)]TJ /F9 11.955 Tf 9.3 0 Td[((d )B =)]TJ /F9 11.955 Tf 9.3 0 Td[(( B )d ,(2)and B= BB A.(2)Substituting, B=0 4 B3( Ad )d d5)]TJ /F4 11.955 Tf 13.15 9.88 Td[( A d3.(2)ByusingNewton'sthirdlaw,wehave: F A=)]TJ /F3 11.955 Tf 9.3 0 Td[(F B.(2)Note:F Acouldhavebeenderivedusing F A=rU=r( AB B),(2)whichyieldsthesameexpressionas( 2 ). 27

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Duetotheconservationofangularmomentum:sincetherearenoexternalforceandtorqueactedonthissystem,thetotalangularmomentumshouldbezero: A+ B+d F B=0 .(2)Therefore,wecandeterminetorqueactingonloopA: A=)]TJ /F4 11.955 Tf 9.3 0 Td[( B)]TJ /F3 11.955 Tf 11.96 0 Td[(d F B.(2)Theabovecompletesthederivationoftheforcesandtorquesinthevectorfar-eldmodelbetweentwocurrentsloops.TheanalysisofthedynamicsofthissystemdetailedinChapter 3 isbasedonthismodel.Thedynamicscanbedescribedfortwocases:2dimensional(2-D)coplanar,co-axialtwist.Thesetwospecialcaseswillleverageforachievingorientationandpositioncontroloftwosatelliteswhichingeneralcouldbeuncontrollableinatleastonedegreeoffreedom.Thefollowingsections, 2.3.2 through 2.3.4 ,providerepresentationsofforceandtorqueforthreedifferentcases.Theconciseformandcomputationalconvenienceofthismodelwillshownaswell. 2.3.22Dimensional(2-D)CoplanarCaseInthe2-Dcoplanarcase,twocurrentloopsAandBarerestrictedtoaplaneasshowinginFigure 2-3 .Thexzcoordinatesystemisinertialcoordinatesystem,FI;xAzAandxBzBarebodyxedcoordinatesystemsFAandFB,coilAandcoilBrespectively;andxRzRisrotatedreferencecoordinatesystem,FR.Forcomputingconvenience,theforceandtorqueisderivedintheFRframe.ReferringtoFigure 2-3 ,thedipolemomentvectorofcoilAandB(alignedwithaxesxAandxB,respectively)arerstalignedwithxR,thenrotate()]TJ /F4 11.955 Tf 12.54 0 Td[(=2)and()]TJ /F4 11.955 Tf 12.54 0 Td[(=2)respectivelyaboutyR. 28

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Figure2-3. 2dimensional(2-D)coplanarcoils. InFR,thedipolemomentcanberepresentedas:R A=Asin^x R+Acos^z R, (2)R B=Bsin^x R+Bcos^z R, (2)Rd =d ^z R. (2)Substituting( 2 )into( 2 ),( 2 ),( 2 )and( 2 )resultsinfollowinginteractionforcesandtorquesactingonthetwocoilsintherotatedreferencecoordinatesystem,FR.F A=30 4AB d4()]TJ /F9 11.955 Tf 9.3 0 Td[((cossin+cossin)^x R+(2coscos)]TJ /F9 11.955 Tf 11.95 0 Td[(sinsin)^z R), (2)F B=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(30 4AB d4()]TJ /F9 11.955 Tf 9.3 0 Td[((cossin+cossin)^x R+(2coscos)]TJ /F9 11.955 Tf 11.95 0 Td[(sinsin)^z R), (2) A=)]TJ /F4 11.955 Tf 12.18 8.08 Td[(0 4AB d3(2sincos+cossin)^y R, (2) B=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 4AB d3(2cossin+sincos)^y R. (2)Fromtheaboveforceandtorqueequations,onecanobservesomeintuitivepropertiesoftwodipoles.AsFigure 2-4 shows,whenandarezero,theforceisonly 29

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alongwiththezRaxis,aswellthetorquebecomestobezero.I.e.attractionorrepulsiveforcesarepresentwhenthetwocoilsareperfectlyaligned.Inthisconguration,forceandtorqueequationsreducetobe: Figure2-4. Alignedconguration. F A=30 4AB d4^z R,F B=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(30 4AB d4^z R, A= B=0 .(2)AsFigure 2-5 shows,thereareasetoflinesduetotheintersectionofthe Asurfacewiththezeroplane.Noticethatforcomputationalconvenience,thefactor0AB 4d3hasbeennormalizedas1.Theselinescorrespondtothevaluesofandforwhenthecontrollabilityof Avanishes.Similarsituationwilloccurfor B.ObservingtheintersectionlinesinFigure 2-6 ,itisfoundthatthecontrolislostsimultaneouslyforboth Aand Bat(,)=f...,()]TJ /F4 11.955 Tf 9.3 0 Td[(,),(0,0),(,)]TJ /F4 11.955 Tf 9.3 0 Td[(),...g,f...,()]TJ /F4 11.955 Tf 9.3 0 Td[(=2,)]TJ /F4 11.955 Tf 9.29 0 Td[(=2),(=2,)]TJ /F4 11.955 Tf 9.3 0 Td[(=2),(=2,=2),...g,andf...,()]TJ /F4 11.955 Tf 9.3 0 Td[(,0),(,0),(0,)]TJ /F4 11.955 Tf 9.3 0 Td[(),...g. 2.3.3Co-AxialTwistCaseInco-axialcase,twocurrentloopsCandDarerestrictedtobeco-axialinzRasshowninFigure 2-7 .ThexRzRcoordinatesystemisrotatedreferencecoordinate 30

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ATorqueactingoncoilA BTorqueactingoncoilBFigure2-5. Torquesvarywithangles. Figure2-6. Intersectinglinesforcontrollostfortorques. system,FR.RefertoFigure 2-7 ,dipolemomentvectorsofcoilCandD(alwaysalignedwithxCandxD,respectively)arerstalignedwithxR.ThenrotateandrespectivelyaboutzR.Whenthetwoanglesarezeros,thecoilsCandDareintheyRzRplane.ThefollowingmathematicalexpressionsrepresentdipolemomentsinFR:R C=Ccos^x R+Csin^y R, (2)R D=Dcos^x R+Dsin^y R, (2)Rd =d^z R. (2) 31

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Figure2-7. Co-axialtwistcase. Substituteto( 2 ),( 2 ),( 2 )and( 2 ),wehave:F C=30cD 4d4cos()]TJ /F4 11.955 Tf 11.96 0 Td[()^z R, (2)F D=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(30cD 4d4cos()]TJ /F4 11.955 Tf 11.96 0 Td[()^z R, (2) C=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(0cD 4d3sin()]TJ /F4 11.955 Tf 11.96 0 Td[()^z R, (2) D=0cD 4d3sin()]TJ /F4 11.955 Tf 11.96 0 Td[()^z R, (2)Somepropertiesoftheforceandtorqueinthiscasecanbeconcluded.First,forcesareonlyonthe^z Raxis.Thismeansthatoncetwocoilsentertheco-axialconguration,theywouldstayinthisconguration.Second,torquesarefunctionsofcDand()]TJ /F4 11.955 Tf 12.01 0 Td[().Systemwouldlosecontroloftorquewhen)]TJ /F4 11.955 Tf 12.12 0 Td[(=0,happens.However,thissituationingeneralsatisesdockingrequirementsofalignment(eitherN,NorN,Saligned).MoredetailsaboutthissingularitywillbeshowninChapter 3 2.3.43Dimensional(3-D)RepresentationInthissectiontheEulerAngleareusedwhichareknownfortheirintuitivenesstorepresenttherotationfromrotatedreferencecoordinatesystemFRtobodyxedcoordinatesystem,FAandFB.3-2-1sequencerotationwillbeused. 32

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ReferringtoFigure 2-8 ,thedipolemomentvectorsofcoilAandB(whicharealwaysalignedwithxAandxB)areinitiallyalignedwithzR,thenrotateandrespectivelyaboutzR,followingbyrotating()]TJ /F4 11.955 Tf 12.94 0 Td[(=2)aboutbodyxedaxisyAand()]TJ /F4 11.955 Tf 12.19 0 Td[(=2)aboutbodyxedaxisyB.Sincetheaxialsymmetryofrings,thelastrotationsaboutxAandxBdonotaffecttheforceandtorquecalculation. Figure2-8. Euleranglerepresentation. ThedirectcosinematrixrepresentsthetransformationfromFRtoFA,couldbewrittenasproductoftwoprinciplerotationmatrices:RR=A=R2()]TJ /F4 11.955 Tf 13.15 8.09 Td[( 2)R3()=266664sin0cos010)]TJ /F9 11.955 Tf 11.29 0 Td[(cos0sin377775266664cossin0)]TJ /F9 11.955 Tf 11.29 0 Td[(sincos0001377775=266664sincossinsincos)]TJ /F9 11.955 Tf 11.29 0 Td[(sincos0)]TJ /F9 11.955 Tf 11.29 0 Td[(coscos)]TJ /F9 11.955 Tf 11.29 0 Td[(cossinsin377775. (2) 33

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InFR,dipolemomentvectorofringAcouldbedescribedas:R A=RR=AA A=266664sincossinsincos)]TJ /F9 11.955 Tf 11.29 0 Td[(sincos0)]TJ /F9 11.955 Tf 11.29 0 Td[(coscos)]TJ /F9 11.955 Tf 11.29 0 Td[(cossinsin377775266664A00377775=A266664sincossinsincos377775. (2)Similarly,wecanderiveR Bouteasily. R B=B266664sincossinsincos377775.(2)Also, Rd =26666400d377775.(2)Substitute( 2 ),( 2 )and( 2 )to( 2 )and( 2 ),wehaveforceandtorqueactonringBrepresentedinreferenceframe:RF B=30AB 4d4266664cossincos+sincoscoscossinsin+sincossinsinsincos()]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F9 11.955 Tf 11.96 0 Td[(2coscos377775, (2)R B=0AB 4d3266664sincossin+2cossinsin)]TJ /F9 11.955 Tf 11.29 0 Td[(sincoscos+2cossincos)]TJ /F9 11.955 Tf 11.3 0 Td[(sinsinsin()]TJ /F4 11.955 Tf 11.95 0 Td[()377775. (2) 34

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2.3.5DipoleLinearSuperpositionWithoutlossofgenerality,assumesatelliteAandBareequippedbythreeorthogonalcurrentrings,thentheelectromagneticforcesactingonringsequippedonsatelliteBcouldbecalculatedbysumminguptheforcesduetothemagneticeldgeneratedbyeachringonsatelliteA. F B=3Xj=13Xi=1f ij,(2)wheref ijistheforceactingonj-thringonsatelliteBduetotheeldgeneratedbyi-thringonsatelliteA.Figure 2-9 showsthelinearsuperposition Figure2-9. Linearsuperposition. Notice,thatthefar-eldmodelforforceandtorqueislinearfunctionofthemagneticdipolemoments.Therefore,threeorthogonaldipolescouldbeconsideredasasetofbasisofaR3space.Thatmeans,onecanconsidercombinationofthesethreedipolevectorsasanewdipolevector.Bychangingthedipolemomentofeachoforthogonalrings,wehavetheabilitytocontrolboththedirectionandmagnitudeoftheoveralldipolevector. =3Xi=1 i.(2)Thelinearsuperpositionsimpliesthedynamicmodelwhenitinvolvesmulti-ringspersatellite. 35

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2.4ModelEvaluationToverifythesuitabilityoftheaboveapproximationsusedinmodelling,theforcesandtorquesofthefar-eldmodelandexactmodelarecompared.Figure 2-10 presentsthepercentageerrorofmagnitudewiththecoilsindifferentcongurations.Figure 2-11 presentstheresultsfortorque. Figure2-10. Comparingthefar-eldforcemodelagainsttheexactmodel. Figure2-11. Comparingthefar-eldtorquemodelagainsttheexactmodel. 36

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Usually,weadmitamodelisvalidwhenthepercentageerrorislessthan10%.Fromthetwogures,wecanseetheerrorvarieswithdifferentcongurationofcoils.Yet,overall,whendistanceisaboveapproximately6timesofthecoilradii,errorstayslessthan10%. 37

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CHAPTER3DYNAMICMODELS 3.1OverviewThepurposeofthischapteristodevelopsystemmodels.Thesemodelswillbeusedforsimulatingandcontrollerdesign.Thischapterwillstartsoffwithsystemdescriptionaswellastheoveralldockingstrategy.Specically,3-Ddocking/operationproblemwillbedecoupledtobeseveralstepsofprincipalbasiccases,suchas2-Dco-planarsinglecoilcaseandco-axialtwistsinglecoilcase.Next,thegeneralrigidbodydynamicmodelingprocedurewillbepresented.Then,developmentofdynamicmodelsofprincipalbasiccasesconstitutesthemainbodyofthischapter.Inthenextchapter,theopenloopbehaviorsofthesecases,andsubsequentcontrollersdesignfortheprincipalbasiccases,willbeinvestigated.Theoverallcontrolschemewillalsobedescribed,detailinghowtheresultingcontrollerisimplementable. 3.2SystemDescriptionTheelectromagneticdocking/proximityoperationsysteminthisthesistypicallyinvolvestwocooperativeentitiessuchassatellites,spacecraftsorassemblypartsinspace.Eachofthemwillbeequippedwiththreeorthogonalcurrentloops.Focusingonthedynamicsandcontrolproblemofthissystem,thedocking/operation/assemblywillbesimpliedasfollowsintosequentialoperationalsteps:Step1,adjusttheattitudeandmaintainaconstantdistanceortrackadesiredtrajectoryrstly;andStep2,maintaintheadjustedrelativeattitudeandtrackadesiredapproachingtrajectory. 3.2.1GeometryofDifferentCoordinateSystemSinceorbitaldynamicsisnottheprimaryobjectiveinthisthesis,severalsimpli-cationswillbemadehere.Consequently,thedenitionsofcoordinatesystemmightbeslightlydifferentfromcommonuses.InertialframeFIissettobetheECIreferenceframewhichhasitsoriginatthecenteroftheEarth,oneaxisaalinedwithnorthpole,onepointstovernalequinox,and 38

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thethirdonecompletesarighthandedaxissystem.OrbitframeFoisdenedassuch,itsoriginissameasECIframe's,oneaxispointstowardstheorbitalplanenormal,oneaxisalignedwiththesemi-majoraxisoftheobit,thethirdonecompletesarighthandedaxissystem.TheFAandFBaredenedasbodyxedcoordinatesystemtorepresenttheorientationsofsatelliteAandB.Figure 3-1 illustratesthesedenitions.Withoutlosinggenerality,setFI=Fo,showedasrightpartofFigure 3-1 .Fortheconvenienceofelectromagneticforceandtorquerepresentation,FRisdenedasrotatedreferencecoordinatesystem,inwhichzRaxisisalignedwithdistancevectord pointsfromoriginofFAtooriginofFB,asshowninFigure 3-2 .OriginofFRisatthecenterofmassofthesystemoftwosatellites. Figure3-1. Geometryofdifferentcoordinatesystem. SinceFRisalwayschangingwiththepositionsofOAandOB,forderivationconvenience,alocalinorbitframeFLisdenedassuch:originistheoriginFR,oneaxisisnormaltotheorbitalplane,secondonetowardsthevelocitydirection,thirdonecompletetherighthandcoordinatesystem.Figure 3-2 alsoshowstherelationshipbetweenFRandFI.MovingFItooriginOAisjustfordrawingconvenience.RotationmatrixfromFRtoFIcouldbeexpressedas: RR=I=R(y,y)R(x,)]TJ /F4 11.955 Tf 9.3 0 Td[(x),(3)where x=tan)]TJ /F8 7.97 Tf 6.58 0 Td[(1dx dz,y=sin)]TJ /F8 7.97 Tf 6.59 0 Td[(1dx d.(3) 39

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Figure3-2. Denitionofrotatedreferencecoordinatesystem. Also,dx,dyanddzarethreecomponentsofvectord .Averyusefulterm,dominantaxis,isdenedtobetheaxisalignedwiththeapproachaxisforthedockingmechanism.AsshowninFigure 3-3 ,xAandxBarethedominantaxesforsatelliteAandBrespectively.Theringperpendiculartodominantaxisisdenedasthedominantring.Bydeningadominantaxis,thecontrolobjectiveofachievingarelativeattitudebetweensatellitesnecessaryfordockingbecomesdetermininghowtoalignthesedominantaxesandtwistabouttheco-axisafterthealignment.Now,startingwithrepresentationoforientationsofAandB,areductiontheofdegreesoffreedomisdiscussedhere.The3-2-1sequentialEuleranglerotationisusedforrepresentingtheorientationsofbodyxedcoordinatesystemFAandFBwithrespecttoFR.ReferringtoFigure 3-3 ,rotationfromFRtoFBhasbeenshownas:FBisinitiallyalignedwithFR,thenrotateaboutzR,followingbyrotatingand)]TJ /F4 11.955 Tf 12.54 0 Td[(=2aboutbodyxedaxisyB,thenrotate aboutbodyxedaxisxB.RotationmatrixfromFRtoFBisexpressedas: RB=R=R(x, )R(y,)]TJ /F4 11.955 Tf 11.95 0 Td[(=2)R(z,).(3) 40

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Figure3-3. Relationshipbetweenbodyxedframeandrotatedreferenceframe. ItissimilartogetFA:FAisinitiallyalignedwithFR,thenrotateaboutzR,followingbyrotatingand)]TJ /F4 11.955 Tf 12.71 0 Td[(=2aboutbodyxedaxisyA,thenrotateaboutbodyxedaxisxA.RotationmatrixfromFRtoFAisexpressedas: RA=R=R(x,)R(y,)]TJ /F4 11.955 Tf 11.96 0 Td[(=2)R(z,).(3)Consideringthatwearedealingwithalignmentproblemoftworings,recallsomecontentsinChapter 2 .PerSection 2.3.4 andbecauseoftheaxialsymmetryofrings,rotationsaboutxAandxBdeterminingdonotaffecttheforceandtorquegeneratedbythesetworings.Consideringthatthealignmentproblemisdealingwiththerelativeattitudebetweentwodominantrings,the3-2sequentialEuleranglerotationcouldbeusedtoidentifytheattitudeofdominantrings.Thethirdrotationaboutthedominantaxiscouldbehandledinthetwiststep.Again,toemphasizethepeculiaritiesofdynamicsbetweencurrentloops,wewillstartfromdeepspaceassumption.Fordeepspacemissions,orbitaldynamics(theinuenceofearthgravityandgeomagneticeld)isusuallyignored.Deepspaceassumptionisthebaseofderivationsofthischapter'sdynamicequations.Sinceindeepspaceassumptiontheorbitdynamicsarenotconsidered,theassumptionthatFLis 41

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alignedwithFIcanbemade.Chapter5willdiscussmoreabouttheimpactofignoringtheorbitaldynamicsandgeomagneticeldontheperformanceofthecontroller. 3.2.2DockingStrategyAsdiscussedinlastsection,thecontrolobjectiveofadjustingtherelativeattitudecanbeseparatedintotwosequentialsteps:aligningthedominantaxeswiththedistancevector(i.e.,thezR),followedbytwistingthesatellitesaboutthealignedaxiszRtothenaldesiredrelativeattitude.Further,thealignmentin3-Drealitycanbeobtainedbybreakingtheprocedureintotwoindependent2-Dcoplanarcases.First,alignonedominantaxiswithzR,thenalignanother.Withoutlosinggenerality,wechoosetorstalignxAwithzR.Detailsabouthowtodecouplethe3-Dcasealignmentcaseinto2-DcoplanarcasesareinSection 3.4 ,includingthedynamicsforeachspecicstep.Oncethedominantaxesareco-axialwithzRtwistingthesatellitesabouttheco-axisisthelaststepforattitudeadjustment.Thisstepcanbeachievedbutrequiresareductioninthecontrolleddegreesoffreedomofthesystemasmentionedinlastsection.Overall,Step1canbedividedinto3sequentialsteps:Step1.A,Step1.B,andStep1.C.OnceStep1isaccomplished,Step2isjusta1-Ddistancecontrolproblem.SeeFigure 3-4 Figure3-4. SubdivisionforStep1. 42

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Oneimportantremark:inStep1.AandStep1.B,takingintoaccounttheproblemofdegreeoffreedom,torquesactonSatelliteAandSatelliteBaresettobecanceledbyreactionwheelsrespectively. 3.3BasicDynamicFundamentalEquationsMomentumofrigidbodyAisdenotedasP A.Ifwechoosethecenterofmasscasreferencepoint,P Aisgivenby: P Ac=mv c,(3)inwhich,misthemassofA,v cisthevelocityofc.Forconvenienceofrepresentation,P AcsometimesissimpliedtobeP A.AngularmomentumofrigidbodyAaboutreferencepointoisdenotedasH Ao.Ifwechoosethecenterofmasscasreferencepoint,H Acisgivenby[ 7 ] H Ac=I! ,(3)where! isangularvelocityofA.Generally,dynamicequationsofrigidbodyAincludetranslationalequationandrotationalequation. _P Ac+[! ]P Ac=f ex,(3)wheref existheexternalforceactonbodyA. _H Ac+[! ]H Ac= ex,(3)where existheexternaltorqueactonbodyA. 3.4DynamicsforSpecicSteps 3.4.1Step1.AInthisstep,alignmentofxBwithd (zR)willbeaccomplished.OrientationofSatelliteAisxed.Comparingtothegeneralcoplanarcase,in3-Dreality,alignmentdoesnothavetohappeninplanexRzR.Therefore,anauxiliarycoordinatesystemFisdened. 43

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AsshownintheFigure 3-5 ,FisinitiallyalignedwithFR,thenrotatesangleaboutaxiszR.Thearrowofthisrotationinthegureisjustforshowingthepositivedirectionof.RotationmatrixfromFRtoFisgivenby: Figure3-5. 3dimensional(3-D)illustrationforStep1.A. R=R=R(z,)=266664cossin0)]TJ /F9 11.955 Tf 11.29 0 Td[(sincos0001377775.(3)Now,itiseasytoseethat,dominantaxisxBanddistancevectorzRbothlieinplanexz.Therefore,Step1.Aisactuallyasimplecoplanarcasewhichoccursinthisplane.InFigure 3-6 ,frontviewofplanexzshowshowthisstepreducestothegeneral2-Dcoplanarcase. 3.4.1.1TranslationaldynamicsThedynamicequationsforStep1.AwillbederivedincoordinatesystemF.PositionvectorofAandBinFcanbeexpressedas: p A=00)]TJ /F6 7.97 Tf 10.49 4.71 Td[(d 2>,p B=00d 2>.(3) 44

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Figure3-6. FrontviewofStep1.A. ChoosingtheseAandBlocationsinFisforcalculationconvenience.Thus,momen-tumsofAandBrepresentedinFaregivenby P A=mA(_p A+! =Ip A),P B=mB(_p A+! =Ip B),(3)inwhich,mAandmBaremassofAandB,! =IistheangularvelocityofFwithrespecttoinertialframe.Consideringthatinthis2-Dcoplanarcasethemotionhappensoccursinplane,itdoesnotchangetherotationmatrixfromFRtoF,R=R,whichmeansangularvelocityofFisthesameof! =I: =I=! R=I.(3)Duetothedeepspaceassumption(FI=FL), =I=! R=I=! R=L=2666640_0377775.(3)Notice,inlowearthorbitcase,! L=Iisnolongerzero.Chapter 5 showsmoreaboutthisinuenceonthemodelingandcontrollerperformance. 45

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Substituting,wehave: P A=mA266664)]TJ /F6 7.97 Tf 10.5 5.12 Td[(d_ 20)]TJ /F8 7.97 Tf 11.83 6.48 Td[(_d 2377775,P B=mB266664d_ 20_d 2377775.(3)Recall( 3 ),wehave: _P A+! =IP A=F A.(3)Substitute( 3 )to( 3 ),wegot: mA264)]TJ /F6 7.97 Tf 10.49 5.11 Td[(d 2)]TJ /F8 7.97 Tf 14.48 6.88 Td[(_d_ 2)]TJ /F8 7.97 Tf 14.49 6.88 Td[(_d_ 20)]TJ /F8 7.97 Tf 13.5 6.48 Td[(d 2+d_2 2375=F A.(3)InF, Aonlygivescomponentsinplanexz, A=266664x0z377775=A266664sin~0cos~377775.(3)Also, BinthisstepisonlygivenbydominantringofB: B=Bx^x B.(3)Substitute Aintoto( 2 ),forceactonAisgivenby: F A=30 4Bx d4266664)]TJ /F4 11.955 Tf 9.3 0 Td[(zsin)]TJ /F4 11.955 Tf 11.95 0 Td[(xcos02zcos)]TJ /F4 11.955 Tf 11.96 0 Td[(xsin377775.(3) 46

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Then,assumemA=mB=3m,wheremisthemassofeachcoil,aftersomereorganization,translationaldynamicsequationsaregivenby:=1 m0 2Bx d5(zsin+xcos))]TJ /F9 11.955 Tf 13.15 8.09 Td[(2_d_ d, (3)d=)]TJ /F9 11.955 Tf 12.34 8.09 Td[(1 m0 2Bx d4(2zcos)]TJ /F4 11.955 Tf 11.96 0 Td[(xsin)+d_2. (3)SinceforthesystemofsatelliteAandB,thereisnoexternalforce,thechangeinmomentumofthiswholesystemequalszero.ThustakingthetimederivativeofP Bwillgiveouttheexacttwoequations. 3.4.1.2RotationaldynamicsSinceinStep1.A,attitudeofsatelliteAisxed,rotationaldynamicequationforBwillbederived.AngularmomentumofBaboutcenterofmassrepresentedinFisasfollows: H Bc=IB! B=I,(3)inwhichIBistheinertiamatrixofBaboutcenterofmassrepresentedinF.Byusingtransformationofinertiamatrix,wehave: IB=R>B=BIBRB=,(3)whereRB=isrotationmatrixfromFtobodyxedframe: RB==R(y,)]TJ /F4 11.955 Tf 13.15 8.09 Td[( 2).(3)Aswell,forthreeringswithradiusr,massm, BIB=2mr2266664100010001377775.(3) 47

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SinceR>B=RB==Iidentity, IB=2mr2266664100010001377775.(3)Note,inthisthesis,onlythemassandinertiaofthethreecoilsisusedintheanalysis.However,infuturesimulations,thesetermscaneasilybeupdatedtoincludethefullsatellitemassandinertia.AngularvelocityofBwithrespecttoinertiaframerepresentedinF,! B=Iisgivenby: B=I=! B=+! =I,(3)where B==2666640_0377775,! =I=2666640_0377775.(3)Substituting,wehave H Bc=IB! B=I=2mBr2B2666640_+_0377775(3)TaketimederivativeofH Bc, _H Bc+[! =I]H Bc= B.(3)Substitute( 3 )and( 3 )into( 3 ),wehaverotationaldynamicsequations: 2mr22666640+0377775= B.(3) 48

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Recall( 2 ),wehave: B=)]TJ /F4 11.955 Tf 12.17 8.09 Td[(0 4B d3(2zsin+xcos).(3)Combine( 3 )and( 3 ),wehave: =)]TJ /F4 11.955 Tf 12.18 8.08 Td[(0 8B d3mr2(2zsin+xcos))]TJ /F9 11.955 Tf 12.68 2.65 Td[(.(3)Then,wehave: =)]TJ /F4 11.955 Tf 12.17 8.09 Td[(0 8B d3mr2(2zsin+xcos))]TJ /F9 11.955 Tf 12.44 8.09 Td[(1 m0 2Bx d5(zsin+xcos)+2_d_ d.(3)Rememberthe Aisgeneratedbythreeorthogonalrings,representationinF.Therefore, AmustbetransferredbacktoFA.IfwecancelthetorqueactingonsatelliteAbymomentummanagement,rotationmatrixfromFRtoFR,RA=R,willbeaconstantmatrix.Also,satelliteBonlyrotatesintheplanexz.Hence, A A=R>A=RR A=R>A=RR>=R A.(3) 3.4.2Step1.BStep1.Bhappensinplanexz,inwhichxAandzRlie.DynamicsequationsforthisstepisderivedincoordinatesystemF.AnauxiliarycoordinatesystemFisdenedasfollows.AsshownintheFigure 3-7 and 3-8 ,FisinitiallyalignedwithFR,thenrotatesangleaboutaxiszR.Thearrowofthisrotationinthegureisjustforshowingthepositivedirectionof.RotationmatrixfromFRtoFisgivenby: R=R=R(z,).(3)DerivationofdynamicequationsissimilartoStep1.A.Remember,orientationofsatelliteBissettobexedinthisstep. 49

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Figure3-7. 3-DillustrationforStep1.B. Figure3-8. FrontviewofStep1.B. 3.4.2.1TranslationaldynamicsBasically,thereisnodifferencefromStep1.A'sdynamicsmodelingexceptforderivingincoordinatesystemFR.Tosavespace,onlyimportantresultswillbeshowninthissection.Translationaldynamicsequationisgivenby: 3m266664)]TJ /F6 7.97 Tf 10.5 5.86 Td[(d 2)]TJ /F8 7.97 Tf 14.49 7.63 Td[(_d_ 2)]TJ /F8 7.97 Tf 14.49 7.63 Td[(_d_ 20)]TJ /F8 7.97 Tf 10.85 6.48 Td[(d 2+d_2 2377775=F A.(3) 50

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InF, Bonlygivescomponentsinplanexz, B=266664x0z377775=B266664sin~0cos~377775(3)Also, AinthisstepisonlygivenbydominantringofA: A=Ax^x A.(3)Substitute Bintoto( 2 ),forceactingonAisgivenby: F A=30 4Ax d4266664)]TJ /F4 11.955 Tf 9.3 0 Td[(xcos)]TJ /F4 11.955 Tf 11.96 0 Td[(zsin02zcos)]TJ /F4 11.955 Tf 11.96 0 Td[(xsin377775.(3)Reorganizing,=1 m0 2Ax d5(xcos+zsin))]TJ /F9 11.955 Tf 13.15 8.09 Td[(2_d_ d, (3)d=1 m0 2Ax d4(2zcos)]TJ /F4 11.955 Tf 11.95 0 Td[(xsin)+d_2. (3) 3.4.2.2RotationaldynamicsSimilarly,onlymainequationswillbeshownforStep1.B. 2mr22666640+0377775= A.(3)Byusing( 2 ),torqueactonAisgivenby: A=2666640)]TJ /F5 7.97 Tf 11.53 5.26 Td[(0 8AB d3(2zsin+xcos)0377775(3) 51

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Therefore =)]TJ /F4 11.955 Tf 12.17 8.09 Td[(0 8Ax d3mr2(2zsin+xcos))]TJ /F9 11.955 Tf 14.99 8.09 Td[(1 m0 2B d5(xcos+zsin)+2_d_ d.(3)Transfer BtoFB: B B=R>B=RR>=R B.(3) 3.4.3Step1.CAfterStep1.AandStep1.B,bothtwodominantaxesarealignedwithdistancevector.ControllingthedistancebetweenofAandBnowreducestoa1-Dattractionorrepulsivecase.Whentwistingothercoilswhichareperpendiculartothedominantringsaboutco-axial,wehave2satevariablesdandangledifference(')]TJ /F4 11.955 Tf 13.01 0 Td[( )withonlyoneinput(AyBy)whichmeansthatdistancecontrolcannotbeguaranteed.However,byintroducinga`roughdocking'approachwhichisdenedasdistancemaintenancebyusingmechanicalcontacts,thetwistingthesatelliteaboutco-axiscouldbeaccomplished.Inthisstep, A A=2666640Ay0377775,B B=2666640By0377775.(3) Figure3-9. 3-DillustrationforStep1.C. 52

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Figure3-10. FrontviewofStep1.C. 3.4.3.1TranslationaldynamicsFromSection 2.3.3 ,wehavetheforcegeneratedbytheAyandBycomponents: F A=30AyBy 4d4cos(')]TJ /F4 11.955 Tf 11.95 0 Td[( )^z R,F B=30AyBy 4d4cos(')]TJ /F4 11.955 Tf 11.96 0 Td[( )^z R.(3)Sumup, F A=30AyBy 4d4cos(')]TJ /F4 11.955 Tf 11.95 0 Td[( )^z R,F B=30AyBy 4d4cos(')]TJ /F4 11.955 Tf 11.96 0 Td[( )^z R.(3)Fortranslationaldynamicsthereisonlyonestatevariable,d.TimederivativeofmomentumforAcouldbeexpressed: )]TJ /F9 11.955 Tf 11.96 0 Td[(3md 2=F A.(3)Reorganizing: d=0AyBy 2md4cos(')]TJ /F4 11.955 Tf 11.96 0 Td[( ).(3) 3.4.3.2RotationaldynamicsFromSection 2.3.3 ,wehavethetorquegeneratedbytheAyandBy: A=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(0AyBy 4d3sin(')]TJ /F4 11.955 Tf 11.95 0 Td[( )^z R, B=0AyBy 4d3sin(')]TJ /F4 11.955 Tf 11.95 0 Td[( )^z R.(3) 53

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TimederivativeofangularmomentumsforAandBarederivedas: 266664002mr2'377775= A,266664002mr2 377775= B.(3)Subtracting: ')]TJ /F9 11.955 Tf 14.13 2.66 Td[( =)]TJ /F4 11.955 Tf 10.5 8.09 Td[(0AyBy 4mr2sin(')]TJ /F4 11.955 Tf 11.96 0 Td[( ).(3) 3.4.4Step2InStep2,therelativeattitudehasbeenadjustedbyStep1.Hence,onlythingneedingtobeaccomplishedinthisstepisgeneratingattractionorrepulsiveforcerepulsivewithdockingmechanism.Sincetheforcedirectionisalongthedistancevector,itisbasicallya1-Dcase.Inthisstep,thedominantringsareusedtogeneratetheforce. A A=266664Ax00377775,B B=266664Bx00377775.(3) Figure3-11. 3-DillustrationforStep2. 54

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FromSection 2.3.2 ,theforceandtorquefor1-Dcasehasbeenderived: F A=30 2AxBx d4^z R,F B=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(30 2AxBx d4^z R,(3)TimederivativeofmomentumforAisderivedas: )]TJ /F9 11.955 Tf 11.96 0 Td[(3md 2=F A.(3)Reorganizing: d=)]TJ /F4 11.955 Tf 14.02 8.08 Td[(0 mAxBx d4.(3)Pertheabove,thesetofdynamicmodelsthatcapturethebehaviorofthetwosatellitesinthepresenteddockingstrategywillnowbeusedforderivingthecontrollerforthemagneticcoildockingsystem.Designdetailsandsimulatedresultsarepresentedinthenextchapter. 55

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CHAPTER4CONTROLLAWSANDSIMULATIONRESULTS 4.1OverviewInthischapter,theoverallcontrolstrategyforthemagneticcoildockingapproachisshown.Detailsofthecontrollerdesignforeachstepinthedockingispresented.Thisisfollowedbysimulationresultsoftheindividualcontrollersanddemonstrationofthesequentialcombinationofthesecontrollers.AllthesimulationsareperformedusingMatlabandSimulink.Importantassumptionsare: Fullrelativepositionandattitudeestimationcanbeachieved,includingrelationshipbetweenFRandbodyxedframesFAandFB.WhichmeansthatbothRB=RandRA=Rarereachable. The3-2-1sequentialEulerangles(,,,,'and )canbedeterminedfromrotationmatrix. Forconvenience,thed,,,,,'and arethestatevariablesdirectlyusedinsimulation. Therelativepositioncanbeusedtodetermineand. Firstordertimederivativeofd,,,,,', ,andcanbemeasured.Determining_,_,_,_,_',and_ isreasonableiftheangularvelocitysensorsisequipped.Relativevelocityestimationbetweentwosatelliteswillgive_d,_,and_.Since,positionandattitudeestimationisawholenewarea,thisthesiswillfocusonthedynamicsandcontrollerdesignforeachstepinthedockingstrategywhereallthestatesandtheirtimederivativesaremeasureddirectlyand/orestimated. 4.2ControlStrategyPerthecontrolstrategymentionedinSection 3.2.2 ,thecorrespondingcontrolowchartisdesignedtomorefullydemonstratetheapproachofthisthesisrefertoFigure 4-3 and 4-4 .Tohelpvisualizingeachstepofthisowchart,Figure 1-3 isrepeatedhere. 56

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Note:inthisthesis,detumblingFRorandisdenedassuch,consideringthesetwosatellitesasasystem,preventingtherotationoftwosatellitesaboutthecenterofmassofthissystem.Note,itisdifferentfromdetumblingasatellitewhichmeansstabilizingtheattitudeofthissatellite.Also,a`roughdocking'approachisdenedasdistancemaintenancebyusingmechanicalcontacts.Figure 4-2 hasshownasuitabledockingmechanism.Thisfoldabledockingmechanism[ 20 ]givesthedistanceconstraintaswellasthecircumferentialconstraint,meanwhile,itleavesonedegreeoffreedomfortwistingabouttheco-axial. Figure4-1. Repeatelectromagneticdockingstrategy. Figure 4-5 showsthecorrespondingcontroldiagramanditsimplementationasusedinsimulation.Thecontrolstrategyisimplementedtothistwosatellitesystembyusingamulti-porttriggerandswitch.Toguaranteetheinputisimplementable,inputsaturationisnecessary.DisturbanceDandDNareintroducedtoexaminetherobustnessofthecontrollers. 57

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Figure4-2. Roughdockingmechanism:foldabledockingMechanism Asindicatedinpriorchapters,thecontrolisachievedbyacombinationofcontrollerswhereeachcontrollerisdesignedusingthecorrespondingdynamicmodelforthespecicstepinthealignmentanddockingprocess.Asshownintheowchartandcontroldiagram,therearesteps1.A.,1.B.and2withtransitionalmodes(e.g.,steps1.A.0,1.A.1/1.B.0,1.B.1)betweencontrollersinvolvingthereactionwheelstates.Thetransitionalmodesuseeachsatellite'sreactionwheelstoconstrainthesatellitebehaviorsuchthatitsdynamicscanbeapproximatedasthe2-DcasespresentedinChapter 3 .Thus,severalcontroltriggerindicesaredenedforjudgingstepscompletedornot,switchingfromonestep'scontrollertothenextcontroller.Criterionsofthesetriggerswillbedenedlaterinthesimulationsection. 4.3ControllerDesignforEachStepReferringto[ 10 ],Lyapunovbasedhighgainrobustcontrollerdesignmethodologyisappliedinfollowingcontrollerdesigns. 58

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Figure4-3. Controlowchartpart1. 59

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Figure4-4. Controlowchartpart2. 60

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Figure4-5. Controldiagram. 4.3.1ControllerforStep1.ARecallthedynamicsequationsforStep1.A:=1 m0 2B d5(zsin+xcos))]TJ /F9 11.955 Tf 13.15 8.08 Td[(2_d_ d, (4)d=)]TJ /F9 11.955 Tf 12.34 8.08 Td[(1 m0 2B d4(2zcos)]TJ /F4 11.955 Tf 11.96 0 Td[(xsin)+d_2, (4)=)]TJ /F4 11.955 Tf 12.17 8.09 Td[(0 8B d3mr2(2zsin+xcos))]TJ /F9 11.955 Tf 15 8.09 Td[(1 m0 2B d5(zsin+xcos)+2_d_ d. (4)Noticethatthereare3statesvariablesinthissystem,yetonlytwoinputsxBandzBcouldbesupplied.Comparingtothecontrolobjectiveinthisstep(showninFigure 3-4 ),controllingisforthepurposeofaligningB'sdominantaxiswithdistancevector,controllingdgivestheabilityofregulatingthedistanceortrackingadesiredtrajectory,controllingisfordetumblingtherotatedreferencecoordinatesystemFR.Toavoidcollision,controllingdshouldalwaystakeprecedenceoverothertwoobjectives.Thus,wedividedthisstepintotwosmallsteps:controldandtoachievethecontrolobjectiveofaligning,setattitudeofsatelliteBtobexedanddrivetobezero.Betweenthesetwosmallsteps,atriggerswitchshouldbedesigned. 61

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4.3.1.1Step1.A.0controlofdandTokeepthecontrollerdesignprocessconcise,replacezBxandxBxwithu1andu2, 264d375=264)]TJ /F8 7.97 Tf 11.79 4.71 Td[(1 m0 1 d4cos1 m0 1 d4sin)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F5 7.97 Tf 7.71 -4.43 Td[(0 41 d3mr2+1 m0 21 d5sin)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F5 7.97 Tf 7.71 -4.43 Td[(0 81 d3mr2+1 m0 21 d5cos375264u1u2375+2642d_2d_2 d375,(4)andreorganizedynamicsmodelassuchaform: q =B(q ,_q )u +G(q ,_q )+D,(4)inwhich q =264d375,_q =264_d_375,q =264d375,u =264u1u2375,B=264)]TJ /F8 7.97 Tf 11.79 4.71 Td[(1 m0 1 d4cos1 m0 1 d4sin)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F5 7.97 Tf 7.72 -4.43 Td[(0 41 d3mr2+1 m0 21 d5sin)]TJ /F11 11.955 Tf 11.3 9.69 Td[()]TJ /F5 7.97 Tf 7.71 -4.43 Td[(0 81 d3mr2+1 m0 21 d5cos375,G=2642d_2d_2 d375,(4)andD 2R2isunknowndisturbancecausedbythemodelinaccuracyorsomeotherfactors.Controlobjectivearesettobe: e =q )]TJ /F3 11.955 Tf 11.96 0 Td[(q des!0 ,_e =_q )]TJ /F9 11.955 Tf 13.59 0 Td[(_q des!0 ,(4)inwhichq desisthedesiretrajectoryforq ,_q desistimederivativeof_q .Beforedesignofthecontroller,somepropertiesshouldbeprovedorassumedasfollows[anewclassofmodularadaptivecontrollers,[ 12 ].TheB)]TJ /F8 7.97 Tf 6.59 0 Td[(1isassumedtoexist.Theq desisdesignedsuchthatn-thorderoftimederivativeofq desexistsandis 62

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bounded.Ifq ,_q 2L1,thenB,G,D2L1(L1meansbeingbounded).FormatrixB,q ,_q 2L1)Bij2L1(refertoAppendix A ),andBij2L1)B.Also,forvectorG,q ,_q 2L1)Gi2L1,thenjGj=p G21+G22isboundedtoo.TheboundednessaboutunknowndisturbanceDmustbeassumed.ToproveB)]TJ /F8 7.97 Tf 6.59 0 Td[(1exists,fromthefar-eldmodelassumptioninSection 2.3.1 wehaverd.Thusr2=d2,ahigherordertermforr=d,r2=d2=0canbeassumed.Then,inmatrixB,thefollowingapproximationcouldbemade: 0 41 d3mr2+1 m0 21 d5sin=0 41 d3mr2sin.(4)Similarly, )]TJ /F11 11.955 Tf 11.96 16.86 Td[(0 81 d3mr2+1 m0 21 d5cos=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 81 d3mr2cos.(4)Then, det(B)1 m0 1 d40 81 d3mr26=0.(4)So,matrixBcanbeconsideredasinvertible.Insteadofusingbacksteppingdesignmethod,lteredtrackingerrorisintroducedhere, r =_e +ce ,(4)wherecisapositiveconstant.Openloopanalysisoferror: _q =q )]TJ /F9 11.955 Tf 12.2 0 Td[(q des+c_e =B(q ,_q )u +G(q ,_q )c_e )]TJ /F9 11.955 Tf 12.2 0 Td[(q des+D.(4)Designu as: u =B)]TJ /F8 7.97 Tf 6.59 0 Td[(1()]TJ /F3 11.955 Tf 9.3 0 Td[(G)]TJ /F3 11.955 Tf 11.96 0 Td[(ce +q des)]TJ /F3 11.955 Tf 11.95 0 Td[(k1r )]TJ /F3 11.955 Tf 11.96 0 Td[(k2r ).(4)Bysuchdesign,sincer ismeasureable,u isimplementable.Also,sinceB,G,D2L1,q ,q des2L1also_q ,_q des2L1,conclusionu 2L1canbemade. 63

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Closedloopanalysisoferror: _r =q )]TJ /F9 11.955 Tf 12.2 0 Td[(q des+c_e =)]TJ /F3 11.955 Tf 9.3 0 Td[(k1r )]TJ /F3 11.955 Tf 11.95 0 Td[(k2r +D,(4)inwhich,k1andk2aredesignedtobepositiveconstant.Forclosedloopstabilityanalysis,Lyapunovstabilityanalysismethodisused.LetVLbeacontinuouslydifferentiablepositivedenitefunctiondenedas: VL=1 2r >r .(4)TaketimederivativeofVL, _VL=1 2r >_r .(4)Substitutingwehave:_VL=R >()]TJ /F3 11.955 Tf 9.3 0 Td[(k1r )]TJ /F3 11.955 Tf 11.96 0 Td[(k2r +D)=)]TJ /F3 11.955 Tf 9.3 0 Td[(k1r >r )]TJ /F9 11.955 Tf 11.95 0 Td[((k2r >r )]TJ /F3 11.955 Tf 11.96 0 Td[(r >D+1 4k2D>D)+1 4k2D>D=)]TJ /F3 11.955 Tf 9.3 0 Td[(k1r >r )]TJ /F3 11.955 Tf 11.95 0 Td[(k2(r )]TJ /F9 11.955 Tf 19.53 8.09 Td[(1 2k2D)>(r )]TJ /F9 11.955 Tf 19.53 8.09 Td[(1 2k2D)+1 4k2D>D)]TJ /F3 11.955 Tf 21.92 0 Td[(k1VL+", (4)Inwhich "=1 4k2D>D.(4)Ifk2canbepickedbigenough,wecanhaveaverysmall".Consequently, VL(t)VL(0)e)]TJ /F6 7.97 Tf 6.59 0 Td[(k1t)]TJ /F4 11.955 Tf 11.95 0 Td[("k1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[(k1t).(4)Thus,globalultimateboundedresulthasbeenproven.Thelargerk2is,thebettercontrolperformanceitdemonstrates. 64

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Recall( 3 )and( 3 )inSection 3.4.1 A=266664x0z377775, B=Bx^x B.(4)ActualcontrolinputarefromthreeorthogonalringsonsatellitesAandB,thefollowingequationgivesouthowtotransfer Abacktobodyxedcoordinatesystem: A A=R>A=RR A=R>A=RR>=R A.(4)Aswell,BxisgeneratedbydominantringofsatelliteB. 4.3.1.2Step1.A.1controldandInthisstep,attitudesofsatelliteAandBaresettobexed.Similartocontrollerdesignfordand,,highgainrobustcontrollerfordandcouldbedesigned.First,reorganizethedynamicequationsintheformofq =B(q ,_q )u +G(q ,_q )+D, q =264d375=2641 m0 21 d5cos1 m0 21 d5cos)]TJ /F8 7.97 Tf 11.79 4.7 Td[(1 m0 1 d4cos1 m0 21 d4sin375264u1u2375+2642d_2)]TJ /F8 7.97 Tf 10.5 6.48 Td[(2_d_2 d375.(4)ThedesignedcontrollerhasthesameformwithStep1.A.0: u =B)]TJ /F8 7.97 Tf 6.59 0 Td[(1()]TJ /F3 11.955 Tf 9.3 0 Td[(G)]TJ /F3 11.955 Tf 11.96 0 Td[(c_e+q des)]TJ /F3 11.955 Tf 11.95 0 Td[(k1r )]TJ /F3 11.955 Tf 11.96 0 Td[(k2r ).(4)Similarityforstabilityanalysisalsoexists. 4.3.2ControllerforStep1.BThereisatrickydifferencebetweenStep1.BandStep1.A.Tosavecontroleffort,detumblingfor_willconcentrateondriving_tozero,thus,afterStep1.Acompleted,therewillbeaconstantstays.However,sinceduringthedetumblingprocess(Step1.A.1),attitudesofsatelliteAandBarestabilizedbyangularmomentum,aconstantwillnotaffecttherelationshipbetweenFA,FB,andFR. 65

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DuetothehighsimilaritybetweenStep1.AandStep1.B,thecontrollerdesignmethodforStep1.BfollowsthesamedesignprocessasStep1.A.Recallthedynamicsequations:=1 m0 2Ax d5(xcos+zsin))]TJ /F9 11.955 Tf 13.15 8.08 Td[(2_d_ d, (4)d=)]TJ /F9 11.955 Tf 12.33 8.08 Td[(1 m0 2Ax d4(2zcos)]TJ /F4 11.955 Tf 11.96 0 Td[(xsin)+d_2, (4)=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 8Ax d3mr2(2zsin+xcos))]TJ /F9 11.955 Tf 15 8.09 Td[(1 m0 2Ax d5(xcos+zsin)+2_d_ d. (4) 4.3.2.1Step1.B.0controldandOrganizethedynamicequations: 264d375=264)]TJ /F8 7.97 Tf 11.79 4.71 Td[(1 m0 1 d4cos1 m0 21 d4sin)]TJ /F11 11.955 Tf 11.29 9.68 Td[()]TJ /F5 7.97 Tf 7.71 -4.42 Td[(0 41 d3mr2+1 m0 21 d5sin)]TJ /F11 11.955 Tf 11.29 9.68 Td[()]TJ /F5 7.97 Tf 7.72 -4.42 Td[(0 81 d3mr2+1 m0 21 d5cos375264u1u2375+264d_22_d_ d375(4)whereu isdenedas: 264u1u2375=264AxzAxx375.(4) 4.3.2.2Step1.B.1controldandOrganizethedynamicequationsas: 264d375=2641 m0 21 d5sin1 m0 1 d4cos)]TJ /F8 7.97 Tf 11.79 4.71 Td[(1 m0 21 d4cos1 m0 21 d4sin375264u1u2375+264d_2)]TJ /F8 7.97 Tf 10.5 5.87 Td[(2_d_ d375(4)ThendesignedinputhasthesameformasintheStep1.A.1.RefertoSection 3.4.2 ,transformationof BtobodyxedframeFB,actualinputB Bisgivenby: B=R>B=RR>=R B.(4)Also,AxisgivenbydominantringofsatelliteA. 66

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4.3.3ControllerforStep1.CRecallthedynamicequationinSection 3.4.3 .Inthisstep,amechanicallatchisintroducedtokeepdistanceaconstant.Thus,theangledifferencebetweenyAandyB,(')]TJ /F4 11.955 Tf 11.95 0 Td[( )istheonlyoneweneedtoconcern.d=0AyBy 2md4cos(')]TJ /F4 11.955 Tf 11.96 0 Td[( ), (4)')]TJ /F9 11.955 Tf 14.14 2.66 Td[( =)]TJ /F4 11.955 Tf 11.38 8.09 Td[(0AyBy 4mr2d3sin')]TJ /F4 11.955 Tf 11.96 0 Td[( (4)Letq=')]TJ /F4 11.955 Tf 11.95 0 Td[( ,u=AyBy: q=)]TJ /F4 11.955 Tf 27.12 8.09 Td[(0u 4mr2d3sinq.(4)ExaminethepropertiesmentionedinSection 4.3.1.1 ,sinqhassingularitywhenq=0,pi.Thismightcausecontrolinputtobeunbounded.Thussaturationforinputuisnecessary.Also,whentheinitialconditionissettobeq=,whichmeansyAandyBareintheoppositedirection,thecontrollerwillbeinvalid.Ifitisthiscase,asymmetricmechanicaldesignfordockingmechanismcouldturnq=tobeavaliddockingattitude.Controllerobjectivecouldbeexpressedas: e=q)]TJ /F3 11.955 Tf 11.96 0 Td[(qdes!0.(4)FollowingthesimilarLyapunovbasedcontrollerdesignmethod,timederivativeoflterederrorr=_e+ce,canbeexpressedas: _r=q+c_e=)]TJ /F4 11.955 Tf 27.12 8.09 Td[(0u 4mr2d3(sinq+c_e).(4)inwhichcisapositiveconstant.Designthecontrolleras: u=)]TJ /F9 11.955 Tf 10.5 8.09 Td[(4mr2d3 sinq0()]TJ /F3 11.955 Tf 9.3 0 Td[(c_e+qdes)]TJ /F3 11.955 Tf 11.96 0 Td[(k_e)]TJ /F3 11.955 Tf 11.95 0 Td[(kcq),(4)wherekisapositiveconstant. 67

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Morediscussionaboutthebehaviorofthiscontrollerwillbeconductedinsimulationsection.Itishardtotellifuisbounded,sincewhenq!0,_e,qdes,and_e!0.Informationontheconvergespeedsareneededtoverifyboundednessofu. 4.3.4ControllerforStep2Recalldynamicequationforthisstep.Noticethat,sincethisstepiscooperatingwiththedockingmechanism,whilethedynamicmodelisonlyfortheidealdockingsituation,anindicativecontrollerisdevelopedhere. d=)]TJ /F4 11.955 Tf 14.02 8.09 Td[(0 mAxBx d4.(4)ExaminethepropertiesmentionedinSection 4.3.1.1 ,assumethenaldwhichdockingmechanismrequiresisdf,then)]TJ /F5 7.97 Tf 18.01 5.25 Td[(0 md4canbeconsideredasboundedandinvertible.Controlobjectiveissettobetrackingadesiredtrajectoryddes,thentheerroris: e=d)]TJ /F3 11.955 Tf 11.96 0 Td[(ddes.(4)Introducinglterederror: r=_e+ce.(4)Again,Tokeepthecontrollerdesignprocessconcise,replaceAxBxwithu,anddesigncontrolinputas: u=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(md4 0()]TJ /F3 11.955 Tf 9.3 0 Td[(kr+ddes)]TJ /F3 11.955 Tf 11.95 0 Td[(c_e),(4)inwhichkischosentobeapositiveconstant.Closedloopstabilityanalysishasnodifferencefromabovecontrollerdesigns. 4.4AssumptionsforSimulationParametersSincethissystemisdesignedfordocking/proximityoperationsofsmallsatellitesorsmallassemblyparts,alltheassumptionsfortheparametersandlimitationsarebasedoncommondesignsforsmallsatellitesormicrosmallsatellites.TakeCubeSatwhichisapopularprogramforsmallsatellitesforexample[ 6 ],thetotalmassofsuchasatellitesliesintheorderof1kg,typicalsizeis0.1m0.1m0.1m,powerbudget 68

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forsatelliteusuallystaysafewwatts(however,newtechnologyinvolvesthemembranelikecongurationofsolarcellsmightgivesmorethan20W(Watts)power).Note,whensatellitesaresetindockingmode,mostofthepowerwillbeusedfordockingmissionincludingactuatingthedockingprocess,measuringandcontrollercomputing.Satellitesinthisthesisaresimpliedtodistributethemassuniformlytothecoils.Basedontheaboveinformation,satellitesmassandcoilradiusaresetas: m=1kg,r=0.1m.(4)Assumptionof4.5Wbudgetfordrivingtheelectromagneticcoilsforeachsatellitecanbefairlymade.Thenforeachcoil,1.5Wisthepowerlimitation.Referringtothedesignofmagnetictorquerin[ 5 ]and[ 8 ],thepowerdissipationinonecoilcanbedeterminedby: P=i2R,(4)whereiiscurrentincoil,andRisresistanceofthecoil.Thecoilresistanceisgivenby: R=2Nr aw,(4)whereNisthenumberorturns,ristheradiusofcoil,=1.5510)]TJ /F8 7.97 Tf 6.58 0 Td[(8mforcopperwire,awissectionareaofthecoil.Foracoilwithradiusof0.1m,aw25mm2isreasonableandsuitableforinertialmatrixassumptioninChapter 3 .Recall( 2 )magneticmomentofacoilisgivenby: =Nr2i=Nr2r Paw 2Nr.(4)ThelargerNis,thelargerthecoilcangenerate.Thus,asmalldiameter(dw)ofcopperwirewithinsulationwhichisavailablehasbeenchosentobe0.15mm.ThenN=4aw d2wcanbeapproximated. 69

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Substituting,alimitationforthemagneticmomentonecoilcangeneratemax=73Am2thencanbedesigned.Additionally,disturbancesandnoisesinsimulationwillbedesignedaswhitenoiseswhichhaveanorderoftenthofinputforcesandtorques.Undertheseassumptions,simulationsforseparatestepsandseveralcompletesimulationresultcombinedwithcontrolstrategyindifferentscenarioswillbeinvestigatedinfollowingsections. 4.5SimulationandResult 4.5.1SeparateSimulationResultforEachControllerBeforecompletesimulationsfordifferentdockingscenariosbegin,vericationforeachcontrollerisessential.Alsotherearesomedetailsaboutthesecontrollerarediscussed.BecauseofthesimilaritybetweenStep1.AandStep1.B,simulationsforStep1.AincludingStep1.A.0andStep1.A.1coulddemonstratetheperformanceofcontrollersinbothofthesetwosteps. 4.5.1.1SimulationforStep1.A.0Consideringthedockingstrategy,aregulationford0and0todesiredconstantddesanddesissimulated.Acontinuoustrajectoryqcfrominitialvalueq0toqdeshasbeendesignedassuch: qc=q0+(qdes)]TJ /F3 11.955 Tf 11.96 0 Td[(q0)tanh(!t)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F9 11.955 Tf 11.96 0 Td[(tanh()]TJ /F4 11.955 Tf 9.3 0 Td[() 1)]TJ /F9 11.955 Tf 11.96 0 Td[(tanh()]TJ /F4 11.955 Tf 9.3 0 Td[(),(4)where!isapositiveconstantfactoradjustingthetimethetrajectorytakestoconvergetoqdes,thelargeritis,thefastertrajectoryconverge,thelargercontroleffortitneeds.RecalltheowchartinFigure 4-3 and 4-4 eachstepneedsacriteriontojudgewhetherthisstepisnishedandtrigcorrespondingcontrollerandangularmomentummanagementbehavior.ForStep1.A.0,itissettobetsA1.Itisdenedtobe1whenthefollowingconditionsaresatised:q)]TJ /F3 11.955 Tf 11.96 0 Td[(qdes0.01,_q0.001otherwise,itstays0.Tofullyinspecttheperformanceofcontroller,3setofinitialconditionshavebeentested,asshowninTable 4-1 70

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ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-6. Simulationresultforinitialcondition1. ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-7. Simulationresultforinitialcondition2. 71

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Table4-1. InitialconditionforsimulationofStep1.A.0 d0(m)_d0(m/s)0_0(1/s) 11.50=3021.50.01=2)]TJ /F9 11.955 Tf 9.3 0 Td[(0.0230.8)]TJ /F9 11.955 Tf 9.3 0 Td[(0.020.01 ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-8. Simulationresultforinitialcondition3. Sincethedockingscenariosareallsettostartfromattitudestableinitialconditions,_0isalwayssettobe0.Also,consideringtheinputlimits,_d0cannotbelarge.Actually,forobjectwithdimensionof0.1m,0.01m/sisareasonableassumption.Desiredconstantddesanddesaresettobe1mand0radrespectively.Simulationresultsfortheseconditionsincludingstatesvariable,errorandinput,aredemonstratedinthefollowinggures:Figure 4-6 ,Figure 4-7 ,Figure 4-8 .Thedesignedsmoothtrajectoriesfromd0and0toddesanddeshavebeendemonstratedinplotAofeachgure.CombiningwithplotB,presentingthetrackingerror,andplotC,presentingtheinputs,onecandrawaconclusionthatwhencontrolsignalsstaysintherangeofinputlimitations(generallyfrom0-10sand20-30s),trackingerrorhasa 72

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goodperformance(under1103).However,eventhesaturationaffectsthetrackingerrors(generallyfrom10-20s),astimegoesby,errorstillcanbedriventozero.TriggerindicatortsA1isturnedonabout32sto35s,whichmeansthetimeconsumptionforthisstepisacceptableconsideringanusualcompletedockingmissionrequires5)]TJ /F1 11.955 Tf 9.3 0 Td[(10minutes[ 19 ].Alsothegoodperformanceontrackingerrorindicatesthatthiscontrollerissuitableforthegeneralreasonableinitialvelocitiesandanglerates. 4.5.1.2SimulationforStep1.A.1Since,thecontrolobjectivesinthissteparedistanceregulatingortracking,anddetumblingtherotatedreferenceframe,themaincontroleffortshouldbespentondistancecontrolanddetumbling(_!0).Whendesignthetrajectory,desiredtrajectoryofusuallyispreferredtodrivetoaconstantnearbyinitialcondition0.Withoutloseofgenerality,controlobjectivearesettoberegulatingtoddes=1anddes=0+.Inwhichisasmallangle,againtosavecontrolenergy,signofisdenedduetoinitialratesof_0.Carefulchoosingofwillreducethetimetoachievecontrolobjective.Inthisstep,sincetheattitudesofsatelliteAandBaresettobestabilizedbyreactionwheels,issettobe0,sois_.Also,duetothetrigconditionofStep1.A.1(q)]TJ /F3 11.955 Tf 11.71 0 Td[(qdes0.01,_q0.001),initialcondition_d0issettobenolargerthan0.001,_0isthevalueof_whenStep1.A.1istriggered.Yettotestifytherobustnessofthecontroller,thefollowing2initialconditionswithfairlylarge_d0and_showninTable 4-2 aretested. Table4-2. InitialconditionsforsimulationofStep1.A.1 d0(m)_d0(m/s)0_0(1/s) 11.2)]TJ /F9 11.955 Tf 9.3 0 Td[(0.01=30.0220.80.02=2)]TJ /F9 11.955 Tf 9.29 0 Td[(0.02 Followingtheguidelineinthebeginningparagraph,des==3+=10anddes==2)]TJ /F4 11.955 Tf 12.33 0 Td[(=10.Thesimulationresultsforcondition1and2aredemonstratedin2gures,Figure 4-9 andFigure 4-10 .SimilarwithStep1.A.0,thetrackingerrorshowsagoodperformancewhensaturationsarenotreached(from10-80s).Detumbling 73

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a0.02rad/slevel_ismoretimeconsumingthandrivingtozeroinStep1.A.0.ReferringtotheplotD,trigconditionsareusuallysatisedataround90s.Nevertheless,inacompletesimulation,Step1.A.1alwayshappensaftercontrolobjectivesofStep1.A.0hasbeenachievedwhichcomesusuallywithasmall_(about0.005rad/s).Ifthisisthecase,thentimeconsumptionwillbereduced.MoredetailsaboutthiscanbereachedincompletesimulationsinSection 4.5.2 ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-9. SimulationresultforStep1.A.1ininitialcondition1. 4.5.1.3SimulationforStep1.CThisstepstartswithroughdocking,whichxedthedistancebetweentwosatellites.Onlytwistingangledifference')]TJ /F4 11.955 Tf 13.12 0 Td[( willbedealtwithinthisstep.Assumeroughdockingkeepsdistancetobedr=0.5,twosetsofinitialconditionswillbetestedforthiscontroller,asshowninTable 4-3 .TriggertsCisdenedtobe1when')]TJ /F4 11.955 Tf 12.51 0 Td[( 0.001,_')]TJ /F9 11.955 Tf 15.53 2.65 Td[(_ 0.001.SimulationresultsfortwoinitialconditionswithinputsaturationareillustratedinFigure 4-11 and 4-12 .TrajectoryiswellfollowedbyobservingtheplotAandB.As 74

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ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-10. SimulationresultforStep1.A.1ininitialcondition2. Table4-3. InitialconditionsforsimulationofStep1.C '0)]TJ /F4 11.955 Tf 11.95 0 Td[( 0(rad)_'0)]TJ /F9 11.955 Tf 15.53 2.66 Td[(_ 0(1/s) 1=2)]TJ /F9 11.955 Tf 9.29 0 Td[(0.012)]TJ /F4 11.955 Tf 9.3 0 Td[(=30.02 discussedinthecontrollerdesignprocedureinSection 4.3.3 ,boundednessofcontrolinputscannotbeguaranteed.ComparingplotAandplotC,singularityofcontrolinputhappenswhenyisclosetozero.However,astimegoesby,onecanobservethissingularityisanonetimeaccident.Afterthat,theinputconvergestoaconstant.And,plotDshowstrigindicatorts2isturnedonaround30s.Comparingwith( 4 ),throughtadditionalsimulations,anunconrmedconclusioncanbeinductedthat: limt!+1u=4k2mr2d3 0.(4)inthetwosimulationsshown,kischosentobe0.1,limt!+1u=12.5Am2Am2.Sumup,saturationforthisstepisnecessaryforthesingularityissue.Tokeeptheachieved 75

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objective,aconstantinputisneeded.Thus,cooperatingwithdockingmechanismisessentialforthisstep.Also,timeconsumptionisacceptable. ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-11. SimulationforStep1.Cininitialcondition1withsaturation. 4.5.1.4SimulationforStep2Initialconditionforthisstepiscomingfromtheroughdockingsetting.Assumeroughdockingkeepsdistancetobedr=0.5.Theactualrequireddistanceformechanicallatchissettobeddes=0.3.SimulationresultisshowninFigure 4-13 .Triggercriterionissatisedaround25s.TrackingerrorinplotBpresentsagoodperformanceofthiscontroller.Timeconsumptionandenergyconsumptionperformwellinthis1-Dattractioncase. 4.5.2CompleteSimulationInthissection,onecompletesimulationisdemonstrated.Withoutlossofgenerality,2setsofinitialconditionsaresetasTable 4-4 .Exceptforintroducingcontrolalgorithminthissection,transformingvirtualcontrolinputsforeachsteptoactualcurrentsofringsisalsoincluded. 76

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ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-12. SimulationforStep1.Cininitialcondition2withsaturation. ADesiredtrajectoryandactualsignal BError CInput DTriggerFigure4-13. SimulationresultforStep2. 77

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Table4-4. Initialconditionsforcompletesimulation d(m)_d(m/s)(rad)(-)(-)(-)'(-) (-) 11.50.01 3)]TJ /F8 7.97 Tf 10.5 4.71 Td[(3 4)]TJ /F5 7.97 Tf 10.5 4.71 Td[( 4 6 5)]TJ /F5 7.97 Tf 10.5 4.71 Td[( 221.70.02)]TJ /F5 7.97 Tf 10.49 4.7 Td[( 3)]TJ /F5 7.97 Tf 10.5 4.7 Td[( 3)]TJ /F5 7.97 Tf 10.5 4.7 Td[( 6 4)]TJ /F8 7.97 Tf 10.49 4.7 Td[(3 4 Observingthevirtualcontrolinputuforeachstep,takeStep1.Aforexample: u =264u1u2375=264zBxxBx375.(4)Calculatethemagnitudeforinput: u=q u21+u22=Bxq 2z+2x.(4)Assumetwosatellitehasthesamepowersupplyability,thenmake Bx=q 2z+2x=p u.(4)Thus,magneticmomentumofArepresentedinF, A,isgivenby: A=266664x0z377775.(4)Transfer AtoFA, A =R>A=RR A=RA=RR=R A.(4)Refertotheequations( 3 )and( 3 )forRA=RandR=RinSection 3.2.1 and 3.4.1 .ItisstraightforwardtocalculateA A.Also,BxisgeneratedbydominantringofB,B Bisobviouslyshownas: B B=266664Bx00377775.(4)Thetransformationsforotherstepsaresimilartothisprocedure. 78

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Thefollowingtwosetofgures(Figure 4-14 and 4-15 andFigure 4-16 and 4-17 )showthed,,,and(')]TJ /F4 11.955 Tf 12.34 0 Td[( )variesduringdifferentsteps,andthecontrolinputs(Ax,Bx,Ay,By,AzandBz)generatedbyeachofthecoilswhichareperpendiculartoaxesxA,xB,yA,yB,zA,andzB,respectively.Also,vetrigindicatorsforStep1.A.1,Step1.B.0,Step1.B.1,Step1.C,Step2anddockingcompletion(tsA1,tsB,tsB1,tsC,ts2andtsf)havebeendemonstrated.RefertoTable 4-5 ,controlobjectivesandtriggertimeforeachstephavebeenshowed. Table4-5. Completesimulationresult StepnameControlobjectiveTriggertimefornextstep(s) Step1.A.0Regulatedto1m,to0tsA1(32,44s)Step1.A.1Detumbling(_!0),keepingdtsB(107,118s)Step1.B.0Regulatedtodr(0.5m),to0tsB1(140,152s)Step1.B.1Detumbling(_!0),keepingdtsC(210,188s)Step1.CRoughdocking,regulate(')]TJ /F4 11.955 Tf 11.95 0 Td[( )to0ts2(247,226s)Step2Drivedistancetoddes(0.3m)tsf(270,249s) Figure4-14. Completesimulationforinitialcondition1:signal. Tosummarize,thecontrollersfordifferentstepsareunitedtogetherandtestedfor2generaldifferentscenariostodemonstratetheeffectivenessofthedockingandcontroloftwosatelliteseachequippedwith3orthogonalmagneticcoils.Theaboveresults 79

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Figure4-15. Completesimulationforinitialcondition1:magneticmoment. Figure4-16. Completesimulationforinitialcondition2:signal. 80

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Figure4-17. Completesimulationforinitialcondition2:magneticmoment. showthewholedockingmissioncostslessthan300swhichisthegeneraltimewithotherdockingpropellentsystemwillcost.RecallSection 4.4 ,thepowerrequirementsforthecasespresentedmatchwiththeinputlimitationanalysis.Aswell,foreachstep,disturbancescausedbyinaccuracyofdynamicsmodel,forceandtorquegeneratedbyuncertainties,andcomponentsofforcesantorquesactingoutofthealignmentplanesoraxeshavebeenaddedtosimulation.TherobustnessofthecontrollersstaysconsistentwiththeresultsshowninseparatesimulationsinSection 4.5.1 .Thus,underdeepspaceassumptions,thisdockingstrategyandcontrollawareapplicableforsmallsatellitesorassemblyparts.Innextchapter,moredetailsaboutexpandingthisdockingstrategytolowEarthorbitscenarioswillbeintroduced. 81

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CHAPTER5THELOWEARTHORBITCHALLENGES 5.1OverviewComparingtodeepspaceassumption,therealityinlowearthorbit(LEO)introducesasetofchallenges,includingtheeffectsoftheEarth'sgravitationaleldandmagneticeld.InSection 5.2 and 5.3 ,thesetwowillbeaddressed.Byasimple2-Dcase,thischapterpresentsthedynamicsequationsinLEOscenario.Aswelldifferencesfromthedeepspacescenarioaredemonstrated.Difcultiesofsolvingthisproblemhavebeendiscussed. 5.2GravitationalFieldWhenitcomestoinorbitcase,twotypesoftermsmainlyaffectthedockingdynamics:oneistheEarthgravity,anotheroneisorbitalangularvelocity.RecallSection 3.2.1 ,thegeometryofdifferentcoordinatesystem,whenthedocking/assemblyareoperatinginsomeorbitinsteadofindeepspace,FI6=FL.Consequently,! L=I6=0 .TakedynamicsforStep1.Aforexample.! =I=! R=L+! L=I, (5)P A=mA(_p A+! =IP A)=mA(_p A+(! R=L+! L=I)P A), (5)_P A+(! R=L+! L=I)P A=F A, (5)inwhichF Aincludebothelectromagneticforceandgravityforce.Similarchangeswillhappentorotationaldynamics.Fordescribingtheorbitalrelativeposition/attitudeproblem,whentheEarthismodeledasaperfectsphere,Clohessy-Wiltshire(CW)orHill'sequationsareusually 82

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used[ 18 ].SinceCWequationsareexpressedinthelocalinorbitframeFLasfollowing,Lx+n2Lx=LFx m, (5)Ly)]TJ /F9 11.955 Tf 11.95 0 Td[(3n2Ly)]TJ /F9 11.955 Tf 11.96 0 Td[(2nL_z=LFy m, (5)Lz+2nLy=LFz m, (5)whereLFistheforceactonthisobjectexceptgravity,nistheorbitalfrequency,givenby: n=r e r3e,(5)wherethegravitationalconstantofEarthe=3.98105km3=s2,reistheorbitalradiusofthesetwosatellites'centerofmass.Whenstudyinginorbitcase,translationaldynamicsmodelispreferredtobederivedinFL.Thecomplexityoftransfermatrixandstatesvariableestimationin3-Drealityisnothelpfulfordemonstratinghoworbitcaseshouldbederived.Thus,asimplenew2-DcaseasshowninFigure 5-1 ispresentedinthissection. Figure5-1. A2-Dcaseconcernsorbitaldynamics. 83

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AssumedipoleAandBaregeneratedbyringAandB,whichhastheidenticalsizeandmass.Inthiscase,byusingCW( 5 ),translationaldynamicsequationsarederivedas:LyA)]TJ /F9 11.955 Tf 11.96 0 Td[(3n2LyA)]TJ /F9 11.955 Tf 11.96 0 Td[(2nL_zA=LFy m, (5)LzA+2nLyA)]TJ /F9 11.955 Tf 11.96 0 Td[(2nL_zA=LFz m. (5)PositionofOArepresentedinFLis(LyA,LzA),misthemassofringA,and 264LFyLFz375=264cossin)]TJ /F9 11.955 Tf 11.3 0 Td[(sincos375264RFAyRFAz375.(5)where cos=LyA p Ly2A+Lz2A,sin=LzA p Ly2A+Lz2A.(5)RecallexpressionsforelectromagneticforceinSection 2.3.2 264RFAyRFAz375=30 4AB d42642cosRcosR)]TJ /F9 11.955 Tf 11.95 0 Td[(sinRsinR)]TJ /F9 11.955 Tf 11.29 0 Td[(cosRsinR)]TJ /F9 11.955 Tf 11.96 0 Td[(cosRsinR375,(5)whered=2p Ly2A+Lz2A.Therotationaldynamicsequationcanbeeasilyderivedsincethisisacoplanarcase:Ix(R+)=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 4AB d3(2sinRcosR+cosRsinR), (5)Iy(R+)=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 4AB d3(2cosRsinR+sinRcosR), (5)whereIxisthexcomponentofinertiaofringA=B.Observingthedynamicsequations,anddnolongerstayindynamicsequationsexplicitly.Eithercontrolobjectiveneedstobetransformedorthedynamicequationshould.Itdependsontheactualsensorsandstatesestimationmethod.Especially,whenitcomesto3-D,thetransformingbecomesmoredifcult.However,thedocking 84

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strategyandcontrollerdesignmethodshowninChapter 4 stillworks.Besides,whentheorbitperiodislongenoughcomparingtotheconsumedtimeondocking,theinuenceoforbitdynamicscouldbeconsideredasdisturbances.Basedonmoreknowledgeonstatesestimation,consideringaboutorbitdynamicswillbecomeeasier.Infuturework,thispartishighlyrecommended. 5.3GeomagneticFieldWhenthissystemisoperatinginEarth'smagneticeld,theelectromagneticcoilsproduceforceandtorqueonthesatellites.Forcegeneratedbygeomagneticeldisafunctionofthegradientofthelocalmagneticeld[ 15 ]: F GM=rBE ,(5)whereBEisthemagnitudeoflocalmagneticeld, isadipolemomentgeneratedbythecoil.Torquegeneratedbygeomagneticeldisgivenby: GM= B E.(5)Byexaminingandthequalitativeanalysis[ 15 ],theamountofdisturbanceforceproducedonsatelliteduetogeomagneticeldislessthantenthsofapercentoftheinteractingforcesbetweentwosatellites.However,thetorqueisonthesameorderofinteractingtorque.Sowhengeomagneticeldisconsidered,thedisturbanceforceisnegligible,whilethetorquemustbetakenintoaccount.TakethesimpleexampleinSection 5.2 forexample,consideringthesmallsizeofcoils,thedirectionandmagnitudeofB Eareassumedxed,presentedinFigure 5-2 .Thetorqueproducedbygeomagneticeldcanbecalculated:L AG=L ALB E=A(cosRBEz)]TJ /F9 11.955 Tf 11.96 0 Td[(sinRBEy), (5)L BG=L BLB E=B(cosRBEz)]TJ /F9 11.955 Tf 11.96 0 Td[(sinRBEy), (5) 85

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Figure5-2. Asimple2-Dcaseconcernsgeomagneticeld. Therotationaldynamicsequationsbecome:Ix(R+)=)]TJ /F4 11.955 Tf 12.17 8.08 Td[(0 4AB d3(2sinRcosR+cosRsinR)+A(cosRBEz)]TJ /F9 11.955 Tf 11.96 0 Td[(sinRBEy), (5)Iy(R+)=)]TJ /F4 11.955 Tf 12.17 8.09 Td[(0 4AB d3(2cosRsinR+sinRcosR)+B(cosRBEz)]TJ /F9 11.955 Tf 11.95 0 Td[(sinRBEy). (5)Thesimilardockingstrategycanbetransplantedtothiscase.Firststep,setattitudeofAxed,regulateRtozero.ThensetBxed,regulateRtozero.Considerbothofthetwodipolesaresteerable,whichmeansanydirectionandmagnitudeofthemagneticmomentcanbegenerated.Taketherststepforexample,reorganizeitas: Ix(R+)=)]TJ /F4 11.955 Tf 12.18 8.09 Td[(0 4A d3(2sinRBy+cosRBz)+A(cosRBEz)]TJ /F9 11.955 Tf 11.96 0 Td[(sinRBEy),(5)inwhichBy=BcosR,Bz=BsinR.Itisinterestingthatthereare3inputs(A,ByandBz).RecallthecontrolobjectiveforStep1.AinSection 4.3.1 :controllingisforthepurposeofaligningB'sdominantaxiswithdistancevector,controllingdgivestheabilityofregulatingthedistanceor 86

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trackingadesiredtrajectory,controllingisfordetumblingtherotatedreferencecoordinatesystemFR.Thedynamicequationshows3degreesoffreedom.SeparatingStep1.Atotwostepsistryingtoavoidtheproblemoflackingcontrolinput.Afterintroducinggeomagnetic,allthe3controlobjectivescanbedoneinonecontroller.Thiscouldalsobeapartoffuturework. 87

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CHAPTER6CONCLUSION 6.1SummaryoftheThesisThisthesisstartsfromtheideathatusingelectromagneticforceandtorquetoadjusttherelativepositionsandattitudesindocking/assemblyandotherproximityoperations.Basedonsomecarefulassumptions,thisthesisinvestigatesacapabledockingstrategytoachievethisgoal.Thekeyobjectivesofthisthesisweretopresentthiselectromagneticdockingideaisfeasiblefromthedynamicsandcontrolpointofview.Inthefollowingparagraphs,thechapter-wisesummaryofthisthesisaredemonstrated:Chapter 1 introducesthebasicideaofelectromagneticdockingandshowstheadvantagesoverconventionalpropulsion.Also,previousrelativeworksandthedifferencesfromotherresearcheshavebeenprovided.Chapter 2 usestheprincipleofmagneticeldtheory,describestheapplicabletheoryofmagneticeldmodelforthisthesis.Exactmodelhasbeenpresentedrstly,andthenlinearizationofthismodelleadstoafareldmodel.Baseonthisfareldmodel,forceandtorqueequationsforseveralprincipalbasiccongurationsofcoilshavebeenderived.Chapter 3 describesthesystemandtheoveralldockingstrategywithdeepspaceassumption.Decoupling3-Dproblemtoseveralstepsofprincipalbasiccasesincluding2-Dco-planarcase,co-axialtwistcaseand1-Ddistancecontrolcase.Thendevelopmentofdynamicmodelsforeachofthesecaseshasbeeninvestigated.Chapter 4 demonstratestheoverallcontrolstrategywhichaccommodateswiththedockingstrategy.Then,thecontrollerdesignandsimulationresultsofeachstephavebeenpresented.Aswell,acompletedockingproceduresimulationhasbeenperformed.Chapter 5 statesthedifcultieswhenthissystemisoperatinginlowearthorbit.Basedonasimple2-Dcase,thischapterpresentshowthegravitationaleldand 88

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geomagneticeldaffectthissystem.Afterdiscussionaboutthedynamicofthissimplecase,suggestionsforfutureworkdirectionshavebeengiven.Thisthesisstartsanewthinkingofintroducingelectromagneticforcesandtorquestodocking/assemblyandproximityoperationsmissions,whichiscombiningthetranslationalandrotationaldegreesoffreedomtogetheranddesigningcontrollawforthem.Otherpresentedworksusuallydecoupledthemtotwoindependentproblembycooperatingwithacompleteattitudecontrolsystem.Though,thisapproachisstraightforwardtothink,italsolosestheinsightaboutthebehavioursofelectromagneticforcesandtorques,abouthowtheyarecoupledwiththerelativepositionandattitude.Byaseriesofsimulations,thedockingstrategyinthisthesishasbeenproventobefeasibleandrobustformulti-scenarios.Also,powerconsumptionandtimecosthavebeendemonstratedsuitableforsmallspacevehicles. 6.2FutureWorkThisthesisconcentratesontheconceptionofelectromagneticdocking/assemblyandproximityoperationsystemfromthebasicdynamicsandcontrolpointofview.Thusnecessaryassumptionsabouttheworkscenario,implementmethodsandequipmentshavebeenmade.Severalrecommendationsaregivenforfuturework:Theinputsaturationinuencesthecontrollerperformanceheavily.Controllerdesignwithsaturationcompensatorisagooddirectiontoimprovetheperformance.Combiningwithstatesestimationresearch,inorbitcasecanbeinvestigatedmoredeep.BasedontheideaintroducedinChapter 5 abouthowtoexploitgeomagneticeldforthissystem,3-Dexpansioncouldbepartofthefuturework. 89

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APPENDIXAMATRIXNORMAmatrixnormisanaturalextensionofthenotionofavectornormtomatrices[ 3 ],[ 21 ].Thep-normforanmmatrixAisdenedas: kAkp=maxx6=0kAxkp kxkp.(A)Inthecaseofp=1,thenormcanbecomputedas: kAk1=max1jnmXi=1jaijj,(A)whichissimplythemaximumabsolutecolumnsumofthematrixA.Incaseofp=1thenormscanbecomputedas: kAk1=max1jmnXj=1jaijj,(A)whichisthemaximumabsoluterowsumofthematrixA.The2-normistheone,usuallyusedinboundingamatrix.ThereisaninequalitiesrelationshipamongkAk2andkAk1andkAk1:1 p nkAk1kAk2p mkAk1, (A)1 p mkAk1kAk2p nkAk1. (A)So,provingmatrixAisbounded,isequivalenttoprovingeveryabsolutecolumnsumofthematrixAisbounded.ItissufcienttohaveeveryabsoluteelementofmatrixAbebounded. 90

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REFERENCES [1] Ahsun,U.Dynamicsandcontrolofelectromagneticsatelliteformations.Ph.D.thesis,MassachusettsInstituteofTechnology.Dept.ofAeronauticsandAstronautics.,2007. [2] Camillo,P.J.andMarkley,F.L.Orbit-averagedbehaviorofmagneticcontrollawsformomentumunloading.JournalofGuidanceandControl3(1980):563. [3] Demmel,J.W.AppliedNumericalLinearAlgebra.SIAM,1997. [4] Dennison,E.Off-AxisFieldDuetoaCurrentLoop.2005.URL http://www.netdenizen.com/emagnet/offaxis/iloopoffaxis.htm [5] Graversen,T.,Frederiksen,M.K.,andVedstesen,S.V.AttitudeControlsystemforAAUCubeSat.Master'sthesis,AalborgUniversity,2002. [6] Hank,H.,Jordi,P.S.,Augustus,S.M.,Schinichi,N.,andRobert,J.T.CubeSat:AnewGenerationofPicosatelliteforEducationandIndustryLow-CostSpaceExperimentation.14thAnnual/USUConferenceonSmallSatellites5(2000). [7] Hughes,P.C.SpacecraftAttitudeDynamics.DoverPublications,2004. [8] Jens,G.DevelopmentofanActiveMagneticAttitudeDeterminationandControlSystemforPicosatellitesonhighlyinclinedcircularLowEarthOrbits.Master'sthesis,SchoolofAerospace,MechanicalandManufacturingEngineering,Science,EngineeringandTechnologyPortfolio,RMITUniversity,2006. [9] Kaneda,R.,Yazaki,F.,Sakai,S.,Hashimoto,T.,andSaito,H.TheRelativePositionControinFormationFlyingSatellitesusingSuper-ConductingMagnets.2ndInternationalSymposiumonFormationFlyingMissionsandTechnologies.2004. [10] Khalil,H.K.NonlinearSystems.PrenticeHall,Inc.,2002,3rded. [11] Lee,D-RandPernicka,H.OptimalControlforProximityOperationsandDocking.InternationalJournalofAeronauticalandSpaceSciences11(2010).3:206. [12] M.Patre,P.,MacKunis,W.,Dupree,K.,andDixon,W.E.Anewclassofmodularadaptivecontrollers,PartI:Systemswithlinear-in-the-parametersuncertainty.AmericanControlConference(2008):1208. [13] Pollack,G.L.andStump,D.R.Electromagnetism.AddisonWesley,2002. [14] Reynerson,C.M.Spacecraftmodulararchitecturedesignforon-orbitservicing.IEEEAerospaceConferenceProceedings4(2000):227238. 91

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[15] Schweighart,S.A.Electromagneticformationightdipolesolutionplanning.Ph.D.thesis,MassachusettsInstituteofTechnology,Dept.ofAeronauticsandAstronautics,2005. [16] Silani,E.andLovera,M.Magneticspacecraftattitudecontrol:asurveyandsomenewresults.ControlEngineeringPractice13(2005):357C371. [17] Sofyali,A.andAslan,A.R.Magneticattitudecontrolofsmallsatellites:asurveyofapplicationsandadomesticexample.Tech.rep.,IstanbulTechnicalUniversity,SpaceSystemsDesignandTestingLaboratuary,2011. [18] Steve,N.ASEN5050SpaceightDynamics.2011.URL http://ccar.colorado.edu/asen5050/ASEN5050/Lectures_files/lecture11.pdf [19] Werts,J.R.andBell,R.AutonomousRendezvousandDockingTechnologies-StatusandProspects.ProceedingsofSPIEAeroSenseSymposium5088(2003):20. [20] Wiens,G.J.andMaldonado,L.AutonomousTorque-ActuatedFoldableDockingMechanismforSmallSpaceVehicles.2011FloridaConferenceonRecentAdvancesinRobotics1(2011):200. [21] Wikipedia,thefreeencyclopedia.Matrixnorm.2010.URL http://en.wikipedia.org/wiki/Matrix_norm [22] Zhang,Y.W.,Yang,L.P.,Zhu,Y.W.,Ren,X.H.,andHuang,H.Self-dockingcapabilityandcontrolstrategyofelectromagneticdockingtechnology.ActaAstronautica69(2011):1073C1081. [23] .Nonlinear6-DOFcontrolofspacecraftdockingwithinter-satelliteelectromagneticforce.ActaAstronautica77(2012):97. 92

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BIOGRAPHICALSKETCH KeHuoreceivedhisBachelorofScienceinAerospaceEngineeringfromtheBeihangUniversity,China,in2009,MasterofScienceinAerospaceEngineeringfromtheUniversityofFloridain2012. 93