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Dynamics and Nonlinear Control of Electromagnetic Docking/Assembly and Proximity Operations
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Title:
Dynamics and Nonlinear Control of Electromagnetic Docking/Assembly and Proximity Operations
Creator:
Huo, Ke
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (93 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering
Mechanical and Aerospace Engineering
Committee Chair:
Wiens, Gloria J
Committee Members:
Barooah, Prabir
Crane, Carl D
Graduation Date:
8/11/2012

Subjects

Subjects / Keywords:
Artificial satellites ( jstor )
Coordinate systems ( jstor )
Docking ( jstor )
Electromagnetism ( jstor )
Magnetic fields ( jstor )
Mathematical vectors ( jstor )
Simulations ( jstor )
Sine function ( jstor )
Torque ( jstor )
Trajectories ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
assembly -- control -- docking -- dynamics -- electromagnetic
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Aerospace Engineering thesis, M.S.

Notes

Abstract:
The use of electromagnetic actuators in attitude control system has been considered as an effective and reliable approach for low Earth orbit (LEO) satellites. One recent application is Electromagnetic Formation Flight (EMFF) which controls the relative translational degrees of freedom between satellites. Compared to the use of traditional thrusters, using an electromagnetic force and torque in multi-spacecrafts missions has some distinct advantages, such as no propellant consumption and plume contamination, as well as continuous controllability. The advantages of electromagnets however come at the cost of highly nonlinear and coupled dynamics. Extending the EMFF approach to rendezvous and docking, this paper focuses on providing small satellites a docking capability in both axial and circumferential directions through the use of two sets of electromagnetic coils. For implementation, a novel control strategy is also presented. The first problem which needs to be solved is the model of electromagnetic field and generated force and torque between two electromagnetic coils. The force and torque are not only functionally related to the characteristics of the coils but are also dependent on the relative position and attitude of the two satellites in proximity of one another. Both a numerically exact model and an analytic far-field model have been built and compared in this paper. This thesis provides a capable electromagnetic docking strategy for two satellites. Each of the satellites in electromagnetic docking system is equipped with reaction wheels and a set of three orthogonal current driven coils. With the assistance of reaction wheels, decoupling the 3-dimensional docking problem to several steps of principal basic cases provides full docking capabilities of the small satellites. Dynamics analysis and nonlinear controller design have been developed. As well, an overall control strategy and a complete simulation have been demonstrated. Power consumption and disturbances due to both gravity and the earth magnetic field are considered in an on-orbit scenario. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2012.
Local:
Adviser: Wiens, Gloria J.
Statement of Responsibility:
by Ke Huo.

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UFRGP
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Copyright Huo, Ke. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
858474083 ( OCLC )
Classification:
LD1780 2012 ( lcc )
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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