Rapid Path-Planning Algorithms for Autonomous Proximity Operations of Satellites

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Title:
Rapid Path-Planning Algorithms for Autonomous Proximity Operations of Satellites
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1 online resource (194 p.)
Language:
english
Creator:
Munoz,Josue David
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University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Fitz-Coy, Norman G
Committee Members:
Rao, Anil
Dixon, Warren E
Burks, Thomas F

Subjects

Subjects / Keywords:
adaptive -- algorithm -- approach -- artificial -- autonomous -- close -- final -- function -- homotopy -- maneuver -- method -- operation -- path -- picard -- planning -- potential -- proximity -- range -- rapid -- real -- rendezvous -- satellite -- spacecraft -- time -- trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Aerospace Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Autonomous proximity operations (APOs) can be bifurcated into two phases: (i) close-range rendezvous and (ii) final approach or endgame. For each APO phase, algorithms capable of real-time path planning provide the greatest ability to react to ?unmodeled? events, thus enabling the highest level of autonomy. This manuscript explores methodologies for real-time computation of APO trajectories for both APO phases. For the close-range rendezvous trajectories, an Adaptive Artificial Potential Function (AAPF) methodology is developed. The AAPF method is a modification of the Artificial Potential Function (APF) methodology which has favorable convergence characteristics. Building on these characteristics, the modification involves embedding the system dynamics and a performance criterion into the APF formulation resulting in a tunable system. Near-minimum time and/or near-minimum fuel trajectories are obtained by selecting the tuning parameter. Monte Carlo simulations are performed to assess the performance of the AAPF methodology. For the final approach or endgame trajectories, two methodologies are considered: a Picard Iteration (PI) and a Homotopy Continuation (HC). Problems in this APO phase are typically solved as a finite horizon linear quadratic (LQ) problem, which essentially are solved as a final value problem with a Differential Riccati Equation (DRE). The PI and HC methods are well known tools for solving differential equations and are utilized in this effort to provide solutions to the DRE which are amenable to real-time implementations; i.e., they provide solutions which are functionals to be evaluated real-time. Several cases are considered and compared to the classical DRE solution.
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In the series University of Florida Digital Collections.
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Includes vita.
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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.
Statement of Responsibility:
by Josue David Munoz.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Fitz-Coy, Norman G.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

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RAPIDPATH-PLANNINGALGORITHMSFORAUTONOMOUSPROXIMITY OPERATIONSOFSATELLITES By JOSUEDAVIDMU NOZ ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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2011JosueDavidMu noz 2

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Tomydearmotherandfather 3

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ACKNOWLEDGMENTS IwouldrstliketothankmyadvisorDr.NormanG.Fitz-Coy,whoIhavehadthe honorofworkingwithsincemyundergraduatecareerandnowseemssolongago. Hehascontinuallychallengedmethroughouttheyearsandhasalsoservedasa role-modelonmanyaccounts.Hispassionforteachingandresearchhasmotivated metopursuethisdegree,andhisconscientiousdemeanorhasmadeitaworthwhile experience. Iwouldliketoexpressmythankstomysupervisorycommittee,Dr.WarrenE. Dixon,Dr.AnilV.Rao,andDr.ThomasF.Burks,whoaresomeofthemostbrilliant mindsIhavehadthepleasureofmeeting.Theknowledgetheyhaveinstilledonmewas pivotalforcompletingthiswork,andforthatIamgrateful. Iwouldalsoliketothanktheagenciesthathaveprovidedmewithresearch and/ornancialsupportthroughouttheyears.Particularly,theUniversityofFlorida, theDefenseAdvancedResearchAgencyProjects,theSouthEastAllianceforGraduate EducationoftheProfessoriateFellowship,theLockheedMartinCorporation,theAir ForceResearchLab/SpaceVehiclesDirectorate,andtheScience,Mathematics,and ResearchTransformationScholarship. Finally,Iwouldliketothankmyfamily,friends,andcolleagueswhohaveprovided mewithmoraleand/orresearchsupportthroughouttheyears.Particularly,mybrothers andsisterswhohaveeducatedmeinmanyways,whethertheyrealizeitornot.Thanks goouttoShawnJohnson,whoImetduringmyrstweekofcollege.Withouthishelp, Iwouldnothavethestudyhabitsthatallowedmetogetthisfarinmycareer.Thanks toErnestodelCastillo,whowasalwaystheretoliftmyspiritswhenIfeltdiscouraged. ThankstoDr.NickMartinson,whoIwasabletobrainstormwithandwasalways availabletodiscussresearchideas.ThankstoDr.FredLeve,whoalwaysprovided adifferentperspective,andourconversationsalwaysprovokednewandinteresting researchideas.ThankstoDr.GeorgeBoyarko,whowasalwaysabletoputthingsinto 4

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perspective,andhelpedbringalotofmyresearchtogether.Andthankstomyfriends andcolleaguesShawnAllgeier,SharanabasaweshwaraAsundi,DanteBuckley,Katie Cason,TakashiHiramatsu,Tzu-YuLin,VivekNagabhushan,KunalPatankar,andBungo Shiotaniforthemanylonghoursspentworkingtogetheronourresearchandacademic endeavors. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................9 LISTOFFIGURES.....................................11 LISTOFABBREVIATIONS................................14 ABSTRACT.........................................16 CHAPTER 1INTRODUCTION...................................18 1.1PreviousMissions...............................21 1.2StateoftheArt.................................25 1.2.1OptimizationMethods.........................27 1.2.2AnalyticMethods............................29 1.3TechnicalChallenges.............................29 1.4ResearchScope................................31 2SYSTEMDYNAMICS................................34 2.1OrbitalMechanics...............................34 2.1.1ZonalHarmonics............................34 2.1.2AerodynamicDrag...........................35 2.1.3SolarDrag................................37 2.1.4ThirdBodyDisturbances........................37 2.2AttitudeDynamics...............................38 2.2.1GravityGradientTorque........................39 2.2.2AerodynamicTorque..........................39 2.2.3SolarTorque..............................39 2.3AttitudeRepresentations............................40 2.3.1EulerAngles..............................40 2.3.2Axis-Angle................................41 2.3.3UnitQuaternion.............................42 3SYSTEMMODELING................................45 3.1Target/ChaserPlant..............................45 3.2ActuatorDynamicsandModel........................47 3.2.1ReactionJets..............................47 3.2.2ReactionWheels............................50 6

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4OPTIMALTRAJECTORIES.............................55 4.1OptimalControlProblem............................55 4.1.1MinimumTimeProblem........................57 4.1.2FixedTimeMinimumControlEffortProblem.............59 4.1.3FiniteHorizonLinearQuadraticProblem...............60 4.2MethodsforSolvingOptimalControlProblems...............63 4.2.1IndirectMethod:Shooting.......................64 4.2.2DirectMethod:Collocation.......................65 4.3OptimalRendezvousTrajectories.......................68 4.3.1MinimumTimeRendezvous......................69 4.3.2FixedTimeMinimumControlEffortRendezvous..........69 4.3.3FiniteHorizonQuadraticCostRendezvous.............71 4.3.4ConstrainedFixedTimeMinimumFuelRendezvous........72 4.4OptimalSlewManeuvers...........................73 4.4.1MinimumTimeSlew..........................76 4.4.2FixedTimeMinimumControlEffortSlew...............76 4.4.3FiniteHorizonQuadraticCostSlew..................78 5ARTIFICIALPOTENTIALFUNCTIONMETHOD.................81 5.1Development..................................81 5.2NumericalExamples..............................83 5.2.1Clohessy-Wiltshire-HillExample....................83 5.2.2SmallAngleApproximationExample.................86 6ADAPTIVEARTIFICIALPOTENTIALFUNCTIONMETHOD...........89 6.1Development..................................89 6.2StabilityAnalysis................................95 6.3NumericalExamples..............................97 6.3.1Clohessy-Wiltshire-HillExample....................97 6.3.2SmallAngleApproximationExample.................100 7FINITEHORIZONLINEARQUADRATICPROBLEMS..............104 7.1Development..................................104 7.2LinearTime-InvariantSystem.........................106 7.3LinearTime-VaryingSystem.........................106 7.3.1PicardIteration.............................106 7.3.2HomotopyContinuation........................108 7.4NumericalExamples..............................110 7.4.1PicardIteration.............................111 7.4.2HomotopyContinuation........................113 7.5Yamanaka-Ankerson-Tschauner-HempelExample.............116 7

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8NUMERICALRESULTS...............................122 8.1MonteCarloSimulations............................122 8.2HighFidelitySimulations............................126 8.2.1MinimumTimeTrajectories......................129 8.2.2FixedTimeMinimumControlEffortTrajectories...........132 8.2.3FiniteHorizonQuadraticCostTrajectories..............136 8.2.4Disturbance-FreeTrajectories.....................140 8.2.5ArticialPotentialFunctionTrajectory.................143 8.2.6AdaptiveArticialPotentialFunctionTrajectory...........143 8.2.7ObstacleAvoidanceTrajectories...................146 8.2.8FinalApproachTrajectory.......................154 9CONCLUSIONS...................................158 APPENDIX ARELATIVEMOTIONMODELS...........................160 A.1RelativeTranslation..............................160 A.1.1Clohessy-Wiltshire-HillEquation...................163 A.1.2Yamanaka-Ankerson-Tschauner-HempelEquation.........164 A.2RelativeOrientation..............................166 BTWOPOINTBOUNDARYVALUEPROBLEMSOLUTION............169 CHOMOTOPYWITHRICCATIEQUATIONL S = S ...............174 DHOMOTOPYWITHTRANSFORMEDRICCATIEQUATIONL X = X .....179 EHOMOTOPYWITHTRANSFORMEDRICCATIEQUATIONL X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(X ..183 REFERENCES.......................................186 BIOGRAPHICALSKETCH................................194 8

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LISTOFTABLES Table page 3-1Reactionwheelsystemparametersandconstraints...............54 4-1Optimalrendezvousparameters..........................69 4-2Optimalslewparameters..............................76 5-1Relativetranslationparameters...........................84 5-2Relativeorientationparameters...........................87 6-1Relativetranslationparameters...........................98 6-2Relativeorientationparameters...........................101 7-1Finitehorizonlinearquadraticparameters.....................110 7-2MaximumdifferenceofRiccatimatrixcomponentsPI..............111 7-3Costsforeachapproximatingmethod.......................113 7-4MaximumdifferenceofRiccatimatrixcomponentsHCcase1.........113 7-5MaximumdifferenceofRiccatimatrixcomponentsHCcase2.........114 7-6MaximumdifferenceofRiccatimatrixcomponentsHCcase3.........116 7-7Finalapproachparameters.............................119 7-8CostsfornalapproachusingHC.........................120 8-1MonteCarlosimulationparametersCWHequation...............123 8-2MonteCarlosimulationparametersSAAequation...............124 8-3Targetspacecraftparameters............................127 8-4Chaserspacecraftparameters...........................127 8-5APFmethodparameters...............................143 8-6AAPFmethodparameters..............................145 8-7Obstaclespacecraftparameters..........................147 8-8ConstrainedAPFmethodparameters.......................149 8-9ConstrainedAAPFmethodparameters......................150 8-10Finalapproachparameters.............................155 9

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B-1Relativeorientationparameters...........................172 B-2Relativeorientationparameters...........................172 10

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LISTOFFIGURES Figure page 1-1IllustrationofJAXA'sightexperiments......................22 1-2IllustrationofDARTmission.............................22 1-3IllustrationoftheXSSmissions...........................23 1-4IllustrationofOEDSmission.............................24 1-5IllustrationofPrismamission............................25 1-6Computationalefciencyvs.optimalitytradeoff..................30 2-1Surfacediscretizationofspacecraftwithunitnormalvectors...........36 3-1HighlevelSimulinkdiagram.............................45 3-2Geometryofspacecraftandreactionjets.....................46 3-3Simulinksatellitemodelblock............................47 3-4Thrustproleofreactionjets............................49 3-5Blockdiagramtoachievethrustprole.......................50 3-6Illustrationofsaturationanddeadbandlimits...................50 3-7Switchtoimposesaturationanddeadbandlimits.................51 4-1Tradeoffbetweennaltimeandcontroleffort...................61 4-2Minimumtimerendezvousresults.........................70 4-3Fixedtimeminimumcontroleffortrendezvousresults..............71 4-4Finitehorizonquadraticcostrendezvousresults.................73 4-5Constrainedxedtimeminimumcontrolefforttrajectories............74 4-6Constrainedxedtimeminimumcontroleffortrendezvousresults........75 4-7Minimumtimeslewresults..............................77 4-8Fixedtimeminimumcontroleffortslewresults...................78 4-9Finitehorizonquadraticcostslewresults.....................80 5-1ArticialpotentialfunctionAPFexample.....................82 5-2APFmethodresultsusingClohessy-Wiltshire-HillCWHequation.......85 11

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5-3APFmethodrunningcostusingCWHequation..................86 5-4APFmethodresultsusingsmallangleapproximationSAAequation......88 6-1Illustrationofadaptiveestimatetrajectory.....................94 6-2ExampleswhereAPFdoesnotguaranteeconvergence.............97 6-3AdaptivearticialpotentialfunctionAAPFmethodresultsusingCWHequation99 6-4AAPFrunningcostusingCWHequation......................100 6-5AAPFadaptiveestimatesusingCWHequation..................100 6-6AAPFresultsusingSAAequation.........................102 6-7AAPFadaptiveestimatesusingSAAequation..................102 7-1PicardIterationresults................................112 7-2Homotopycontinuationcase1results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 20 ...............115 7-3Homotopycontinuationcase2results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 2 ................117 7-4Homotopycontinuationcase3results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 2 ................118 7-5Homotopycontinuationresultsfornalapproachexample h = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 10 .....120 8-1Initialrelativeposition/orientationsusedinMonteCarlosimulations.......125 8-2CostsandsettlingtimesdatafromMonteCarlosimulations...........126 8-3Minimumtimetrajectorytrackingresults.....................131 8-4Minimumtimeforcetrackingresults........................133 8-5Fixedtimeminimumcontrolefforttrajectorytrackingresults..........135 8-6Fixedtimeminimumcontroleffortforcetrackingresults.............137 8-7Finitehorizonquadraticcosttrajectorytrackingresults.............139 8-8Finitehorizonquadraticcostforcetrackingresults...............141 8-9Fixedtimeminimumcontrolefforttrajectorytrackingresultsnodisturbances142 8-10Finitehorizonquadraticcostforcetrackingresultsnodisturbances.....144 8-11APFmethodresults.................................145 8-12AAPFmethodresults................................146 8-13Constrainedxedtimeminimumcontrolefforttrajectories............148 12

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8-14Constrainedxedtimeminimumcontrolefforttrajectorytrackingresults...149 8-15ConstrainedAPFmethodtrajectories.......................151 8-16ConstrainedAPFmethodresults..........................152 8-17ConstrainedAAPFmethodtrajectories.......................153 8-18ConstrainedAAPFmethodresults.........................154 8-19Finalapproachresults................................156 A-1Relativeorientationbetweenchaserandtarget..................167 B-1Twopointboundaryvalueproblemexamples...................173 13

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LISTOFABBREVIATIONS AAPFAdaptivearticialpotentialfunction ACLAutonomouscontrollevel ANGELSAutonomousNanosatelliteGuardianforEvaluatingLocalSpace APFArticialpotentialfunction APOAutonomousproximityoperation CDMCelldecompositionmethod CMCenterofmass CWHClohessy-Wiltshire-Hill DARTDemonstrationofAutonomousRendezvousTechnology DOFDegrees-of-freedom DREDifferentialRiccatiequation ETSEngineeringTestSatellite FRENDFront-endRoboticEnablingNear-termDemonstration G&CGuidanceandcontrol GASGloballyasymptoticallystable HCHomotopycontinuation HTVH-IItransfervehicle ISSInternationalSpaceStation KKTKarush-Kuhn-Tucker LCFLyapunovcandidatefunction LEOLowEarthorbit 14

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LQLinearquadratic LTILineartimeinvariant LTVLineartimevarying MCSMonteCarlosimulation MSEMaclaurinseriesexpansion NLPNonlinearprogram OCMOrthogonalcollocationmethod OCPOptimalcontrolproblem OEDSOrbitalExpressDemonstrationSystem PIPicarditeration PIDProportional-integral-derivative PVTPrimervectortheory SAASmallangleapproximation SSASpacesituationalawareness STMStatetransitionmatrix SUMOSatellitefortheUniversalModicationofOrbits TPBVPTwopointboundaryvalueproblem TSETaylorseriesexpansion UAVUnmannedaerialvehicle XSSExperimentalSatelliteSystem YATHYamanaka-Ankerson-Tschauner-Hempel 15

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy RAPIDPATH-PLANNINGALGORITHMSFORAUTONOMOUSPROXIMITY OPERATIONSOFSATELLITES By JosueDavidMu noz August2011 Chair:NormanG.Fitz-Coy Major:AerospaceEngineering AutonomousproximityoperationsAPOscanbebifurcatedintotwophases:i close-rangerendezvousandiinalapproachorendgame.ForeachAPOphase, algorithmscapableofreal-timepathplanningprovidethegreatestabilitytoreactto unmodeledevents,thusenablingthehighestlevelofautonomy.Thismanuscript exploresmethodologiesforreal-timecomputationofAPOtrajectoriesforbothAPO phases. Fortheclose-rangerendezvoustrajectories,anAdaptiveArticialPotentialFunction AAPFmethodologyisdeveloped.TheAAPFmethodisamodicationoftheArticial PotentialFunctionAPFmethodologywhichhasfavorableconvergencecharacteristics. Buildingonthesecharacteristics,themodicationinvolvesembeddingthesystem dynamicsandaperformancecriterionintotheAPFformulationresultinginatunable system.Near-minimumtimeand/ornear-minimumfueltrajectoriesareobtainedby selectingthetuningparameter.MonteCarlosimulationsareperformedtoassessthe performanceoftheAAPFmethodology. Forthenalapproachorendgametrajectories,twomethodologiesareconsidered: aPicardIterationPIandaHomotopyContinuationHC.ProblemsinthisAPOphase aretypicallysolvedasanitehorizonlinearquadraticLQproblem,whichessentially aresolvedasanalvalueproblemwithaDifferentialRiccatiEquationDRE.The PIandHCmethodsarewellknowntoolsforsolvingdifferentialequationsandare 16

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utilizedinthisefforttoprovidesolutionstotheDREwhichareamenabletoreal-time implementations;i.e.,theyprovidesolutionswhicharefunctionalstobeevaluated real-time.SeveralcasesareconsideredandcomparedtotheclassicalDREsolution. 17

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CHAPTER1 INTRODUCTION Theparadigmofspacetechnologyistransitioningtoenableautonomouson-orbit operations.[1,2]OnesignofthistransitionistheimminentretirementoftheSpace Shuttle.TheSpaceShuttlehascarriedmanysatellitesandastronautsintospace, whichresultedinunprecedentedmissionsandtechnologicaladvancements.TheSpace Shuttlehasalsoprovidedlogisticstovaluablespaceassetswhichwouldotherwisehave beendecommissioned.[3]TheretirementoftheSpaceShuttleisaresultofmanyyears ofhighoperationandmaintenancecosts.Autonomousspacecraftpresentanalternative forperformingproximityoperationsthatwerepreviouslydonemanually.[4]Thiswould alsoallowmannedspaceightmissionstofocusonotherspacesciencemissionsrather thanmundaneresupplymissions. Anotherexampleofthetransitioninspacetechnologyistherapidlyincreasing numberofobjectsanddebrisinorbit.Asspacetechnologycontinuestomature, itisbecomingeasiertodevelopandlaunchspacesystemsintoorbit.Inaddition, collisionssuchastheChineseASATtestin2007,theUSA193interceptin2008,and theIridium33/Cosmos2251collisionin2009contributegreatlytoorbitaldebristhat couldpotentiallybeharmfultoneighboringspaceassets.[5,6]Currently,collisions areavoidedbyplanningaseriesofmaneuversfromagroundstationanduplinking thecommands.However,asthenumberofobjectsinorbitincreases,itisbecoming increasinglydifculttokeeptrackofallobjectsandtosolveforcollisionavoidance maneuverswithmultipleconstraintsinatimelymanner.TheIridium33/Cosmos2251 collisionisaprimeexampleofthis,wheretheIridium33satellitewasafunctional satellite,yetacollisionwasnotavoided.Thisexampleshowswhyspacesituational awarenessSSAisbecomingcriticaltoprotectvaluablespaceassetsandtoperform successfulcollisionavoidancemaneuvers.TosupplementtheSSAcapabilitiesfrom 18

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groundstations,satellitescapableofperformingautonomousproximityoperations APOscanbeusedtoaccuratelytrackobjectsinspaceorremoveorbitaldebris.[7] Theresponsivespaceinitiativeandsmallsatellitetechnologyalsorequiretheneed forautonomy.Oneoftherequirementsofresponsivespacesystemsisautonomysuch thattheyspacecraftisabletochangeitsorbitorperformorbitalstationkeepingonboard. [8,9].Likewise,smallsatellitesorgroupsoffractionatedsmallsatellitescansupplement orreplacecertainspaceassets.[10,11]SmallsatellitescanbeusedtoperformAPOs onotherspaceassetsaswell.[12,13] APOsincludebutarenotlimitedtoclose-rangerendezvous,inspection, interception,docking,payloadtransfer,orbitalstation-keeping,formationying,and on-orbitassembly.Forthediscussionsinthismanuscript,APOsaregeneralizedand decomposedintotwophases:iclose-rangerendezvousandiinalapproachor endgame.BeingabletoaccomplishAPOswiththehighestlevelofautonomyispivotal for: EnablingAPOswithspacecraft GivingheritagetotechnologydevelopedforAPOs StandardizationformissionsperformingAPOs Reductioninsupervisionneededforperformingcertainmissions Increasedrobustnesstocommunicationconstraints Enableservicingofspacecraftinorbit AdistinguishingfeatureofAPOsisautonomy.Thus,itisnecessarytodenewhat ismeantbyautonomyinthecontextofAPOs.Aproposeddenitionforanautonomous systemisasystemthat mustperformwellundersignicantuncertaintiesintheplant andtheenvironmentforextendedperiodsoftimeanditmustbeabletocompensate forsystemfailureswithoutexternalintervention. [14]Thisdenitionprovidesinsight, yetitistoobroadforthepurposeofAPOs.Adifferentdenitionofautonomyhasbeen proposedbythearticialintelligence,robotics,andintelligentsystemscommunityusing threecategorizationsofmodel-basedarchitectures:[15] 19

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1.Model-basedarchitecture:Abilitytoachieveprescribedobjectives,all knowledgebeingintheformofmodels,asinthemodel-basedarchitecture. 2.Adaptivemodel-basedarchitecture:Abilitytoadapttomajorenvironmental changes.Thisrequiresknowledgeenablingthesystemtoperformstructure reconguration,i.e.,itneedsknowledgeofstructuralandbehavioral alternativesascanberepresentedinthesystementity. 3.Generativemodel-basedarchitecture:Abilitytodevelopitsownobjectives. Thisrequiresknowledgetocreatenewmodelstosupportthenewobjectives, thatisamodelingmethodology. Thesethreecategoriesareusefulforhigh-levelanalysis,yetasystematicapproach ofcharacterizingautonomyinspacesystemsisdesired.[16,17]Theapproachtaken bytheAirForceResearchLabtocharacterizeautonomyofUnmannedAerialVehicles UAVsusesAutonomousControlLevelsACLsrangingbetween0-10.[18]The lowestACLisLevel0,whichcorrespondstoavehiclethatisremotelypilotedand hasnodecisionmakingcapabilities.ThehighestACLisLevel10andisdescribed asHuman-like.Theintermediatelevelsaredenedfordifferentrangesofabilities todetectandtrackothervehiclesincloseproximityandhavingdecisionmaking capabilitieson-boardwithdifferentamountsofexternalsupervision.Ifasimilar classicationofautonomyisusedforspacecraft,thenbeingabletotrackobjectsin closeproximityandrapidlyplanoptimizedtrajectorieson-boardwithminimalexternal supervisionorinterventionisthelevelofautonomydesiredforAPOs. Ontheotherhand,performingtasksforproximityoperationmissionsofinetendsto havehighoperationcosts.[19]Planningandschedulingofineiscomputationallycostly andtheadditionalcommunicationconstraintspresentsproblemsduetolow-bandwidth, high-latencycommunicationlinks.[20]Anexampleofautonomoussystemwiththese capabilitiesistheMarsExplorationRoverwhichlandedsuccessfullyonMarsin2004. Thelatencyofthecommunicationrangingfrom 8 to 42 minutesrequiredthatalarge numberoftasksbeplannedanduplinkedintermittentlyandexecutedwithouthuman monitoringorconrmation.[21]Whilethedelaysandcommunicationconstraints withaspacecraftinorbitarelesssevere,thisexampleillustrateshowscheduling 20

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andplanningautonomoustasksofinecanbecumbersome.Thecommunication constraintsalsovarydependingontheorbitofthespacecraft.Thespacecraftcould beavailableforcommunicationseveraltimesadayatbest,whileatothertimesitmay notbeaccessibleforlongperiodsoftimes.Multiplegroundstationscanbeusedto alleviatethecommunicationconstraints,however,thereislittlefreedominstrategically placinggroundstationsduetomonetarylimitsandpropertyrights.Ontheotherhand, increasingthelevelofautonomywouldreducetheneedforconstantcommunication withaspacecraft. 1.1PreviousMissions OneoftherstmissionstoattempttodemonstrateAPOswasJAXA'sEngineering TestSatelliteVIIETS-VIIin1998.TheETS-VIIisshowninFigure1-1Aandconsists ofapassivetargetsatelliteORIHIMEandanactivesatelliteHIKOBOSHI;both initiallyinadockedstate.Themainobjectivewastodemonstraterelativeapproach, nalapproach,anddockingbetweenHIKOBOSHIandORIHIME.Therstexperiment requiredthatbothspacecraftdetach,yinformationatadistanceoftwometers,and thendemonstrateautonomousdocking.[22,23]Theexperiencegainedfromthis experimentwouldlaterbeusedontheH-IITransferVehicleHTVin2009,which provideslogisticstotheInternationalSpaceStationISS.[24]TheHTVisshown inFigure1-1B.TheETS-VIIwasalsotodemonstratearelativeapproachtrajectory betweenthetwosatellites,however,technicaldifcultieswereexperienceddueto attitudedeviations.Inshort,everytimeanorbitalmaneuverwasexecuted,thethrusters wouldchangetheorientationofthechasersatellitewhichdidnotallowthechaserto trackitspredeterminedtrajectory.Inordertoalleviatethisproblem,amodicationhadto bemadetotheightsoftware.[23]Nevertheless,thelessonslearnedfromtheETS-VII missionledtothesuccessoftheHTV.Thesetwomissionsdemonstratehowcritical itistoperformanAPOparticularlyrendezvousanddockingwithhighprecisionand collisionavoidance,especiallywhendockingwithavaluedspaceassetliketheISS. 21

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AArtistdepictionofETS-VII BArtistdepictionofHTV Figure1-1.IllustrationofJAXA'sightexperiments AnothermissionaimedattestingtheabilitytoperformAPOswastheDemonstration forAutonomousRendezvousTechnologyDARTprojectin2005.[25]Themain objectivewastodemonstratelong-rangerendezvous,close-rangerendezvous,and collisionavoidancewiththedecommissionedMUBLCOMsatellitewhichhadbeenin orbitsince1999.[26]BoththeDARTspacecraftandMUBLCOMsatelliteareshownin Figure1-2.TheDARTmissionresultedinamishap,whereasoftcollisionbetween DARTandMUBLCOMoccurredafterDARTexhaustedallofitspropellant.Propellant managementalongwithnavigationandcollisionavoidancemalfunctionscausedthetwo spacecrafttocollide,however,bothspacecraftsurvived.[27] Figure1-2.IllustrationofDARTmission 22

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AconcurrentmissionwithDARTwastheExperimentalSatelliteSystem-11 XSS-11in2005whichfolloweditspredecessor,theXSS-10.TheXSS-11missionwas tofurtherdemonstratecapabilitiesforperformingautonomousrendezvousandother APOswiththeupperstageofitsMinotaurIlaunchvehicle.[28,29]BothXSS-10and XSS-11aredepictedinFigure1-3BandFigure1-3A,respectively.TheXSS-11mission wassuccessfulincompletingrendezvousand75naturalmotioncircumnavigations. TheXSS-11spacecraftalsoconductedAPOswithseveralUS-owneddecommissioned satellites,however,theseresultsarenotreadilyavailableinthepublicdomain.[12,29] AIllustrationofXSS-10 BIllustrationofXSS-11 Figure1-3.IllustrationoftheXSSmissions TheOrbitalExpressDemonstrationSystemOEDSmissionin2007was successfulandalandmarkforAPOs.Thismissionwasabletodemonstrateautonomous roboticpayloadtransferandrecongurationofsatellites.[30]Themissionconsistedof aservicingsatelliteASTROandaserviceablesatelliteNextSat;bothinitiallyina dockedstate.BoththeASTROandNextSatsatellitesareshowninFigure1-4.ASTRO wasabletosuccessfullyperformAPOsonNextSatincludingrendezvousatdifferent ranges,capture,propellantandhardwaretransfer,whichwasamajormilestonefor 23

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autonomoustechnologyinspace.However,theAPOswereperformedatalowlevelof autonomy,whereseveralground-basedcommandswererequiredforOEDStocomplete anoperation.[30,31] Figure1-4.IllustrationofOEDSmission ThemostrecentmissionisthePrismasatellites,whichweredevelopedbythe SwedishSpaceCorporationandlaunchedonJune15,2010.[3234]ThePrisma satellitesconsistofanactivesatelliteMANGOandapassivesatelliteTANGO;both initiallyinadockedstate.MANGOandTANGOareshowninFigure1-5.Thismission willattempttoexperimentallyvalidatecertainalgorithmsandhardwareforAPOs.The maindirectiveistoperformaseriesofformationyingandrendezvousmaneuversat differentranges.Thespacecraftwillvalidatecollisionavoidancemaneuverswhenboth areinproximityofeachother.Thismissionwillalsogiveightheritagetothesmallest thrustersdevelopedtodate.[32] TherearefuturemissionsliketheSatellitefortheUniversalModicationof Orbits/Front-endRoboticEnablingNear-termDemonstrationSUMO/FREND,the AutonomousNanosatelliteGuardianforEvaluatingLocalSpaceANGELS,and themostrecentlyproposedventure,ViviSat.[3537]SUMO/FRENDisparticularly interestingsincedetailsofthetrajectoryplanningalgorithmsthatwillbeusedhavebeen publishedinthepublicdomain.[38]Eachspacecraftwillcarryoutspecicmissionsfor 24

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Figure1-5.IllustrationofPrismamission spaceapplicationpurposes,asopposedtotestingconceptsforAPOs.Thesemissions willalsobemilestonesfordeterminingthemostappropriateAPOmethodologies. 1.2StateoftheArt Despiterecentandfuturemissions,themosteffectivepath-planningmethodology forAPOsisstillundetermined.Whilethemethodsusedinthemissionsdiscussedare undisclosed,itisclearthatthedesiredlevelofautonomyhasnotbeendemonstrated. Toachievethislevelofautonomy,therearedifferentframeworksthathavebeen suggested.Oneframeworksuggestsdevelopingfastermicroprocessorssuchthat existingalgorithmsthatwouldnormallybeexecutedofinewouldbesimpliedand executedon-board.Anotherframeworksuggestsdevelopingnewalgorithmsthatare capableofbeingexecutedinhardwarethatisreadilyavailable.Whilebothframeworks arevalid,verifyingandvalidatingeitherrequiresightheritage. Giventhesuccessesand/orshortcomingsofthemissionsdiscussedintheprevious section,thequestionofwhetherrapidpath-planningalgorithmsneedtobedeveloped musttobeinvestigated.Whilealevelofautonomyhasbeendemonstratedtobe feasibleforin-spaceoperations,forthcomingdemandsrequirethatcertainmissions havethiscapabilityofrapidpath-planningforhigherlevelsofautonomy.Thisis particularlytruewiththeemergenceofsmallsatellitetechnologyandtheresponsive spaceinitiative.Withbothofthesetechnicalareas,successismeasuredfroma 25

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cheaper,faster,andgetsthejobdoneperspective.Sincesmallsatellitetechnology canberelativelycheaper,thiswouldbeanavenuethatcouldbetakentoultimately determinethemosteffectivepath-planningmethodsforAPOs. Ingeneral,path-planningforthetranslationalmotionofspacecraftisreferredto asorbitaltransfers.Forthiscase,adistinctionismadebetweenmaneuverplanning andtrajectoryplanning.Maneuverplanningreferstoaseriesofcontrolactionsfor pathswithlongtransfertimesontheorderoftheorbitalperiod.Thecontrolactions aretypicallyassumedtobeimpulsivesincetheengineburntimeisasmallfractionof thetransfertime.[39,40]Inaddition,thetrajectoryobtainedisnotasimportantasthe terminalconditionsofthetrajectory.Trajectoryplanningreferstoplanningaseriesof controlactionsforpathswithshorttransfertimesontheorderoffractionsoftheorbital period.Forthiscase,thecontrolactionsarenotnecessarilyassumedimpulsiveand thetrajectoryobtainedisrelevanttothemission.Path-planningfortherotationalmotion ofspacecraftistypicallysynonymouswithtrajectoryplanningsinceslewmaneuvers haverelativelyshorttransfertimes.Inthismanuscript,path-planningandtrajectory planningareusedinterchangeablysincethetransfertimesforAPOsaresmall. Tothisend,severalalgorithmshavebeenused/proposedforpath-planningof APOs.Thesealgorithmsarebifurcatedintotwogroups:optimizationmethodsand analyticmethods.Optimizationmethodsaremethodologiesderivedfromoptimal controltheoryandhistoricallyhavebeenstudiedtoagreaterextent.Someexamples ofoptimizationmethodsincludePrimerVectorTheoryPVT,celldecomposition methodsCDM,orthogonalcollocationmethodsOCM,theInverseDynamicsinthe VirtualDomainIDVDmethod,theGuidanceusingAnalyticSolutionGASmethod,to nameafew.Foranalyticmethods,thewordanalyticisusedinthesensethatthese methodshavelowcomplexityandthesolutionsareobtainedusingaxednumberof computations.Twoprominentanalyticmethodsaretheglideslopemethodandarticial potentialfunctionAPFmethod. 26

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1.2.1OptimizationMethods Optimalcontroltheoryisthedefaultapproachforcomputingoptimaltrajectories. [4143]Usingthecalculusofvariations,onecanmathematicallydeterminethe necessaryandsufcientconditionsforanoptimaltrajectorybasedontheconstraints onthesystemi.e.,dynamics,time,boundary,path,controleffort, etc ..Thestudyof optimalcontroltheoryappliedtoastrodynamicsparticularlyorbittransfersiscalled PrimerVectorTheoryPVT.[41,42]Theprimervectorisnothingmorethanthecostate associatedwiththevelocitystateofthesatellite.Itturnsoutthatsolutionstoseveral relevantoptimaltrajectoriesdependonthesolutiontotheprimervector.Despiteits elegance,PVTaswellasgeneraloptimalcontrolproblemsstillyieldsatwopoint boundaryvalueproblemTPBVPwithboththestatesandcostates,whichcanbe difculttosolve.[42,44]Thecoupledtranslationalandrotationalmotionhasalsobeen posedasanoptimalcontrolproblemOCPusingavariationalapproach.[38,45] CelldecompositionmethodsattempttofacilitatesolvingtheOCPbydiscretizing thestatespace.Asaresult,anexhaustivetreesearchisperformedtodeterminethe optimaltrajectory.[4648]Imposingadditionalsystemconstraintsisfairlyeasyandcan bedonewithoutaddingmuchcomplexitytothealgorithm.However,thecomputational costincreasesdramaticallyasthenumberofstatesincreases.Inaddition,itisdifcult todeterminewhetherthesolutionobtainedistheglobaloptimizer.Althoughthese exhaustivesearchmethodshavehighcomputationalcost,theyalwaysconvergetoa solutioniftheproblemiswell-posed. Oneofthemosteffectiveandcomputationallyefcientwaystosolveanoptimal controlproblemisusingorthogonalcollocationmethodsOCM.Thesemethods essentiallytranscribetheinnitedimensionalOCPtoanequivalentnitedimensional nonlinearprogramNLP.[49,50]ReducingthedimensionalityofanOCPusing anOCMgreatlyreducesthecomputationneededtoobtainasolutiontoanOCP. However,convergencetimesarestillindeterminateandthesolutionisonlyknownatthe 27

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collocationpoints.Moreover,itisdifculttodeterminewhetherthesolutionobtainedis theglobaloptimizeroftheoriginalOCP. TheIDVDmethodissimilartoanOCMexceptthedevisedNLPinvolvessolving forcoefcientsofasetofuser-denedbasisfunctions.Thisapproachgreatlyreduces thedimensionalityoftheNLPand,inturn,thetimeneededtocomputeasolution. Thesebasisfunctionsarechosenbasedonknownbehaviorofthedynamicalsystem andknownstructureoftheoptimalcontrol.Thetrajectoriesareobtainedinthevirtual domainusuallyanafnetransformofthetimedomainandthenmappedbacktothe timedomain.WhiletheIDVDmethoddoesnotprovideanexactsolutiontoanOCP, theshapeofthetrajectoriesobtainedaresimilartotheshapeofthetrajectoriesthat wouldbeobtainedfromsolvinganOCPassumingappropriatebasisfunctionsare chosen.Ithasalsobeenshownthatthesetrajectoriesarenear-optimalbasedontheir performanceindexvalues.[51] Aframeworkthatcanbeimplementedforanyoptimizationmethodisthereceding horizonapproach.[52]Thisapproachattemptstosegmenttheproblembyconsidering acertainhorizonoftimeratherthansolvingtheOCPfortheentiretimehorizon.As aresult,theOCPissolvedsequentiallyuntiltheterminalconditionsaremetatthe naltime.Theintentofthisapproachistoreduceconvergencetimes.However,since theproblemisbeingsolvedsequentially,itissuboptimalaccordingtothePrincipleof Optimality.[44] TheGASmethodisanadhocmethodthatreducesthedimensionalityofanOCP byemployingtherecedinghorizonframeworkandperforminganoptimizationonlyinthe timedomain.Thismethodrequiresthatananalyticsolutionbeknownforthedynamics ofthesystem.[53]Thisapproachgreatlyreducesthedimensionalityoftheproblem. However,anoptimizationproblemstillneedstobesolvediteratively.Whilethismethod obtainsasolutionwithrelativelysmallcomputationalload,theendresultissuboptimal duetotherecedinghorizonframework. 28

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1.2.2AnalyticMethods Therstandmostwidelyusedanalyticmethoddiscussedistheglideslope algorithm.[54]Thismethodiscommonlyusedsinceitcaneasilybeimplemented duetoitslowcomplexity.Thecornerstoneofthisalgorithmistheclassictwo-impulse rendezvoussolution.[55]Thisalgorithmiseffectiveyettheperformanceobtainedis suboptimal,sincethereisnoconsiderationofacostmetric.Thereisalsonocollision avoidancelogicinthealgorithmwhichdoesnotmakeitfavorableforAPOs. AnotheranalyticmethodistheAPFmethod,whichcanbethoughtofasan automatedwayofcomputingmaneuversfortheglideslopemethod.TheAPFmethod hasalsobeenconsideredasasingle-stepsolutiontoalocaloptimizationproblemi.e., innitesimallysmallrecedinghorizon,sinceonlylocalgradientinformationisusedto planamaneuver.[52]Ifthearticialpotentialisdenedwithcertaincharacteristics,then thesemaneuvershavebeenshowntoyieldfavorableconvergenceproperties.[5661]A collisionavoidancelogiccanbeincludedbyaugmentingtheAPFwitharticialpotentials representingavoidanceregions.TheAPFmethod,however,doesnotincludesystem dynamicsnoraperformancemetricandisthussuboptimal.Asaresult,thetrajectories generatedforAPOsarenotwelldened,whichisimportantwhenbeingusedona conservativesystemsuchasaspacecraft. 1.3TechnicalChallenges Certainconsiderationsneedtobetakenwiththealgorithmdevelopmentfor APOs.ThepriorityforanyAPOshouldbesafetyandconservationofthespace assetstheirresources.Therefore,analgorithmusedforAPOsshouldberobustto unmodeledeventsi.e.,rapidlygenerateneworcorrectedtrajectoriesandhavea collisionavoidancelogic.Inaddition,thealgorithmshouldoptimizethetrajectoriestobe cost-effectivei.e.,withrespecttocontroleffort,power,time,computation, etc. Achallengewithwhichoneisfacedwhendevelopingalgorithmsisthetradeoff betweencomputationalefciencyandoptimalityillustratedinFigure1-6.Ideally, 29

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analgorithmwouldhavethecomputationalefciencyofananalyticmethodwhile maintainingtheperformancecharacteristicsfromoptimizationmethods.Giventhat computationalefciencyplaysaroleinAPOs,itisdifculttocharacterizedifferent algorithmsbasedonasingleperformanceindex.Thus,whenconsideringalgorithms forAPOs,aposteriorancillaryperformanceindexmustbeconsideredwhichquanties thecomputationsperformedandtheconvergencetimetoobtainasolution.Measuring oatingpointoperationsisnotpracticalforoptimizationmethodssincetheirconvergence timesareindeterminate.[52,62]Moreover,theconservativecomputationenvironment onighthardwarerequiresthatpowerandsystemresourcesrequiredbetakeninto account.Inadditiontotheconstraintsonradiation-hardenedhardware,onlyafraction ofthecomputationresourceswouldbeavailablesinceotherightoperationsmustbe executedaswell. Figure1-6.Computationalefciencyvs.optimalitytradeoff Thechallengeofdevelopinganalgorithmthatisbothoptimalandcomputationally efcientisdaunting.ThedefaultsolutionisdeferringtoMoore'sLawandstandingby forighthardwarethatguaranteesreal-timeexecutionofexistingalgorithms.However, implementingexistingalgorithmson-boardighthardwarewouldresultinexcess computationssincetrajectorieswouldcontinuouslyneedtobeupdated.Usinganalytic methodologiesismorefavorablesincetrajectoriesareobtainedusingaxednumber 30

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ofcomputationsi.e.,functionalevaluations;whichisreadilyimplementablewith existinghardware.However,thetrajectoriesobtainedcanstillbeoptimizedandtheir robustnessneedstobeveried.Differentavenuesfordevelopingalgorithmsmustbe continuouslyexploredtodeterminethemostsuitablealgorithmsforAPOs.Theneedfor accurateabsoluteandrelativenavigationisanadditionalchallenge.[63]Itisessential tohavecontinuousknowledgeofthestatesandobjectsinproximityoftheautonomous spacecrafttoensureautonomy. 1.4ResearchScope Thedynamicsforbothtranslationalandrotationalmotionofarigidbodyorbiting theEartharepresentedinChapter2.Forbothtypesofmotion,thegoverningequations arepresentedalongwiththeenvironmentaldisturbancesexperiencedinorbit.Different parameterizationsfordescribingtheorientationofarigidbodyarethendiscussedalong withthekinematicsforeachparameterization. ThedevelopmentofahighdelitysimulationenvironmentisdiscussedinChapter3. ThisincludestheSimulinkmodelbasedonthedynamicmodelsdiscussedinChapter2. Detailedactuatormodelsaredevelopedforreactionjetsandreactionwheels.The reactionjetsarethelinearmomentumexchangedevicesusedtoeffectanorbital maneuver.Reactionwheelsaretheangularmomentumexchangedevicesusedto effectachangeinorientation.ThesimulationenvironmentislaterusedinChapter8 tocharacterizeoptimaltrajectoriesandtodeterminehowtheAPFandAAPFmethods performunderhigherdelitydynamics. Chapter4discussessomeofthepropertiesassociatedwiththepertinentoptimal trajectoriesforAPOs.Namely,thesetrajectoriesareminimumtimetrajectories, xedtimeminimumcontrolefforttrajectories,andnitehorizonlinearquadraticLQ trajectories.ItisalsoshowninChapter4thatsolvinganitehorizonLQproblem reducestosolvinganalvalueproblemwiththeDifferentialRiccatiEquationDRE. TheoptimaltrajectoriesforrendezvousandslewmaneuversarecomputedinChapter4. 31

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Arendezvousmaneuverwithapathconstraintrepresentativeofanobstacleisalso computed. TheAPFalgorithmisdiscussedinChapter5andtheadaptivearticialpotential functionAAPFmethodispresentedinChapter6.TheAAPFmethod,whichisa modicationoftheAPFmethod,isdevelopedtoexploitthecomputationalefciency oftheAPFmethodandincreaseitsoptimalitybychoosingatimedependentformof thearticialpotentials.AstabilityanalysisisperformedforboththeAPFandAAPF methodsinChapter6.TwonumericalexamplesareprovidedforboththeAPFand AAPFmethodstodemonstratebothalgorithms'effectiveness. Chapter7discussesmethodsforsolvingnitehorizonLQOCPs.Thesolution foranLQOCPwithalineartimeinvariantLTIHamiltonianmatrixisobtainedusinga statetransitionmatrixSTMrepresentation.Twonewmethodologiesaredeveloped forobtainingthesolutiontoaLQOCPwithalineartimevaryingLTVHamiltonian matrixusing:thePicardIterationPIandtheHomotopyContinuationHC.A numericalexamplewithaLTVsystemisthensolvedusingthePIandthreedifferent HCmappings.Anexamplerepresentinganalapproachscenarioissolvedusingthe Yamanaka-Ankerson-Tschauner-HempelYATHrelativemotionmodelusingtheHC. Chapter8presentstheresultsfromthenumericalanalysesperformed.First, theresultsoftwoMonteCarlosimulationsarepresentedtoverifythattheAAPFhas improvedperformanceandconvergencecharacteristicsovertheAPFmethod.Next, theresultsobtainedfromthesimulationsperformedusingthehighdelitymodel arepresented.TheoptimaltrajectoriesobtainedinChapter4forboththerotational andtranslationalproblemsaretrackedinthehighdelitysimulationenvironmentto determinewhetherthetrajectoriesarefeasibleandtocharacterizeanyperformance degradation.TheAPFandAAPFmethodsarealsoimplementedinthehighdelity simulationenvironmenttodeterminehowthesemethodsperforminahighdelity model.Asetofresultsforrendezvouswithobstacleavoidanceisalsopresented,where 32

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trackingatrajectoryandtheAPFandAAPFmethodsareimplemented.Lastly,anal approachscenarioissimulatedbyusinganitehorizonLQcontrollawinthehigh delitymodel.Finally,Chapter9presentsconclusionsfromtheanalysesdoneinthe previouschapters. 33

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CHAPTER2 SYSTEMDYNAMICS Thedynamicsequationsgoverningthecoupledsixdegrees-of-freedomDOF motionofaspacecraftinorbitarediscussedinthischapter.Thespacecraftisassumed tobearigidbody,thusthemotionofarigidbodycanbebifurcatedintotranslational motionandrotationalmotion.Thedynamicsassociatedwithbothmotionsarediscussed alongwiththeenvironmentaldisturbancesthataffectanobjectinorbit.Thedynamics equationsareusedintheChapter3tomodelthespacecraftanditsenvironmentandin Chapter4asthedynamicsconstrainttoobtainoptimaltrajectories. 2.1OrbitalMechanics Assumingasphericallyhomogeneouscentralbodyi.e.,Earth,whichissignicantly largerthantheorbitingbodyi.e.,spacecraft,themotionofthesmallerbodywith respecttothelargerbodycanbemodeledas r + k r k 3 r = f + a d where r isthepositionofthespacecraftrelativetothecenterofmassCMoftheEarth, istheEarth'sgravitationalparameter, f isthecontrolactioni.e.,specicthrust appliedbythesatellite,and a d isthesumofdisturbingaccelerationsactingonthe satellite.When a d = 0 ,thisisknownastheKeplerianmodel.[39,40]Thecoupling betweenrotationalmotionandtranslationalmotionisintroducedin f and a d since thesetermsdependontheorientationofthespacecraft.Thedisturbingaccelerations arecausedbytheenvironmentexperiencedinorbit.Inparticular,thedisturbances discussedarehigherordergravitationaleffects,atmosphericdrag,solardrag,andthird bodyeffectsi.e.,gravitationaleffectsfromtheMoonandSuninparticular. 2.1.1ZonalHarmonics ModelingtheEarthasadistributedmasssystemwhichisnotspherically homogeneous,thegravityeffectoftheEarthcanbedeterminedasthenegative 34

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gradientofgravitypotentials.Usingsphericalharmonicszonal,sectoral,andtesseral tosectiontheEarth,thegravitypotentialofthezonalharmonicswhichhavethe dominatingdisturbingeffectis V r = k r k 1 )]TJ/F29 7.9701 Tf 16.355 14.944 Td [(1 X k =2 J k R k r k k P k cos gc # where J k istheempiricallydeterminedconstantforthe k th zonalharmonic, R isthe equatorialradiusofEarth, P k isthe k th orderLegendrepolynomial,and gc isthe geocentriclatitudeofthesatellite.[39,40]Thenegativegradientofthispotentialyields theeffectivegravitationalspecicforcefromthezonalharmonicsonthespacecraft. Notethatthepotentialtermoutsideoftheseriesyieldsthegravitytermfortherestricted two-bodymodeli.e.,sphericallyhomogeneousEarth.Thesectoralandtesseral harmonicsareomittedsincethezonalharmonicshavethelargestdisturbingeffect i.e.,thesphericallyhomogeneousEarthtermandthe J 2 effect.Infact,the J 2 effect isupto 1000 timesgreaterthanthenextmostdominatingeffect.[39]Thedecreasein gravitationaleffectcanalsobeseeninthesubsequentzonaltermssincewhen k !1 then R k r k k 0 .The J 2 effectrepresentedintheEarth-CenteredEarth-FixedECEF frameis a J 2 = )]TJ/F22 11.9552 Tf 10.494 8.448 Td [(3 J 2 R 2 2 r 5 r I 1 )]TJ/F23 7.9701 Tf 13.15 6.479 Td [(5 r 2 K r 2 r J 1 )]TJ/F23 7.9701 Tf 13.151 6.479 Td [(5 r 2 K r 2 r K 3 )]TJ/F23 7.9701 Tf 13.151 6.479 Td [(5 r 2 K r 2 T where r I r J ,and r K arethecomponentsof r representedintheECEFframe.Forthe purposesofmodeling,therstsixzonalharmonicsareusedpp.550-552of[39]. 2.1.2AerodynamicDrag Thesurfacesofthespacecraftarediscretizedinto n facestodeterminethe aerodynamicdrag.Foraconvexspacecraftstructure,theaggregateaerodynamic dragis a ad = )]TJ/F31 11.9552 Tf 11.291 20.443 Td [( 1 2 h m k v rel k 2 n X i =1 i A i C D i ^ n i ^ v rel ^ v rel 35

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where m isthemassofthespacecraft, h istheairdensityasafunctionofaltitude h A i istheareaofface i C D i isthedragcoefcientofface i n i istheunitvectornormalto face i ,and i isdenedbelow.[39,64] i = 8 > > < > > : 1 if ^ n i ^ v rel > 0 0 otherwise Anillustrationofthevectors n i isshowninFigure2-1foraspacecraftwithacube structure.Thecouplingbetweentranslationalandrotationalmotionisevidentinthis disturbancesincetheorientationofthespacecraftdeterminestheaerodynamicdrag. Likewise,theaerodynamicdragmaycauseamomentwhichaffectstherotational motionofthespacecraft.Theparameter ^ v rel istheunitvectorparallelto v rel ,whichisthe relativevelocitybetweenthesatelliteandtheEarth'satmosphere.Itisassumedthatthe atmospherehasthesamerotationalvelocityastheEarth.Asaresult, v rel isdenedas v rel =_ r )]TJ/F46 11.9552 Tf 11.955 0 Td [(! r where istherotationrateoftheEarth. Figure2-1.Surfacediscretizationofspacecraftwithunitnormalvectors 36

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Differentmodelsexistfordescribingtheairdensityasafunctionofheight.The modelusedistheExponentialAtmosphereModel,whichissimpleyetfairlyaccuratefor altitudes h 1000 km.Usingthismodel,theairdensityisdenedas h = 0 exp )]TJ/F40 11.9552 Tf 10.494 8.088 Td [(h )]TJ/F40 11.9552 Tf 11.955 0 Td [(h 0 H where h 0 0 ,and H aretabulatedparametersandalsodependonthethealtitude. Valuesfortheseparameterscanbefoundin[39]. 2.1.3SolarDrag Usingthesamediscretizationofthesurfacesofthespacecraft,theaggregatesolar dragis a sd = )]TJ/F31 11.9552 Tf 11.291 20.443 Td [( SP m n X i =1 i A i C R i ^ n i ^ r sun ^ r sun where SP =4.57 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(6 N = m 2 isthemeansolarpressure, ^ r sun istheunitvectorpointing fromtheSuntothesatellite'sCM, C R i isthecoefcientofreectivityofface i ,and i is denedasbelow.[39,64] i = 8 > > < > > : 1 if ^ n i ^ r sun > 0 0 otherwise Thecouplingbetweentranslationalandrotationalmotionisevidentinthisdisturbance aswell.Itshouldbenotedthatthisdisturbanceisonlyactivewhenthespacecraftisnot ineclipse.[39,64]Theconditionfordeterminingwhetheraspacecraftisineclipseis sin )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 R k r k cos )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r k r k ^ r sun 2.1.4ThirdBodyDisturbances Thirdbodydisturbancesaregravitationaleffectscausedbyobjectsotherthanthe Earth.Inparticular,theobjectsthathavethemostinuenceonanEarthspacecraftare theSunandtheMoon.[39]ThegravitationaleffectoftheSunonabodyorbitingthe 37

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Earthis a = R )]TJ/F45 11.9552 Tf 11.955 0 Td [(r k R )]TJ/F45 11.9552 Tf 11.955 0 Td [(r k 3 )]TJ/F45 11.9552 Tf 21.594 8.088 Td [(R k R k 3 where isthegravitationalparameteroftheSunand R isthepositionoftheSun relativetotheEarth'scenter.ThegravitationaleffectoftheMoononabodyorbitingthe Earthis a $ = $ R $ )]TJ/F45 11.9552 Tf 11.955 0 Td [(r k R $ )]TJ/F45 11.9552 Tf 11.955 0 Td [(r k 3 )]TJ/F45 11.9552 Tf 21.594 8.541 Td [(R $ k R $ k 3 where $ isthegravitationalparameteroftheMoonand R $ isthepositionofthemoon relativetotheEarth'scenter.ThemodelsusedfordeterminingthepositionoftheSun andMoonasafunctionoftimearepresentedin[39]. 2.2AttitudeDynamics TherotationalmotionofarigidbodycanbemodeledusingEuler'ssecondlaw. [64,65]TheangularmomentumofarigidbodyaboutitsCMis H = J where J isthecentroidalinertiadyadicoftherigidbodyand istheangularvelocity relativetoaninertialreferenceframe.Thus,thetimerateofchangeoftheangular momentumisequaltothesumofexternaltorques d H dt = J + J = + d where isthecontroltorqueand d isthesumofexternaldisturbingtorquesacting onthesatellite.Duringight,aspacecraftissubjectedtoexternaldisturbancetorques whichaffectitsmotion.Forcompleteness,thedisturbingtorquesdiscussedarethe gravitygradienttorque,aerodynamictorque,andsolartorque.Adetailedexpos eof thesedisturbancescanbefoundin[39,64]. 38

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2.2.1GravityGradientTorque Agravitygradienttorqueisexperiencedwhenabodyisnotsymmetricaboutany axisanddoesnothaveahomogeneousmassdistribution.Thegravitygradienttorque isdenedas gg =3 k r k 3 ^ c J ^ c where ^ c istheunitvectorinthenadirdirection. 2.2.2AerodynamicTorque Anaerodynamictorqueresultsfromtheaerodynamicdraginequation2,which doesnotactalongtheCMofthespacecrafti.e.,theCMandgeometriccenterare distinct.Forthesecases,theaerodynamictorqueismodeledas at = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(1 2 h m k v rel k 2 n X i =1 i A i C D i ^ n i ^ v rel 2 ^ v rel r cp i where r cp i isthevectorfromtheCMtothecenterofpressureofface i .Specically,this vectorisdenedas r cp i = l i ^ n i )]TJ/F45 11.9552 Tf 11.955 0 Td [(r cm where l i isthedistancefromthegeometriccentertothecenterofpressureofface i and r cm isthepositionofthespacecraft'sCMrelativetothegeometriccenterofthe spacecraft.[39,64] 2.2.3SolarTorque Solarwindhasaneffectsimilartothatofatmosphericwind.Thus,solardrag producesatorquewhenthecenterofpressureisdistinctfromtheCMForthesecases, thesolartorqueismodeledas st = )]TJ/F40 11.9552 Tf 10.494 8.088 Td [(SP m n X i =1 i A i C R i ^ n i ^ r sun 2 ^ r sun ^ r cp i 39

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2.3AttitudeRepresentations Numerousparameterizationsexisttodescribetheorientationofarigidbody. [66]First,theclassicalEuleranglesarediscussed.TheEuleranglerepresentation isusedinAppendixAtoderiveasetofequationsforrelativerotationalmotion.Next, theaxis-anglerepresentationandtheunitquaternionarediscussed.Theseattitude representationsareintimatelyrelatedandareusedtodenerelativeorientationsand thescalarmetrictomeasureforrelativeorientationerrors. 2.3.1EulerAngles Theorientationofonecoordinatesystemrelativetoanothercanbedescribed usingatmostasequenceofthreeEulerrotations.[66,67]AnEulerrotationisarotation aboutoneoftheaxesthatdenestheorthonormalbasisofthecoordinatesystem.This rotationisrepresentedbyamatrixoperation.Thematrixthatrotatesavectoraboutthe rstaxisis C = 2 6 6 6 6 4 100 0cos sin 0 )]TJ/F22 11.9552 Tf 11.291 0 Td [(sin cos 3 7 7 7 7 5 aboutthesecondaxisis C = 2 6 6 6 6 4 cos 0 )]TJ/F22 11.9552 Tf 11.291 0 Td [(sin 010 sin 0cos 3 7 7 7 7 5 andaboutthethirdaxisis C = 2 6 6 6 6 4 cos sin 0 )]TJ/F22 11.9552 Tf 11.291 0 Td [(sin cos 0 001 3 7 7 7 7 5 Consequently,arotationmatrixrelatingvectorrepresentationsinframes A and B isdenedbyarotationsequenceaboutthe i j ,and k axeswithangles ,and 40

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respectivelyasseeninequation2.[6567] C BA = C k C j C i Thus,avectorrepresentedinframe A cannowberepresentedinframe B as B v = C BA A v ThekinematicsoftheEuleranglescanbederivedbysummingalltherotationrates representedinthesamecoordinatesystem.[66]Thisresultsintheexpression = S , 2 6 6 6 6 4 _ 3 7 7 7 7 5 wherethedenitionof S , dependsontherotationsequenceused. 2.3.2Axis-Angle Thesamerotationmatrixcanbederivedbyperformingasinglerotationaboutaunit vector ^ e i.e.,eigenaxisbyanangle i.e.,eigenangle,andisdenedas C BA = C ^ e =cos I + )]TJ/F22 11.9552 Tf 11.955 0 Td [(cos ^ e ^ e T )]TJ/F22 11.9552 Tf 11.955 0 Td [(sin ^ e where e istheskewoperatorandisdenedforanarbitrarycolumnmatrix a = a 1 a 2 a 3 T below.[65,66] a = 2 6 6 6 6 4 0 )]TJ/F40 11.9552 Tf 9.298 0 Td [(a 3 a 2 a 3 0 )]TJ/F40 11.9552 Tf 9.299 0 Td [(a 1 )]TJ/F40 11.9552 Tf 9.299 0 Td [(a 2 a 1 0 3 7 7 7 7 5 41

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Thekinematicsforthisparameterizationaregivenbelow.[66] =^ e ^ e = 1 2 ^ e )]TJ/F22 11.9552 Tf 11.955 0 Td [(cot 2 ^ e ^ e Thisrepresentationisusefulsincetheanglecanbeusedasascalarmetricforthe relativeorientationbetweentwoorientations. 2.3.3UnitQuaternion Aunitquaternionisasetoffourparametersusedtorepresenttheorientationofa rigidbody q = 2 6 4 3 7 5 where isthevectorcomponentand isthescalarcomponent.Theunityconstraint requiresthat q T q =1 .Unitquaternionsarerelatedtotheaxis-anglerepresentationby =sin 2 ^ e =cos 2 TherelationshipbetweenEuleranglesandquaternionsisnotdirectandcanbefoundin [67]or[66]. Givenaquaternionthatrelatescoordinatesystem A tocoordinatesystem B ,the rotationmatrixusingthisquaternionisdenedas C BA q = T q q 42

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where q and q aredenedbelow.[66,68] q = 2 6 4 I )]TJ/F46 11.9552 Tf 11.956 0 Td [( )]TJ/F46 11.9552 Tf 9.298 0 Td [( T 3 7 5 q = 2 6 4 I + )]TJ/F46 11.9552 Tf 9.298 0 Td [( T 3 7 5 Afavorableattributeofquaternionsisthattheinverserelationshipbetweentherotation matrixandthequaternionisnonsingular.Thismeansthatauniquequaternioncanbe extractedfromtherotationmatrixwhichrepresentstheoriginalorientation.[66]Another favorableattributeisthatthequaternionkinematicsarebilinearinthequaternionand theangularvelocityandarerepresentedas q = 1 2 q = 1 2 q where = 2 6 4 )]TJ/F46 11.9552 Tf 9.298 0 Td [(! )]TJ/F46 11.9552 Tf 9.299 0 Td [(! T 0 3 7 5 Quaternionsareparticularlyusefulsinceanerrorquaternionbetweentwo orientationscanbedened.Todeneanerrorquaternion,thequaternionproduct operationisrstdenedas q 1 q 2 = q 1 q 1 q 2 = q 2 q 2 q 1 Asaresult,thequaternioninverseisdenedas q )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = 2 6 4 )]TJ/F46 11.9552 Tf 9.299 0 Td [( 3 7 5 43

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suchthatthequaternionproductofaquaternionanditsinverseisthezeroquaternion 0 T 1 T .Thisway,theerrorquaternionis q e = q 1 q )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 sincethezeroquaternionisobtainedwhen q 1 and q 2 areequivalent.Anerrorangleis alsodenedbasedontherelationshipbetweentheaxis-anglerepresentationandunit quaternions e =2cos )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e where e isthescalarcomponentoftheerrorquaternion.Thiserroranglerepresentsa scalarmetricfortherelativeorientationbetweentwoorientationsdescribedby q 1 and q 2 Inconclusion,thedynamicsassociatedwithbothtranslationalandrotational motionofaspacecraftinorbitarepresentedinthischapter.Thetranslationalmotion includesdisturbancesfromtheEarth'soblateness,aerodynamicandsolardrag,and thirdbodygravitationaleffects.Therotationalmotionincludesdisturbancesfromthe gravitygradientandaerodynamicandsolardrag.Differentsetsofparameterizations forrepresentinganorientationarealsodiscussed.Thedynamicsdiscussedinthis chapterareusedtodevelopahighdelitymodelofaspacecraft.Theyarealsousedfor obtainingoptimaltrajectorieswhileneglectingdisturbancesforrendezvousandslew maneuvers. 44

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CHAPTER3 SYSTEMMODELING ThehighdelitysimulationenvironmentdevelopedinSimulinkisdiscussedinthis chapter.Thesimulationenvironmentisusedforcharacterizingperformanceofdifferent algorithmsbyapplyingtheminthehighdelitymodelofthespacecraft.Thesimulation environmentincludesmodelsfortwospacecrafti.e.,chaserandtarget,whereeach spacecraftismodeledusingthedynamicsdiscussedinChapter2.Thechasersatellite hasthrustersfortranslationalcontrolandreactionwheelsforattitudecontrol,wherea modelforthesedevicesisalsodiscussedinthischapter.Thesemodelsareimportant sincetheactuatordynamicsdeterminewhetherthecommandsfromacontrollerare realizable.ThehighlevelSimulinkdiagramofthesimulationenvironmentisshownin Figure3-1. Figure3-1.HighlevelSimulinkdiagram 3.1Target/ChaserPlant Thesatellitegeometryforboththetargetandchaserisbasedonacubestructure withsidelength l =1 m.Thestructureisimportantbecauseitdictatestheeffectsof thedisturbancesexperienced.Theactuatorgeometryassumesunidirectionalthrusters oneachfacethathavealineofactionalongthecenterofmassCMofthespacecraft. 45

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Figure3-2illustratesthearrangementofthesixthrustersinthebodyframe.The reactionwheelsystemissuchthatthespinaxisofeachreactionwheelisalignedwith eachaxisofthebodyframe. Figure3-2.Geometryofspacecraftandreactionjets Itshouldbenotedthatthemassisnotmodeledasavariablequantity.Inreality,the massisvariablesincethereactionjetsexpendfueltoproduceaforce.[69]Thisextra degreeoffreedomdoeshaveconsiderableeffectonthedynamics,however,itisdifcult todeterminehowtomodelthiswithouthavingapreliminarydesignofthespacecraft. Havingavariablemassandinertiamatrixaffectsthemagnitudeofsomedisturbing forcesandtheattitudedynamics.Thereactionjetsarealsoaffectedsincetheywould havealimitedsupplyoffueltoburn.[69] Themotionofbothspacecraftismodeledusingtheequationsdiscussedin Chapter2.Particularly,thetranslationalmotionismodeledusingtherestrictedtwo-body modelwhileincludingdisturbances.AmodeloftheSunandMoonpositionisgivenin [39]asafunctionoftheJulianDate,whichdeterminesthesolardrag,solartorque,and thirdbodygravitationaleffects.TherotationalmotionismodeledusingEuler'ssecond lawwhileincludingthedisturbingtorques.Asaresult,thesatelliteplantisgroupedinto 46

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oneblockasshowninFigure3-3,wheretheonlyinputsaretheforceandtorquefrom theactuators. Figure3-3.Simulinksatellitemodelblock 3.2ActuatorDynamicsandModel Theactuatorschosenforthechaserarelow-thrustreactionjetsfororbital maneuversandreactionwheelsforattitudemaneuvers.Thedynamicsandconstraints forboththereactionjetsandthereactionwheelsarediscussedincludinganydelays, saturationlimits,and/ordeadbandlimits.Modelingactuatorsassistindetermining whetherthecommandedactionsbyaparticularcontrollerarerealizable. 3.2.1ReactionJets Reactionjetsarelinearmomentumexchangedevicesthatgeneratetheforce requiredforperformingorbitalmaneuvers.Dependingonthecongurationofthejets onthespacecraft,theycanimpartaforceand/ormoment.Ingeneral,therelationship betweeneachindividualreactionjetforce,andtheresultantforceandmomentcanbe expressedas 2 6 4 F M 3 7 5 = Lf where F and M aretheforceandmomentexpressedinthebodyxedframe, respectively, L isthecongurationmatrixthatdetermineseachindividualreaction jet'scontributiontotheforceandmoment,and f isacolumnmatrixcontainingtheforce magnitudesfromeachreactionjet. 47

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ThecongurationofthereactionjetschosenisillustratedinFigure3-2.This congurationischosenforsimplicityandtoavoidcontrollabilityissues.Thematrix L for thiscongurationmatrixisdecomposedas L = 2 6 4 L 1 L 2 3 7 5 where L 1 = 2 6 6 6 6 4 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(10000 001 )]TJ/F22 11.9552 Tf 9.299 0 Td [(100 00001 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 3 7 7 7 7 5 and L 2 = 0 sincethereactionjetforcesactalongtheCManddonotproducea moment.Thus,givenacommandedforce F comm ,theindividualreactionjetforcescanbe determinedusingthemin-normsolution f = L T 1 L 1 L T 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 F comm Toenforcetheunidirectionalconstrainti.e.,reactionjetscanonlyproducepositive forces,thefollowingconditionsareenforcedonthemin-normsolution f 2 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = 8 > > < > > : f 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + j f 2 i j if f 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 0 and f 2 i 0 0 otherwise for i =1,2,3 f 2 i = 8 > > < > > : f 2 i + j f 2 i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 j if f 2 i 0 and f 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 0 0 otherwise for i =1,2,3 Thus,theactualforcefromthereactionjetsolution f = f 1 f 2 f 3 f 4 f 5 f 6 T 48

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is F act = L 1 f Theinherentdelaysanddynamicsassociatedwiththechemical,electrical,and mechanicalprocessesofthejetsaremodeledaswell.Anexampleofatypicalthrust proleisshowninFigure3-4.[64]First,aforceiscommandedattime t 0 .However, duetotheprocessesrequiredforturningonthereactionjetthereisadelayandis insteadturnedonattime t 1 .Next,theforcefromthereactionjetmustrampuptothe commandedvaluewhichisachievedattime t 2 .Thesamedelayisseenwhentheforce commandedischangedattime t 3 yetthechangebeginsattime t 4 .Thethrustthen rampsdowntothenewcommandedvalueat t 5 .Thevaluesforthesedelays,growth, anddecaytimesrangefromafewmillisecondstohundredsofmilliseconds.[64]The blockdiagraminFigure3-5illustrateshowthisthrustproleisachieved,wherethedelay usedis 1 msandthetransferfunctionusedis G s = 100 s +100 whichyieldsagrowthanddecaytimeofabout 100 ms. Figure3-4.Thrustproleofreactionjets 49

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Figure3-5.Blockdiagramtoachievethrustprole Asaturationanddeadbandlimitisimposedoneachreactionjetaswell.Figure3-6 illustratesagenericcommandedsignalthatcontainsboththeselimitsandtheresulting achievablethrustprole.Inordertoachievethisthrustprole,athree-partswitchis usedtodistinguishbetweenthethreedifferentcasesandisillustratedinFigure3-7. FromFigure3-7,therstcasecorrespondstothesaturationlimitbeingactive,the secondcaseiswhenthedeadbandlimitisactive,andthethirdcaseiswhenneitheris active.Figure3-7alsoshowsthedelayandthetransferfunctiondiscussedpreviously. Thesaturationlimitanddeadbandlimitsusedare f max = p 3 Nand f db =0.01 N, respectively. Figure3-6.Illustrationofsaturationanddeadbandlimits 3.2.2ReactionWheels Reactionwheelsareangularmomentumexchangedevicesthatdistributeangular momentumofthesatellitebyspinningupordownthereactionwheels.Initially,these 50

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Figure3-7.Switchtoimposesaturationanddeadbandlimits devicescontainaportionofthesatellite'sangularmomentum.Sinceangularmomentum isconserved,whenthereactionwheelsarespunupordown,theangularmomentum oftheremainingcomponentsofthesatellitehastobeadjustedtomaintainatotal constantvalue.[64,70]Asaresult,theorientationofthesatellitechanges.Ingeneral, theangularmomentumofareactionwheeldevicecanbeexpressedasthesumof angularmomentaofthereactionwheels h = N X i =1 C i C i C i 2 6 6 6 6 4 I fw i i 0 0 3 7 7 7 7 5 where i istheseparationangle, i istheinclinationangle, i isthegimbalangle, I fw i isthemomentofinertiaofthereactionwheelaboutitsspinaxis,and i istheangular velocityofthereactionwheel.Therotationmatricesexpresstheangularmomentum ofeachreactionwheelinthebodyxedframe.Theonlytime-varyingquantityinthis expressionistheangularvelocitiesofthereactionwheels.Thus,thetimederivativeof 51

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theangularmomentumofthereactionwheeldeviceis h = N X i =1 C i C i C i 2 6 6 6 6 4 I fw i i 0 0 3 7 7 7 7 5 where i istheangularaccelerationofthereactionwheelwhichproducestheeffectofa torque. Typically,thegeometryofthereactionwheelsystemusedhaseachreaction wheel'sspinaxisalignedwitheachprincipalaxisoftheprincipalbodyxedframe i.e.,threereactionwheels.Ifaredundantwheelisused,itistypicallyskewedsuch thatitsdirectionisalongthediagonaloftheprincipalbodyxedframedirections.The congurationusedinthesimulationenvironmentisathreereactionwheelattitude controlsystemwitheachreactionwheel'sspinaxisalignedwitheachoftheaxesof thebodyxedframe.Asaresult,theangularmomentumofthereactionwheelsystem expressedinthechaser'sbodyxedframeis h = 2 6 6 6 6 4 I fw ,1 00 0 I fw ,2 0 00 I fw ,3 3 7 7 7 7 5 2 6 6 6 6 4 1 2 3 3 7 7 7 7 5 h = I fw Likewise,thederivativeoftheangularmomentumisexpressedas h = I fw Toderivethecontrolmethodologytorealizeacommandedtorqueusingareaction wheelsystem,therotationalequationsofmotionarerevisited.Theangularmomentum ofaspacecraftaboutitsCMisexpressedas H = J + h 52

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wherethecentroidalinertiamatrix J doesnotincludethereactionwheels.Assuming noexternaltorquesareactingonthesatellitei.e.,nodisturbingtorques,thenthetotal angularmomentumisconserved.Thus,theinertialtimederivativeofthetotalangular momentumis H = J + h + J + h = 0 J + J = )]TJ/F22 11.9552 Tf 10.909 2.656 Td [(_ h )]TJ/F46 11.9552 Tf 11.955 0 Td [(! h Coulombandviscousfrictionandstictionareconsiderabletorquesthataffectthe performanceofthereactionwheels.[64]Stictionisnoteasilymodeledandisignored sinceitonlyaffectsthereactionwheelswhentheyareatlowspinrates.Thefriction modelusedonthereactionwheelsisthesumofcoulombandviscousfriction friction = c sgn + v where c istheCoulombfrictioncoefcientand v istheviscousfrictioncoefcient. Recallingequation3,theinternaltorquefromthereactionwheelsisobtainedas comm )]TJ/F46 11.9552 Tf 11.955 0 Td [( friction = )]TJ/F22 11.9552 Tf 10.909 2.657 Td [(_ h )]TJ/F46 11.9552 Tf 11.955 0 Td [(! h where comm isthedesiredtorquefromthereactionwheelsystem.Givenanattitude controller comm andfeedbackofreactionwheelangularvelocities,thereaction wheels'angularaccelerationthatwouldproducethecommandedtorqueisfound usingequation3andequation3,andis = I )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 fw )]TJ/F46 11.9552 Tf 9.299 0 Td [( comm + friction )]TJ/F46 11.9552 Tf 11.955 0 Td [(! I fw Theparametersandconstraintsusedforthereactionwheeldeviceareshownin Table3-1.Theseincludesaturationlimitsplacedontheangularvelocityandangular accelerationofthereactionwheels. 53

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Table3-1.Reactionwheelsystemparametersandconstraints ParameterValueUnits I fw i 0.625 kg m 2 c 7.06 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 N m v 1.16 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 N m = rad = s max 100 rad = s 2 max 524 rad = s Inthischapter,asimulationenvironmentdevelopedtomodeltwospacecraftin closeproximityisdiscussed.Includedisaplantmodelforboththechaserandtarget spacecrafts.Thecoupledsixdegrees-of-freedomdynamicsdiscussedinChapter2 areusedforbothspacecraft.Thedynamicsandconstraintsassociatedwiththe actuatorsonthechaserarealsodiscussed.Thismodelislaterusedtodetermine whetheroptimaltrajectoriesarerealizableandtocharacterizedifferencesbetween theoreticalperformanceandactualperformance.Itisalsousedtodeterminewhether thealgorithmsdevelopedusinglinearizedmodelsarestillvalidinahigherdelitymodel. 54

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CHAPTER4 OPTIMALTRAJECTORIES SolutionstoanoptimalcontrolproblemOCPcanbeusedforobtainingoptimal trajectories.Inthischapter,ageneralformofanOCPispresented.TheOCPsof particularinterestthatarediscussedaretheminimumtimeOCP,xedtimeminimum controleffortOCP,andnitehorizonlinearquadraticLQOCP.Minimumtimeandxed timeminimumcontrolefforttrajectoriesaretypicallyusedforclose-rangerendezvous. FinitehorizonLQtrajectoriesaretypicallyusedforthenalapproachorendgameof aproximityoperation.ItisshownthatsolvinganitehorizonLQOCPisequivalent tosolvinganalvalueproblemwiththeDifferentialRiccatiEquationDRE.Finally, thethreepertinentoptimaltrajectoriesforboththetranslationalandrotationalmotion arecomputed.Inaddition,axedtimeminimumcontroleffortproblemissolvedwhich includesanobstacle. 4.1OptimalControlProblem Withoutlossofgenerality,thecontinuoustimeOCPisposedas min u 2 R m J = x t 0 x t f t 0 t f + Z t f t 0 L x t u t t dt subjectto x t = f x t u t t x t 0 x t f t 0 t f = 0 c x t u t t 0 where : R n R n R R R isthecostincurredfromtheboundaryconditions, L : R n R m R R istheLagrangian, f : R n R m R R n isthederivativeconstraint i.e.,dynamicson x : R n R n R R R k isthecolumnmatrixofconstraintsat theboundaryconditions,and c : R n R m R R l l m isthecolumnmatrixof stateand/orcontrolconstraints.[41,43,44]Itisimportanttonotethatthelessthanor 55

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equaltoin c x t u t t 0 andwhenusedwithvectorsormatricesintherestof thischapterisacomponent-wiseoperator. Thecalculusofvariationsisusedtodeterminethenecessaryandsufcient conditionstosolveanOCP.[41,44]First,anaugmentedcostfunctionisdenedas J a =+ T + Z t f t 0 L + T f + T c dt where 2 R k 2 R n ,and 2 R l arethecostatesassociatedwiththeconstraints f and c ,respectively.Thecostate hastheproperty 0 if c = 0 = 0 if c 0 Asaresult,theHamiltonianisdenedastheintegrandoftheaugmentedcostas H = L + T f Thenecessaryconditionsforoptimalityaregiveninequation4andthetransversality conditionsaregiveninequation4.[43,44] x = @ H @ T = )]TJ/F31 11.9552 Tf 11.291 16.857 Td [( @ H @ x T 0 = @ H @ u T t 0 + @ @ x t 0 T + @ @ x t 0 T = 0 if x t 0 isunspecied )]TJ/F46 11.9552 Tf 9.298 0 Td [( t f + @ @ x t f T + @ @ x t f T = 0 if x t f isunspecied 56

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H t 0 + @ @ t 0 + @ @ t 0 T =0 if t 0 isunspecied H t f + @ @ t f + @ @ t f T =0 if t f isunspecied. Notethatthereisnoexplicitconditionforthecostateassociatedwiththestateand/or controlconstraintsi.e., .Thiscostateisdeterminedbyequation4whenthe constraintisinactivei.e., c < 0 orbyequation4andequation4simultaneously whentheconstraintisactivei.e., c = 0 .[43] Whenstateand/orcontrolconstraintsareimposedand/orwhentheHamiltonian isafneinthecontrol,theMinimumPrincipleofPontryaginMPPmustbeusedsince thenecessaryconditionsforoptimalitycannotbeemployed.[43,44]Inessence,the MPPprovidesanadditionalnecessaryconditiontodeterminetheoptimalcontrol.This principleisstatedas H x , u t H x , u t 8 u 2U where U = f u 2 R m j c x u t 0 g isthesetofadmissiblecontrolinputs.The superscripthereandhenceforthdenotestheoptimalsolutionofthevariable. UsingthecalculusofvariationsapproachtosolvetheOCPincreasesthe dimensionalitybyintroducingnewvariablesi.e.,costatesforeachsetofconstraints. [44]Solvingforthecostatesisnecessarysincetheydeterminetheoptimalcontrolas wellasunknownboundaryconditions.Also,thetransversalityconditionsshowthatthe boundaryconditionsforthestatesandcostatesarenotknownatthesameboundary, makingitdifculttosolveforbothstatesandcostatessimultaneously. 4.1.1MinimumTimeProblem Theminimumtimeproblemcanbestatedas min u 2U J = Z t f t 0 1 dt = t f )]TJ/F40 11.9552 Tf 11.956 0 Td [(t 0 57

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subjectto x t = a x t t + B x t t u t x t 0 x t f t 0 arespecied u min u i u max for i =1,2,..., m where a : R n R R n and B : R n R R n m .[44]Notethatthedynamics constraintisafneinthecontrol,whichisafairassumptionformostdynamicalsystems. Itisinherentthatthenaltimebeunspeciedandthatsaturationlimitsbeplacedon thecontrol.Otherwise,thesolutionwouldyieldaninnitecontrolactiontoreachthe terminalconditionimmediately.Also,giventheinitialconditionandthecontrolconstraint, thenalcondition x f musttobereachablefromtheinitialcondition x 0 .[44,71] TheHamiltonianforthisproblemisdenedas H =1+ T [ a + Bu ] TheMPPmustbeappliedandresultsinthefollowingcondition T B x t u T B x t u Ifthematrix B iswrittenas B x t = b 1 x t b 2 x t b m x t where b i 2 R n 1 arecolumnmatrices,thentheproductinequation4canbe expressedas T B x t u = m X i =1 T b i x t u i 58

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Substitutingequation4intoequation4,theoptimalcontrolthatsatisesthe MPPisgivenas u i = 8 > > > > > > < > > > > > > : u max if T b i x t < 0 u min if T b i x t > 0 undeterminedif T b i x t =0 for i =1,2,..., m Notethatthisproblemresultsinabang-bangcontrolstructure.[43,44]This meansthatthecontrolliesontheboundaryof U throughoutthetrajectoryuntilthe terminalconditionsarereached.Theoptimalcontroldependsonthevaluesofthe costates,thusthecostatesstillneedtobedetermined. 4.1.2FixedTimeMinimumControlEffortProblem Thexedtimeminimumcontroleffortproblemcanbestatedas min u 2U J = Z t f t 0 k u t k dt subjectto 3 Notethatthenaltimeisspeciedinthisproblem.Ifthenaltimewasnotspecied, thenthesolutionwouldbetoapplyaninnitesimalcontrolactionoveraninniteamount oftime.[44]Again,sincethereisaconstraintonthecontrol,thenalcondition x f must bereachablefromtheinitialcondition x 0 .[44,71] TheHamiltonianforthisproblemisdenedas H = k u k 1 + T [ a + Bu ] TheMPPmustbeappliedandresultsinthefollowingcondition k u k 1 + T B x t u k u k 1 + T B x t u 59

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Usingequation4andthedenitionofthe 1 -norm,therighthandsidecanbewritten as k u k 1 + T B x t u = m X i =1 j u i j + T b i x t u i TheoptimalcontrolthatsatisestheMPPinequation4isobtainedas u i = 8 > > > > > > > > > > < > > > > > > > > > > : u max if T b i x t < )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 0 if )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 < T b i x t < 1 u min if T b i x t > 1 undeterminedif T b i x t = 1 for i =1,2,..., m Notethatthisproblemresultsinabang-off-bangcontrolstructure.[43,44]An interestingresultthatalsoarisesfromthisproblemisthetradeoffbetweenthenal timespeciedandthetotalcontroleffortasdepictedinFigure4-1.[44]Asnotedin thegure,thereisalowerboundonthenaltimespeciedduetothedynamicsand controlconstraints.Forlinearsystems,thisboundcanbedeterminedandisafunction oftheinitialconditions.[44]Thisboundexistsfornonlinearsystems,butitisnotas easilydetermined.Anotherinterestingresultisiftherearenopathconstraints,thenthe solutioncanbeapproximatedusingtwoimpulsesattheinitialandnaltime.[53,72] Thisapproximation,alongwiththetradeoffbetweenthexedtimeandcontroleffort,is usedlaterinthealgorithmdevelopmentinChapter6. 4.1.3FiniteHorizonLinearQuadraticProblem ThenitehorizonLQproblemcanbestatedas min u 2U J = 1 2 x T t f S f x t f + 1 2 Z t f t 0 x T t Q t x t + u T t R t u t dt 60

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Figure4-1.Tradeoffbetweennaltimeandcontroleffort subjectto x t = A t x t + B t u t x t 0 t 0 t f arespecied, where A t 2 R n n B t 2 R n m S f 2 R n n issymmetricpositive-semidenite, Q t 2 R n n issymmetricpositive-semidente,and R t 2 R m m issymmetric positive-denite.ThisLQproblemisentitlednitehorizonsincethenaltimeis speciedandnite,linearsincethedynamicsconstraintislinear,andquadratic sincethecostfunctionisaquadraticfunction.Itisalsorequiredthatthepair A B be controllableandthepair A Q beobservable.[43,44,71] TheHamiltonianforthisproblemisdenedas H = 1 2 x T Qx + u T Ru + T Ax + Bu 61

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Oneofthenecessaryconditionsforoptimalitystates @ H @ u T = 0 Ru + B T = 0 andleadingtotheoptimalcontrollaw u = )]TJ/F45 11.9552 Tf 9.299 0 Td [(R )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T NotethattheMPPdidnothavetobeemployedheresincetherearenocontrol constraintsandtheHamiltonianisnotafneinthecontrol.Fromthetransversality conditions,theboundaryconditionofthecostateatthenaltimeisgivenby t f = S f x f Theothertwonecessaryconditionsforoptimalityyield x = @ H @ T = Ax + Bu = )]TJ/F31 11.9552 Tf 11.291 16.857 Td [( @ H @ x T = )]TJ/F45 11.9552 Tf 9.299 0 Td [(Qx )]TJ/F45 11.9552 Tf 11.955 0 Td [(A T Substitutingequation4forthecontrolandwritinginstatespaceformyieldsthe followingsystem 2 6 4 x 3 7 5 = 2 6 4 A )]TJ/F45 11.9552 Tf 9.298 0 Td [(BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T )]TJ/F45 11.9552 Tf 9.298 0 Td [(Q )]TJ/F45 11.9552 Tf 9.299 0 Td [(A T 3 7 5 2 6 4 x 3 7 5 2 6 4 x t 0 t f 3 7 5 = 2 6 4 x 0 S f x f 3 7 5 wherethestatematrixistheHamiltonianmatrix. Giventhetransversalityconditioninequation4,theexistenceofacontinuous linearmappingthatrelatesthecostatestothestatesisexplored.Thismappingis assumedtobeoftheform t = S t x t 62

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where S t f = S f .Thetimederivativeofequation4yields = Sx + S x Substitutingtheexpressionsforthederivativesinequation4intoequation4 andusingtherelationshipinequation4yields S + Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S + SA + A T S x = 0 Thus,thenontrivialsolutioni.e., x 6 = 0 isgivenby S + Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S + SA + A T S = 0 S t f = S f wherethisequationistheDRE.TheDREisanonlinearmatrixdifferentialequationand ananalyticsolutioncannotbedeterminedingeneral.Also,sinceitonlyhasaboundary conditionspeciedatthenaltime,theDREneedstobeintegratedbackwardsintime. [43,68]Solvingthisequationalsoallowsfortheoptimalcontrollawtobewritteninstate feedbackformas u = )]TJ/F45 11.9552 Tf 9.298 0 Td [(R )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T Sx 4.2MethodsforSolvingOptimalControlProblems ThissectiondiscussestheshootingmethodwhichisusedforsolvinganOCP indirectlyi.e.,viathecalculusofvariationsapproach.Thismethodrequiresthatthe necessaryconditionsforoptimalitybesetup,andthataninitialguessfortheunknown boundaryconditionsbeguessed.Acollocationmethodwhichisusedforsolvingan OCPdirectlyi.e.,withoutintroducingcostatesisalsodiscussed.Thismethodentails performingatranscriptionofthecontinuoustimeOCPtoanitedimensionalnonlinear programNLPwhichmakessolvingtheoptimizationproblemmoretractable. 63

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4.2.1IndirectMethod:Shooting Recallthatthetransversalityconditionsassertthatifthestateand/ortimeis speciedatboundary,thenthecostateand/orHamiltonianisnotspeciedatthat boundary.Additionally,notethatiftheHamiltonianisnotanexplicitfunctionoftime,then itisconstantsince d H dt = @ H @ t + @ H @ x x + @ H @ d H dt =0+ @ H @ x @ H @ T )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(@ H @ @ H @ x T d H dt =0 Todescribetheprocessofapplyingtheshootingmethod,anexampleisusedfora generalHamiltonianboundaryvalueproblemHBVPwheretheinitialstateandtimeare speciedandthenalstateandtimeareunspecied.Asimilarprocesscanbeapplied ifadifferentsetofboundaryconditionsareknown.First,theHBVPisestablishedbased onthenecessaryconditionsforoptimalityas 2 6 6 6 6 4 x _ H 3 7 7 7 7 5 = 2 6 6 6 6 4 )]TJ/F26 7.9701 Tf 6.675 -4.977 Td [(@ H @ T )]TJ/F31 11.9552 Tf 11.291 9.684 Td [()]TJ/F26 7.9701 Tf 6.675 -4.977 Td [(@ H @ x T 0 3 7 7 7 7 5 u=u with x 0 and t 0 specied. Theprincipalstepoftheshootingmethodisthataguessismadefortheunknown initialvaluesof t 0 and H t 0 .Subsequently,equation4isintegratedforward intimeuntilthevaluesobtainedfor t f and H t f fromthetransversalityconditions areobtained.Iftheyarenotobtained,thenanewguessisneededandtheprocess shouldberepeated.Aroot-ndingschemecanbeemployedtomakerenedguesses forsubsequentiterations.However,ifapoorinitialguessismade,thenitisunlikelythat theshootingmethodconverges.Unfortunately,thereisnospecicwayofchoosinga 64

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goodinitialguess.Inaddition,eachiterationoftheshootingmethodrequiresnumerical integration,whichbecomescomputationallycostlyifmultipleiterationsarerequired.[73] 4.2.2DirectMethod:Collocation Differentapproachesforcollocationhavebeenproposedandstudiedforsolving OCPs.[49,50,74]Thefoundationoftheseapproachesliesinchoosinganitesetof collocationpointsandsettingupanoptimizationproblemequivalenttotheOCP,where theconstraintsoftheOCPareenforcedatthecollocationpoints.Thisisdonebyrst approximatingthestateatthe N collocationpointsusingafunctionapproximation x t N X i =1 L i t x t i where t i arethecollocationpoints, x t i isthevalueofthestateatthecollocationpoint, and L i t istheLagrangepolynomial L i t = N Y j =1 j 6 = i t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t j t i )]TJ/F40 11.9552 Tf 11.955 0 Td [(t j Thecollocationpointsareusuallydeterminedasrootsofaspecialkindofpolynomial. Oneapproachistousetherootsofanorthogonalpolynomial;thisiscalledan orthogonalcollocationmethodOCM.[49,50] Differentiatingequation4yields x t N X i =1 L i t x t i = N X i =1 f x t i u t i t i Thedynamicsconstraintisnowenforcedusinganalgebraicequationateachcollocation point.Next,amatrixisdenedwherethevaluesofthestatewhicharestilltobe 65

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determinedareconcatenatedas X = 2 6 6 6 6 6 6 6 4 x T t 1 x T t 2 . x T t N 3 7 7 7 7 7 7 7 5 Similarly,amatrixisdenedwherethevaluesofthecontrolwhicharestilltobe determinedareconcatenatedas U = 2 6 6 6 6 6 6 6 4 u T t 1 u T t 2 . u T t N 3 7 7 7 7 7 7 7 5 andthevaluesofthedynamicconstraintsareconcatenatedas F X U t 1 t 2 ,..., t N = 2 6 6 6 6 6 6 6 4 f T x t 1 u t 1 t 1 f T x t 2 u t 2 t 2 . f T x t N u t N t N 3 7 7 7 7 7 7 7 5 Thus,thedynamicconstraintateachcollocationpointsiswrittenas DX = F X U t 1 t 2 ,..., t N wherethematrix D includesthederivativesoftheLagrangianpolynomialsevaluatedat thecollocationpoints.Thismatrixisnotuniqueanddependsonthecollocationscheme used.[49,50]Thestateand/orcontrolconstraintsareemployedinasimilarmanner, wheretheconstraintisappliedateachcollocationpoint.Thismeansthestateand/or 66

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controlconstraintsarewrittenas C X U t 1 t 2 ,..., t N = 2 6 6 6 6 6 6 6 4 c T x t 1 u t 1 t 1 c T x t 2 u t 2 t 2 . c T x t N u t N t N 3 7 7 7 7 7 7 7 5 0 Inaddition,theintegralpartofthecostfunctionalisnowapproximatedusinga quadratureruleas Z t f t 0 L x u t dt N X i =1 w i L x t i u t i t i where w i arethequadratureweights.[49,50,73]Now,thecostfunctionaland constraintsareallfunctionsofthenitesetofcollocationpoints.Theresultisanite dimensionaloptimizationproblemi.e.,NLPoftheform min X 2 R N n U 2 R N m J = J X U subjectto DX = F X U t 1 t 2 ,..., t N x t 0 x t N t 0 t N = 0 C X U t 1 t 2 ,..., t N 0 ThesolutiontoaNLPissuchthatparameters X and U satisfytheKarush-Kuhn-Tucker KKTconditions.[75]However,sincetheKKTconditionsareonlyenforcedatthe collocationpoints,globaloptimalitywithrespecttotheoriginalOCPcannotbe guaranteed.Also,collocationmethodsstillrequireaninitialguessateachofthe collocationpoints,wheretheconvergencetimetosolvetheNLPdependsonthequality oftheinitialguess.[49,50,75] 67

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Apowerfultoolthatisavailablefree-of-chargeandworksintheMatlabenvironment inconjunctionwiththeSparseNonlinearOptimizationSNOPTsolveristheGauss PseudospectralOptimalControlSoftwareGPOPS.[76,77]TheGPOPSsoftware allowstheusertoeasilytranscribeacontinuousOCPtoaNLPandsolvetheNLPusing SNOPT.GPOPSisusedtoobtaintheoptimaltrajectoriesinthefollowingsection.These trajectoriesarealsousedinChapter8todeterminewhetherthereareperformance differencesinahigherdelitymodel. 4.3OptimalRendezvousTrajectories Fourexamplesofoptimalrendezvoustrajectoriesarepresentedinthissection. Therendezvousprobleminvolvesapassivespacecrafti.e.,targetandanactive spacecrafti.e.,chaser.Thetranslationalmotionforbothspacecraftisgovernedbythe Keplerianmodelinequation2i.e.,nodisturbingaccelerations.Theobjectiveofthe rendezvousproblemis r rel = r c )]TJ/F45 11.9552 Tf 11.955 0 Td [(r t 0 r rel =_ r c )]TJ/F22 11.9552 Tf 12.528 0 Td [(_ r t 0 where r c and r t arethepositionofthechaserandtarget,respectively.Theparameters usedaregiveninTable4-1.Theinitialconditionsofthetargetplaceitinacircularorbit witha 600 kmaltitude.Theinitialconditionsofthechaserarechosensuchthatthe relativepositionbetweenthechaserandtargetinthetarget'sorbitalframeis r t 0 = 500 )]TJ/F22 11.9552 Tf 9.299 0 Td [(500500 T m andtherelativevelocityiszero.Thefollowingpathconstraintisalsoimposedonthe chasertoensurethatitstaysinlowEarthorbit. 200 km k r c k)]TJ/F40 11.9552 Tf 20.589 0 Td [(R 1000 km 68

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Similarly,thecontrolconstraintbelowisimposed j f i j f max for i =1,2,3 whichisrepresentativeofacontrolconstraintforreactionjets.Themarkersintheplots foreachexamplerepresentthecollocationpointsusedbyGPOPS. Table4-1.Optimalrendezvousparameters ParameterValueUnits r c t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.3394116816.939420904.891745 T km v c t 0 0.175776 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.9637767.494102 T km = s r t t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.9593486817.396210904.495486 T km v t t 0 0.175776 )]TJ/F22 11.9552 Tf 9.299 0 Td [(0.9637767.494102 T km = s f max 0.001 km = s 2 4.3.1MinimumTimeRendezvous Theminimumtimeproblemspeciedinequation4requiresthatboththeinitial andnalstatesbespecied.Theinitialstatesarespecied,yetthenalstatesforthe individualspacecraftarefree.Whatisspecied,isarelativepositionandvelocityofzero atthenaltime.Therelativetrajectory,relativeposition,relativevelocity,andcontrol historyareshowninFigure4-2.TherelativetrajectoryshowninFigure4-2Aindicates howthechaserapproachesthetarget.Therelativepositionandrelativevelocityplotsin Figure4-2BandFigure4-2C,respectively,indicatethatthenalboundaryconditionsare satised.ThecontrolhistoryplotinFigure4-2Dshowsthatacontrolstructuresimilarto bang-bangisobtained.Recallthattheoptimalcontrolisunspeciedwhenthecontrol isswitched.Theminimumtimewhichisalsothecostforrendezvousis t f =49.86 s. 4.3.2FixedTimeMinimumControlEffortRendezvous Theminimumcontroleffortrendezvousproblemspeciedinequation4requires thatboththeinitialandnalstatesbespeciedalongwithanaltime.Theinitial statesarespecied,butthenalstatesfortheindividualspacecraftarefree.What isspeciedisarelativepositionandvelocityofzeroatthenaltime.Thenaltime 69

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ARelativepositiontrajectory BRelativepositionhistory CRelativevelocityhistory DOptimalcontrolhistory Figure4-2.Minimumtimerendezvousresults chosenforthisproblemis t f =500 s.Therelativetrajectory,relativeposition,relative velocity,andcontrolhistoryareshowninFigure4-3.Therelativetrajectoryshownin Figure4-3Aindicateshowthechaserapproachesthetarget.Therelativepositionand relativevelocityplotsinFigure4-3BandFigure4-3C,respectively,indicatethatthenal boundaryconditionsaresatisedatthenaltime.ThecontrolhistoryplotinFigure4-3D showsthatasimilarcontrolstructuretobang-off-bangisobtained.Recallthatthe optimalcontrolisunspeciedwhenthecontrolswitches.Thecostobtainedortotal controleffortforthisproblemis J =6.61 m = s.Itshouldbenotedthatthisproblem 70

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requiredlesscollocationpointssincethetimerangewaslongerandtheoptimalcontrol waszeroforamajorityofthecollocationpoints. ARelativepositiontrajectory BRelativepositionhistory CRelativevelocityhistory DOptimalcontrolhistory Figure4-3.Fixedtimeminimumcontroleffortrendezvousresults 4.3.3FiniteHorizonQuadraticCostRendezvous Thequadraticproblemspeciedinequation4requiresthatonlytheinitial statesbespeciedalongwithanaltime.Instead,acostisaddedfortheinitialand terminalboundaryconditionsdesired.Thetimehorizonchosenisthesameasthe minimumfuelproblemi.e., t f =500 s.Thevaluesusedforthematricesthatdenethe 71

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costfunctionalare S f = Q = 2 6 4 10 4 I0 0 10 2 I 3 7 5 2 R 6 6 R =10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 I 2 R 3 3 Therelativetrajectory,relativeposition,relativevelocity,andcontrolhistoryare showninFigure4-4.TherelativetrajectoryshowninFigure4-4Aindicateshow thechaserapproachesthetarget.Therelativepositionandrelativevelocityplotsin Figure4-4BandFigure4-4C,respectively,indicatethatthedesirednalstatesare virtuallyobtainedatthenaltime.ThecontrolhistoryplotinFigure4-4Dshowsthatthe controldecaysastheterminalconditionsareapproached. 4.3.4ConstrainedFixedTimeMinimumFuelRendezvous Thesamecostisusedasthexedtimeminimumcontroleffortproblemwiththe addedpathconstraintof k r c t )]TJ/F45 11.9552 Tf 11.955 0 Td [(r o t k 100 m, where r o t isthepositionofanobstacleandisgovernedbythetwobodyrelative motionmodelinequation2neglectingdisturbances.Theinitialconditionsusedfor theobstacleare r o t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.6494976817.167778904.693252 T km v o t 0 = 0.175775 )]TJ/F22 11.9552 Tf 9.299 0 Td [(0.9637767.494102 T km = s. Therelativepositiontrajectoriesofthechaserareshownatdifferentinstances intimeinFigure4-5.Theseguresindicatehowthechaserisabletoavoidthe obstacleandreachthetarget.Therelativeposition,relativevelocity,andcontrol historyareshowninFigure4-6.Therelativepositionandrelativevelocityplotsshown inFigure4-6AandFigure4-6B,respectively,indicatethatthenalboundaryconditions 72

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ARelativepositiontrajectory BRelativepositionhistory CRelativevelocityhistory DOptimalcontrolhistory Figure4-4.Finitehorizonquadraticcostrendezvousresults aresatisedatthenaltime.ThecontrolhistoryinFigure4-6Cthatabang-off-bang controlstructureisobtained,withamidcoursecorrectioninthe z direction. 4.4OptimalSlewManeuvers Threeexamplesofoptimalslewmaneuversarepresentedinthissection. Neglectingdisturbances,theequationsofmotionusedarethequaternionkinematicsin equation2andtheequationsofmotioninequation2.Theparametersusedfor 73

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A t =130 s B t =250 s C t =380 s D t =500 s Figure4-5.Constrainedxedtimeminimumcontrolefforttrajectories 74

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ARelativepositionhistory BRelativevelocityhistory COptimalcontrolhistory Figure4-6.Constrainedxedtimeminimumcontroleffortrendezvousresults theexamplesaregiveninTable4-2.Theobjectiveoftheslewmaneuveris ! 0 1 Theseparametersdenea 180 slewmaneuverabouttherstbodyaxis.Thecontrol constraintshownbelowisenforced j i j max for i =1,2,3 75

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whichisrepresentativeofacontrolconstraintforareactionwheelattitudecontrol system.Themarkersintheguresforeachexamplerepresentthecollocationpoints usedbyGPOPS. Table4-2.Optimalslewparameters ParameterValueUnits J 2 4 3002010 201000 100200 3 5 kg m 2 q t 0 1000 T t 0 000 T rad = s max 1 N m 4.4.1MinimumTimeSlew Theminimumtimeproblemspeciedinequation4requiresthattheinitialand nalstatesbespecied,thusthenalstatesaregivenbythecontrolobjective.The quaternion,angularvelocity,andcontrolhistoryareshowninFigure4-7.Thequaternion andangularvelocityplotsinFigure4-7AandFigure4-7B,respectively,indicatethatthe nalboundaryconditionsareobtained.ThecontrolhistoryplotinFigure4-7Cshows thatacontrolstructuresimilartobang-bangisobtained.Theresultingminimumtime whichisalsothecostfortheslewmaneuveris t f =48.70 s. 4.4.2FixedTimeMinimumControlEffortSlew Theminimumcontroleffortproblemisspeciedinequation4hadthesame boundaryconditionsastheminimumtimeslewmaneuver.Thenaltimespeciedfor thisproblemis t f =100 s.Thequaternion,angularvelocity,andcontrolhistoryare showninFigure4-8.ThequaternionandangularvelocityplotsinFigure4-8Aand Figure4-8B,respectively,indicatethatthenalboundaryconditionsareobtainedatthe naltimespecied.ThecontrolhistoryplotinFigure4-8Cshowsthatasimilarcontrol structuretobang-off-bangisobtained.Itisinterestingtonotethatalargenumberof collocationpointswererequired,andasaresultthesolutiontookalongtimetoobtain. Thismaybeduetothefactthattheoptimalcontrolisunspeciedattheswitchingtimes. 76

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AQuaternionhistory BAngularvelocityhistory COptimalcontrolhistory Figure4-7.Minimumtimeslewresults Theintervalwherethecontrolisunspeciedalsohassomenitewidth.Theoptimal controlsolutionalsohadsmallvaluesinbetweentheinitialandnalswitchesinthe control.Thenalcostandalsothecontroleffortobtainedis J =24.37 N m. 77

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AQuaternionhistory BAngularvelocityhistory COptimalcontrolhistory Figure4-8.Fixedtimeminimumcontroleffortslewresults 4.4.3FiniteHorizonQuadraticCostSlew Thenitehorizonquadraticcostrequiresthatonlytheinitialconditionsonthestate andthenaltimebespecied.Thecostusedforthisproblemis J = 1 2 T t f S f ,1 t f + s f )]TJ/F25 11.9552 Tf 11.955 0 Td [( t f 2 + T t f S f ,2 t f +... 1 2 Z t f t 0 T t Q 1 t + q )]TJ/F25 11.9552 Tf 11.955 0 Td [( t 2 + T t Q 2 t + T t R t dt 78

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wherethevaluesusedforthematricesare S f ,1 = S f ,2 = Q 1 = Q 2 = I 2 R 3 3 R =10 I 2 R 3 3 s f = q =1 Thenaltimespeciedisthesameastheminimumcontroleffortproblemi.e., t f = 100 s.Thequaternion,angularvelocity,andcontrolhistoryareshowninFigure4-9. ThequaternionandangularvelocityplotsinFigure4-9AandFigure4-9B,respectively, indicatethatthedesirednalconditionsarevirtuallyobtained.Thecontrolhistoryplotin Figure4-9Cindicatesthatthecontrolissaturatedfortheinitialonsetofthemaneuver, andthenrampeddown.Thisisattributedtotheconservativesaturationlimitsonthe torquegiventhesizeofthespacecraft. Inconclusion,thetheoryrelatedtogeneratingoptimaltrajectoriesispresented inthischapter.Inparticular,thepertinentOCPsforclose-rangerendezvousarethe minimumtimeproblemandminimumcontroleffortproblem,andthepertinentOCPfor thenalapproachorendgameofaproximityoperationisthenitehorizonLQproblem. AsetoftrajectoriesforeachOCPisobtainedforrendezvousandslewmaneuvers. Thesetrajectoriesarelaterusedasbaselinecomparisonsforthealgorithmsdeveloped. Itisalsolaterdeterminedwhetherthesetrajectoriesarerealizedusingahighdelity model. 79

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AQuaternionhistory BAngularvelocityhistory COptimalcontrolhistory Figure4-9.Finitehorizonquadraticcostslewresults 80

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CHAPTER5 ARTIFICIALPOTENTIALFUNCTIONMETHOD Usinganarticialpotentialisaneffectivemethodforderivingguidancelaws. Justasapotentialshapescertainforceeldse.g.,gravity,electromagnetism,etc., anarticialpotentialcanbedenedtoshapeanarticialforceeldfordetermininga controllaw.Thearticialpotentialmusthavecertainpropertiessuchthattheresulting controllawprovidesstableresults.ThischapterdiscussestheAPFmethodology.Two setsofexamplesarepresentedwheretheAPFmethodisappliedtotheClohessy-Wiltshire-Hill CWHequationandthesmallangleapproximationSAAequationderivedin AppendixA. 5.1Development Anarticialpotentialisdenedbysuperimposingattractiveandrepulsivepotentials. Thepurposeoftheattractivepotentialistoimposeaglobalminimumatthedesired terminalposition x f .Therepulsivepotentialscreateregionsofhighpotentialatpositions x i ofthestatespacethataretobeavoided.[5659,78] Differentformsofarticialpotentialshavebeenexplored.Oneformistouse harmonicfunctionsi.e.,satisfyLaplace'sequationaspotentialssincethesefunctions donoexhibitlocalminimawhensuperimposed.[7981]However,thegradientofsinks andsourceswhicharetypicalharmonicfunctionsusedforattractiveandrepulsive potentials,respectivelyisundenedattheircenterandisnearzeroawayfromtheir center.Thus,scalingthesinksandsourcescanbedifcult. Anothertypicalchoiceforanattractivepotentialfunctionandtheoneusedinthis workisaquadraticfunction a x = 1 2 x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f T P x )]TJ/F45 11.9552 Tf 11.956 0 Td [(x f where P isasymmetricpositive-deniteweightingmatrixthatshapestheattractive potential.Atypicalchoicefordeningrepulsivepotentialsandtheoneusedinthis 81

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workisusingGaussianfunctions r x = N X i =1 i exp )]TJ/F22 11.9552 Tf 10.494 8.088 Td [( x )]TJ/F45 11.9552 Tf 11.956 0 Td [(x i T N i x )]TJ/F45 11.9552 Tf 11.956 0 Td [(x i i where i and i aretheheightandwidthparameters,respectively,and N i isa symmetricpositive-deniteweightingmatrixthatshapesthe i th repulsivepotential. [57,59].Usingthesefunctionsmightyieldlocalminimainthearticialpotential dependingontheparameterschosenfortherepulsivepotential,whichmaycause convergenceproblems.[82,83]RegardlessofthechoiceofAPFsused,thetotal articialpotentialisdenedasthesumoftheattractiveandrepulsivepotentials. = a + r AnexampleofanAPFconsistingofaquadraticattractivepotentialandtwoGaussian repulsivepotentialsforatwodegreeoffreedomsystemisillustratedinFigure5-1. A3Dsurfaceplotwithcontouroutline B2Dcontourplot Figure5-1.ArticialpotentialfunctionAPFexample SeveralcombinationsofthetypesofAPFshavebeenusedtoalleviatesome ofthedrawbacksassociatedwitheachofthem.[56,58,60,8486]Systematically resolvingthelocalminimaproblemisbeyondthescopeofthiswork.However,since theclose-rangerendezvousworkspaceissparse,thereisaconsiderableseparation betweenobstaclesandthegoal.Thus,localminimumareavoidedformostsetsof 82

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obstacleparameters.Ifthelocalminimaproblemcannotbeavoided,thendifferentforms ofpotentialsshouldbeused. AfeedbackcontrollercanbedevelopedfromthedenitionoftheAPF,whereaforce intheoppositedirectionofthelocalgradientofthearticialpotentialisappliedwhenthe rateofchangeofthearticialpotentialispositive.Thisfeedbackcontrollerisdenedas u x ,_ x = 8 > > < > > : )]TJ/F40 11.9552 Tf 9.299 0 Td [(k r x )]TJ/F22 11.9552 Tf 11.955 -0.166 Td [( x )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x f if 0 0 otherwise where k isapositivegain,and r x = @ @ x T r x = P x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f )]TJ/F41 7.9701 Tf 17.297 14.944 Td [(N X i =1 2 i i exp )]TJ/F22 11.9552 Tf 10.494 8.088 Td [( x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i T N i x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i i N i x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i = @ @ x d x dt + @ @ x f d x f dt + N X i =1 @ @ x i d x i dt = x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f T P x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x f )]TJ/F41 7.9701 Tf 17.297 14.944 Td [(N X i =1 2 i i x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i T exp )]TJ/F22 11.9552 Tf 10.494 8.087 Td [( x )]TJ/F45 11.9552 Tf 11.956 0 Td [(x i T N i x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i i N i x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x i 5.2NumericalExamples TwosetsofnumericalexamplesarepresentedtodemonstratetheAPFmethod.In therstexample,theAPFmethodisappliedtotheCWHequationwithagridof14static obstaclesinthestatespace.Inthesecondexample,theAPFmethodisappliedtothe SAAequationwithatumblingsatellite. 5.2.1Clohessy-Wiltshire-HillExample TheparametersusedinthisexamplearegiveninTable5-1.Thegoalofthechaser initiallylocatedattherelativeposition r 0 istoreachthetargetlocatedattheorigin whileavoidingtheobstacles.Asetof14staticobstaclesareplacedinthestatespaceat thepositionsshownbelow. 83

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r 1 = 50000 T r 2 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(50000 T r 3 = 05000 T r 4 = 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5000 T r 5 = 00500 T r 6 = 00 )]TJ/F22 11.9552 Tf 9.298 0 Td [(500 T r 7 = 250250250 T r 8 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250 T r 9 = 250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 10 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 11 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250250 T r 12 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 T r 13 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 14 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T Thesepositionsdeneanobstaclegridthatdoesnotyieldlocalminima,yetstillclutters thestatespace.Staticobstaclesarepracticalforavoidingspacecraftthataretethered oryinginformation.Otherwise,theseobstacleswouldnotbestaticandwoulddrift. However,staticobstaclesareusedforsimplicityandtodemonstratethecollision avoidanceabilitiesoftheAPFalgorithm. Table5-1.Relativetranslationparameters ParameterValueUnits n 0.0012 rad = s r 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(750500 )]TJ/F22 11.9552 Tf 9.298 0 Td [(750 T m v 0 0.5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(20 T m = s r f 000 T m f max 1 m = s PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 N i I 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 i 1.5 10 5 i 1.0 10 4 m 2 k 0.002 Forthisexample,thecontrolconstraintshownbelowisenforced. k f k 2 f max Settlingtimeisusedasaperformancemetric,wherethetrajectoryobtainedis consideredsettledwhen k r k 10 m.Therunningcontroleffortcostisalsousedas 84

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aperformancemetricandisdenedas V t = Z t t 0 k u k 1 d TheresultsobtainedaredisplayedinFigure5-2.ThetrajectoryplotinFigure5-2A showsthatanobstacleisencounteredandthechaserisabletomaneuveraroundit. ThepositionandvelocityplotsinFigure5-2BandFigure5-2C,respectively,indicate thatthechaserisabletoreachthetargetwithzerovelocity.Thecorrespondingcontrol historyisgiveninFigure5-2D. A3Dplotoftrajectory BPositionhistories CVelocityhistories DControlhistories Figure5-2.APFmethodresultsusingClohessy-Wiltshire-HillCWHequation 85

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Thesettlingtimeobtainedis 2620 sandthecontroleffortatthesettlingtimeis 16.94 m = s 2 .TherunningcostisshowninFigure5-3whichillustrateshowthecost increaseswitheverycontrolaction. Figure5-3.APFmethodrunningcostusingCWHequation 5.2.2SmallAngleApproximationExample TheparametersusedinthisexampleareshowninTable5-2.Thegoalofthe chaserwithinitialrelativeorientation 0 tothetargetistoreacharelativeorientation of = 0 radwiththetargetthatistumblingattherate t .Thecontrolconstraintshown belowisenforced. j i j max for i =1,2,3 Settlingtimeisusedasperformancemetricwherethetrajectoryobtainedisconsidered settledwhen k k 0.01 rad.Therunningcontroleffortcostisalsousedasa performancemetricandisdenedas V t = Z t t 0 k k 1 d TheresultsforthisexamplearedisplayedinFigure5-4.Therelativeorientation andrelativeangularvelocityplotsinFigure5-4AandFigure5-4B,respectively,indicate thatboththeseparametersconvergetozero.Thecorrespondingcontrolhistoryis 86

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Table5-2.Relativeorientationparameters ParameterValueUnits J 2 4 3002010 201000 100200 3 5 kg m 2 t 0.25 )]TJ/F22 11.9552 Tf 9.299 0 Td [(0.250.25 T rad = s 0 0.3 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.15 T rad 0 000 T rad = s f 000 T rad max 20 N m PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 k 2 giveninFigure5-4C.Thesettlingtimeobtainedis 22.7 s.Notethattherunningcostin Figure5-4Dlinearlyincreaseswithtimesincethecontrollercontinuouslycompensates forthetumblingsatellite. Inconclusion,theAPFmethodisdiscussedinthischapter.Theprocedurefor deninganAPFandthecorrespondingcontrollawispresented.Twoexamplesof theAPFmethodarealsopresentedusingtheCWHandSAAequations.Fromthe discussionandexamplesinthischapter,theAPFmethodisausefultoolforcomputing guidancelawsaslongasitsrestrictionsareunderstood. 87

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APositionhistories BVelocityhistories CControlhistories DRunningcost Figure5-4.APFmethodresultsusingsmallangleapproximationSAAequation 88

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CHAPTER6 ADAPTIVEARTIFICIALPOTENTIALFUNCTIONMETHOD ThearticialpotentialfunctionAPFmethoddiscussedinChapter5isamethod thatcanbeusedinarapidpath-planningframework.ItwasshownthattheAPFmethod iswell-suitedforclose-rangerendezvousscenariosandrelativeorientationproblems. However,adrawbackoftheAPFmethodisthatithasnoconsiderationofthedynamics oraperformanceindex.TheadaptivearticialpotentialfunctionAAPFmethodis amodicationoftheAPFmethodwiththeintentofembeddingthedynamicsand performanceindexintheformulation.TheAPFframeworkisadoptedbecauseofits lowcomplexityandfavorableconvergencecharacteristics.Thedevelopmentofthe AAPFmethodandtheadaptiveupdatelawusedtodeterminetheweightingparameters oftheAPFisdiscussedinthischapter.Astabilityanalysisisalsopresentedwhich showsthatstabilityisobtainedforunconstrainedcasesi.e.,noobstacleregions, butthatstabilitycannotbeprovenforconstrainedcasesi.e.,withobstacleregions. Finally,twonumericalexamplesarepresentedwheretheAAPFmethodisappliedto theClohessy-Wiltshire-HillCWHequationandthesmallangleapproximationSAA equationderivedinAppendixA. 6.1Development ItwasshowninChapter5thatdifferentformsofAPFsexist.However,thereisno systematicapproachfordeninganAPFforaparticularsetofdynamicsorperformance criteria.TheparametersthatdeneAPFsarechosenadhoc,thusitisdifcultto determinetheparameterswhichyieldthebestperformancewithleastcontroleffort. Additionally,sincethereisnoconsiderationofthedynamics,thetrajectoriesobtained fromtheAPFmethodarenotdenedsuchthatthegoalisreachedataspeciedtime. Thesefactorsareparticularlyimportantwithclose-rangerendezvousofspacecraft, wherecontroleffort,power,andtimemustbeconserved.Therefore,path-planning algorithmsforclose-rangerendezvousmusttakethesefactorsintoaccount.Toremedy 89

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thedrawbacksofusingtheAPFmethodforclose-rangerendezvous,theAAPFmethod isdevelopedbyembeddingthedynamicsandaperformancecriteriaintheformulation. ToembedthedynamicsintheformulationoftheAAPFmethod,therelativemotion modelsdiscussedinAppendixAareused.Thesemodelsareusedsincethetwopoint boundaryvalueproblemTPBVPsolutionsareobtainedanalyticallyandtheTPBVP solutionisusedintheformulationtoincludethedynamics.Todothis,considerthesame formoftheattractivepotentialinequation5exceptlettheattractiveweightingmatrix betimedependenti.e., P = P t .Toenforcethesymmetricpositive-denitecondition, aCholeskyfactorization P t = R T t R t isused,where R t istheuppertriangular matrixtermedtheCholeskyfactorandisdenedas R t = 2 6 6 6 6 4 11 t 12 t 13 t 0 22 t 23 t 00 33 t 3 7 7 7 7 5 Duetotheadaptation,thetimederivativeoftheattractivepotentialnowcontains additionaltermsasshownbelow. a = @ a @ t + @ a @ x d x dt + @ a @ x f d x f dt a = 1 2 x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f T R T R + R T R x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f + x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f T R T R x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x f Atimedependentattractiveisusedsincethisiswhatgovernshowthegoalisapproached. Usingatimedependentrepulsivepotentialsimplychangestheshapeandsizeofthe obstacleregionanddoesnotnecessarilyimproveperformance.Asaresult,theformof therepulsivepotentialremainsthesame. SincetheformoftheAPFisthesame,theformofthecontrollawalsoremainsthe sameasshowninequation5.Thedifferenceisthattheweightsin R t mustbe choseneffectivelytoimproveperformance.Thisisachievedbyallowingthenegative gradientoftheattractivepotentialtoadapttothevelocityproleoftheTPBVPsolution. 90

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TheformulationoftheTPBVPvelocityproleispresentedinAppendixB.Thevelocity proleobtainedinequationBmustthenbewritteninafeedbackformbylettingthe time t 0 = t andtheinitialcondition x 0 = x .[40] x d x t = 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T x + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] Notethat x t isnowtimedependentandisshownbelow. x t = Z T t 12 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d Fortherelativetranslationproblem,thestatesinquestionarethecomponentsof therelativepositioni.e., x = r .Toobtainthevelocityprole,equation6isusedand isdenedas v d = 22 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T r )]TJ/F45 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T r f where 12 22 areblocksoftheSTMfortheCWHequationand T isthetransfertime oftheTPBVPsolutionfortheCWHequations.Notethat r t = 0 since u = 0 i.e.,no constantforcingterm.Also,notethatthevelocityproledependson T ,wherethis parameterischosensuchthat )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T exists. Therelativeorientationproblemissimilarwherethestatesinquestionarethe componentsoftherelativeorientationi.e., x = .Toobtainthevelocityprole, equation6isusedandisdenedas d = 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T f where 12 22 areblocksoftheSTMfortheSAAequationand T isthetransfertime oftheTPBVPsolutionfortheSAAequations.Notethat t = 0 whichcorrespondsto alwayshavingacompensationfortheconstantforcingterm u = )]TJ/F46 11.9552 Tf 9.298 0 Td [(! t J t .Also,note thatthevelocityproledependsonthechoiceof T ,wherethisparameterischosen suchthat )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T exists.ThistransfertimeandthetransfertimeoftheTPBVPsolution 91

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fortheCWHequationsisdistinctsincetheSTMsaredifferent.Fortheremainderof thederivation,anarbitrarystate x isusedsincethevelocityproleforboththeCWH equationandtheSAAequationisofthesameform. Theadaptiveupdatelawisnowdevelopedtochoosetheweightsoftheattractive articialpotential.First,anerrorvariableisdenedas e =_ x d )]TJ/F22 11.9552 Tf 11.956 -0.167 Td [( )]TJ/F49 11.9552 Tf 9.299 0 Td [(r x a e = 22 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T x + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T x f + R T R x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f Next,thetimederivativeofequation6yields e = 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T x + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T x f + R T R + R T R x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f + R T R x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x f whichislinearlyparameterizedas e = 22 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T x + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T x f + Y + Z + R T R x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x f where Y = 2 6 6 6 6 4 1 00000 0 1 0 2 00 00 1 0 2 3 3 7 7 7 7 5 Z = 2 6 6 6 6 4 11 x 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,1 11 x 2 )]TJ/F40 11.9552 Tf 11.956 0 Td [(x f ,2 11 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 000 12 x 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,1 12 x 2 )]TJ/F40 11.9552 Tf 11.956 0 Td [(x f ,2 12 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 22 x 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,2 22 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 0 13 x 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,1 13 x 2 )]TJ/F40 11.9552 Tf 11.956 0 Td [(x f ,2 13 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 23 x 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,2 23 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 33 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 3 7 7 7 7 5 = 11 12 13 22 23 33 T 92

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Theterms i for i =1,2,3 aredenedbelow. 1 = 11 x 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,1 + 12 x 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,2 + 13 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 2 = 22 x 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,2 + 23 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 3 = 33 x 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(x f ,3 Anadaptiveupdatelawisthendenedfortheweightsas = )]TJ/F22 11.9552 Tf 9.299 0 Td [( Y + Z # 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T x + )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T x f + R T R x )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x f + e where Y + Z # = Y + Z T Y + Z Y + Z T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 whichdrives e 0 as t !1 .Notethatthematrix Y + Z isfullrankexceptwhen x = x f .However,thisonlyoccursat t = 1 intheAPFframeworksincethecontroller resultsin x x f asymptotically.Therefore, Y + Z isnonsingularexceptat t = 1 .This indicatesthatthisalgorithmshouldonlybeusedforclose-rangerendezvousorintercept oruntilthechaseriswithinathresholdof x f .Inaddition,anonzeroinitialconditionfor isneededtoinitiallyhaveapositive-deniteweightingmatrix. Sinceanadaptationisbeingdonewithatime-varyingsignali.e,thevelocity prole,adiscontinuousprojectionalgorithmisemployedtoensurethattheadaptive estimatesarebounded.[87,88]First,ifcloseproximitysituationsareonlyconsidered, thenthereexists 2 R suchthat k x t )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f k 8 t 2 [ t 0 1 Asaresult,thevelocityproleinequation6isalsobounded.Thus,thereexistsan upperandlowerbound i i 2 R for i =1,2,...,6 ,suchthataconvexsetcanbedened 93

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foreachestimateas i = f 2 R j i i g for i =1,2,...,6 Theprojectionoperatorisdenedbasedonthedenitionoftheconvexsetas ^ i = Proj i = 8 > > > > > > > > > > < > > > > > > > > > > : i if i 2 i i if i = i and i 0 i if i = i and i 0 0 otherwise for i =1,2,...,6 where i for i =1,2,...,6 isdenedinequation6.Figure6-1showsanillustration oftheresultantadaptiveestimatetrajectoryusingtheprojectionalgorithm.Using theadaptiveupdatelawinequation6withthearticialprojectionalgorithmin equation6,theAAPFcontrollawis u x ,_ x = 8 > > < > > : )]TJ/F45 11.9552 Tf 9.299 0 Td [(R T R x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f )]TJ/F40 11.9552 Tf 11.955 0 Td [(k r x r )]TJ/F22 11.9552 Tf 11.955 -0.166 Td [( x )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x f if 0 0 otherwise wherethematrix R containstheadaptiveestimates. Figure6-1.Illustrationofadaptiveestimatetrajectory 94

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6.2StabilityAnalysis ToanalyzethestabilityoftheAPFandAAPFmethod,theunconstrainedcasei.e., obstacle-freeisrstconsidered.ConsideraLyapunovcandidatefunctionLCFthat hasthesameformasthequadraticattractivepotential V = 1 2 T where = x )]TJ/F45 11.9552 Tf 12.004 0 Td [(x f 0 isthecontrolobjective.GivenaLCF,thesufcientconditionsfora globallyasymptoticallystableGASequilibriumpointat = 0 are:[89] 1. V 0 =0 2. V > 0 8 x 2 R n )-222(f 0 g 3. V isradiallyunboundedi.e., V !1 as k k!1 4. V < 0 8 2 R n )-222(f 0 g TherstthreeconditionsaresatisedbasedonthechoiceoftheLCF.Forthefourth condition,thederivativeoftheLCFis V = T x )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x f Ifthecontrolactionisimpulsive,thenthevelocityatanyinstantintimecanbedescribed asthevelocityplusthemagnitudeoftheimpulsegiventheimpulseisnonzeroat thatinstantintime.[71]Toensurethefourthconditionissatised,animpulsive controlactioni.e., x isdenedaccordingtothecontrollawinequation5or equation6thatoccurswhen 0 .Thus,thevelocityaftertheimpulsiveaction occursisdenedas x )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x f +_ x = )]TJ/F40 11.9552 Tf 9.298 0 Td [(k r x = )]TJ/F40 11.9552 Tf 9.299 0 Td [(k P where k isapositivescalar.[57]Notethat r x = P r x V .Thus,deningthecontrollerin equation5orequation6ensuresthevelocityisascalarmultipleofthenegative 95

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gradientoftheAPFandLCFwhen 0 .ThederivativeoftheLCFthenbecomes V = 8 > > < > > : )]TJ/F40 11.9552 Tf 9.299 0 Td [(k T P if 0 T x )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x f if < 0 < 0 andsatisesthefourthcondition.Therefore,alltheconditionsaremetandthepoint = 0 isaGASequilibriumpoint. Whenconsideringobstacleregions,thesamestabilityanalysiscannotbeused.For one,choosingaLCFbecomesdifcultsincethesameformoftheAPFdoesnotsatisfy thersttwoconditionsforaGASequilibriumpoint.Onewaytoremedythisistodene theAPFas = 1 2 x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f T P + N X i =1 i exp )]TJ/F22 11.9552 Tf 10.494 8.088 Td [( x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i T N i x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x i i I # x )]TJ/F45 11.9552 Tf 11.955 0 Td [(x f anduseaLCFofthisformaswell.However,thisstilldoesnotresolvethelocalminima problem.[90]Infact,itisknownthatstabilitycannotbeprovenwithageneralsetof obstacleregionsandparameters.[82]Consideratwodegrees-of-freedomsystem exampleillustratedinFigure6-2A.Iftheobstaclesarearrangedasillustratedandthe obstacleparametersaredenedsuchthattherepulsivepotentialsarenotsenseduntil thechaserisnearthebarricadeofobstacles,thenthechaserwouldnotbeableto escapethebarricadeofobstacles.AnotherexampleshowninFigure6-2Boccurs whenthedynamicsofthesystemarenegligiblee.g.,thedoubleintegratorandthe obstacleisalongthelineofsightofthetarget.Inthisscenario,iftheobstacleregionis denedsymmetricallyanditliesalongthelineofsightofthechasertothetarget,then theattractiveandrepulsivepotentialseventuallycounteracteachotherandthecontrolis nullatthispoint. Regrettably,thestabilityforanobstacleavoidanceproblemusingtheAPForAAPF methodisbasedonintuition.TheintuitivenatureoftheAPFmethodgivesanotionthat bydeningtheAPFappropriately,thesolutionconvergestothegoalposition.Thetwo 96

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ALocalminimumexample BCounteractingobstacleexample Figure6-2.ExampleswhereAPFdoesnotguaranteeconvergence examplesdemonstratehowtheAPFmethoddoesnotyieldconvergenceforanarbitrary dynamicalsystemwitharbitraryarticialpotentialparameters.Thesetypesofproblems canbedealtwithinanadhocmannerwheretheappropriateparametersareused suchthatlocalminimaareavoided.[53,84]TheAAPFmethod,however,considers thedynamicswhichalleviatessomeofthelocalminimaproblems.Inaddition,the close-rangerendezvousscenariohasfree-oatingtargetandobstacleswhichreduces theprobabilityoflocalminimaaswell. 6.3NumericalExamples Twosetsofnumericalexamplesarepresentedinthissectiontodemonstratethe AAPFmethod.Intherstexample,theAAPFmethodisappliedtotheCWHequation withagridof14staticobstacleinthestatespace.Inthesecondexample,theAAPF methodisappliedtotheSAAequationwithatumblingsatellite. 6.3.1Clohessy-Wiltshire-HillExample TheparametersusedinthisexamplearegiveninTable6-1.Thegoalofthechaser initiallylocatedattherelativeposition r 0 istoreachthetargetlocatedattheorigin whileavoidingtheobstacles.Asetof14staticobstaclesareplacedinthestatespaceat thepositionsshownbelow. 97

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r 1 = 50000 T r 2 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(50000 T r 3 = 05000 T r 4 = 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5000 T r 5 = 00500 T r 6 = 00 )]TJ/F22 11.9552 Tf 9.298 0 Td [(500 T r 7 = 250250250 T r 8 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250 T r 9 = 250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 10 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 11 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250250 T r 12 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 T r 13 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 14 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T Forthisexample,thecontrolconstraintshownbelowisenforced. k f k f max Settlingtimeisusedasaperformancemetricwherethetrajectoryobtainedis consideredsettledwhen k r k 10 m.Therunningcontroleffortcostisalsousedas aperformancemetricandisdenedas V t = Z t t 0 k u k 1 d Table6-1.Relativetranslationparameters ParameterValueUnits n 0.0012 rad = s r 0 7500 )]TJ/F22 11.9552 Tf 9.299 0 Td [(500 T m v 0 0.520 T m = s r f 000 T m f max 1 m = s 2 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 N i I 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 i 1.5 10 5 i 1.0 10 4 m 2 k 0.002 T 1000 s 98

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TheresultsobtainedaredisplayedinFigure6-3.ThetrajectoryplotinFigure6-3A showsthatanobstacleisencounteredandthechaserisabletomaneuveraroundit. ThepositionandvelocityplotsinFigure6-3BandFigure6-3C,respectively,indicate thatthechaserreachestheoriginwithzerovelocity.Thecorrespondingcontrolhistoryis giveninFigure6-3D. A3Dplotoftrajectory BPositionhistories CVelocityhistories DControlhistories Figure6-3.AdaptivearticialpotentialfunctionAAPFmethodresultsusingCWH equation Thesettlingtimeobtainedis 1715 sandthecontroleffortatthesettlingtimeis 8.63 m = s.TherunningcostisshowninFigure6-4,whichillustrateshowthecost increaseswitheverycontrolaction. 99

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Figure6-4.AAPFrunningcostusingCWHequation TheplotoftheadaptiveestimatesisshowninFigure6-5.Theseplotsindicatethat theadaptiveestimatescontinuetochangedependingonthechaser'spositioninthe statespace.Thisreinforcesthefactthatthemagnitudeofthecommandedforceshould dependonthedynamicsandthelocationofthechaser. Figure6-5.AAPFadaptiveestimatesusingCWHequation 6.3.2SmallAngleApproximationExample TheparametersusedinthisexampleareshowninTable6-2.Thegoalofthe chaserwithinitialrelativeorientation 0 tothetargetistoreacharelativeorientation of = 0 radwiththetargetthatistumblingattherate t .Thecontrolconstraintshown 100

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belowisenforced. j i j max for i =1,2,3 Settlingtimeisusedasaperformancemetricwherethetrajectoryobtainedis consideredsettledwhen k k 0.01 rad.Therunningcontroleffortcostisalsousedas performancemetricandisdenedas V t = Z t t 0 k k 1 d Table6-2.Relativeorientationparameters ParameterValueUnits J 2 4 3002010 201000 100200 3 5 kg m 2 t 0.25 )]TJ/F22 11.9552 Tf 9.299 0 Td [(0.250.25 T rad = s 0 0.3 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.15 T rad 0 000 T rad = s f 000 T rad max 20 N m 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 T 5 s TheresultsforthisexamplearedisplayedinFigure6-6.Therelativeorientation andrelativeangularvelocityplotsinFigure6-6AandFigure6-6B,respectively,indicate thatboththeseparametersconvergetozero.Thecorrespondingcontrolhistoryis giveninFigure6-6C.Thesettlingtimeobtainedisapproximately 20.1 s.Notethatthe runningcostinFigure6-6Dincreaseslinearlywithtimesincethecontrollercontinuesto compensateforthetumblingsatellite. TheplotsoftheadaptiveestimatesareshowninFigure6-7.Theseplotsindicate thattheadaptiveestimatescontinuetochangedependingonthespacecraftsrelative orientation.Thisreinforcesthefactthatthemagnitudeofthecommandedtorqueshould dependonthedynamicsandtherelativeorientation. 101

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APositionhistories BVelocityhistories CControlhistories DRunningcost Figure6-6.AAPFresultsusingSAAequation Figure6-7.AAPFadaptiveestimatesusingSAAequation 102

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Inconclusion,amodicationtotheAPFmethodispresentedinthischapter. Thismodicationisdoneinanefforttoimplementaperformancemetricfortuning theAPFmethod.Asaresult,anadaptiveupdatelawisusedtovarytheweightson theattractiveAPF.ThevelocityproleoftheTPBVPisusedinthederivationsince thissolutionisagoodapproximationtothexedtimeminimumcontroleffortproblem. Astabilityanalysisisalsopresented,whereitisdemonstratedthatconvergenceis onlyguaranteedforobstacle-freeunconstrainedcases.Finally,twoexamplesare performedtodemonstratetheAAPFmethod.Therstisarelativetranslationproblemin aclutteredenvironmentandthesecondisarelativeorientationproblemwithatumbling spacecraft. 103

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CHAPTER7 FINITEHORIZONLINEARQUADRATICPROBLEMS Inthepreviouschapter,theclose-rangerendezvoustrajectorieswereaddressed bydevelopingtheadaptivearticialpotentialfunctionAAPFmethod.However,the AAPFmethodisnotwell-suitedforthenalapproachorendgameofanautonomous proximityoperationAPO.TrajectoriesobtainedfromsolvingalinearquadraticLQ optimalcontrolproblemsOCPsaretypicallyusedforthenalapproachorendgame. Moreover,solvinganitehorizonLQOCPaugmentstheproblemwithatimeconstraint whichisnecessaryforcertainAPOsi.e.,intercept,rendezvouswithuncooperative target,docking, etc. .ThischapterdiscussesthemethodsforsolvingnitehorizonLQ problems.First,theDifferentialRiccatiEquationDREisadoptedfromequation4. Itisshownthatthereexistsasystemoflinearmatrixdifferentialequationswhich areintimatelyrelatedtotheDRE.Furthermore,whentheHamiltonianmatrixistime invariant,theDREissolvedusingusingastatetransitionmatrixSTMrepresentationof thelinearsystemofmatrixdifferentialequations.WhentheHamiltonianistimevarying, thesolutioncannotbeobtainedusingaSTMrepresentation.Asaresult,twomethods ofsolvingaredevelopedusing:PicardIterationPIandHomotopyContinuationHC. Anumericalexampleispresentedtodemonstratetheeffectivenessofsolvingthe DREforaLTVsystemusingPIandHC.Finally,analapproachtrajectoryisobtained usingHCwiththeYamanaka-Ankerson-Tschauner-HempelYATHequationderivedin AppendixA. 7.1Development TrajectoriesforthenalapproachorendgameofanAPOaretypicallyobtainedby solvingaLQoptimalcontrolproblemOCP.Thisisbecausealinearizedmodelserves asagoodapproximationinthislevelofproximityandusingalinearizedmodelsimplies thesolutionprocess.Moreover,thecontrollawobtainedfromanLQproblemisinthe statefeedbackformwhichmakesitcompliantforimplementation.Itwasshownin 104

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Chapter4thatsolvinganitehorizonLQOCPinequation4amountstosolvinga nalvalueproblemwiththeDREinequation4.ThedifcultyhereisthattheDRE isanonlinearequation,whichingeneral,cannotbeanalyticallysolved.Thus,solvinga nalvalueproblemwiththeDRErequiresthattheDREbeintegratedbackwardsintime everytimeanewsolutionisneeded. AninterestingpropertyoftheDREisthatitisintimatelyrelatedtothesystemof linearmatrixdifferentialequations 2 6 4 U V 3 7 5 = 2 6 4 A )]TJ/F45 11.9552 Tf 9.299 0 Td [(BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T )]TJ/F45 11.9552 Tf 9.299 0 Td [(Q )]TJ/F45 11.9552 Tf 9.298 0 Td [(A T 3 7 5 2 6 4 U V 3 7 5 2 6 4 U t f V t f 3 7 5 = 2 6 4 I S f 3 7 5 where V U 2 R n n .[91,92]NotethatthestatematrixistheHamiltonianmatrixfrom equation4.If U )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 exists,thenthesolutiontotheDREcanbewrittenusingthe variabletransformation S = VU )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 Thisisseenifequation7issubstitutedintothetimederivativeofequation7as shownbelow. S = VU )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(VU )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 UU )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 S = )]TJ/F45 11.9552 Tf 9.298 0 Td [(QU )]TJ/F45 11.9552 Tf 11.955 0 Td [(A T V U )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(VU )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 AU )]TJ/F45 11.9552 Tf 11.955 0 Td [(BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T V U )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 S = )]TJ/F45 11.9552 Tf 9.299 0 Td [(Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(A T )]TJ/F45 11.9552 Tf 5.479 -9.683 Td [(VU )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F31 11.9552 Tf 11.955 9.683 Td [()]TJ/F45 11.9552 Tf 5.48 -9.683 Td [(VU )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 A )]TJ/F31 11.9552 Tf 11.955 9.683 Td [()]TJ/F45 11.9552 Tf 5.48 -9.683 Td [(VU )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T )]TJ/F45 11.9552 Tf 5.479 -9.684 Td [(VU )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 S = )]TJ/F45 11.9552 Tf 9.299 0 Td [(Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(A T S )]TJ/F45 11.9552 Tf 11.955 0 Td [(SA + SBR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S NotethattheboundaryconditionontheDREisalsosatisedusingthevariable transformationinequation7asshownbelow. S t f = V t f U )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t f = S f 105

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Therefore,ifthesolutionfor U and V canbeobtained,thentheoptimalcontrollaw isdenedusingthevariabletransformationinequation7.Asaresult,theoptimal controllawfromequation4isdenedas u = )]TJ/F45 11.9552 Tf 9.298 0 Td [(R )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T VU )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 x 7.2LinearTime-InvariantSystem WhentheHamiltonianmatrixistimeinvariant,equation7canbesolvedusinga STMrepresentation.[9294]Inparticular,fortheconstantHamiltonianmatrix H c = 2 6 4 A )]TJ/F45 11.9552 Tf 9.298 0 Td [(BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T )]TJ/F45 11.9552 Tf 9.298 0 Td [(Q )]TJ/F45 11.9552 Tf 9.299 0 Td [(A T 3 7 5 thesolutiontoequation7iswrittenas 2 6 4 U t V t 3 7 5 =exp H c t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f 2 6 4 I S f 3 7 5 = t t f 2 6 4 I S f 3 7 5 where t t f istheSTMofthesystemoflinearmatrixdifferentialequationsin equation7. 7.3LinearTime-VaryingSystem WhentheHamiltonianmatrixistimevarying,theSTMisnoteasilysolvedforin general.Inprinciple,theSTMdoesexist,however,thedifcultyisinobtainingtheactual formoftheSTM.Toaddressthisdifculty,twomethodsareinvestigatedforsolvingthe DRE:PIandHC.BothPIandHCareknowntobeconvergentingeneral.[95,96]Thus, obtainingthesolutionusingPIand/orHCcanbeusedratherthanhavingtointegratethe DREbackwardsintime. 7.3.1PicardIteration ThePIisaxedpointiterativetechniqueforsolvingdifferentialorintegral equations.[95]Notethatthenalvalueproblemwithequation7isadifferential equation.Thus,iftheHamiltonianmatrixistimevarying,thenthenalvalueproblemin 106

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equation7canbesolvedusingPI.First,thevariable X isdenedas X = 2 6 4 U V 3 7 5 sothatequation7iswrittenas X = H t X X t f = X f where H t isthetimevaryingHamiltonianmatrix.UsingPI,thesolutiontothe k +1 th iterationiswrittenas X k +1 t = X f + Z t t f H t X k d ThePIrequiresaninitialguessi.e.,thexedpoint.Using X = 0 astheinitialguess, thesolutionas k !1 is X t = I + Z t t f H t 1 d 1 + Z t t f H t 2 Z 2 t f H t 1 d 1 d 2 +... Z t t f H t 3 Z 3 t f H t 2 Z 2 t f H t 1 d 1 d 2 d 3 + # X f X t = t t f X f Theintegraltermscanbecomputedanalyticallyusingasymbolicmanipulator oraquadratureruleforanumericalapproximation.However,analyticexpressions arefavorablesincethequadraturerulewouldhavetobedoneateachtimestepand recursivelyforeachadditionalterm.NotethatiftheHamiltonianmatrixisconstant, thenthesolutionsimplybecomestheTaylorSeriesExpansionTSEofthematrix exponentialabout t = t f whichisthesolutioninequation7.However,ifthe Hamiltonianisnotconstant,thenthePicardIterationprovidesasolutionfortheSTM. 107

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7.3.2HomotopyContinuation Givenasmoothnonlinearoperator N x ,theHCcanbeusedtosolve N x = 0 whichcannotbeanalyticallysolvedingeneral.[97]Thisisdonebydeninga homotopy,whichisacontinuoustopologicalmapthatinjects L x N x asthe embeddingparameter p 1 .Theideaistocontinuouslyembedthesolutionof L x = 0 whichisknowntothesolutionof N x = 0 .[96,97]ThebenetthatHChasoverother perturbationtechniquesisthattheradiusandrateofconvergencecanbeincreased dependingontheparameterschoseninthesolutionformulation.[96]Itisknownthat HCisaneffectivetoolforsolvingawiderangeofproblemse.g.,algebraic,ordinary differential,partialdifferential,boundaryvalue, etc. .[96]Theapproachforsolvingthe DREusingHCisdiscussedinthissection. First,theconvexhomotopyisdened H x p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p [ L x )]TJ/F40 11.9552 Tf 11.955 0 Td [(L y 0 ] )]TJ/F40 11.9552 Tf 11.955 0 Td [(ph t [ N x ] = 0 where p 2f 2 R j 0 1 g istheembeddingparameter, h 6 =0 istheauxiliary parameter, t 6 = 0 istheauxiliaryfunction,and y 0 isaninitialguess.[96]Notethatthe homotopyhastheproperty H x ,0= L x )]TJ/F40 11.9552 Tf 11.955 0 Td [(L y 0 = 0 H x ,1= N x = 0 Next,theparameter x iswrittenasaMaclaurinSeriesExpansionMSEinthe embeddingparameter x = x 0 + p x 1 + p 2 x 2 + 108

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wheretheparameters x 0 x 1 x 2 ,... aretobedetermined.Notethatthesolutionto N x = 0 cannowbewrittenas x =lim p 1 x 0 + p x 1 + p 2 x 2 + = 1 X i =0 x i Therefore,solvingequation7amountstosolvingfortheparameters x 0 x 1 x 2 ,... Theseparametersaresolvedforbysubstitutingequation7intoequation7. AppendixCdiscussesthesolutionderivationusingtheoriginalDREastheoperator N S = S + Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S + A T S + SA = 0 whilechoosing L S = S AppendixDdiscussesthesolutionderivationusingtheoperator N X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X = 0 whichisthesystemoflinearmatrixdifferentialequationsrelatedtotheDRE,and choosing L X = X AppendixEdiscussesthesolutionderivationusingequation7astheoperatorand choosing L X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(X Basedontheformulation,therearefreeparametersthatcanbemanipulated. Namely,thesearetheoperator L x ,theinitialguess y 0 ,theauxiliaryparameter h ,and theauxiliaryfunction t .Theseparameterscanbechosensuchthattheradiusand rateofconvergenceisimproveddependingontheproblembeingsolved.[96]Inthe 109

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solutionderivationsintheappendices,theseparametersarechosenbydefaultexcept fortheauxiliaryparameter h .However,athoroughanalysisshouldbeperformedto determinethesetofparametersthatyieldthedesiredresults. AfeatureofthePIandHCistheparametricsolutionstructureoftheRiccati matrix.Sincethesolutionsarefunctionsofthetimehorizonandthestate,input,and LQmatrices,thenthesolutioniseasilyupdated.Thismakesthesolutionstructure amenabletochangesinthetimehorizon,dynamicsmodel,and/orperformanceindex. 7.4NumericalExamples AnexampleispresentedtodemonstratehowwellthesolutiontotheDREcanbe approximatedusingPIandtheHC.Thisexampleisusedsincelinearizedmodelsof dynamicalsystemstendtohaveoscillatorycomponentsintheirdynamics.Inparticular, thisistrueformodelsofrelativetranslationalmotioninorbitasshowninAppendixA. [98,99] Atwodegrees-of-freedomlinearsystemisusedwiththedynamics 2 6 4 x 1 x 2 3 7 5 = 2 6 4 01 sin t 0 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 0 1 3 7 5 u 2 6 4 x 1 t 0 x 2 t 0 3 7 5 = 2 6 4 2 1 3 7 5 x = A t x + B u x t 0 = x 0 TheparametersandmatricesthatareassociatedwiththenitehorizonLQproblemare giveninTable7-1. Table7-1.Finitehorizonlinearquadraticparameters ParametersValueUnits S f I 2 R 2 2 QI 2 R 2 2 RI 2 R 1 1 t 0 0 s t f 20 s 110

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7.4.1PicardIteration UsingthePIapproachtosolvetheDRE,theresultsareshowninFigure7-1.The resultswerecomputedfor 0 th to 7 th orderapproximations.Thefourcomponentsof theRiccatimatrix S areshowninFigure7-1AthruFigure7-1D,where S i j indicates theoptimalsolution.Theplotsindicatethataddinganorderofapproximationdoes notnecessarilyyieldabetterresult.Infact,itisshownthatonlytheevenordered approximationsyieldedfavorableestimates.Themaxdifferencebetweenthecomponents of S k and S isshowninTable7-2.Itisshownthatincreasingtheorderofapproximation didimprovethesolutionabouttheboundaryconditionat t = t f Table7-2.MaximumdifferenceofRiccatimatrixcomponentsPI Approximationorder S 1,1 S 1,2 S 2,2 S 2,2 02 282 292 291 35 146 0444 3645 6643 82 22 212 102 051 19 322 9914 8820 5213 07 42 111 841 891 03 56 35 e 34 44 e 36 09 e 34 26 e 3 62 121 801 731 04 790 4568 8080 3861 00 Theresultingsolutionswerethenusedtodeterminethecontrol,andinturn theresponseofthestates.ThestatesareshowninFigure7-1Eandthecontrolis showninFigure7-1F,where x and u indicatetheoptimalstatesandoptimalcontrol, respectively.Thestatesconvergetotheoptimalsolution,butatthecostofalarge initialcontrolvalue.Thesetofstatesthatconvergedfasterwerefortheevennumbered approximationssincetheRiccatimatricesforthesecaseswerefairapproximations. TheresultingcostsobtainedforeachapproximationaregiveninTable7-3where J indicatestheoptimalcost.Thelowestcostwasobtainedforthe 3 rd orderapproximation, however,thisisstillfarfromtheoptimalcost.Notethatthe 5 th orderapproximation resultedinalargecost.Itwasdeterminedthatthiswascausedbythe U becoming ill-conditioned. 111

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A S 11 B S 12 C S 21 D S 22 EStates FControl Figure7-1.PicardIterationresults 112

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Table7-3.Costsforeachapproximatingmethod CostPIHCcase1HCcase2HCcase3 J 8 448 448 448 44 J 223 18223 18223 18223 18 J 92 34229 2031 9315 67 J 91 35240 0241 1811 13 J 29 78243 158 8011 70 J 59 12250 0311 7311 73 J 1 25 e 32282 329 3611 41 J 43 76369 908 90 -.J 57 43 -.--.--.7.4.2HomotopyContinuation ThreedifferentcasesofHCwereattemptedforsolvingtheDRE.Therstcaseis forthehomotopydenedbytheoperatorsinequation7andequation7.The resultsobtainedareshowninFigure7-2,wherethevaluefortheauxiliaryparameter usedis h = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 20 .Theresultswerecomputedforthe 0 th to 6 th orderapproximations. ThefourcomponentsoftheRiccatimatrix S areshowninFigure7-2AthruFigure7-2D. TheresultsindicatethatthisHCapproachdoesnotimproveastheorderofthe approximationincreases.Also,theresultsobtaineddonotseemtohavethesame trendastheoptimalsolution.Themaxdifferencebetweenthecomponentsof S k and S isshowninTable7-4. Table7-4.MaximumdifferenceofRiccatimatrixcomponentsHCcase1 Approximationorder S 1,1 S 1,2 S 2,2 S 2,2 02 282 292 291 35 11 971 672 291 35 21 682 502 281 34 31 923 392 261 33 42 994 092 241 32 54 084 682 221 36 65 165 212 221 51 Theresultingsolutionsarethenusedtodeterminethecontrol,andinturnthe responseofthestates.ThestatesareshowninFigure7-2Eandthecontrolisshown inFigure7-2F.Theresultsconvergeforeachcase,butdonotnecessarilyimprove 113

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astheorderofapproximationincreases.Thiscanbeimprovedbychoosingabetter combinationoftheoperator L S ,theauxiliaryparameter h ,theauxiliaryfunction t ,andtheinitialapproximation Y 0 .Theresultingcostsobtainedforeachorder approximationaregiveninTable7-3.Thebestresultswereobtainedforthe 1 st order approximationsincethisgavethebestapproximationfortheRiccatimatrixatthe initialtime.NoticethatsincetheDREisbeingsolveddirectly,thereisnoconcernfor ill-conditionedmatrices. Thesecondcaseisforthehomotopydenedbytheoperatorsinequation710 andequation7.TheresultsobtainedareshowninFigure7-3wherethevalue fortheauxiliaryparameterusedis h = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 2 .Theresultswerecomputedforthe 0 th to 6 th orderapproximations.ThefourcomponentsoftheRiccatimatrix S areshownin Figure7-3AthruFigure7-3D.TheresultsindicatethatthisHCapproachdoeshave consistentimprovementastheorderoftheapproximationincreases.Theresultstend tohavethesametrendastheoptimalsolutionaswell.Themaxdifferencebetweenthe componentsof S k and S isshowninTable7-5. Table7-5.MaximumdifferenceofRiccatimatrixcomponentsHCcase2 Approximationorder S 1,1 S 1,2 S 2,2 S 2,2 02 282 292 291 35 112 9212 1712 5411 67 22 131 881 840 98 31 030 990 990 48 41 551 161 130 39 51 121 000 920 46 61 050 840 720 26 Theresultingsolutionswerethenusedtodeterminethecontrol,andinturnthe responseofthestates.ThestatesareshowninFigure7-3Eandthecontrolisshownin Figure7-3F.Theresultsdoconvergeandstayneartheoptimalsolutionastheorderof theapproximationincreases.Theresultingcostsobtainedforeachorderapproximation aregiveninTable7-3.Thebestresultswereobtainedforthe 3 rd orderapproximation. Thisagainmightbecoincidentalsincethisapproximationmighthaveyieldedagood 114

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A S 11 B S 12 C S 21 D S 22 EStates FControl Figure7-2.Homotopycontinuationcase1results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 20 115

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approximationfortheinitialvalueoftheRiccatimatrix.Despitethis,itisshownthatthe homotopyusedyieldsagoodapproximationtothesolutiontotheDRE. Thethirdcaseisforthehomotopydenedbytheoperatorsinequation7 andequation7.TheresultsobtainedareshowninFigure7-4wherethevalue fortheauxiliaryparameterusedis h = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 2 .Theresultswerecomputedforthe 0 th to 5 th orderapproximations.ThefourcomponentsoftheRiccatimatrix S areshown inFigure7-4AthruFigure7-4D.TheresultsindicatethatthisHCsolutionshows improvementimmediatelyafterthe 0 th approximation.However,theimprovementswith thesubsequentapproximationsseemtoplateau.However,theresultsdohavethesame trendastheoptimalsolution.Themaxdifferencesbetweenthecomponentsof S k and S areshowninTable7-6. Table7-6.MaximumdifferenceofRiccatimatrixcomponentsHCcase3 Approximationorder S 1,1 S 1,2 S 2,2 S 2,2 02 282 292 291 35 11 531 541 541 01 21 341 161 140 63 31 441 181 160 55 41 441 191 160 56 51 411 171 140 56 Theresultingsolutionswerethenusedtodeterminethecontrol,andinturnthe responseofthestates.ThestatesareshowninFigure7-4Eandthecontrolisshownin Figure7-4F.Theresultsdoconvergeandstayneartheoptimalsolutionastheorderof theapproximationincreases.Theresultingcostsobtainedforeachorderapproximation aregiveninTable7-3.Thebestresultswereobtainedforthe 2 nd orderapproximation. Thisagainmightbecoincidentalsincethisapproximationmighthaveyieldedagood approximationfortheinitialvalueoftheRiccatimatrix. 7.5Yamanaka-Ankerson-Tschauner-HempelExample AnexampleusingtheYATHequationderivedinAppendixAispresentedinthis section.Basedontheexampleintheprevioussection,theHCCase2yieldedthe 116

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A S 11 B S 12 C S 21 D S 22 EStates FControl Figure7-3.Homotopycontinuationcase2results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 2 117

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A S 11 B S 12 C S 21 D S 22 EStates FControl Figure7-4.Homotopycontinuationcase3results h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 2 118

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closestresulttotheoptimalsolution.Inthissection,theHCCase2approachshownin AppendixDisusedwherethesolutionisobtaineduptothefourthorderapproximation. Thevalueoftheauxiliaryparameterusedis h = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 = 10 .Theparametersassociated withtheLQproblemaregiveninTable7-7.Notethattheindependentvariableforthis setofequationsisthetrueanomalyofthetarget. Table7-7.Finalapproachparameters ParametersValueUnits S f 10 3 I0 0 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 I 2 R 6 6 Q 10 3 I0 0 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 I 2 R 6 6 R 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 I 2 R 3 3 e 0.15 k 0.0256 p rad = s r 0 10 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1010 T m v 0 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(00 T m = s 0 0 deg f 10 deg TheresultsobtainedareshowninFigure7-5.Therelativepositionhistoryisshown inFigure7-5Awheretheblacklinerepresentstheoptimalsolutionandtheotherlines representthedifferentapproximationorders.Thisgureshowsthatthesolutionsbegin toconvergeaftertherstorderapproximation.Inaddition,thesolutiongetscloseto theoptimalsolutionastheorderoftheapproximationincreases.Therelativevelocity historiesareshowninFigure7-5B.Theseresultsindicatethatthevelocityconverges tozeroaftertherstorderapproximationaswell.Thecorrespondingcontrolhistories areshowninFigure7-5CandazoomedviewofthisplotisshowninFigure7-5D.These plotsshowthattheapproximationsrequirearelativelylargeinitialcontrolaction.Thisis likelyduetoapoorapproximationtotheRiccatimatrixattheinitialtime.Despitethis, thetrajectorieswereshowntoconvergetotheorigin.Thecostsassociatedwitheach trajectoryareshowninTable7-8.Thistableshowsthecostcontinuestodecreaseas theorderoftheapproximationincreases. 119

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APositionhistories BVelocityhistories CControlhistories DControlhistorieszoomedview Figure7-5.Homotopycontinuationresultsfornalapproachexample h = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 = 10 Table7-8.CostsfornalapproachusingHC CostHCcase2 J 6 74 e 3 J 1 78 e 5 J 7 72 e 4 J 7 41 e 4 J 7 13 e 4 J 7 08 e 4 120

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Inthischapter,themethodsforsolvinganitehorizonLQproblemforbothanLTI andLTVsystemsarediscussed.ForthecasewheretheHamiltonianmatrixisconstant, itisshownthatsolutionisdeterminedbyobtainingtheSTMoflinearsystemofmatrix differentialequationsrelatedtotheDRE.WhentheHamiltonianmatrixisnotconstant, twomethodsarepresentedforobtainingthesolution.Forthenumericalexampleshown inthischapter,thebestresultisobtainedbyusingtheHCapproachwiththemapping betweenequation7andequation7.Thismappingisthenusedtosolve thenalapproachproblemwiththeYATHequationspresentedinAppendixA.Itis shownthattheaftertherstorderapproximation,thesolutionsconvergetotheorigin.In addition,astheorderoftheapproximationincreased,thesolutionmatchesmoreclosely totheoptimalsolution. 121

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CHAPTER8 NUMERICALRESULTS Theresultsfromthenumericalanalysesperformedarepresentedanddiscussed inthischapter.First,theresultsoftheMonteCarlosimulationsMCSsperformed characterizetheperformanceandconvergencecharacteristicsfortheArticialPotential FunctionAPFandAdaptiveArticialPotentialFunctionAAPFmethods.The MCSsareperformedwithboththeClohessy-Wiltshire-HillCWHandsmallangle approximationSAAequations.Next,theresultsofimplementingtheoptimalsolutions obtainedinChapter4inthehighdelitymodelarepresented.Twomethodsoftracking areused:trajectorytrackingandforcetracking.Theaimofthisstudyistodetermine howwellthespacecraftisabletotrackthetrajectoriesobtainedandtocharacterize thediscrepanciesbetweentheoptimaltrajectoryandtherealizedtrajectory.TheAPF andAAPFmethodsarealsoimplementedinthehighdelitymodeltodeterminehow thesemethodsperforminahigherdelitymodel.Next,aclose-rangerendezvous scenariowithobstacleavoidanceispresented.Thescenarioissimulatedwithaplanned trajectory,theAPFmethod,andtheAAPFmethod.Lastly,analapproachscenariois presentedwhichusesthenitehorizonLQsolutionusingtheCWHequationandthe statetransitionmatrixSTMsolutionpresentedinChapter7. 8.1MonteCarloSimulations ItwasshowninChapter6thatastabilityprooffortheAPFandAAPFmethodsonly existsfortheobstacle-freecase.Moreover,thereisnomethodologytocharacterize theperformanceofthesemethods.Asaresult,MCSsareperformedtodemonstrate performanceandconvergencecharacteristicsoftheAPFandAAPFmethods.Two MCSsareperformedforrelativetranslationi.e.,CWHequationandrelativeorientation i.e.,SAAequation. IntherelativetranslationMCS,theinitialconditionsofthechaserarevaried.The initialconditionsarechosenasnormallydistributedrandomvectorswith r 0 N 0 ,500 I 122

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v 0 N 0 I ,andwiththeconstraints 250 m k r 0 k 1000 m k r 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [(r i k 50 mfor i =1,2,3,...,14 Theseconstraintsareenforcedtoensurethatthereisaconsiderableseparation initiallybetweenthechaser,target,andobstacleswhilemaintainingthecloseproximity assumption.Asetof14staticobstaclesareplacedinthestatespaceatthepositions shownbelow. r 1 = 50000 T r 2 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(50000 T r 3 = 05000 T r 4 = 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5000 T r 5 = 00500 T r 6 = 00 )]TJ/F22 11.9552 Tf 9.298 0 Td [(500 T r 7 = 250250250 T r 8 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250 T r 9 = 250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 10 = 250 )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 11 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250250250 T r 12 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 T r 13 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(250250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T r 14 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 )]TJ/F22 11.9552 Tf 9.299 0 Td [(250 T TheinitialpositionsandobstaclesareillustratedinFigure8-1Awherethebluecircles representtheinitialpositionsandtheblackdotsrepresentthestaticobstacles.The parametersusedinthisMCSaregiveninTable8-1. Table8-1.MonteCarlosimulationparametersCWHequation ParameterAPFValueAAPFValueUnits n 0.00120.0012 rad = s r f 00 m f max 11 m = s PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 N i I 2 R 3 3 I 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 i 1.5 10 5 1.5 10 5 i 1.0 10 4 1.0 10 4 m 2 k 0.0020.002 T 1000 s 123

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IntherelativeorientationMCS,theinitialconditionsofthechaserandtumbling rateofthetargetarevaried.Theinitialconditionswerechosenasnormallydistributed randomvectorswith 0 N 0 ,0.1 I 0 N 0 ,0.1 I t N 0 ,0.05 I ,andthe constraints 0.1 rad k 0 k 0.25 rad k 0 k 0.25 rad = s k t k 0.25 rad = s. Theconstraintsareenforcedtomaintainthesmallangleassumptionyethaveadistinct initialrelativeorientationandtumblingratebetweenthechaserandtarget.Obstacles arenotincludedinthisMCS.However,theycouldhavebeenincludedtorepresent keepoutzonesintherelativeorientationstates.Theinitialorientationsusedare illustratedinFigure8-1B.TheparametersusedinthisMCSaregiveninTable8-2. Table8-2.MonteCarlosimulationparametersSAAequation ParameterAPFValueAAPFValueUnits f 00 rad max 2020 m = s PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 k 2 T 5 s Settlingtimeisusedtocharacterizeconvergencewherethesolutionisassumedto havesettledwhen r 10 mfortherelativetranslationcaseand 0.01 radforthe relativeorientationcase.Performanceischaracterizedbasedonthecontroleffortcost J = Z t f t 0 k u k 1 dt where u isthecontrolparameterand t f isthesettlingtime. TheresultsobtainedfortherelativetranslationMCSareshowninFigure8-2.The costdataisshowninFigure8-2AwheretheaveragecostobtainedusingtheAPF 124

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ACWHinitialpositionsandobstaclepositions BSAAinitialorientations Figure8-1.Initialrelativeposition/orientationsusedinMonteCarlosimulations methodis 8.60 m = sandtheaveragecostobtainedusingtheAAPFmethodis 4.12 m = s. ThesettletimedataisshowninFigure8-2Bwheretheaveragesettlingtimeobtained withtheAPFmethodis 1315 sandtheaveragesettlingtimeobtainedusingtheAAPF methodwas 1209 s.TheseresultsindicatethattheAAPFmethodcommandsless controleffortthantheAPFmethodintherelativetranslationscenario.Inaddition,the AAPFmethodisabletospecifyatimeconstraintbasedonthelargelyskewedtothe rightdatainFigure8-2BfortheAAPFmethod.Alargemajorityofthedatapoints centeredaroundthe 1000 sisaresultofchoosingthetransfertime T =1000 s.The resultsoftheMCSindicatethattheAAPFmethodcommandslesscontroleffortglobally whilebeingabletochooseatimecriterioninthetransfertime T TheresultsobtainedfortherelativeorientationMCSareshowninFigure8-2.The costdataisshowninFigure8-2CwheretheaveragecostobtainedusingtheAPF methodis 207.7 N mwhiletheaveragecostobtainedusingtheAAPFmethodis 65.03 N m.ThesettletimedataisshowninFigure8-2Dwheretheaveragesettlingtime obtainedusingtheAPFmethodis 6.85 sandtheaveragesettlingtimeobtainedusing theAAPFmethodis 8.97 s.TheseresultsindicatethattheAAPFmethodcommands lesscontroleffortthantheAPFmethodintherelativeorientationscenario.However,a 125

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rmtimeconstraintisnotseenbasedonthetransfertimechosenfortheAAPFmethod. Thisisduetothelinearizedmodeli.e.,SAAequationandtheconstantcompensation forthetumblingratesofeachsamplepoint.Althoughtheaveragesettlingtimewas slightlylargerfortheAAPFmethod,thecontroleffortwasconsiderablylower. ACWHcostdata BCWHsettlingtimedata CSAAcostdata DSAAsettletimedata Figure8-2.CostsandsettlingtimesdatafromMonteCarlosimulations 8.2HighFidelitySimulations Thesimulationresultspresentedinthissectionaregeneratedusingthehigh delitymodeldiscussedinChapter3.Theparametersusedforthetargetandchaser spacecraftaregiveninTable8-3andTable8-4,respectively.Inthesimulations,the optimaltrajectoriesobtainedinChapter4areimplementedusingtwomethodsof 126

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tracking:trajectorytrackingandforcetracking.Sincethetrajectoriesobtainedin Chapter4areonlydenedatthecollocationpoints,interpolationisusedforboth trackingmethodstoobtainacontinuousreferencesignal. Table8-3.Targetspacecraftparameters ParameterValueUnits m t 100 kg l t 3 m r t t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.9593486817.396210904.495486 T km v t t 0 0.175776 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.9637767.494102 T km = s J t 2 4 3002010 201000 100200 3 5 kg m 2 q t t 0 0001 T t t 0 000 T rad = s Table8-4.Chaserspacecraftparameters ParameterValueUnits m c 100 kg l c 3 m r c t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.3394116816.939420904.891745 T km v c t 0 0.175776 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.9637767.494102 T km = s f max p 3 m = s 2 f db 0.01 m = s 2 J c 2 4 3002010 201000 100200 3 5 kg m 2 q c t 0 1000 T c t 0 000 T rad = s max 5000 rad = s max 100 rad = s 2 Thetrajectorytrackingmethodinvolvestrackingtheoptimalstatesobtainedfrom theoptimalsolution.Acubicsplineisusedtointerpolatebetweenthecollocationpoints oftheoptimaltrajectorysincecubicsplinesaresmoothi.e.,ensurethederivativeof theinterpolatingpolynomialiscontinuousanddonothavetheoscillatorybehaviorof 127

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high-degreepolynomialinterpolation.[73]Totrackaposition,theerrorisdenedas e d = r c )]TJ/F45 11.9552 Tf 11.955 0 Td [(r c where r c istheoptimalpositionofthechaser.Asaresult,theproportional-integral-derivative PIDcontroller f = k p e d + k i Z t t 0 e d d + k d e d isused,where k p =0.6 k i =0.06 ,and k d =1.5 arethetunedPIDcontrollergains.[71] ThePIDcontrollerdenestheforcerequiredtotracktheoptimalchaserpositionbased onthetruechaserposition.Totrackanorientation,theerrorquaternionisdened accordingtoequation2 q e = q c q )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 c where q c istheoptimalattitudetrajectoryofthechaser.Asaresult,theeigenaxis attitudecontroller = )]TJ/F40 11.9552 Tf 9.299 0 Td [(k J c e )]TJ/F40 11.9552 Tf 11.955 0 Td [(c J c c + c J c c isused,where e isthevectorcomponentoftheerrorquaternionand k =9.54 c =5.5 aregains.[100]Theeigenaxiscontrollergeneratesthetorquerequiredtoperforman eigenaxisslewi.e.,minimumeigenangleslewtowardstheoptimalchaserorientation basedonthetruechaserorientation.[100] Theforcetrackingmethodinvolvestrackingtheoptimalcontrolandtorqueproles usingtherespectiveactuators.ApiecewisecubicHermiteinterpolatingpolynomial isusedtointerpolatebetweenthecollocationpointsofthecontrol.Thismethodof interpolationischosensincetheoptimalcontrolissometimesdiscontinuousand piecewisepolynomialscanhandlethisphenomenon.[73]Asaresult,theinterpolated optimalcontrolisfedforwardtotheactuatorsasthecommandedcontrolaction. 128

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TheAPFandAAPFmethodsarealsoimplementedinthehighdelitymodel. ThesesimulationsareperformedtodeterminehowwelltheAPFandAAPFmethods performinahighdelitymodel.Bothmethodsaredesignedtore-plantrajectories untiltheterminalconditionsaremet.Aclose-rangerendezvousscenariowithobstacle avoidanceisalsosimulated.ThetrajectoryobtainedinChapter4isimplementedusing trajectorytracking.TheAPFandAAPFmethodsarealsoimplementedtodemonstrate close-rangerendezvouswithobstacleavoidance. Thelastsetofresultspresentedarethenalapproachscenario.Thenalapproach isdoneusingthenitehorizonLQsolutionusingtheCWHequation.Thefeedback controllawforthisapproachisdevelopedinChapter7.Thiscontrollerisimplementedto demonstratehowusingalinearizedmodelcanbeusedtodevelopafeedbackcontroller whichisimplementableinhigherdelitydynamics. 8.2.1MinimumTimeTrajectories Figure8-3showstheresultsofimplementingtheminimumtimetrajectoriesusing trajectorytrackinginthehighdelitymodel.Themagnitudeoftherelativeposition isshowninFigure8-3Awherethebluelinerepresentstherelativepositionbetween thechaserandtargetandthegreenlinerepresentstherelativepositionbetweenthe chaserandtheoptimaltrajectory.Thisgureindicatesthattheoptimaltrajectoryis trackedwithin 5 mandthedistancebetweenthechaserandtheoptimaltrajectoryis approximately 0.5 matthenaltime.However,thedistancebetweenthechaserandthe targetisapproximately 10 m.Thisdeviationissmallandisduetodisturbancesand/or inaccuraciesintheoptimalsolution. TheorientationerrorisshowninFigure8-3Bwherethebluelinerepresentsthe erroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandoptimaltrajectory.Thisgureindicatesthattheattitude trajectorycanbetrackedwithina 10 errorangle.Theerroranglebetweenthechaser 129

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andtargetandbetweenthechaserandoptimaltrajectoryisvirtuallythesamei.e., about 1 atthenaltime.Thedeviationisdueinaccuraciesintheoptimalsolution. TherelativepositionplotsareshowninFigure8-3Candthequaterniontrajectories areshowninFigure8-3D.Bothmatchcloselytotheoptimaltrajectories,whichindicates thattherespectivedynamicmodelsusedareaccurateforminimumtimetrajectories. TheforceplotisshowninFigure8-3EandthetorqueplotisshowninFigure8-3F. Theforceplotindicatesthattrajectorytrackingleadstoacontrolhistorysimilarto theoptimalcontrolhistory.Thetorqueplotindicatesthattrajectorytrackingdoesnot necessarilyleadtoacontrolhistorysimilartotheoptimalcontrolhistory.Inparticular, therstcomponentofthetorquematchestheoptimalcontrol;however,thesecondand thirdcomponentsdonot. Next,Figure8-4showstheresultsofimplementingtheminimumtimetrajectories usingforcetrackingwiththehighdelitymodel.Therelativepositionmagnitudeis showninFigure8-4Awherethebluelinerepresentstherelativepositionbetweenthe chaserandtargetandthegreenlinerepresentstherelativepositionbetweenthechaser andtheoptimaltrajectory.Thisgureindicatesthattheoptimalforcedoesnotyieldthe sameresponseinahigherdelitymodel.Atthenaltime,therelativepositionbetween thechaserandtheoptimaltrajectoryisapproximately 10 mandtherelativeposition betweenthechaserandthetargetisapproximately 3 m.Itisequivocalastowhyforce trackingyieldsabetterresponseinthehighdelitymodel. TheorientationerrorplotisshowninFigure8-4Bwherethebluelinerepresents theerroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandtheoptimaltrajectory.Thisgureindicatesthattheerror anglebetweenthechaserandtargetandbetweenthechaserandtheoptimaltrajectory arevirtuallythesameapproximately 5 atthenaltime.Thisisevidencethatthe optimalcontroldoesnotyieldthesameresponseinthehighdelitymodelandasa resulttheterminalconditionsarenotsatised.Thisislikelyduetoadriftintheattitude 130

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-3.Minimumtimetrajectorytrackingresults 131

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ofthetargetcausedbydisturbances.Anothersourceoferrormightbethebang-bang controlstructure.Thiscontrolstructurerequiresthatadiscontinuousjumpbemadein thecontrolwhichisnotphysicallyrealizablewiththereactionwheels. TherelativepositionplotsareshowninFigure8-4Candthequaterniontrajectories areshowninFigure8-4D.Bothplotsindicatethattheresultanttrajectoryissimilar totheoptimaltrajectory.Thisindicatesthatthemodelsusedforgeneratingthe optimalsolutionsareaccurateforminimumtimeproblems.Theforceplotisshown inFigure8-4EandthetorqueplotisshowninFigure8-4F.Theseplotsindicatethatthe optimalcontrolcanbetrackedaccuratelybytheactuators. 8.2.2FixedTimeMinimumControlEffortTrajectories Figure8-5showstheresultsofimplementingthexedtimeminimumcontroleffort trajectoriesusingtrajectorytrackingwiththehighdelitymodel.Themagnitudeof therelativepositionisshowninFigure8-5Awherethebluelinerepresentstherelative positionbetweenthechaserandtargetandthegreenlinerepresentstherelative positionbetweenthechaserandtheoptimaltrajectory.Thisgureindicatesthatthe optimaltrajectoryistrackedaccuratelyi.e.,withina 1 merror.Theplotshowsahigh frequencylowamplitudetrackingsignalwhichimpliesthePIDcontrollerisaccurately trackingthetrajectory.However,thepositionofthechaserisnotnearlythesameasthe positionofthetargetapproximately 1 kmawayatthenaltime.Thisisduetosecular disturbingeffectswhichwerenotmodeledinthereferencetrajectories. TheorientationerrorisshowninFigure8-5Bwherethebluelinerepresentsthe erroranglebetweenthechaserandtargetandthegreenlinerepresentstheerrorangle betweenthechaserandoptimaltrajectory.Thiserrorplotisevidencethattheoptimal attitudetrajectorycanbetrackedwithina 1 errorangleandthatthenalorientation matchestheresultsfromthehigherdelitymodelwithina 0.1 errorangleatthenal time. 132

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-4.Minimumtimeforcetrackingresults 133

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TherelativepositionplotsshowninFigure8-5Cillustratetheseculardisturbing effects.Thetrajectoriesareinitiallycoincidentandbegintodriftafterapproximately 100 s.Sincethedisturbancesaffectthechaserandthetarget,theerrorgoesunaccounted forinbothspacecraft.Thisresultismotivationforaneedtore-planforunmodeled effects. ThequaterniontrajectoriesshowninFigure8-5Dmatchcloselywhichagrees withtheresultsinFigure8-5B.Thisindicatesthatthemodelusedforrotationalmotion isaccurate.TheforceplotisshowninFigure8-5Eandthetorqueplotisshownin Figure8-5F.Boththeforceandtorqueplotsindicatethattrajectorytrackingleadstoa controlhistorysimilartotheoptimalcontrolhistory. Next,Figure8-6showstheresultsofimplementingthexedtimeminimumcontrol efforttrajectoriesusingforcetrackingwiththehighdelitymodel.Therelativeposition magnitudeisshowninFigure8-6Awherethebluelinerepresentstherelativeposition betweenthechaserandtargetandthegreenlinerepresentstherelativeposition betweenthechaserandtheoptimaltrajectory.Thegreenlineindicatesthatoptimal controldoesnotyieldthesameresponseinahigherdelitymodel.Thisisduetothe seculardisturbingeffectsactingonthechaser.However,thechaserisclosertothe targetatthenaltimeapproximately 300 m.Thisisaresultofthebang-off-bang controlstructurebeingfullycapturedwhenusingforcetracking. TheorientationerrorplotisshowninFigure8-6Bwherethebluelinerepresents theerroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandtheoptimaltrajectory.Thisgureindicatesthatthe errorbetweenthechaserandtargetandtheerrorbetweenthechaserandtheoptimal trajectoryareaboutthesameapproximately 10 atthenaltime.Thisisevidence thattheoptimalcontrolhistorydoesnotyieldthedesiredterminalconditionsina higherdelitymodel.Thismaybeduetoadriftintheorientationofthetargetorthe bang-off-bangcontrolstructure.Sincethereisanitewidthinbetweentheswitching 134

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-5.Fixedtimeminimumcontrolefforttrajectorytrackingresults 135

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pointsofthecontrol,theinterpolatedvaluesattheswitchingpointsaddanerrortothe response. TherelativepositionplotsshowninFigure8-6Cillustratetheseculardisturbing effects.Itisshownthatthetrajectoriesareinitiallycoincidentandbegintodriftafter approximately 100 s.Thisgurealsoillustratesthattheterminalconditionobtained usingforcetrackingarebetterthanthoseobtainedusingtrajectorytracking.However, theerroratthenaltimeisstilllarge.ThequaternionplotshowninFigure8-6D indicatesthatthethetrajectoriesaresimilar,yetthereisa 10 errorattheterminal conditionwhichisevidentinFigure8-6B.TheforceplotisshowninFigure8-6Eand thetorqueplotisshowninFigure8-6F.Theseplotsindicatethattheoptimalcontrol historiesaretrackedaccuratelybytheactuators. 8.2.3FiniteHorizonQuadraticCostTrajectories Figure8-7showstheresultsofimplementingthenitehorizonquadraticcost trajectoriesusingtrajectorytrackingwiththehighdelitymodel.Themagnitudeof therelativepositionisshowninFigure8-7Awherethebluelinerepresentstherelative positionbetweenthechaserandtargetandthegreenlinerepresentstherelative positionbetweenthechaserandtheoptimaltrajectory.Thisgureindicatesthatthe optimaltrajectoryistrackedaccuratelyi.e.,undera 0.01 merror.However,therelative positionbetweenthechaserandtargetisnotclosetozeroapproximately 1 kmatthe naltime.Thisisagainduetoseculardisturbingeffects. TheorientationerrorisshowninFigure8-7Bwherethebluelinerepresentsthe erroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandoptimaltrajectory.Thisplotisevidencethattheoptimal trajectorycanbetrackedwithina 1 errorangle,andthatthenalorientationmatches theresultsfromthehigherdelitymodelwithina 0.1 erroratthenaltime.Thisresult indicatesthatdisturbanceshavelesseffectontherotationalmotion. 136

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-6.Fixedtimeminimumcontroleffortforcetrackingresults 137

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TherelativepositionplotsshowninFigure8-7Cillustratehowthechaserdriftsas ittrackstheoptimaltrajectory.Initiallythetrajectoriesarecoincidentandbegintodrift afterapproximately 100 s.ThequaterniontrajectoriesshowninFigure8-7Dmatch closely,whichisreinforcedbytheresultsinFigure8-7B.Theforceplotisshownin Figure8-7EandthetorqueplotisshowninFigure8-7F.Theforceplotindicatesthat trajectorytrackingleadstoacontrolhistorysimilartotheoptimalcontrolhistory.The torqueplotisanotherexamplewheretrajectorytrackingdoesnotnecessarilyleadtoa controlhistorysimilartotheoptimalcontrolhistory. Next,Figure8-8showstheresultsofimplementingthenitehorizonquadratic costtrajectoriesusingforcetrackingwiththehighdelitymodel.Therelativeposition magnitudeisshowninFigure8-8Awherethebluelinerepresentstherelativeposition betweenthechaserandtargetandthegreenlinerepresentstherelativeposition betweenthechaserandtheoptimaltrajectory.Thegreenlineindicatesthatoptimal controldoesnotyieldthesameresponseinahigherdelitymodel.However,thechaser isclosertothetargetatthenaltimeapproximately 300 m.Thisisagainduetothe seculardisturbingeffects. TheorientationerrorplotisshowninFigure8-8Bwherethebluelinerepresents theerroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandtheoptimaltrajectory.Thisgureindicatesthatthe errorbetweenthechaserandtargetandtheerrorbetweenthechaserandtheoptimal trajectoryareaboutthesameabout 10 atthenaltime.Thisisevidencethatthe optimalcontrolhistorydoesnotyieldtheterminalconditionsinahigherdelitymodel. Thisisduetoadriftintheorientationofthetargetandtheinitialbang-bangcontrol structurewhenthecontrolissaturated. TherelativepositionplotsshowninFigure8-8Cillustratetheseculardisturbing effects.Thetrajectoriesareinitiallycoincidentandbegintodriftafterapproximately 100 s.ThequaternionplotsshowninFigure8-8Dindicatethatthetrajectories 138

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-7.Finitehorizonquadraticcosttrajectorytrackingresults 139

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aresimilar.TheforceplotisshowninFigure8-8Eandthetorqueplotisshownin Figure8-8F.Theseplotsindicatethattheoptimalcontrolhistoriescanbetracked accuratelybytheactuators. 8.2.4Disturbance-FreeTrajectories Todeterminetheextenttowhichdisturbances,inaccuraciesinoptimalsolutions, andthecontrolstructurecontributetothedriftingtrajectories,twoadditionalexamples areperformedwhiledisablingdisturbances.Therstexampleisimplementingthexed timeminimumcontrolefforttrajectoriesusingtrajectorytracking.Figure8-9shows theresultsobtainedforthisexample.Themagnitudeoftherelativepositionisshownin Figure8-9Awherethebluelinerepresentstherelativepositionbetweenthechaserand targetandthegreenlinerepresentstherelativepositionbetweenthechaserandthe optimaltrajectory.Thisgureindicatesthattheoptimaltrajectoryistrackedaccurately i.e.,undera 0.1 merror.Therelativepositionbetweenthechaserandtargetand betweenthechaserandoptimaltrajectoryarevirtuallythesameapproximately 0.3 m atthenaltime.Figure8-9Calsosupportsthissincebothtrajectoriesaresimilarand theybothareneartheoriginatthenaltime. TheorientationerrorisshowninFigure8-7Bwherethebluelinerepresentsthe erroranglebetweenthechaserandtargetandthegreenlinerepresentstheerror anglebetweenthechaserandoptimaltrajectory.Thequaternionplotsareshownin Figure8-9D.Theeffectsofremovingdisturbancesisnotevidentintheseplotssince disturbancesdidnothavealargeeffectontherotationalmotionintherstplace.The forceplotisshowninFigure8-9EandthetorqueplotisshowninFigure8-9F.The controlhistoryhereissimilartothecontrolhistorywithdisturbances. Thenextexampleisimplementingthenitehorizonquadraticcosttrajectories usingforcetracking.Figure8-10showstheresultsforthisexample.Theresults indicatethattheforcetrackingmethoddoesnotgivefavorableresultswithout disturbances.Figure8-10AandFigure8-10Cshowthattheterminalconditionsare 140

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-8.Finitehorizonquadraticcostforcetrackingresults 141

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-9.Fixedtimeminimumcontrolefforttrajectorytrackingresultsno disturbances 142

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notobtained.Moreover,Figure8-10BandFigure8-10Dshowresultssimilartothe resultsinFigure8-8whichincludedisturbances.Thus,deviationsusingforcetracking arecausedbyinaccuraciesintheoptimalsolutions.Theseinaccuraciesareduetothe interpolationsincethesolutionsareonlyknownatthecollocationpoints.Thisresultis motivationfortheneedtore-planwhenthereareinaccuraciesinasolution. 8.2.5ArticialPotentialFunctionTrajectory ThetargetandchaserparametersusedarethesameasgiveninTable8-3 andTable8-4,respectively.TheAPFparametersusedforthisexamplearegivein Table8-5.Figure8-11showstheresultsofimplementingtheAPFmethodinthehigh delitymodel.Figure8-11Aisaplotofthemagnitudeoftherelativepositionversus time.Thisgureindicatesthatthemagnitudeoftherelativepositioncontinuesto decreasewitheverymaneuver.Figure8-11Bshowstherelativepositionhistoryandthe correspondingcontrolhistoryisshowninFigure8-11C.Theseresultsindicatethatthe chasercontinuouslyapproachesthetargetwitheverymaneuver. Table8-5.APFmethodparameters ParameterValueUnits r f 0 m PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 k 0.002 8.2.6AdaptiveArticialPotentialFunctionTrajectory ThetargetandchaserparametersusedarethesameasgiveninTable8-3and Table8-4,respectively.TheAAPFparametersusedforthisexamplearegivein Table8-6.Figure8-12showstheresultsofimplementingtheAAPFmethodinthe highdelitymodel.Figure8-12Aisaplotofthemagnitudeoftherelativeposition versustime.Thisgureindicatesthatthemagnitudeoftherelativepositioncontinues todecreaseandthatamaneuverisexecutedapproximatelyevery 200 s.Thisisno coincidencesincethetransfertimechosenwas T =200 s.Figure8-12Bshowsthe relativepositionhistoryandthecorrespondingcontrolhistoryisshowninFigure8-12C. 143

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APositiontrackingerror BAngletrackingerror CRelativeposition DErrorquaternion EForcehistory FTorquehistory Figure8-10.Finitehorizonquadraticcostforcetrackingresultsnodisturbances 144

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APositiontrackingerror BRelativeposition CControlhistory Figure8-11.APFmethodresults Theseresultsindicatethatthechasercontinuouslyapproachesthetargetwithevery maneuver.Moreover,theAAPFmethodissuitableforrendezvousinahighdelity modeldespiteitsdevelopmentusingalinearizedmodel. Table8-6.AAPFmethodparameters ParameterValueUnits r f 0 m 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 T 200 s 145

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APositiontrackingerror BRelativeposition CControlhistory Figure8-12.AAPFmethodresults 8.2.7ObstacleAvoidanceTrajectories Theresultsobtainedfortheclose-rangerendezvousscenariowithobstacle avoidancearepresentedinthissection.Thechaserandtargethavethesame parametersshowninTable8-4andTable8-3,respectively.Theobstaclehasthe parametersshowninTable8-7andismodeledthesameasthechaserandtarget.The optimalsolutionobtainedinChapter4isimplementedusingtrajectorytracking. Theresultsobtainedfromimplementingtheconstrainedxedtimeminimumcontrol efforttrajectoryusingtrajectorytrackingareshowninFigure8-13andFigure8-14. Therelativepositiontrajectoriesofthechaserareshownatdifferentinstancesin 146

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Table8-7.Obstaclespacecraftparameters ParameterValueUnits m o 100 kg l o 3 m r o t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1182.6494976817.167778904.693252 T km v o t 0 0.175775 )]TJ/F22 11.9552 Tf 9.298 0 Td [(0.9637767.494102 T km = s J o 2 4 3002010 201000 100200 3 5 kg m 2 q o t 0 0001 T o t 0 000 T rad = s Figure8-13AthruFigure8-13D.Theseguressupporttheresultsthatwereshown previously.Thatis,whiletheoptimaltrajectorycantrackedaccurately,thereisdisparity betweentheoptimaltrajectoryandtheactualtrajectory.Inthiscase,trackingthe trajectorydidnotresultinacollisionbetweenthechaserandthetarget.However,ifthe chaserdidapproachtheobstacleduetodriftingeffects,thenacorrectionwouldmustbe madetothetrajectorytoavoidacollision. ThemagnitudeoftherelativepositionisshowninFigure8-14Awheretheblue linerepresentstherelativepositionbetweenthechaserandthetargetandthegreen linerepresentstherelativepositionbetweenthechaserandtheoptimaltrajectory. Thisgureindicatesthatthetrajectorycanbetrackedaccurately.Therelativeposition plotsshowninFigure8-14Billustratethedriftingthatoccursfromtrackingaplanned trajectory.TheforceplotisshowninFigure8-14C.Thecommandedforceissimilarto theoptimalforcesolutionwhichisthesameresultfortheprevioustrajectorytracking resultspresented. TheparametersusedforimplementingtheAPFmethodforclose-rangerendezvous withobstacleavoidanceareshowninTable8-8.Recallthatthepathconstraintfor theobstaclerequiredthatthechaserbemorethat 100 mfromtheobstacleatall times.Thus,theAPFparametersarechosensuchthattherepulsivepotentialroughly representsaspherewithradiusof 100 m. 147

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A t =125 s B t =250 s C t =375 s D t =500 s Figure8-13.Constrainedxedtimeminimumcontrolefforttrajectories 148

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APositiontrackingerror BRelativeposition CForcehistory Figure8-14.Constrainedxedtimeminimumcontrolefforttrajectorytrackingresults Table8-8.ConstrainedAPFmethodparameters ParameterValueUnits r f 0 m PI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 NI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 2.5 10 6 2.5 10 3 m 2 k 0.004 149

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TheresultsobtainedfromimplementingtheAPFmethodareshowninFigure8-15 andFigure8-16.Therelativepositiontrajectoriesofthechaserareshownatdifferent instancesinFigure8-15AthruFigure8-15D.Theseguresindicatethatthechaseris abletorendezvouswiththetargetwhileavoidingtheobstacle.Figure8-15Ashowsthat thechaserencounterstheobstacleafterapproximately 250 sandperformsamaneuver toavoidit. Themagnitudeoftherelativepositionbetweenthechaserandthetargetisshown inFigure8-16A.Thisgureshowsthatthechasercontinuestoapproachthetarget andgetswithin 8 mofthetargetafter 1000 s.Therelativepositionplotsareshownin Figure8-16BandthecorrespondingcontrolhistoriesareshowninFigure8-16C.These forceplotshowstheavoidancemaneuverthatoccurredatapproximately 150 s.Asa result,therelativepositionplotshowsthatthisavoidancemaneuverkeepsthechaser fromapproachingthetargetuntilthenextmaneuver. TheparametersusedforimplementingtheAAPFmethodforclose-rangerendezvous withobstacleavoidanceareshowninTable8-9.Thesameparametersfortherepulsive potentialareusedforthissimulation.Thegain k issmallersinceitisonlyappliedtothe gradientoftherepulsivepotentialandthescalingbetweentheattractiveandrepulsive potentialsisdifferentthantheAPFmethod. Table8-9.ConstrainedAAPFmethodparameters ParameterValueUnits r f 0 m 0 0.002 100101 T s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 T 200 s NI 2 R 3 3 s )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 2.5 10 6 2.5 10 3 m 2 k 0.001 TheresultsobtainedfromimplementingtheAAPFmethodareshowninFigure8-17 andFigure8-18.Therelativepositiontrajectoriesofthechaserareshownatdifferent instancesinFigure8-17AthruFigure8-17D.Theseguresshowthatthechaserisable 150

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A t =250 s B t =500 s C t =750 s D t =1000 s Figure8-15.ConstrainedAPFmethodtrajectories 151

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APositiontrackingerror BRelativeposition CForcehistory Figure8-16.ConstrainedAPFmethodresults torendezvouswiththetargetwhileavoidingtheobstacle.Inthiscase,theobstacleis sensedsoonerthantheAPFmethod.Thisisduetothedifferentscalingbetweenthe attractiveandrepulsivepotentials.Smallervaluesof and canbeusedtoreducethe effectoftherepulsivepotentialsandtoallowthechasertoconvergetothetargetata fasterrate. Themagnitudeoftherelativepositionbetweenthechaserandtargetisshownin Figure8-18A.Thisgureshowsthatthechaserapproachesthetargetatafasterrate thantheAPFmethod.Thisisexpectedsinceitwasshownthatconvergencetimesare improvedwiththeAAPFmethod.TherelativepositionplotsareshowninFigure8-18B 152

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A t =250 s B t =500 s C t =750 s D t =1000 s Figure8-17.ConstrainedAAPFmethodtrajectories 153

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andthecorrespondingcontrolhistoriesareshowninFigure8-18C.Theforceplotshows thattheavoidancemaneuveroccursatapproximately 100 s.Asaresult,therelative positionplotshowsthattheobstacleisavoidedyetthechasercontinuestoapproachthe target. APositiontrackingerror BRelativeposition CForcehistory Figure8-18.ConstrainedAAPFmethodresults 8.2.8FinalApproachTrajectory Theresultsobtainedforthenalapproachscenarioarepresentedinthissection. ThenalapproachisperformedusingthenitehorizonLQcontrollaw.TheCWH equationfromAppendixAisusedintheLQproblemandtheRiccatimatrixisobtained usingtheSTMrepresentation.TheparametersusedaregiveninTable8-10which 154

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includetheparametersdeningtheRiccatimatrixandanewsetofinitialpositionsand velocitiesforthechaserandtarget. Table8-10.Finalapproachparameters ParameterValueUnits r t t 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1005.4483035813.4800843725.077734 T km v t t 0 0.697715 )]TJ/F22 11.9552 Tf 9.298 0 Td [(3.9762596.387082 T km = s r t t 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1005.4539765813.5104113725.067550 T km v t t 0 0.697738 )]TJ/F22 11.9552 Tf 9.298 0 Td [(3.9764986.386375 T km = s S f 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 I0 0 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 I 2 R 6 6 Q 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 I0 0 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 I 2 R 6 6 R 10 2 I 2 R 3 3 t f 50 s TheresultsofimplementingthenitehorizonLQfeedbackcontrollawareshownin Figure8-19.Themagnitudeoftherelativepositionbetweenthechaserandthetarget isshowninFigure8-19A.Thisgureshowsthatthechaserisabletoperformthenal approachwiththetargetandgetswithin 0.2 m.Therelativepositionandrelativevelocity plotsareshowninFigure8-19BandFigure8-19C,respectively.Theseplotsshowthat usingalinearizedmodeltodevelopacontrollawforthenalapproachisvalid.The correspondingcontrolhistoriesareshowninFigure8-19D.Theeffectsofthedeadband constraintsontheactuatorareseeninthisgurewhenthecontrolvaluesarenearzero. ThisisalsoseeninFigure8-19Asincethedeadbandlimitsdonotallowtherelative positiontodecreasebelow 0.2 m. Inconclusion,theMCSspresentedinthischaptersuggestthattheAAPFmethod developedinChapter6ismoreeasilytunedthantheAPFmethod.Asinglescalar parameterisrequiredtotunetheAAPFmethodwherethisparameterbalancesthe transfertimeversusthecontroleffort.Also,fasterconvergencetimeswithlesscontrol effortareseensincethedynamicsareembeddedintheformulationoftheAAPF method. 155

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APositiontrackingerror BRelativeposition CRelativevelocity DForcehistory Figure8-19.Finalapproachresults Thehighdelitysimulationresultssuggestthatusingimplementingplanned trajectoriesisonlysuitableforacertainclassesofproblems.Therotationalmotionin theclose-rangerendezvousscenarioyieldedfavorableresultsforthethreepertinent problemsatmosta 10 error.Thetranslationalmotiononlyyieldedfavorableresults fortheminimumtimeproblemssincethetransfertimewasshort.Thecaseswithalong transfertimee.g.,minimumcontroleffortandnitehorizonquadraticcostexamples experiencedseculardisturbingeffectswhichgreatlyaffectedtheperformanceand terminalconditions.Moreover,inaccuraciesareintroducedsincetheoptimalsolution isonlyknownatthecollocationpoints.Aninterpolationschemeisusedtoobtaina 156

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continuoussolution;however,theinterpolatedpointsarenotnecessarilypartofthe optimalsolution. TheAPFandAAPFmethodsarealsoimplementedinthehighdelitymodel.The resultsindicatethatthesemethodscanbeusedforclose-rangerendezvousscenarios withandwithoutobstacles.Inparticular,theAAPFmethodisabletoconvergefaster thantheAPFmethoddespiteitbeingdevelopedusingalinearizedmodel.Anal approachscenarioisalsosimulatedusingthenitehorizonLQcontrollaw.Thismethod alsoperformedfavorablydespitethecontrollawbeingdevelopedusingalinearized model. 157

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CHAPTER9 CONCLUSIONS Thetechnologyforenablingpath-planningonboardspacesystemsforautonomous proximityoperationsAPOsisdevelopedinthismanuscript.Proximityoperations arebifurcatedintotwophases:iclose-rangerendezvousandiinalapproach orendgame.Themotivationfordevelopingalgorithmsthatcomputetrajectoriesin real-timeisdevelopedsincethismustbetodonetoaccountforunmodeledevents.As aresult,autonomousspacesystemsmusthavethecapabilityofplanningorcorrecting trajectorieson-demandwhileoptimizingthesolutionswithrespecttoaperformance index. TheadaptivearticialpotentialfunctionAAPFmethodwasdevelopedforthe close-rangerendezvousscenario.Thismethodisamodicationtothearticialpotential functionAPFmethodwhichembedsthedynamicsandaperformancecriterioninthe formulation.ThismodicationwasdonesincetheAPFmethodisnotwell-suitedfor aconservativesystemwithrespecttoresourcessuchasaspacecraft.TheAAPF methodwasshowntohaveimprovedperformanceandconvergencetimesovertheAPF methodthroughMonteCarlosimulationsMCSs. Itisalsoshownthroughextensivenumericalanalysesthattrackingplanned trajectoriesisineffectiveforAPOswhenthetransfertimesofthetrajectoriesarelong. Thisimpliesthatusingshorttimehorizonstoplantrajectoriesmustbedonetoaccount foralldisturbancesorunmodeledevents.Moreover,theshorttimehorizonframework lendsitselfwelltousingtheAAPFmethodsinceitisdevelopedusingalinearizedmodel whichisaccurateforshorttimehorizons.Thiswasdemonstratedinsimulationwhich provesthattheAAPFmethodcanbeusedwithahighdelitymodel. Finalapproachtrajectoriesareconventionallyobtainedbysolvinganitehorizon linearquadraticLQproblemwhichessentiallyissolvedasanalvalueproblem withaDifferentialRiccatiequationDRE.ItisshownthatiftheHamiltonianmatrixis 158

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lineartimeinvariantLTI,thenthesolutionisobtainedusingastatetransitionmatrix. However,thiscannotbedoneingeneralwhentheHamiltonianmatrixislineartime varyingLTV.Asaresult,twomathematicaltoolswereutilizedtoavoidsolvinganal valueproblemandinturnobtainthesolutiontotheDREinreal-time.Namely,these toolsarethePicardIterationPIandtheHomotopyContinuationHC.Theparametric solutionstructureobtainedusingthePIandtheHCallowforupdatestobeeasilymade inthetimehorizon,dynamicsmodel,and/orperformanceindex.Anumericalexample waspresentedtodemonstratehowsolutionsareobtainedusingthePIandHC.In addition,anexampleofanalapproachscenarioinanellipticorbiti.e.,usingthe Yamanaka-Ankerson-Tschauner-HempelequationwaspresentedwiththeHCmethod. Lastly,analapproachtrajectoryisimplementedinthehighdelitymodelusinganLQ controllawwhichdemonstratedhowusingalinearmodelisvalidforthisscenario. Inconclusion,themethodsdevelopedinthismanuscriptarewell-suitedfor autonomousspacesystems.Themethodsdevelopedcomputeoptimizedtrajectories throughfunctionalevaluationsratherthanthroughiterativetechniques.Tofurther increasetherangeofapplicability,differentmodelscanbeusedinthedevelopmentlike thosedescribedin[98,99,101]whichareamenablewiththemethodsdeveloped.An instructionaleffectfromdevelopingthesemethodologiesisthatpath-planningcanbe doneasthepropertiesofthedynamicsbecomemorewell-established. 159

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APPENDIXA RELATIVEMOTIONMODELS Themodelstodescriberelativemotionoftwospacecraftarediscussedinthis appendix.Thetranslationalrelativemotionmodelisderivedusingthetwobodymodel onboththechaserandtarget.Usingthecloseproximityassumption,themodelresults inalinearform.Twosetsofequationsarepresentedtodescriberelativetranslational motionwhenthetarget'sorbitis:icircularandiieccentric.Therotationalrelative motionmodelisderivedusingasmallangleapproximationSAAwhichalsoresultsina linearform. A.1RelativeTranslation Thepositionofthechaserrelativetothetargetis r = r c )]TJ/F45 11.9552 Tf 11.956 0 Td [(r t where r c isthepositionofthechaserand r t isthepositionofthetarget.Ifboth spacecraftaregovernedbythetwobodyrelativemotionmodel r t = )]TJ/F25 11.9552 Tf 16.25 8.088 Td [( k r t k 3 r t r c = )]TJ/F25 11.9552 Tf 16.668 8.087 Td [( k r c k 3 r c + f andthechaseristheonlyspacecraftwithcontrolthrust f ,thentherelativepositionis differentiatedtwicetoyield r = r c )]TJ/F22 11.9552 Tf 11.134 0 Td [( r t r = )]TJ/F25 11.9552 Tf 16.669 8.088 Td [( k r c k 3 r c + f + k r t k 3 r t r = )]TJ/F25 11.9552 Tf 25.873 8.088 Td [( k r + r t k 3 r + r t + f + k r t k 3 r t .A 160

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Notethatthevalue k r + r t k canbesimpliedas k r + r t k = r + r t T r + r t 1 2 = k r t k 2 +2 r T r t + k r k 2 1 2 k r + r t k 3 = k r t k 3 1+2 r T r t k r t k 2 + k r k 2 k r t k 2 3 2 ThisidentityissubstitutedintoequationAtoyield r = )]TJ/F25 11.9552 Tf 16.25 8.088 Td [( k r t k 3 1+2 r T r t k r t k 2 + k r k 2 k r t k 2 )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(3 2 r + r t + f + k r t k 3 r t .A Moreover,thepolynomialexpansion + x k =1+ kx + k k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2! x 2 + isusedtoexpand 1+ 2 r T r t k r t k 2 + k r k 2 k r t k 2 )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(3 2 =1 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(3 2 2 r T r t k r t k 2 + k r k 2 k r t k 2 + Usingthecloseproximityassumptioni.e., r t r andthepolynomialexpansionabove, equationAissimpliedas r = )]TJ/F25 11.9552 Tf 16.25 8.087 Td [( k r t k 3 r + r t )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 r T r t k r t k 2 r t + f + k r t k 3 r t r = )]TJ/F25 11.9552 Tf 16.25 8.088 Td [( k r t k 3 r )]TJ/F22 11.9552 Tf 11.956 0 Td [(3 r T r t k r t k 2 r t + f .A ThisrelativemotionmodelinequationAexpressedinthetarget'sorbitalframeis r = )]TJ/F40 11.9552 Tf 9.298 0 Td [(n 2 r )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 r T r t k r t k 2 r t + f )]TJ/F22 11.9552 Tf 14.724 0 Td [(_ o r )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 o r )]TJ/F46 11.9552 Tf 11.955 0 Td [(! o o r .A wherethederivativesarederivatesinthetarget'srotatingorbitalframe.Thetarget positioncanbeexpressedinthetarget'sorbitalframeas r t = r t 00 T 161

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where r t isthedistancefromtheEarth'scentertothetarget.Inaddition,theangular velocityofthetarget'sorbitalframeisexpressedinthetarget'sorbitalframeas o = 00 T where isthetarget'sorbitalrate.Lettingtherelativepositionvectorbedenedas r = xyz T thenthecrossproducttermsaresimpliedas o r = )]TJ/F25 11.9552 Tf 9.299 0 Td [(! y x 0 T o r = )]TJ/F22 11.9552 Tf 11.414 0 Td [(_ y x 0 T o o r = )]TJ/F25 11.9552 Tf 9.299 0 Td [(! 2 x 2 y 0 T r )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 r T r k r k 2 r t = )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 xyz T Fromtheorbitalangularmomentumrelationship r 2 t = h thesimplicationcanbemade k r t k 3 = h 3 2 3 2 = k 3 2 where k = h 3 2 162

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isaconstant.Asaresult,equationAsimpliesto x )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 k 3 2 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 2 x )]TJ/F22 11.9552 Tf 14.071 0 Td [(_ y )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 y = f x y + k 3 2 y )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 2 y +_ x +2 x = f y A z + k 3 2 z = f z where f x f y f z arethecomponentsofthecontrolacceleration f inthetarget'sorbital frame. A.1.1Clohessy-Wiltshire-HillEquation Assumingthetargetisinacircularorbit,then =0 k = 1 2 SimplifyingequationAresultsintheClohessy-Wiltshire-HillCWHequationwhich describestherelativetranslationalmotionbetweentwospacecraft[39,55] x )]TJ/F22 11.9552 Tf 11.955 0 Td [(3 n 2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n y = f x y +2 n x = f y A z + n 2 z = f z where n = isthemeanmotionofthetarget'sorbitalframe.EquationAcanbe writteninstatespaceform 2 6 4 r v 3 7 5 = 2 6 4 0I A 21 A 22 3 7 5 2 6 4 r v 3 7 5 + 2 6 4 0 I 3 7 5 u 163

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where r = xyz T v = x y z T u = f x f y f z T and A 21 A 22 areblocksofthestatematrixandaredenedas A 21 = 2 6 6 6 6 4 3 n 2 00 000 00 )]TJ/F40 11.9552 Tf 9.298 0 Td [(n 2 3 7 7 7 7 5 A 22 = 2 6 6 6 6 4 02 n 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(2 n 00 000 3 7 7 7 7 5 A.1.2Yamanaka-Ankerson-Tschauner-HempelEquation Byrelaxingthecircularorbitassumption,adifferentsetofequationscanbe obtained.[98,99]First,theindependentvariableischangedfromtimetothetrue anomalyofthetargeti.e., .Throughthechainrule,thetimederivativeofanarbitrary scalarparameter a nowbecomes da dt = da d d dt da dt = a 0 164

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andthesecondtimederivativebecomes d 2 a dt 2 = d dt a 0 + da 0 dt d 2 a dt 2 = d d d dt a 0 + da 0 d d dt d 2 a dt 2 = !! 0 a 0 + 2 a 00 Usingtheserelationships,equationAcanbewrittenas 2 x 00 + !! 0 x 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 k 3 2 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 2 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(!! 0 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 2 y 0 = f x 2 y 00 + !! 0 y 0 + k 3 2 y )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 2 y + !! 0 x +2 2 x 0 = f y A 2 z 00 + !! 0 z 0 + k 3 2 z = f z Recallthatthepositionmagnitudeintermsofthetrueanomalycanbewrittenas r t = p 1+ e cos where p isthesemi-latusrectumofthetarget'sorbitand e istheeccentricityofthe target'sorbit.[40]Similarly,fromtheconservationoforbitalangularmomentum,the semi-latusrectumcanbewrittenas p = h 2 .A Asaresult,theorbitalrateofthetargetcanbewrittenas = h r 2 t = h p 2 + e cos 2 = k 2 + e cos 2 ,A andthederivativeoftheorbitalratewithrespecttothetrueanomalyis 0 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 k 2 e sin + e cos .A SubstitutingequationAandequationAintoequationAanddividingby 2 yieldstheYamanaka-Ankerson-Tschauner-HempelYATHequationswhichdescribethe 165

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relativetranslationalmotionbetweentwospacecraft x 00 )]TJ/F31 11.9552 Tf 11.956 16.857 Td [( 1+ 2 1+ e cos x + 2 e sin 1+ e cos y )]TJ/F22 11.9552 Tf 21.951 8.088 Td [(2 e sin 1+ e cos x 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 y 0 = 1 k 4 + e cos 4 f x y 00 )]TJ/F22 11.9552 Tf 21.951 8.088 Td [(2 e sin 1+ e cos x )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( 1 )]TJ/F22 11.9552 Tf 36.85 8.088 Td [(1 1+ e cos y +2 x 0 )]TJ/F22 11.9552 Tf 21.951 8.088 Td [(2 e sin 1+ e cos y 0 = 1 k 4 + e cos 4 f y z 00 + 1 1+ e cos z )]TJ/F22 11.9552 Tf 21.951 8.088 Td [(2 e sin 1+ e cos z 0 = 1 k 4 + e cos 4 f z Thesesetofequationscanbewritteninstatespaceform 2 6 4 r 0 v 0 3 7 5 = 2 6 4 0I A 21 A 22 3 7 5 2 6 4 r v 3 7 5 + 1 k 4 + e cos 4 2 6 4 0 I 3 7 5 f where r = xyz T v = x 0 y 0 z 0 T f = f x f y f z T and A 21 A 22 areblocksofthestatematrixandaredenedas A 21 = 1 1+ e cos 2 6 6 6 6 4 3+ e cos )]TJ/F22 11.9552 Tf 9.298 0 Td [(2 e sin 0 2 e sin e cos 0 00 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 3 7 7 7 7 5 A 22 = 1 1+ e cos 2 6 6 6 6 4 2 e sin 20 )]TJ/F22 11.9552 Tf 9.299 0 Td [(22 e sin 0 002 e sin 3 7 7 7 7 5 A.2RelativeOrientation AsmallangleapproximationSAAcanbeusedtodescribetherelativeorientation ofthechaserwithrespecttothetarget.Theunderlyingassumptionsherearesmall 166

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angulardisplacementsbetweenthetwobodyframesandapassivelytumblingtarget targetwithconstantangularvelocity t relativetoaninertialframe.[65]FigureA-1 illustratesthesmallangulardisplacementassumption,where f b t ,1 b t ,2 b t ,3 g isthebasis associatedwiththetarget'sbodyframe B t and f b c ,1 b c ,2 b c ,3 g isthebasisassociated withthechaser'sbodyframe B c .UsinganantisymmetricEulersequencetorelatethe orientationofthetargetbodyframetothechaserbodyframe,therotationmatrixto relatetworeferenceframesisapproximatedas C B c B t I )]TJ/F46 11.9552 Tf 11.956 0 Td [( where = 1 2 3 T and 1 2 3 aretheanglesassociatedwiththerotationsintheantisymmetricEuler sequence.[65]Asaresult,therelativeangularvelocityis andtheangularvelocityof thechaserbodyframerelativetoaninertialframeisapproximatedas c =_ + I )]TJ/F46 11.9552 Tf 11.955 0 Td [( t FigureA-1.Relativeorientationbetweenchaserandtarget 167

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ApplyingEuler'ssecondlaw,thegoverningequationsforsmallangulardisplacements ofthechaserbodyframerelativetothetargetbodyframeis J )]TJ/F22 11.9552 Tf 7.27 -9.684 Td [( )]TJ/F22 11.9552 Tf 15.141 0 Td [(_ t + )]TJ/F22 11.9552 Tf 8.665 -9.684 Td [(_ + )]TJ/F45 11.9552 Tf 10.461 -9.684 Td [(I )]TJ/F46 11.9552 Tf 11.955 0 Td [( t J )]TJ/F22 11.9552 Tf 8.664 -9.684 Td [(_ + I )]TJ/F46 11.9552 Tf 11.956 0 Td [( t = ,A where J istheinertiamatrixofthechaser.Consideringonlytherstorderterms, equationAreducesto J = t T t J )]TJ/F46 11.9552 Tf 11.955 0 Td [(! t J t )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F46 11.9552 Tf 5.479 -9.684 Td [(! T t J t I + J t )]TJ/F46 11.9552 Tf 11.955 0 Td [(! t J )]TJ/F45 11.9552 Tf 11.955 0 Td [(J t )]TJ/F46 11.9552 Tf 11.955 0 Td [(! t J t + A EquationAcanbewritteninstatespaceform 2 6 4 3 7 5 = 2 6 4 0I A 21 A 22 3 7 5 2 6 4 3 7 5 + 2 6 4 0 J )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 3 7 5 u where u = )]TJ/F46 11.9552 Tf 11.956 0 Td [(! t J t and A 21 A 22 areblocksofthestatematrixandaredenedas A 21 = J )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t T t J )]TJ/F46 11.9552 Tf 11.955 0 Td [(! t J t )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F46 11.9552 Tf 5.479 -9.684 Td [(! T t J t I A 22 = J )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 J t )]TJ/F46 11.9552 Tf 11.955 0 Td [(! t J )]TJ/F45 11.9552 Tf 11.955 0 Td [(J t Itisinterestingtonotethataconstantforcetermappearsinthissetofequationsdueto theconstanttumblingrateofthetarget. 168

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APPENDIXB TWOPOINTBOUNDARYVALUEPROBLEMSOLUTION ThetwopointboundaryvalueproblemTPBVPisstatedas: Givenadynamical system x = f x u t ,aninitialcondition x t 0 = x 0 ,andadesirednalcondition x t f = x f obtainedattime t = t f ,determinethecontrolaction u t thatsatisesthese conditions. Theproblemofinterestistodetermineaninitialchangeinvelocity x i to reachanalposition x f intime T witharbitrarynalvelocity.SolvingaTPBVPcan becumbersome,butthisisnotthecaseforlinearsystems.Sincetherelativemotion equationsarelinear,theycanbewrittenusingablockformofthestatetransitionmatrix STMrepresentationas[71] 2 6 4 x t x t 3 7 5 = 2 6 4 11 t 12 t 21 t 22 t 3 7 5 2 6 4 x 0 x 0 3 7 5 + Z t t 0 2 6 4 11 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(" 12 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(" 21 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(" 22 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(" 3 7 5 Bu d .B Thus,thesolutionoftheTPBVPofinterestisobtainedbyletting x 0 =_ x 0 +_ x i andsolvingasystemofalgebraicequationsfor x i TheTPBVPsolutionisobtainedforboththeCWHequationandSAAequation. SincetheSAAequationhasaconstantforcingterm,theassociatedTPBVPproblem includesageneralconstantforcingterm u = )]TJ/F46 11.9552 Tf 9.298 0 Td [(! t J t .Also,notethatforbothrelative motionequationstheinputmatrixisoftheform B = 2 6 4 0 B 2 3 7 5 Thus,consideringtherstrowofequationB,theinitialchangeinvelocityrequiredto reachanalposition x f intime T issolvedforas x i = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.956 0 Td [(x t )]TJ/F45 11.9552 Tf 11.955 0 Td [( 11 T x 0 ] )]TJ/F22 11.9552 Tf 13.2 0 Td [(_ x 0 ,B 169

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where x t = Z T t 0 12 T )]TJ/F40 11.9552 Tf 11.955 0 Td [(t B 2 u dt Thisisfortheapplicationofinterceptionsincetheonlyconditionisfor x x f as t T SubstitutingequationBintoequationBandconsideringthesecondrowof equationByieldsthevelocityprole x d t = 12 t x 0 + 22 t x 0 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t )]TJ/F45 11.9552 Tf 11.955 0 Td [( 11 T x 0 )]TJ/F22 11.9552 Tf 13.201 0 Td [(_ x 0 ... + Z t t 0 22 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d x d t = 21 t )]TJ/F45 11.9552 Tf 11.955 0 Td [( 22 t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T 11 T x 0 + 22 t )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] ... + Z t t 0 22 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d .B RecalltheSTMproperty[71] t 1 t 2 = t 1 + t 2 UsingtheblockformoftheSTMandletting t 1 = T and t 2 = )]TJ/F40 11.9552 Tf 9.299 0 Td [(T ,the ,2 blockofthis STMpropertyis 11 T 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T + 12 T 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T = 0 Multiplyingontheleftby )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T andmultiplyingontherightby )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T ,theequation reducesto )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T 11 T = )]TJ/F45 11.9552 Tf 9.299 0 Td [( 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T .B 170

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UsingthesameSTMpropertyandletting t 1 = t and t 2 = )]TJ/F40 11.9552 Tf 9.298 0 Td [(T ,the ,2 blockofthe STMpropertyforthiscaseis 21 t 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T + 22 t 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T = 22 t )]TJ/F40 11.9552 Tf 11.955 0 Td [(T 22 t 22 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T = 22 t )]TJ/F40 11.9552 Tf 11.955 0 Td [(T )]TJ/F45 11.9552 Tf 11.955 0 Td [( 21 t 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T .B SubstitutingequationBintoequationBresultsinequationB3 x d t = 21 t )]TJ/F45 11.9552 Tf 11.955 0 Td [( 22 t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 22 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T x 0 + 22 t )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] ... + Z t t 0 22 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d SubstitutingequationBintothisequationresultsin x d t = 21 t )]TJ/F22 11.9552 Tf 11.956 -0.166 Td [( 22 t )]TJ/F40 11.9552 Tf 11.955 0 Td [(T )]TJ/F45 11.9552 Tf 11.955 0 Td [( 21 t 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T x 0 + 22 t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] ... + Z t t 0 22 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d x d t = 22 t )]TJ/F40 11.9552 Tf 11.955 0 Td [(T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.298 0 Td [(T x 0 + 22 t )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] + Z t t 0 22 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(" B 2 u d B Thisvelocityprolegivesthevelocityhistoryofthetrajectoryfromtheinitialconditionto thenalposition. EquationBcanalsobeusedtodenethevelocitychange x f neededat position x f sothatthevelocityattime T iszeroedout.Thisvelocitychangeisthe negativeofthevelocityproleattime t = T x f = )]TJ/F45 11.9552 Tf 9.298 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F40 11.9552 Tf 9.299 0 Td [(T x 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [( 22 T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 12 T [ x f )]TJ/F45 11.9552 Tf 11.955 0 Td [(x t ] )]TJ/F31 11.9552 Tf 11.955 16.272 Td [(Z T t 0 22 T )]TJ/F40 11.9552 Tf 11.955 0 Td [(t B 2 u dt Thisisfortheapplicationofrendezvoussince x x f x 0 as t T Obtainingthesetwovelocitychangesisparticularlyusefulwhenconsideringa minimumcontroleffortOCP.Thesolutiontotheminimumcontroleffortoptimalcontrol 171

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problemOCPcanbeapproximatedasthetwoburnsolution,withoneburnatthe initialtimeandoneburnatthenaltime.[72]Thus,thecostforthisproblemisalso approximatedas J k x i k 1 + k x f k 1 Itshouldbenotedthatthematrix 12 T canbesingularforcertainvaluesof T .Toillustratethis,twointerceptionexamplesareperformedusingtheCWHand SAAequations.TheparametersusedfortheCWHexampleareinTableB-1,andthe parametersfortheSAAexampleareinTableB-2. TableB-1.Relativeorientationparameters ParameterValueUnits n 0.001 rad = s r 0 500500500 T m v 0 000 T m = s r f 000 T m TableB-2.Relativeorientationparameters ParameterValueUnits J 2 4 30000 01000 00200 3 5 kg m 2 t 0.10.10.1 T rad = s 0 0.10.10.1 T rad 0 000 T rad = s f 000 T rad FigureB-1AshowsasetofTPBVPtrajectoriesfortheCWHequationwithdifferent valuesof T .FigureB-1Bshowsaplotof k v i k 1 fordifferentvaluesof T anddepicts thevalueswhere 12 T becomessingulari.e.,theverticalasymptotes.Forthis example,thematrix 12 T becomingsingularphysicallysigniesaninniteimpulse thatispointeddirectlyatthedesiredpositionorintheoppositedirection. 172

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FigureB-1CshowsasetofTPBVPtrajectoriesfortheSAAequationwithdifferent valuesof T .FigureB-1Dshowsaplotof k i k 1 fordifferentvaluesof T anddepicts thevalueswhere 12 T becomessingulari.e.,theverticalasymptotes.Forthis example,thematrix 12 T becomingsingularphysicallysigniesaninniteimpulseto reachthedesiredorientation. ACWHTPBVPtrajectories BCWHTPBVPimpulsemagnitudes CSAATPBVPtrajectories DSAATPBVPimpulsemagnitudes FigureB-1.Twopointboundaryvalueproblemexamples 173

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APPENDIXC HOMOTOPYWITHRICCATIEQUATIONL S = S Thederivationforsolvingthenonlinearoperator N S = S + Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S + A T S + SA = 0 S t f = S f ,C usingtheHomotopyContinuationHCmethodispresentedinthisAppendix.First,the convexhomotopyisdenedas H S p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p [ L S )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 ] )]TJ/F40 11.9552 Tf 11.955 0 Td [(ph t [ N S ] = 0 ,C where p 2f 2 R j 0 1 g istheembeddingparameter, h 2 R )-233(f 0 g istheauxiliary parameter, t 6 = 0 8 t istheauxiliaryfunction, Y 0 isaninitialguess,and L S = S Next,theMaclaurinSeriesExpansionMSEof S isobtainedas S = S 0 + p S 1 + p 2 S 2 + ,C SubstitutingequationCintoequationCyields, H S 0 S 1 S 2 ,..., p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p L S 0 + p S 1 + p 2 S 2 + )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(... ph t N S 0 + p S 1 + p 2 S 2 + C Theoperator L S iseasilydecomposedas L S 0 + p S 1 + p 2 S 2 +...= L S 0 + pL S 1 + p 2 L S 2 + ,C sinceitisalinearoperator.Theoperator N S denedinequationCisnonlinear andmustbedecomposedmanually.DirectsubstitutionoftheMSEintoequationC 174

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resultsin N S 0 S 1 S 2 ,..., p = S 0 + p S 1 + p 2 S 2 + + Q )]TJ/F22 11.9552 Tf 11.955 0 Td [(... S 0 + p S 1 + p 2 S 2 + BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 0 + p S 1 + p 2 S 2 + +... A T S 0 + p S 1 + p 2 S 2 + + S 0 + p S 1 + p 2 S 2 + A = 0 Theresultisrewrittensuchthatcoefcientsaregroupedforthedifferentpowersofthe embeddingparameter, N S = G 0 + p G 1 + p 2 G 2 + C where G 0 = S 0 + Q )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 0 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S 0 + A T S 0 + S 0 A G 1 = S 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 0 BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 1 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S 0 + A T S 1 + S 1 A G 2 = S 2 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 0 BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 2 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 1 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S 1 )]TJ/F45 11.9552 Tf 11.956 0 Td [(S 2 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S 0 + A T S 2 + S 2 A G 3 = S 3 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 0 BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 3 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 1 BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 2 )]TJ/F45 11.9552 Tf 11.956 0 Td [(S 2 BR )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 B T S 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(S 3 BR )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 B T S 0 + A T S 3 + S 3 A . . EquationCandequationCaresubstitutedintoequationCtoobtain, H S 0 S 1 S 2 ,..., p = )]TJ/F40 11.9552 Tf 11.956 0 Td [(p L S 0 + pL S 1 + p 2 L S 2 + )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F40 11.9552 Tf 11.955 0 Td [(p L Y 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(... ph t G 0 + p G 1 + p 2 G 2 + = 0 Thecoefcientsforthedifferentpowersoftheembeddingparameteraregrouped,which resultsin 175

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p 0 terms : L S 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 p 1 terms : L S 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L S 0 + L Y 0 p 2 terms : L S 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L S 1 p 3 terms : L S 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L S 2 . . Thetrendforobtainingthesubsequentgroupoftermsisthesame.Thecoefcientsare individuallysettozerotosolvefortheparameters S 0 S 1 S 2 ,... .Theresultisasystem ofdifferentialequations.ForthederivationpresentedinthisAppendix,theauxiliary functionusedis t = I ,however,thisfunctionisfreetochoose. Theequationforthezerothordertermsuggests L S 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 = 0 S 0 )]TJ/F22 11.9552 Tf 14.76 2.657 Td [(_ Y 0 = 0 Themosteasilyobtainablesolutionistolet S 0 = Y 0 where Y 0 isfreetochoose.The chosenparameteris Y 0 = C 0 = S f sincethiswouldsatisfytheequation L Y 0 = 0 andtheboundarycondition Y 0 t f = S f .Thus,thesolutionforthezerothparameteris S 0 = C 0 = S f Theequationfortherstordertermsuggests L S 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L S 0 + L Y 0 = 0 S 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 0 = 0 176

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Notethattheterm G 0 onlydependson S 0 whichhasalreadybeensolvedfor.Tosolve fortheparameter S 1 ,thehomogeneoussolutionisrstobtained S 1, h = C 1 where C 1 isaconstant.Theparticularsolutionisdeterminedbyemployingthevariation ofparameterstechnique.Thatis,theintegrationconstantisassumedtobetime dependenti.e., C 1 = C 1 t t f andthehomogeneoussolutionissubstitutedintothe originaldifferentialequation.Thus,theresultingdifferentialequationfortheparticular solutionis C 1 = h G 0 Noticethatthisequationonlydependson S 0 ,whichhasalreadybeensolvedfor.The solutionfortheconstantis C 1 t t f = h Z t t f G 0 d Also,notethat C 1 t f t f = 0 inordertomaintaintheboundaryconditionof S t f = S f Thus,thesolutionfortherstparameteris S 1 = C 1 t t f Theequationforthesecondordertermsuggests L S 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L S 1 = 0 S 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G 1 )]TJ/F22 11.9552 Tf 13.864 2.656 Td [(_ S 1 = 0 Noteagainthattheterm G 1 onlydependson S 0 and S 1 whichhavealreadybeensolved for.Itshouldbenotedthatthesubsequentdifferentialequationstobesolvedareofthe 177

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sameform S i +1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h G i )]TJ/F22 11.9552 Tf 13.865 2.657 Td [(_ S i = 0 for i =1,2,3,... Thus,fortheremainingderivationisdoneforthe i +1 th term.Thehomogeneous solutionof S i +1 isrstfoundandisoftheform S i +1, h = C i +1 for i =1,2,3,... where C i +1 isaconstant.Thevariationofparameterstechniqueisagainemployed tosolvefortheparticularsolution.Theresultingdifferentialequationfortheparticular solutionis C i +1 = C i + h G i for i =1,2,3,... Thisequationisintegratedtoobtainthesolution C i +1 t t f = C i + h Z t t f G i d for i =1,2,3,... Noteagainthat C i +1 t f t f = 0 tomaintaintheboundaryconditionof S t f = S f .The solutionforthe i +1 th parameteris S i +1 = C i +1 t t f for i =1,2,3,... Asaresult,the k th orderapproximationofthehomotopysolutionisgivenby S k = k X i =0 S i 178

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APPENDIXD HOMOTOPYWITHTRANSFORMEDRICCATIEQUATIONL X = X ThederivationforsolvingthelineartimevaryingLTVdifferentialequation N X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X = 0 X t f = X f usingtheHomotopyContinuationHCmethodispresentedinthisAppendix.First,the convexhomotopyisdenedas H X p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p [ L X )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 ] )]TJ/F40 11.9552 Tf 11.956 0 Td [(ph t [ N X ] = 0 ,D where p 2f 2 R j 0 1 g istheembeddingparameter, h 2 R )-233(f 0 g istheauxiliary parameter, t 6 = 0 8 t istheauxiliaryfunction, Y 0 isaninitialguess,and L X = X Next,theMaclaurinSeriesExpansionMSEof X isobtainedas X = X 0 + p X 1 + p 2 X 2 + ,D SubstitutingequationDintoequationDyields H X 0 X 1 X 2 ,..., p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p L X 0 + p X 1 + p 2 X 2 + )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(... ph t N X 0 + p X 1 + p 2 X 2 + .D Notethatboth L X and N X arelinearoperatorsandareeasilydecomposedas L S 0 + p S 1 + p 2 S 2 +...= L S 0 + pL S 1 + p 2 L S 2 + D N S 0 + p S 1 + p 2 S 2 +...= N S 0 + pN S 1 + p 2 N S 2 + .D 179

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EquationDandequationDaresubstitutedintoequationDtoobtain H X 0 X 1 X 2 ,..., p = )]TJ/F40 11.9552 Tf 11.955 0 Td [(p L X 0 + pL X 1 + p 2 L X 2 + )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F40 11.9552 Tf 11.955 0 Td [(p L Y 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(... ph t N X 0 + pN X 1 + p 2 N X 2 + = 0 Thecoefcientsforthedifferentpowersoftheembeddingparameteraregroupedwhich resultsin p 0 terms : L X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 p 1 terms : L X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h t N X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L X 0 + L Y 0 p 2 terms : L X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h t N X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L X 1 D p 3 terms : L X 3 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h t N X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L X 2 . . Thetrendforobtainingthesubsequentgroupoftermsisthesame.Thecoefcientsare individuallysettozerotosolvefortheparameters X 1 X 2 X 2 ,... .Theresultisasystem ofdifferentialequations.ForthederivationpresentedinthisAppendix,theauxiliary functionusedis t = I ,however,thisfunctionisfreetochoose. Theequationforthezerothordertermsuggests L X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 = 0 X 0 )]TJ/F22 11.9552 Tf 14.759 2.657 Td [(_ Y 0 = 0 Themosteasilyobtainablesolutionistolet X 0 = Y 0 where Y 0 isfreetochoose.The chosenparameteris Y 0 = C 0 = S f sincethiswouldsatisfytheequation L Y 0 = 0 andtheboundarycondition Y 0 t f = X f .Thus,thesolutionforthezerothparameteris S 0 = C 0 = X f .D 180

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Theequationfortherstordertermsuggests L X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(hN X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L X 0 + L Y 0 = 0 X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X 0 = 0 Tosolvefortheparameter X 1 ,thehomogeneoussolutionisrstobtained X 1, h = C 1 where C 1 isaconstant.Theparticularsolutionisdeterminedbyemployingthevariation ofparameterstechnique.Thatis,theintegrationconstantisassumedtobetime dependenti.e., C 1 = C 1 t t f andthehomogeneoussolutionissubstitutedintothe originaldifferentialequation.Thus,theresultingdifferentialequationfortheparticular solutionis C 1 = )]TJ/F40 11.9552 Tf 9.299 0 Td [(h H t C 0 Noticethatthisequationonlydependson S 0 ,whichhasalreadybeensolvedfor.The solutionfortheconstantis C 1 t t f = )]TJ/F40 11.9552 Tf 9.298 0 Td [(h Z t t f H t C 0 d Notethat C 1 t f t f = 0 inordertomaintaintheboundaryconditionof X t f = X f .Thus, thesolutionfortherstparameteris X 1 = C 1 t t f Theequationforthesecondordertermsuggests L X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(hN X 1 )]TJ/F40 11.9552 Tf 11.956 0 Td [(L X 1 = 0 X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X 1 )]TJ/F22 11.9552 Tf 14.596 2.656 Td [(_ X 1 = 0 181

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Noteagainthattheequationonlydependson X 0 and X 1 whichhavealreadybeen solvedfor.Itshouldalsobenotedthatthesubsequentdifferentialequationstobesolved areofthesameform, X i +1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X i )]TJ/F45 11.9552 Tf 11.956 0 Td [(H t X i )]TJ/F22 11.9552 Tf 14.595 2.657 Td [(_ X i = 0 for i =1,2,3,... Thustheremainingderivationisdoneforthe i +1 th term.Thehomogeneoussolution of X i +1 isrstfoundandisoftheform X i +1, h = C i +1 for i =1,2,3,... where C i +1 isaconstant.Thevariationofparameterstechniqueisagainemployed tosolvefortheparticularsolution.Theresultingdifferentialequationfortheparticular solutionis C i +1 = h +1 C i )]TJ/F40 11.9552 Tf 11.956 0 Td [(h H t C i for i =1,2,3,... Thisequationisintegratedtoobtainthesolution C i +1 t t f = h +1 C i )]TJ/F40 11.9552 Tf 11.956 0 Td [(h Z t t 0 H t C i d for i =1,2,3,... Noticeagainthat C i +1 t f t f = 0 tomaintaintheboundaryconditionof X t f = X f Thereforethesolutionforthe i +1 th parameteris X i +1 = C i +1 t t f for i =1,2,3,... Asaresult,the k th orderapproximationofthehomotopysolutionisgivenby X k = k X i =0 X i 182

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APPENDIXE HOMOTOPYWITHTRANSFORMEDRICCATIEQUATIONL X = X )]TJ/F45 11.9552 Tf 11.956 0 Td [(X ThederivationforsolvingthelineartimevaryingLTVdifferentialequation N X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X = 0 X t f = X f usingtheHomotopyContinuationHCmethodispresentedinthisAppendix.First,the convexhomotopyinequationDisdenedwith L X = X )]TJ/F45 11.9552 Tf 11.955 0 Td [(X Next,theMaclaurinSeriesExpansionMSEof X inequationDisobtained. SubstitutingequationDintoequationDyieldsequationD.Since L X and N X arelinearoperators,theycaneasilybedecomposedasshowninequationD andequationD,respectively.Asaresult,thecoefcientsforthedifferentpowersof theembeddingparameteraregroupedandresultinthesameformasequationD. ForthederivationpresentedinthisAppendix,theauxiliaryfunctionusedis t = I however,thisfunctionisfreetochoose. Theequationforthezerothordertermsuggests L X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L Y 0 = 0 X 0 )]TJ/F22 11.9552 Tf 14.759 2.657 Td [(_ Y 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H c [ X 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [(Y 0 ] = 0 Themosteasilyobtainablesolutionistolet X 0 = Y 0 where Y 0 isfreetochoose.The chosenparameteris Y 0 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C 0 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I X f sincethiswouldsatisfytheequation L Y 0 = 0 183

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andtheboundarycondition Y 0 t f = X f .Thus,thesolutionforthezerothparameteris X 0 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C 0 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I X f Theequationfortherstordertermssuggests, L X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(hN X 0 )]TJ/F40 11.9552 Tf 11.955 0 Td [(L X 0 + L Y 0 = 0 X 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H c X 1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X 0 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X 0 = 0 Tosolvefortheparameter X 1 ,thehomogeneoussolutionisrstobtained X 1, h =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C 1 where C 1 isaconstant.Theparticularsolutionisdeterminedbyemployingthevariation ofparameterstechnique.Thatis,theintegrationconstantisassumedtobetime dependenti.e., C 1 = C 1 t t f andthehomogeneoussolutionissubstitutedintothe originaldifferentialequation.Thus,theresultingdifferentialequationfortheintegration constantis exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f C 1 = h [ exp t )]TJ/F40 11.9552 Tf 11.956 0 Td [(t f I C 0 )]TJ/F45 11.9552 Tf 11.956 0 Td [(H t exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C 0 ] C 1 = h [ I )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t ] C 0 Notethatthematrixexponentialtermisadiagonalterm,thusitcommuteswiththe matrixmultiplicationoperation.Thesolutionfortheconstantis C 1 t t f = h Z t t f [ I )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t ] C 0 d Notethat C 1 t f t f = 0 inordertomaintaintheboundaryconditionof X t f = X f .Thus, thesolutionfortherstparameteris X 1 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(tf I C 1 t t f 184

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Theequationforthesecondordertermsuggests L X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(hN X 1 )]TJ/F40 11.9552 Tf 11.956 0 Td [(L X 1 = 0 X 2 )]TJ/F45 11.9552 Tf 11.955 0 Td [(X 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t X 1 )]TJ/F31 11.9552 Tf 11.955 9.684 Td [( X 1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(X 1 = 0 Itshouldbenotedthatthesubsequentdifferentialequationstobesolvedareoftheform X i +1 )]TJ/F45 11.9552 Tf 11.955 0 Td [(X i +1 )]TJ/F40 11.9552 Tf 11.955 0 Td [(h X i )]TJ/F45 11.9552 Tf 11.956 0 Td [(H t X i )]TJ/F31 11.9552 Tf 11.955 9.684 Td [( X i )]TJ/F45 11.9552 Tf 11.955 0 Td [(X i = 0 for i =1,2,3,... Theremainingderivationaredoneforthe i +1 th term.Thehomogeneoussolutionof X i +1 isrstfoundandisoftheform X i +1, h =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C i +1 for i =1,2,3,... where C i +1 isaconstant.Thevariationofparameterstechniqueisagainemployed tosolvefortheparticularsolution.Theresultingdifferentialequationfortheparticular solutionis exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C i +1 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C i + h exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I )]TJ/F45 11.9552 Tf 5.479 -9.683 Td [(C i + C i )]TJ/F45 11.9552 Tf 11.956 0 Td [(H t exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C i C i +1 =+ h C i + h [ I )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t ] C i for i =1,2,3,... Thisequationisintegratedtoobtainthesolution C i +1 t t f =+ h C i + h Z t t f [ I )]TJ/F45 11.9552 Tf 11.955 0 Td [(H t ] C i d for i =1,2,3,... Noticeagainthat C i +1 t f t f = 0 tomaintaintheboundaryconditionof X t f = X f Therefore,thesolutionforthe i +1 th parameteris X i +1 =exp t )]TJ/F40 11.9552 Tf 11.955 0 Td [(t f I C i +1 for i =1,2,3,... Asaresult,the k th orderapproximationofthehomotopysolutionisgivenby X k = k X i =0 X i 185

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REFERENCES [1]NationalSpacePolicyoftheUnitedStatesofAmerica,June2010. [2]Space-Travel.com,TugboatsinSpace, http://www.space-travel.com/reports/Tugboats in Space 999.html,April 2011. [3]Lanzerotti,L.J., AssessmentofOptionsforExtendingtheLifeoftheHubble SpaceTelescope:FinalReport ,NationalAcademyPress,2005. [4]Polites,M.E.,TechnologyofAutomatedRendezvousandCaptureinSpace, JournalofSpacecraftandRockets ,Vol.36,1999,pp.280. [5]Kelso,T.S.,Vallado,D.A.,Chan,J.,andBuckwalter,B.,ImprovedConjunction AnalysisViaCollaborativeSpaceSituationalAwareness, AAS/AIAASpaceFlight MechanicsMeeting ,2008. [6]Kelso,T.S.,AnalysisoftheIridium33Cosmos2251Collision,Tech.rep.,Center forSpaceStandards&Innovation,2009. [7]Carrico,T.,Langster,T.,Carrico,J.,Alfano,S.,Loucks,M.,andVallado,D., ProximityOperationsforSpaceSituationalAwarenessSpacecraftRendezvous andManeuveringusingNumericalSimulationsandFuzzyLogic, TheAdvanced MauiOpticalandSpaceSurveillanceTechnologiesConference ,2006. [8]Cebrowski,A.K.,OperationallyResponsiveSpace:ANewDefenseBusiness Model,Tech.rep.,U.S.ArmyWarCollege,2005. [9]Richards,M.G.,Viscito,L.,Ross,A.M.,andHastings,D.E.,Distinguishing AttributesfortheOperationallyResponsiveSpaceParadigm, ResponsiveSpace Vol.6,2008. [10]Brown,O.,Fractionatedspacearchitectures:Avisionforresponsivespace, Tech.rep.,DefenseAdvancedResearchProjectsAgency,2006. [11]Siewert,S.,McClure,L.H.,andHansen,E.,ASystemArchitecturetoAdvance SmallSatelliteMissionOperationsAutonomy, AIAAConferenceonSmall Satellites ,1995. [12]Tatsch,A.R.,Fitz-Coy,N.,andGladun,S.,On-OrbitServicing:ABriefSurvey, PerformanceMetricsforIntelligentSystemsConference ,2006. [13]Gralla,E.L.,Shull,S.,anddeWeck,O.,On-OrbitAssemblyStrategiesfor Next-GenerationSpaceExploration, The57thInternationalAstronautical Congress ,2006. 186

PAGE 187

[14]Antsaklis,P.J.,Passino,K.M.,andWang,S.J.,TowardsIntelligentAutonomous ControlSystems:ArchitectureandFundamentalIssues, JournalofIntelligentand RoboticSystems ,Vol.1,No.4,1989,pp.315. [15]Zeigler,B.P.,HighAutonomySystems:ConceptsandModels, Proceedingsin AI,Simulation,andPlanninginHighAutonomySystems ,1990,pp.2. [16]Huang,H.M.,Messina,E.,andAlbus,J.,TowardaGenericModelforAutonomy LevelsforUnmannedSystemsALFUS, PerformanceMetricsforIntelligent Systems ,2003. [17]Huang,H.M.,Pavek,K.,Albus,J.,andMessina,E.,AutonomyLevelsfor UnmannedSystemsALFUSFramework:AnUpdate, Proceedingsofthe2005 SPIEDefenseandSecuritySymposium ,2005. [18]Clough,B.T.,Metrics,Schmetrics!HowtheHeckDoYouDetermineaUAV's AutonomyAnyway,Tech.rep.,AirForceResearchLab/Wright-PattersonAFB, OH,2002. [19]Lavery,D.,PerspectivesonFutureSpaceRobotics, AerospaceAmerica ,Vol.32, No.5,1994,pp.32. [20]Fehse,W., AutomatedRendezvousandDockingofSpacecraft ,Cambridge,2003. [21]Bajracharya,M.,Maimone,M.W.,andHelmick,D.,AutonomyforMarsRovers: Past,Present,andFuture, IEEEComputerSociety ,Vol.41,No.12,2008, pp.44. [22]Yamanaka,K.,RendezvousStrategyoftheJapaneseLogisticsSupportVehicle totheInternationalSpaceStation, SpacecraftGuidance,Navigation,andControl Systems ,Vol.381,1997,p.103. [23]Kawano,I.,Mokuno,M.,Kasai,T.,andSuzuki,T.,ResultofAutonomous RendezvousDockingExperimentofEngineeringTestSatellite-VII, Journalof SpacecraftandRockets ,Vol.38,No.1,2001,pp.105. [24]Miki,Y.,Abe,N.,Matsuyama,K.,Masuda,K.,Fukuda,N.,andSasaki,H., DevelopmentoftheH-IITransferVehicleHTV, MitsubishiHeavyIndustries TechnicalReview ,Vol.47,No.1,2010,pp.58. [25]Gladun,S.A., InvestigationofClose-ProximityOperationsofanAutonomous RoboticOn-OrbitServicerUsingLinearizedOrbitMechanics ,Master'sthesis, UniversityofFlorida,2005. [26]Wofnden,D.C.andGeller,D.K.,NavigatingtheRoadtoAutonomous OrbitalRendezvous, JournalofSpacecraftandRockets ,Vol.44,No.4,2007, pp.898. 187

PAGE 188

[27]Croomes,S.,DemonstrationofAutonomousRendezvousTechnologyMishap InvestigationBoardReview,Tech.rep.,NationalAeronauticsandSpace Administration,2006. [28]Davis,T.M.andMelanson,D.,XSS-10Micro-SatelliteFlightDemonstration, PaperNo.GT-SSEC.D.3 ,2005. [29]AFRL,XSS-11Micro-Satellite, http://www.kirtland.af.mil/shared/media/document/AFD-070404-108.pdf, December2005. [30]Weismuller,T.andLeinz,M.,GNCTechnologyDemonstratedbytheOrbital ExpressAutonomousRendezvousandCaptureSensorSystem, AASGuidance andControlConference ,2006. [31]SatNews.com,OrbitalExpressConductsHistory'sFirstSatellite-to-Satellite HardwareTransfer,http://www.satnews.com/stories2007/4315/,April2007. [32]Prisma,PrismaSatellites,http://www.prismasatellites.se/?sid=9028. [33]SSC,SwedishSpaceCorporation,http://www.ssc.se/?id=7611. [34]Bodin,P.,Larsson,R.,Nilsson,F.,Chasset,C.,Noteborn,R.,andNylund, M.,PRISMA:AnIn-OrbitTestBedforGuidance,Navigation,andControl Experiments, JournalofSpacecraftandRockets ,Vol.46,No.3,2009. [35]Bosse,A.B.,Barnds,W.J.,Brown,M.A.,Creamer,N.G.,Feerst,A.,Henshaw, C.G.,Hope,A.S.,Kelm,B.E.,Klein,P.A.,Pipitone,F.,etal.,SUMO:Spacecraft fortheUniversalModicationofOrbits, ProceedingsofSPIE ,2004. [36]Space.com,AirForceANGELS:SatelliteEscortstoTakeFlight, http://www.space.com/1816-air-force-angels-satellite-escorts-ight.html, November2007. [37]SpaceDaily.com,ViviSatLaunched, http://www.spacedaily.com/reports/ViviSat Launched 999.html,January 2011. [38]Henshaw,C.G.andSanner,R.M.,VariationalTechniqueforSpacecraft TrajectoryPlanning, JournalofAerospaceEngineering ,Vol.23,No.2,2010, pp.147. [39]Vallado,D.A.andMcClain,W.D., FundamentalsofAstrodynamicsandApplications ,KluwerAcademicPublishers,2001. [40]Battin,R.H., AnIntroductiontotheMathematicsandMethodsofAstrodynamics AIAA,1999. [41]Lawden,D.F., OptimalTrajectoriesforSpaceNavigation ,Butterworths,1963. 188

PAGE 189

[42]Marec,J.P., OptimalSpaceTrajectories ,Elsevier,1979. [43]Bryson,A.E.andHo,Y.C., AppliedOptimalControl ,JohnWiley&SonsInc, 1975. [44]Kirk,D.E., OptimalControlTheory:AnIntroduction ,DoverPublications,2004. [45]Henshaw,C.G., AVariationalTechniqueforSpacecraftTrajectoryPlanning ,Ph.D. thesis,UniversityofMaryland,2003. [46]Rosell,J.andIniguez,P.,PathPlanningUsingHarmonicFunctionsand ProbabilisticCellDecomposition, IEEEInternationalConferenceonRobotics andAutomation ,2005,pp.1803. [47]Phillips,J.,Kavraki,L.,andBedrossian,N.,ProbabilisticOptimizationAppliedto SpacecraftRendezvousandDocking, AdvancesintheAstronauticalSciences Vol.114,2003,pp.261. [48]Phillips,J.M.,Bedrossian,N.,andKavraki,L.E.,GuidedExpansiveSpaces Trees:ASearchStrategyforMotion-andCost-ConstrainedStateSpaces, IEEEInternationalConferenceonRoboticsandAutomation ,Vol.4,2004,pp. 3968. [49]Benson,D., AGaussPseudospectralTranscriptionforOptimalControl ,Ph.D. thesis,MassachusettsInstituteofTechnology,2005. [50]Huntington,G.T., AdvancementandAnalysisofaGaussPseudospectralTranscriptionforOptimalControlProblems ,Ph.D.thesis,MassachussetsInstituteof Technology,2007. [51]Boyarko,G.A., SpacecraftGuidanceStrategiesforProximityManeuveringand CloseApproachwithaTumblingObject ,Ph.D.thesis,NavalPostgraduateSchool, 2010. [52]Henshaw,C.G.,AUnicationofArticialPotentialFunctionGuidanceand OptimalTrajectoryPlanning,Tech.rep.,NavalCenterforSpaceTechnology,U.S. NavalResearchLaboroatory,2005. [53]Martinson,N.S., ObstacleAvoidanceGuidanceandControlforAutonomous Satellites ,Ph.D.thesis,UniversityofFlorida,2009. [54]Hablani,H.B.,Tapper,M.L.,andDana-Bashian,D.J.,GuidanceandRelative NavigationforAutonomousRendezvousinaCircularOrbit, JournalofGuidance, Control,andDynamics ,Vol.25,No.3,2002,pp.553. [55]Clohessy,W.H.andWiltshire,R.S.,TerminalGuidanceSystemforSatellite Rendezvous, JournaloftheAerospaceSciences ,Vol.27,No.9,1960, pp.653. 189

PAGE 190

[56]Khatib,O.,Real-TimeObstacleAvoidanceforManipulatorsandMobileRobots, TheInternationalJournalofRoboticsResearch ,Vol.5,No.1,1986,pp.90. [57]Lopez,I.andMclnnes,C.R.,AutonomousRendezvousUsingArticialPotential FunctionGuidance, JournalofGuidance,Control,andDynamics ,Vol.18,No.2, 1995,pp.237. [58]Ge,S.S.andCui,Y.J.,DynamicMotionPlanningforMobileRobotsUsing PotentialFieldMethod, AutonomousRobots ,Vol.13,No.3,2002,pp.207. [59]McInnes,C.R.,AutonomousPath-planningforOn-orbitServicingVehicles, JournaloftheBritishInterplanetarySociety ,Vol.53,No.1/2,2000,pp.26. [60]Huang,W.H.,Fajen,B.R.,Fink,J.R.,andWarren,W.H.,VisualNavigation andObstacleAvoidanceUsingaSteeringPotentialFunction, Roboticsand AutonomousSystems ,Vol.54,No.4,2006,pp.288. [61]Gazi,V.,SwarmAggregationsUsingArticialPotentialsandSliding-Mode Control, IEEETransactionsonRobotics ,Vol.21,No.6,2005,pp.1208. [62]Strizzi,J.,Ross,I.M.,andFahroo,F.,TowardsReal-TimeComputationof OptimalControlsforNonlinearSystems, AIAAGuidance,Navigation,andControl Conference ,2002. [63]Wertz,J.R.andBell,R.,AutonomousRendezvousandDockingTechnologiesStatusandProspects, ProceedingsofSPIE ,Vol.5088,2003,pp.20. [64]Wertz,J.R., SpacecraftAttitudeDeterminationandControl ,KluwerAcademic Publishers,1978. [65]Hughes,P.C., SpacecraftAttitudeDynamics ,JohnWiley&SonsInc,1986. [66]Shuster,M.D.,SurveyofAttitudeRepresentations, JournaloftheAstronautical Sciences ,Vol.41,1993,pp.439. [67]Kuipers,J.B., QuaternionsandRotationSequences ,PrincetonUniversityPress, 1999. [68]Crassidis,J.L.andJunkins,J.L., OptimalEstimationofDynamicSystems Chapman&Hall,2004. [69]Zipfel,P.H., ModelingandSimulationofAerospaceVehicleDynamics ,AIAA, 2000. [70]Wie,B., SpaceVehicleDynamicsandControl ,AIAA,1998. [71]Ogata,K., ModerncontrolEngineering ,PrenticeHall,2009. 190

PAGE 191

[72]Lembeck,C.A.andPrussing,J.E.,OptimalImpulsiveInterceptwithLow-Thrust RendezvousReturn, JournalofGuidance,Control,andDynamics ,Vol.16,1993, pp.426. [73]Atkinson,K.E., AnIntroductiontoNumericalAnalysis ,JohnWiley&SonsInc, 1989. [74]Benson,D.A.,Huntington,G.T.,Thorvaldsen,T.P.,andRao,A.V.,Direct TrajectoryOptimizationandCostateEstimationViaanOrthogonalCollocation Method, JournalofGuidance,Control,andDynamics ,Vol.29,No.6,2006, pp.1435. [75]Bertsekas,D.P.,Hager,W.W.,andMangasarian,O.L., NonlinearProgramming AthenaScientic,1999. [76]Rao,A.V.,Benson,D.A.,Darby,C.,Patterson,M.A.,Francolin,C.,Sanders,I., andHuntington,G.T.,Algorithm902:GPOPS,AMATLABSoftwareforSolving Multiple-PhaseOptimalControlProblemsUsingtheGaussPseudospectral Method, ACMTransactionsonMathematicalSoftware ,Vol.37,No.2,2010, pp.1. [77]Rao,A.V.,Benson,D.A.,Huntington,G.T.,Origin,B.,Seattle,L.,Francolin,C., Darby,C.L.,Patterson,M.A.,andSanders,I., UsersManualforGPOPSVersion 3.0:AMATLABRSoftwareforSolvingMultiple-PhaseOptimalControlProblems UsingPseudospectralMethods ,2010. [78]Badawy,A.andMcInnes,C.R.,On-OrbitAssemblyUsingSuperquadricPotential Fields, JournalofGuidance,Control,andDynamics ,Vol.31,No.1,2008, pp.30. [79]Kim,J.O.andKhosla,P.K.,Real-TimeObstacleAvoidanceUsingHarmonic PotentialFunctions, IEEETransactionsonRoboticsandAutomation ,Vol.8, No.3,1992,pp.338. [80]Rimon,E.andKoditschek,D.E.,ExactRobotNavigationUsingArticialPotential Functions, IEEETransactionsonRoboticsandAutomation ,Vol.8,No.5,1992, pp.501. [81]Waydo,S.andMurray,R.M.,VehicleMotionPlanningUsingStreamFunctions, IEEEInternationalConferenceonRoboticsandAutomation ,2003. [82]Koditschek,D.,ExactRobotNavigationbyMeansofPotentialFunctions:Some TopologicalConsiderations, IEEEInternationalConferenceonRoboticsand Automation ,1987. [83]Mabrouk,M.H.andMcInnes,C.R.,SolvingthePotentialFieldLocalMinimum ProblemUsingInternalAgentStates, RoboticsandAutonomousSystems Vol.56,No.12,2008,pp.1050. 191

PAGE 192

[84]Tatsch,A.R., ArticialPotentialFunctionGuidanceforAutonomousIn-Space Operations ,Ph.D.thesis,UniversityofFlorida,2006. [85]Izzo,D.andPettazzi,L.,AutonomousandDistributedMotionPlanningfor SatelliteSwarm, JournalofGuidance,Control,andDynamics ,Vol.30,No.2, 2007,pp.449. [86]McCamish,S.B., DistributedAutonomousControlofMultipleSpacecraftDuring CloseProximityOperations ,Ph.D.thesis,NavalPostgraduateSchool,2007. [87]Dixon,W.E.,Behal,A.,Dawson,D.M.,andNagarkatti,S.P., NonlinearControlof EngineeringSystems:ALyapunov-BasedApproach ,Birkhauser,2003. [88]Yao,B.andTomizuka,M.,SmoothRobustAdaptiveSlidingModeControlof ManipulatorswithGuaranteedTransientPerformance, IEEEAmericanControl Conference ,Vol.1,1994,pp.1176. [89]Khalil,H.K.andGrizzle,J.W., Nonlinearsystems ,PrenticeHall,2002. [90]Martinson,N.,Munoz,J.D.,andWiens,G.J.,ANewMethodofGuidance ControlforAutonomousRendezvousinaClutteredSpaceEnvironment, AIAA Guidance,Navigation,andControlConference ,2007. [91]Reid,W.T., RiccatiDifferentialEquations ,AcademicPress,1972. [92]Engwerda,J.C., LQDynamicOptimizationandDifferentialGames ,JohnWiley& SonsInc,2005. [93]Jodar,L.andNavarro,E.,ClosedAnalyticalSolutionofRiccatiTypeMatrix DifferentialEquations, IndianJournalofPureAppliedMathematics ,Vol.23,No.3, 1992,pp.185. [94]Razzaghi,M.,SolutionoftheMatrixRiccatiEquationinOptimalControl, InformationSciences ,Vol.16,No.1,1978,pp.61. [95]Bellman,R., StabilityTheoryofDifferentialEquations ,McGraw-Hill,1953. [96]Liao,S.J., BeyondPerturbation:IntroductiontotheHomotopyAnalysisMethod, Vol.2ofCRCSeries:ModernMechanicsandMathematics ,Chapman&Hall, 2004. [97]Allgower,E.L.andGeorg,K., NumericalContinuationMethods:AnIntroduction Springer,1990. [98]Tschauner,J.andHempel,P.,OptimaleBeschleunigeungsprogrammefurdas Rendezvous-Manover, AstronauticaActa ,Vol.10,1964,pp.296. [99]Yamanaka,K.andAnkersen,F.,NewStateTransitionMatrixforRelativeMotion onanArbitraryEllipticalOrbit, JournalofGuidance,Control,andDynamics Vol.25,No.1,2002,pp.60. 192

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[100]Wie,B.,Weiss,H.,andArapostathis,A.,QuaternionFeedbackRegulatorfor SpacecraftEigenaxisRotations, JournalofGuidanceandControl ,Vol.12,No.3, 1989,pp.375. [101]Schweighart,S.A.andSedwick,R.J.,High-FidelityLinearizedJ2Modelfor SatelliteFormationFlight, JournalofGuidance,Control,andDynamics ,Vol.25, No.6,2002,pp.1073. 193

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BIOGRAPHICALSKETCH JosueDavidMu nozwasborninGuatemalaCity,Guatemalain1982.Hemoved toMiami,Floridaattheripeageofveyearsoldandwasraisedthere.Hereceived hisBachelorsofScienceinaerospaceengineeringfromtheUniversityofFloridain December2005andgraduatedCumLaude.HewasthenacceptedtothePh.D.direct programattheUniversityofFloridaandobtainedhisMastersofScienceinaerospace engineeringinDecember2008.HealsohadthehonorofbeingawardedtheSouthEast AllianceforGraduateEducationoftheProfessoriateFellowshipaswellastheScience, Math,andResearchTransformationScholarship. Hehadthepleasureofbeingagraduateteachingassistantaswellasagraduate researchassistant,workingonprojectsforagencieslikeDefenseAdvancedResearch AgencyProjectsandtheLockheedMartinCorporation.HewasalsoSpaceScholarat theAirForceResearchLab/SpaceVehiclesDirectoratefortheSummersof2009and 2010.HewasalsoabletobepartoftheStudentTemporaryEmploymentProgramat theAirForceResearchLabfortheFall2009semester,wherehewasabletocontribute withhisresearch.HewasamemberoftheSpaceSystemsGroupandSmallSatellite DesignClubattheUniversityofFlorida,andisamemberoftheAmericanInstituteof AeronauticsandAstronautics,andAmericanAstronauticsSociety. 194