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Obstacle Avoidance Guidance and Control for Autonomous Satellites

Permanent Link: http://ufdc.ufl.edu/UFE0041033/00001

Material Information

Title: Obstacle Avoidance Guidance and Control for Autonomous Satellites
Physical Description: 1 online resource (131 p.)
Language: english
Creator: Martinson, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: nonlinear, obstacle, optimal, rendezvous
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The guidance and control techniques that are to be developed must be able to minimize fuel usage as well as ensure obstacle avoidance with other neighboring satellites, space debris, and the satellites they are intending to rendezvous. In this framework, the new methods of guidance and control are compared with previous state of the art (i.e., Artificial Potential Function Guidance (APFG)). Within this scope, three methods are developed which demonstrate improvement in the areas of fuel usage and ensuring obstacle avoidance while maintaining acceptable limits of computational efficiency. The first method uses a switching control between rendezvous and obstacle avoidance. Using the CW maneuver for rendezvous is an advantage compared to APFG schemes because APFG does not have the ability to directly compensate for the Coriolis effect. A statistical analysis of this method shows that both the number of impulsive thrusts and fuel usage was reduced as compared to APFG. The second guidance algorithm uses a 3-D search of the time-of-flight and final relative position to steer trajectories away from obstacles and toward a desired relative position. A beneficial property of the guidance and control method is that the states and control are analytically derived. The guidance and control method is constrained to avoid obstacles and penalize fuel use. A feasible set of TOF developed from the norm of the control encloses a single minimum for which any guess of the TOF within these bounds can return the correct solution. This optimization grades trajectories over small horizons of time to reduce the complexity of the problem, allow for real-time implementation, incorporate feedback to mitigate disturbances and noise, and enable multiple impulsive maneuvers to be planned separately over the entire TOF. The last guidance and control algorithm developed is for orbits which are elliptical and the form of the control is based on the power series solution to the restricted two-body problem. Optimization is done in a similar manner to the GAS method. In summary, the three methods presented in this dissertation indicate that obstacles can be avoid safely and require less fuel use than APFG methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Nicholas Martinson.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Wiens, Gloria J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041033:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041033/00001

Material Information

Title: Obstacle Avoidance Guidance and Control for Autonomous Satellites
Physical Description: 1 online resource (131 p.)
Language: english
Creator: Martinson, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: nonlinear, obstacle, optimal, rendezvous
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The guidance and control techniques that are to be developed must be able to minimize fuel usage as well as ensure obstacle avoidance with other neighboring satellites, space debris, and the satellites they are intending to rendezvous. In this framework, the new methods of guidance and control are compared with previous state of the art (i.e., Artificial Potential Function Guidance (APFG)). Within this scope, three methods are developed which demonstrate improvement in the areas of fuel usage and ensuring obstacle avoidance while maintaining acceptable limits of computational efficiency. The first method uses a switching control between rendezvous and obstacle avoidance. Using the CW maneuver for rendezvous is an advantage compared to APFG schemes because APFG does not have the ability to directly compensate for the Coriolis effect. A statistical analysis of this method shows that both the number of impulsive thrusts and fuel usage was reduced as compared to APFG. The second guidance algorithm uses a 3-D search of the time-of-flight and final relative position to steer trajectories away from obstacles and toward a desired relative position. A beneficial property of the guidance and control method is that the states and control are analytically derived. The guidance and control method is constrained to avoid obstacles and penalize fuel use. A feasible set of TOF developed from the norm of the control encloses a single minimum for which any guess of the TOF within these bounds can return the correct solution. This optimization grades trajectories over small horizons of time to reduce the complexity of the problem, allow for real-time implementation, incorporate feedback to mitigate disturbances and noise, and enable multiple impulsive maneuvers to be planned separately over the entire TOF. The last guidance and control algorithm developed is for orbits which are elliptical and the form of the control is based on the power series solution to the restricted two-body problem. Optimization is done in a similar manner to the GAS method. In summary, the three methods presented in this dissertation indicate that obstacles can be avoid safely and require less fuel use than APFG methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Nicholas Martinson.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Wiens, Gloria J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041033:00001


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OBSTACLEAVOIDANCEGUIDANCEANDCONTROLFORAUTONOMOUSSATELLITESByNICHOLASS.MARTINSONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2009

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c2009NicholasS.Martinson 2

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Idedicatethisdissertationtomyparentsforpushingmetoreachmygoals. 3

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ACKNOWLEDGMENTS First,IwouldliketothankmyanceeJenniferAltyforallherloveandsupport.IwouldliketothankJosueMunoz,FrederickLeve,andTakashiHiramatsuforallowingmetobounceideasoffthemandtakingtheirtimetodiscusscontrolsrelatedtopics.Ithoroughlyenjoyedthetimespentsittingarounddiscussingresearchandinfrontofthewhiteboarddescribingourproblemsandpossiblesolutions.IamgreatlyappreciativeoffundingthroughtheDefenseAdvancedResearchProjectsAgency,theUniversityofFloridaAlumnifellowship,theUniversityofFlorida'sMechanicalandAerospaceEngineeringDepartment'sTeachersAssistantship,andfromtheLockheedMartinCorporation.IwouldalsoliketothankmycommitteemembersProfessor'sWarrenDixon,CarlCrane,andJacobHammerfortheirinputthroughthisprocess.Inaddition,IwouldpersonallylikethankDr.Fitz-Coyinhisguidancethroughthisprocess.Finally,Iwouldliketothankmyadvisor,ProfessorGloriaJ.Wiens,forwithouthernoneofthiswouldbepossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Motivation .................................... 13 1.2MissionConceptAndDetails ......................... 15 1.2.1TrajectoryPlanning ........................... 16 1.2.2SystemHierarchy ............................ 17 1.3LiteratureReview ................................ 19 1.4ResearchScope ................................ 23 1.5DissertationOutline .............................. 25 2SYSTEMDYNAMICS ................................ 26 2.1CWManeuver ................................. 27 2.2RestrictedTwo-BodyProblem ......................... 28 2.3ClassicalOrbitalElements ........................... 30 3ARTIFICIALPOTENTIALFUNCTIONGUIDANCE ................ 32 3.1TheoreticalDevelopment ........................... 32 3.1.1Limitations ................................ 37 3.1.2SpacecraftRendezvousUsingAPFGAlgorithm ........... 40 3.1.3Conclusions ............................... 42 3.2APFGWithSwitchingControl ......................... 44 3.2.1Stability ................................. 49 3.2.2SpacecraftRendezvousUsingAPFGSC ............... 49 3.2.3Conclusion ............................... 51 4GUIDANCEUSINGANALYTICSOLUTIONOFDYNAMICEQUATIONS .... 54 4.1GASMethod .................................. 54 4.1.1MinimizationOfCWManeuver .................... 55 4.1.2OptimizationStructure ......................... 58 4.1.3ObstacleAvoidance .......................... 60 4.1.4InitializingGASAlgorithm ....................... 61 4.1.5OptimalityConditions .......................... 61 4.1.6SensitivityAnalysis ........................... 64 5

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4.1.7Stability ................................. 66 4.1.8SpacecraftRendezvousUsingGASMethod ............. 66 4.1.9Conclusion ............................... 79 4.2GuidanceUsingAnalyticSolutionOfTheTwo-BodyProblem ....... 81 4.2.1OptimizationStructure ......................... 86 4.2.2ObstacleAvoidance .......................... 88 4.2.3InitializingGASTBAlgorithm ...................... 89 4.2.4Stability ................................. 89 4.2.5SpacecraftRendezvousUsingGASTBMethod ........... 90 4.2.6Conclusion ............................... 100 4.3TerminalGuidanceStrategy .......................... 102 5CONCLUSION .................................... 107 APPENDIX ANECESSARYANDSUFFICIENTCONDITIONSANDSENSITIVITYANALYSIS 110 BPOWERSERIESSOLUTIONOFTHERESTRICTEDTWO-BODYPROBLEM 124 REFERENCES ....................................... 126 BIOGRAPHICALSKETCH ................................ 131 6

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LISTOFTABLES Table page 3-1PerformanceofAPFGversusAPFGSC ...................... 51 4-1PerformanceofGASvs.APFG ........................... 76 4-2PerformanceofGASTBwithobstaclesandAPFGwithoutobstacles ...... 97 7

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LISTOFFIGURES Figure page 1-1Controlarchitectureforsatelliterendezvous .................... 18 2-1RSWframedenedsuchthatxrelistheR-bardirectionandyrelistheV-bardirection.ThecomponentsIx,Iy,Izdescribeaninertialcoordinatesystem. 27 2-2ECIcoordinatesystem ................................ 29 3-1Constraintforxedvs.xandy-positions ..................... 39 3-2TrajectoryforsatelliterendezvoususingAPFGalgorithm ............. 41 3-3_duringrendezvoususingAPFGalgorithm .................... 42 3-4R-bar,V-bartrajectoriesandimpulsesduringrendezvoususingAPFGalgorithm 43 3-5TrajectoryforsatelliterendezvoususingAPFGalgorithmwithoneobstacle .. 44 3-6_duringrendezvoususingAPFGalgorithmwithoneobstacle .......... 45 3-7R-bar,V-bartrajectoriesandimpulsesduringrendezvoususingAPFGalgorithmwithoneobstacle ................................... 46 3-8RendezvoususingCWmaneuver ......................... 47 3-9RendezvoususingCWmaneuver ......................... 48 3-10SimulationofAPFGSC(red)versusAPFG(green) ................ 50 3-11HistogramplotofAPFGandAPFGSCcontrollers ................. 52 4-11-normand2-normofvversusTOF ....................... 57 4-2BlockdiagramforsatelliteGN&CusingGASmethod ............... 60 4-3RendezvouswiththreeobstaclesusingGAScontroltechnique ......... 67 4-4Impulse,v,forGASG&Cmethodforrendezvous. ................ 68 4-5Firstobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. ................ 69 4-6Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps ....................................... 70 4-7Secondobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. ................ 71 8

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4-8Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps ....................................... 72 4-9Thirdobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. ................ 73 4-10Plotsshowingobstacleavoidanceforthirdobstacleencounteredforsubsequenttimesteps ....................................... 74 4-11Plotsshowingobstacleavoidanceforthirdobstacleencounteredforsubsequenttimesteps ....................................... 75 4-12Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[110]Tkm ................................. 77 4-13Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[1)]TJ /F5 11.955 Tf 11.95 0 Td[(10]Tkm ................................ 77 4-14Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm ................................ 78 4-15Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm ............................... 78 4-16Histogramplotsoffuelusev(m/s)forGASmethodinscenariowithsixobstacles 80 4-17SolutionofvforrangeofTOFtothepowerseriessolutionoftherestrictedtwo-bodyproblem .................................. 83 4-18Measureofsatisfactionofrst-orderoptimalityconditionsforr(t))]TJ /F3 11.955 Tf 12.47 0 Td[(F(t)r0)]TJ /F3 11.955 Tf -394.04 -14.45 Td[(G(t)v0=0 ...................................... 85 4-19BlockdiagramforsatelliteGN&CusingGASTBmethod ............. 88 4-20RendezvouswithoneobstacleusingGASTBcontroltechnique. ......... 92 4-21Impulse,v,forGASTBG&Cmethodforrendezvous. .............. 93 4-22Differencesintrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. ...................... 94 4-23Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps ....................................... 95 4-24Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps ....................................... 96 9

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4-25Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7179.1km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,kek=0.101 .. 98 4-26Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7178.3km,==80056rad,kek=0.101 ... 99 4-27Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,kek=0.101 .. 99 4-28Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=20056rad,kek=0.101 .. 100 4-29MagnitudeoffuelusewithESRinitialconditionsofkrk=7179.1km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,kek=0.101andinitialobstaclelocationofkrk=7177.48km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=24056rad,kek=0.101 ........................... 101 4-30MagnitudeoffuelusewithESRinitialconditionsofkrk=7178.3km,==80056rad,kek=0.101andinitialobstaclelocationofkrk=7178.26km,=0rad,kek=0.101 ................................ 102 4-31MagnitudeoffuelusewithESRinitialconditionsofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,kek=0.101andinitialobstaclelocationofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=60056rad,kek=0.101 ........................... 103 4-32MagnitudeoffuelusewithESRinitialconditionsofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=20056rad,kek=0.101andinitialobstaclelocationofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=19056rad,kek=0.101 ........................... 104 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOBSTACLEAVOIDANCEGUIDANCEANDCONTROLFORAUTONOMOUSSATELLITESByNicholasS.MartinsonDecember2009Chair:GloriaWiensMajor:AerospaceEngineering Theguidanceandcontroltechniquesthataretobedevelopedmustbeabletominimizefuelusageaswellasensureobstacleavoidancewithotherneighboringsatellites,spacedebris,andthesatellitestheyareintendingtorendezvous.Inthisframework,thenewmethodsofguidanceandcontrolarecomparedwithpreviousstateoftheart(i.e.,ArticialPotentialFunctionGuidance(APFG)).Withinthisscope,threemethodsaredevelopedwhichdemonstrateimprovementintheareasoffuelusageandensuringobstacleavoidancewhilemaintainingacceptablelimitsofcomputationalefciency.Therstmethodusesaswitchingcontrolbetweenrendezvousandobstacleavoidance.UsingtheCWmaneuverforrendezvousisanadvantagecomparedtoAPFGschemesbecauseAPFGdoesnothavetheabilitytodirectlycompensatefortheCorioliseffect.AstatisticalanalysisofthismethodshowsthatboththenumberofimpulsivethrustsandfuelusagewasreducedascomparedtoAPFG.Thesecondguidancealgorithmusesa3-Dsearchofthetime-of-ightandnalrelativepositiontosteertrajectoriesawayfromobstaclesandtowardadesiredrelativeposition.Abenecialpropertyoftheguidanceandcontrolmethodisthatthestatesandcontrolareanalyticallyderived.Theguidanceandcontrolmethodisconstrainedtoavoidobstaclesandpenalizefueluse.AfeasiblesetofTOFdevelopedfromthenormofthecontrolenclosesasingleminimumforwhichanyguessoftheTOFwithintheseboundscanreturnthecorrectsolution.Thisoptimizationgradestrajectoriesoversmallhorizons 11

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oftimetoreducethecomplexityoftheproblem,allowforreal-timeimplementation,incorporatefeedbacktomitigatedisturbancesandnoise,andenablemultipleimpulsivemaneuverstobeplannedseparatelyovertheentireTOF.Thelastguidanceandcontrolalgorithmdevelopedisfororbitswhichareellipticalandtheformofthecontrolisbasedonthepowerseriessolutiontotherestrictedtwo-bodyproblem.OptimizationisdoneinasimilarmannertotheGASmethod.Insummary,thethreemethodspresentedinthisdissertationindicatethatobstaclescanbeavoidsafelyandrequirelessfuelusethanAPFGmethods. 12

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CHAPTER1INTRODUCTION Inthisdissertation,thebasicpurposeoftheresearchistoexplorealgorithmsforclose-proximityrendezvousofsatellitesinaclutteredenvironment.Ahighdelitymodelisusedtodemonstrateobstacleavoidance,rendezvous,andperformanceoftrajectoriesbasedontheamountoffuelused,v,forasmallclassofsatellite(i.e.,microorsmaller).Thischapterprovidesmotivationforthisparticularmission,areviewofpreviousmethodsusedforthisresearch,andthescopeofthisdissertation. 1.1Motivation Ifspaceistobecomemoreaccessibleforcommercial,military,andgovernmentpurposes,thenserviceswillneedtobeperformedinspacetoreduceoperationalcostoffutureandcurrentspaceassets.Typically,thetwotypesofmissionsidentiedtoreduceoperationalcostsareassemblyandinspection.Thisdissertationprimarilyfocusesontheproximitymaneuveringforassemblyandrendezvoustypemissionsprovidingservicestospaceassets.Forassemblyandrendezvoustypemissions,minimizingfueluseandensuringobstacleavoidanceareofparamountimportance.Examplesofservicesprovidedthroughassemblyandrendezvoustypemissionsincludestructureassembly,refuel/recharge,repair,andspacetug. Commercially,therehavebeenseveralproposedmissionstoachieverendezvousanddockingtodemonstratethefeasibilityofthesemissions.AproposedmissioncalledOLEV[ 1 ]isacollaborationbetweenindustryinEuropeancountriesandtheEuropeanSpaceAgency(ESA).ThemissionforOLEViswhatistermedasaspacetugwhereoldersatellites,thathaveslowlydecayedfromtheiroriginalorbits,aremovedtoanorbitforwhichtheycanfunction.AsimilarmethodtoOLEVcalledFREND-SUMO[ 3 ]isindevelopmentbyDARPA.AlthoughDARPAisprimarilyamilitaryorganizationitisalsoattemptingtodemonstratespacetug.Fromtheliterature,itisnotknownthedegreeofautonomythesemissionsachieve.Itis,nevertheless,knownthatteleoperationfor 13

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guidanceandcontrol(G&C)isdifcultduetothelatencyincommunicationbetweenabasestationandthesatelliteandcostlyduetodirecthumaninvolvement.Giventhesedifculties,itisexpectedthatthehigherthedegreeofautonomyforthesemissionsthebetterasfarasensuringsuccessinthepresenceofdisturbances,noise,adynamicallyevolvingenvironment,anddecreasingmonetarycosts. ProposedmilitarymissionsincludeDART[ 2 ],XSS10andXSS11[ 4 ],FREND-SUMO,OrbitalExpress[ 5 ],ETS-VII[ 6 ],andANGELS[ 7 ].Thesevariousmissionshavemetdifferentlevelsofsuccess.DARTcouldnotdemonstraterendezvousduetoasystemfailurewhichcausedthechaservehicletocollidedwiththetargetvehicle.OrbitalExpresswassuccessfulinachievingrendezvousalthoughithadsomenotablesystemfailures.ETS-VIIdemonstratedsuccessinitsmission,however,itisnotknownthelevelofautonomyachievedanditsprimaryobjectiveswereV-barandR-barapproachesanddocking.Inaddition,itcompletedcooperativecontrolofaroboticarmaswellasdemonstrationofsatelliteattitudecontrol,visualinspection,equipmentexchange,andrefueling.Althoughthesedifferentmissionshavemetacertainlevelofsuccesswithrespecttocollisionavoidance,itisclearfromallofthesemissionsthatautonomousrendezvousinclutteredenvironmentsandthuscollisionavoidanceisaveryimportantsafetycriticalissue.Further,providinglogisticstospaceassetswillcausethespaceinwhichthesesatellitesoperatetobecomemoreandmorecluttered.ItisthereforedesiredtodevelopG&Calgorithmswhichcansolvetrajectories,withinclutteredenvironments,thatavoidobstacles. Inspectionforspaceapplicationsconsistsofexaminingspacevehicleswhileinorbit.Oneexampleofthisistoexaminetheshuttletileswhileinspace.AnotherisinspectionoftheInternationalSpaceStation(ISS)becauseitissusceptibletocollisionswithspacedebris.Recently,NASA'sJohnsonSpaceCenter,hasundertakenaprojectforvisualinspectionoftheISScalledminiAutonomousExtravehicularRoboticCamera(miniAERCam)[ 8 ],[ 9 ].Althoughthisprojectisnotspaceready,simulationshavebeen 14

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developedthatvalidateitsapplicability.Inaddition,atestbedhasbeendevelopedtodemonstrate2-Dtranslationalcontroland1-Dattitudecontrol.OneoftheprimaryreasonsfordevelopingtheminiAERCamistofreeastronautsfromtheworkloadofcontrollingthesesatellitesforclose-proximitymissions.TheISSInspector[ 10 ]isajointprojectthroughDaimler-ChryslerandtheEuropeanSpaceAgencytoprovideinspectioncapabilitiesfortheISS.Itsmissionobjectivesforinspectionarestructuralvibrationobservation,leakdetection,thermalimaging,andothers.Thesetypesofmissionsare,however,safetycriticalwhichistheunderlyingconcernformanyagencies(i.e.,ESA,NASA,etc.).ArecentpapersuggestedthatthebestwaytodemonstrateautonomywithhighsuccessrateistoperformtheGEOmaintenanceforexistingandfuturespaceassets[ 11 ].Itistheauthorsbeliefthatopenlooporbitchangescanbedesignedaprioritoaccomplishthesetypesofmissions. Inconcludingremarksitshouldbenoted,withfewexceptions,thatthereislimitedpubliclyavailableliteratureonthemethods,feasibility,andperformanceanalysisforcurrentmissionsinvolvingspacecraftrendezvousinclutteredenvironments. 1.2MissionConceptAndDetails Thegeneralmissionstatementistoachieverendezvousofsatellitesforclose-proximityoperationsinaclutteredenvironment.Thekeyaspectsinconsideringatheoreticaldevelopmentforthismissionareprimarily: Safety: Itisimperativethatcollisionisavoidedandthatthereisasufcientmarginofsafetytothiseffect. Fuel: Inordertoreducemissioncostsitisdesirabletominimizetheamountoffuelused,v. Autonomy: Generally,spacesystemshavelimitedprocessingperformancebecausespacecertiedhardwaredoesnothavethesameperformancewhichiscommerciallyavailableforland,air,andseaapplications.Also,powerstorageandconsumptionishighlylimitedinadditiontosizeandweightconstraints.Inordertoreduce 15

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missioncostsbyreducinghuman-in-the-loopviateleoperationtheG&Cshouldbedoneautonomouslywithcurrenttechnologies. Withinthisscopeitisconsideredthatthetargetandobstacle'sparametersareknownapriori.Further,itisassumedthatthetargetisnotactivelymovingunderitsownpropulsion.Thecasewheretargetsarenon-cooperativewillnotbeconsideredinthisinvestigation,butleftforfuturework. Itisdesiredtoinvestigateasmallclasssizeofsatellites(i.e.,microorsmaller)inordertoreducemissioncosts.Inordertofulllmultiplemissionswithinasmallclasssatellite,andthereforeanindividuallylimitingroboticplatform,thecollaborationofteamsoftheserobotsareconsidered.Tothisend,itisenvisionedthataclusterorgroupingofmultiplesatelliteswithspeciccapabilitiesthatdependondesirablemissionplanswouldberequired.Ingeneral,onecanenvisionthebroadscopeofassembly,repair,andinspectionmissionsamongotherchoices.Althoughthesedifferenttypesofmissionswillhavetoincorporatemethodsfordocking,sensing(i.e.,imageacquisitionandprocessing,relativedistancemeasure,etc.),roboticmanipulatorcontrol,andothers,itisassumedthatthesecapabilitiescanbedevelopedforthisparticularclassofsatellite. 1.2.1TrajectoryPlanning Pastmissionshaveprimarilyusedhuman-in-the-loopforG&Corpredesignedopenlooporbitmaneuvers.Inordertoreducemissioncostsitisdesiredtoincreasetheautonomyforfuturesatellitemissions.AnessentialpartinincreasingautonomyconsistsoftrajectoryplanningthroughaG&Csystemthatenablessafemaneuversthroughaparticularenvironment.Thisenvisionedenvironmentwouldincludebothmaneuveringand/ornonmaneuveringobstacles.Typicaltypesofnonmaneuveringobstaclescouldbegeneralspacedebrisorothersatelliteswhichmayhappentobeincloseorbitwithrespecttothetargetlocation.Typesofmaneuveringobstaclesare,specically,othercooperativeservicingsatellitesperformingtheirowntaskinadecentralizedtypemanner.Allobstaclelocationsinthecongurationspacearenotnecessarilyassumed 16

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tobeknowninitiallynoristheirsubsequentmotion.Whatisassumedisthatsensorsonboardcandetectthemwithinacertainradiusandastheautonomoussatellitemovesitwoulddetectotherobstaclesthatmoveinsidethisradiusofthesatellite.Theautonomoussatelliteshouldthenhavethecapabilityofupdatingitstrajectory,inreal-time,basedonobstaclesenteringandleavingitsdetectableregionaswellasontheirmovement. Theprimaryinvestigationofthisresearchistodeterminewhetheroptimizedandreal-timeimplementableG&Calgorithmsforpathplanningtoachievedesirablemissionplansarepossible.Thespecictaskofthealgorithmistoensuresafetyinplanningtranslationalmotioninthepresenceofobstaclesandotherconstrainedregions,minimizefueluse,maintainacceptablelimitsofcomputationtime,andachieveboundaryconditionsatthenaltimeandtargetlocation.TheresearchpresentedinthisdissertationisananalysisbetweennewmethodsdevelopedandotherpreviouslyestablishedG&Cmethods. 1.2.2SystemHierarchy Indevelopingmissionplans,orwhatthespecicsofeachservicingsatelliteistoaccomplish,theremustbeasystemhierarchypassingthisinformationtoeachsatellite.ThishierarchyiscomposedofthreesegmentswhichencapsulatealloftheG&CfortranslationalmotionasillustratedinFigure 1-1 .Inthetoplevel,themissionplanislaidoutthroughdirecthumaninvolvementwherethemissionspecicsincludingwhichsatellitefromacollectionofsatellitesistoachieveaparticulartaskandwherethetargetislocated.Ingeneral,thenextleveltakesthesemissiondenitionsandplansatrajectorybasedonsystemdynamics,pathconstraints,andtheboundaryconditionsattheterminaltime.Trajectorygenerationforspacecraftrendezvousmissionsisbasedonsatisfyingsafetyconstraints,minimizingfueluse,andmaintainingthatthealgorithm'scomputationaltimeiswithinacceptablelimits.Thenextlevelisthetrackingcontrolofthetrajectorybasedonthesystemsactuatordynamics,andthedisturbancesandnoise 17

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Figure1-1. Controlarchitectureforsatelliterendezvous 18

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inthesystem.Thetrackingcontrolcaneithertrackwaypointsfromadiscretizedformofthetrajectoryorananalyticfunctionoftimedependingontheoutputfromthesecondtier. Themissionplannerreceivesfeedbackfromthetrajectoryplannerasthepathistraversedinthesenseofhowmanyobstaclesarepresentataparticularinstanceintime.Thiscanallowforredenitionofthegoalorpossiblyrequiringthesatellitetomaneuvertoasaferegionincaseofemergency.Thetrajectoryplannerreceivesfeedbackfromthetrackingcontrollerateverysamplinginstancesothatthetrajectoryplannercanupdatebasedoncurrentlymeasuredpositions.Inthiswaythedifferentlevelsofguidance,navigation,andcontrolinteractwitheachothertoachieveautonomousrendezvousaccordingtomissionplansdevelopedbyhumanoperators. Essentially,thisarchitecturecanbeseenasbreakingdowntheoverallproblemofrendezvousintomanageabletasksforeachpartofmissionplanning,guidance,andtrackingcontrol.Themissionplanningportionallowsforpracticalaspectsfromahumanpointofviewtobeincorporated.Theguidanceportionallowsforminimizingfueluseandobstacledenitionstobeincorporatedandprovidethetrackingcontrolwithasequenceofwaypointstofollowforwhichitcanrejectdisturbancesanduncertaintyinthemodel.TheinterconnectionofthesesystemscanbevisualizedinFigure 1-1 .Theinvestigationofthisresearchfocusesontheguidanceportionforminimizingfueluseandobstacleavoidance. 1.3LiteratureReview Therearenumerousmethodstosolvethepathplanningprobleminliterature.Theadvantagesanddisadvantagesofexistingapproacheswillbedescribedastohowtheyapplytotherequirementsofthetrajectoryplanningproblemathand.Itisdesiredtofocusthisreviewonmethodsthatsolvethetrajectoryplanningprobleminthepresenceofpathconstraintsorinotherwords,obstacles.Itisnottheintentoftheauthortogivea 19

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thoroughsurveyofthesemethodsbuttohighlighttheimportantaspectsofeachsoastoprovidereasoningfortheG&Ctechniquesdeveloped. AnexcellentmonographofmotionplanningtechniquesisprovidedbyLatombe[ 12 ]inwhichhediscusses,usingexplicitmathematicaldenitions,theconceptsandcomplexitiesofthemotionplanningproblem.AbookonroboticsrecentlypublishedbySpong[ 13 ]coversmanytechniquesintheeldsofguidance,controls,kinematics,anddynamics.AmoredetailedbookofrecentworkonmotionplanningisprovidedbyLaValle[ 14 ].TherearemanygreatsurveysofmotionplanninggivenbyHwang[ 15 ],Rao[ 16 ],Schwartz[ 17 ],andLatombe[ 18 ].TherstapproachtakenintheArticialIntelligencecommunityisdynamicprogrammingasdescribedbyBellman[ 19 ]inwhichagraphsearchisusedtosolvethemotionplanningproblem.Graphsearchmethodsincludecelldecomposition,Dijkstra'salgorithm,depth-rst,A*searchandmanyothers.Inthesemethodstheentirecongurationspaceissearchedinordertondafeasiblesolutionbasedonacostfunction.Oftentimesitisthennecessarytosmooththepathinordertooptimizewithrespecttoobstacleclearancedistanceorothermeasurablecostsinordertoincreasethedelityofthesolution.Thedifcultyofthisapproachisthatitrequirescomputationoftheentirecongurationspacetobesearchedwhichuseslargeamountsmemoryandthereforethealgorithmsrequirelargecomputationtimes. Skeletons,whichareoftentermedroadmaporhighwayapproaches,connect1-Dshortestlinesbetweenedgesofobstacleswithotherobstacles,thetarget,androbotlocation.Ageneralsearchingmethodisthenusedtondapathwithdesirablecharacteristics.Thismethodrequiresasmallamountofmemoryandiftheobstaclesaremodeledaccuratelythenthismethodcouldbetractableforcurrentspacemissions.Infact,aversionofthismethodisbeingproposedfortheAERcamproject[ 20 ]. MethodshavebeendevelopedfromMcInnes[ 32 ]-[ 35 ]andTatsch[ 36 ]-[ 38 ]inwhichapotentialfunctionisdevelopedtosteertowardsadesiredlocationandanothertosteerawayfromcertainregions.IntheapproachbyMcInnes[ 32 ]theadditionofthese 20

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twopotential'sgradientismaintainedtobenegativebyusingthesecondmethodofLyapunov.IntheapproachbyTatsch[ 36 ]anovelweightingfunctionisdevelopedtoefcientlyavoidobstaclesusingpotentialfunctions.IntheapproachbyMcInnes[ 32 ]thetimeofreandthemagnitudeanddirectionoftheimpulseischosenautomaticallybythecontrollawinwhichlocaloptimaltrajectoriesresult.Thereareknownissueswiththismethod,includinganeedforanapproachtochoosetheweightingforobstaclepotentialstoensurethatobstaclescanbeavoided.Also,thismethodmightrequirethatthemagnitudeoftheimpulsiveinputbelocatedoutsidepracticallimits.Forinstance,thevelocityoftheobstaclerelativetotheESR'svelocitymayrequiretheimpulse'smagnitudetobelarge.Thisproblemisfurtherexacerbatedbythefactthatitisassumedobstaclesarestaticbecausesensormeasurementsarefedintothecontrollawatahighfrequency.Timedelaysandlossofsignalfromsensorsarealsoproblematicwhenassumingthatobstaclesarestatic.Thereisonlyalimitedabilitytodealwithdisturbances;theactualsystemmaydifferfromthemodeledsysteminwhichitisthendifculttochoosegainstomeetcertainperformancemeasures.IntheapproachbyTatsch[ 36 ],obstaclesareavoidedefcientlybychoosingtimeofreusingsmallintervals.Thetotalfueluse,however,canbelarge.Inthismethod,thereisanexactformdescribingtheboundariesforobstaclesaswellasconstraintsonthetimeofretoensurethatavoidancecanbeachievedgivenanyvelocitiesoftheobstacleandchasersatelliteandwithinlimitsontheactuationofthechasersatellite.Inbothofthesemethods,thefueluserequiredisnotdirectlyminimizedandsominimumfueltrajectoriesarenotfound.Also,inthesemethodsthepotentialfunctionsmaycontainlocalminimumforwhichtheguidanceproceduremayterminateprematurely.However,theguidanceprocedureisanalyticinformandthereforereal-timeimplementable. TherehasbeennumerousmethodsdevelopedusinganoptimalcontrolframeworkasdevelopedbyChen[ 21 ],Herman[ 22 ],andRichards[ 23 ].Onedifcultywiththesemethodsisthatthecontrolformulationisopenloop.Motivatedbymethodsusedfrom 21

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industryinchemicalengineeringknownasnonlinearModelPredictiveControl(MPC),Richards[ 42 ]-[ 43 ]hasdevelopedannonlinearMPCtrajectoryplanningmethodformanydifferenttypesofautonomousvehicleswhichincorporatesafeedbackmechanism.Asaconsequenceofthisfeedbackmechanism,suboptimalperformanceisinherentaswellasfeasibilityissuesexist.ItshouldbenotedthatthenonlinearMPCmethoddevelopedbyRichards[ 43 ]hasnotbeenproventobereal-timeimplementableforcurrentspacecertiedhardware. TheMPCisatechniquethatusestherepeatedsolutionofanoptimalcontrolproblemoverahorizonoftime,calledthepredictionhorizon,whichissmallerthanthedifferencebetweenthenalandinitialtime.Asmallerhorizonoftime,calledthecontrolhorizon,ofthesolutionisimplementedandtheproblemissolvedagainbeginningattheendofthepreviouscontrolhorizon.ItcanbeviewedthatMPCcombinesbothfeedforwardandfeedbackcontrol.AbenetofnonlinearMPCisthatsolutionprocessisbasedonoptimalcontroltheory.However,thesolutionisonlyapproximate,optimalitycannotbeguaranteedforon-lineimplementation,andthereisasampling-timerestriction.Furthermore,optimalitycannotbeguaranteedforon-lineimplementationbecausethecontrolproblemistypicallydiscretizedforadesirednumberofpointsthatresultsinanonconvexoptimizationproblemforwhichtheiterativeproceduremightterminateatalocalminimum.Asaconsequence,poorperformanceandeveninstabilitymayresultbyimplementingthecontrolsequencereturnedfromtheon-lineoptimizer.Thecomputationaltimetoyieldasolutiontotheoptimizationproblemtypicallyrequiresalongsamplingtimeduetothecomplexityresultingfromalargesetofconstraints.ForthesereasonsitisbelievedthatnonlinearMPCisnotreal-timeimplementableforsatellitesystems. TherehasbeensomeworkdoneusingMPCoptimizationswhereparametricsolutionshavebeenfoundwhichdramaticallydecreasecomputationtimeasshownbyBemporad[ 39 ],andSakizlis[ 40 ].Inaddition,aclosed-formapproachbyChen[ 41 ] 22

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usingaTaylor-seriesexpansionoftheoutputisshowntobestableforappropriatetimesofthepredictionhorizonandtherelativedegreeoftheapproximation.Also,thechoiceofcostfunctionsintheapproachbyChen[ 41 ]islimitedtothesquareoftheerrorandcontroleffortcannotbedirectlyminimized.Finally,thereisnoabilitytohandleconstrainedproblemsusingtheanalyticmethodbyChen[ 41 ]. 1.4ResearchScope Theanalyticsolutiontothelinearizedequationsofmotion,knownastheCWHequations(Clohessy-Whiltshire-Hill),withtheinputforcemodeledasanimpulsecanbeusedtodriveasatellitefromonerelativepositiontoanother.AssumptionsfortheCWHequationsarethatbothbodiesareconsideredtobepointmasses,theorbitsarecircularandtheyarerelativelyclosetogether.ThismaneuveriscalledaCWmaneuver.TwoG&CtechniquesaredevelopedusingtheCWmaneuverbecauseitisdeemednecessarytodevelopaplanningschemeinwhichnonlocalbaseddecisionsareused.Thereasonforthisistoavoidthepossibilityofearlyterminationintheguidanceprocedure,knownaslocalminimum,whichisaninfeasiblesolutionandtominimizefueluse.UsingaCWmaneuvercanbeviewedasanonlocalbaseddecisionsincethetrajectoryisknownanalyticallyasafunctionoftimeassuminganinitialimpulsivethrust.ThetwoG&Ctechniquesdevelopeddifferbasedonwhatcourseofactiontotakeinthepresenceofobstacles.Inotherwords,adecisionattheinitialtimecanbemadetoalterthetrajectorybetweentheinitialandnalpositionsbasedonintermediatepathconstraints.ItisbelievedthatiftheCWmaneuversareusedforguidance,betterdecisionscanbemadeintermsofavoidingobstaclesandreducingfuelusage.Inoneapproach,apotentialfunctionmethodisusedwithaswitchingcontroltoswitchbetweenrepulsive,orobstacleavoidancemaneuvers,andattractivemaneuvers.Forattractivemaneuvers,theCWmaneuverisusedinplaceofderivingthecontrolfromapotentialeld.ItisshownbyusingaswitchingcontrolandCWmaneuverinthismannertheamountoffuelrequiredisreducedascomparedtoAPFGmethod.Inthesecond 23

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approach,theCWmaneuverissolelyutilizedtoachieverendezvousandobstacleavoidancemaneuversbyusingthetimeofight(TOF)andnalrelativepositioninordertosteerthetrajectory.Theoptimalvaluesaredeterminedbyuseofanumericaloptimization.Theproblemistospecicallydetermineanoptimaloneimpulsemaneuverwhichminimizesacostfunctionsubjecttoconstraintsoverasmallhorizonoftime.ThisoptimizationproblemincludesconstraintsontheTOF,nalrelativeposition,minimumandmaximummagnitudeofthethrust,andobstaclesoversmallhorizonsoftime.ItisshownthatconstraintsontheTOFandintervallengthsforobstacledetectionsetsmallenoughcanbeimposedthatguaranteeadesiredminimumcanbefoundbyanalyzingthenormofthecontrolasafunctionoftheTOF.Thehorizonsareconsideredtobesmallcomparedtotheentirelengthoftimeforthemaneuver.Optimizingatrajectoryoversmallhorizonsoftimeallowsforalesscomplexoptimizationproblemwhichinturnreducestheamountofcomputationaltimetocomputeasinglemaneuver.Thesinglemaneuveroptimizationproblemislesscomplexbyconsideringsmallerhorizonsoftimebecauselessobstacleconstraintdenitionsneedbeincluded.ItisshownthatusingthismethodreducesfuelrequirementsascomparedtoAPFGandcomputationtimescanbetintoareal-timeframeworkforcurrentground-basedhardware. Anothermethodisdevelopedwhichusestheanalyticsolutiontothedynamicsoftherestrictedtwo-bodyproblem.Thesolutiontotherestrictedtwo-bodyproblemdescribeseithercircular,elliptical,andhyperbolicorbits.Thismethodutilizesanumericaloptimizationmethodtodetermineimpulsiveinputsthatminimizeparticularcostfunctions,avoidobstacles,maintainlimitsonactuation,andotherconstraintstobediscussedinChapter4inacomputationaltimewhichisfastenoughforreal-timeimplementationongroundbasehardware.AprocessisdescribedinwhichonecanensurethatadesiredminimumcanbefoundbyappropriatelyboundingtheTOFandsettingtheintervallengthforobstacledetectiontobesmall.Itisshownthatusingthis 24

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methodreducesfuelrequirementsascomparedtoAPFG.ItshouldbenotedthatthismethodissimilartotheGASmethod. 1.5DissertationOutline Thedissertationisorganizedinthefollowingmanner.Chapter1describesanintroductiontotheproblemandmissionswhichhavebeenproposedtodemonstratecurrentcapabilitiesfortranslationalmotionofsatellitesystems.Itdescribesthescopeoftheresearchaswellasageneralmissionplanforasmallclasssizeofsatellites.Chapter2discussestheknownmethodsformodelingsystemdynamicsforspacevehiclesanditsapplicabilityforclose-proximityoperations.InChapter3,amotionplanningtechniqueknownintheeldofroboticsisdiscussed.ThismethodisknownasArticialPotentialFunctionGuidance(APFG)whichisthebasecontrollerforwhichnewmethodswillbecomparedagainstintermsoftheamountofcomputationtimerequired,fueluseandobstacleavoidance.ThischapteralsoincludesanewpotentialfunctionmethodwhichusesaswitchingcontrolandtheCWmaneuvertermedArticialPotentialFunctionGuidanceusingSwitchingControl(APFGSC).InChapter4,theGuidanceusingAnalyticSolution(GAS)isdevelopedwhereaminimizationisusedtosolveforaTOFandnalrelativeposition,r(T),thatensuresoptimalityaswellasobstacleavoidance.AnothermethodtermedGuidanceusingAnalyticSolutiontotheTwoBodyproblem(GASTB)methodisusedforthemoregeneralcasewheresatellitesareinellipticorbits.Inthenalchaptertheoutcomesofthesecontroltechniquesisdiscussedintermsoftheiradvantagesanddisadvantages. 25

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CHAPTER2SYSTEMDYNAMICS Indescribingthedynamics,thetargetsatellitewillbetermedastheResidentSpaceObject(RSO)andthechasersatelliteastheExpertSpaceRobot(ESR).WhenconsideringtherelativemotionbetweenthecenterofmassoftwobodiesincloserangewhileorbitingtheEarth,itiscustomarytousetheCWHequations.Theassumptionfortheseequationsarethatthetargetistravelinginacircularorbitandthetwobodiesarerelativelyclosetogether.Foranin-depthanalysisfortheapplicabilityoftheCWHequations,forrendezvousmissionsinpathconstrainedenvironments,seeStern[ 24 ],andSoileau[ 25 ].Usingtheseassumptions,thegoverningequationfortheESR'srelativemotioncanbedenedbyEq.( 2 )accordingtoVallado[ 26 ]andFehse[ 29 ]intheRSWcoordinatesystemdenedinFig. 2 wherenisthemeanmotion,andfisaforceappliedtotheESRwithRSWcomponents(fxRSO,fyRSO,fzRSO). xrel)]TJ /F5 11.955 Tf 11.95 0 Td[(2n_yrel)]TJ /F5 11.955 Tf 11.95 0 Td[(3n2xrel=fxRSOyrel+2n_xrel=fyRSOzrel+n2zrel=fzRSO(2)Equation( 2 )canberewritteninstatespaceformasshowninEq.( 2 ) _xss=Axss+Bu(2) where xss(0)=xss0,u(t)=(t)w(2) andxss0isthecolumnvectorcontainingtheinitialpositionsxrel0,yrel0,zrel0andinitialvelocities_xrel0,_yrel0,_zrel0.AsshowninEq.( 2 ),Eq.( 2 )canberewritteninstatespaceformsuchthatxss=[xrelyrelzrel_xrel_yrel_zrel]Twithamodicationthatthe3by1inputvectoru(t)isanidealimpulsethrust,Aisa6by6statematrix,andBisthe6by3inputmatrix.FromEq.( 2 )theexpression(t)istheidealimpulseandwisthedirectionandmagnitudeofthethrust.ThesolutiontoEq.( 2 )isshowninEq.( 2 )wherexss 26

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Figure2-1. RSWframedenedsuchthatxrelistheR-bardirectionandyrelistheV-bardirection.ThecomponentsIx,Iy,Izdescribeaninertialcoordinatesystem. representsthestates(i.e.,relativepositionandvelocityandxss0aretheinitialstatesoftheESR). xss(t)=exp(At)[xss0+Bw]=(t)[xss0+Bw](2) FromEq.( 2 ),itisimportanttonotethatforanimpulseresponse,theinitialconditionsxss0areaugmentedbytheproductBw.ThisisconvenientsincewhenperformingaCWmaneuver,thispropertybecanusedtoderivetherequiredimpulseforaCWmaneuver. 2.1CWManeuver ACWmaneuvercanbedenedasthesingleimpulsethatwouldberequiredfortheESRtoreachaprescribedrelativeposition.TheimpulserequiredforaCWmaneuvercanbedeterminedbyrstrewritingEq.( 2 ),thesolutionoftheimpulseresponsetotheCWequations,asshowninEq.( 2 ) 8><>:rrel_rrel9>=>;=2641,1(t)1,2(t)2,1(t)2,2(t)3752648><>:rrel0_rrel09>=>;+Bw375(2) 27

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wherexss=rTrel_rTrelT.The6by3matrixB=[0I]Tcontainsthe3by3nullandidentitymatrices,0andI.NowdenotetheimpulsevectorvastheidentitymatrixwithinthematrixBmultipliedbyw,v=Iw.Theexpression(t)isshowninitsentiretyinEq.( 2 ). (t)=26666666666666644)]TJ /F5 11.955 Tf 11.95 0 Td[(3cos(nt)00sin(nt) n2 n(1)]TJ /F5 11.955 Tf 11.95 0 Td[(cos(nt))06(sin(nt))]TJ /F3 11.955 Tf 11.96 0 Td[(nt)10)]TJ /F8 7.97 Tf 10.56 4.71 Td[(2 n(1)]TJ /F5 11.955 Tf 11.95 0 Td[(cos(nt))1 n(4sin(nt))]TJ /F5 11.955 Tf 11.95 0 Td[(3nt)000cos(nt)001 nsin(nt)3nsin(nt)00cos(nt)2sin(nt)0)]TJ /F5 11.955 Tf 9.3 0 Td[(6n(1)]TJ /F5 11.955 Tf 11.95 0 Td[(cos(nt))00)]TJ /F5 11.955 Tf 9.3 0 Td[(2sin(nt)4cos(nt))]TJ /F5 11.955 Tf 11.96 0 Td[(3000)]TJ /F3 11.955 Tf 9.3 0 Td[(nsin(nt)00cos(nt)3777777777777775(2) FromtherstrowofEq.( 2 )onecandeterminethenalrelativepositionasstrictlyafunctionoftheTOFandtheimpulsevectorvasshowninEq.( 2 ).Thus,ifonechoosesaTOFandconsiderstheinitialvelocitytobetheinitialvelocitybeforetheimpulseplustheimpulsevector,thenonecansolvefortheimpulsevectorforthedesiredrelativepositionasshowninEq.( 2 ). rrel(T)=1,1(T)rrel0+1,2(T))]TJ /F5 11.955 Tf 6.49 -9.68 Td[(_r)]TJ /F4 7.97 Tf -1.38 -8.27 Td[(rel+v(2) v=)]TJ /F8 7.97 Tf 6.59 0 Td[(11,2(T)(rrel(T))]TJ /F5 11.955 Tf 11.95 0 Td[(1,1(T)rrel0))]TJ /F5 11.955 Tf 12.96 0 Td[(_r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(rel(2) ForEq.( 2 )and( 2 ),TistheTOFofthemaneuver,rrel(T)istheprescribednalrelativeposition,andrrel0,_r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(relarethepositionandvelocitybeforetheimpulse. 2.2RestrictedTwo-BodyProblem TheEarth-CenteredInertial(ECI)coordinatesystemhasitsoriginattheearth'scenterbutisnotxedthere.Thiscoordinatesystemisusedtomeasuretherelativemotionoftheearthandanorbitingsatellite,inthiscasetheESRorRSO.ThetwobodiesMandmaswellastherelativepositionvectorrareshowninFigure 2-2 whereMeisthemassoftheearthandmeisthemassofthesatellite. 28

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Figure2-2. ECIcoordinatesystem TherelativemotionisdescribedbyBate[ 28 ]andisalsoshowninEq.( 2 ) r=)]TJ /F3 11.955 Tf 9.3 0 Td[(G(Me+me) krk3r(2) whereGistheuniversalgravitationalconstant.ItisassumedthatthemassofthesatelliteisnegligibleincomparisontothemassoftheearthandthereforeEq.( 2 )canberewrittentoformthatisknownastherestrictedtwobodyproblemasshowninEq.( 2 ) r+ krk3r=0(2) where=GMe.ThesolutionofEq.( 2 )isshowninBattin[ 27 ]andisapproximatedbypowerseriessolutionasshowninEq.( 2 ). r(t)=r0+(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)dr dt0+(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0)2 2!d2r dt20+(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0)3 3!d3r dt30+(2) ThepartialderivativesinEq.( 2 )involvehigherderivativesoftheterm=krk3whichcanbemoreefcientlycalculatedbysubstitutionintodifferentexpressions.Wedene = krk3,(2) 29

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d dt=)]TJ /F5 11.955 Tf 9.3 0 Td[(3 krk4d dt(krk)=)]TJ /F5 11.955 Tf 9.29 0 Td[(31 krkd dt(krk),(2) =1 krkd dt(krk)=rTv krk2,(2) d dt=1 krk2vTv+rTdv dt)]TJ /F5 11.955 Tf 11.96 0 Td[(2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(rTv1 krk3d dt(krk)=v2 krk2)]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F5 11.955 Tf 11.96 0 Td[(22,(2) =v2 krk2=vTv krk2,(2) and d dt=2 krk2vTdv dt)]TJ /F5 11.955 Tf 11.96 0 Td[(2v21 krk3d dt(krk)=)]TJ /F5 11.955 Tf 9.3 0 Td[(2(+ ).(2) TobeginthesolutionbypowerseriesonestartswithEq.( 2 )andsubstituteusingEqs.( 2 )-( 2 ). dr dt=v,dv dt=)]TJ /F6 11.955 Tf 9.3 0 Td[(r(2) Thismethodiscontinueduntilthedesiredorderoftheapproximationisreached.Thesolutionofr(t)uptothe11thtermisshowninitsentiretyinAppendixB. AnotherconvenientcoordinatesystemsthatcanbeusedtodescribetherelativemotionoftwobodiesisthePerifocalCoordinateSystem(PCS)asreferencedbyBate[ 28 ].Inthiscoordinatesystemitisassumedthatthex!)]TJ /F3 11.955 Tf 10.41 0 Td[(y!directionslieinthesatellite'sorbitwherex!pointstowardperiapsis;they!axisrotated 2radiansinthedirectionoftheorbitalmotion;thez!axiscompletestheright-handedcoordinatesystem.Theunitvectorsforthedirectionsx!,y!,z!areP,Q,W. 2.3ClassicalOrbitalElements Thelistofclassicalorbitalelements,asreferencedbyBate[ 28 ],aredenedas 1. a,thesemi-majoraxis,isaconstantwhichdenesthesizeofaconicorbit. 2. kek,theeccentricity,isaconstantdeningtheshapeofaconicorbit. 3. i,theinclination,istheanglebetweenthez-axisoftheECIcoordinatesystemandtheangularmomentumvector. 30

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4. ,longitudeoftheascendingnode,istheanglebetweenthex-axisoftheECIcoordinatesystemandtheascendingnodewhichiswhentherotatedx-axis,aboutthez-axis,crossestheorbitalplane. 5. !,argumentoftheperiapsis,theangle,intheplaneofthesatellite'sorbit,betweentheascendingnodeandthelocationofperiapsis,directedalongthesatellite'smotion. 6. Tpp,timeofperiapsispassage,thetimewhenthesatellitewasatperiapsis. Theangularmomentum,forcalculationsusedinthisdocument,isthevectorcrossproductofthepositionofthesatelliteanditsvelocitymeasuredintheECIcoordinatesystem. 31

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CHAPTER3ARTIFICIALPOTENTIALFUNCTIONGUIDANCE Recognizingthemanymethodsdevelopedtosolvethepathplanningproblemandthosespecicallydevelopedfororbitalrendezvous,ArticialPotentialFunctionGuidance(APFG)ischosenasthebaselineforcomparison.ThereasonforthisselectionisthatAPFGiscomputationallyfeasibleanditsmathematicaldevelopmentleadsdirectlytoanexpressionforeasilydelineatingstability.ThissatisessafetyandautonomywhicharetwoofthethreekeyaspectsidentiedearlierforasatisfactoryG&Calgorithm. 3.1TheoreticalDevelopment InAPFGmethodsbyMcInnes[ 32 ]andTatsch[ 36 ],stabilityandconvergencewithrespecttoadesiredgoallocationpdandobstacleavoidanceisachievedbyapotentialfunctionwithtwoterms.Thispotentialfunctioncontainsabowlshapedtermforattractionandabumpshapedtermforrepulsion.Morespecically,stabilityandconvergencecanbestatedasfollows: Denition1(Stable). Theequilibriumpoint=eisstableiffor>09=()>0suchthat k(0))]TJ /F6 11.955 Tf 11.95 0 Td[(ek<)k(t))]TJ /F6 11.955 Tf 11.95 0 Td[(ek<,8tt0.(3) Denition2(Convergent). Theequilibriumpoint=eisconvergentif91: k(0))]TJ /F6 11.955 Tf 11.96 0 Td[(ek<1)limt!1(t)=e(3) Twoparticularformsofstabilityforasystemwithcontrol,u,asgenerallydescribedforanautonomoussysteminEq.( 3 )andshownbyKhalil[ 30 ]areseeninTheorem1and2. _p=q(p,v).(3) 32

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Theorem3.1. Letp=0beanequilibriumpointforEq.( 3 )andD0inD)-221(f0g(3) _V(p)0inD(3) kpk!1)V(p)!1(3) Then,p=0isstable.If _V(p)<0inD)-222(f0g(3) thenp=0isGloballyAsymptoticallyStable. Furthermore,thesystemsresponsecanbedenedinaglobalsensebyDenition3. Denition3(GlobalUniformUltimateBound). Forallan2(0,1)9someconstantbandsometimeT=T(an,b)0independentoft00 kp(t0)kan)kp(t)kb,8tt0+T.(3) Todenoteobstacles,Denition4providesafunctionalform. Denition4(Obstacleavoidance). fckk(p)0g,kk=1:N(3) forthenumberofobstaclesN.Inthiscaseckk(p)isatypeofdistancefunctiondenedforeachobstaclekk. Next,theformofthepotentialfunctioncanbeshowninEq.( 3 ) =attractive+repulsive.(3) Typically,inusingtheabovetheoremsanddenitions,achangeofvariableisintroducedsuchthatthedesiredpositiontotrackbecomespd=0.ItcanbenotedfromEq.( 2 ) 33

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thatthisisindeedthecaseduetothefactthattherelativemotionequationsareusedtodescribethemissiontasks.Also,forallofChapter3theobstacleswillbeconsideredtobestaticrelativetothetarget. ThefollowinganalysisforAPFGwillmainlybedrawnfromthecontributionbyMcInnes[ 32 ].Tobegin,thepotentialfunctioncanbewrittenasapositivedenitefunctionandthereforehastheformasaLyapunovcandidatefunctionasshowninEq.( 3 ) =rTrelPrrel+Xjjexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[((rrel)]TJ /F5 11.955 Tf 11.58 0 Td[(~rj)TMj(rrel)]TJ /F5 11.955 Tf 11.57 0 Td[(~rj) j(3) wheretheobstaclepositionsrelativetothetargetpositionaredenotedaserj=[exj,eyj,ezj]T.ItcanbeseeninEq.( 3 )that(rrel)>08rrel2D)-307(f0gbutthat(0)6=0.Since(0)isequaltoapositiveconstantthestabilitycanbedeterminedwithrespecttoboundsonrrel(t).Next,thecontrollawisshowninEq.( 3 ) f=8><>:0if_<0(t)vif_09>=>;(3) where _rrel(t)=_r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(rel+v=)]TJ /F3 11.955 Tf 9.3 0 Td[(krrrel,(3)nisthenumberofimpulsivethrusterrings,andrrrelisthepartialderivativeofwithrespecttorrel.InEq.( 3 )(t)isanidealimpulse,visdeterminedbyEq.( 3 ),Pisaknownpositivedenitematrix,_r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(relisthevelocitythemomentbeforeimpulse,j,andjarepositiveconstantswhichshapetheobstaclespotentialfunction,andkisaconstantgain.ItcanbeshownthatataninstantthevelocityisasshowninEq.( 3 ) _rrel(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(krrrel(3) andsothetimederivativeofisasshowninEq.( 3 ). _(rrel(t))=)]TJ /F3 11.955 Tf 9.3 0 Td[(k(rrrel)T(rrrel)(3) 34

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Obstaclesaretreatedasstationarybycalculatingthe_condition,asshowninEq.( 3 ),insmallenoughintervalstoensureobstacleavoidance.TheseintervalscanbechosenasafunctionofhowclosetheESRisallowedtocometoobstaclesaswellashowfasttheobstaclesaremovingrelativetotheESR.DeningPtobeapositivedenitediagonalmatrixasshowninEq.( 3 ) P=266664p1000p2000p3377775(3) andMtobeapositivedenitediagonalmatrixasshowninEq.( 3 ) M=266664m1000m2000m3377775(3) thenrcanbedenedasshowninEq.( 3 ) r=2666642xrelp1)]TJ /F5 11.955 Tf 11.95 0 Td[(212yrelp2)]TJ /F5 11.955 Tf 11.95 0 Td[(222zrelp3)]TJ /F5 11.955 Tf 11.95 0 Td[(23377775(3) where 1=Pjm1 j(xrel)]TJ /F11 11.955 Tf 11.96 .5 Td[(exj)jexp)]TJ /F8 7.97 Tf 10.49 6.11 Td[((rrel)]TJ /F8 7.97 Tf 6.3 0 Td[(~rj)TMj(rrel)]TJ /F8 7.97 Tf 6.29 0 Td[(~rj) j2=Pjm2 j(yrel)]TJ /F11 11.955 Tf 12.06 .5 Td[(eyj)jexp)]TJ /F8 7.97 Tf 10.49 6.11 Td[((rrel)]TJ /F8 7.97 Tf 6.3 0 Td[(~rj)TMj(rrel)]TJ /F8 7.97 Tf 6.29 0 Td[(~rj) j3=Pjm3 j(zrel)]TJ /F11 11.955 Tf 11.92 .5 Td[(ezj)jexp)]TJ /F8 7.97 Tf 10.49 6.11 Td[((rrel)]TJ /F8 7.97 Tf 6.3 0 Td[(~rj)TMj(rrel)]TJ /F8 7.97 Tf 6.29 0 Td[(~rj) j.(3) Therefore,itcanbeshownthatthetimederivativeofisasshowninEq.( 3 ). _=)]TJ /F5 11.955 Tf 9.3 0 Td[(4k(xrelp1)]TJ /F6 11.955 Tf 11.95 0 Td[(1)2+(yrelp2)]TJ /F6 11.955 Tf 11.96 0 Td[(2)2+(zrelp3)]TJ /F6 11.955 Tf 11.96 0 Td[(3)2(3) FromEq.( 3 )itcanbeseenthat_isnegativesemideniteandnotequaltozerowhenrrel=0.Tondtheequilibriumpointsetrrrel=[000]Tandsolvefortherelativedistancesxrel,yrel,andzrel.Theequilibriumpositionsshallbedenotedasre=[xeyeze]T 35

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andthereforerrel(t)willhaveabound,b,asshowninEq.( 3 ) b=re)]TJ /F5 11.955 Tf 11.95 0 Td[([000]T=re.(3) OnecanthenconcludefromDenition3thatbecausethisbounddoesnotdependoninitialconditionsoraninitialtimeandsorrel(t)isGloballyUniformlyUltimatelyBounded(GUUB). Itshouldbenotedthatsincethecontrolisatthelevelofvelocity,theaboveanalysiscannotbeusedtoconcludeanythingaboutthebehaviorofthevelocityast!1.However,takingtheEuclideannormofEq.( 3 )onecanseethat k_rrelk=2k(xrelp1)]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 21)2+(yrelp2)]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 22)2+(zrelp3)]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 23)21 2(3) whichgenerallydecreasesasthesystemgoestoitsequilibriumposition. Ageneralassumptionisusuallymadethatnoobstaclesareconsiderednearthetargetlocation(i.e.,krrelk<10m)orthatadifferentG&CalgorithmwouldbeusedforthisphaseofrendezvoussuchasthoseshownbyHablani[ 47 ],andMcInnes[ 32 ].Therefore,ifadifferentG&Calgorithmwereusedinaproximitywhichisclosetothetarget,abetterconvergenceresultcanbeobtained. ItcanbeseenfromDenition4thatthesystemwithcontrollawasshowninEq.( 3 )isGUUB.However,therepulsivepotentialfunctionswhichinturndescribetheactiontotakeinthepresenceofobstaclescanformlocalminimatowhichthesystemmightconverge.Itisthereforenotpossibletosayexactlytowhichequilibriumposition,8rrel(t0)andpossibleobstaclecongurations,thesystemwillconverge.Thewaylocalminimacanpracticallyoccurforthisparticulartypeofcontrolisthattheforcecommanded,whenthesystemisnearanobstacle,actstomovethesystemawayandthenasmallamountoftimelaterthecontrolactstomovethesystembacktowardthegoal.Ifitisthecasethatthesystemmovestowardtheobstacle(s)againalocalminimacanexistbywhichthismotionrepeatsitselfandthesystembecomestrapped.Itshould 36

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alsobenotedthatthisproblemisinherentinthetypeofG&CusedbyAPFG.APFGusesonlytheinformationoftherelativedistancetothetargetlocationandrelativedistancefromobstaclesandsotheG&Ciscompletelylocalinnature. ItshouldbenotedthatinEq.( 3 )ananalyticandthecomputationtimetocomputethecontrolissmallenoughtoguaranteereal-timeimplementation.TheG&Conlyrequiresrelativedistancemeasurementsandthereforethissignalismeasurable.Theimpulsivecontrolisimplementablealthoughisrealisticallyonlyanapproximationofthethrusterringeventthatoccursforsatellitesystems.AlthoughlocalminimaexistinanypotentialfunctionthatcanbecreatedasreferenceinKoditschek[ 31 ]theyaregenerallyunstablesaddlepoints.Duetothefactthatlocalminimaaretypicallyunstablesaddlepointsandthateachsatelliteandobstacleareindifferentorbitsitisnotlikelythesystemwillremaintrappedforalltime.Itisthusexpectedthatforsimplecongurationsofobstaclesthisisarealisticandsafecontrolstrategy. 3.1.1Limitations Inordertoprovethatobstacleavoidanceisachievedthepotentialfunctionmustbedenedappropriately.ThisrequiresthatjandjinEq.( 3 )havevaluesthatensureisnonincreasingandthatthesystemdoesnotcrosstheboundariesofobstacles.Oneapproachtomeetthisrequirementistoforce,asshowninEq.( 3 ),alongtheentiresurfaceoftheobstacle((rsurface))tobethesameaswhenthesystemisatitsinitialcondition(i.e.,(rrel(0))).Inordertoimplementthisidea,onemustdenethepotentialfunctiontorepresentthegeometryoftheobstacle'ssurface.Ageneral,albeitconservativeapproachistorepresenttheobstacleasacirclewherethediameteristhemaximumdistancebetweenallpointsontheobstacle.Now,itisneededtoconsideratwhatpositionrelativetothetarget(i.e.,rsurface)onthecircletocalculatejandjasdescribedpreviouslyandshownbyTatsch[ 36 ](i.e.,atwhatpositiontoenforcetheconstraintmentionedabove).Thereasonforthisisthatifonechoosesanarbitrarypositionlocatedonthecirclerepresentingtheobstacleandset(rsurface)=(rrel(0)) 37

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andsomehowcalculatevaluesforjandj,thenifonechecksvaluesofEq.( 3 )atdifferentpositionsitmightbethat(rsurface)<(rrel(0)).Thisisclearlyaprobleminprovingobstacleavoidance. Ifitisassumedthatoneknowsthegeometryoftheobstacleandcancalculatethemaximumdistancebetweenallpointsontheobstacletowhichthediameterofourcircularobstacleregionisdenedthenonecanconstructtheminimumandmaximumdistancesbetweentargetandtheobstacleregion.Thiscanbedonebyutilizingthesimplegeometryoftheproblem.ThegeometryoftheproblemletsonedeterminethepositionoftheminimumandmaximumdistancesasshowninEq.( 3 )andEq.( 3 ). rmin=er)]TJ /F3 11.955 Tf 11.96 0 Td[(Robser kerk(3) rmax=er+Robser kerk(3) ItshouldbenotedthatRobsistheradiusoftheobstacleregion.ConsideringtheconstraintshowninEq.( 3 ) (rrel)=(rrel(0)),(3) whererrelisapositionontheobstacledenitionssurface.Onecansolveforjorjbyassigningonetoaconstantandsolvingfortheother. Inthefollowingexamplejisassignedtoaconstantandjissolvedusingtheprecedingmethodforeachofthefollowingpositionsrmin,rmax,rrel=[20]T,andrrel=[22]Taslocatedontheobstaclesurface.Theobstacleitselfislocatedat~r=[21]Twitharadiusof1.OnecanthenobtaintheplotasshowninFigure 3-1 where=10and=58.9,55.9,51,48whenviewedfromtoptobottominbothsubplots.Thelargestandsmallestvaluesofrepresentthevaluesobtainedfromusingrminandrmaxandtheothersarejustintermediatevalues.UponinspectionofFigure 3-1 onecannot 38

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ensurethat(rsurface)willsatisfy (0))]TJ /F6 11.955 Tf 11.95 0 Td[((rsurface)0(3) whenjandjischoseninthisfashion.Ingeneral,however,itispossibletochoosejandthenjinordertosatisfyEq.( 3 ),butaminimizationsubjecttothenonlinearconstraintoftheformgivenbyEq.( 3 )wouldbenecessary.UponsimulationitwasalsofoundthatitisnotpossibletoensureconvergenceoftheminimizationforanyparticularcostfunctionJ(,)andinitialguessofand. Figure3-1. Constraintforxedvs.xandy-positions IthasnowbeendemonstratedthatitisnotpossibletoensureobstacleavoidancethroughtheconstraintofEq.( 3 )withobstaclepotentialfunctionsdenedinEq.( 3 ).Anotherproblemwiththeformofrepulsivefunctionisthatitdoesnotexactlyconformtothesizeoftheobstacle.TheuseoftheGaussianfunctionforobstacle 39

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avoidanceisproblematicnotonlybecauseonecannotensureobstacleavoidancebutalsobecausethepotentialeldextendsoutsideoftheobstacleregion.Practically,thisbecomesproblematicwhenobstaclesarerelativelyclosetogethertheseeldswilloverlapandtheirpotentialswilladdtogetherasagainseenbyEq.( 3 ).Thiscancauseaproblemintermsofperformancebecausethisregionmaynowbeavoidedbecauseoftheincreasedrepulsivepotential.TherehavebeenproposedmethodsintheliteraturetodealwiththisproblemasshownbyRimon[ 44 ]andConn[ 45 ].Thismethod,however,musttakeintoaccountallobstaclesgeometryandpositioninordertocalculatethetotalpotentialeld.AnothermethodofobstacledenitionusingexteriorpenaltyfunctionswhichcanoffersmallcomputationtimesfordeningrepulsivepotentialsaswellasensuringobstacleavoidanceisgivenbyGalicki[ 46 ]. 3.1.2SpacecraftRendezvousUsingAPFGAlgorithm Inthissection,anexamplecaseispresentedfortheAPFGalgorithmperformingrendezvoustotheRSO.Inordertodemonstratethisalgorithm,theresultingtrajectoryisshownrstforthecasewherenoobstaclesarepresent.Theinitialpositionis1kmintheR-barand1kmintheV-bardirectionandtheinitialvelocityiszeroinbothdirections.Figure 3-2 showsthetrajectorybasedintheLVLHframewhereitcanbenotedthatsharpcornersexistwhichindicatetheimpulsivecontrolpushingthesysteminthedirectionofthenegativegradientofthepotentialfunction.ThetrajectoryveersawayfromtheRSOaftereachimpulsivethrustduelargelyinparttogravity.Figure 3-3 showsthatthecontrolEq.( 3 )maintains_0andthereforeisnonincreasing.Figure 3-4 showstheR-barandV-bartrajectoriesaswellastheimpulsiveforcesintheR-barandV-bardirection.Itcanbeshownthatthevexpenditureuseisapproximately14.57m/sandthetotaltimeforrendezvouswas4310s(71.85minutes).TheimpulsivevelocitycontrolinFigure 3-4 showsrelativelyevenlyspacedthrustsandatotalof12impulseswererequired.TheR-barandV-bartrajectoriesshowstabilitywithrespecttotheequilibriumposition,rrel=0,withonlyasmallamountofovershoot.Now,the 40

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Figure3-2. TrajectoryforsatelliterendezvoususingAPFGalgorithm APFGalgorithmisdemonstratedwithobstaclespresentinthecongurationspace.Anobstaclewasplacedatthepositioner1=[600)]TJ /F5 11.955 Tf 11.95 0 Td[(121.50]Tmwheretheabovemethodofobstacledenitionwasusedwith=1000and wascalculatedtobe =3,092,361.TheresultsforthetrajectoryintheLVLHframeareshowninFigure 3-5 .InordertotryandmaintaintobenonincreasingbyusingthemethodsmentionedpreviouslythespaceforwhichtheresultingtrajectoryoftheESRcannotenterisenlarged.Thefuelusewasv=17.82m/sandthetotaltimedurationofthemaneuverwas75.33minuteswithatotalof14impulses.Aplotof_isshowninFigure 3-6 indicatesthat_0ismaintained.ThetrajectoriesintheR-barandV-bardirectionsaswellastheimpulsesrequiredshowninFigure 3-7 indicatethatsmallovershootaswellasindividuallysmallandrelativelyevenlyspacedoutimpulseswererequired.Obstacledetectionwas 41

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Figure3-3. _duringrendezvoususingAPFGalgorithm checkedthroughthecontrollawasshowninEq.( 3 )byupdatingthepositionsandvelocitiesoftheESRandobstaclesevery0.1s. 3.1.3Conclusions FortheAPFGalgorithmitwasfoundthatananalyticformoftheguidanceexistsbutthattherepulsivepotentialfunctionparametersjandjmustbedenedtoensureobstacleavoidance.HavingananalyticalformfortheG&Cishighlydesirableforautonomousspacecraftmissionsbecausethecomputationtimestocomputethecontroltwithinareal-timeframeworkandthusfulllsoneofthe3requirements.AlthoughaGUUBresultwasdeterminedforthetrajectoryrrel(t)itisexpectedthataGloballyAsymptoticallyStable(GAS)resultcanbeobtainedwithaterminalguidancealgorithmasshowninMcInnes[ 32 ]whentheESRcomesinverycloseproximity(i.e.,krrel(t)k10m)andbecauseofthefactthatlocalminimaduetoobstaclesareunstable 42

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a)b) c)d)Figure3-4. R-bar,V-bartrajectoriesandimpulsesduringrendezvoususingAPFGalgorithm saddlepoints.AsectionwaspresentedthatshoweditisnotpossibletoensurethesystemdoesnotcrosstheboundariesofanobstaclebysatisfyingtheconstraintinEq.( 3 ).However,theexampleshowedthatobstacleavoidancemaneuversoccurredwellbeforethesystemreachestheboundaryofanobstacle.ThesimulationdemonstratedthatalthoughfuelusewaslowtherewasnodirectcompensationforgravitationalforceintheG&Cformulation.OneadvantageoftheAPFGmethodisthataTOFneednotbespeciedinwhichrendezvousistobeachieved.IfthenumberofobstaclesaswellastheirpositionsandvelocitiesarenotknownaheadoftimethenhavingtheTOFbeunspeciedcanprovidetheG&Cmoreexibilityinndingadesirablesolution.Itshould 43

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Figure3-5. TrajectoryforsatelliterendezvoususingAPFGalgorithmwithoneobstacle benotedthattheAPFGalgorithmrequiresthatthepositionsoftheobstaclesandESRinturnwiththeconditionforthecontrollawbeupdatedfrequently.Thefrequencyofupdatesfortheconditiononthecontroldependsonthenumberofobstaclesaswellastheirpositionsandvelocitiestoensurethatobstacleavoidanceisachieved. 3.2APFGWithSwitchingControl TomotivatethissectiononestartswiththeapproachthatitwillnotrequirethatthereexistsanenergyequationsuchasaLyapunovcandidatefunctionwhichmustbenonincreasing.Todemonstratethisidea,aCWmaneuverisplottedasshowninFigure 3-8 usingTOFoftheprevioussection'srendezvousexampleandtheAPFGalgorithm.Thetotalfuelusedforthismaneuverwasv=2.44m/s.UsingthesameformforasinEq.( 3 ),withnoobstacles,onecanseethat_hasthefollowinginterestingbehaviorasshowninFigure 3-9 .Itisclearthatfuelusageissmallerandyet 44

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Figure3-6. _duringrendezvoususingAPFGalgorithmwithoneobstacle thetrajectoryrrel(t)doesnotstaywithinsomeboundswhichdependonanenergyequationasseeninEq.( 3 ). Theideaistodesignacontrollerthatswitchesbetweenagoaloriented,orattractionmode,andrepulsive,orobstacleavoidancemode,inordertoachievebothrendezvousandavoidingobstacles.TheattractivepartoftheguidancecoulduseaTOFprovidedbythemissionplanneroritcouldarbitrarilyuseaTOFwhichwouldmaintainaLyapunovcandidatefunctiontobemonotonicallydecreasing.ThiscanbedonebyconsideringthefollowingLyapunovcandidatefunction =1 2rTrelrrel(3) 45

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a)b) c)d)Figure3-7. R-bar,V-bartrajectoriesandimpulsesduringrendezvoususingAPFGalgorithmwithoneobstacle whereitsderivativecanbecalculatedtoasshowninEq.( 3 ). _=_rTrelrrel(3) InordertondthelimitsforkeepingthetrajectorymonotonicallydecreasingEq.( 2 )canbeusedtorewriteEq.( 3 )attheinitialtimet0andsetequaltozeroasshowninEq.( 3 ) )]TJ /F5 11.955 Tf 5.48 -9.69 Td[(21(t0,t0)rrel0+22(t0,t0))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(rel0+v(T)T(11(t0,t0)rrel0+12(t0,t0))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(r)]TJ /F4 7.97 Tf -1.37 -8.27 Td[(rel0+v(T)=0 (3) 46

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Figure3-8. RendezvoususingCWmaneuver whereTistheTOFforthemaneuver.Now,Eq.( 2 )isreplacedforvinEq.( 3 )andaTOFissolvedforwhichsatisestheequation.InordertokeepthetrajectorystrictlydecreasingonecanchooseaTOFwhichisslightlylessthantheTdeterminedsuchas0.95T.ChoosingasmallerTOFthanTdecreasestheamountoftimethecoriolisforceeffectsthesystem.Thisresulteffectstheshapeofthetrajectoryandthereforetheinitialvelocity.DecreasingtheTOFfromTgenerallymakestheanglebetweenthepositionandvelocityvectorgreaterthan 2whileincreasingtheTOFgenerallymakestheanglelessthan 2.InordertomaintainobstacleavoidanceaLyapunovcandidatefunctionischosensuchthatitstimederivativeshowsthatitwillnotallowthesystemtocrosstheboundariesoftheobstacle.TheLyapunovfunctionis 47

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Figure3-9. RendezvoususingCWmaneuver denedasshowninEq.( 3 ). rep=Xjjexp)]TJ /F5 11.955 Tf 10.5 8.09 Td[((rrel)]TJ /F5 11.955 Tf 11.57 0 Td[(~rj)TMj(rrel)]TJ /F5 11.955 Tf 11.57 0 Td[(~rj) j(3) InordertoensureobstacleavoidancethederivativeofEq.( 3 )mustbelessthanathresholdvalue.Itisnotdesiredtosolvedirectlyforavaluethatwillworkingeneral,however,avalueforwillbefoundfromsimulationtodetermineaneffectivethreshold.Thecontrollawduringtheobstacleavoidancecontrolmodethenbecomes f=8><>:0if_rep<(t)vif_rep9>=>;(3) _r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(rel+v=)]TJ /F3 11.955 Tf 9.3 0 Td[(krrrel.(3) 48

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ItshouldbenotedthatEq.( 3 )isusedjusttodenewhenanobstacleisencountered,howevertheviscalculatedfromthenegativegradientofthesum()oftheattractive(attractive)andrepulsive(repulsive)functions.ThiswasobservedtobeimportantbecausethevcalculatedfromEq.( 3 )hastheeffectofsimultaneouslymovingthesysteminadirectionawayfromtheobstaclewhilefuturecontrolintheattractivemodeisnotcontinuallybeingobstructedbyanobstacle.Thereasonfordoingtheobstacledetectioninthiswaywastoincorporateheuristicsfromsimulationsorothermeanstomorecloselyreectwhenthesystemshouldmaneuverawayfromanobstacle. InordertoallowforasmoothtransitionbetweenthetwomodestheESRisallowedtocoastasufcientdistanceawayfromtheobstaclebeforeswitchingbacktotheattractivecontrol.Thisisdonebyrequiringtherepulsivemodetobeactiveforasetamountoftimeafterthelastmaneuverhasbeencommanded. 3.2.1Stability Thismethodofcontroldoesnotallowonetoconcludethatlocalminimawillnotbeencountered,however,theredoesexistanabilitytocompensatefortheeffectsofgravityintrackingadesiredrelativeposition.Itcanbeshownthroughsimulation,tobediscussedlater,thatobstacleavoidancecanbeensured.OnecanseefromtheformofEq.( 2 )thatwhenthecontrolmodeisinattractionmodethatEq.( 2 )isastabilizingcontrol.ThisisbecauseEq.( 2 )ensuresrendezvouswillbeachievedandthroughtherepulsivemodeobstacleavoidancecanbeachieved.Inaddition,itisassumedthatoncetheESRhascomewithinasmallrelativedistanceoftheRSOanemotioncontrollercantakeoverwhereobstaclesarenotassumedtobepresent.AcommonandprovenmethodfornemotioncontrolistheglideslopemethodasshownbyHablani[ 47 ]whichdrivesboththerelativepositionandvelocitytozero. 3.2.2SpacecraftRendezvousUsingAPFGSC AsimulationwasperformedcomparingAPFGandAPFGSCtoseeifanyperformancecharacteristicscanbeshown.Inordertoeffectivelydetermineperformance 49

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Figure3-10. SimulationofAPFGSC(red)versusAPFG(green) betweenAPFGandAPFGSCascenariowascreatedwithboundariesofrrel2[22]Tkm,and50randomlyplacedobstaclesforwhich3differentsizeswererandomlyselectedtollthesepositions.Thesimulationwasrun50timesforeachinitialconditionoftheESRasrrel=[110]Tkm,rrel=[1)]TJ /F5 11.955 Tf 13.17 0 Td[(10]Tkm,rrel=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm,rrel=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(10]Tkm. AresultfromthissimulationisshowninFigure 3-10 wheretheinitialconditionwasrrel=[110]Tkm.ThedifferencesbetweenthesetwoG&Capproachesisshownin 50

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Table3-1. PerformanceofAPFGversusAPFGSC Method Meanvtotalm/s MeanTOFs Mean#ofimpulses APFG 9.11 2787 7.1 APFGSC 5.3 1680 2 Initialposition:rrel0=[110]Tkm Method Meanvtotalm/s MeanTOFs Mean#ofimpulses APFG 15.22 2568 7.34 APFGSC 11.74 2302 3.59 Initialposition:rrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm Method Meanvtotalm/s MeanTOFs Mean#ofimpulses APFG 9.3 2783 7.04 APFGSC 4.94 1661 2.6 Initialposition:rrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm Method Meanvtotalm/s MeanTOFs Mean#ofimpulses APFG 15.37 2545 7.59 APFGSC 13.36 2448 3.84 Initialposition:rrel0=[1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm Table 3-1 bycomparingtheaveragev,TOF,andnumberofimpulsesforeachinitialconditionandcontrolapproach. InFigure 3-11 itcanbeseenthattheAPFGalgorithmtendstohaveamorenormaldistributionintheoccurrenceofdifferentmagnitudesofvthanAPFGSC.ThisiscontributedtoAPFGSChavingthehighestnumberofincidenceswherenoobstacleswereencountered.IntheAPFGcase,itisfrequentthatanobstacleactuallysteersthesatellitetowardtheRSOandthereforeaidsrendezvous.ThisobservationindicatesthatwhenusingAPFGthenoobstaclecaseisnotthelowestfuelusecase. 3.2.3Conclusion Inconclusion,itcanbeseenthatAPFGandAPFGSCachievethesamegeneralstabilityresultalthoughitisdifculttodeterminetheexactboundeachcontrollercanbeexpectedachieveduetolocalminima.Ingeneral,APFGSCoutperformsAPFGinboththelessnumberofimpulsesrequiredandtheloweramountfuel,v,required.Itshouldbenotedthatinordertoincludeaterminthecontrolthataccountsfortheforceduetogravity,intermsoftrackingaposition,andrequiresperformanceintermsoffuel 51

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a)b)Figure3-11. HistogramplotofAPFGandAPFGSCcontrollers use(v)itmayberequiredthatthedistancetravelledwillinitiallyincreaseforacertainamountoftimeuntilarapiddecreaseoccurstowhichthesystemwillnallyreachthetargetposition.ItisalsopossiblethatthedistancebetweenESRandRSOmayincreasewhenanobstacleisencountereduntiltheattractionmodecanbeactivated.Thiswillallowforfuelefcientmaneuveringaroundobstacles.ThisiscertainlyproblematicfortrackingproblemsusingLyapunovtypemethodswhichseekanonincreasingLyapunovfunction. Fromthehistogramplots(Figure 3-11 )comparingAPFGandAPFGSCitshouldbenotedthatthereasonperformanceintermsof(v)andnumberofimpulsesrequiredisbetterforAPFGSCthenAPFGisbecausetherewerealargenumberofinstanceswhennoobstacleswereencounteredatall.Thisdevelopmentleadstothenextsection 52

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whereatrueplannerisdevelopedwhichusesthedynamicmodeltopropagatethesystem'strajectoryadvanceintimetoseewhetherpathconstraintswillbeviolatedinordertomakedecisionsaheadoftime. 53

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CHAPTER4GUIDANCEUSINGANALYTICSOLUTIONOFDYNAMICEQUATIONS ThischapterpresentsamethodofG&CwhichutilizesthesolutionofthedynamicequationsgiveninEq.( 2 )tochooseanoptimizedtrajectorywhichsatisesbothadesiredmaneuverandobstacleavoidancesimultaneously.TheCWmaneuver,whichisdeterminedfromthesolutiontotheCWHequationsofmotion,solvesforanimpulsivecontrolthrustgivenadesirednalrelativeposition,rrelT,andTOF.Tomotivatethiscontroltechniqueanoverviewisprovidedwhichdescribeshowthepreviouscontroltechnique,showninsection3.1,works.Fromsection3.1,onecanseefromtheAPFGcontrollaw(Eq.( 3 ))that_>0ifoneoftwoscenariosarise: 1. ThesystemdriftsawayfromtheRSO:krrelkincreasesbetweentwoinstancesoftime 2. Thesystemencountersanobstacle Whenthecondition_>0occursacontrolthrustiscommandedwhichcanachievebothrendezvousandobstacleavoidance.ThestabilityresultshowsthattheLyapunovfunctionisnonincreasing.Thereforethestabilityisrobusttodisturbancesandnoises.InordertoeffectivelyusetheCWmaneuverforcontrolthenactiverepetitionoftheCWmaneuver,areneededforthisformofcontrolbecauseofinherentdisturbances,noises,andobstaclesthesystemwillencounter.Aplanneristhereforeneededtoupdateatcertaintimeincrementstocompensateforthesemodeledandunmodeledeffects. 4.1GASMethod Inordertoattenuatetheeffectsofdisturbancesandnoisesonthesystemsperformanceitisdesiredtoincorporateatypeoffeedbackmechanism.Thismechanismcanbedoneinamultitudeofways.Inthisdissertationaconstanttimeintervalinwhichtoplantoavoidobstaclesaswellasreducefuelisused.Therefore,planningisdoneoverahorizonoftimewhichthenmovesaftereachintervaliscompleteduntilboundaryconditionsaremet.Inordertoincreasetheperformanceandredundancyofthisscheme,theG&Cwillbeplannedforaninitialpredictionhorizon,Tp,andthen 54

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ashortercontrolhorizon,Tc,willbeimplemented.Thisessentiallymeansaplanninghorizonwillbeusedtoavoidobstaclesandminimizefuelusebutwillbeimplementedforamuchshortertimeduration.Thisshortertimeforwhichthecontrolisimplementedwilldependonthenumberofobstaclesandtheirpositionsandvelocities,therateatwhichsensorinformationcanbeupdated,andtheamountoftimetocomputethenextimpulsivethrust. Optimizationisdonebyperforminga3-DsearchoftheTOFanddesirednalrelativeposition,rrelT,withtheconstraintthatmaneuversareoftheformofaCWmaneuverdenedbyEq.( 2 ).TodeterminewhereallofthelocalminimaoccurintermsoftheTOF,itisnecessarytoinvestigatethepropertiesofEq.( 2 ).ThisisimportantbecauseaninitialguessoftheTOFneedstobecorrectlychosenthatwillalwayssatisfytheconstraintsforaparticularproblemandguaranteethatthecorrectminimumisfound. Theobjectiveforautonomoussatellitemaneuversistominimizefueluseaswellasitisdesiredtoreducethecomplexityoftheoptimizationproblem.Otherwise,theoptimizationwouldneedtobeintermsofthethreeindependentvariablesinEq.( 2 ),thatis,v,rrelT,andTOF.Therefore,thecostfunctionmustcontainvfromEq.( 2 ). 4.1.1MinimizationOfCWManeuver OnecannowanalyzetheCWmaneuvergivenbyEqs.( 2 )-( 2 )byexpandingasshowninEq.( 4 ). v(T)=2666666666666664n()]TJ /F8 7.97 Tf 6.59 0 Td[(4sin(nT)+3nT)(rrelx(T))]TJ /F8 7.97 Tf 6.59 0 Td[((4)]TJ /F8 7.97 Tf 6.59 0 Td[(3cos(nT))rrelx(0)) )]TJ /F8 7.97 Tf 6.59 0 Td[(8+3nTsin(nT)+8cos(nT)+)]TJ /F8 7.97 Tf 6.58 0 Td[(2n()]TJ /F8 7.97 Tf 6.59 0 Td[(1+cos(nT))(rrely(T))]TJ /F8 7.97 Tf 6.59 0 Td[((6(sin(nT))]TJ /F4 7.97 Tf 6.59 0 Td[(nT)rrelx(0)+rrely(0))) )]TJ /F8 7.97 Tf 6.59 0 Td[(8+3nTsin(nT)+8cos(nT))]TJ /F5 11.955 Tf 12.96 0 Td[(_r0x2n()]TJ /F8 7.97 Tf 6.58 0 Td[(1+cos(nT))(rrelx(T))]TJ /F8 7.97 Tf 6.59 0 Td[((4)]TJ /F8 7.97 Tf 6.58 0 Td[(3cos(nT))rrelx(0)) )]TJ /F8 7.97 Tf 6.59 0 Td[(8+3nTsin(nT)+8cos(nT)+)]TJ /F4 7.97 Tf 6.59 0 Td[(nsin(nT)(rrely(0))]TJ /F8 7.97 Tf 6.59 0 Td[((6(sin(nT))]TJ /F4 7.97 Tf 6.58 0 Td[(nT)rrelx(0)+rrely(0))) )]TJ /F8 7.97 Tf 6.58 0 Td[(8+3nTsin(nT)+8cos(nT))]TJ /F5 11.955 Tf 12.97 0 Td[(_r0yn(rrelz(T))]TJ /F8 7.97 Tf 6.59 0 Td[(cos(nT)rrelz(0)) sin(nT))]TJ /F5 11.955 Tf 12.97 0 Td[(_r0z3777777777777775(4) 55

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ItcanbeseenfromEq.( 4 )thatthexandycomponentsofvaredividedbythetermshowninEq.( 4 )whichwillbedenotedasforconvenience.TheexpressionshowninEq.( 4 ) =)]TJ /F5 11.955 Tf 9.3 0 Td[(8+3nTsin(nT)+8cos(nT)(4) isperiodicandisequaltozeroinaninnitenumberoflocations.Forexample,therstinstancewhenthisoccursisatT=5336secondsifacircularorbitisassumedwitharadiusof6600km.ThismeansthatthexandycomponentsofEq.( 4 )willbecomelargeas!0. Tofurtherexplore,theplotoftheminimumfuelcostfunctionisshowninFig. 4-1 forxednalpositionrrel(T)=[000]Tandfourdifferentinitialconditions,rrel0,intheR-barandV-bardirectionsas[1)]TJ /F5 11.955 Tf 12.35 0 Td[(10]Tkm,[110]Tkm,[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm,[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 12.34 0 Td[(10]Tkm.Itisnotedthattheinitialconditions[1)]TJ /F5 11.955 Tf 12.45 0 Td[(10]Tkmand[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkmproducethesameplotforboththe1-normand2-normofvaswellastheinitialconditions[110]Tkmand[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 12.31 0 Td[(10]TkmasindicatedinFig. 4-1 .Theminimumfuelcostfunctionissimplydenedasthe1-normofvandisshowninEq.( 4 ). J(T)=jv(T)jx+jv(T)jy+jv(T)jz(4) WhatcanobservedfromFig. 4-1 isthatthefuelcostdecreasesasthetimeofightincreasesuntiltheregionswheretheTOFcauses!0.Inaddition,onecanobservefromFig. 4-1 thatthereisalocalminimaforthetwoinitialconditions[1)]TJ /F5 11.955 Tf 12.57 0 Td[(10]T,[)]TJ /F5 11.955 Tf 9.29 0 Td[(110]TkmbetweenTOF=0andtheTOFwhichrstcauses!0becausethe1-normisapiecewisefunction. J(T)=v(T)Tv(T)(4)Onewaytoalleviatethelocalminimaproblemisbyusingacostfunctionoftheminimumenergyformwhichisthe2-normofv,showninEq.( 4 ).Plotsoftheminimumenergycostfunctionforxednalpositionrrel(T)=[000]Tandfourdifferent 56

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Figure4-1. 1-normand2-normofvversusTOF initialconditions,rrel0,intheR-barandV-bardirectionsgivenas[1)]TJ /F5 11.955 Tf 12.16 0 Td[(10]Tkm,[110]Tkm,[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm,[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 12 0 Td[(10]TkmarealsoshowninFig. 4-1 .Thepreviouslymentionedlocalminimaisavoidedbecausethe2-normisnotapiecewisecontinuousfunction.FromFig. 4-1 itcanbeseenthatbeforetherstsingularity,theTOFsassociatedwiththeminimumoftheminimumfuelandenergyfunctionsdonotdiffersignicantly.Furthermore,theTOFassociatedwiththeminimumoftheminimumenergyfunctioninFig. 4-1 hasacorrespondingfuelcostwhichisnotsignicantlydifferentfromtheminimumoftheminimumfuelcostfunction. OnecanobservefromthedynamicsinEq.( 2 )thatthex-ycomponentsaredecoupledfromthezcomponent.ThismeansthatforonlytheCWmaneuverdoinga3-Dorout-of-planemaneuverwillthevcostbedependentonallthreecomponentsx-yandz.OnecanimmediatelyseefromtheexpandedformofvinEq.( 4 )that 57

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thecostduetothezcomponentwillbecomelargeassin(nT)!0.Uponinspectionofthisfunction,sin(nT)!0withaperiodof2668sforacircularorbitwithradiusof6600km.Ifout-of-planemotionisbeingconsiderthenTOFslessthanT=2668sshouldbeutilized.Ifmotioniskeptin-planethenitcanbeseenfromL'HospitalsruleasshownbySteward[ 48 ]thatasTgoesto2668sthelimitonthezcomponentgoestozero.Therefore,in-planemotionisnoteffectedbythez-componentofv. 4.1.2OptimizationStructure Theobjectiveistominimizev,thedifferencerrelT)]TJ /F3 11.955 Tf 12.7 0 Td[(rrel(T),andthedifferencebetweentheTOFanddesiredTOFoverthepredictionhorizon,Tp.Ingeneral,rrelTisthenaldesiredrelativeposition;butinthefollowingdiscussionwithoutlossofgeneralitythisvariablewillbeavectorofzeros.Inthissense,oneisminimizingafunctionofv,rrel(T),andTforthelengthofTp.Inthefollowing,whenevera;occursinafunctionargument,thefollowingsymbolshouldbeviewedasanadditionalparameter. ThestructureoftheoptimizationcanbedenedwiththecostasshowninEq.( 4 ) J(T;Tp)=vTQv+rrel(T)TRrrel(T)+c1(Tf)]TJ /F3 11.955 Tf 11.95 0 Td[(T)(4) andsubjecttotheconstraints v(T)=)]TJ /F5 11.955 Tf 9.29 0 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(11,2(T)(rrel(T))]TJ /F5 11.955 Tf 11.95 0 Td[(1,1(T)rrel0))]TJ /F5 11.955 Tf 12.97 0 Td[(_r)]TJ /F8 7.97 Tf -1.37 -7.98 Td[(0(4) jjv(T)jjvmax(4) jjv(T)jjvmin(4) 266666666664)]TJ /F5 11.955 Tf 11.29 -.17 Td[((rrel(1))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(1))TP1(rrel(1))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(1))+10...)]TJ /F5 11.955 Tf 11.29 -.17 Td[((rrel(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(s))TP1(rrel(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(s))+10...)]TJ /F5 11.955 Tf 11.29 -.16 Td[((rrel(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~rj(s))TPj(rrel(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~rj(s))+10377777777775(4) TTmin(4) 58

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TTmax(4) jjrrel(T)jjrmax(4) Forthecostfunction,asshowninEq.( 4 ),QandRarediagonalmatriceswhosecomponentsaregainsforthecomponentsofvandrrel(T),c1isaconstantgain,andTfisadesiredTOF.ItisnotedthattheconstraintinEq.( 4 )isnotactuallyimplementedasanequalityconstraintbutratherissubstituteddirectlyintothecostfunctionforv. InEq.( 4 )thepathconstraintsarecheckedatsconstantintervalsoverthelengthofthepredictionhorizon,s,andforeachofthejobstacles.Thebeginningoftheseintervalsaredenotedas1,...,s.Knowledgeoftheobstaclelocationsatanygiventimeareassumedtobeknown.ThebasisofthisassumptionisthattheobstaclesmotioncanbemodeledasmovingwithrespecttotheRSOforwhichtheCWequationscanbeusedtodescribethemotion.InEq.( 4 )itisassumedthattheTOFscannotbenegativeandinEq.( 4 )amaxTOFisusedsothesingularitiesintheCWmaneuverareavoided.ConstrainingtheimpulsesasinEq.( 4 )andEq.( 4 )requiresthatbeforeeachmaneuverthesatellitemustbeabletoreorientitself.Thetimeforthesatellitetoorientitselfcangenerallybeseenasaconstraintforthelengthofthecontrolhorizonsothatwhenthesatelliteiscoastingitcanalignitselfforthenextmaneuver. TheG&Calgorithmessentiallyhasthreestageswhichhandlenewmeasurementdata,control,andprediction.Thepredictionhorizonisthetimeoverwhichthemodelisusedtopredictthedynamicbehaviorofthesystemandplananoptimizedtrajectory.Acontrolhorizonwhichislessthanorequaltothepredictionhorizonisthetimeintervalforwhichthecontrolisimplemented.Onceimplementationofthecontrolhorizonbegins,newinformationfromsensorsisreceivedatsamplingtime-unitsforwhichtheopen-loopcontrollerisupdated.ThenewsensorinformationincludesupdatesontherelativepositionoftheESRandlocationofobstacles.ThisisdoneinordertoincorporatefeedbackintotheG&Calgorithm.Oncetheoptimizationiscompletedthe 59

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Figure4-2. BlockdiagramforsatelliteGN&CusingGASmethod G&Creturnsavtotheplant.AblockdiagramillustratinghowtheG&CisincorporatedinthesystemisshowninFig. 4-2 4.1.3ObstacleAvoidance ThefunctionusedtodetectiftheESR'strajectoryisintersectingobstacleboundariesassumesanellipticalshapeforobstacles.Althoughothershapescouldbechosenitisassumedthatmostobjectsorbitingtheearthcanbeapproximatedbyellipses.Withoutlossofgeneralityandforbrevity,thisfunctionwillbeshownforthe2-Dcasewhereitcanbeeasilyextendedtothe3-Dcase.ObstacleconstraintdenitionsshowninEq.( 4 )arenowwrittenintermsofjusttheirgeometryasshowninEq.( 4 )andEq.( 4 ). (r)]TJ /F5 11.955 Tf 11.58 0 Td[(~r)TP(r)]TJ /F5 11.955 Tf 11.57 0 Td[(~r))]TJ /F5 11.955 Tf 11.96 0 Td[(1=0(4) P=2666641=a20001=b20000377775(4) 60

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InEq.( 4 )aandbaretheequatorialradii.Thez-component(polarradius)hasbeenomittedsinceonlythe2-Dcaseisbeingconsidered. 4.1.4InitializingGASAlgorithm Tobegin,theinitialguessoftheTOFisdenotedasTiandtheoptimizedTOFisdenoted,pereachpredictionhorizon,asTiwhereiistheithhorizon.Foreachpredictionhorizon,aninitialguessoftheTOF,Ti,anddesiredrelativeposition,rrelT,needstobeprovidedtotheoptimization.ThenaldesiredrelativepositionisalwaysinitializedasrrelT=[000]TwhichisthelocationoftheRSO.TheinitialguessoftheTOFisdonebyalwaysusingthelastsolutionoftheTOFwhichisTi)]TJ /F8 7.97 Tf 6.59 0 Td[(1andsubtractingthelengthofthecontrolhorizonasshowninEq.( 4 ). Ti=Ti)]TJ /F8 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc(4) Toinitializethisalgorithm,onecaneitherstartwithaCWmaneuverwithadesirableTOFthatwillnotbeaffectedbylocalminimumsinthevfunctionasshowninFig. 4-1 .ThiscanbedonebyprovidingasmallTOF(e.g.,10s)whichisbeforetherstsingularity. 4.1.5OptimalityConditions Todetermineifasolutiontotheminimizationproblemiscorrect,onecanuserstordernecessaryandsecondordersufcientconditionsforoptimality.TherstordernecessaryconditionsforoptimalityareattributedtoKarush-Kuhn-Tucker(KKT)andarereproducedhereastheyweredevelopedbyBertsekas[ 49 ]and[ 50 ]. Theorem4.1. Letxbealocalminimumoftheproblemminf(x) subjecttoh1(x)=0,...,hm(x)=0g1(x)0,...,gr(x)0 61

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wheref,hi,gjarecontinuouslydifferentiablefunctionsfrom
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aswellasrg1(x),...,rgr(x) arelinearlyindependent. Thoughtheminimizationproblemthatisbeingconsiderincludesbothequalityandinequalityconstraints,theproofforsecondordersufciencyconditionfortheinequalityconstrainedproblemusesconceptsfromtheequalityconstrainedandunconstrainedproblems.Thesecondordersufciencyconditionsfortheunconstrainedoptimizationproblemisshownrst. Theorem4.2. Letf:0and>0suchthat f(x)f(x)+ 2kx)]TJ /F3 11.955 Tf 11.96 0 Td[(xk2,8xwithkx)]TJ /F3 11.955 Tf 11.96 0 Td[(xk<.(4) Wenowshowthesecondordersufciencyconditionsfortheequalityconstrainedoptimizationproblem. Theorem4.3. Letxbealocalminimumoftheproblemminf(x) subjecttoh1(x),...,hm(x)=0. DenetheLagrangianfunctionL(x,)tobeL(x,)=f(x)+Th(x). 63

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Assumef,hiaretwicedifferentiablefunctions,andletx20,8y6=0withrh(x)Ty=0. Theorem4.4. Letxbealocalminimumoftheproblemminf(x) subjecttoh1(x),...,hm(x)=0.g1(x),...,gr(x)0. DenetheLagrangianfunctionL(x,,)tobeL(x,,)=f(x)+Th(x)+Tg(x). Assumef,hi,gjaretwicedifferentiablefunctions,andletx20,8j2A(x),j=0,8j=2A(x),yTr2xxL(x,,)y>0, forally6=0suchthatrhi(x)Ty=0,8i=1,...,m,rgj(x)Ty=0,8j2A(x). 4.1.6SensitivityAnalysis ItcanbeshowntheLagrangemultipliersprovideanimportantinterpretationinthecontextofthisproblem.TheLagrangemultiplierscanbeviewedastheratesofchange 64

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oftheoptimalcostasthelevelofconstraintchanges.ThefollowingTheoremisusedtoprovetheassertion. Theorem4.5. Letxand(,)bealocalminimumandLagrangemultipliers,respec-tively,oftheproblem minf(x)s.t.h1(x)=0,...,hm(x)=0, (4) g1(x)0,...,gr(x)0, satisfyingthesecondordersufciencyconditionsfortheinequalityconstrainedproblemgiveninTheorem5.Assumethatthegradientsrhi(x),i=1,...,m,rgj(x),j2A(x),arelinearlyindependent.Considerthefamilyofproblems minf(x)s.t.h(x)=u,g(x)v, (4) parameterizedbythevectorsu2Remandv2
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4.1.7Stability SincetheCWmaneuverrequiresthatthetrajectorypassesthroughtheRSOlocationanysolutionreturnedfromtheminimizationofcostasshowninEq.( 4 )andconstraintsasshowninEq.( 4 )-( 4 )willreturnastablecontrol.Aswitchingofcontrolisnecessary,oncetheESRiswithinacloseproximitytotheRSO,toensureboththerelativepositionandvelocityapproachzero.Assumingthattherearenoobstaclespresentinthisregion,onecanusewellestablishedapproachestosimultaneouslydecaytherelativepositionandvelocity.MethodstodecaytherelativepositionandvelocityincludetheglideslopemethodandthesolutionoftheCWequationsfromwhichonecancommandadesiredrelativevelocityasshowninEq.( 4 ). v(T)=)]TJ /F5 11.955 Tf 9.3 0 Td[((T))]TJ /F8 7.97 Tf 6.58 0 Td[(12,2)]TJ /F5 11.955 Tf 6.49 -9.69 Td[(_r(T))]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.58 0 Td[(11,2(T)rrel0)]TJ /F5 11.955 Tf 12.97 0 Td[(_r)]TJ /F8 7.97 Tf -1.38 -7.97 Td[(0(4) 4.1.8SpacecraftRendezvousUsingGASMethod TostudytheperformanceoftheGASmethod,asimulationforsatelliterendezvouswithinitialconditionsof1kmintheR-barand-1kmintheV-bardirectionswasperformed.AtotalofsixobstacleswereincludedinthesimulationwherethreewereintentionallyplacedsothattheESRwouldhavetoavoidthem.Theinitiallocationofthesethreeobstaclesare~r1=[575)]TJ /F5 11.955 Tf 11.96 0 Td[(11750]Tkm,~r2=[300)]TJ /F5 11.955 Tf 11.96 0 Td[(13000]Tkm,and~r3=[)]TJ /F5 11.955 Tf 9.29 0 Td[(215)]TJ /F5 11.955 Tf 11.96 0 Td[(6950]Tkmandtheshapeparameterswerea1=50,b1=50,a2=25,b2=25,a3=25,andb3=25.Thelocationoftheotherthreeobstacleswere~r4=[425)]TJ /F5 11.955 Tf 11.96 0 Td[(2250]Tkm,~r5=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1075)]TJ /F5 11.955 Tf 11.95 0 Td[(10250]Tkm,and~r6=[225)]TJ /F5 11.955 Tf 11.96 0 Td[(4500]Tkmandtheshapeparameterswerea4=25,b4=25,a5=25,b5=25,a6=25,andb6=25. inFig. 4-3 ,theresultingtrajectoryisshowninblue,thepositionswherethrusterringoccurredisshownasgreenstars,andtheshapeoftheobstaclescanbevisualized.Forthisexample,ThemagnitudesoftheimpulsewhichareafunctionoftheTOFcanbeseeninFig. 4-4 .Thepredictionhorizonwas80sandpathconstraintswerecheckedat5sintervals.Thehorizonforwhichcontrolactionwastakenoccurred 66

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Figure4-3. RendezvouswiththreeobstaclesusingGAScontroltechnique every25s.Thetotalmagnitudeofthevforthisscenariois5.49m=sandtheTOFwas3270s. Next,itisshownhowthetrajectoryisaffectedbythecontrolasseeninFig. 4-5 .ThemagentatrajectoryasshowninFig. 4-5 iswhatwouldhaveresultedifnoactionweretakenandtheblueisafterimpulsivering.ItcanbeseenfromFig. 4-5 thatthemagentatrajectoryshowsatrajectorywhichisclosertotheobstaclethanthebluetrajectory.InFig. 4-5 thedirectionoftheESRmovesfromtoplefttobottomrightwhiletheobstaclemovesfrombottomrighttotopleft.AftertheESRhaspassedtheobstacle,itmaneuverstoachieverendezvouswiththeRSOlocationasindicatedinthesubgurewithsimulationtimeofT=200sasshowninFig. 4-5 .Itcanbealsoseen,thatforeachtimestep,obstacleavoidancewasachievedfortherstobstacleinFig. 4-6 .Thecontrolinthepresenseofthesesecondobstacledemonstratessimilarperformanceasthecase 67

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Figure4-4. Impulse,v,forGASG&Cmethodforrendezvous. withtherstobstacle,thecontrollermaneuverstheESRawayfromtheobstacles,pereachhorizon,asshowninFig. 4-7 .ItcanbeseenfromFig. 4-8 thatobstacleavoidancewasachievedforeverytimestepofthesecondobstacleencounter.InFig. 4-7 thedirectionoftheESRmovesfromtoplefttobottomrightwhiletheobstaclemovesfrombottomrighttotopleft.ThethirdobstacledemonstratessimilarcontrollerperformancecharacteristicsasthersttwoobstaclesintermsoftheG&CmaneuveringtheESRawayfromtheobstacle,pereachhorizon,asindicatedinFig. 4-9 .InFig. 4-9 theESRismovingfromtoplefttobottomrightandtheobstacleismovingfromtoplefttobottomright.ItcanbeseenfromFig. 4-10 and 4-11 thatobstacleavoidancewasachievedforeverysimulationtimestepofthethirdobstacleencounter. ForanalysisdoneinthisdissertationthesimulationsweredoneinMatlabwitha1.8GHzprocessorand1GbRAM.Theaverageamountoftimetondanoptimized 68

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a)b) c)d) e)Figure4-5. Firstobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. 69

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a)b) c)d) e)Figure4-6. Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps 70

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a)b) c)d) e)Figure4-7. Secondobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. 71

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a)b) c)d) e)Figure4-8. Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps 72

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a)b) c)d) e)Figure4-9. Thirdobstacleencountertrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. 73

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a)b) c)d)Figure4-10. Plotsshowingobstacleavoidanceforthirdobstacleencounteredforsubsequenttimesteps solutionwas0.92swithamaximumamountoftimetondasolutionof3.92s.Tofurtherquantifytheperformance,theresultsusingtheGASmethodshowninTable 4-1 weregeneratedfromscenarioswheretheESRhasfourdifferentinitialconditionsandobstaclelocationswerevaried,foreachinitialrelativepositionoftheESR.Inthischapter,thevariationsaredoneinanincrementalfashionstartingfromaninitialcongurationwheretheG&Calgorithmwouldhavetoavoidtheobstacles.Theobstacleinitiallocationswerevariedbyassumingthatthereexistsaninitialrelativepositionofan 74

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a)b) c)Figure4-11. Plotsshowingobstacleavoidanceforthirdobstacleencounteredforsubsequenttimesteps obstaclesuchas~r=[q1q2q3]T wheretheG&Calgorithmwouldhavetomaneuvertoavoidhittingtheobstacle.Theinitialobstaclepositionwasthenvariedas ~r=[q1+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn;q2+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn;q3]T(4) wherecnwasincrementedfromone,byone,totwenty.ObstaclelocationswereimplementedtotwentybecausethatwasthesituationwheretheG&Cnolongerhad 75

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Table4-1. PerformanceofGASvs.APFG Method Meanvtotal,m s MeanTOF,s Mean#ofimpulses APFG 14.57 4310 11 GAS 3.34 2563 5.95 Initialposition:rrel0=[1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm APFG 8.97 4529 11 GAS 2.64 2781 9.45 Initialposition:rrel0=[110]Tkm APFG 14.77 4311 11 GAS 3.35 2464 6.58 Initialposition:rrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm APFG 8.98 4520 11 GAS 2.18 2262 7.15 Initialposition:rrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm toavoidobstacles.Movingobstacle'sintialpositioninthiswayprovidedasimplewaytosimulatetheeffectivenessoftheGASmethodencounteringobstaclesfromdifferentdirectionsandvelocities. AcomparisonoftheGASmethodbetweenAPFGandAPFGSCcanbemadebycomparingtheresultsobtainedinTable 3-1 and 4-1 .Itisunderstoodthateventhoughthecongurationsofobstaclesaredifferent,itcanbeseenhowwelltheGAS,APFG,andAPFGSCmethodscompareintermsoffueluse,TOF,andnumberofimpulses.Ingeneral,fromTable 4-1 itcanbeseenthatthefuelusefortheGASmethodislower,theTOFisneitherbothgreaterorsmaller,andthenumberofimpulsesisneitherbothgreaterorsmallerincomparisontoAPFGwithobstaclesasshowninTable 3-1 .Inaddition,Table 4-1 showsacomparisonbetweenGASandAPFGwhereAPFGissimulatedwiththesameinitialconditionsbutwithnoobstacles.FromTable 4-1 wecanseethatthefueluseislower,theTOFislower,andthenumberofimpulsesislowerfortheGASmethodwithobstaclesascomparedtoAPFGwithoutobstacles. ItshouldbenotedthatfortheinitialESRrelativepositionrrel0=[)]TJ /F5 11.955 Tf 9.29 0 Td[(110]Ttherewasasimulationinwhichtheoptimizationcouldnotconvergetoasolution.Thereasonforthisfailureisbecausethetherewasnotenoughinformationabouttheobstacle's 76

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a)b)Figure4-12. Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[110]Tkm a)b)Figure4-13. Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm trajectorycrossingpathswiththeESR.Theobstaclewhichgeneratedthefailurehadaninitialpositionof~r=[)]TJ /F5 11.955 Tf 9.3 0 Td[(51012950]km.Thesimulationwasrerunwithobstacledetectiondoneevery2sinsteadof5sandthelengthofthepredictionhorizonwaschangedto48sinsteadof80s.Withthesenewparametersfortheoptimizationitterminatedsuccessfullywithatotalv=13.61m/s,ameantimeforconvergenceof1.24s,andamaxtimeforconvergenceof4.11s. 77

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a)b)Figure4-14. Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]Tkm a)b)Figure4-15. Meanandmaximumtimeforoptimizationtoconvergeforinitialrelativepositionofrrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(10]Tkm Ifthecongurationofsixobstaclesisagainconsideredinwhichtheirpositionsareincrementallymoved,asdiscussedpreviouslyforthefourdifferentinitialrelativepositionsoftheESR,ahistogramplotofthemeanandaveragetimesfortheoptimizationtoconvergeareshowninFig. 4-12 4-15 .Itcanbeseenthatthemeantimefortheoptimizationtoconvergeissmall(i.e.,lessthan1s)andthemaximumtimestoconvergearewithinthepredictionhorizonlengthof25s. 78

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ThedistributionofimpulsesbetweentheminimumandmaximumthrusterconstraintsareshowninFig. 4-16 .Theminimumandmaximumthrusterconstraintschosenwerevmin=0.1m/sandvmax=10m/s.TheminimumandmaximumconstraintmagnitudeswerechosenbasedonthemagnitudesofaCWmaneuverrequiredfromtheselectedinitialrelativepositions.FromFig. 4-16 thedistributionisconcentratedtowardstheminimumthrustervaluewhicharetypicallysmallcorrectionstoensurerendezvousafteravoidinganobstacle.ThehighestthrustervalueinFig. 4-16 wastheinitialringforrendezvousandnotanobstacleavoidancemaneuver.TheminimumandmaximumconstraintmagnitudesfortheTOFgiveninEq.( 4 )and( 4 )wereselectedfromFig. 4-1 tobeTmin0.1andTmax5200.TheminimumTOFmagnitudewasselectedtobeasmallpositiveconstantwhilethemaximumTOFmagnitudewasselectedtobesmallerthantherstsingularityinthetwonormofvfunctionwhichwas5336s.Therewerenoconstraintschosentoboundthevariablerrel(T). 4.1.9Conclusion TheresultsoftherendezvousexampleshowedthattheGASmethodrequiredlessfuelusagethanboththeAPFGandtheAPFGSCmethods.TheGASmethodessentiallyallowsthecontrollertosteerthesatellitearoundanobstaclebychangingtheTOFandnalrelativepositionrrel(T).ThecontrollerincludesboundsontheTOF,asseeninEq.( 4 )andEq.( 4 ),duetosingularitieswhicharepresentintheformofcontrolusedasshowninEq.( 4 ).ItisalsoshownthatthattheCWmaneuverwrittenasshowninEq.( 4 )neednotbeincludedintheconstraintshowninEq.( 4 )asanequalityconstraintintheoptimizationbutcanratherbedirectlysubstitutedintothecostforv.Thiseffectivelylimitsthechoicesofcostfunctionsthatcanbechosenbymakingthembeafunctionofvbutthismeetstheobjectiveofndingvoptimizedmaneuversthatcanachieveobstacleavoidance.IfEq.( 4 )wereincludedasanequalityconstraint 79

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a)b) c)d)Figure4-16. Histogramplotsoffuelusev(m/s)forGASmethodinscenariowithsixobstacles thenvwouldneedtobevariabletherebyincreasingtheamountindependentvariablesbythreeforout-of-planemaneuversandtwoforin-planemaneuvers. FromFigs. 4-12 4-15 ,itcanbeseenthatthemeantimefortheoptimizationtoconvergeissmall(i.e.,lessthan1s)andthemaximumtimesarewithinthepredictionhorizonof25s.ItisapparentthatthemaximumtimerequiredfortheoptimizationtoconvergeislargerfortheinitialESRpositionsofrrel0=[1)]TJ /F5 11.955 Tf 11.95 0 Td[(10]Tandrrel0=[)]TJ /F5 11.955 Tf 9.3 0 Td[(110]TasshowninFigs. 4-13 4-14 thanthepositionsrrel0=[110]Tandrrel0=[)]TJ /F5 11.955 Tf 9.29 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(10]TasshowninFigs. 4-12 and 4-15 80

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ItcanbeseenfromthesimulationwithincrementallychangedpositionsoftheobstaclesandresultsshowninTable 4-1 ,thatobstacledetectionintervalsaswellaspredictionhorizonlengthsneedtobechosenappropriatelyinorderfortheoptimizationtoconvergetoasolution.Intheexamplesshownearlierforrendezvouswithobstacleavoidance,theobstacleswereincrementallymovedfromaninitialcongurationinordertodetermineappropriateintervalsforobstacledetectionaswellaspredictionhorizons. Inimplementations,theG&CalgorithmwouldterminatewithinasetdesireddistancefromtheRSOsuchas10or20maway.Fromhere,itisexpectedthatacloserangecontroller,suchasaglideslopemethod,couldtakeovertocompletetherendezvous.Thisisseenassatisfactorybecauseobstacleobstructionisnotproblematicinthisregionaswellasitisdesiredtohaveaprecisionmotioncontrolwithinthiscloseproximity.Thiscloserangecontrollerisdescribedinfulldetaillaterinthischapter.ThetransitionbetweenthesecontrollersisseamlesssincethecontrolformisthesamebecausebothuseCWmaneuvers. 4.2GuidanceUsingAnalyticSolutionOfTheTwo-BodyProblem ThisG&CapproachextendsbeyondthecapabilityoftheGASmethodbyconsideringsatellitemaneuversbetweenellipticalorbits,notjustcircularoneswheretheCWequationsareapplicable.ItisassumedthatthegoverningequationsarebasedintheECIcoordinatesystem.ThederivationoftheequationsofmotionanddepictionofthecoordinatesystemareshowninChapter2.Inthissection,itwillbeshownthattheanalyticsolutiontotherestrictedtwo-bodyproblemcanbeusedinasimilarmanneraswasperformedwiththeCWmaneuver.Theanalyticsolution,showninChapter2,canbeexpressedasapowerseriessolutionoftherestrictedtwo-bodyproblemasshowninEq.( 2 ) r(t)=rrel0+(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0)vj0)]TJ /F5 11.955 Tf 13.15 7.92 Td[((t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)2 2rj0+(t)]TJ /F3 11.955 Tf 11.96 0 Td[(t0)3 3!(3r)]TJ /F6 11.955 Tf 11.95 0 Td[(v)j0+(4) 81

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wherethesolutionuptothe11thorderisshowninEq.( B ),inAppendixB.Theapproximationofthesolutionwastakenuptothe11thtermbecauseitwasdeemednecessaryfromobservationofthecorrespondingG&Csimulations.Theapproximatesolution,asshowninEq.( 4 )canberewritteninamoregeneralformasshowninEq.( 4 ). r(t)=F(t)r0+G(t)v0(4) FromEq.( 4 )theinitialvelocitycanbemodiedbyanimpulsev.TheTOF,Tandnaldesiredposition,rrelT,canbespeciedinordertomaneuverasatellitefromonepositiontoanother.AdifcultywithEq.( 4 )isthatF(t)andG(t)areafunctionoftheinitialvelocityv0inwhichisalsoafunctionoftheimpulsev.SinceitisnotpossibletoexplicitlyexpressvinEq.( 4 ),aniterativeschememustbeemployedwhichrequiresaninitialguessofv.Itisnecessarytoshowthatforacertainrangeofinitialguesses,ofvaminimumcanbefoundwhichgivesavalidsolutiontotheG&Cproblem.UnliketheGASmethod,itisnotpossibletoxvaluesfortheTOFandnalrelativepositionr(T)andobtainanexactsolutionfortheimpulse.Tosolveforv,onemustrstprovideaguessofvthatsatisesEq.( 4 ) r(t))]TJ /F3 11.955 Tf 11.96 0 Td[(F(t)r0)]TJ /F3 11.955 Tf 11.96 0 Td[(G(t)v0=0(4) anduseanumericalmethodtondanapproximatesolution.Itispossible,however,toxvaluesfortheinitialpositionr0,nalpositionr(T),andthenvarytheTOFtoseeifsingularitiesariseinthenumericalsolutionforv. UsingEq.( 4 )aplotofvasafunctionofTOFisshowninFig. 4-17 .TheplotofvasafunctionofTOFinFig. 4-17 issimilartotheresultsobtainedusingtheCWmaneuverasshowninFig. 4-1 .TheplotofthemagnitudeofvasafunctionofTOFinFig. 4-17 issimilartotheresultsfromtheCWmaneuverbecausenearzeroTOFthemagnitudeofvislargefollowedbyadecreasetoapointinwhichanincreaseisobservedforaregioninwhichasingularityoccurs.Theinitialposition 82

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Figure4-17. SolutionofvforrangeofTOFtothepowerseriessolutionoftherestrictedtwo-bodyproblem andvelocityoftheRSOintheECIcoordinatesystem,forthisexample,wasrRSO=[5075.6895075.6698.57]Tkmand_rRSO=[)]TJ /F5 11.955 Tf 9.3 0 Td[(5.5265.5260.025]Tkm/s.TheinitialpositionandvelocityoftheESRintheECIcoordinatesystem,forthisexample,wasrESR=[5075.6325076.0108.571]Tkmand_rESR=[)]TJ /F5 11.955 Tf 9.3 0 Td[(5.5295.5280.026]Tkm/s.TheorbitoftheRSOandESR,forthisexample,wascreatedbyconsideringanorbitcoordinatizedinthePCSsystem.InthePCSsystem,onecanexpressthescalarmagnitudeofthepositionasshowninEq.( 4 ) krk=p 1+kekcos(4) wherepistheparameterand,calledthetrueanomaly,istheangleintheplaneofthesatellite'sorbitbetweenperiapsisandthepositionofthesatelliteataparticulartime. 83

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FromEq.( 4 )onecanchoosethemagnitudekrkiftheeccentricityisxedaswellasthetrueanomaly.TheeccentricityandtrueanomalyoftheRSOwerechosentobeeRSO=0.1andRSO=0radians.TheeccentricityandtrueanomalyoftheESRwerechosentobeeESR=0.101andESR= 80056radians.ThemagnitudeofthepositionfortheRSOwaschosenaskrRSOk=7178.1kmwhilefortheESRitwaskrESRk=7178.3km.ThevelocitiesgivenearlierforthisexampleweredevelopedfromthevelocityequationgiveninEq.( 4 ). v=r p[)]TJ /F5 11.955 Tf 11.29 0 Td[(sinP+(e+cos)Q](4) TheanglesassociatedwithconversionfromthePCScoordinatesystemtotheECIsystemwereselectedasi= 900,!= 9,and=5 36.ThevaluesofRSO,ESR,i,!,andwerechosenarbitrarily.TheeccentricitieseRSOandeESRwherechosensuchthattheirvalueswerebetween0and1.ThisdistancesoftheRSOandESR,measuredfromthecenteroftheearth,krRSOkandkrESRkwherechosentobeapproximatelythesamefortheexamplesdoneusingtheGASmethod. ThistypeofanalysisfortheTOF,aswasdoneusingFig. 4-17 ,wouldberequiredforeachprobleminordertoavoidamaneuverwhichwouldrequiretoolargeofav.Specically,itisdesiredtoavoidasingularityforaparticularTOFasshowninFig. 4-17 .Althoughitisnotknowniftheminimumsfoundarecorrect,onecanchecktherst-orderoptimalityconditionsfortheminimization.Checkingthemeasureofsatisfactionoftherst-orderoptimalityconditionsofEq.( 4 ),onecanobservethattheyaresatisedwhentheTOFislessthanthevalueatwhichthesingularityoccurs,butnotsatisedwhentheTOFisgreaterthanthisvalueasshowninFig. 4-18 .Checkingtherst-orderoptimalityconditionscanbedonebysettingthegradientoftheLagrangianequaltozeroasshowninTheorem5.FromFig. 4-17 onecanseethatasingularityarisesatapproximately2100s. 84

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Figure4-18. Measureofsatisfactionofrst-orderoptimalityconditionsforr(t))]TJ /F3 11.955 Tf 11.96 0 Td[(F(t)r0)]TJ /F3 11.955 Tf 11.96 0 Td[(G(t)v0=0 AsignicantdifferencebetweentheGASandGASTBmethodisthattheGASmethodusesequationsofmotionthatdescribetherelativemotionofonesatellitetoanotherwhiletheGASTBmethodusesequationsofmotionfortwobodies(i.e.,theearthandasatellite).WhenusingtheGASTBmethodonemustusetherestrictedtwo-bodyproblemtomodelthemotionofboththeESRandRSOwithrespecttotheearth.Therefore,iftheTOFischangedwithintheoptimizationitchangeswherethelocationoftheRSOisrelativetothepreviousTOF.VaryingtheTOFmakestheoptimizationmoredifcultifitisdesiredtomaneuverfromonelocationtoanother,minimizefueluse,andavoidobstaclesusingtherestrictedtwo-bodyproblemtomodelthemotionoftheESRandRSOascomparedtoxingtheTOF.Asimplerproblemwhichcanstillmaneuverfromonelocationtoanother,minimizefueluse,andavoidobstaclesistoxtheTOFforthemaneuverandvarythenalposition,r(T).Basedonaparticularmaneuver'sinitialdistancetothetargetlocationarangeofappropriate 85

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TOFsistypicallyknown.IfarangeofTOFsforaparticularmaneuverisnotknown,ananalysissimilartotheoneshowninFig. 4-17 canbeutilized. Whenusingthepowerseriessolutionoftherestrictedtwo-bodyproblemtocontrolaspacecraftfromonepositiontoanother,theorbitmustbeelliptical(i.e.,kek<1)andnopointontheorbitbelessthanathreshold,krmink,whichismeasuredfromthecenteroftheearth.Atanygiveniterationintheoptimization,theinitialpositionoftheESR,rESR0;initialvelocitybeforeimpulse,_r)]TJ /F4 7.97 Tf -1.37 -8.28 Td[(ESR0;andtheimpulsevisknown.TheeccentricityvectorecanbecomputedasshowninEq.( 4 ). e=1 kvk2)]TJ /F6 11.955 Tf 18.36 8.08 Td[( krkr)]TJ /F11 11.955 Tf 11.95 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(rTvv(4) Giventhisinformationtheeccentricity,kek,aswellasanapproximationforapositionontheorbitr(t)canbecalculated. 4.2.1OptimizationStructure TheobjectiveistominimizevandthedifferencerESR(T))]TJ /F3 11.955 Tf 12.85 0 Td[(rRSO(T)overthepredictionhorizon(i.e.,Tp).TheexpressionrRSO(T)isthedesirednalpositionoftheRSOforthemaneuver.ThestructureoftheoptimizationcanbedenedwiththecostasshowninEq.( 4 ) J(rESR(T),_rESR0;Tp)=vTQv+(rESR(T))]TJ /F3 11.955 Tf 11.95 0 Td[(rRSO(T))TR(rESR(T))]TJ /F3 11.955 Tf 11.95 0 Td[(rRSO(T))(4) andsubjecttotheconstraintsasshowninEq.( 4 )-( 4 ). rESR(T)=F(T)rESR0+G(T)_rESR0(4) jjv(T)jjvmax(4) jjv(T)jjvmin(4) kek1(4) 86

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266664krESR(1)k+krmink0...krESR(s)k+krmink0377775(4) 266666666664)]TJ /F5 11.955 Tf 11.29 -.17 Td[((rESR(1))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(1))TP1(rESR(1))]TJ /F5 11.955 Tf 11.58 0 Td[(~r1(1))+10...)]TJ /F5 11.955 Tf 11.29 -.17 Td[((rESR(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(s))TP1(rESR(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~r1(s))+10...)]TJ /F5 11.955 Tf 11.29 -.17 Td[((rESR(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~rj(s))TPj(rESR(s))]TJ /F5 11.955 Tf 11.57 0 Td[(~rj(s))+10377777777775(4) jjrESR(T)jjkrmaxk(4) InEq.( 4 )pathconstraintsarecheckedatconstantintervalsforthelengthofthepredictionhorizon,s,andforeachofthejobstacles.InEq.( 4 )thetrajectoryischeckedtodetermineifitisbelowtheallowabledistance,measuredfromthecenteroftheearth,asdenotedaskrmink.TheconstraintshowninEq.( 4 )maintainsthateachneworbitbeellipticalinshape.ForthecostfunctionasshowninEq.( 4 ),QandRarediagonalmatriceswhosecomponentsaregainsforthecomponentsofvandrESR(T)vectors.ItisnotedthattheconstraintinEq.( 4 )isafunctionofvandnalposition,rESR(T),oftheESR. InEq.( 4 )and( 4 ),thetrajectoryischeckedalongstimeintervalsforwhichthersttimeinstancesoftheseintervalsaredenotedas1,...,s.Itisassumedthatknowledgeofobstaclelocationsisknownatanygiventimebecausetheirmotioncanbemodeled,usingtherestrictedtwo-bodydynamicsasshowninEq.( 2 ),asmovingfromaninitialmeasuredpositionandvelocityintheECIcoordinatesystem.ConstrainingtheimpulsesasshowninEq.( 4 )andEq.( 4 )requiresthatbeforeeachmaneuverthesatellitemustbeabletoreorientitself.Thetimeforthesatellitetoorientitselfcangenerallybeseenasaconstraintforthelengthofthecontrolhorizonsothatwhenthesatelliteiscoastingitcanalignitselfforthenextmaneuver. 87

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Figure4-19. BlockdiagramforsatelliteGN&CusingGASTBmethod TheG&Calgorithmessentiallyhasthreestageswhichhandlenewmeasurementdata,control,andprediction.Thepredictionhorizonisthetimeoverwhichtherestrictedtwo-bodyproblemisusedtopredictthedynamicbehaviorofthesystemandplananoptimizedtrajectory.Acontrolhorizonwhichislessthanorequaltothepredictionhorizonisthetimeintervalforwhichthecontrolisimplemented.Onceimplementationofthecontrolhorizonbegins,newinformationfromsensorsisreceivedatsamplingtime-unitsforwhichtheopen-loopcontrolproblemcanthenbererun.ThisisdoneinordertoincorporatefeedbackintotheG&Calgorithm.AblockdiagramofhowtheG&CisincorporatedinthesystemisshowninFig. 4-19 .ThemaneuverlistedinG&CblockofFig. 4-19 istheimpulsereturnedfromsolutionofEq.( 4 ). 4.2.2ObstacleAvoidance ThefunctionusedtodetectiftheESR'strajectoryisintersectingobstacleboundariesassumesanellipticalshapeforobstacles.Althoughothershapescouldbechosenitisassumedthatmostobjectsorbitingtheearthcanbeapproximatedbyellipses.ObstacleconstraintdenitionsareshowninEq.( 4 )butarenowshownin 88

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termsofjusttheirgeometryinEq.( 4 )andEq.( 4 ). (r)]TJ /F5 11.955 Tf 11.58 0 Td[(~r)TP(r)]TJ /F5 11.955 Tf 11.57 0 Td[(~r))]TJ /F5 11.955 Tf 11.96 0 Td[(1=0(4) P=2666641 a20001 b20001 c2377775(4) InEq.( 4 )aandbaretheequitorialradiiandcisthez-component(polarradius). 4.2.3InitializingGASTBAlgorithm FortheGASTBmethodaTOFisxedfortheproblemandthenalpositionoftheESR,rESR(T),isvaried.TheinitialguessofthenalpositionoftheESRisthepredictedpositionoftheRSObasedontheTOF,initialposition,andinitialvelocityoftheRSO.ThepredictedpositionoftheRSOiscalculatedbythepowerseriessolutiontothetwo-bodyproblemasshowninEq.( B ).TheTOFisinitiallyxedasT0andeachsubsequentTOFisthepreviousTOFminusthecontrolhorizonasTi=Ti)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 12 0 Td[(Tcwhereiistheithhorizon. 4.2.4Stability OnecancommandamaneuvertopassthroughtheRSOlocation,soanysolutionreturnedfromtheminimizationofthecostasshowninEq.( 4 )andconstraintsasshowninEq.( 4 )-( 4 )willreturnastablecontrolaslongasthevelocitydifferencebetweenESRandRSOiszero.Aswitchingofcontrolisnecessary,oncetheESRiswithinacloseproximitytotheRSO,toensureboththerelativepositionandvelocityapproachzero.Assumingthattherearenoobstaclespresentinthisregiononecanusetheapproximatesolutionofthepowerseriessolutionoftherestrictedtwo-bodyproblemtocommandadesiredrelativevelocityasshowninEq.( 4 ). v(T)=Ft(T)r0+Gt(T)v0(4) 89

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InEq.( 4 )thetermsFtandGtaresimplythetimederivativesofFandGasshowninEq.( 4 ). 4.2.5SpacecraftRendezvousUsingGASTBMethod AsimulationwasdoneforrendezvouswherethepositionandvelocityoftheRSO,ESR,andtheobstaclesweregeneratedinamannersimilartothemethodoutlinedinSection4.2.TheorbitoftheRSOandESR,forthisexample,wascreatedbyconsideringanorbitcoordinatizedinthePCSsystem.InthePCSsystem,onecanexpressthescalarmagnitudeofthepositionasshowninEq.( 4 ).TheeccentricityandtrueanomalyoftheRSOwerechosentobeeRSO=0.1andRSO=0radians.TheeccentricityandtrueanomalyoftheESRwerechosentobeeESR=0.101andESR= 80056radians.ThemagnitudeofthepositionfortheRSOwaschosenaskrRSOk=7178.1kmwhilefortheESRitwaskrESRk=7178.3km.ThevelocitycannowbecalculatedfromEq.( 4 ).TheanglesassociatedwithconversionfromthePCScoordinatesystemtotheECIsystemwereselectedasi= 900,!= 9,and=5 36.Onecannowobtainvaluesofthepositionandvelocity,intheECIcoordinatesystem,whichcanbeusedtosimulatethemotionofESR,RSO,andtheobstaclesbyintegratingtherestrictedtwo-bodyproblemequationsofmotion.TheinitialpositionandvelocityoftheRSO,roundedofftotheseconddecimalplace,wasrRSO=[5075.695075.678.57]Tkmand_rRSO=[)]TJ /F5 11.955 Tf 9.3 0 Td[(5.535.530.026]Tkm/s.Itisnotedthatthenumericalvaluesoftheinitialpositionsandvelocities,coordinatizedintheECIsystem,oftheESR,RSO,andobstaclesareroundedofftotheseconddecimalplaceforbrevity.Thedifference,roundedofftotheseconddecimalplace,betweentheinitialESRandinitialRSOposition,thatisrESR)]TJ /F3 11.955 Tf 12.4 0 Td[(rRSO,was[)]TJ /F5 11.955 Tf 9.3 0 Td[(57.76340.61.16]Tm.AtotalofsixobstacleswereincludedinthesimulationbutonlyonewasintentionallyplacedsothattheESRwouldhavetoavoidit.Theeccentricityandtrueanomalyofthisobstaclewaschosentobek~ek=0.101and~=0radians.Themagnitudeofthepositionforthisobstaclewaschosenask~rk=7178.26km.TheanglesassociatedwithconversionfromthePCS 90

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coordinatesystemtotheECIsystemwereselectedasi= 900,!= 9,and=5 36.Thedifference,roundedofftotheseconddecimalplace,betweentheinitialobstacleandinitialRSOpositionwas~r1=[113.14113.140.19]Tmandtheshapeparameterswerea1=50m,b1=50m.Theeccentricityandtrueanomalyofeachoftheotherveobstacleswerechosentobe0.101and0radians.Themagnitudeofthepositionfortheotherveobstacleswerechosenask~r2k=7177.2km,k~r3k=7177.68km,k~r4k=7177.58km,k~r5k=7177.48km,k~r6k=7177.38km.Thedifference,roundedofftotheseconddecimalplace,betweentheotherveinitialobstacle'spositionandtheinitialRSOpositionwere~r2=[)]TJ /F5 11.955 Tf 9.3 0 Td[(296.99)]TJ /F5 11.955 Tf 11.95 0 Td[(296.98)]TJ /F5 11.955 Tf 11.96 0 Td[(0.5]Tm,~r3=[)]TJ /F5 11.955 Tf 9.29 0 Td[(367.7)]TJ /F5 11.955 Tf 11.96 0 Td[(367.69)]TJ /F5 11.955 Tf 11.95 0 Td[(0.62]Tm,~r4=[)]TJ /F5 11.955 Tf 9.29 0 Td[(438.41)]TJ /F5 11.955 Tf 11.96 0 Td[(438.41)]TJ /F5 11.955 Tf 11.96 0 Td[(0.74]Tm,~r5=[)]TJ /F5 11.955 Tf 9.3 0 Td[(509.12)]TJ /F5 11.955 Tf 11.95 0 Td[(509.12)]TJ /F5 11.955 Tf 11.95 0 Td[(0.86]Tm,~r6=[)]TJ /F5 11.955 Tf 9.3 0 Td[(636.4)]TJ /F5 11.955 Tf 11.95 0 Td[(636.4)]TJ /F5 11.955 Tf 11.96 0 Td[(1.07]Tmandtheshapeparameterswerea2=25m,b2=25m,a3=25m,b3=25m,a4=25m,b4=25m,a5=25m,b5=25m,a6=25m,b6=25m. Theresultingtrajectoryforthisexampleofsatelliterendezvousisshowninblue,thepositionswherethrusterringoccurredareshownasgreenstars,andtheshapeoftheobstaclesareshowninFig. 4-20 .ThetrajectoriesshowninFig. 4-20 areforthedifferencebetweenthepositionoftheESRandRSO,intheECIcoordinatesystem,thatisrESR)]TJ /F3 11.955 Tf 12.07 0 Td[(rRSO.Themagnitudesoftheimpulseforthisexample,asafunctionoftheTOF,areshowninFig. 4-21 .Thepredictionhorizonusedforobstacledetectionwas80sandpositionswerecheckedat5sintervals.Thehorizonforwhichcontrolactionwastakenoccurredevery50s.Thetotalvforthisscenariois6.07m=sandtheTOFwas900s. Next,thetrajectoryisshownwhichiseffectedbythecontrolasseeninFig. 4-22 wherethemagentatrajectoryiswhatwouldhaveresultedifnoactionweretakenandtheblueisafterimpulsivering.ThetrajectoriesshowninFig. 4-22 areforthedifferencebetweenthepositionoftheESRandRSO,intheECIcoordinatesystem,thatisrESR)]TJ /F3 11.955 Tf 12.46 0 Td[(rRSO.ThetrajectoryoftheESRshowninFig. 4-22 movesfromtoprighttobottomleftandtheobstaclemovesfrombottomrighttotopleft.Thebluetrajectory, 91

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Figure4-20. RendezvouswithoneobstacleusingGASTBcontroltechnique. asshowninFig. 4-22 a),forthegureshowingthetrajectoriesuptoaTOFof47.9992sindicatesthattheGASTBmethodcommandedanimpulsemaneuverthatmovedtheESRtowardtheRSOlocationbutalsoawayfromtheobstacle.Themagentatrajectory,asshowninFig. 4-22 a),forthegureshowingthetrajectoriesuptoaTOFof47.9992sdoesnotmodifytheinitialvelocitywithanimpulseandsotheESRwouldhavedriftedawayfromtheRSO.Thebluetrajectory,asshowninFig. 4-22 b),forthegureshowingthetrajectoriesuptoaTOFof99.4657sindicatesthattheGASTBmethodcommandedanimpulsemaneuverthatmovedtheESRtowardtheRSOlocationbutalsoawayfromtheobstacle.Themagentatrajectory,asshowninFig. 4-22 b),forthegureshowingthetrajectoriesuptoaTOFof99.4657sdoesnotmodifytheinitialvelocitywithanimpulseandsotheESRwouldhavedriftedawayfromtheRSOandalsointersectedtheobstacleboundary.Thebluetrajectory,asshowninFig. 4-22 c),forthegureshowing 92

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Figure4-21. Impulse,v,forGASTBG&Cmethodforrendezvous. thetrajectoriesuptoaTOFof150sindicatesthattheGASTBmethodcommandedanimpulsemaneuverthatmovedtheESRtowardtheRSOlocation.Themagentatrajectory,asshowninFig. 4-22 c),forthegureshowingthetrajectoriesuptoaTOFof150sdoesnotmodifytheinitialvelocitywithanimpulseandsotheESRwouldhavedriftedawayfromtheRSO. Itcanalsobeshownthat,pereachtimestep,thelevelofsatisfactionofthepathconstraintsfortheobstacleasshowninFig. 4-23 and 4-24 .ItcanbeseeninFig. 4-23 and 4-24 thatobstacleavoidanceisnotstrictlyensuredastheendoftrajectorylieswithintheobstacleregion.TheinequalityconstraintsinEq.( 4 )arebasedontheapproximatesolutiontotherestrictedtwo-bodyproblemandthereforewillnotingeneralagreewiththetrajectorygeneratedfromintegration.ItcanalsobeseenfromFig. 4-23 93

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a)b) c)Figure4-22. Differencesintrajectories,foralengthoftimeofthecontrolhorizon,foractuatedsysteminblueandunactuatedsystemmodelinmagenta.Thegreenstarsindicatewherethrusterringsoccured. c)-d)and 4-24 a)thatthedistancewithintheobstaclethatthetrajectoryliesisnotlargeincomparisontothesizeoftheobstacle. ThesimulationwasdoneinMatlabwitha1.8GHzprocessorand1GbRAM.Fortheabovesimulationtheaverageamountoftimetondanoptimizedsolutionwas2.99swithamaximumamountoftimetondasolutionof18.53s.TheintegratorusedwasMatlab'sODE45withrelativeandabsolutetolerancesof1e)]TJ /F5 11.955 Tf 12.41 0 Td[(15.TofurtherevaluatetheGASTBperformance,theresultsoftheGASTBmethodshowninTable 4-2 weregeneratedfromscenarioswheretheESRhasfourdifferentinitialconditions 94

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a)b) c)d)Figure4-23. Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps andobstaclelocationswerevaried,foreachinitialrelativepositionoftheESR,inanincrementedfashionfromtherstinitialcongurationcasechosenwheretheG&Calgorithmwouldhavetoavoidtheobstacle.Thatis,theobstaclelocationswerevariedbyassumingthatthereexistsaninitialrelativeposition,coordinatizedinthePCSsystem,ofanobstacleasshowninEq.( 4 ) ~r=[kq1kcos(q2)kq1ksin(q2)]T(4) 95

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a)b) c)Figure4-24. Plotsshowingobstacleavoidanceforrstobstacleencounteredforsubsequenttimesteps wheretheG&Calgorithmwouldhavetomaneuvertoavoidhittingtheobstacle.Theexpressionkq1kisthemagnitudeofthevectororiginatingatthecenteroftheearthtotheobstacleandtheexpressionkq2kisthetrueanomaly.TheobstaclepositionwasthenvariedbyincrementingcnasshowninEq.( 4 ) ~r=h(q1+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn)cosq2+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn (q1+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn)sinq2+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)cncn iT(4) 96

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Table4-2. PerformanceofGASTBwithobstaclesandAPFGwithoutobstacles Method Meanvtotal,m s MeanTOF,s Mean#ofimpulses APFG 5.86 1121 2 GASTB 3.56 1000 3.7 Initialcondition:krk=7179.1km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=60056rad,kek=0.101 APFG 6.55 755 1 GASTB 3.86 1000 2.15 Initialcondition:krk=7178.3km,==80056rad,kek=0.101 APFG 6.75 993 2 GASTB 3.82 1000 1.65 Initialcondition:krk=7177.5km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=80056rad,kek=0.101 APFG 6.53 1091 2 GASTB 3.71 1000 1.8 Initialcondition:krk=7177.5km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=20056rad,kek=0.101 whereisaconstanttobechosen.Movingobstacle'sintialpositioninthiswayprovidedasimplewaytosimulatetheeffectivenessoftheGASTBmethodencounteringobstaclesfromdifferentdirectionsandvelocities. Ifthecongurationofsixobstaclesisagainconsideredinwhichtheirinitialpositionsareincrementallymoved,asdiscussedpreviouslyforthefourdifferentinitialrelativepositionsoftheESR,ahistogramplotofthemeanandmaximumtimesfortheoptimizationtoconvergeareshowninFig. 4-25 4-28 .TherstsimulationwasfortheESRinitialconditionsofkrESRk=7179.1km,ESR=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.48km,~=)]TJ /F6 11.955 Tf 9.29 0 Td[(=24056rad,k~ek=0.101.ThevaluesofcnfromEq.( 4 )wasindexedfromone,byone,totenandwaschosenas1e7.ThehistogramplotofthemeanandmaximumtimesfortheoptimizationtoconvergeareshowninFig. 4-25 .ThenextsimulationwasfortheESRinitialconditionsofkrESRk=7178.3km,ESR==80056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7178.26km,~=0rad,k~ek=0.101.ThevaluesofcnfromEq.( 4 )wasindexedfromone,byone,totwentyandwaschosenas5e6.ThehistogramplotofthemeanandmaximumtimesfortheoptimizationtoconvergeareshowninFig. 4-26 .ThenextsimulationwasfortheESRinitialconditionsofkrESRk=7177.5km, 97

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a)b)Figure4-25. Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7179.1km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,kek=0.101 ESR=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.5km,~=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,k~ek=0.101.ThevaluesofcnfromEq.( 4 )wasindexedfromone,byone,totwentyandwaschosenas5e6.ThehistogramplotofthemeanandmaximumtimesfortheoptimizationtoconvergeareshowninFig. 4-27 .ThelastsimulationwasfortheESRinitialconditionsofkrESRk=7177.5km,ESR=)]TJ /F6 11.955 Tf 9.3 0 Td[(=20056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.5km,~=)]TJ /F6 11.955 Tf 9.3 0 Td[(=19056rad,k~ek=0.101.ThevaluesofcnfromEq.( 4 )wasindexedfromone,byone,totwentyandwaschosenas9e5.ThehistogramplotofthemeanandmaximumtimesfortheoptimizationtoconvergeareshowninFig. 4-28 .FromtheresultsofFig. 4-25 4-28 ,itisobservedthatthemeantimefortheoptimizationtoconvergeissmall(i.e.,lessthan5s)andthemaximumtimestoconvergearewithinthepredictionhorizonlengthof50s. ThedistributionofimpulsesbetweentheminimumandmaximumthrusterconstraintsfortheESRinitialconditionsofkrESRk=7179.1km,ESR=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.48km,~=)]TJ /F6 11.955 Tf 9.3 0 Td[(=24056rad,k~ek=0.101areshowninFig. 4-29 .ThedistributionofimpulsesbetweentheminimumandmaximumthrusterconstraintsfortheESRinitialconditionsofkrESRk=7178.3km, 98

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a)b)Figure4-26. Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7178.3km,==80056rad,kek=0.101 a)b)Figure4-27. Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,kek=0.101 ESR==80056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7178.26km,~=0rad,k~ek=0.101areshowninFig. 4-30 .ThedistributionofimpulsesbetweentheminimumandmaximumthrusterconstraintsfortheESRinitialconditionsofkrESRk=7177.5km,ESR=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.5km,~=)]TJ /F6 11.955 Tf 9.29 0 Td[(=60056rad,k~ek=0.101areshowninFig. 4-31 .Thedistributionofimpulsesbetweentheminimumandmaximumthrusterconstraintsforthe 99

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a)b)Figure4-28. Meanandmaximumtimeforoptimizationtoconvergeforinitialcondition,inPCScoordinatesystem,krk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=20056rad,kek=0.101 ESRinitialconditionsofkrESRk=7177.5km,ESR=)]TJ /F6 11.955 Tf 9.29 0 Td[(=20056rad,keESRk=0.101andinitialobstaclelocationofk~rk=7177.5km,~=)]TJ /F6 11.955 Tf 9.3 0 Td[(=19056rad,k~ek=0.101areshowninFig. 4-32 .Theminimumandmaximumthrusterconstraintschosenwerevmin=0.1m/sandvmax=10m/s.Inallcases,theminimumandmaximumconstraintmagnitudeswerechosenbasedonthemagnitudesofasinglemaneuverrequiredtomeetadesiredrelativepositionfromaninitialrelativeposition.FromFigs. 4-29 4-32 ,itsisobservedthatthedistributionisconcentratedtowardstheminimumthrustervaluewhicharetypicallysmallcorrectionstoensurerendezvousafteravoidinganobstacle.ThehighestthrustervalueinFigs. 4-29 4-32 wastheinitialringforrendezvousandnotanobstacleavoidancemaneuver.TherewerenoconstraintschosentoboundthevariablerESR(T). 4.2.6Conclusion Fromtheaboveanalysisandresults,thecontrollermustincludeboundsontheTOFduetosingularitieswhicharepresentintheformoncontrolusedasshowninFig. 4-17 .SincetheGASTBmusthaveaxedTOF,numericalvalueschosenforthetimeofthemaneuvertobecompletedmustbelessthantherstsingularity.Thisistobedonetoensurethatnolocalminimaarefoundbythenumericaloptimizer.Itisshownthatthev 100

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Figure4-29. MagnitudeoffuelusewithESRinitialconditionsofkrk=7179.1km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,kek=0.101andinitialobstaclelocationofkrk=7177.48km,=)]TJ /F6 11.955 Tf 9.29 0 Td[(=24056rad,kek=0.101 requiredtomaneuvertheESRfromonepositiontoanotherisachievedbysolvingforavthatsatisesEq.( 4 ).Equation( 4 )isincludedasanequalityconstraintintheoptimizationproblembecausevcannotbeexplicitlydened.FromFigs. 4-25 4-28 itcanbeseenthatthemeantimefortheoptimizationtoconvergeissmall(i.e.,lessthan5s)andthemaximumtimesarewithinthepredictionhorizonof50s. Itcanbeseenfromthesimulation,withresultsshowninTable 4-2 ,thatobstacledetectionintervalsaswellaspredictionhorizonlengthsneedtobechosenappropriatelyinorderfortheoptimizationtoconvergetoasolution.Intheexamplesshownearlierforrendezvouswithobstacleavoidance,theobstacleswereincrementallymovedfromaninitialcongurationinordertodetermineappropriateintervalsforobstacledetectionaswellaspredictionhorizons. 101

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Figure4-30. MagnitudeoffuelusewithESRinitialconditionsofkrk=7178.3km,==80056rad,kek=0.101andinitialobstaclelocationofkrk=7178.26km,=0rad,kek=0.101 Inimplementation,theG&CalgorithmwouldterminatewithinasetdesireddistancefromtheRSOsuchas10or20maway.Fromhereitisexpectedthatamaneuverwouldbeimplementedtostoptherelativevelocitybetweenthetwosatellites.ThemaneuvertochangetherelativevelocityoftheESRcanbeachievedasshowninEq.( 4 ). 4.3TerminalGuidanceStrategy Thestrategyusedfortheterminalphaseforautomatedrendezvousisthegeneralmultipulseglideslopetransfer[ 47 ].AnassumptionforusingthisstrategyisthatinaregionsufcientlyclosetothetargettheESRisnotexpectedtointeractwithotherservicetypesatellitesorotherobstacles.Therefore,onecanusethismethodasaphysicallypracticalguidancestrategytodriveboththerelativepositionandvelocitytozero. 102

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Figure4-31. MagnitudeoffuelusewithESRinitialconditionsofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=80056rad,kek=0.101andinitialobstaclelocationofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=60056rad,kek=0.101 Thisguidancestrategyassumesthatthereisaknownrelativeposition,rrelT,referencedfromthetargetsatellitewherewewanttheESRtorendezvousandtheinitialrelativepositionoftheESRisdenotedasr0.Thechasersatelliteisassumedtostartwithaninitialvelocityof_r)]TJ /F8 7.97 Tf -1.38 -7.98 Td[(0andwillcompletethemaneuverwithoftimeofightT.Apathforthistransferisgeneratedfromr0torrelTandwillbedenotedas.ThevectoremanatesfromrrelTwiththeboundaryconditions0=r0)]TJ /F3 11.955 Tf 12.31 0 Td[(rrelT,(T)=0,and,atanytimet,(t)=rc(t))]TJ /F3 11.955 Tf 12.13 0 Td[(rrelTwherercistherelativepositionofthechaser.Giventheinitial 103

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Figure4-32. MagnitudeoffuelusewithESRinitialconditionsofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=20056rad,kek=0.101andinitialobstaclelocationofkrk=7177.5km,=)]TJ /F6 11.955 Tf 9.3 0 Td[(=19056rad,kek=0.101 andnalconditionsonthedirectioncosinesofaregivenas cos()=(x0)]TJ /F3 11.955 Tf 11.95 0 Td[(xT)=k0kcos()=(y0)]TJ /F3 11.955 Tf 11.96 0 Td[(yT)=k0kcos()=(z0)]TJ /F3 11.955 Tf 11.95 0 Td[(zT)=k0k(4) Thedirectionofthispathisthengivenby u=cos()cos()cos()T(4) wherethescalardistancebformsthepathofthevectoras=bu.Specifyingthedistancetogobasafunctionoftime,b(t),sothattheboundaryconditionscanbemet 104

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withatimeofight,T,andaterminalvelocityof_bTu.Itisassumedthat_bT,lessthanzero,isalsopredetermined. Inorderfor_btodiminishaswithbthefollowinglinearrelationshipisproposed _b=ab+_bT(4) wheretheparameteraistheslope_b=b.Itisalsoknownorcanbespeciedthatforb(t)attheboundaryconditions b(0)=b0_b(0)=_b0<0b(T)=bT_b(T)=_bT<0(4) Theslopeacanthenbedenedas a=)]TJ /F5 11.955 Tf 6.76 -6.7 Td[(_b0)]TJ /F5 11.955 Tf 13.23 2.99 Td[(_bT=b0(4) ThesolutiontoequationEq.( 4 )is b(t)=b0eat+)]TJ /F5 11.955 Tf 6.75 -6.69 Td[(_bT=a)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(eat)]TJ /F5 11.955 Tf 11.95 0 Td[(1(4) andthetimeofight T=(1=a)`n)]TJ /F5 11.955 Tf 6.75 -6.7 Td[(_bT=_b0(4) Inordertofollowthispath,letthenumberofthrusterringsbeNandauniformintervalbetweenthrusterringsbet=T=N.Theimpulsiveringtimesarethendenedastm=mt(m=1,2,...,N) andsothepositionsattheseringtimesis rm=rrelT+mu(4) bm=b0eatm+)]TJ /F5 11.955 Tf 6.76 -6.69 Td[(_bT=a)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(eatm)]TJ /F5 11.955 Tf 11.95 0 Td[(1(4) 105

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Theimpulsivethrusterringcanthenbedeterminedas v=)]TJ /F8 7.97 Tf 6.59 0 Td[(11,2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(tm)(rm+1)]TJ /F5 11.955 Tf 11.95 0 Td[(1,1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(tm)rm))]TJ /F5 11.955 Tf 12.96 0 Td[(_rm(4) andthetrajectorycanbedeterminedas r(t)=1,1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(tm)rm+1,2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(tm)_r+m(4) ItcanthenbeseenthatsincethetimeintervalbetweenimpulsesisthesamethespacecraftwillslowasitapproachesthedestinationrrelT. 106

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CHAPTER5CONCLUSION APFGwasusedasabaselinecomparisonbecauseitutilizesacontrollawwhichcanappropriatelydenethemagnitudeanddirectionofanimpulse,atimetoimpulsetoachieverendezvouswithatargetlocation,andobstacleavoidance.Thisisimportantbecausemanymethodsintheroboticsliteratureassumethatanonzeroforcemayberequiredfortheentiredurationofamaneuverwhich,forsatellitesystems,maynotbeimplementableforcurrentthrustersaswellasrequirelargefueluse.Alargefueluserequirementwouldonlyoccurifanoptimizationforthemaneuverwasnotperformed,however,thesemethodsarenotreal-timeimplementableforcurrentspacecertiedhardware. AnewmethodtermedAPFGSCusestheCWmaneuverinplaceoftheattractiveformofcontrolinAPFG.ItwasshownthroughsimulationthatusingtheCWmaneuverasincorporatedintheAPFGSCmethodreducesfueluserequirementsforanyinitialpositionoftheESRandobstacles.Theimpulsemagnitudeanddirectionareobtainedanalytically,derivedfromtheCWmaneuverforcontrolinattractivemodeandfromanarticialpotentialfunctionbymaintainingitstimederivativebelessthananappropriatelydenedthresholdforcontrolintherepulsivemode.Avarietyofdifferentgeometricalshapesofobstaclesmaybeincorporatedbymanipulatingthepotentialshape.BothAPFGandAPFGSCsufferfromissueswithlocalminimaduetocertainarrangementsofobstacles.Thisisbecausethepotentialeldsarenotshapedtoavoidtheproblemoflocalminima. ThegoaloftheothernewG&Cmethodsdevelopedistoprovideoptimizedtrajectoriesthatcanbeimplementedoncurrentorfuturespacecertiedhardware.Itisspecicallydesiredtonotprovidegloballyoptimalsolutionsbutforsolutionswhichareoptimaloversmallhorizonsoftimeandforaparticularformofcontrol.TheCWmaneuvercanbeusedasaparticularformofcontrolwhichcansteeratrajectoryby 107

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changingtheTOFand/ornalrelativepositionofthemaneuver.AfeasiblesetofTOFwasdevelopedfromthenormofthecontrolthatenclosesasingleminimumforwhichanyguessoftheTOFwithintheseboundscanreturnthecorrectsolution.ThismethodusestheanalyticsolutiontotheCWequationstopredictasatellitesmotionandsothepredictionhorizonallowstheESRtobeginmaneuveringfarenoughinadvancetoavoidobstacleswithinitsthrusterconstraints.ThepredictionhorizontimealsoallowsfortheESRtoreorientitselfafterathrusterring.ThismethodallowsforperformanceintermsofattainingadesiredrelativepositionatacertaintimebyxingtheTOForbypenalty.ItwasshownthefueluserequirementscouldbeloweredbyusingthismethodascomparedtoboththeAPFGandAPFGSCmethods.ForautonomouscontrolofsatellitesusingtheGASmethod,itisnecessarytoknowspecicsoftheproblemwhichincludethepositionandvelocityoftheESRandRSOaswellasthepositionandvelocityofobstacles.Thisisnecessarytodeterminethepredictionhorizon,controlhorizon,andhowoftenobstacledetectionistobedoneinordertoguaranteetheoptimizationcanconvergetoasolution.Althoughitwasshowninanexamplethattheoptimizationmayfailtoreachasolution,theobstacledetectionintervallengthwasdecreasedandtheoptimizationrunagain,acorrectsolutionwasfound.Thiswouldneedtobedoneinpracticetoensuresafetyintermsofavoidingobstacles.Anumberofhistogramplotswereshownthatindicatedtheoptimizationtimestwellwithinthecontrolhorizontimefortheground-basedtestbed.Thisindicatesthatthismethodisimplementableatleastforcurrentground-basedsystems.Anotherinterestingnoteaboutthismethodisthatifanallowableamountoftimeforthenumericaloptimizationtoreachitsrstorderoptimalityconditionshavebeenmet(i.e.,thecontrolhorizon)therestillmayexistafeasiblesolutionthatwasfoundbutjustdoesnotsatisestheoptimalityconditions.ThisisbecausethecontrolusedisanalyticinformguaranteeingacertainrelativepositionwillbereachedforaparticularTOF.Therefore,thissolutioncanstillbeusedtoreachtheobjectivesofthemissionandtheprocessbecontinued. 108

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TheGASTBmethodxestheTOFforamaneuverbutvariesthenalrelativepositionoftheESRinordertosteertheESR.ThismethoddiffersfromtheGASmethodbecausetheGASmethodassumesthatorbitsmustbycircularwhiletheGASTBmethodassumesorbitstobeelliptical.ItwasshownthatafeasiblesetofTOFforparticularinitialconditionsanddesirednalpositionwasdevelopedfromthenormofthecontrolthatencloseasingleminimumforwhichanappropriateTOFcanbechosen.AswiththeGASmethod,itisimportanttoprovidetheoptimizationproblemwithsmallenoughintervalsfordetectingobstaclessothattheoptimizationproblemhasenoughinformationtoconvergetoasolution.Itwasshownthattheoptimizationmayfailtoreachasolutionbutiftheobstacledetectionintervallengthwasdecreasedandtheoptimizationrunagain,acorrectsolutionwasfound.ItwasshownthatthefueluserequirementsarelowerthanthatforAPFGwithoutobstacles.Anumberofhistogramplotswereshownthatindicatedtheoptimizationtimestwithinthecontrolhorizontimefortheground-basedtestbed.Thisindicatesthatthismethodisimplementableatleastforcurrentground-basedsystems. FutureworkinthisareaistoformulatetheGASmethodasalinearprogramtobesolvedbyaMixedIntegerLinearProgrammingsolver.ThismayhavethebenetofreducingtheamountoftimeforthenumericaloptimizationsolvertoconvergetoasolutionthanwasshownfortheGASmethod.Additionalworkistomodeltheinputtothesystemasanpulseratherthananimpulse.Thishasanadditionalparameterforthelengthofburnthanassuminganimpulsiveinputbutisabetterapproximationtotherealsystem. 109

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APPENDIXANECESSARYANDSUFFICIENTCONDITIONSANDSENSITIVITYANALYSIS TheproofoftherstordernecessaryconditionsknownastheKKTconditionsaredevelopedusingapenaltyapproachandarereproducedhereastheyweredevelopedbyBertsekas[ 49 ]and[ 50 ].ThepropositionisgiveninChapter4asTheorem2.Firstafeasiblevectorisdenedasavectorthatsatisestheconstraintsofthegivenproblem. Proof:Theoriginalconstrainedproblemisapproximatedbyanunconstrainedoptimizationproblemthatinvolvesapenaltyforviolationoftheconstraints.Inparticularfork=1,2,...,thecostfunctionisintroducedasFk(x)=f(x)+k 2kh(x)k2+ 2kx)]TJ /F3 11.955 Tf 11.96 0 Td[(xk2+k 2kg+(x)k2 subjecttox2Swhereisaxedpositivescalar,S=fxjkx)]TJ /F3 11.955 Tf 12.19 0 Td[(xkg,>0issuchthatf(x)f(x)forallfeasiblexwithx2S,andg+j(x)=maxf0,gj(x)g,j=1,...,r wherewecanwriteg+j(x)asacolumnvectorg+forallj=1,...,r.Fromtherstordernecessaryconditionswehaveforsufcientlylargek 0=rFk(x)=rf(xk)+krh(xk)h(xk)+(xk)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+krg(xk)g+(xk).(A) Sincerh(x)hasrankm,thesameistrueforrh(xk)ifkissufcientlylarge.Forsuchk,rh(xk)Trh(xk)isinvertibleandbypremultiplying A by)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rh(xk)Trh(xk))]TJ /F8 7.97 Tf 6.59 0 Td[(1rh(xk)Tweobtain kh(xk)=)]TJ /F11 11.955 Tf 11.3 9.69 Td[()]TJ /F2 11.955 Tf 5.47 -9.69 Td[(rh(xk)Trh(xk))]TJ /F8 7.97 Tf 6.58 0 Td[(1rh(xk)T)]TJ /F2 11.955 Tf 5.47 -9.69 Td[(rf(xk)++(xk)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+krg(xk)g+(xk). (A) Bytakingthelimitask!1andxk!x,weseethatfkh(xk)gconvergestothevector =)]TJ /F11 11.955 Tf 11.29 9.68 Td[()]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rh(x)Trh(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1rh(x)T)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rf(x)+krg(x)g+(x)(A) 110

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Wecansimilarlypremultiply A by)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rg(xk)Trg(xk))]TJ /F8 7.97 Tf 6.59 0 Td[(1rg(xk)Tweobtain kg+(xk)=)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rg(xk)Trg(xk))]TJ /F8 7.97 Tf 6.59 0 Td[(1rg(xk)T)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rf(xk)+(xk)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+kh(xk)rh(xk).(A) Bytakingthelimitask!1andxk!x,weseethatfkg+(xk)gconvergestothevector =)]TJ /F11 11.955 Tf 11.29 9.68 Td[()]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rg(x)Trg(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1rg(x)T(rf(x)+kh(x)rh(x)).(A) Substituting A into A and A into A wehave =)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rh(x)Trh(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1rh(x)T(rf(x)+rg(x))(A) and =)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rg(x)Trg(x))]TJ /F8 7.97 Tf 6.58 0 Td[(1rg(x)T(rf(x)+rh(x)).(A) Finally,substituting A and A andtakingthelimitask!1in A weobtain rf(x)+rh(x)+rg(x)=0(A) whichprovestherstorderoptimalityconditionsinTheorem2.Byusingthesecondorderunconstrainedoptimalityconditionweseethat r2Fk(xk)=r2f(xk)+krh(xk)rh(xk)T+kmXi=1hi(xk)r2hi(xk)++krg(xk)rg(xk)T+krXj=1g+j(xk)r2gj(xk)+I (A) ispositivesemidenite,forallsufcientlylargekandforall>0.NowconsiderafunctionV(x)whichisthesubspaceofrstorderfeasiblevariationsas V(x)=fyjrh(x)Ty=0,rg(x)Ty=0g.(A) 111

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Fixanyy2V(x),andletykbetheprojectionofyonthenullspaceofrh(xk)T+rg(xk)T,thatis, yk=y)]TJ /F11 11.955 Tf 11.96 9.69 Td[()]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rh(xk)+rg(xk)h)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rh(xk)+rg(xk)T)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rh(xk)++rg(xk))]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rh(xk)+rg(xk)Ty (A) Sincerh(xk)Tyk=0,rg(xk)Tyk=0andr2Fk(xk)ispositivesemidenite,wehave 0ykTr2Fk(xk)yk=ykT r2f(xk)+kmXi=1hi(xk)r2hi(xk)++krXj=1g+j(xk)r2gj(xk)!yk+kykk2. (A) Sincekhi(xk)!i,kgj(xk)!j,andfrom A togetherwithxk!x,rh(x)Ty=0,rg(x)Ty=0,wehaveyk!yandobtain 0yT r2f(x)+mXi=1ir2hi(x)+rXj=1jr2gj(x)!y+kyk2,(A)8y2V(x).Sincecanbetakenarbitrarilyclosetozero,weobtain 0yT r2f(x)+mXi=1ir2hi(x)+rXj=1jr2gj(x)!y(A) whichisthesecondorderLagrangemultipliercondition.Q.E.D. ThenextproofisforsecondordersufciencyconditionsforequalityconstrainedproblemsgiveninTheorem4.Webeginwiththesecondordersufciencyconditionsforanunconstrainedoptimization.LetusrstintroducetwoLemma's. Lemma1. Asquarematrixisnonnegativedeniteifandonlyifallofitseigenvaluesarenonnegativedenite. Proof:LetandxbeaneigenvalueandacorrespondingrealnonzeroeigenvectorofasymmetricnonnegativedenitematrixA.Then0xTAx=xTx=kxk2,whichprovesthat0.Fortheconverseresult,letybeanarbitraryvectorin
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correspondingsetofnonzero,real,andorthogonaleigenvectors.Letusexpressyintheformy=Pni=1ixi.ThenyTAy=)]TJ 5.48 -.72 Td[(Pni=1ixiT)]TJ 5.48 -.72 Td[(Pni=1iixi.Fromtheorthogonalityoftheeigenvectors,thelatterexpressionisequaltoPni=12iikxik20,whichprovesthatAisnonnegativedenite.Q.E.D. Lemma2. LetAbeasymmetricnnmatrix,let1nbeitseigenvalues,andletx1,...,xnbeassociatedorthogonaleigenvectors,normalizedsothatkxik=1foralli.Then1kyk2yTAynkyk2forally2
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ItisseenthatEquation 4 issatisedforany>0and>0suchthat 2+o)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kdk2 kdk2 2,8dwithkdk<. WenowprovethefollowingLemma. Lemma3. LetPandQbetwosymmetricmatrices.AssumethatQispositivesemidef-initeandPispositivedeniteonthenullspaceofQ,thatis,xTPx>0forallx6=0withxTQx=0.Thenthereexistsascalar csuchthatP+cQ:positivedenite,8c> c. Proof:Assumethecontrary.Thenforeveryintegerk,thereexistsavectorxkwithkxkk=1suchthatxkTPxk+kxkTQxk0. Sincefxkgisbounded,thereisasubsequencefxkgk2Kconvergingtosome x,andsincekxkk=1forallk,wehavek xk=1.Takingthelimitsuperiorintheaboveinequality,weobtain xTP x+limk!1,k2Ksup(kxkTQxk)0.(A) Since,bythepositivedenitenessofQ,xkTQxk0,weseethatfxkTQxkgKmustconvergetozero,forotherwisethelefthandsideoftheaboveinequalitywouldbe1.Therefore, xTQ x=0andsincePispositivedenite,weobtain xTP x>0.Thiscontradicts A .Q.E.D. Proof:LetusnowintroducetheaugmentedLagrangianfunctionLc(x,)=f(x)+Th(x)+c 2kh(x)k2, wherecisascalar.ThisistheLagrangianfunctionfortheproblemminf(x)+c 2kh(x)k2s.t.h(x)=0, 114

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whichhasthesamelocalminimaasouroriginalproblemofminimizingf(x)subjecttoh(x)=0.ThegradientandHessianofLcwithrespecttoxarerxLc(x,)=rf(x)+rh(x)(+ch(x)),r2xxLc(x,)=r2f(x)+mXi=1(i+chi(x))r2hi(x)+crh(x)rh(x)T. Inparticular,ifxandsatisfytheconditions,wehave rxLc(x,)=rf(x)+rh(x)(+ch(x))=rxL(x,)=0,(A)r2xxLc(x,)=r2f(x)+mXi=1ir2hi(x)+crh(x)rh(x)T==r2xxL(x,)+crh(x)rh(x)T. Byassumption,wehavethatyTr2xxL(x,)y>0forally6=0suchthatyTrh(x)rh(x)Ty=0,sobyapplyingLemma1withP=r2xxL(x,)andQ=rh(x)rh(x)T,itfollowsthatthereexistsa csuchthat r2xxLc(x,):positivedenite,8c> c(A) Usingnowthestandardsufcientoptimalityconditionforunconstrainedoptimization,weconcludefromEquations A and A ,thatforc> c,xisanunconstrainedlocalminimumofLc(x,).Inparticular,thereexist>0and>0suchthatLc(x,)Lc(x,)+ 2kx)]TJ /F3 11.955 Tf 11.95 0 Td[(xk2,8xwithkx)]TJ /F3 11.955 Tf 11.95 0 Td[(xk<. Sinceforallxwithh(x)=0wehaveLc(x,)=f(x>0),rL(x,)=h(x)=0,itfollowsthatf(x)f(x)+ 2kx)]TJ /F3 11.955 Tf 11.95 0 Td[(xk2,8xwithh(x)=0,andkx)]TJ /F3 11.955 Tf 11.95 0 Td[(xk<. Thusxisastrictlocalminimumoffoverh(x)=0.Q.E.D. 115

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Wenowintroducetheproofforthesecondordersufciencyconditionsforinequalityconstrainedproblems. Proof:Weprovethisresultbyusingatransformationtoanequalityconstrainedproblemaswellastheproofofthesecondordersufciencyconditionsforequalityconstrainedproblems.Considertheequivalentequalityconstrainedproblem minf(x)s.t.h1(x)=0,...,hm(x)=0, (A) g1(x)+z21=0,...,gr(x)+z2r=0, whichisanoptimizationprobleminvariablesxandz=(z1,...,zr).Considerthevector(x,z),wherez=(z1,...,zr),zj=()]TJ /F3 11.955 Tf 9.3 0 Td[(gj(x))1=2,j=1,...,r. Wewillshowthat(x,z)and(,)satisfythesufciencyconditionsfortheequalityconstrainedproblem,thusshowingthat(x,z)isastrictlocalminimumproblem A ,provingthatxisastrictlocalminimumoftheoriginalinequalityconstrainedproblem. Let L(x,z,,)betheLagrangianfunctionforthisproblem,i.e., L(x,z,,)=f(x)+mXi=1ihi(x)+rXj=1j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(gj(x)+z2j. Wehaver(x,z) L(x,z,,)T=rx L(x,z,,)T,rz L(x,z,,)T=rxL(x,,)T,21z1,...,2rzr=[0,0], 116

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wherethelastequalityfollowssince,byassumption,wehaverxL(x,,)=0,andj=0forallj=2A(x),whereaszj=()]TJ /F3 11.955 Tf 9.3 0 Td[(gj(x))1=2=0forallj2A(x).Wealsohaver(,) L(x,z,,)T=h1(x),...,hm(x),)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(g1(x)+(z1)2,...,)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(gr(x)+(zr)2=[0,0]. Hencetherstorderconditionsofthesufciencyconditionsforequalityconstrainedproblemsaresatised. Wenextshowthatforall(y,w)6=(0,0)satisfying rh(x)Ty=0,rgj(x)Ty+2zjwj=0,j=1,...,r,(A) wehave (yTwT)0BBBBBBBBBB@r2xxL(x,,)00210...0022...0............00...2r1CCCCCCCCCCA0B@yw1CA>0.(A) Thelefthandsideoftheprecedingexpressioncanalsobewrittenas yTr2xxL(x,,)y+2rXj=1jw2j.(A) Let(y,w)6=(0,0)beavectorsatisfyingEquation A .Wehavethatzj=0forallj2A(x),soitfollowsfromEquation A thatrhi(x)Ty=0,8i=1,...,mrgj(x)Ty=0,8j2A(x). Hence,ify6=0,itfollowsbyassumptionthatyTr2xxL(x,,)y>0, 117

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whichimplies,byEquation A andtheassumptionj0forallj,that(y,w)satisesEquation A ,provingourclaim. Ify=0,itfollowsthatwk6=0forsomek=1,...,r.Inthiscase,byusingEquation A ,wehave2zjwj=0,j=1,...,r, fromwhichweobtainthatzkmustbeequalto0,andhencek2A(x).Byassumption,wehavethatj>0,8j2A(x). Thisimpliesthatkw2k>0,andtherefore2rXj=1jw2j>0, showingthat(y,w)satisesEquation A ,completingtheproof. WenowproveTheorem6whichassertsthatLagrangemultiplierscanbeviewedastheratesofchangeoftheoptimalcostasthelevelofconstraintchanges.Webeginbyrstprovingtheresultfortheequalityconstrainedproblem. TheoremA.1. LetxandbealocalminimumandLagrangemultiplier,respectively,satisfyingthesecondordersufciencyconditions,andassumethatthegradientsrhi(x),i=1,...,m,arelinearlyindependent.Considerthefamilyofproblems minf(x)s.t.h(x)=u, (A) parameterizedbythevectoru2
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addtion,forallu2Swehaverp(u)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u), wherep(u)istheoptimalcostparameterizedbyu,thatis,p(u)=f(x(u)). Proof:Considerthesystemofequations rf(x)+rh(x)=0,h(x)=u.(A) Foreachxedu,thissystemrepresentsn+mequationswithn+munknowns-thevectorsxand.Foru=0thesystemhasthesolution(x,).Thecorresponding(n+m)(n+m)Jacobianmatrixwithrespectto(x,)isgivenbyJ=0B@r2xxL(x,)rh(x)rh(x)T01CA. LetusshowthatJisnonsingular.Ifitwerenot,somenonzerovector(yT,zT)TwouldbelongtothenullspaceofJ,thatis, r2xxL(x,)y+rh(x)z=0,(A) rh(x)Ty=0.(A) PremultiplyingEquation A byyTandusingEquation A ,weobtainyTr2xxL(x,)y=0. InviewofEquation A ,itfollowsthaty=0,forotherwiseoursecondordersufciencyassumptionwouldbeviolated.Sincey=0,Equation A yieldsrh(x)z=0,whichinviewofthelinearindependenceofthecolumnsrhi(x),i=1,...,m,ofrh(x),yieldsz=0.Thus,weobtainy=0,z=0,whichisacontradiction.Hence,Jisnonsingular. 119

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Returningnowtothesystem A ,itfollowsfromthenonsingularityofJandtheImplicitFunctionTheoremthatforalluinsomeopensphereScenteredatu=0,thereexistx(u)and(u)suchthatx(0)=x,(0)=,thefunctionsx(u)and(u)arecontinuouslydifferentiable,and rf((x(u))+rh((x(u))(u)=0, (A) h(x(u))=u. (A) Forusufcientlycloseto0,thevectorsx(u)and(u)satisfythesecondordersufciencyconditionsforproblem A ,sincetheysatisfythembyassumptionforu=0.Thisisstraightforwardtoverifybyusingourcontinuityassumptions.[Ifitwerenottrue,therewouldexistasequencefukgwithuk!0,andasequencefykgwithkykk=1andrh)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(x(uk)Tyk=0forallk,suchthatykTr2xxL)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(x(uk),(uk)yk0,8k. Bytakingthelimitalongaconvergentsubsequenceoffykg,wewouldobtainacontradictionofthesecondordersufciencyconditionat(x,).]Hence,x(u)and(u)arealocalminimumLagrangemultiplierpairforproblem A Thereremainstoshowthatrp(u)=ruff(x(u))g=)]TJ /F6 11.955 Tf 9.3 0 Td[((u).BymultiplyingEquation A byrx(u),weobtainrx(u)rf(x(u))+rx(u)rh(x(u))(u)=0. Bydifferentiatingtherelationh(x(u))=u,itfollowsthat I=rufh(x(u))g=rx(u)rh(x(u)),(A) whereIisthemmidentitymatrix.Finally,byusingthechainrule,wehaverp(u)=ruff(x(u))g=rx(u)rf(x(u)). 120

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Combiningtheabovethreerelations,weobtain rp(u)+(u)=0,(A) andtheproofiscomplete.Q.E.D. Wenextusetheprecedingresulttoshowthecorrespondingresultforinequalityconstrainedproblems.Weassumethatxand(,)arealocalminimumandLagrangemultiplier,respectively,oftheproblem minf(x)s.t.h1(x)=0,...,hm(x)=0, (A) g1(x)0,...,gr(x)0, andtheysatisfythesecondordersufciencyconditions.Wealsoassumethatthegradientsrhi(x),i=1,...,m,rgj(x),j2A(x)arelinearlyindependent,i.e.,xisregular.Weconsidertheequalityconstrainedproblem minf(x)s.t.h1(x)=0,...,hm(x)=0, (A) g1(x)+z21=0,...,gr(x)+z2r=0, whichisanoptimizationprobleminvariablesxandz=(z1,...,zr).Letzbeavectorwithzj=()]TJ /F3 11.955 Tf 9.3 0 Td[(gj(x))1=2,j=1,...,r. Itcanbeseenthat,sincexand(,)satisfythesecondorderassumptionsoftheinequalityconstrainedproblem,(x,z)and(,)satisfythesecondorderassumptionsoftheequalityconstrainedproblem,thusshowingthat(x,z)isastrictlocalminimumofproblem A .Itisalsostraightforwardtoseethatsincexisregularforproblem A ,(x,z)isregularforproblem A .Weconsiderthefamilyof 121

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problems minf(x)s.t.hi(x)=ui,i=1,...,m, (A) gj(x)+z2j=vj,j=1,...,r, parameterizedbyuandv. RecallingthatthereexistsanopensphereScenteredat(u,v)=(0,0)suchthatforevery(u,v)2Sthereisanx(u,v)20forj=2A(x(u,v)),whereA(x(u,v))=fjjgj(x(u,v))=vjg, thelastequationcanalsobewrittenasj(u,v)=0,8j=2A(x(u,v)). 122

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Thus,toshow(u,v)and(u,v)areLagrangemultipliersforproblem A ,thereremainstoshowthenonnegativityof(u,v).Forthispurposeweusethesecondordernecessaryconditionfortheequivalentequalityconstrainedproblem A .Ityields (yTwT)0B@r2xxL(x(u,v),(u,v),(u,v))001CA0B@yw1CA0,(A) where=0BBBBBBB@21(u,v)0...0022(u,v)...0............00...2r(u,v)1CCCCCCCA forally2
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APPENDIXBPOWERSERIESSOLUTIONOFTHERESTRICTEDTWO-BODYPROBLEM Thecompleteformoftherestrictedtwo-bodyproblemisshowninEq. B uptothe11thorderterm. r(t)=1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2t2+1 2t3+1 24)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.29 0 Td[(152+3 )]TJ /F5 11.955 Tf 11.96 0 Td[(22t4+1 120)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1053)]TJ /F5 11.955 Tf 11.95 0 Td[(45 +302t5+1 720)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(6302 )]TJ /F5 11.955 Tf 11.96 0 Td[(9454)]TJ /F5 11.955 Tf 11.96 0 Td[(42022)]TJ /F5 11.955 Tf 11.96 0 Td[(45 2+662 )]TJ /F5 11.955 Tf 11.96 0 Td[(223t6+1 5040(630023+7563+103955)]TJ /F5 11.955 Tf 11.95 0 Td[(22682 )]TJ /F5 11.955 Tf 11.96 0 Td[(94503 +1575 2t7+1 40320)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.29 0 Td[(20160321351356)]TJ /F5 11.955 Tf -248.89 -26.9 Td[(10395024)]TJ /F5 11.955 Tf 11.96 0 Td[(425252 2+1559254 +6048022 +1575 3)]TJ /F5 11.955 Tf 11.96 0 Td[(36182 2+26283 )]TJ /F5 11.955 Tf 11.96 0 Td[(5844t8+1 362880)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(51282033+189189025+355204+20270257)]TJ /F5 11.955 Tf 11.96 0 Td[(28378355 +10914753 2+2232902 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1598403 )]TJ /F5 11.955 Tf 11.95 0 Td[(153846023 )]TJ /F5 11.955 Tf 11.96 0 Td[(99225 3t9+1 3628800)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(3541863 2+1703044 +43659002 3+567567006 )]TJ /F5 11.955 Tf -369.51 -26.9 Td[(283783504 2)]TJ /F5 11.955 Tf 11.95 0 Td[(344594258)]TJ /F5 11.955 Tf 11.95 0 Td[(3783780026)]TJ /F5 11.955 Tf 11.95 0 Td[(152328042)]TJ /F5 11.955 Tf 11.96 0 Td[(1324323034+685476032 )]TJ /F5 11.955 Tf 11.95 0 Td[(967626022 2+3972969024 )]TJ /F5 11.955 Tf 11.96 0 Td[(283845)]TJ /F5 11.955 Tf 11.95 0 Td[(99225 4+3114902 3t10+1 39916800)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(37297260023 2)]TJ /F5 11.955 Tf 11.95 0 Td[(106702596025 )]TJ /F5 11.955 Tf -354.98 -26.9 Td[(26216190033 +7662154505 2)]TJ /F5 11.955 Tf 11.95 0 Td[(12405393007 +82702620027+5825820043+35567532035+6547290759+26751125)]TJ /F5 11.955 Tf -326.77 -26.89 Td[(1702701003 3)]TJ /F5 11.955 Tf 11.96 0 Td[(302346002 3+337879083 2)]TJ /F5 11.955 Tf 11.96 0 Td[(160506724 +9823275 4t11rrel0+ (B) 124

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t)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6t3+1 4t4+1 120)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(452+9 )]TJ /F5 11.955 Tf 11.95 0 Td[(82t5+1 720()]TJ /F5 11.955 Tf 9.3 0 Td[(180 +4203+1502t6+1 5040)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(252022)]TJ /F5 11.955 Tf -208.5 -28.11 Td[(47254+31502 +3962 )]TJ /F5 11.955 Tf 11.95 0 Td[(225 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1723t7+1 40320)]TJ /F5 11.955 Tf 5.47 -9.69 Td[(65523+623705+9450 2)]TJ /F5 11.955 Tf 11.95 0 Td[(158762 )]TJ /F5 11.955 Tf 11.95 0 Td[(567003 +4410023t8+1 362880)]TJ /F2 11.955 Tf 5.47 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(289442 2+250563 +11025 3)]TJ /F5 11.955 Tf 11.95 0 Td[(71364)]TJ /F5 11.955 Tf 11.96 0 Td[(19278032)]TJ /F5 11.955 Tf 11.96 0 Td[(83160024)]TJ /F5 11.955 Tf 11.95 0 Td[(9459456)]TJ /F5 11.955 Tf -378.28 -26.89 Td[(2976752 2+10914754 +48384022 t9+1 3628800)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(227026805 +87318003 2)]TJ /F5 11.955 Tf 11.95 0 Td[(16729203 +20096102 2+162162007+4566004+1702701025+537768033)]TJ /F5 11.955 Tf 11.96 0 Td[(793800 3)]TJ /F5 11.955 Tf 11.95 0 Td[(1384614023 t10+1 39916800)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(4849845)]TJ /F5 11.955 Tf 11.95 0 Td[(40367163 2+31149002 3)]TJ /F5 11.955 Tf 11.96 0 Td[(893025 4+22998244 +39729690024 +5108103006 )]TJ /F5 11.955 Tf 11.95 0 Td[(2070288042)]TJ /F5 11.955 Tf 11.95 0 Td[(37837800026)]TJ /F5 11.955 Tf -360.33 -26.9 Td[(3101348258)]TJ /F5 11.955 Tf 11.96 0 Td[(15135120034)]TJ /F5 11.955 Tf 11.95 0 Td[(9676260022 2+7823574032 )]TJ /F5 11.955 Tf -358.98 -26.89 Td[(2554051504 2+392931002 3t11_rrel0 125

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[26] D.A.Vallado,FundamentalsofAstrodynamicsandApplications,KluwerAcademic,Boston,MA,2001. [27] R.H.Battin,AnIntroductiontotheMathematicsandMethodsofAstrodynamics,Revisededition,AmericanInstituteofAeronauticsandAstronautics,Inc.,Reston,VA,1999. [28] R.R.Bate,D.D.Muller,J.E.White,FundamentalsofAstrodynamics,DoverPublications,Inc.,NewYork,1971. [29] W.Fehse,AutomatedRendezvousandDockingofSpacecraft,CambridgeUniversityPress,2003. [30] H.K.Khalil,NonlinearSystems,PrenticeHall,UpperSaddleRiver,NJ,2002. [31] D.E.Koditschek,ExactRobotNavigationbyMeansofPotentialFunctions:SomeTopologicalConsiderations,ProceedingsoftheInstitudeofElectricalandElectronicsEngineersInternationalConferenceonRoboticsandAutomation,March1987,Raleigh,NC,1-6. [32] I.Lopez,andC.R.McInnes,AutonomousRendezvousUsingArticialPotentialFunctionGuidance,AIAAJournalofGuidance,Control,andDynamics,0731-5090,Vol.18,AIAA,Washington,DC,1995,pp.237-241. [33] C.R.McInnes,Autonomouspathplanningforon-orbitservicingvehicles,JournalofBritishInterplanetarySociety,53(1/2),London,UK,January-February,2000,pp.26-38. [34] A.Badawy,C.R.McInnes,On-OrbitAssemblyUsingSuperquadraticPotentialFields,AIAAJournalofGuidance,Control,andDynamics,0731-5090,Vol.31,No.1,Washington,DC,January-February2008,pp.30-43. [35] A.Badawy,C.R.McInnes,Free-FlyerManoeuvringroundaspacestation,12thInternationalConferenceonAerospaceSciencesandAviationTechnology(ASAT-12),Cairo,29-31May2007. [36] A.R.Tatsch,ArticialPotentialFunctionGuidanceforAutonomousIn-SpaceOperations,PhDDissertation,DepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida,2005. [37] A.R.Tatsch,Y.Xu,N.Fitz-Coy,ANonlinearControllerviaArticialPotentialFunctionsforImpulsiveSpacecraftManeuvers,28thAASGuidanceandControlConference,Breckenridge,CO,Feb.2005,AAS05-008. [38] A.R.Tatsch,N.Fitz-Coy,W.Edmonson,ArticialPotentialFunctionGuidanceforOn-orbitInspection,SpaceAutomationandRoboticsSymposium,NavalResearchLaboratory,March30-31,2005,CD-ROM. 128

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BIOGRAPHICALSKETCH NicholasS.MartinsonwasborninWinterPark,Florida.HereceivedaBachelorofSciencedegreeinMechanicalEngineeringinAugustof2005fromtheUniversityofCentralFlorida'sDepartmentofMechanical,Materials,andAerospaceEngineering.InAugustof2005hewasawardedtheAlumniFellowshipforpursuingadoctoraldegreeinAerospaceEngineeringattheUniversityofFlorida.HecompletedtheMasterofSciencedegreeinspringof2008andwasthenadmittedtocandidacyforPhD.HisprofessionalexperienceincludesbeingagraduateteachingassistantattheUniversityofFloridafromJanuary2007-December2008;graduateresearchassistantattheUniversityofFloridafromAugust2005-August2009;engineeringintern,NationalAeronauticsandSpaceAdministration,fromJune-August2008;engineeringintern,LaserPathTechnologiesLLC,fromDecember2003-March2004;engineeringintern,NavalAirWarfareTrainingSystemsDivision,fromOctober2001-December2003.HeisamemberofTauBetaPi,TheEngineeringHonorSociety,amemberoftheGoldenKeyInternationalHonourSociety,andPiTauSigma,theInternationalMechanicalEngineeringHonorSociety. 131