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Trajectory Optimization of Low-Thrust Orbit Transfers

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Title:
Trajectory Optimization of Low-Thrust Orbit Transfers
Creator:
Graham, Kathryn F
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (152 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering
Mechanical and Aerospace Engineering
Committee Chair:
RAO,ANIL
Committee Co-Chair:
FITZ-COY,NORMAN G
Committee Members:
CONKLIN,JOHN
HAGER,WILLIAM WARD
Graduation Date:
12/18/2015

Subjects

Subjects / Keywords:
Apocenter ( jstor )
Initial guess ( jstor )
Optimal control ( jstor )
Orbitals ( jstor )
Orbits ( jstor )
Periapsis ( jstor )
Semimajor axis ( jstor )
Spacecraft ( jstor )
Textual collocation ( jstor )
Trajectories ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
low-thrust -- orbit-transfers -- trajectory-optimization
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Aerospace Engineering thesis, Ph.D.

Notes

Abstract:
Since the world's first artificial satellite, the Sputnik 1, completed its first orbit around the Earth, we have persistently been developing new technology with an emphasis on cost-effective ways to put spacecraft into space. NASA has identified low-thrust propulsion as a critical technology that would greatly reduce the cost of orbital insertion of spacecraft. To date, low-thrust propulsion has been studied in extensive detail and used mostly for interplanetary missions. Little attention, however, has been applied to the usage of low-thrust technology for orbital insertion. This research seeks to develop an approach to solve for optimal trajectories of various orbital transfers using two different types of low-thrust propulsion models. The first low-thrust propulsion model assumes the spacecraft has an on-board energy source to power the propulsion system such as a battery. This allows for continuous thrust throughout the entire transfer. The second low-thrust propulsion model assumes the spacecraft is powered solely by solar panels such that the spacecraft cannot thrust unless it is directly in the line of sight of the Sun. In other words, the spacecraft will have zero thrust when traveling through the Earth's shadow. Using both of the low-thrust propulsion models, the problem of determining high-accuracy minimum-time Earth-orbit transfers is considered. The orbit transfer is posed as a constrained nonlinear optimal control problem and is solved using a variable-order Legendre-Gauss-Radau quadrature orthogonal collocation method. For the first propulsion model, the orbit transfer is posed as a single-phase optimal control problem. An initial guess for the optimal control problem is obtained by solving a sequence of modified optimal control problems where the final true longitude is constrained and the mean square difference between the specified terminal boundary conditions and the computed terminal conditions is minimized. It is found that solutions to the minimum-time low-thrust optimal control problem are only locally optimal in that the solution has essentially the same number of orbital revolutions as that of the initial guess. A search method is then devised that enables computation of solutions with an even lower cost where the final true longitude is constrained to be different from that obtained in the original locally optimal solution. A numerical optimization study is then performed to determine optimal trajectories and control inputs for a range of initial thrust accelerations and constant specific impulses. The key features of the solutions are then determined, and relationships are obtained between the optimal transfer time and the optimal final true longitude as a function of the initial thrust acceleration and specific impulse. Finally, a detailed post-optimality analysis is performed to verify the close proximity of the numerical solutions to the true optimal solution. For the second propulsion model, the orbit transfer is posed as a multiple-phase optimal control problem where the spacecraft can thrust only during phases where it has line of sight to the Sun. Event constraints, based on the geometry of a penumbra shadow region, are enforced between the phases and determine the amount of time spent in an eclipse. An initial guess generation method is developed that constructs an intelligent guess by solving a series of single-phase optimal control problems and analyzing the resulting trajectory to approximate where the spacecraft enters and exits the Earth's shadow. To demonstrate the effectiveness of the approach developed in this research, an optimal transfer trajectory is computed for an Earth-orbit transfer found in the literature. In addition to the comparison case studied, a separate Earth-orbit transfer is examined and solutions are presented for four different departure dates. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: RAO,ANIL.
Local:
Co-adviser: FITZ-COY,NORMAN G.
Statement of Responsibility:
by Kathryn F Graham.

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Source Institution:
UFRGP
Rights Management:
Copyright Graham, Kathryn F. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2015 ( lcc )

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TRAJECTORYOPTIMIZATIONOFLOW-THRUSTORBITTRANSFERSByKATHRYNFRANCESGRAHAMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015KathrynFrancesGraham

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TomybelovedOrneryBear

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ACKNOWLEDGMENTSFirstandforemost,itisessentialthatIacknowledgemyadvisor,Dr.AnilRao,forgivingmetheopportunitytopursueaPh.D.Isincerelythankhimforhisunfailingpatienceandcondenceinmethroughoutthepastveyears.Heinspiredmetoworkhardandalwaysemphasizedqualityoverquantity.Itwouldhavebeenimpossibletocompletethisresearchwithouthisdedicationandexpertguidance.Next,Iwouldliketoextendmygratitudetothoseservingonmycommittee,Dr.JohnConklin,Dr.NormanFitz-Coy,andDr.WilliamHager.Theirinvaluablesuggestionsandencouragingremarksweregreatlyappreciated.IwouldalsoliketothanktheNationalAeronauticsandSpaceAdministrationfortheirvariousformsofnancialsupportwhichmadethisworkpossible.Finally,Iamgratefultomyfamilyfortheirunconditionalsupportthroughoutthisexperience. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 1.1BackgroundInformation ............................. 14 1.2TrajectoryOptimizationofLow-ThrustOrbitTransfers ............. 16 1.3ResearchMotivationandObjectives ....................... 20 1.4OrganizationofDissertation ........................... 23 2MATHEMATICALBACKGROUND .......................... 25 2.1Continuous-TimeBolzaOptimalControlProblem ............... 25 2.1.1First-OrderNecessaryOptimalityConditions ............... 26 2.1.2Pontryagin'sPrinciple .......................... 29 2.2NumericalMethodsforSolvingOptimalControlProblems ........... 30 2.3KeyConceptsforNumericallySolvingOptimalControlProblems ....... 31 2.3.1NumericalSolutionofDierentialEquations ............... 31 2.3.1.1Time-marchingmethods .................... 32 2.3.1.2Collocationmethods ...................... 34 2.3.2NonlinearOptimization .......................... 37 2.4CommonDirectMethodsforSolvingOptimalControlProblems ........ 38 2.4.1DirectShootingMethod ......................... 39 2.4.2DirectMultipleShootingMethod .................... 40 2.4.3DirectCollocationMethod ........................ 41 3LEGENDRE-GAUSS-RADAUCOLLOCATIONANDADAPTIVEMESHREFINEMENT 43 3.1NotationandConventions ............................ 43 3.2Continuous-TimeBolzaOptimalControlProblem ............... 44 3.2.1TransformedContinuous-TimeBolzaOptimalControlProblem ..... 45 3.2.2MultipleIntervalTransformedContinuous-TimeOptimalControlProblem 46 3.3Variable-OrderLegendre-Gauss-RadauCollocationMethod ........... 46 3.4phAdaptiveMeshRenement .......................... 49 3.4.1RelativeErrorEstimateinEachMeshInterval .............. 49 3.4.2ReningtheMeshUsingthephMethod ................ 50 3.4.3SummaryofphMeshRenementMethod ................ 51 3.5OrbitTransferExampleandComparison .................... 52 5

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3.5.1ProblemDescription ........................... 52 3.5.2InitialGuessGeneration ......................... 55 3.5.3ResultsandDiscussion .......................... 56 3.6NumericalSolutionofOptimalControlProblem ................ 58 4DESCRIPTIONOFORBITTRANSFERPROBLEM ................. 63 4.1EquationsofMotion ............................... 63 4.2BoundaryConditionsandPathConstraints ................... 65 4.3ModelforSolarElectricPropulsion ....................... 67 4.3.1DenitionofaShadowRegion ...................... 67 4.3.2EventConstraintsDeningEntranceandExitofaShadowRegion ... 69 5ORBITTRANSFERSTUDY1:NON-ECLIPSING .................. 72 5.1OptimalControlProblem ............................ 72 5.2InitialGuessGeneration ............................. 72 5.3SearchMethodtoAssistinObtainingGloballyOptimalSolution ........ 74 5.4ResultsandDiscussion .............................. 75 5.4.1OptimalGTOtoGEOTransfers ..................... 76 5.4.2OptimalLEOtoGEOTransfers ..................... 78 5.4.3OptimalSSTOtoGEOTransfers .................... 79 5.4.4EstimationofMinimum-TimeTransferTime .............. 81 5.4.5EstimationofMinimum-TimeFinalTrueLongitude ........... 82 5.4.6CoecientofDeterminationtoAssessQualityofRegressions ..... 83 5.4.7Post-OptimalityAnalysis ......................... 84 6ORBITTRANSFERSTUDY2:ECLIPSING ..................... 111 6.1Multiple-PhaseOptimalControlProblem .................... 111 6.2InitialGuessGeneration ............................. 112 6.3ResultsandDiscussion .............................. 114 6.3.1OptimalGTOtoGEOTransferwithEclipsing .............. 114 6.3.2OptimalLEOtoGEOTransferwithEclipsing .............. 117 6.3.3OptimalSSTOtoGEOTransferswithEclipsing ............. 118 7CONCLUSIONS .................................... 139 7.1SummaryofNon-EclipsingOrbitTransferResearch ............... 139 7.2SummaryofEclipsingOrbitTransferResearch ................. 139 7.3FutureWork ................................... 140 7.3.1Third-BodyGravitationalPerturbationsandSolarRadiationPressure . 140 7.3.2OptimalityofEclipsingOrbitTransfers ................. 141 APPENDIX:RELATIONSHIPBETWEENCOSTATEINMODIFIEDEQUINOCTIALELEMENTSANDCLASSICALORBITALELEMENTS ....................... 142 REFERENCES ........................................ 145 6

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BIOGRAPHICALSKETCH ................................. 152 7

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LISTOFTABLES Table page 2-1Adams-Bashforthcoecientsfororderp=0,1,2,3. ................. 34 2-2Adams-Moultoncoecientsfororderp=0,1,2,3. ................. 34 3-1Physicalconstantsforexampleorbittransfer. ..................... 58 3-2Meshrenementiterationresultsforph-(4,6)algorithm. ............... 58 3-3Meshrenementiterationresultsforvarioushandphmethodswithrelativeerrortolerancesetto10)]TJ /F6 7.97 Tf 6.59 0 Td[(7. ................................. 59 3-4SolutionsummaryforLGR-ph,SOS,andPSOPT. ................. 59 4-1Physicalproperties. .................................. 70 5-1Orbittypes. ...................................... 87 5-2Regressioncoecientsfor^tf. ............................. 100 5-3Regressioncoecientsfor^Lf. ............................. 100 5-4CoecientofdeterminationR2for^tfand^Lf. .................... 101 5-5GTOtoGEOpost-optimalityresultsforT=m0=2.5010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2. ....... 102 5-6LEOtoGEOpost-optimalityresultsforT=m0=4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2. ........ 102 5-7SSTOtoGEOpost-optimalityresultsforT=m0=3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2. ....... 103 6-1Missioncharacteristics. ................................ 121 6-2NumericalresultsforSSTOtoGEOtransferswithvariousdeparturedates. ..... 122 8

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LISTOFFIGURES Figure page 2-1FamilyofLegendre-Gaussorthogonalcollocationpoints. ............... 42 3-1Examplelow-thrusttransferoptimaltrajectory. .................... 59 3-2Meshrenementiterationhistoryforph-(4,6)algorithm. ............... 60 3-3Examplelow-thrusttransferstatevariablecomparison. ................ 61 3-4Examplelow-thrusttransfercontrolvariablecomparison. ............... 62 4-1Penumbraconegeometry. ............................... 71 4-2Projectedspacecraftlocationgeometry. ........................ 71 5-1Locallyoptimalsolutionsobtainedwithfreeandboundednaltruelongitudevs.Lf=(2)forGTOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,3000s). ........................................... 87 5-2OptimaltrajectoryforGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). .................................... 88 5-3Semi-majoraxistimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. ................................... 89 5-4EccentricitytimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. ...................................... 89 5-5InclinationtimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. ...................................... 90 5-6ControltimehistoryforGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). .................................... 91 5-7ControlforselectedsegmentsofGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). ................................. 92 5-8SelectedsegmentsofGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). .................................... 93 5-9OptimaltrajectoryforLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s). .................................... 94 5-10Semi-majoraxistimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s. ................................... 95 5-11InclinationtimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s. ...................................... 95 9

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5-12ControltimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s. ......................................... 96 5-13LongitudeoftheascendingnodetimehistoryforLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s). ............................. 97 5-14ControlforselectedsegmentsofLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s). ................................. 97 5-15OptimaltrajectoryforSSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). .................................... 98 5-16Semi-majoraxistimehistoryforSSTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. ................................... 99 5-17EccentricitytimehistoryforSSTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. ................................... 99 5-18InclinationtimehistoryforSSTOtoGEOtransferusingvariousvaluesofT=m0andIsp=3000s. ................................... 100 5-19ControltimehistoryforSSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). .................................... 101 5-20SSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s)duringtherstfeworbitalrevolutions. .............................. 102 5-21SSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s)duringthelastfeworbitalrevolutions. .............................. 103 5-22tfvs.Isp. ........................................ 104 5-23RegressioncoecientAvs.T=m0. .......................... 105 5-24RegressioncoecientBvs.T=m0. .......................... 106 5-25RegressioncoecientCvs.T=m0. .......................... 107 5-26Lf=(2)vs.tf. .................................... 108 5-27RegressioncoecientDvs.Isp. ............................ 109 5-28RegressioncoecientEvs.Isp. ............................ 110 6-1TrajectoryforGTOtoGEOtransferwitheclipsing. .................. 122 6-2OrbitalelementtimehistoryforGTOtoGEOtransferwitheclipsing. ........ 123 6-3ControltimehistoryforGTOtoGEOtransferwitheclipsing. ............ 124 6-4GTOtoGEOtransferwitheclipsingduringtherstfeworbitalrevolutions. ..... 125 10

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6-5uvs.=(2)duringregionofGTOtoGEOtransferwitheclipsingwhenur0. .. 125 6-6SegmentofGTOtoGEOtransferwitheclipsingwhenuhbecomes-1duringtheGTOtoGEOtransferwitheclipsing. ......................... 126 6-7GTOtoGEOtransferwitheclipsingduringthenalfeworbitalrevolutions. ..... 127 6-8TrajectoryforLEOtoGEOtransferwitheclipsing. .................. 128 6-9OrbitalelementhistoryforLEOtoGEOtransferwitheclipsing. ........... 129 6-10ControltimehistoryforLEOtoGEOtransferwitheclipsing. ............. 130 6-11LEOtoGEOtransferwitheclipsingwhen=[67,72]. ............... 131 6-12TrajectoryforSSTOtoGEOtransferwithVernalEquinoxdeparture. ........ 132 6-13TrajectoryforSSTOtoGEOtransferwithSummerSolsticedeparture. ....... 132 6-14TrajectoryforSSTOtoGEOtransferwithAutumnalEquinoxdeparture. ...... 133 6-15TrajectoryforSSTOtoGEOtransferwithWinterSolsticedeparture. ........ 133 6-16Semi-majoraxistimehistoryforSSTOtoGEOtransferwithvariousdepartures. .. 134 6-17EccentricityandinclinationtimehistoryforSSTOtoGEOtransferwithVernalEquinoxdeparture. ....................................... 135 6-18ControltimehistoryforSSTOtoGEOtransferwithVernalEquinoxdeparture. ... 136 6-19uvs.=(2)forSSTOtoGEOtransferduringtherstfeworbitalrevolutions. .. 137 6-20SSTOtoGEOtransferwithVernalEquinoxdepartureduringtherstfeworbitalrevolutions. ....................................... 138 6-21uvs.=(2)forSSTOtoGEOtransferduringthenalfeworbitalrevolutionswithVernalEquinoxdeparture. ............................... 138 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTRAJECTORYOPTIMIZATIONOFLOW-THRUSTORBITTRANSFERSByKathrynFrancesGrahamDecember2015Chair:AnilV.RaoMajor:AerospaceEngineeringSincetheworld'srstarticialsatellite,theSputnik1,completeditsrstorbitaroundtheEarth,wehavepersistentlydevelopednewtechnologywithanemphasisoncost-eectivewaystoplaceobjectsinspace.NASAhasidentiedlow-thrustpropulsionasacriticaltechnologythatwouldgreatlyreducethecostoforbitalinsertionofspacecraft.Todate,low-thrustpropulsionhasbeenstudiedinextensivedetailandusedmostlyforinterplanetarymissions.Littleattention,however,hasbeenappliedtotheusageoflow-thrusttechnologyfororbitalinsertion.Thisresearchseekstodeviseapproachesforobtaininghigh-accuracyminimum-timetrajectoriesforvariousEarth-orbittransfersusingtwodierenttypesoflow-thrustpropulsionmodels.Therstpropulsionmodelassumesthespacecrafthasanenergysourcethroughouttheentiretransfer.Thisallowsforcontinuousthrustthroughouttheentiretransfer.Thistypeofpropulsionmodelleadstoanon-eclipsingorbittransfer.ThesecondpropulsionmodelassumesthespacecrafthasanenergysourceonlywhenithasdirectlineofsighttotheSun.Inotherwords,thespacecraftwillhavezerothrustwhentravelingthroughtheEarth'sshadow.Thistypeofpropulsionmodelleadstoaneclipsingorbittransfer.Forthenon-eclipsingorbittransferstudy,thetrajectoryoptimizationproblemwasposedasasingle-phaseoptimalcontrolproblem.Aninitialguessfortheoptimalcontrolproblemwasobtainedbysolvingasequenceofmodiedoptimalcontrolproblemswherethenaltruelongitudewasconstrainedandthemeansquaredierencebetweenthespeciedterminalboundaryconditionsandthecomputedterminalconditionswasminimized.Itwasfoundthat 12

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solutionstotheminimum-timelow-thrustoptimalcontrolproblemwereonlylocallyoptimalinthatthesolutionhasessentiallythesamenumberoforbitalrevolutionsasthatoftheinitialguess.Asearchmethodwasthendevisedthatenablescomputationofsolutionswithanevenlowercostwherethenaltruelongitudeisconstrainedtobedierentfromthatobtainedintheoriginallocallyoptimalsolution.Anumericaloptimizationstudywasthenperformedtodetermineoptimaltrajectoriesandcontrolinputsforarangeofinitialthrustaccelerationsandconstantspecicimpulses.Thekeyfeaturesofthesolutionsweredetermined,andrelationshipswereobtainedbetweentheoptimaltransfertimeandtheoptimalnaltruelongitudeasafunctionoftheinitialthrustaccelerationandspecicimpulse.Adetailedpost-optimalityanalysiswasperformedtoverifythecloseproximityofthenumericalsolutionstothetrueoptimalsolution.Fortheeclipsingorbittransferstudy,thetrajectoryoptimizationproblemwasposedasamultiple-phaseoptimalcontrolproblemwherethespacecraftcanthrustonlyduringphaseswhereithaslineofsighttotheSun.Eventconstraints,basedonthegeometryofapenumbrashadowregion,wereenforcedbetweenthethrustphasesanddeterminetheamountoftimespentinaneclipse.Aninitialguessgenerationmethodwasdevelopedthatconstructsanintelligentguessbysolvingaseriesofsingle-phaseoptimalcontrolproblemsandanalyzingtheresultingtrajectorytoapproximatewherethespacecraftentersandexitstheEarth'sshadow.Todemonstratetheeectivenessoftheapproachdeveloped,anoptimaltransfertrajectorywascomputedforanEarth-orbittransferfoundintheliterature.Inadditiontothecomparisoncasestudied,twoadditionalorbittransferswereexaminedandsolutionswerepresentedforvariousdeparturedates. 13

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CHAPTER1INTRODUCTION 1.1BackgroundInformationTheUCSSatelliteDatabaselistsmorethan1,200operationalsatellitescurrentlyinorbitaroundtheEarth.AsofJune2014,512ofthesesatellitesareoperatedbytheUnitedStatesbelongingtothecivil,commercial,government,andmilitarysectors.Satelliteshelpusobserveandprovidevaluableinformationabouttheworldwelivein.Television,telephone,andradiotransmissionscanbesentanywhereintheworldviacommunicationsatellites.WerelyonnavigationsatellitestotellusourpositiononorabovetheEarth.Reconnaissancesatellitesobserveothercountries,monitoringhostilemilitantgroupsanddetectingmissilelaunches.Furthermore,satellitesareextensivelyusedtostudytheEarth.Weathersatellitescanbeusedtomonitorcloudmovementallowingmeteorologiststobetterpredictweathereventsandviolentstorms.Othersatellitesallowustoobservedangerousorinaccessiblelocationstostudyanimalmigrationpatterns,theeectsofdeforestation,orhelpsearchandrescueteamslocateaircrafts,ships,orindividualsneedingassistance.Asasociety,werelyheavilyonsatellitestonotonlykeepusinformed,buttokeepussafe.Undoubtedly,thecostofplacingasatelliteinanoperationalorbitisveryexpensive.First,anexpendablelaunchvehicledeliversthesatellitefromthesurfaceoftheEarthtoaparkorbit.Traditionally,aparkorbitiseitheralow-Earthorbit(LEO)orageostationarytransferorbit(GTO).SpaceXcharges$61.2milliontolaunchaFalcon9fromCapeCanaveralintheyear2015.TheFalcon9iscapableofdeliveringupto13,150kgtoLEOor4,850kgtoGTO[ 1 ].Giventhatthepredictedaveragemasspercommercialsatelliteatlaunchin2015is4,380kg[ 2 ],onlyonesatellitecanbeplacedintoGTO.Oncethesatelliteissuccessfullyinsertedintoaparkorbit,thesatellitemustthentraveltothedesiredoperationalorbit.Finalorbitinsertionistypicallycarriedoutbyachemicalpropulsionsystemknownasaliquidapogeeengine(LAE)thatappliesaseriesofimpulsiveburns.Consequently,thesatellitemassatlaunchmustinclude 14

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thenecessaryfueltoachieveorbitinsertionandadditionalfuelforstationkeepingwhichinturnlimitsthepayloadofthesatellitethatcanbedeliveredtothedesiredoperationalorbit.Low-thrustelectricpropulsion(EP)hasbeenidentiedasapotentiallycheaperalternativetothetraditionalchemicalpropulsionmethods.EPischaracterizedasapropulsionsystemthatuseselectricitytoincreasetheexhaustvelocityofapropellant.ThesetypesofthrusterswererstusedoperationallybyRussiatoperformstation-keepingdutiesonacommunicationssatellitelaunchedin1971.Thepropellantusedinthesethrustersistypicallyxenonbecauseitissafetohandle,canbeeasilystoredinsmalltanksathighpressure,andhasalargemasscomparedtootherinertgasesallowingthethrustertogeneratehigherthrustsforagiveninputpower.Thesethrusterstypicallyhavespecicimpulsesofthousandsofsecondstotensofthousandsofseconds,producethrustontheorderofafractionofanewton,andtheiroperationalpowerisintherangeofhundredsofwattstotensofkilowatts.Recently,NASAbegantheSolarElectricPropulsion(SEP)projectwhichseekstodevelopcriticaltechnologiesinaneorttoreducecostsandextendthecapabilitiesofsatellites[ 3 ].Electricallypropelledsystemspoweredbysolarpanelswilluseconsiderablylesspropellantthanthetraditionalchemicalpropulsionsystems.Inmanycases,thepropellantmassneededfororbitinsertionisgreaterthanthedrymassofthesatellite[ 2 ].ByutilizingSEPtechnologyfororbitinsertion,thelaunchmassofthesatellitecanbegreatlyreducedsinceelectricpropulsionrequiressignicantlylesson-boardfuelthanchemicalpropulsion.Bykeepingthemassofthesatellitelow,asmallerlaunchvehiclecanbeusedtodeliverthesatellite.Alternatively,twosatellitescanbelaunchedtogetherorthepayloadtotheoperationalorbitcanbeincreased,extendingthecapabilitiesofthesatellite.WhileSEPincreasesthetimerequiredfororbitinsertionfromdaystomonthscomparedtotraditionalpropulsionsystems,theprojectedsavingsinlaunchingsmallersatellitessignicantlyoutweighstheadditionaltimerequiredforachievingthenalorbitalposition[ 2 ]. 15

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1.2TrajectoryOptimizationofLow-ThrustOrbitTransfersWiththegrowinginterestofEPfororbitalmaneuvers,trajectoryoptimizationoflow-thrustorbittransfershasbecomeapopularresearchtopic.Insteadofusingaseriesofimpulsiveburns,orbittransfersperformedviaEParecharacterizedbycontinuousthrust.Inordertoobtaintheoptimaltrajectoryitisthennecessarytosolveacontinuous-timeoptimalcontrolproblemwhichisqualitativelydierentfromtheimpulsivecase.Thegoaloftheassociatedoptimalcontrolproblem(orthetrajectoryoptimizationproblem)istodeterminethestateandcontrolofthespacecraftthatoptimizesaspeciedperformanceindex,suchasighttimeorfuelexpenditure,subjecttodynamicconstraints,boundaryconditions,andpathconstraints.Solvinganoptimalcontrolproblemisnotaneasytask.Analytictechniquesforsolvingoptimalcontrolproblemsinvolvendingthenecessaryconditionsforoptimality,usuallybymeansofcalculusofvariations.Exceptforspecialcases,mostoptimalcontrolproblemscannotbesolvedusingapurelyanalyticapproach.Thus,optimalcontrolproblemsaremostcommonlysolvedusingnumericalmethods.Numericalmethodsforapproximatingsolutionstooptimalcontrolproblemsfallintotwobroadcategories:indirectanddirectmethods[ 4 { 6 ].Inanindirectmethod,calculusofvariationstheory[ 7 , 8 ]isappliedtodeterminetherst-ordernecessaryoptimalityconditionsforanoptimalsolution.Byapplyingcalculusofvariationstheory,theoptimalcontrolproblemistransformedintoaHamiltonianboundaryvalueproblem(HBVP).Themainbenetofusinganindirectmethodisthatahighlyaccurateapproximationtotheoptimalsolutioncanbeobtainedandtheproximityofthatapproximationtotheoptimalsolutioncanbeestablished.Unfortunately,anindirectmethodhasseveraldisadvantages.Inparticular,toimplementanindirectmethod,thecomplicatedrst-ordernecessaryoptimalityconditionsmustbederived.Inaddition,toachieveconvergencewithanindirectmethod,averygoodinitialguessoftheunknownboundaryconditionsmustbeprovided.Inparticular,determininganinitialguessforthecostate,anon-intuitiveandnon-physicalquantity,isespeciallydicult.Anotherdisadvantageofanindirectmethodisthatwhenevertheproblemismodied,suchasaddingor 16

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removingaconstraint,thenecessaryconditionsmustbereformulated.Furthermore,itcanbeextremelychallengingtosolveanHBVPoveralargetimeinterval.Theearliestresearchinsolvinglow-thrustorbittransfersutilizedthebranchofmathematicscalledcalculusofvariationstosolveforsimplecontinuousthrusttransfers[ 9 { 24 ].Oneoftheearliestpioneersintheareaoflow-thrustorbittransferswasEdelbaum[ 10 , 11 ],whoderivedananalyticexpressionforthemaximumchangeininclinationbetweentwogivencircularorbitsusingcontinuous,constantaccelerationandaxedtransfertime.Edelbaumalsoderivedananalyticexpressionforthetotalchangeinvelocity(V)requiredtocarryoutthetransferbetweeninclinedcircularorbitsusingaconstantyawprole.WieselandAlfano[ 16 ]createdamissionplanningtoolformany-revolutionorbittransfersbyextendingEdelbaum'stheorytoallowforvariationintheyawangleovereachrevolution.TheanalyticsolutionsobtainedwerepresentedingraphicalformmakingitpossibletoeasilyobtaintheVrequiredforanycircle-to-circletransferofinterest.AlfanoandThorne[ 17 ]furtherexploitedoptimalcontroltheoryforcoplanarcircle-to-circleorbittransfersdevelopingatoolfororbitalmissionplannersthatwouldrelatevehicledesignparameterstoorbitdesignparameters.Inaneorttominimizefuelconsumption,Matogawa[ 18 ]introducedcoastingphasesineachorbitalrevolution,thenoptimizedforminimum-timeovereachrevolution.BreakwellandChanal[ 19 ]evaluatedtheVrequiredforminimum-fueltransfersfromlowparkingorbitswithinclinationsofzeroand28.5degtoanequatorialgeosynchronousorbit.AnothersignicantcontributiontotheliteraturewasmadebyFernandes[ 20 { 22 ]whoformulatedtheorbittransferasaMayeroptimalcontrolproblem,appliedthePontryaginmaximumprinciple[ 25 , 26 ]todeterminetheoptimalthrustacceleration,andobtainedsimpleanalyticsolutionsforlong-timetransfersusingHori'smethodforgeneralizedcanonicalsystems.InRef.[ 21 ],Fernandesconsideredtransfersbetweenquasi-circularorbitsaroundanoblateplanetandlaterextendedthetechniqueinRef.[ 22 ]tostudytransfersbetweenneighboringellipticnon-equatorialorbits.Employingorbitalaveraging,Kechichian[ 23 ]derivedthenecessaryconditionsforoptimalityusingasetofnon-singularequinoctialelementsforaminimum-fuel,time-xed,thrust-boundedorbittransferwith 17

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continuouslyvaryingspecicimpulse.Kechichian[ 24 ]laterreformulatedEdelbaum'stime-xedlow-thrusttransferbetweentwoinclinedorbitstoaminimum-timetransfer[ 10 ],derivingasingleanalyticexpressionforthemaximumchangeinorbitalinclinationthatisuniformlyvalidforalltransfersbetweentwocircularorbits.Inadirectmethod,thecontinuousfunctionsoftime(thatis,thestateand/orthecontrol)oftheoptimalcontrolproblemareapproximatedandtheproblemistranscribedintoanite-dimensionalnonlinearprogrammingproblem(NLP).TheNLPisthensolvedusingwelldevelopedalgorithmsandsoftware[ 27 { 29 ].Inthecasewhereonlythecontrolisapproximated,themethodiscalledacontrolparameterizationmethod.Whenboththestateandcontrolareapproximated,themethodiscalledastateandcontrolparameterizationmethod.Directmethodsovercomemanyofthedisadvantagesofindirectmethods.Themainadvantageofadirectmethodisthattherst-ordernecessaryoptimalityconditionsdonotneedtobederived.Sincetheoptimalityconditionsarenotderived,theoptimalcontrolproblemcanbemodiedrelativelyeasily.Furthermore,aninitialguessoftheunknown,non-intuitivecostateisnotneeded,nordoestheinitialguessneedtobeasgoodasthatrequiredbyanindirectmethodinordertoachieveconvergence.Ontheotherhand,directmethodsareoftenlocallyoptimalandarethusnotasaccurateasindirectmethods.Verifyingtheoptimalityofasolutionrequiresmorework,sincemanydirectmethodsdonotprovideanyinformationaboutthecostate.HargravesandPariswereamongthersttoemployadirectmethodtosolvetrajectoryoptimizationproblems[ 30 ].Theyparameterizedthestateandcontrolusingpiecewisepolynomialsandusedasequentialquadraticprogramming(SQP)methodtosolvetheresultingNLP.EnrightandConway[ 31 ]developedaRunge-Kuttaparallel-shootingmethodasawaytodecreasethesizeoftheNLP.Implementingtheparallel-shootingmethod,ScheelandConway[ 32 ]useddirecttranscriptionwithcontrolparameterizationtosolvecontinuous,very-low-thrustorbittransfersfromplanarlow-Earthorbit(LEO)togeosynchronousorbit(GEO)andGEOtoanouterorbittransferofspeciedradius.Whilethedimensionsizeoftheproblemwasreducedsomewhat,additionalNLPdesignvariableswererequiredtodetermine 18

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thecontrolsteering.Betts[ 33 ]demonstratedthecapabilityofadirectcollocationmethodtosolveaminimum-fuellow-thrustnear-polarEarth-orbittransferwithover578revolutions,whileRossetal.[ 34 ]developedananti-aliasingmethodutilizingdirectcollocationtoobtainsolutionstosimplelow-thrusttrajectoryoptimizationproblems.HigherordercollocationasderivedbyHerman[ 35 ]wasusedbyHermanandSpencer[ 36 ]toexaminevariousLEOtoGEOtransferswitharangeofinitialthrustaccelerationvalues.Yang[ 37 ]utilizedcontrolparameterizationandadirectmultipleshootingmethodtosolveawiderangeofEarth-orbittransfers.Someauthorssoughtwaystocombinethebenetsofbothindirectanddirectmethodsbycollocatingthecostateinadditiontothestateand/orcontrol.Employingorbitalaveraging,Kluever[ 38 ]usedcostatenodalvaluesasthedesignparametersinsteadofdirectlyparameterizingthethrustandspecicimpulseproles,resultinginareducedSQPdesignspace.Usingasimilarsetofhybridcontrolvariablesarisingfromthederivationofthecostates,FalckandDankanich[ 39 ]usedcollocationandorbitalaveragingtocalculateGTOtoGEOandLEOtoGEOtransfers.Whilemanyauthorshaveproposedtechniquestoaddresstheissueofsolvinglow-thrusttrajectoryoptimizationproblems,onlyafewhavesolvedfortrajectoriesthatincludeeclipseconstraints.Eclipseconstraintsarisewhensolarcellsarethemainsourceofpowerforthepropulsionsystem.Theresultingoptimalcontrolproblemisfundamentallydierentasitcontainsmultiple-phasesandisoftenextremelydiculttosolve.GeroyandEpenoy[ 40 ]utilizedanorbitalaveragingtechniquetoexamineminimum-timeandfuel-savingstrategiesundertheinuenceofenvironmentalconstraintssuchastheshadoweect,whichwasdeterminedusingEdelbaum'scylindricalshadowmodel[ 13 ].Similarly,Kluever[ 41 ]employedadirectoptimizationandorbitalaveragingapproachtoobtainnear-optimalEarth-orbittransferswithEarthshadowingeects.InKluever'sapproach,thethrustdirectionwasdeterminedbyextremalcontrollawsthatmaximizetheratesofchangesintheorbitalelements.Theshadowexitandentrancelongitudesforagivenorbitandepochdatewerecomputedassumingacylindricalshadowmodel.Furthermore,Kechichian[ 42 { 44 ]analyzedavarietyoforbittransfers 19

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allinthepresenceofacylindricalEarthshadow.In[ 42 ],Kechichianusedapurelyanalyticapproachtodescribetheproblemoftangentialthrustingalongsmall-to-moderateeccentricityorbits.Usingsimplesteeringlawsconsistingofpiecewiseconstantyawangle,Kechichianalsoaddressedlow-thrustzero-eccentricity-constrainedorbitraisinginacircularorbit[ 43 ]andexaminedlow-thrustinclinationchangesinanear-circularorbit[ 44 ].FerrierandEpenoy[ 45 ]lateremployedashootingalgorithmtosolveneighboringproblemsthatwereconstructedfromasimpliedcylindricaleclipsemodel.Theneighboringproblemsdevelopedavoidedthediscontinuousbehaviorassociatedwiththespacecraftenteringandexitingashadowregion.Morerecently,Betts[ 46 ]formulatedanapproachthatsolvesaseriesoflarge-scalemultiple-phaseoptimalcontrolproblemswithaburn-coast-burnphasesequence.InBetts'approach,theshadowregionisdenedusingpenumbralconegeometryandtheinitialguesswasconstructedbyarecedinghorizonalgorithm. 1.3ResearchMotivationandObjectivesFromtheliterature,afewshortcomingsareidentiedandprovidethemotivationforthisresearch.First,manyauthorsemployanorbitalaveragingtechniquetoinsurerapidcalculationofthetrajectories.Whileorbitalaveragingreducesthecomputationalloadofpropagatingtheorbitbyutilizingeachelement'smeantimerate-of-change[ 47 ],theaccuracyoftheobtainedsolutionsuers.Inadditiontoorbitalaveraging,manyauthorshavemadeassumptionsabouttheproblemormodiedtheproblemsuchthatitiseasiertosolveandnear-optimalsolutionsareobtained.Forexample,manyofthegeometricshadowmodelsseenintheliteratureassumeacylindricalshadowregion.Cylindricalmodelsareagreatapproximationandworkwellforshorttrajectories,buttheoptimaltrajectoryisexpectedtotakemorethanafewdayswithmultiplerevolutionsthattravelthroughashadowregion.SincethepositionoftheSunrelativetotheEarthisconstantlymoving,amoreadvancedshadowmodelthatcanaccuratelydeterminetheshadowregionasafunctionoftimeisdesirable.Theoverallgoalofthisdissertationistodeveloparobustapproachforsolvinghigh-accuracylow-thrustorbittransfers.Twoapproachesaredeveloped,oneforsolvingnon-eclipsingorbit 20

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transfersandanotherforsolvingeclipsingorbittransfers.Anon-eclipsingorbittransferassumesthespacecrafthaspowerthroughouttheentiretransferandrequiressolvingasingle-phaseoptimalcontrolproblem.AneclipsingorbittransferassumesthespacecrafthaspoweronlywhenthespacecrafthaslineofsighttotheSunandinthisresearchwillrequiresolvingamultiple-phaseoptimalcontrolproblem.Thehigh-accuracysolutionswillbeobtainedbyemployingavariable-orderLegendre-Gauss-Radau(LGR)orthogonalcollocationmethod[ 48 , 49 ].OneoftheadvantagesofusingLGRcollocationisthatthesupportpointsusedarethepointsobtainedfromtherootsofaLegendrepolynomialanditsderivatives.ThesepointsareknowntohaveadistributionthatminimizesRungephenomenon.Furthermore,thedirectcollocationapproachchoseninthisresearchdoesnotrequiretherst-orderoptimalityconditions.Inaddition,solutionstotheproblemsconsideredhereareobtainedwithouthavingtoreplacetheequationsofmotionwithaveragedapproximationsovereachorbitalrevolutionsuchasinusinganorbitalaveragingtechnique.Thisresearchalsoemploysaphmeshrenementmethod,orvariable-ordermethod.Usingthismethod,thesolutiondomainisdividedintoamesh,andthedegreeoftheapproximatingpolynomial(thatis,thenumberofLGRcollocationpoints)isallowedtovaryineachinterval.Usingaph-method,itispossibletodividetheproblemintointervalssuchthatthesolutionineachintervalissmooth.Convergenceisthereforeachievedbyeitherincreasingthedegreeoftheapproximatingpolynomialinanintervalordividinganintervalintosub-intervals.Inthismanner,itispossibletoachieveahighaccuracysolutionwhilekeepingthesizeoftheNLPsmallerthanwhatmightbepossiblewithapurelyh-method.(Inanh-method,convergenceisachievedbydividinganintervalintosub-intervalsandassigningaxednumberofcollocationpointstoeachsub-interval.)Therstphaseofthisresearchexaminesnon-eclipsingorbittransfers.Theapproachforsolvingthesecontinuous,constantmagnitudethrusttransfersincludesthedevelopmentofaninitialguessgenerationmethodandasearchmethodtodeterminethesolutionthathasthelowestcostamidarangeoflocallyoptimalsolutions.Threenon-trivialorbittransferswere 21

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consideredandtheresultsforawiderangeofinitialthrustaccelerationandspecicimpulsevaluesarepresented.Regressionanalyseswerethenperformedtodeterminethetransfertimeasafunctionoftheinitialthrustaccelerationandthespecicimpulse,andtodeterminethenaltruelongitudeasafunctionofthetransfertime.Theobtainedregressionsareusefulforpredictingthenaltimeandnalnumberoforbitalrevolutionsforvaluesofinitialthrustaccelerationandspecicimpulsethatwerenotconsidered.Finally,tovalidatethesolutionsobtained,apost-optimalityanalysiswasperformed.Inthesecondphaseofthisresearch,eclipsingorbittransferswereconsidered.Ashadowmodeliscreatedbasedonthepenumbraconicalprojections[ 50 ].Aninitialguessmethodwasdevelopedthatconstructstheguessbysolvingaseriesofsingle-phaseoptimalcontrolproblemsandanalyzingtheresultingtrajectorytoapproximatewherethespacecraftentersandexitstheEarth'sshadow.Themultiple-phaseoptimalcontrolproblemwasformulatedsuchthatitconsistsofthrustphasesonly.Eventconstraintsareemployedthataccountforthechangeintheorbitfromtheentrancetotheexitofeachshadowregion,wheretheshadowregionsaredeterminedfromthegeometryoftheproblem.Theadvantageofutilizingtheseeventconstraintsisthattheconstraintseliminatetheneedtoincludecoastphasesandconsequentlykeepsthesizeofthemultiple-phaseoptimalcontrolproblemdown.Optimaltrajectorieswerecomputedforthesamethreeorbittransferspresentedintherstphaseoftheresearch.Todemonstratetheeectivenessoftheapproachdeveloped,threeorbittransfersweresolvedwithavarietyofdeparturedates.Oneoftheorbittransferexampleswastakenfromtheliterature,whichallowedforacomparisonofatraditionalorbitalaveragingapproachandtheapproachdevelopedinthisresearch.Tosummarize,thecontributionsofthisresearchareasfollows.First,theframeworkforsolvingnon-eclipsinglow-thrustorbittransfersispresented.Theframeworkincludesaninitialguessgenerationmethodandasimplesearchmethodforobtainingthebestlocallyoptimalsolution.Todemonstratetheapproach,solutionsarepresentedforthreedierentorbittransfersusingavarietyofinitialthrustaccelerationandspecicimpulsevalues.Regression 22

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analysesarepresentedthatcanbeusedtopredictthenaltimeandnalnumberoforbitalrevolutionsforgivenvaluesoftheinitialthrustaccelerationandthespecicimpulse.Thepost-optimalityanalysisperformedveriesthatthesolutionsobtainedareinfactveryclosetooptimal.Second,anapproachforsolvingeclipsinglow-thrustorbittransferispresented.TheapproachincludesaninitialguessgenerationmethodandanadvancedshadowmodeltopredictwhenthespacecraftentersandexitstheEarth'sshadow.Theproblemisformulatedasamultiple-phaseoptimalcontrolproblemconsistingofeventconstraintsthatareconstructedbasedontheshadowmodel.Theeectivenessoftheapproachisdemonstratedfordierentorbittransfersandavarietyofdeparturedates. 1.4OrganizationofDissertationThedissertationisorganizedasfollows.Chapter2providesthenecessarymathematicalbackgroundforunderstandingmethodsforsolvingoptimalcontrolproblemsnumerically.Afterreviewingbasicoptimalcontroltheory,indirectanddirectmethodsfornumericallysolvingoptimalcontrolproblemsarediscussed.Emphasisisthenplacedonadirectmethodapproachandtwoimportantnumericalconceptsforunderstandingdirectmethodsaregiven.Usingthenumericalconcepts,threecommondirectmethodapproachesareoutlined.Chapter3presentstheLGRcollocationmethodusedinthisresearch.ThechapteroutlinesthetranscriptionprocessfromageneraloptimalcontrolproblemtoanNLPusingthevariable-orderLGRcollocationmethod.Thephmeshrenementalgorithmemployedisalsosummarized.Finally,thechapterpresentsanexampleproblemtodemonstratetheeectivenessoftheLGRcollocationmethodforsolvingasimplelow-thrustorbittransfer.Thesolutionobtainedisthencomparedtootheravailablesolutionsthatwereobtainedusingwell-establisheddirectcollocationmethods.Finally,thegeneralpurposeoptimalcontrolsoftwareusedforsolvingtheorbittransfersinthisresearchisgiven.InChapter4,theminimum-timelow-thrustorbittransferproblemofinterestisdescribed.First,theequationsofmotionaregiven,followedbyadescriptionoftheboundaryconditions 23

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andpathconstraints.Then,amodelforsolarelectricpropulsiontobeusedintheeclipsingorbittransferstudyisdescribed.Afterdeningashadowregion,theeventconstraintsthatdeterminetheentranceandexitofashadowregionareformulated.Chapter5presentstherstorbittransferstudywhichexaminesnon-eclipsingtransfers.Inthischapter,thesingle-phaseoptimalcontrolproblemisstatedandtheinitialguessgenerationmethodisdescribed.Sincelocallyoptimalsolutionsaretypicallyobtainedwhenusingadirectmethodapproach,asearchmethodusedtoidentifytheapproximatelocationofthegloballyoptimalsolutionispresented.Theresultsforthreeorbittransfersarethenpresentedindetail.Theresultsincludeadescriptionofthekeyfeaturesofeachorbittransfer,followedbytheregressionanalysesperformedtoestimatethenaltransfertimeandthenalnumberoforbitalrevolutionsforagiveninitialthrustaccelerationandspecicimpulse.Finally,thepost-optimalityanalysisisdescribedandtheresultsarepresentedverifyingthatthenumericalsolutionsobtainedareincloseproximitytothetrueoptimalsolution.InChapter6,thesecondorbittransferstudywhichexaminedeclipsingorbittransfersispresented.First,themultiple-phaseoptimalcontrolproblemisstated,followedbyadescriptionoftheinitialguessgenerationmethod.Next,theapproachdevelopedisappliedtoatransferfoundintheliterature.Inaddition,twootherorbittransfersareexamined.Thekeyfeaturesoftheoptimaltrajectoriesobtainedareexploredindetail,includingthelocationsandtheeectsoftheshadowregionsonthetrajectories.Finally,Chapter7providesasummaryofthecontributionsofthisdissertation. 24

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CHAPTER2MATHEMATICALBACKGROUNDThegoalofthischapteristoprovideanoverviewofthedierentmethodsforsolvingtrajectoryoptimizationproblems.First,thegeneralformofanoptimalcontrolproblemisstated,followedbythederivationoftherst-ordernecessaryoptimalityconditionsusingbasicoptimalcontroltheory.Next,numericalmethodsforsolvingoptimalcontrolproblemsareoutlined.Thesemethodsaredividedintotwocategories:indirectanddirectmethods.Thefocusisthenplacedondirectmethodsandtwokeyconceptsfornumericallysolvingoptimalcontrolproblemsusingadirectapproacharediscussed.Therstconceptisthenumericalsolutionofdierentialequationsandthesecondconceptisnonlinearoptimization.Afteroutliningthesekeyconcepts,threecommondirectmethodapproachesusedtosolveoptimalcontrolproblemsaregiven. 2.1Continuous-TimeBolzaOptimalControlProblemTheobjectiveofanoptimalcontrolproblemistodeterminethestateandcontrolthatoptimizeaperformanceindexsubjecttodynamicconstraints,boundaryconditions,andpathconstraints[ 7 , 8 ].Withoutlossofgenerality,considerthefollowinggeneraloptimalcontrolproblem.Determinethestate,y(t)2Rny,thecontrol,u(t)2Rnuy,theinitialtime,t0,thenaltime,tf,ontheintervalt2[t0,tf]thatminimizethecostfunctional J=(y(t0),t0,y(tf),tf)+Ztft0\(y(t),u(t),t)dt(2{1)subjecttothedynamicconstraints dy(t) dt=a(y(t),u(t),t),(2{2)theboundaryconditions b(y(t0),t0,y(tf),tf)=0,(2{3)andtheinequalitypathconstraints c(y(t),u(t),t)0.(2{4) 25

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ThecostfunctionaliscomposedoftheMayercost,,andtheLagrangian,)]TJ /F1 11.955 Tf 6.78 0 Td[(.Thefunctions,)]TJ /F1 11.955 Tf 6.77 0 Td[(,a,b,andcaredenedbythefollowingmappings: :RnyRRnyR!R,)-839(:RnyRnuR!R,a:RnyRnuR!Rny,b:RnyRRnyR!Rnb,c:RnyRnuR!Rnc.(2{5) 2.1.1First-OrderNecessaryOptimalityConditionsUsingthebranchofmathematicscalledcalculusofvariations,therst-ordernecessaryoptimalityconditionsforanextremalsolutionoftheoptimalcontrolproblemarederived[ 7 , 8 ].First,theaugmentedcostfunctional,Ja,isformulatedbyaugmentingtheboundaryconditionsandconstraintsoftheoptimalcontrolproblemtothecostfunctionalas Ja=(y(t0),t0,y(tf),tf))]TJ /F13 11.955 Tf 11.95 0 Td[(Tb(y(t0),t0,y(tf),tf)+Ztft0\(y(t),u(t),t))]TJ /F13 11.955 Tf 11.96 0 Td[(T(t)_y(t))]TJ /F9 11.955 Tf 11.96 0 Td[(a(y(t),u(t),t))]TJ /F13 11.955 Tf 11.95 0 Td[(T(t)c(y(t),u(t),t)dt(2{6)where(t),,and(t)aretheLagrangemultipliersassociated,respectively,withthedynamicconstraintsofEq.( 2{2 ),theboundaryconditionsofEq.( 2{3 ),andthepathconstraintsofEq.( 2{4 ).Thequantity(t)iscalledthecostateortheadjoint.ItisnowconvenienttointroducetheaugmentedHamiltonianfunctionHwhichisdenedas H(y(t),u(t),(t),(t),t)=\(y(t),u(t),t)+T(t)a(y(t),u(t),t))]TJ /F13 11.955 Tf 9.99 0 Td[(T(t)c(y(t),u(t),t).(2{7) 26

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SubstitutingtheaugmentedHamiltonianintotheaugmentedcostfunctionalofEq.( 2{6 )yieldstheexpression Ja=(y(t0),t0,y(tf),tf))]TJ /F13 11.955 Tf 11.95 0 Td[(Tb(y(t0),t0,y(tf),tf)+Ztft0H(y(t),u(t),(t),(t),t))]TJ /F13 11.955 Tf 11.95 0 Td[(T(t)_y(t)dt.(2{8)Next,takingtherstvariationoftheaugmentedcostfunctionalwithrespecttoallthefreevariables(i.e.,y(t),(t),u(t),y(t0),y(tf),t0,tf,,(t)),weobtain Ja=@ @y(t0)x0+@ @t0t0+@ @y(tf)xf+@ @tftf)]TJ /F8 11.955 Tf 11.95 0 Td[(Tb)]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @y(t0)y0+@b @t0t0+@b @y(tf)yf+@b @tftf+H(tf)tf)]TJ /F13 11.955 Tf 11.96 0 Td[(T(tf)_y(tf)tf)-222(H(t0)t0+T(t0)_y(t0)t0+Ztft0@H @y(t)y(t)+@H @u(t)u(t)+@H @(t))]TJ /F4 11.955 Tf 13.29 .01 Td[(_yT(t)(t)+@H @(t)(t))]TJ /F13 11.955 Tf 11.95 0 Td[(T(t)_y(t)dt.(2{9)Itisnotedthaty(t0)isthedierencebetweeny(t),anextremalcurve,andy(t)evaluatedatt0,whereasy0isthedierencebetweeny(t)andy(t)evaluatedatthebeginningofeachcurve.Asimilarinterpretationistruefory(tf)andyf.Therst-orderapproximationstoy0andyfarethereforegivenas y0=y(t0)+_y(t0)t0,yf=y(tf)+_y(tf)tf.(2{10)Next,thetermT_yinEq.( 2{9 )isintegratedbypartstoyieldanexpressionintermsofy(t0),y(tf),andyasfollows Ztft0T_ydt=T(tf)y(tf))]TJ /F13 11.955 Tf 11.95 0 Td[(T(t0)y(t0)+Ztft0_Tydt.(2{11) 27

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SubstitutingEqs.( 2{10 )and( 2{11 )intoEq.( 2{9 ),therstvariationofJaisnowwrittenas Ja=@ @y(t0))]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @y(t0)+T(t0)y0+@ @y(tf))]TJ /F13 11.955 Tf 11.95 0 Td[(T@b @y(tf))]TJ /F13 11.955 Tf 11.95 0 Td[(T(tf)yf+@ @t0)]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @t0)-222(H(t0)t0+@ @tf)]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @tf+H(tf)tf)]TJ /F8 11.955 Tf 11.95 0 Td[(Tb+Ztft0@H @y(t)+_Ty(t)+@H @(t))]TJ /F4 11.955 Tf 13.29 0 Td[(_yT(t)(t)dt+Ztft0@H @u(t)u(t)+@H @(t)(t)dt.(2{12)BysettingthevariationofJaequaltozerowithrespecttoeachfreevariableintheoptimalcontrolproblem,asetofrst-ordernecessaryconditionsforoptimalityareobtained.Therst-orderoptimalityconditionsareexpressedas y:)]TJ /F4 11.955 Tf 11.57 2.66 Td[(_T(t)=@H @y(t):_yT(t)=@H @(t)u:@H @u(t)=0y0:T(t0)=)]TJ /F8 11.955 Tf 19.72 8.09 Td[(@ @y(t0)+T@b @y(t0)yf:T(tf)=@ @y(tf))]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @y(tf)t0:H(t0)=@ @t0)]TJ /F13 11.955 Tf 11.96 0 Td[(T@b @t0tf:H(tf)=)]TJ /F8 11.955 Tf 11.8 8.09 Td[(@ @tf+T@b @tf:b=0:i(t)=0whenci(y(t),u(t),t)<0i(t)0whenci(y(t),u(t),t)=0,(i=1,...,nc).(2{13)Whenci(t)<0,thepathconstraintinEq.( 2{4 )isinactive.Therefore,bysettingi(t)=0theconstraintwillbeignoredintheaugmentedcost.Furthermore,thenegativevalueof 28

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i(t)whenci(t)=0isinterpretedsuchthatthecostcanonlyimprovebyviolatingthepathconstraint[ 7 ].TheoptimalityconditionsofEq.( 2{13 )deneasetofnecessaryconditionsthatmustbesatisedforanextremalsolutionofanoptimalcontrolproblemwhichcanbeamaxima,minima,orsaddle.Thesecond-ordersuciencyconditionsmustbeimplementedtoconrmwhichoftheextremalsolutionsisaglobalminima.Foralocalminima,however,theparticularextremalwiththelowestcostischosen. 2.1.2Pontryagin'sPrincipleThePontryagin'sprinciple[ 7 , 8 , 25 ]isusedtodeterminetheconditionsforobtainingtheoptimalcontrol.Iftheoptimalcontrolisinteriortothefeasiblecontrolset,therst-ordernecessaryconditionrelatedtothecontrolgiveninEq.( 2{13 )as @H @u(t)=0(2{14)isusedtoobtaintheoptimalcontrol.TheexpressioninEq.( 2{14 )isknownasthestrongformofPontryagin'sprinciple.If,however,thesolutionliesontheboundaryofthefeasiblecontrolandstateset(\bang-bang"control),thestrongformofthePontryagin'sprinciplecannotbeusedtocomputetheoptimalcontrol.Also,iftheHamiltoniandenedinEq.( 2{7 )islinearinthecontrol,thatis, H(y(t),u(t),(t),(t),t)=\(y(t),t)+T(t)a(y(t),t)+T(t)u(t))]TJ /F13 11.955 Tf 11.95 0 Td[(T(t)c(y(t),t),(2{15)thestrongformofPontryagin'sprinciplewillnotprovideanyinformationabouttheoptimalcontrol.Forsuchproblems,theweakformofthePontryagin'sprinciplemustbeusedtodeterminetheoptimalcontrolthatminimizestheaugmentedHamiltonianinEq.( 2{7 ).IfUisthesetofpermissiblecontrol,theweakformofthePontryagin'sprincipleisstatedas[ 25 ] u(t)=argminu2UH(y(t),u(t),(t),(t),t).(2{16) 29

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Inotherwords,theweakformofPontryagin'sprinciplestatesthattheoptimalcontrol,u2U,satisesthecondition H(y(t),u(t),(t),(t),t)H(y(t),u(t),(t),(t),t),8u2U.(2{17) 2.2NumericalMethodsforSolvingOptimalControlProblemsItisoftenverydiculttoobtainanalyticsolutionstooptimalcontrolproblems.Asaresult,itnecessarytoemploynumericalmethodsinordertosolveoptimalcontrolproblems.Numericalmethodsforsolvingoptimalcontrolproblemstraditionallyfallintotwobroadclasses:indirectmethodsanddirectmethods.Anindirectmethodappliesthecalculusofvariationstoderivetherst-ordernecessaryoptimalityconditionsofEq.( 2{13 ).Theindirectapproachresultsinamultiple-pointboundary-valueproblemwhichissolvedtodeterminetheextremalsolution.Incontrast,adirectmethoddoesnotrequirederivationoftheoptimalityconditionsandinturn,attemptstondaminimumoftheobjectivefunction.Inadirectmethod,thestateand/orcontroloftheoptimalcontrolproblemare/isreplacedwithanappropriatefunctionapproximation(e.g.,polynomialapproximation,piecewiseconstantparameterization)andtheproblemistranscribedtoanonlinearprogrammingproblem(NLP),whichisthensolvedusingwell-knownoptimizationtechniques.Inanindirectapproach,theoptimalcontrolproblemissolvedindirectlybyconvertingtheoptimalcontrolproblemtoaboundary-valueproblem.Whileanindirectmethodistypicallyeasytounderstandandiscapableofproducinghighlyaccuratesolutionswhenconvergenceisachieved,indirectmethodshaveafewdrawbacks.First,inanindirectmethodtheoptimalityconditionsmustbederived.Formostapplications,theoptimalcontrolproblemiscomplexmakingitextremelydiculttoconstructtheseexpressions.Derivingtheoptimalityconditionscanbeespeciallytedious,subjecttoerrors,andattimesevenimpossible.Anotherdisadvantageofanindirectmethodisthatthemethodisverysensitivetosmallchangesintheunspeciedboundaryconditions.Sincethecostate(oradjoint)variablesareunknownandnotphysicalquantities,itisdicultfortheusertoprovideagoodinitialguessthatguarantees 30

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convergence.Furthermore,anindirectapproachisnotexibleinthattheoptimalityconditionsmustbederivedandconstructedagainiftheoptimalcontrolproblemismodied.Duetothepracticallimitationsofindirectmethods,theremainingsectionsofthischapterwillfocusondirectmethods.Inadirectmethod,thecontinuousfunctionsoftimeoftheoptimalcontrolproblemareapproximatedandtheproblemistranscribedtoanNLP.Inthecasewhereonlythecontrolfunctionoftimeisapproximated,themethodiscalledacontrolparameterizationmethod,whereaswhenboththestateandcontrolfunctionsoftimeareapproximated,themethodiscalledastateandcontrolparameterizationmethod.Directmethodsovercomemanyofthedisadvantagesofindirectmethods.Oneofthemainadvantagesofadirectmethodisthatitdoesnotrequiresthederivationoftheoptimalityconditions.Sincetheoptimalityconditionsarenotderived,theoptimalcontrolproblemcanbemodiedrelativelyeasily.Moreover,theuserdoesnotneedtoprovideaninitialguessfortheunspecied,non-intuitivecostate,nordoestheinitialguessprovidedneedtobeasgoodasthatrequiredbyanindirectmethodinordertoachieveconvergence.Whileverifyingtheoptimalityofasolutionobtainedusingadirectmethodmayrequiremorework,theadvantagesofusingadirectmethodtypicallyoutweightheadvantagesofanindirectmethod. 2.3KeyConceptsforNumericallySolvingOptimalControlProblemsInordertosolveanoptimalcontrolproblemwithadirectmethod,itisimportanttoaddresstwokeynumericalconcepts:(1)thenumericalsolutionofdierentialequationsand(2)nonlinearoptimization.Toobtainasolutiontotheoptimalcontrolproblem,itisnecessarytosolveasetofdierentialequations.Therefore,variousintegrationschemesarepresentedforsolvingdierentialequations.Finally,nonlinearoptimizationisimportantbecauseinadirectmethodtheoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP).ThegeneralformofanNLPisstated,followedbycommonmethodsforsolvingNLPs. 2.3.1NumericalSolutionofDierentialEquationsFundamentaltosolvingoptimalcontrolproblemsisthenumericalsolutionofaninitialvalueproblem(IVP)forordinarydierentialequations.TheIVPcanbestatedasfollows. 31

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Given _y=f(y(t),t),y(ti)=yi,(2{18)determinethevalueofy(ti+1)whereti+1>ti.IntegratingEq.( 2{18 )fromtitoti+1yieldstheexpression y(ti+1)=y(ti)+Zti+1ti_y(s)ds=y(ti)+Zti+1tif(y(s),s)ds(2{19)orequivalently, y(ti+1))]TJ /F9 11.955 Tf 11.95 0 Td[(y(ti)=Zti+1tif(y(s),s)ds.(2{20)IntegrationschemesforsolvingtheIVPareclassiedaseithertime-marchingorcollocationmethods. 2.3.1.1Time-marchingmethodsThemostwell-knowntime-marchingmethodsbelongtothefamilyofRunge-Kuttaschemes[ 51 { 53 ],whichhavethefollowingform yi+1=yi+hikXj=1jfij,(2{21)where fij=f" yi+hikXl=1jlfil!,(ti+hij)#,1jk(2{22)wherekisreferredtoasthestage.InEqs.( 2{21 )and( 2{22 ),theconstantsj,j,andjlareknownandjhastheproperty0121.UsingthefollowingformoftheButcherarray[ 54 , 55 ] 1 111k... ......k k1kk 1k(2{23)thecoecientsforthemostcommonRunge-Kuttaschemes,k=1,2,3,4,aregivenas: 32

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Euler(k=1): 0 0 1(2{24) Trapezoidal(k=2): 0 001 1=21=2 1=21=2(2{25) Hermite-Simpson(k=3): 0 0001=2 5=241=3)]TJ /F4 11.955 Tf 9.3 0 Td[(1=241 1=62=31=6 1=62=31=6(2{26) ClassicalRunge-Kutta(k=4): 0 00001=2 1=20001=2 01=200 1=61=31=31=6(2{27)Iftheconstantjl=0forallljinEq.( 2{22 ),theschemeiscalledexplicit,otherwisetheschemeiscalledimplicit.Inanexplicitmethod,thevaluey(ti+1)appearsonlyontheleft-handsideoftheequation,whereasinanimplicitmethod,thevaluey(ti+1)appearsontheleft-handandright-handsidesoftheequation.Whenemployinganimplicitmethod,thesolutionatti+1isobtainedusingapredictor-corrector,wherethepredictorisanexplicitmethodandthecorrectorisanimplicitmethod.Otherwidelyusedtime-marchingmethodsbelongtothefamilyofAdamsmethods[ 51 ].Adamsmethodsaretypicallyusedasapredictor-correctorinwhichtheerroriscontrolledbyadjustingthestepsizehandtheorderpofthemethod.InanAdamsmethod,apolynomialisusedtointerpolate_y(t)=f(y(t),t).Theinterpolatingpolynomialisthenintegratedover[ti,ti+1]toobtainanapproximationtoyi+1.ThetwomostcommonAdamsmethodsaretheAdams-Bashforthmethod[ 56 ]andtheAdams-Moultonmethod[ 57 ].ToderivetheAdams-Bashforthformulas,apolynomialofdegreepisusedtointerpolate_y(t)atti)]TJ /F7 7.97 Tf 6.59 0 Td[(p,...,ti.UsingtheNewtonbackwarddierenceinterpolationformula 33

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expandedabouttiastheinterpolatingpolynomial,theAdams-Bashforthmethodtakesthegeneralform yi+1=yi+hpXj=0jf(yj,tj))]TJ /F9 11.955 Tf 11.95 0 Td[(f(yj)]TJ /F6 7.97 Tf 6.58 0 Td[(1,tj)]TJ /F6 7.97 Tf 6.59 0 Td[(1),(2{28)wheretheAdams-BashforthcoecientsaregiveninTable 2-1 . Table2-1. Adams-Bashforthcoecientsfororderp=0,1,2,3. 0123 11=25=123=8 Inasimilarmanner,theAdams-Moultonformulasarederived,butinsteaduseapolynomialofdegreep0tointerpolate_y(t)atp+1pointsti+1,...,ti)]TJ /F7 7.97 Tf 6.58 0 Td[(p+1.ThegeneralformoftheAdams-Moultonmethodisexpressedas yi+1=yi+hpXj=0jf(yj+1,tj+1))]TJ /F9 11.955 Tf 11.95 0 Td[(f(yj,tj),(2{29)wheretheAdams-MoultoncoecientsaregiveninTable 2-2 . Table2-2. Adams-Moultoncoecientsfororderp=0,1,2,3. 0123 1)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2)]TJ /F4 11.955 Tf 9.3 0 Td[(1=12)]TJ /F4 11.955 Tf 9.3 0 Td[(1=24 Usingthepreviouslystateddenitionsofanexplicitmethodandanimplicitmethod,theAdams-Bashforthmethodisclassiedasanexplicitmethod,whiletheAdams-Moultonmethodisimplicit.Implicitmethodsaremorestablethanexplicitmethods[ 6 , 58 ],butanimplicitmethodrequiresadditionalcomputationsateachstepduetotheneedtoimplementapredictoralongwiththecorrector. 2.3.1.2CollocationmethodsWhiletheRunge-KuttafamilyandAdamsmethodsaretypicallyconsideredtime-marchingmethods,theycanalsobeimplementedascollocationmethods.Inacollocationmethod,thevaluesofthestateateachcollocationpointareobtainedsimultaneouslyandarethussaidtoimplicitlysimulatethedynamicsofthesystem[ 6 ].Inacollocationmethod,thestateisapproximatedusingaspeciedfunctionalform.Forexample,thestatecanbeapproximated 34

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usingthefollowingK-thdegreepiecewisepolynomialgivenas Y(t)KXk=0ai(t)]TJ /F3 11.955 Tf 11.96 0 Td[(ti)k(2{30)wherethecoecientsoftheapproximatingpiecewisepolynomial(a0,...,aK)arechosentomatchthevalueofthefunctionatthebeginningofthestepas Y(ti)=y(ti).(2{31)Theapproximationofthestatederivativeisthensetequaltotheright-handsideofthedierentialequationevaluatedateachofthecollocationpoints _y(j)=f(y(j,j),(j=1,...,K)(2{32)wherethecollocationpoints(1,...,K)aredeterminedbydividingtheinterval[ti,ti+1]intoKsubintervals[j,j+1]suchthat j=ti+hij,(j=1,...,K),hi=ti+1)]TJ /F3 11.955 Tf 11.96 0 Td[(ti.(2{33)Inordertosolvethedierentialequationateachcollocationpointsimultaneously,alloftheunknownparametersmustbedeterminedatthesametime.ThisisdonebywritingthecollocateddynamicsofEq.( 2{32 )asdefectconstraintswhichhavetheform j=_Y(j))]TJ /F9 11.955 Tf 11.95 0 Td[(f(y(j),j).(2{34)Theobjectiveisthentodeterminethevaluesofthecoecientsoftheapproximatingpiecewisepolynomial,(a0,...,aK),thatmakethedefectconstraintsequaltozero.Collocationmethodshaveasignicantadvantageoverthetime-marchingmethods[ 6 ].Inatime-marchingmethod,thevalueofthestateateachpointissolvedsequentiallyanddependingonthemethodmayrequiremorecomputationateachstep(e.g.,predictor-correctormethods).Usingacollocationmethodeliminatestheneedforapredictor-corrector,becausea 35

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collocationmethoddeterminesthevaluesofthestateateachdiscretizationpointatthesametime.Animportantsubsetofcollocationmethodsareorthogonalcollocationmethods,wherethecollocationpointsareselectedtobetherootsofanorthogonalpolynomial.Afeatureofthisorthogonalpolynomialisthatitalsohasanassociatedquadratureruleforapproximatingthedeniteintegralofaknownfunction[ 52 , 59 ].ThemostwidelyusedpolynomialsforobtainingorthogonalcollocationpointsareChebyshevpolynomialsorLegendrepolynomials[ 51 ].Typically,thestateinanorthogonalcollocationmethodisapproximatedoverthetimeinterval2[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,1]as y()Y()=NXk=1ckLk()(2{35)whereLk()isabasisofLagrangepolynomials.UsingthefactthatLagrangepolynomialssatisfytheisolationproperty Lk(i)=(1,k=j0,k6=j(2{36)itfollowsfromEq.( 2{35 )that ck=y(k).(2{37)ThemostcommonorthogonalcollocationmethodsbelongtothefamilyofLegendre-Gausscollocationpoints[ 48 ].Legendre-Gauss(LG)collocationpointslieontheopeninterval2()]TJ /F4 11.955 Tf 9.3 0 Td[(1,1),Legendre-Gauss-Radaucollocationpointslieonthehalfopeninterval2[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,1)or2()]TJ /F4 11.955 Tf 9.3 0 Td[(1,1],andLegendre-Gauss-Lobatto(LGL)collocationpointslieontheclosedinterval2[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,1].DenotingNasthenumberofcollocationpointsandPN()astheN-thdegreeLegendrepolynomialdenedas PN()=1 2NN!dN[(2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)N] dN,(2{38)theLGpointsaretherootsofPN(),theLGRpointsaretherootsofPN)]TJ /F6 7.97 Tf 6.58 0 Td[(1()+PN(),andtheLGLpointsaretherootsof_PN)]TJ /F6 7.97 Tf 6.58 0 Td[(1()alongwiththepoints)]TJ /F4 11.955 Tf 9.3 0 Td[(1and+1.Aschematicof 36

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theLegendre-GausscollocationpointsisshowninFig. 2-1 ,whereitisseenthatthesetsofcollocationpointsarenotequallyspaced.Anadvantageofusingnon-equallyspacedpointsisthatnon-equallyspacedpointsareknowntohaveadistributionthatminimizesRungephenomenon[ 60 { 62 ].Anotherbenetthatorthogonalcollocationhasoverstandardcollocationisthatthequadratureapproximationofthedeniteintegralishighlyaccuratewhenthecollocationpointsarechosentobeorthogonal.Forexample[ 6 ],considertheapproximationoftheintegralofascalarfunctionf()usinganN-pointLGquadrature Z+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1f()dNXk=1wkf(k)(2{39)wherekareLGpointswhicharetherootsofthePN()polynomialdenedinEq.( 2{38 )andwkaretheassociatedLGweights.IfNLGpointsareusedinthequadratureapproximation,Eq.( 2{39 )willbeexactiff()isapolynomialofdegree2N)]TJ /F4 11.955 Tf 12.84 0 Td[(1orless.Usingasimilarapproach,NLGRpointsareexactforapolynomialofdegree2N)]TJ /F4 11.955 Tf 12 0 Td[(2orlessandNLGLpointsareexactforapolynomialofdegree2N)]TJ /F4 11.955 Tf 12.89 0 Td[(3orless.Inthisresearch,theoptimalcontrolproblemwasdiscretizedusingLGRcollocationpoints. 2.3.2NonlinearOptimizationAnotherimportantaspectofnumericallysolvingoptimalcontrolproblemsusingdirectmethodsisnonlinearoptimization[ 63 { 66 ].Whenemployingadirectmethod,thecontinuous-timeoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP).TheobjectiveofanNLPistodeterminethedecisionvariablesthatminimizesomecostfunctionthatissubjecttoasetofalgebraicconstraints.AnNLPhasthefollowinggeneralform.Determinethevectorofdecisionvariableszthatminimizethecostfunction f(z)(2{40)subjecttothealgebraicequalityconstraints g(z)=0,(2{41) 37

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andthealgebraicinequalityconstraints h(z)0(2{42)wherez2Rn,g(z)2Rm,andh(z)2Rp.AnNLPiscategorizedaseitherdenseorsparse.AdenseNLPcontainsmanynon-zeroderivativesoftheobjectivefunctionandconstraintfunctionswithrespecttothecomponentsofthedecisionvector(e.g.,asindirectshooting),whereasasparseNLPcontainsalargepercentageofderivativesthatarezero(e.g.,asinadirectcollocationmethod).Itmaybecounter-intuitivethatsolvingalargesparseNLPwouldbepreferredoversolvingtheboundary-valueproblemofanindirectmethod,butinpractice,thesparseNLPcanbeeasiertosolve,anddoesnotrequireasaccurateofaninitialguess.Today'sNLPsolversareverycapableofhandlingcomplexproblemsbecausetheyaredesignedtobeextremelycomputationallyecientbyexploitingthesparsityofthederivativesintheobjectivefunctionandtheconstraints.ManynumericalalgorithmsexistforsolvingNLPssuchasNewton'smethod,conjugatedirectionmethods,andgradient-basedmethods.Gradient-basedmethodsarethemostpopularandincludesequentialquadraticprogramming(SQP)andinterior-pointmethods.Examplesofwell-knownsoftwarethatemploySQPmethodsincludethedenseNLPsolverNPSOL[ 67 ]andthesparseNLPsolversSNOPT[ 27 ]andSOS[ 68 ].Otherwell-knownsparseNLPsolversuseinterior-pointmethodssuchasIPOPT[ 29 ]andKNITRO[ 28 ].Inthisresearch,theinterior-pointNLPsolverIPOPTischosentosolvetheNLP. 2.4CommonDirectMethodsforSolvingOptimalControlProblemsThissectionpresentsthethreemostcommondirectmethodsfornumericallysolvingoptimalcontrolproblems.Whilethissectionhighlightsdirectmethods,itisnotedthatthenumericalmethodsdescribedherecanbeappliedtoanindirectproblemformulation.Forexample,themultipleshootingmethodcanbeappliedtobothanindirectoradirectproblemformulation. 38

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2.4.1DirectShootingMethodThedirectshootingmethod[ 5 , 6 , 51 , 52 , 69 ]isthemostbasicdirectmethodforsolvingoptimalcontrolproblems.Inadirectshootingmethod,onlythecontrolisparameterizedusingaspeciedfunctionaloftheform u(t)^u(t)=pXi=1ii(t),(2{43)where1,...,pareasetofcontrolparameterstobedeterminedfromtheoptimizationand1(t),...,p(t)areknownfunctions.Usingadirectshootingmethod,theBolzaoptimalcontrolproblemofEqs.( 2{1 )-( 2{4 )istransformedtohavethefollowingform.Minimizethecostfunctional J(y(t0),t0,y(tf),tf)+Ztft0\(y(t),^u(t),t)dt,(2{44)subjecttotheboundaryconditions b(y(t0),t0,y(tf),tf)=0,(2{45)andthepathconstraints c(y(t),^u(t),t)0,(2{46)wherethestateisgivenby y(t)=y(t0)+Ztft0a(y(t),^u(t),t)dt.(2{47)ThestateinEqs.( 2{47 )isdeterminedbyintegratingthedierentialequationsusingatime-marchingalgorithmsuchasEuler,Trapezoidal,Hermite-Simpson,orClassicalRunge-KuttafoundinSection 2.3.1.1 .Similarly,thecostfunctionofEq.( 2{44 )isfoundusingaquadratureapproximationthatisconsistentwiththenumericalintegratorusedtosolvethedierentialequations.Theoptimizationproblemisthensolvedtoobtaintheoptimalvaluesoftheunknowncontrolparameters1,...,pthatminimizethecostsubjecttoanypathandinterior-pointconstraints. 39

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2.4.2DirectMultipleShootingMethodInadirectmultipleshootingmethod[ 5 , 6 , 51 , 52 , 69 ],thetimeinterval[t0,tf]isdividedintoKintervals,whereeachtimeintervalisonthedomain[sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,sk],(k=1,...,K)andt0=s0
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wherethestateisgivenby y(k)(t)=y(k)(sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+Zsksk)]TJ /F6 5.978 Tf 5.76 0 Td[(1a(y(k)(t),^u(k)(t),t)dt,k=1,...,K)]TJ /F4 11.955 Tf 11.96 0 Td[(1.(2{54)Similartothedirectshootingmethod,thestateinEqs.( 2{54 )isdeterminedbyintegratingthedierentialequationsusingatime-marchingalgorithmsuchasEuler,Trapezoidal,Hermite-Simpson,orClassicalRunge-KuttafoundinSection 2.3.1.1 andthecostfunctionof( 2{44 )isfoundusingaquadratureapproximationthatisconsistentwiththenumericalintegratorusedtosolvethedierentialequations.Bysolvingtheoptimizationproblem,theoptimalvaluesoftheunknowncontrolparameters1,...,pandtheunknowncomponentsoftheinitialstateineachsubintervalthatminimizethecostsubjecttoanypathandinterior-pointconstraintsaredetermined. 2.4.3DirectCollocationMethodDirectcollocationmethods[ 5 , 6 , 31 , 35 , 69 { 74 ]areveryappealingforcomplicatedapplicationssincetheyareversatileandrobust.Furthermore,directcollocationmethodshaveanadvantageoverashootingapproachinthatthereisnopropagationoferrorwhenintegratingthedynamicssincetheoptimizationisperformedsimultaneously.Adirectcollocationmethodisastateandcontrolparameterizationmethod,wherethestateandcontrolareapproximatedusingaspeciedfunctional,whichtypicallytaketheformofChebyshevorLagrangepolynomials.ThecollocationordiscretizationisthenperformedatthechosenpointsusingRunge-Kuttamethodsorcollocationmethods(Sections 2.3.1.1 and 2.3.1.2 ).ThecostfunctionofEq.( 2{1 )isapproximatedusingaquadraturethatisconsistentwiththenumericalmethodusedtosolvethedierentialequations.Thetwomostcommonformsofdirectcollocationarelocalcollocationandglobalcollocation.Inlocalcollocation,thedegreeoftheapproximatingpolynomialisxedandthenumberofsegmentsisvaried.Ontheotherhand,inaglobalcollocationmethod,thenumberofsegmentsisxedandthedegreeoftheapproximatingpolynomialisvaried.Inbothlocalandglobalcollocationmethods,thetimeinterval[t0,tf]isdividedintoKsubintervals,whereeachsubintervalisonthedomain 41

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[sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,sk],(k=1,...,K)andt0=s0
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CHAPTER3LEGENDRE-GAUSS-RADAUCOLLOCATIONANDADAPTIVEMESHREFINEMENTTheobjectiveofthischapteristodescribeindetailthedirectcollocationmethodandmeshrenementalgorithmutilizedinthisresearchfornumericallysolvingthelow-thrustoptimalcontrolproblems.Afterintroducingthenotationandconventionsusedthroughoutthechapter,thegeneralformofthecontinuous-timeoptimalcontrolproblemisstated.Next,theoptimalcontrolproblemismodiedsuchthattheindependentvariableiscontainedonaxeddomainandthedomainisthendividedintoameshcreatingamultipleintervaloptimalcontrolproblem.ThemultipleintervaloptimalcontrolproblemisthendiscretizedusingtheLegendre-Gauss-Radau(LGR)orthogonalcollocationmethod.Adetaileddescriptionofthephmeshrenementalgorithmisthenpresented.Finally,theeectivenessoftheLGRcollocationmethodandthephmeshrenementmethodaredemonstratedonanexamplelow-thrustorbittransfer. 3.1NotationandConventionsThefollowingnotationandconventionswillbeemployedthroughoutthisdissertation.Allscalarswillberepresentedbylowercasesymbols(e.g.,y,u).Allvectorfunctionswillbetreatedasrowvectorsandwillbedenotedbylowercaseboldsymbols.Therefore,ify(t)2Rnisavectorfunctionoftime,theny(t)=y1(t)yn(t).Anyvectorthatisnotafunctionoftimewillbedenotedasacolumnvector.Forexample,ifz2Rnisastaticvector,thenz=z1znT.Next,matriceswillbedenotedbyuppercaseboldsymbols.Thus,Y2RNnisamatrixofNn.Furthermore,f(y),f:Rn!Rm,isafunctionthatmapstherowvectorsy2Rntorowvectorsf(y)2Rm.Theresultofevaluatingf(y)atthepoints(y1,...,yN)isthematrixF2RNnisdenedas F1Nf(yk)1N=266664f(y1)...f(yN)377775.(3{1) 43

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Thesinglesubscript,i,attachedtoamatrixdenotesaparticularrowofthematrix.Forexample,Yiisthei-throwofthematrixY.Thedoublesubscript,i,j,attachedtoamatrixdenotestheelementlocatedinrowiandcolumnjofthematrix.Forexample,Yi,jisthe(i,j)-thelementofthematrixY. 3.2Continuous-TimeBolzaOptimalControlProblemConsiderthefollowinggeneraloptimalcontrolproblem.Determinethestate,y(t)2Rny,thecontrol,u(t)2Rnuy,theinitialtime,t0,thenaltime,tf,ontheintervalt2[t0,tf]thatminimizethecostfunctional J=(y(t0),t0,y(tf),tf)+Ztft0\(y(t),u(t),t)dt(3{2)subjecttothedynamicconstraints dy(t) dt=a(y(t),u(t),t),(3{3)theboundaryconditions bminb(y(t0),t0,y(tf),tf,q)bmax,(3{4)andtheinequalitypathconstraints cminc(y(t),u(t),t)cmax.(3{5)Thefunctions,)]TJ /F1 11.955 Tf 6.78 0 Td[(,a,b,andcaredenedbythefollowingmappings: :RnyRRnyR!R,)-839(:RnyRnuR!R,a:RnyRnuR!Rny,b:RnyRRnyR!Rnb,c:RnyRnuR!Rnc,(3{6)whereallvectorfunctionsoftimearetreatedasrowvectors. 44

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3.2.1TransformedContinuous-TimeBolzaOptimalControlProblemTheoptimalcontrolproblemgiveninEqs.( 3{2 )-( 3{5 )isdenedonthetimeintervalt2[t0,tf]wheretimeistheindependentvariable.Forthemethodutilizedinthisresearch,itisusefultomodifytheoptimalcontrolproblemsothattheindependentvariableiscontainedonaxeddomain.Theindependentvariabletismappedtoanewindependentvariable,2[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1],viatheanetransformation =2t tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0+tf+t0 tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0.(3{7)Theoptimalcontrolproblemintermsofthenewindependentvariable,,isdenedasfollows.Determinethestate,y()2Rny,thecontrol,u()2Rnu,theinitialtime,t0,thenaltime,tf,onthetimeinterval=[)]TJ /F4 11.955 Tf 9.29 0 Td[(1,+1]thatminimizethecostfunctional J=(y()]TJ /F4 11.955 Tf 9.29 0 Td[(1),t0,y(+1),tf)+tf)]TJ /F3 11.955 Tf 11.96 0 Td[(t0 2Z+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1\(y(),u(),;t0,tf)d(3{8)subjecttothedynamicconstraints dy() d=tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0 2a(y(),u(),;t0,tf),(3{9)theboundaryconditions bminb(y()]TJ /F4 11.955 Tf 9.3 0 Td[(1),t0,y(+1),tf)bmax,(3{10)andthepathconstraints cminc(y(),u(),;t0,tf)cmax.(3{11)ItisnotedthattheoptimalcontrolproblemofEqs.( 3{8 )-( 3{11 )canbetransformedfromthetimeinterval=[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1]tothetimeintervalt2[t0,tf]viatheanetransformation t=tf)]TJ /F3 11.955 Tf 11.96 0 Td[(t0 2+tf+t0 2.(3{12) 45

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3.2.2MultipleIntervalTransformedContinuous-TimeOptimalControlProblemSupposethatthetimeinterval=[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1]isdividedintoameshconsistingofKmeshintervalsSk=[sk)]TJ /F6 7.97 Tf 6.58 0 Td[(1,sk],(k=1,...,K),where(s0,...,sK)arethemeshpoints.Themeshpointshavethepropertythat)]TJ /F4 11.955 Tf 9.3 0 Td[(1=s0
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ineachmeshintervalSk,(k=1,...,K)as y(k)()Y(k)()=Nk+1Xj=1Y(k)jL(k)j()(3{17)where L(k)j()=Nk+1Yl=1,l6=j)]TJ /F8 11.955 Tf 11.95 0 Td[((k)l (k)j)]TJ /F8 11.955 Tf 11.95 0 Td[((k)l,(j=1,...,Nk+1)(3{18)isabasisofLagrangepolynomials,=[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1],and((k)1,...,(k)Nk)aretheLGRcollocationpointsinmeshintervalSk.ThemeshintervalSkisdenedonthesub-interval=[sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,sk)andNk+1=sKwhichisanon-collocatedpoint.ThederivativeofY(k)()inEq.( 3{17 )withrespectto,isthengivenas dY(k)() d=Nk+1Xj=1Y(k)jdL(k)j() d.(3{19)ThecostfunctionalofEq.( 3{13 )isthenapproximatedusingamultiple-intervalLGRquadratureas JJ=(Y(1)1,t0,Y(K)NK+1,tf)+KXk=1NkXi=1tf)]TJ /F3 11.955 Tf 11.96 0 Td[(t0 2w(k)i\(Y(k)i,U(k)i,(k)i;t0,tf),(i=1,...,Nk,k=1,...,K)(3{20)wherew(k)iaretheLGRquadratureweights[ 79 ]inmeshintervalSk,U(k)iaretheapproximationsofthecontrolattheNkLGRpointsinmeshintervalSk,Y(1)1istheapproximationofy(s0),andY(K)Nk+1istheapproximationofy(sK).(Recallthats0=)]TJ /F4 11.955 Tf 9.3 0 Td[(1andsK=+1.)CollocatingthedynamicconstraintsofEq.( 3{14 )attheNkLGRpointsineachmeshintervalSkusingEq.( 3{19 )yieldsthealgebraicdefectconstraints (k)i=Nk+1Xj=1D(k)ijY(k)j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(tf)]TJ /F3 11.955 Tf 11.96 0 Td[(t0 2a(Y(k)i,U(k)i,(k)i;t0,tf)=0,(i=1,...,Nk,j=1,...,Nk+1,k=1,...,K),(3{21) 47

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where D(k)ij=dL(k)k((k)i) d,(i=1,...,Nk,j=1,...,Nk+1,k=1,...,K),(3{22)istheNk(Nk+1)LGRdierentiationmatrix[ 48 ]formeshintervalSk.Thedynamicscanalsobecollocatedusingtheequivalentimplicitintegralform[ 48 , 61 , 75 , 76 ].TheimplicitintegralformoftheLGRcollocationmethodisgivenas (k)i=Y(k)i+1)]TJ /F9 11.955 Tf 11.95 0 Td[(Y(k)1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0 2NkXj=1I(k)ija(Y(k)i,U(k)i,(k)i;t0,tf)=0,(i=1,...,Nk,j=1,...,Nk+1,k=1,...,K),(3{23)whereI(k)ijistheNkNkLGRintegrationmatrixformeshintervalSk.TheLGRintegrationmatrixisobtainedbyinvertingasubmatrixofthedierentiationmatrixformedbycolumns2throughNk+1andisgivenas I(k)=D(k)2D(k)Nk+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1,(k=1,...,K)]TJ /F4 11.955 Tf 11.95 0 Td[(1).(3{24)Forcompleteness,itisnotedthatI(k)D(k)1=)]TJ /F9 11.955 Tf 9.3 0 Td[(1,where1isacolumnvectorofalloneswithalengthofNk.Next,theboundaryconditionsofEq.( 3{15 )areapproximatedas bminb(Y(1)1,t0,Y(K)Nk+1,tf)bmax(3{25)andthepathconstraintsofEq.( 3{16 )areapproximatedas cminc(Y(k)i,U(k)i,(k)i;t0,tf)cmax,(i=1,...,Nk,k=1,...,K).(3{26)Continuityinthestateattheinteriormeshpointsisenforcedviathecondition Y(k)Nk+1=Y(k+1)1,(k=1,...,K)]TJ /F4 11.955 Tf 11.96 0 Td[(1).(3{27)Computationally,theconditioninEq.( 3{27 )iseliminatedfromtheproblembyusingthesamevariableforbothY(k)Nk+1andY(k+1)1.TheNLPthatarisesfromdiscretizationoftheoptimal 48

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controlproblemusingtheLGRcollocationmethodisstatedasfollows.MinimizethecostfunctionofEq.( 3{20 )subjecttothealgebraicconstraintsofEqs.( 3{21 ),( 3{25 ),and( 3{26 ). 3.4phAdaptiveMeshRenementInthissection,thepreviouslydevelopedphadaptivemeshrenementmethod[ 49 ]employedinthisresearchisbrieydescribed.Inanhmethod,thenumberofintervalsisincreasedtoachieveconvergence,whereasinapmethod,thenumberofcollocationpointsisincreasedtoachieveconvergence.Withthedesiretocombinethebenetsofbothhandpmethods,phmethodshavebeendeveloped.ForthephmethoddevelopedfortheLGRcollocationmethod,arelativeerrorestimateiscalculatedineachinterval.Dependingonthevalueoftheerrorestimate,thedegreeoftheapproximatingpolynomial(thatis,thenumberofLGRcollocationpoints)ineachintervalisincreasedortheintervalisdividedintomoreintervals.Themethodforestimatingtherelativeerrorinacurrentsolutionispresentedrst,followedbyadescriptionofthepthenhstrategyusedforreningthemesh. 3.4.1RelativeErrorEstimateinEachMeshIntervalTherelativeerrorestimateisformulatedbasedonthestate,sincethestateistheonlyquantityintheLGRcollocationmethodforwhichauniquelydenedfunctionapproximationisavailable.Theerrorestimateisobtainedbycomparingtwoapproximationsofthestate,oneofwhichhasahigheraccuracythantheother.Thekeyideaisthatforaproblemwhosesolutionissmooth,anincreaseinthenumberofLGRcollocationpointsshouldyieldastatethatmoreaccuratelysatisesthedynamics.Hence,thedierencebetweenthesolutionassociatedwiththeoriginalsetofLGRcollocationpointsandtheapproximationassociatedwithanincreasednumberofLGRcollocationpointsshouldyieldanestimatefortheerrorinthestate.AssumethattheNLPofEqs.( 3{20 )-( 3{26 ),whichcorrespondstothediscretizedoptimalcontrolproblem,hasbeensolvedonthemeshSk=[sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,sk],(k=1,...,K),withNkLGRcollocationpointsineachmeshintervalSk.SupposethatwewanttoestimatetheerrorinthestateatasetofMk=Nk+1LGRcollocationpoints(^(k)1,...,^(k)Mk),where^(k)1=(k)1=sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1and^(k)Mk+1=sk.Supposefurtherthatthevaluesofthestateapproximationgivenin 49

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Eq.( 3{17 )atthepoints(^(k)1,...,^(k)Mk)aredenotedas(Y(^(k)1),...,Y(^(k)Mk).Next,letthecontrolbeapproximatedinmeshintervalSkusingtheLagrangeinterpolatingpolynomial U(k)()=Xj=NkU(k)j^L(k)j(),^L(k)j()=NkYl=1,l6=j)]TJ /F8 11.955 Tf 11.96 0 Td[((k)l (k)j)]TJ /F8 11.955 Tf 11.96 0 Td[((k)l,(3{28)andletthecontrolapproximationat^(k)ibedenotedasU(^(k)i),1iMk.Next,thevalueoftheright-handsideofthedynamicsat(Y(^(k)i),U(^(k)i),^(k)i)isusedtoconstructanimprovedapproximationofthestate.Let^Y(k)beapolynomialofdegreeofatmostMkthatisdenedonthemeshintervalSk.Ifthederivativeof^Y(k)matchesthedynamicsateachoftheLGRcollocationpoints^(k)i,1iMk,then ^Y(k)(^(k)j)=Y(k)(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0 2MkXl=1^I(k)jla(Y(k)(^(k)l),U(k)(^(k)l),^(k)l),(j=2,...,Mk+1),(3{29)where^I(k)jlistheMkMkLGRintegrationmatrixcorrespondingtotheLGRcollocationpointsdenedby(^(k)1,...,^(k)Mk).UsingthevaluesofY(^(k)l)and^Y(^(k)l),theabsoluteandtherelativeerrorsinthei-thcomponentofthestateat(^(k),...,^(k)Mk+1)arethendenedrespectively,as E(k)i(^(k)l)=^Y(k)i(^(k)l)]TJ /F3 11.955 Tf 11.95 0 Td[(Y(k)i(^(k)l),(i=1,...,ny,l=1,...,Mk+1),(3{30) (k)i(^(k)l)=E(k)i(^(k)l 1+maxj2[1,...,Mk+1]Y(k)i(^(k)j),(i=1,...,ny,l=1,...,Mk+1).(3{31)ThemaximumrelativeerrorinmeshintervalSkisthendenedas (k)max=maxi2[1,...,ny],l2[1,...,Mk+1](k)i(^(k)l).(3{32) 3.4.2ReningtheMeshUsingthephMethodAssumeagainthattheNLPofEqs.( 3{20 )-( 3{26 )hasbeensolvedonthemeshSk=[sk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,sk],(k=1,...,K).IneachmeshintervalSk,itisdesiredtosatisfyauser-speciedrelativeerroraccuracytolerancedenotedas".Ifthetolerance"isnotmet 50

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inatleastonemeshinterval,thenthecurrentmeshisrenedbyeitherdividingthemeshintervalorincreasingthedegreeoftheapproximatingpolynomial(thatis,thenumberofLGRcollocationpoints)withinthemeshinterval.SupposethattheintervalSkemploysNkLGRcollocationpointsandthattherelativeerrorestimate(k)maxcomputedasdescribedinSection 3.4.1 isfoundtobelargerthanthedesiredrelativeerrortolerance".BasedontheconvergencetheorysummarizedinRef.[ 80 ],theerrorshouldbemultipliedbythefactor"=(k)max.ThereductionisachievedbyincreasingNkbyPkwherePkischosensuchthatN)]TJ /F7 7.97 Tf 6.59 0 Td[(Pkk="=(k)max,orequivalently, NPkk=(k)max ".(3{33)SolvingforPkyields Pk=logNk(k)max ".(3{34)BecauseEq.( 3{34 )maynotbeaninteger,theresultisroundeduptoobtain Pk=logNk(k)max ".(3{35)ItisnotedthatPk0sincePkisonlycalculatedwhen(k)max>".UsingEq.( 3{35 ),thepredictednumberofLGRcollocationpointsrequiredinthemeshintervalSkis~Nk=Nk+Pk.LetNminandNmaxbeuser-speciedminimumandmaximumboundsonthenumberofLGRcollocationpointswithinanymeshinterval.If~NkNmax,thenNkisincreasedto~NkinmeshintervalSk.Ontheotherhand,if~Nk>Nmax,thenthemeshintervalSkmustbedividedintosub-intervals.Thenumberofsub-intervals,Bk,intowhichthemeshintervalSkisdividediscomputedas Bk=max~Nk Nmin,2.(3{36) 3.4.3SummaryofphMeshRenementMethodAsummaryoftheadaptivemeshrenementalgorithmemployedinthisresearchappearsbelow.ThemeshnumberisdenotedbyMandthemeshnumberincreasesby1ineachloop 51

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ofthealgorithm.ThealgorithmterminatesinStep4whenthedesirederrortoleranceissatisedorwhenMreachesauser-speciedmaximumnumberofrenementsMmax.Foramoredetaileddescriptionofthephmeshrenementmethodemployedinthisresearch,seeRefs.[ 49 , 69 , 81 ]. Step1:SetM=0andsupplyinitialmeshS=SKkSk=[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1],whereTKkSk=;. Step2:SolveNLPofEqs.( 3{20 )-( 3{26 )oncurrentmeshS. Step3:Computescalederror(k)maxinSk,(k=1,...,K),usingmethoddescribedinSection 3.4.1 . Step4:If(k)max"forallk2[1,...,K]orM>Mmax,thenquit.Otherwise,proceedtoStep5. Step5:UsingthemethodofSection 3.4.2 ,modifyallmeshintervalsSkforwhich(k)max>". Step6:SetM=M+1andreturntoStep2. 3.5OrbitTransferExampleandComparisonThepurposeofthissectionistodemonstratetheeectivenessoftheLGRcollocationmethodandthephmeshrenementmethodforsolvingalow-thrustorbittransfer.First,theexampleproblemandinitialguessgenerationmethod,takendirectlyfromRefs.[ 68 ]and[ 82 ],isdescribed.Next,resultsarepresentedforselectph-(Nmin,Nmax)methods.Solutionsfromtwowell-establishedoptimalcontrolsoftware,SOSandPSOPT,wereavailableforcomparison. 3.5.1ProblemDescriptionConsiderthefollowinglow-thrustorbittransferoptimalcontrolproblemtakendirectlyfromRef.[ 68 ].Thestateofthesystemisgiveninmodiedequinoctialelementswhilethecontrolisgiveninradial-transverse-normalcoordinates.Thegoalistodeterminethestate y=(p,f,g,h,k,L,w),(3{37)thecontrol u=(ur,u,uh),(3{38) 52

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andtheconstantthrottleparameter,,thattransferthespacecraftfromaninitialorbittoanalorbitwhilemaximizingthenalweightofspacecraft.Thespacecraftstartsinacircularlow-Earthorbitwithinclinationi(t0)=28.5degandterminatesinahighlyellipticlowperiapsisorbitwithinclinationi(tf)=63.4deg.Thecontinuous-timeoptimalcontrolproblemcorrespondingtothisorbittransferproblemcanbestatedasfollows.Minimizethecostfunctional J=)]TJ /F3 11.955 Tf 9.3 0 Td[(w(tf)(3{39)subjecttothedynamicconstraints _y=A(y)+b,(3{40) _w=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(T(1+0.01) Isp,(3{41)thepathconstraint jjujj=1,(3{42)theparameterconstraint )]TJ /F4 11.955 Tf 11.96 0 Td[(500,(3{43)andtheboundaryconditions p(t0)=21837080.052835ft,p(tf)=40007346.015232ft,f(t0)=0,p f2(tf)+g2(tf)=0.73550320568829,g(t0)=0,p h2(tf)+k2(tf)=0.61761258786099,h(t0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.25396764647494,f(tf)h(tf)+g(tf)k(tf)=0,k(t0)=0,g(tf)h(tf))]TJ /F3 11.955 Tf 11.96 0 Td[(k(tf)f(tf)0,L(t0)=rad,w(t0)=1lbm,i(t0)=28.5deg,i(tf)=63.4deg.(3{44) 53

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ThematrixA(y)inEq.( 3{40 )isgivenas A=26666666666666666402p qq p 0q p sin(L)q p 1 q(q+1)cos(L)+f)]TJ /F12 11.955 Tf 9.3 12.38 Td[(q p g qhsin(L))]TJ /F3 11.955 Tf 11.96 0 Td[(kcos(L))]TJ /F12 11.955 Tf 9.3 12.38 Td[(q p cos(L)q p 1 q(q+1)sin(L)+gq p f qhsin(L))]TJ /F3 11.955 Tf 11.95 0 Td[(kcos(L)00q p s2cos(L) 2q00q p s2sin(L) 2q00q p hsin(L))]TJ /F3 11.955 Tf 11.96 0 Td[(kcos(L)377777777777777775(3{45)whilethevectoris b=00000p p(q p)2T,(3{46)where q=1+fcos(L)+gsin(L),r=p=q,2=h2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2,=p h2+k2,s2=1+2.(3{47)Thespacecraftaccelerationismodeledas =g+T,(3{48)wheregistheaccelerationduetotheoblatenessoftheEarthwhileTisthethrustspecicforce.TheaccelerationduetoEarthoblatenessisexpressedinrotatingradialcoordinatesas g=QTrg,(3{49)whereQristhetransformationfromrotatingradialcoordinatestoEarthcenteredinertialcoordinates.ThematrixQrisgivencolumn-wiseas Qr=iriih,(3{50) 54

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wherethebasisvectorsir,i,andiharegivenas ir=r jjrjj,ih=rv jjrvjj,i=ihir.(3{51)Furthermore,thevectorgisdenedas g=gnin)]TJ /F8 11.955 Tf 11.95 0 Td[(grir,(3{52)whereinisthelocalNorthdirectionandisdenedas in=en)]TJ /F4 11.955 Tf 11.96 0 Td[((eTnir)ir jjen)]TJ /F4 11.955 Tf 11.96 0 Td[((eTnir)irjj(3{53)anden=(0,0,1).Theoblateearthperturbationsarethenexpressedas gr=)]TJ /F8 11.955 Tf 12.19 8.09 Td[( r24Xk=2(k+1)Re rkPk(s)Jk,(3{54) gn=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(cos( ) r24Xk=2Re rkP0k(s)Jk,(3{55)whereReistheequatorialradiusofearth,Pk(s),s2[)]TJ /F4 11.955 Tf 9.3 0 Td[(1,+1]isthek-thdegreeLegendrepolynomial,P0kisthederivativeofPkwithrespecttos,s=sin( ),andJkrepresentsthezonalharmoniccoecientsfork=(2,3,4).Next,theaccelerationduetothrustisgivenas T=geT(1+0.01) wu.(3{56)Finally,thephysicalconstantsusedintheproblemaregiveninTable 3-1 . 3.5.2InitialGuessGenerationTheinitialguesswascalculatedinamannersimilartotheapproachgiveninRefs.[ 82 ]and[ 83 ].Specically,theinitialguesswasobtainedusingavariablestepordinarydierentialequationsolverbypropagatingthemodiedequinoctialdynamicsfromtheinitialconditionstoanaltimeof9104swithathrottleparameterof=)]TJ /F4 11.955 Tf 9.29 0 Td[(25.Incomputingtheinitialguess, 55

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thecontroldirectionwasassumedtoliealongthedirectionofinertialvelocity,thatis, u=QTrv jjvjj.(3{57)Thevaluesatthedesiredcollocationpointswerefoundusinginterpolationtoformtheinitialguess.ThecollocationpointsgeneratedfromaninitialmeshofMintervalseachcontainingaspeciednumberofLGRcollocationpoints.ThenumberofintervalMwascalculatedby M=ceil(Lf))]TJ /F4 11.955 Tf 11.95 0 Td[(1.(3{58)TheintervalswerethenevenlyspacedonthedomainofthetruelongitudeLas Lm=Li+m)]TJ /F4 11.955 Tf 11.95 0 Td[(1 M)]TJ /F4 11.955 Tf 11.96 0 Td[(1(Lf)]TJ /F3 11.955 Tf 11.96 0 Td[(Li),(m=1,...,M)(3{59)whereLiandLfaretheinitialandterminalvaluesofthetruelongitudeL.Finally,theevenlyspacedintervalsonthetruelongitudedomainwereconvertedtocorrespondingintervalsonthetimetdomain.Thedesirednumberofcollocationpointswerethenusedwithineachmeshinterval.Itisnotedthattheapproachdescribedhereforobtainingtheinitialmeshprovidesanincreaseddensityofcollocationpointsnearperiapsisofthetrajectory. 3.5.3ResultsandDiscussionThelow-thrustorbittransferproblemwassolvedusingtheLGRcollocationmethodasdescribedinSection 3.3 andthevariable-orderphadaptivemeshrenementalgorithmasdescribedinSection 3.4 .Theph-(Nmin,Nmax)methodwasemployed,whereNminandNmaxrepresenttheminimumandmaximumnumberofLGRcollocationpointsthatareallowedineachmeshinterval.Intherstpartofthisexample,Nmin=4andNmax=6(thatis,aph-(4,6)methodwasused).UsingtheinitialguessasdescribedinSection 3.5.2 ,theoptimalnalweight,transfertime,andthrottleparameterwerefound,respectively,tobew(tf)=0.2201805609lb,tf=86815.6s,and=)]TJ /F4 11.955 Tf 9.3 0 Td[(9.09679.AviewinEarth-centeredinertialCartesian(ECI)coordinates(x,y,z)oftheoptimalorbittransferisshowninFig. 3-1 .Asummaryoftheph-(4,6)meshrenementiterationhistoryisshowninTable 3-2 ,whereall 56

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computationswereperformedona2.3GHzIntelCorei7MacBookProrunningMacOS-X10.9.5with16GBofRAM.ExaminingthemeshrenementevolutionshowninFig. 3-2 ,itisnotedthatthemeshdensityincreaseswhenthespacecraftisnearperiapsis.AsthespacecrafttrajectorymovesawayfromtheEarth,lesscollocationpointsareneededsincethevelocityofthespacecraftisdecreasing.Computationswerealsoperformedusingvariousxedandvariable-orderphmeshrenementapproachesandareshowninTable 3-3 .Allofthemethodstestedreturnedsimilaroptimalvaluesforthenalweight,transfertime,andthrottleparameter.Whilecomputationalperformancewasessentiallythesameusingbothhandphmethods,thephadaptivemethodrequiredlesscollocationpointsthanthecorrespondinghalgorithm.Solutionstotheproblemusingthegeneral-purposeoptimalcontrolsoftwareSparseOpti-mizationSuite(SOS)inRef.[ 68 ]andtheoptimalcontrolsoftwarePSOPTinRef.[ 82 ]wereavailableforcomparison.SOSandPSOPTbothutilizeatrapezoidalthenHermite-Simpsondiscretizationmethodandanautomaticmeshrenementalgorithmtosolvetheproblem.ThesolutionsgeneratedbythesetwosolversarepresentedwiththesolutionobtainedusingtheLGRcollocationmethodandphmeshrenementmethodinTable 3-4 .ItisclearthattheLGRcollocationmethodreturnsasolutionthatisessentiallyequivalent.ThestatevariablesandcontrolvariablesfoundareplottedtogetherwiththesolutionobtainedusingSOSinFigs. 3-3 and 3-4 ,respectively.Theseguresclearlydemonstrateastrongagreementbetweenthesolutions.ThesolutionobtainedusingtheLGRcollocationmethodhasbeenveriedthroughcomparisonwithsolutionsgeneratedfromotherwell-establisheddirectcollocationmethods.TheresultsofthisstudydemonstratethecomputationaleciencyoftheLGRcollocationmethodtosolveasimplelow-thrustorbittransfer.ItisalsoevidentfromFigs. 3-3 and 3-4 thattheLGRcollocationmethodpairedwiththephmeshrenementmethodisapotentiallymoreecientapproachsinceitrequireslesscollocationpointsthanthemethodemployedinSOS. 57

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3.6NumericalSolutionofOptimalControlProblemAlloptimalcontrolproblemsthataredescribedintheremainingchaptersofthisdissertationweresolvedusingtheMATLABoptimalcontrolsoftwareGPOPS)]TJ /F11 11.955 Tf 11.96 0 Td[(II[ 49 ].GPOPS)]TJ /F11 11.955 Tf 11.95 0 Td[(IIemploysthevariable-orderLGRcollocationmethodpreviouslydescribedinSection 3.3 totranscribetheoptimalcontrolproblemintoalargesparseNLP.TheNLParisingfromtheLGRcollocationmethodwassolvedtoatoleranceof10)]TJ /F6 7.97 Tf 6.59 0 Td[(7usingtheopen-sourceNLPsolverIPOPT[ 29 ].Furthermore,usingtheph-(Nmin,Nmax)adaptivemeshrenementmethoddescribedpreviouslyinSection 3.4 ,theNLPwassolvedonsuccessivemeshestoarelativeaccuracytoleranceof10)]TJ /F6 7.97 Tf 6.59 0 Td[(7withNmin=4andNmax=8.Firstandsecondderivativeswereobtainedanalyticallyusingtheopen-sourcealgorithmicdierentiationpackageADiGator[ 84 ]orapproximatedusingasparsenite-dierencemethod[ 49 , 78 ]. Table3-1. Physicalconstantsforexampleorbittransfer. ConstantValue Isp450sT4.44661810)]TJ /F6 7.97 Tf 6.58 0 Td[(3lbfge32.174fts)]TJ /F6 7.97 Tf 6.58 0 Td[(2Re20925662.73fte1.4076457941016ft3s)]TJ /F6 7.97 Tf 6.58 0 Td[(2J21082.63910)]TJ /F6 7.97 Tf 6.58 0 Td[(6J3)]TJ /F4 11.955 Tf 9.3 0 Td[(2.56510)]TJ /F6 7.97 Tf 6.59 0 Td[(6J4)]TJ /F4 11.955 Tf 9.3 0 Td[(1.60810)]TJ /F6 7.97 Tf 6.59 0 Td[(6 Table3-2. Meshrenementiterationresultsforph-(4,6)algorithm. IterationNRelativeErrorCPUTime(s) 1448.910610)]TJ /F6 7.97 Tf 6.59 0 Td[(22.103421081.790510)]TJ /F6 7.97 Tf 6.59 0 Td[(36.175131711.286410)]TJ /F6 7.97 Tf 6.59 0 Td[(43.355841913.380610)]TJ /F6 7.97 Tf 6.59 0 Td[(51.056951961.004010)]TJ /F6 7.97 Tf 6.59 0 Td[(50.669061979.458410)]TJ /F6 7.97 Tf 6.59 0 Td[(60.5066Total13.8669 58

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Figure3-1. Examplelow-thrusttransferoptimaltrajectory. Table3-3. Meshrenementiterationresultsforvarioushandphmethodswithrelativeerrortolerancesetto10)]TJ /F6 7.97 Tf 6.59 0 Td[(7. MethodRenementsNRelativeErrorCPUTime(s) h-(3)59819.470310)]TJ /F6 7.97 Tf 6.59 0 Td[(823.4846ph-(3,4)58599.471410)]TJ /F6 7.97 Tf 6.59 0 Td[(823.2272ph-(3,6)57739.469910)]TJ /F6 7.97 Tf 6.59 0 Td[(823.1240ph-(3,8)57138.739310)]TJ /F6 7.97 Tf 6.59 0 Td[(819.9378ph-(3,10)65015.171810)]TJ /F6 7.97 Tf 6.59 0 Td[(822.5306h-(4)56929.566110)]TJ /F6 7.97 Tf 6.59 0 Td[(818.0760ph-(4,6)54859.566210)]TJ /F6 7.97 Tf 6.59 0 Td[(817.6091ph-(4,8)54429.566210)]TJ /F6 7.97 Tf 6.59 0 Td[(818.3661ph-(4,10)73759.566210)]TJ /F6 7.97 Tf 6.59 0 Td[(819.6352 Table3-4. SolutionsummaryforLGR-ph,SOS,andPSOPT. Solverw(tf)(lb)tf(s)RelativeError LGR-ph0.220180686815.6)]TJ /F4 11.955 Tf 9.29 0 Td[(9.096799.510)]TJ /F6 7.97 Tf 6.59 0 Td[(6SOS0.220179186810.0)]TJ /F4 11.955 Tf 9.29 -.01 Td[(9.090816.110)]TJ /F6 7.97 Tf 6.59 0 Td[(6PSOPT0.220340386844.6n/a3.510)]TJ /F6 7.97 Tf 6.59 0 Td[(4 59

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Figure3-2. Meshrenementiterationhistoryforph-(4,6)algorithm. 60

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A B C D E FFigure3-3. Examplelow-thrusttransferstatevariablecomparison.A)pvs.t.B)fvs.t.C)gvs.t.D)hvs.t.E)kvs.t.F)Lvs.t. 61

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A B CFigure3-4. Examplelow-thrusttransfercontrolvariablecomparison.A)urvs.t.B)uvs.t.C)uhvs.t. 62

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CHAPTER4DESCRIPTIONOFORBITTRANSFERPROBLEMThepurposeofthischapteristoprovidethenecessarydetailsforsolvingtheproblemoftransferringaspacecraftfromaninitialEarth-orbittoanalEarth-orbitusinglow-thrustpropulsion.First,theequationsofmotion,boundaryconditions,andpathconstraintsaredescribedwhichareusedinboththenon-eclipsingandeclipsingorbittransferstudies.Next,themodelforsolarelectricpropulsiontobeusedintheeclipsingorbittransferstudyonlyispresented.Finally,thedetailsofthesoftwareemployedtosolvetheoptimalcontrolproblemsinthisresearcharegiven. 4.1EquationsofMotionThedynamicsofthespacecraft,modeledasapointmass,aredescribedusingmodiedequinoctialelements(p,f,g,h,k,L)wherepisthesemi-parameter,fandgareelementsthatdescribetheeccentricity,handkareelementsthatdescribetheinclination,andListhetruelongitude[ 85 ].Modiedequinoctialelementswerechosenbecausetheseelementsdonotexhibitsingularitieswhentheeccentricityorinclinationisequaltozero.Thestateofthespacecraftisthengivenas(p,f,g,h,k,L,m),wheremisthevehiclemass.Thecontrolisthethrustdirection,u,whereuisexpressedinrotatingradialcoordinatesasu=(ur,u,uh).Afourth-orderoblategravitymodelisusedandthethrustmagnitudeisassumedconstantwhenthespacecraftisnotinaneclipse.Thedierentialequationsofmotionofthespacecraftarethengivenas dp dt=r p e2p qFp,(4{1) df dt=r p esinLr+1 q(q+1)cosL+f)]TJ /F7 7.97 Tf 13.15 5.03 Td[(g qhsinL)]TJ /F3 11.955 Tf 11.95 0 Td[(kcosLhFf,(4{2) dg dt=r p e)]TJ /F4 11.955 Tf 11.29 0 Td[(cosLr+1 q(q+1)sinL+g+f qhsinL)]TJ /F3 11.955 Tf 11.95 0 Td[(kcosLhFg,(4{3) dh dt=r p es2cosL 2qhFh,(4{4) 63

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dk dt=r p es2sinL 2qhFk,(4{5) dL dt=r p ehsinL)]TJ /F3 11.955 Tf 11.95 0 Td[(kcosLh+p epq p2FL,(4{6) dm dt=)]TJ /F3 11.955 Tf 16.81 8.09 Td[(T geIspFm,(4{7)where q=1+fcosL+gsinL,r=p=q,2=h2)]TJ /F3 11.955 Tf 11.95 0 Td[(k2,s2=1+p h2+k2.(4{8)Inthisresearch,thetruelongitudeisusedastheindependentvariableinsteadoftime.Denotingdierentiationwithrespecttotruelongitudeby()0,thedierentialequationfortisgivenas t0=dt dL=1 FL=F)]TJ /F6 7.97 Tf 6.59 0 Td[(1LGt,(4{9)whiletheremainingsixdierentialequationsfor(p,f,g,h,k,m)thatdescribethedynamicsofthespacecraftaregivenas (p,f,g,h,k,m)0=F)]TJ /F6 7.97 Tf 6.59 0 Td[(1L(Fp,Ff,Fg,Fh,Fk,Fm)(Gp,Gf,Gg,Gh,Gk,Gm).(4{10)Next,thespacecraftacceleration,=(r,,h),ismodeledas =g+T,(4{11)wheregisthegravitationalaccelerationduetotheoblatenessoftheEarthandTisthethrustspecicforce.TheaccelerationduetoEarthoblatenessisexpressedinrotatingradialcoordinatesas g=QTrg,(4{12)whereQr=iriihisthetransformationfromrotatingradialcoordinatestoEarth-centeredinertialcoordinatesandwhosecolumnsaredenedas ir=r krk,ih=rv jjrvjj,i=ihir.(4{13) 64

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Furthermore,thevectorgisdenedas g=gnin)]TJ /F8 11.955 Tf 11.95 0 Td[(grir(4{14)whereinisthelocalNorthdirectionandisdenedas in=en)]TJ /F4 11.955 Tf 11.96 0 Td[((eTnir)ir jjen)]TJ /F4 11.955 Tf 11.96 0 Td[((eTnir)irjj(4{15)anden=(0,0,1).Theoblateearthperturbationsarethenexpressedas gr=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(e r24Xk=2(k+1)Re rkPk(sin)Jk, (4{16) gn=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(ecos r24Xk=2Re rkP0k(sin)Jk, (4{17) whereReistheequatorialradiusoftheearth,Pkisthekth-degreeLegendrepolynomial,P0kisthederivativeofPkwithrespecttosin,andJkrepresentsthezonalharmoniccoecientsfork=(2,3,4).Next,thethrustspecicforceisgivenas T=T mu.(4{18)Whenusingsolarelectricpropulsion,thethrustisgivenas T=2P geIsp(4{19)wherePisthepowerandistheeciency.Finally,thephysicalconstantsusedinthisstudyaregiveninTable 4-1 . 4.2BoundaryConditionsandPathConstraintsTheboundaryconditionsfortheorbittransferaredescribedintermsofbothclassicalorbitalelementsandmodiedequinoctialelements.Theinitialorbitisspeciedintermsof 65

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classicalorbitalelements[ 86 ]as a(L0)=a0,(L0)=0,e(L0)=e0,!(L0)=!0,i(L0)=i0,(L0)=0,(4{20)whereaisthesemi-majoraxis,eistheeccentricity,iistheinclination,isthelongitudeoftheascendingnode,!istheargumentofperiapsis,andisthetrueanomaly.Equation( 4{20 )canbeexpressedequivalentlyintermsofthemodiedequinoctialelementsas p(L0)=a0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e20),h(L0)=tan(i0=2)sin0,f(L0)=e0cos(!0+0),k(L0)=tan(i0=2)cos0,g(L0)=e0sin(!0+0),L0=0+!0+0.(4{21)Theterminalorbitusedinthisresearchisspeciedinclassicalorbitalelementsas a(Lf)=af,(Lf)=Free,e(Lf)=ef,!(Lf)=Free,i(Lf)=if,(Lf)=Free.(4{22)Equation( 4{22 )canbeexpressedequivalentlyintermsofthemodiedequinoctialelementsas p(Lf)=af(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e2f),f2(Lf)+g2(Lf)1=2=ef,h2(Lf)+k2(Lf)1=2=tan(if=2).(4{23)Finally,duringthetransferthethrustdirectionmustbeavectorofunitlength.Thus,theequalitypathconstraint kuk2=u2r+u2+u2h=1(4{24)isenforcedthroughoutallphasesofthetransferwherethespacecraftisallowedtothrust. 66

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4.3ModelforSolarElectricPropulsionAmodelforsolarelectricpropulsionisdescribedfortheorbittransfercasestudywitheclipseconstraints.ThemodelassumesthatthespacecraftappliesmaximumthrustwhenithaslineofsighttotheSunandapplieszerothrustwhenlineofsighttotheSunislost.Themodelconsistsoftwoparts.Therstpartofthemodeldenesthegeometryoftheshadowregion(aneclipse)wherethelineofsighttotheSunislost.Thesecondpartofthemodeldenestheeventconstraintsoftheoptimalcontrolproblembasedonthedenitionoftheshadowregiongeometry.Theeventconstraintsdescribetheconditionsatastartandaterminusofashadowregionalongwiththechangeinthelocationofthespacecraftasitmovesfromanentrancetoanexitofashadowregion. 4.3.1DenitionofaShadowRegionBycalculatingthepositionoftheSun,s,throughoutthetransfer,thepenumbraandumbrashadowregionscanbedened.LowprecisionformulaswereemployedtoecientlydeterminethecoordinatesoftheSuninequatorialrectangularcoordinates[ 87 ].(ThelowprecisionformulasgivethecoordinatesoftheSuntoaprecisionof1/100degandtheequationoftimetoaprecisionof3.5sbetween1950and2050.)Ifasolarpoweredspacecrafttravelsthroughapenumbraregion,thepropulsionsystemwillreceivelimitedsolarenergy,whilepassagethroughanumbraregionwillresultinacompletelossofsolarenergy.InamannersimilartoRef.[ 50 ],thecelestialbodiesareassumedtobesphericalwhichgeneratesshadowregionsthatareconicalprojections.Inthisapproach,onlyapenumbrashadowregionisconsideredandthethrustofthespacecraftisassumedtobezerowhentravelingthroughtheshadowregion.ThepenumbraconegeometryisshowninFig. 4-1 ,wherethedistancebetweenpointPandthecenteroftheearthisdenedas p=Redp Re+Rs(4{25)andtheanglepisdenedas p=sin)]TJ /F6 7.97 Tf 6.58 0 Td[(1Re p.(4{26) 67

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Theprojectedspacecraftlocationisthenutilizedtolocatethepenumbraconeterminators.Figure 4-2 showstheprojectedspacecraftlocationgeometry,wherethelocationofthespacecraft,r,isdenedas r=266664r s2(cosL+2cosL+2hksinL)r s2(sinL)]TJ /F8 11.955 Tf 11.95 0 Td[(2sinL+2hkcosL)2r s2(hsinL)]TJ /F3 11.955 Tf 11.95 0 Td[(kcosL)377775.(4{27)Thesolarunitvector ^s=s jjsjj(4{28)isutilizedtodeterminetheprojectionofthespacecraftalong^swhichisdenedas rp=(r^s)^s.(4{29)Thedistancebetweenthecenterofthepenumbraconeandthespacecraftattheprojectionpointisdenedas =r)]TJ /F9 11.955 Tf 11.95 0 Td[(rp,(4{30)andthedistancebetweenthecenterofthepenumbraconeandthepenumbraterminatorpointattheprojectedspacecraftlocationisdenedas =(p+jjrpjj)tanp.(4{31)Acomparisonofthedistancewiththemagnitudeofthevectordeterminestheshadowterminatorpointsasdescribedbelow: 1. Shadowterminatorpointsareonlyfeasibleifr^s<0.However,ifjjjj>,thespacecraftisstillinsunlight. 2. Penumbraterminatorpointsoccurwhenjjjj=andthespacecraftisinthepenumbraconeifjjjj<. 68

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4.3.2EventConstraintsDeningEntranceandExitofaShadowRegionThefollowingeventconstraintsdenetheentranceandexitofashadowregion.First,becausethedurationofcoastingightduringaneclipseislessthanonethirdofanorbitalperiod,itisassumedduringaneclipsethatthespacecraftisundertheinuenceofcentralbodygravity(thatis,Keplerianmotion).Asaresult,therstvemodiedequinoctialelements(p,f,g,h,k)alongwithmareconstantduringaneclipseandtheonlymodiedequinoctialelementthatchangesisthetruelongitude,Lalongwitht.Next,letL(r)beoptimizationparametersthatdenethechangeintruelongitudeduringthertheclipse(wherer=1,...,R)]TJ /F4 11.955 Tf 12.34 0 Td[(1andRisthenumberofphases).Becausethedurationoftheeclipseislessthanonethirdofanorbitalperiod,thefollowingmoderateconstraintisplacedonL(r): 0
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whiletheinitialpointsofalltheintermediatephasesandterminalphasemustsatisfy jj(L(r+1)0)jj)]TJ /F8 11.955 Tf 21.25 0 Td[(=0,(r=1,...,R)]TJ /F4 11.955 Tf 11.96 0 Td[(1).(4{37)Finally,becausethespacecraftisassumedtobeundertheinuenceofcentralbodygravitationduringaneclipse,thefollowingconstraintsareimposedonthestatecomponents(p,f,g,h,k,m)atthestartandterminusofaneclipse: p(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(p(r)(L(r)f)=0,f(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(f(r)(L(r)f)=0,g(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(g(r)(L(r)f)=0,h(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(h(r)(L(r)f)=0,k(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(k(r)(L(r)f)=0,m(r+1)(L(r+1)0))]TJ /F3 11.955 Tf 19.26 0 Td[(m(r)(L(r)f)=0.(4{38)ItisemphasizedagainthatKeplerianmotionduringaneclipseisassumedbecausethedurationofaneclipseisonlyasmallfractionofanorbitalperiod. Table4-1. Physicalproperties. ConstantValue ge9.80665ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2Re6378140me3.98600471014m3s)]TJ /F6 7.97 Tf 6.58 0 Td[(2Rs695500000ms1.327124400181020m3s)]TJ /F6 7.97 Tf 6.59 0 Td[(2J21082.63910)]TJ /F6 7.97 Tf 6.58 0 Td[(6J3)]TJ /F4 11.955 Tf 9.3 0 Td[(2.56510)]TJ /F6 7.97 Tf 6.59 0 Td[(6J4)]TJ /F4 11.955 Tf 9.3 0 Td[(1.60810)]TJ /F6 7.97 Tf 6.59 0 Td[(6 70

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Figure4-1. Penumbraconegeometry. Figure4-2. Projectedspacecraftlocationgeometry. 71

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CHAPTER5ORBITTRANSFERSTUDY1:NON-ECLIPSINGInthisstudy,orbittransferswithouteclipseconstraintswereconsidered.Thespacecraftwasassumedtothrustcontinuouslywithaconstantmagnitudethroughouttheentiretransfer.Thesingle-phaseoptimalcontrolproblemisstatedrst,followedbythedescriptionofanapproachforgeneratinginitialguessesnecessaryforsolvingtheoptimalcontrolproblem.Next,becausethesolutionsobtainedfromtheNLPareonlylocallyoptimal,amotivationisprovidedfordevelopingasearchmethodtoobtainsolutionsthatareclosertotheglobaloptimal.Thesearchmethoddevelopedisthendescribed.Afterdescriptionoftheapproachdeveloped,threeorbittransfersaresolvedforarangeofinitialthrustaccelerationandspecicimpulsevaluesandtheresultsareanalyzed.Finally,apost-optimalityanalysisisperformed. 5.1OptimalControlProblemSincethespacecraftcontinuouslythruststhroughouttheentiretransfer,thetrajectoryoptimizationproblemwasformulatedasasingle-phaseoptimalcontrolproblem.Theoptimalcontrolproblemisstatedasfollows.Determinethetrajectory(p(L),f(L),g(L),h(L),k(L),m(L),t(L))andthecontrol(ur(L),u(L),uh(L))thatminimizethecostfunctional J=tf,(5{1)subjecttothedynamicconstraintsofEqs.( 4{9 )and( 4{10 ),theinitialconditionsofEq.( 4{21 ),theterminalconditionsofEq.( 4{23 ),andthepathconstraintsofEq.( 4{24 ).Finally,itisnotedthat=1=86400istheconversionfactorfromunitsofsecondstounitsofdays. 5.2InitialGuessGenerationInordertosolvethesingle-phaseoptimalcontrolproblemdescribed,itwasnecessarytoprovideaninitialguessfromwhichtheNLPsolverwillconvergetoasolution.Becausethisresearchisfocusedonsolvingaproblemwhosesolutionwillresultinalargenumberoforbitalrevolutions,theinitialguessmustitselfcontainanumberoforbitalrevolutionsthatis 72

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reasonablyclosetotheactualnumberoforbitalrevolutionsofthesolutionobtainedbytheNLPsolver.Therefore,anovelinitialguessprocedurewasdevisedwhereasequenceofoptimalcontrolsub-problemsweresolved.Thegoalofeachsub-problemwastodeterminethestateandcontrolthattransferthespacecraftfromtheinitialorbittotheterminalconditionsthatminimizethefollowingmeansquarerelativedierence: J=p(Lf))]TJ /F3 11.955 Tf 11.95 0 Td[(pd 1+pd2+f2(Lf)+g2(Lf))]TJ /F3 11.955 Tf 11.95 0 Td[(e2d 1+e2d2+"h2(Lf)+k2(Lf))]TJ /F4 11.955 Tf 11.96 0 Td[(tan2(id 2) 1+tan2(id 2)#2.(5{2)Inotherwords,theobjectiveoftheoptimalcontrolsub-problemistoattainasolutionthatisascloseinproximitytothedesiredterminalsemi-parameter,pd,eccentricity,ed,andinclination,id.Eachsub-problemisevaluatedatmostoveronetruelongitudecycleusingtheterminalstateoftheprevioussub-problemastheinitialstateofthecurrentsub-problem.Thecontinuous-timeoptimalcontrolsub-problemisthenstatedasfollows.MinimizethecostfunctionalofEq.( 5{2 )subjecttothedynamicconstraintsofEqs.( 4{9 )and( 4{10 ),thepathconstraintofEq.( 4{24 ),andtheboundaryconditions p(r)(L(r)0)=p(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.58 0 Td[(1)f),p(r)(L(r)f)=Free,f(r)(L(r)0)=f(r)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f),f(r)(L(r)f)=Free,g(r)(L(r)0)=g(r)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f),g(r)(L(r)f)=Free,h(r)(L(r)0)=h(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f),h(r)(L(r)f)=Free,k(r)(L(r)0)=k(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f),k(r)(L(r)f)=Free,m(r)(L(r)0)=m(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(L(r)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f),m(r)(L(r)f)=Free,L(r)0=L(r)]TJ /F6 7.97 Tf 6.58 0 Td[(1)f,L(r)fL(r)]TJ /F6 7.97 Tf 6.58 0 Td[(1)f+2,(5{3)forr=1,...,RwhereRrepresentsthetotalnumberoftruelongitudecycles.Theinitialconditionsfortherstcycle,whenr=1,aresimplytheinitialconditionsstatedinEq.( 4{21 ).Oncethedesiredterminalconditions,asstatedinEq.( 4{23 ),areobtainedwithinauserspeciedtolerance,thesub-problemsolutionsarethencombinedtoformtheinitialguess.The 73

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initialmeshiscomprisedofintervalsbasedonthetotalnumberoftruelongitudecyclesandanarbitrarilychosennumberofcollocationpointsassignedtoeachinterval. 5.3SearchMethodtoAssistinObtainingGloballyOptimalSolutionItisgenerallythecasethatgradient-basedoptimizationmethodsconvergetolocallyoptimalsolutionsasopposedtogloballyoptimalsolutions.Inthecaseofthelow-thrustorbitaltransferproblemssolvedinthisresearch,itwasfoundthattheoptimalsolutiontypicallycontainedthesamenumberoftruelongitudecyclesasthatoftheinitialguessduetothefactthattheinitialguesswasveryclosetosatisfyingtheterminalconstraints.Thus,whiletheNLPsolverconvergeswiththeinitialguessprovided,thesolutionisusuallynottheglobaloptimal.Inordertoobtainasolutionwithanumberoftruelongitudecyclesdierentfromthatoftheinitialguess,thenaltruelongitudewasboundedwithinaspeciedcycle(thatis,toliewithinaspeciedintervalof2).Byboundingthenaltruelongitude,theNLPsolverisforcedtodeviatefromtheinitialguessandpotentiallymoveclosertoaglobalsolution.Figure 5-1 showsthecostobtainedwhenthenaltruelongitudeisboundedasdescribedaboveandwhenthenaltruelongitudeisfreefortheGTOtoGEOorbittransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).ItisseenfromFig. 5-1 thatthesolutionobtainedusingtheprovidedinitialguesswithafreeterminaltruelongitudedoesnothavethelowestcostsolution.Instead,thelowestcostliessomewherebetween95and105truelongitudecycles.BasedonthestructureshowninFig. 5-1 ,thefollowingsimplesearchmethodisusedtoidentifytheapproximatelocationofthegloballyoptimalsolution.First,aninitialguessisgeneratedasdescribedinSection 5.2 andthenaltruelongitudethatisobtainedfromthisinitialguess,denotedL(0)f,isthestartingpointforthesearchmethod.Aniterationonthenaltruelongitudeisthenperformedasfollows,whereKistheiterationnumberandK=0correspondstoL(0)f.TheoptimalcontrolproblemissolvedforanaltruelongitudeLf2[L(K)f)]TJ /F4 11.955 Tf 12.13 0 Td[(2,L(K)f]=I(K)landthecostobtainedfromthissolutionisdenotedJ(K)l.Next,inordertodeterminewhichdirectiontosearchforalowercostsolution,theoptimalcontrolproblemissolvedagainforanaltruelongitudeLf2[L(K)f,L(K)f+2]=I(K)randthecost 74

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associatedwiththissolutionisdenotedJ(K)r.Thecyclethatcontainsthelowestcostisthenobtainedusingthefollowingiterativeprocess: SetK!K+1. Case1(minimumliestotherightoftheinitialization):IfJ(K)]TJ /F6 7.97 Tf 6.58 0 Td[(1)l>J(K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)r,thensetL(K)f=L(K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)f+2,I(K)l=I(K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)r,J(K)l=J(K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)r,andI(K)r=[L(K)f,L(K)f+2].ThensolvetheoptimalcontrolproblemagainforLf2I(K)randthecostobtainedisdenotedJ(K)r.RepeatuntilJ(K)l
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dierentialequationsforthesemi-majoraxis,eccentricity,andinclination[ 88 ](wherenisthemeanmotionandx=p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e2): da dt=2esin nxur+2ax nru, (5{4) de dt=xsin naur+x na2ea2x2 r)]TJ /F3 11.955 Tf 11.95 0 Td[(ru, (5{5) di dt=rcos(+!) na2xuh. (5{6) ExaminingEq.( 5{4 ),thesemi-majoraxiswillincreasemostrapidlywhenthecontrolpointseitherinthepositiveudirection,radiallyoutwardnear==2(halfwaybetweenperiapsisandapoapsis),orradiallyinwardnear=3=2(halfwaybetweenapoapsisandperiapsis).ThecyclicbehaviorofurasitrelatestoEq.( 5{4 )increasesapoapsisanddecreasesperiapsiswhen2[0,]anddecreasesapoapsisandincreasesperiapsiswhen=[,2].Furthermore,thrustingradiallyinthismannerincreasesnotonlythesemi-majoraxis,butalsoincreasestheeccentricity.Ontheotherhand,fromEq.( 5{5 )theeccentricitywilldecreasemostrapidlywhenthecontrolpointseitherinthepositiveudirection,radiallyinwardnear==2,orradiallyoutwardnear=3=2.ThecyclicbehaviorofurasitrelatestoEq. 5{4 decreasesapoapsisandincreasesperiapsiswhen2[0,]andincreasesapoapsisanddecreasesperiapsiswhen=[,2].Thrustingradiallyinthismannerdecreasesthesemi-majoraxis,alongwithdecreasingtheeccentricity.Finally,fromEq. 5{6 ,theinclinationwilldecreasewhencos(+!)uhisnegative. 5.4.1OptimalGTOtoGEOTransfersFigure 5-2 showsaviewinEarth-centeredinertialCartesian(ECI)coordinates(x,y,z)ofatypicaloptimalGTOtoGEOtrajectoryforthecase(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,3000s).Theoptimalsolutionhasa8.67%changeinmass,adurationof88.56days,and119.13orbitalrevolutions.ItisseenfromFig. 5-3 thatthesemi-majoraxisincreasesrapidlynearthestartofthetransferandincreasesmoreslowlytowardstheendofthetransfer.Furthermore,Fig. 5-4 showsthattheeccentricitydecreasesslowlynearthestartofthetransfer 76

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anddecreasesmorerapidlytowardstheendofthetransfer.Also,Fig. 5-5 showsthattheinclinationdecreasesatanapproximatelylinearratethroughouttheentiretransfer.Therateofchangeforeachelementthea,e,andivariesonlyslightlyasafunctionofT=m0.FurtherinsightintothebehavioroftheoptimalGTOtoGEOtransfersisobtainedbyexaminingthecontrolu=(ur,u,uh)alongdierentsegmentsoftheoptimalsolution.ThetypicaloverallbehaviorofuisshowninFig. 5-6 for(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,3000s).Acloserexaminationofurevealsthatthefollowingfoursegmentsidentifythekeyfeaturesoftheoptimalcontrol:(1)therstfeworbitalrevolutionsofthetransfer,(2)theregionwhereur0,(3)theregionwhereuhbecomes)]TJ /F4 11.955 Tf 9.29 0 Td[(1,and(4)thenalrevolutionsofthetransfer.First,Fig. 5-7A showstheoptimalcontrolintherstfeworbitalrevolutionsofthetransfer.Inthissegment,urispositivenear==2andnegativenear=3=2,whileu1alongall.Eventhoughurincreasesthesemi-majoraxis,itsimultaneouslyincreaseseccentricity.Thissmalleectofurincreasingeccentricity,however,isnegatedbythefactthatuliespredominantlyinthepositiveu-direction,therebyresultinginanoverallincreaseinsemi-majoraxisanddecreaseineccentricity.Lastly,fromFig. 5-8A thelargestnegativeslopeindi=dtisatapoapsis.ThisisconsistentwithFig. 5-8B whichshowsthatuhismostpositiveandcos(+!)=)]TJ /F4 11.955 Tf 9.29 0 Td[(1when=.Next,Fig. 5-7B showsuinthesegmentofanoptimalGTOtoGEOtransferwhereur0.Foreveryorbitalrevolutionontheoptimalsolutionbeyondwhereurbecomeszero(thatis,allvaluesbeyond=(2)=38.5asshowninFig. 5-7B ),upointsradiallyinwardnear==2suchthatapoapsisdecreasesandperiapsisincreaseswhen2[0,]andpointsradiallyoutwardnear=3=2suchthatapoapsisincreasesandperiapsisdecreaseswhen2[,2].Thrustingradiallyinthismannerdecreasesboththesemi-majoraxisandeccentricity.Althoughurdecreasesthesemi-majoraxis,thethrustdirectionliespredominantlyinthepositiveudirection,therebyincreasingthesemi-majoraxisanddecreasingtheeccentricity.Finally,becauseuhismostpositivenear=andcos(+!)=)]TJ /F4 11.955 Tf 9.3 0 Td[(1,theinclinationdecreasesmostrapidlynearapoapsis. 77

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Next,Fig. 5-7C showsuinthesegmentofanoptimalGTOtoGEOtransferwhereuhdropsto)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Inthissegment,urcontinuestopointinwardnear==2andoutwardnear=3=2,decreasingboththesemi-majoraxisandtheeccentricity.Moreover,unolongerdominatesthethrustdirection,reachingitsmostpositivevalueatapoapsisandagraduallydecreasingvalueatperiapsis.Thrustinginthismannerintheurandudirectionsraisesperiapsiswhen2[0,]andlowersapoapsiswhen2[,2].Consequently,thesemi-majoraxisincreaseswhileeccentricitydecreases.Lastly,fromFig. 5-8D ,uhattainsitsmostnegativevaluenear=andcos(+!)=)]TJ /F4 11.955 Tf 9.3 0 Td[(1andnear=2andcos(+!)=+1.Whiletheinclinationcontinuestodecreasesignicantlynearapoapsisofthetransfer,Fig. 5-8C showsthatdi=dtisalsonegativenearperiapsis.Finally,Fig. 5-7D showsuduringthenalfewrevolutionsofanoptimalGTOtoGEOtransfer.Itisseenthaturisnegativenear==2andpositivenear=3=2,whileuispositivenear=2andisnegativenear=.Consequently,bythrustinginthismannerperiapsisincreasesandapoapsisdecreases,therebyresultinginalargersemi-majoraxisandasmallereccentricity.Finally,uhisnegativenear=2andispositivenear=.Becausetheorbitisnearlycircularneartheendofthetransfer,therateatwhichinclinationdecreasesisessentiallythesamenearperiapsisandapoapsis. 5.4.2OptimalLEOtoGEOTransfersFigure 5-9 showsathree-dimensionalviewinECIcoordinates(x,y,z)ofatypicaloptimalLEOtoGEOtrajectoryforthecase(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,1000s).Theoptimaltrajectoryhasa43.91%changeinmass,takes126.61days,andcontains790.12revolutions.Thesemi-majoraxisandinclinationforallvaluesofT=m0andIsp=1000sareshowninFigs. 5-10 and 5-11 ,respectively.Itisseenthatthesemi-majoraxisincreasesataslowerrateatthebeginningofthetransferandincreasesmorerapidlytowardstheendofthetransfer.Theinclinationdecreasesataslowerrateatthebeginningofthetransferanddecreasesmorerapidlytowardstheendofthetransfer.Theeccentricityisroughlyzerothroughouttheentiretransfer. 78

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ThestructureoftheoptimalLEOtoGEOtransfersisexaminedingreaterdetailbystudyingthecomponentsofthecontrolalongtheoptimalsolution.Thetypicaloverallbehaviorofthecontrolu=(ur,u,uh)isshowninFig. 5-12 for(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,1000s).Asexpected,urstaysnearzero,whereasuanduharenon-zerothroughouttheentiretransfer.Greaterinsightintothestructureoftheoptimalcontrolisnowobtainedbyexaminingthecontrolnearthestartandtheterminusofthetransfer.Figure 5-14A showsuasafunctionofnearthestartofthetransfer,where=L)]TJ /F4 11.955 Tf 12.07 0 Td[()]TJ /F8 11.955 Tf 12.08 0 Td[(!LbecauseL,!,and,areapproximatelyzero(Fig. 5-13 ).ItisseenfromFig. 5-14A thatupointsinthepositiveudirectiontoincreasethesemi-majoraxis(Eq.( 5{4 )),whileuhattainsitsmostpositivevaluenearapoapsisanditsmostnegativevaluenearperiapsistodecreasetheinclination(Eq.( 5{6 )).Neartheterminusofthetransfer,=L)]TJ /F4 11.955 Tf 12.02 0 Td[()]TJ /F8 11.955 Tf 12.02 0 Td[(!L+sinceapproachesanapproximatevalueof)]TJ /F8 11.955 Tf 9.3 0 Td[((Fig. 5-13 )and!isassumedtobezero.Fig. 5-14B showsuasafunctionofneartheendofthetransferwhereitisseenthatupointsinthepositiveudirectionanduhattainsitsmostpositivevaluenearapoapsisanditsmostnegativevaluenearperiapsis.ThusitisclearthroughouttheLEOtoGEOtransferthattheoptimalthrustdirectionincreasesthesemi-majoraxisanddecreasestheinclinationwhiletheeccentricityremainsrelativelyunchangednearzero. 5.4.3OptimalSSTOtoGEOTransfersFigure 5-15 showsathree-dimensionalviewinECIcoordinates(x,y,z)ofatypicaloptimalSSTOtoGEOtrajectoryforthecase(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).Theoptimaltrajectoryhasa7.11%changeinmass,takes72.59days,andcontains60.21revolutions.Thesemi-majoraxis,eccentricity,andinclinationforallvaluesofT=m0andIsp=3000sareshowninFigs. 5-16 , 5-17 ,and 5-18 ,respectively,andthetypicaloverallbehaviorofthecontrolu=(ur,u,uh)isshowninFig. 5-19 for(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,3000s).Thesemi-majoraxisandeccentricitybothsteadilydecreasethroughouttheentiretransfer.Theinclination,however,decreasesrapidlyduringthebeginninghalfofthe 79

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transferanddecreasesataslowerratefortheremainderofthetransfer.FromFig. 5-16 ,thehighlyoscillatorybehaviorofthesemi-majoraxisisclearlyseen.Greaterinsightintothestructureoftheoptimaltrajectoryisnowobtainedbyexaminingthecontrolnearthestartandtheterminusofthetransfer.Thecontrolcomponents,semi-majoraxis,eccentricity,andinclinationduringtherstfeworbitalrevolutionsareshowninFig. 5-20A .Itisseenthaturisnegativewhen=[0,]andispositivewhen=[,2].Thiscyclicbehaviorofurdecreasesthesemi-majoraxisandtheeccentricity.Thesemi-majoraxisisshowntodecreaseonlywhen=[0,3=4]and=[5=4,3=2]asseeninFig. 5-20B .Ontheotherhand,thesemi-majoraxisincreaseswhen=[3=4,5=4]and=[3=2,2].Theincreaseobservedinthesemi-majoraxisislargelyduetouwhichispositivewhen=[=2,3=2].Since=[=2,3=2]enclosesapogee,asignicantamountoftimeisspentdecreasingtheeccentricityandconsequentlyincreasingthesemi-majoraxis.Theneteectofuranduisnochangeinthesemi-majoraxisandadecreaseintheeccentricity.Thelargestdecreaseintheinclinationisseenatapoapsiswhenuhismostpositiveandcos(+!)=)]TJ /F4 11.955 Tf 9.29 0 Td[(1,similartotheobservationsmadeinbothofthepreviousorbittransferexamples.Sincetheinitialorbithasalargesemi-majoraxisandhigheccentricity,aninclinationchangeismoreecientatthebeginningofthetransfer.Therefore,theemphasisofthecontroleortintherstfeworbitalrevolutionsistodecreasetheeccentricityandtheinclination.Thecontrolcomponents,semi-majoraxis,eccentricity,andinclinationduringthenalfeworbitalrevolutionsareshowninFig. 5-21 .Thebehaviorofurisverysimilartothebehavioritdidduringtherstfeworbitalrevolutions,whereurisnegativewhen=[0,]andispositivewhen=[,2].Thebehaviorofu,however,haschangedslightlyinthatispositiveforashorteramountoftimewhen=[3=4,5=4]insteadof=[=2,3=2].Whilethesemi-majoraxisstillhasanoscillatorybehavior,theoverallneteectofuranduisnowtodecreasethesemi-majoraxisasseeninFig. 5-21B .Inadditiontodecreasingthesemi-majoraxis,urandualsodecreasetheeccentricity.Sincetheorbitisnearlycircular,therateat 80

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whichtheeccentricityandinclinationdecreaseisessentiallylinearlysinceperiapsisandapoapsisareapproximatelythesame. 5.4.4EstimationofMinimum-TimeTransferTimeAkeyfeatureoftheresultsistheabilitytoestimatetheoptimaltransfertimeasafunctionofinitialthrustaccelerationandspecicimpulse.Figure 5-22 showsforeachoftheorbittransfersthenaltimeoftheorbittransferasafunctionofthespecicimpulseforeachoftheinitialthrustaccelerationvaluesexamined.ForeachvalueofT=m0,tfincreasesslightlyasIspincreasesinamannersimilartothatofapowerfunction tf=AIBsp+C,(5{7)wherethecoecientsA,B,andCarefunctionsofT=m0becauseeachvalueofT=m0hasanassociatedpowerfunctionexpressionfortfintermsofIsp.ThecoecientsA,B,andCaredeterminedasfollows.Figure 5-23 showsthecoecientAasafunctionofT=m0fortheGTOtoGEO,LEOtoGEO,SSTOtoGEOtransfers,respectively.ItisseenthattherelationshipbetweenAandT=m0hastheform A=a1(T=m0)b1,(5{8)wherea1andb1areconstantcoecients.Figure 5-24 showsthecoecientBasafunctionofT=m0forthethreeorbittransfersconsidered.BecauseBhasnosignicantchangeasafunctionofT=m0,itisassumedthatBisconstant,andforanyparticularorbitaltransferthisconstantistheaveragevalueofBoverallvaluesofT=m0andIspforthattransfer.Figure 5-25 showsthecoecientCasafunctionofT=m0forthethreeorbittransfersconsidered.ItisseenthattherelationshipbetweenCandT=m0isgivenas C=a2(T=m0)b2,(5{9) 81

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wherea2andb2areconstants.Therefore,theestimatedtransfertime,^tf,canbewrittenasafunctionofbothIspandT=m0andisgivenas ^tf=a1(T=m0)b1(Isp)B+a2(T=m0)b2.(5{10)Valuesforthecoecientsa1,b1,B,a2,andb2areshowninTable 5-2 forallthreeorbittransfers.Equation 5{10 makesitpossibletoestimatethenaltransfertime,tf,forvaluesofIspthataredierentfromthoseobtainedinthisstudywithouthavingtore-solvetheoptimalcontrolproblem. 5.4.5EstimationofMinimum-TimeFinalTrueLongitudeAnotherkeyfeatureoftheresultsistheabilitytoapproximateL(tf)asafunctionofthenaltransfertimeandspecicimpulse.Figure 5-26 showsLfasafunctionoftfforallthreeorbittransfersconsidered.ItisseenthatL(tf)increaseslinearlyasafunctionoftfforeachvalueofIsp,thatis, Lf=Dtf+E,(5{11)whereDandEarefunctionsofIsp.ExpressionsforcoecientsDandEaredeterminedasfollows.Figure 5-27 showsthecoecientDasafunctionofIspforthethreeorbittransfersconsidered.ItisseenthatastheIspincreases,Ddecreasesinamannersimilartothatofanexponentialfunction D=a3eb3Isp+c3ed3Isp,(5{12)wherea3,b3,c3,andd3areconstantcoecients.Figure 5-28 showsthecoecientEasafunctionofIspfortheorbittransfersconsidered.ItisseenthatastheIspincreases,Eissmall.Forexample,Eisnomorethanhalfatruelongitudecycleforanyoftheorbittransfersexamined.Consequently,Eistreatedasaconstantandforanyorbitaltransfer,thisconstantistheaveragevalueofEoverallvaluesofT=m0andIspforthattransfer.UsingthederivedexpressionsforthecoecientsDandE,LfcanbewrittenasafunctionofbothIspandtfand 82

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isgivenby ^Lf=a3eb3Isp+c3ed3Isptf+E(5{13)wherea3,b3,c3,d3,andEaretheregressionconstantsand^LfdenotestheestimateofLf.ValuesfortheregressionconstantsareshowninTable 5-3 forallthreeoftheorbittransfers.TogetherwithEq. 5{10 ,Eq. 5{13 makesitpossibletoquicklyestimateLfatpointsforwhichtheresultswerenotobtained.Equations 5{10 and 5{13 areextremelyusefulforgeneratinginitialguessesandprovideanexcellentstartingpointforthesimplesearchmethodasdescribedinSection 5.3 . 5.4.6CoecientofDeterminationtoAssessQualityofRegressionsTodemonstratethequalitywithwhichtheregressionmodelsfor^tfand^Lfttheobserveddata,thecoecientsofdetermination,R2,arecalculated[ 89 ].Thecoecientofdeterminationisavaluebetween0and1,whereavalueofunityindicatesaperfectt.Thecoecientofdeterminationfor^tfiscalculatedasfollows.Lettand^tbetheobservedandpredictedvalues,respectively,oftheithvalueoftf.Then,thesumofthesquaresoftheerror,denotedS,iscalculatedas S=nXi=1t)]TJ /F4 11.955 Tf 11.54 1.61 Td[(^t2.(5{14)Next,letSyydenotethetotalsumofsquares Syy=nXi=1t)]TJ /F4 11.955 Tf 11.55 1.61 Td[(tf2,(5{15)wheretfisthemeanoftheobservedvaluesoftf.Finally,thecoecientofdeterminationiscalculatedusing R2=1)]TJ /F3 11.955 Tf 17.2 8.08 Td[(S Syy.(5{16)Thecoecientofdeterminationfor^Lfiscalculatedinasimilarmannerto^tf.AllcoecientsofdeterminationforthethreeorbittransfersconsideredareshowninTable 5-4 ,whereitisseenthatR2isclosetounityinallcases. 83

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5.4.7Post-OptimalityAnalysisItisknownfrompreviousresearchthattherst-orderoptimalityconditionsoftheNLParisingfromdiscretizationofacontinuousoptimalcontrolproblemviatheLGRcollocationmethodareadiscreteapproximationoftherst-ordercalculusofvariationsoptimalityconditionsoftheoptimalcontrolproblem[ 48 , 76 ].Moreover,thecostateoftheoptimalcontrolproblemcanbeobtainedviaasimplelineartransformationoftheLagrangemultipliersoftheNLParisingfromtheLGRcollocation.InadditiontotheequivalencebetweentheNLPandcalculusofvariationsoptimalityconditions,ithasalsobeenproventhatthesolutionobtainedusingthevariable-order(hp)LGRcollocationmethodconvergesexponentially(thatis,thestate,control,andcostateassociatedwiththeLGRcollocationmethodallconverge)attheconvergencerategiveninRef.[ 80 ].Consequently,bysolvingtheNLParisingfromtheLGRcollocationmethodonanappropriatemesh,anaccurateapproximationtothesolutionoftheoptimalcontrolproblemisobtainedtoboththeprimalvariables(thatis,thestateandcontrol)andthedualvariable(thatis,thecostate).Thereforebysolvingthevariable-orderLGRNLPonasucientlyaccuratemesh,itispossibletoverifytheextremalityofthesolutionswith-outhavingtosolvetheHamiltonianboundary-valueproblemthatarisesfromthecalculusofvariations.Inotherwords,byobtainingthesolutiontothevariable-orderLGRNLPonanappropriatemeshtheoptimalityofthesolutioncanbeveriedwithouthavingtoresorttosolvingtheoptimalcontrolproblemusinganindirectmethod.Inthissectiontheproximityofthenumericalsolutionstothetrueoptimalsolutionsisinvestigatedbyexaminingvariousaspectsoftherst-ordercalculusofvariationsconditions.Inthisanalysistherst-ordervariationalconditionsarepresentedintermsoftheclassicalorbitalelements(asopposedtothemodiedequinoctialelementswhichwereusedtosolvetheoptimalcontrolproblem),wheretherst-orderoptimalityconditionsareobtainedintermsoftheclassicalorbitalelementsasfollows.First,thediscreteapproximationofthecostateintermsofthemodiedequinoctialelementsareobtainedusingthetransformationoftheNLPLagrangemultipliersasdescribedinRefs.[ 48 ]and[ 76 ](whereitisnotedthatGPOPS)]TJ /F11 11.955 Tf 11.95 0 Td[(II 84

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performsthiscostatecomputationaftertheNLPissolved).Next,thecostateapproximationintermsofthemodiedequinoctialelementsobtainedfromtheLGRcollocationmethodistransformedtothecostateintermsofclassicalorbitalelementsusingtherelationshipbetweenthemodiedequinoctialelementcostateandtheclassicalorbitalelementcostateasderivedintheAppendix.Then,usingthefactthatthecostateisthesensitivityofthecostwithrespecttothestatealongtheoptimalsolution,thecostateintermsofclassicalorbitalelementsattheinitialtimeisalsoapproximatedbysolvingtheoptimalcontrolproblemataperturbedinitialorbitalelementandtakingtheratioofthechangeincosttothechangeintheorbitalelementofinterest(forexample,ifoneisinterestedincomputingthecostateassociatedwiththeeccentricity,thentheratioofthechangeinthecosttoaperturbationintheeccentricityattheinitialpointiscomputed).Thecostatesassociatedwiththeclassicalorbitalelementsofinterestwereveriedbyresolvingtheoptimalcontrolproblemwithasmallperturbationintheinitialsemi-majoraxis,initialeccentricity,andinitialinclination.Foraperturbationintheinitialsemi-majoraxis,thechangeincostfromtheoptimalcostisapproximatedas JJ+@J @a(L0)a(L0))]TJ /F3 11.955 Tf 11.96 0 Td[(a(L0)(5{17)whereJandJdenotethecostontheperturbedandoptimalsolutions,respectively.Therefore,theestimatedsemi-majoraxiscostateatL=L0isapproximatedby @J @a(L0)J)]TJ /F3 11.955 Tf 11.96 0 Td[(J a(L0))]TJ /F3 11.955 Tf 11.96 0 Td[(a(L0)=J a(5{18)whichisthencomparedtothederivedcostatevalue,a(L0).Foraperturbationintheinitialeccentricityorinitialinclination,theestimatedcostatevalueiscalculatedinasimilarmanner(thatis,replacethesemi-majoraxis,a,witheithertheeccentricity,e,ortheinclination,i,inEqs.( 5{17 )and( 5{18 )).Inthisanalysis,theperturbationsintheinitialsemi-majoraxis,initialeccentricity,andinitialinclinationwerea=1000m,e=0.0001andi=0.00017453rad(=0.01deg),respectively.Tables 5-5 , 5-6 , 5-7 showthecostate 85

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approximations(a(L0),e(L0),i(L0))alongsidetheratiosofthecosttotheperturbations,(J=a,J=e,J=i)intheorbitalelementsfortheGTOtoGEOcasewithT=m0=2.5010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,fortheLEOtoGEOcasewithT=m0=4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2,andfortheSSTOtoGEOcasewithT=m0=3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2.ForallthreecasestheLGRcostateapproximationscloselymatchtheestimatedchangeincostduetoaperturbationintheclassicalorbitalelementofinterest,andthecostateapproximationsareconsistentwiththeexpectedbehavior(forexample,increasingtheinitialsemi-majoraxisforeitherorbittransferdecreasesthecost,whileincreasingtheinitialeccentricityincreasesthecost).Moreover,itisseenthatperturbingtheinitialeccentricitysignicantlyincreasesthecostfortheGTOtoGEOcasebutincreasesthecostmuchlessfortheLEOtoGEOcase.Also,inallcasesincreasingtheinitialinclinationincreasesthecost.Finally,inallcasesthemagnitudeofcostsensitivityincreasesasthespecicimpulseincreases.Thislastresultisconsistentwiththefactthattheeciencyoftheengineincreasesasthespecicimpulseincreases.Asafurthervericationofthecloseproximityofthenumericalsolutionstothetrueoptimalsolution,thenalcolumnofTables 5-5 , 5-6 ,and 5-7 showthemaximumabsolutevalueofHu@H=@u=(Hur,Hu,Huh)onL2[L0,Lf]givenasmaxL2[L0,Lf](jHurj,jHuj,jHuhj),whereHiscomputedasgivenintheAppendixusingthecostateapproximationobtainedfromtheLGRcollocationmethodasdescribedinRef.[ 76 ].Becausethecontrolliesontheinterioroftheallowablecontrolsetfortheproblemstudiedinthisdissertation,itisknowntheoreticallythatHuiszeroalongtheoptimalsolution.CommensuratewiththisknownvalueofHu,Tables 5-5 , 5-6 ,and 5-7 showthatHuisextremelysmall,furthersubstantiatingthecloseproximityofthenumericalsolutiontothetrueoptimalsolution. 86

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Figure5-1. Locallyoptimalsolutionsobtainedwithfreeandboundednaltruelongitudevs.Lf=(2)forGTOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). Table5-1. Orbittypes. Orbita,mei,deg,deg!,deg GTO243640000.730628.500LEO69270000.000128.500SSTO515256400.870522.500GEO4216400000FreeFree 87

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Figure5-2. OptimaltrajectoryforGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). 88

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Figure5-3. Semi-majoraxistimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. Figure5-4. EccentricitytimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. 89

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Figure5-5. InclinationtimehistoryforGTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. 90

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A B CFigure5-6. ControltimehistoryforGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).A)urvs.t.B)uvs.t.C)uhvs.t. 91

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A B C DFigure5-7. ControlforselectedsegmentsofGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).A)uvs.=(2)duringtherstfewrevolutions.B)uvs.=(2)whenur0.C)uvs.=(2)whenuhbecomes)]TJ /F4 11.955 Tf 9.3 0 Td[(1.D)uvs.=(2)duringthelastfewrevolutions. 92

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A B C DFigure5-8. SelectedsegmentsofGTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).A)ivs.=(2)duringtherstfewrevolutions.B)uhandcos(+!)vs.=(2)duringtherstfewrevolutions.C)ivs.=(2)whenuhbecomes)]TJ /F4 11.955 Tf 9.3 0 Td[(1.D)uhandcos(+!)vs.=(2)whenuhbecomes)]TJ /F4 11.955 Tf 9.3 0 Td[(1. 93

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Figure5-9. OptimaltrajectoryforLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s). 94

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Figure5-10. Semi-majoraxistimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s. Figure5-11. InclinationtimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s. 95

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A B CFigure5-12. ControltimehistoryforLEOtoGEOtransferwithvariousvaluesofT=m0andIsp=1000s.A)urvs.t.B)uvs.t.C)uhvs.t. 96

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Figure5-13. LongitudeoftheascendingnodetimehistoryforLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s). A BFigure5-14. ControlforselectedsegmentsofLEOtoGEOtransferwith(T=m0,Isp)=(4.0010)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,1000s).A)uvs.=(2)duringtherstfewrevolutions.B)uvs.L=(2)duringthelastfewrevolutions. 97

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Figure5-15. OptimaltrajectoryforSSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s). 98

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Figure5-16. Semi-majoraxistimehistoryforSSTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. Figure5-17. EccentricitytimehistoryforSSTOtoGEOtransferwithvariousvaluesofT=m0andIsp=3000s. 99

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Figure5-18. InclinationtimehistoryforSSTOtoGEOtransferusingvariousvaluesofT=m0andIsp=3000s. Table5-2. Regressioncoecientsfor^tf. Transfer a1 b1 B a2 b2 GTOtoGEO -1.3465 -1.0024 -8.425010)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2.927010)]TJ /F6 7.97 Tf 6.58 0 Td[(2 -1.0078 LEOtoGEO -1.9876 -1.0030 -6.823910)]TJ /F6 7.97 Tf 6.58 0 Td[(1 6.813610)]TJ /F6 7.97 Tf 6.58 0 Td[(2 -1.0001 SSTOtoGEO -1.0902 -1.0020 -8.699010)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2.488710)]TJ /F6 7.97 Tf 6.59 0 Td[(2 -1.0017 Table5-3. Regressioncoecientsfor^Lf. Transfer a3 b3 c3 d3 E GTOtoGEO 1.4110)]TJ /F6 7.97 Tf 6.59 0 Td[(1 )]TJ /F4 11.955 Tf 9.3 0 Td[(2.4810)]TJ /F6 7.97 Tf 6.59 0 Td[(3 1.36 )]TJ /F4 11.955 Tf 9.3 0 Td[(1.7010)]TJ /F6 7.97 Tf 6.59 0 Td[(6 2.3610)]TJ /F6 7.97 Tf 6.58 0 Td[(2 LEOtoGEO 2.76 )]TJ /F4 11.955 Tf 9.3 0 Td[(2.0310)]TJ /F6 7.97 Tf 6.59 0 Td[(3 6.02 )]TJ /F4 11.955 Tf 9.3 0 Td[(6.9210)]TJ /F6 7.97 Tf 6.59 0 Td[(3 )]TJ /F4 11.955 Tf 9.29 0 Td[(3.8410)]TJ /F6 7.97 Tf 6.58 0 Td[(2 SSTOtoGEO 8.4810)]TJ /F6 7.97 Tf 6.59 0 Td[(1 )]TJ /F4 11.955 Tf 9.3 0 Td[(3.3410)]TJ /F6 7.97 Tf 6.59 0 Td[(6 )]TJ /F4 11.955 Tf 9.3 0 Td[(2.9410)]TJ /F6 7.97 Tf 6.59 0 Td[(2 )]TJ /F4 11.955 Tf 9.3 0 Td[(2.6810)]TJ /F6 7.97 Tf 6.59 0 Td[(4 )]TJ /F4 11.955 Tf 9.29 0 Td[(5.6110)]TJ /F6 7.97 Tf 6.58 0 Td[(2 100

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A B CFigure5-19. ControltimehistoryforSSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s).A)urvs.t.B)uvs.t.C)uhvs.t. Table5-4. CoecientofdeterminationR2for^tfand^Lf. Transfer ^tf ^Lf GTOtoGEO 0.999987 0.999956 LEOtoGEO 0.999984 0.999982 SSTOtoGEO 0.999998 0.999879 101

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A BFigure5-20. SSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s)duringtherstfeworbitalrevolutions.A)uvs.=(2).B)a,e,andivs.L=(2). Table5-5. GTOtoGEOpost-optimalityresultsforT=m0=2.5010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2. Isp a(t0) J=a e(t0) J=e i(t0) J=i maxL2[L0,Lf](jHurj,jHuj,jHuhj) 500 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6610)]TJ /F6 6.974 Tf 6.23 0 Td[(6 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6610)]TJ /F6 6.974 Tf 6.23 0 Td[(6 29.88 29.90 44.38 44.39 6.7810)]TJ /F6 6.974 Tf 6.23 0 Td[(8 1000 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.1910)]TJ /F6 6.974 Tf 6.23 0 Td[(6 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.1810)]TJ /F6 6.974 Tf 6.23 0 Td[(6 39.59 39.61 58.47 58.49 7.0310)]TJ /F6 6.974 Tf 6.23 0 Td[(8 3000 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.5010)]TJ /F6 6.974 Tf 6.23 0 Td[(6 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.5010)]TJ /F6 6.974 Tf 6.23 0 Td[(6 46.99 47.01 69.30 69.31 6.5110)]TJ /F6 6.974 Tf 6.23 0 Td[(8 5000 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.7110)]TJ /F6 6.974 Tf 6.23 0 Td[(6 )]TJ /F4 9.963 Tf 7.75 0 Td[(2.7210)]TJ /F6 6.974 Tf 6.23 0 Td[(6 49.51 49.42 72.80 72.75 7.0010)]TJ /F6 6.974 Tf 6.23 0 Td[(8 Table5-6. LEOtoGEOpost-optimalityresultsforT=m0=4.0010)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.58 0 Td[(2. Isp a(t0) J=a e(t0) J=e i(t0) J=i maxL2[L0,Lf](jHurj,jHuj,jHuhj) 500 )]TJ /F4 9.963 Tf 7.74 0 Td[(4.5910)]TJ /F6 6.974 Tf 6.22 0 Td[(6 )]TJ /F4 9.963 Tf 7.74 0 Td[(4.5810)]TJ /F6 6.974 Tf 6.23 0 Td[(6 )]TJ /F4 9.963 Tf 7.75 0 Td[(0.05 )]TJ /F4 9.963 Tf 7.75 0 Td[(0.05 38.12 38.12 5.951510)]TJ /F6 6.974 Tf 6.23 0 Td[(8 1000 )]TJ /F4 9.963 Tf 7.74 0 Td[(8.2210)]TJ /F6 6.974 Tf 6.22 0 Td[(6 )]TJ /F4 9.963 Tf 7.74 0 Td[(8.2010)]TJ /F6 6.974 Tf 6.23 0 Td[(6 0.01 0.01 68.10 68.11 4.672010)]TJ /F6 6.974 Tf 6.23 0 Td[(8 3000 )]TJ /F4 9.963 Tf 7.74 0 Td[(1.2010)]TJ /F6 6.974 Tf 6.22 0 Td[(5 )]TJ /F4 9.963 Tf 7.74 0 Td[(1.1910)]TJ /F6 6.974 Tf 6.23 0 Td[(5 0.10 0.10 100.02 100.04 4.383610)]TJ /F6 6.974 Tf 6.23 0 Td[(8 5000 )]TJ /F4 9.963 Tf 7.74 0 Td[(1.3010)]TJ /F6 6.974 Tf 6.22 0 Td[(5 )]TJ /F4 9.963 Tf 7.74 0 Td[(1.2910)]TJ /F6 6.974 Tf 6.23 0 Td[(5 0.13 0.13 108.11 108.14 4.551810)]TJ /F6 6.974 Tf 6.23 0 Td[(8 102

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A BFigure5-21. SSTOtoGEOtransferwith(T=m0,Isp)=(3.3310)]TJ /F6 7.97 Tf 6.58 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2,3000s)duringthelastfeworbitalrevolutions.A)uvs.=(2).B)a,e,andivs.L=(2). Table5-7. SSTOtoGEOpost-optimalityresultsforT=m0=3.3310)]TJ /F6 7.97 Tf 6.59 0 Td[(4ms)]TJ /F6 7.97 Tf 6.59 0 Td[(2. Isp a(t0) J=a e(t0) J=e i(t0) J=i maxL2[L0,Lf](jHurj,jHuj,jHuhj) 500 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.4610)]TJ /F6 6.974 Tf 6.23 0 Td[(7 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.4610)]TJ /F6 6.974 Tf 6.23 0 Td[(7 61.72 61.74 16.08 16.08 1.1310)]TJ /F6 6.974 Tf 6.23 0 Td[(7 1000 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.7210)]TJ /F6 6.974 Tf 6.23 0 Td[(7 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.7210)]TJ /F6 6.974 Tf 6.23 0 Td[(7 76.63 76.66 20.01 20.02 1.1510)]TJ /F6 6.974 Tf 6.23 0 Td[(7 3000 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6710)]TJ /F6 6.974 Tf 6.23 0 Td[(7 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6710)]TJ /F6 6.974 Tf 6.23 0 Td[(7 86.44 86.48 22.69 22.70 1.1210)]TJ /F6 6.974 Tf 6.23 0 Td[(7 5000 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6910)]TJ /F6 6.974 Tf 6.23 0 Td[(7 )]TJ /F4 9.963 Tf 7.75 0 Td[(1.6910)]TJ /F6 6.974 Tf 6.23 0 Td[(7 88.75 88.79 23.31 23.31 2.1310)]TJ /F6 6.974 Tf 6.23 0 Td[(7 103

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A B CFigure5-22. tfvs.Isp.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 104

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A B CFigure5-23. RegressioncoecientAvs.T=m0.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 105

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A B CFigure5-24. RegressioncoecientBvs.T=m0.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 106

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A B CFigure5-25. RegressioncoecientCvs.T=m0.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 107

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A B CFigure5-26. Lf=(2)vs.tf.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 108

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A B CFigure5-27. RegressioncoecientDvs.Isp.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 109

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A B CFigure5-28. RegressioncoecientEvs.Isp.A)GTOtoGEOtransfer.B)LEOtoGEOtransfer.C)SSTOtoGEOtransfer. 110

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CHAPTER6ORBITTRANSFERSTUDY2:ECLIPSINGInthischapter,orbittransferswitheclipseconstraintsareconsidered.Thespacecraftisassumedtothrustonlywhenpowerisavailable,thatis,whenthespacecrafthaslineofsighttotheSun.Furthermore,thespacecraftisassumedtothrustcontinuouslywithaconstantmagnitudethroughouttheregionsofthetransferwherepowerisavailable.Whiletheequationsofmotion,boundaryconditions,andpathconstraintsremainthesameasthatofthenon-eclipsingorbittransferstudyofSection 5 ,theEarthshadowmodeldescribedinSection 4.3 isemployedtodeterminetheentranceandexitoftheshadowregions.Takingintoaccounteclipsingleadsnaturallytodecomposingthespacecraftmotionintophases.Usingthisapproach,thetrajectoryoptimizationproblemisformulatedasamultiple-phaseoptimalcontrolproblem.Thischapterisorganizedasfollows.First,themultiple-phaseoptimalcontrolproblemisstated.Next,theinitialguessgenerationmethodisdescribed.Finally,theresultsofthethreeorbittransfersforvariouslaunchdatesarepresentedandanalyzed. 6.1Multiple-PhaseOptimalControlProblemThestructureofthemultiple-phaseoptimalcontrolproblemisasfollows.Therstphasestartswiththespacecraftinitsinitialorbitandterminateswhenthespacecraftreachesthestartoftherstshadowregion.Eachintermediatephaseofthemultiple-phaseoptimalcontrolproblemcorrespondstoasegmentofthetrajectorythatbeginsatanexitpointofashadowregionandendsattheentrypointofthenextshadowregion.Thenalphasecorrespondstothetrajectorysegmentthatbeginsatthelastshadowexitpointandendswhenthedesiredterminalorbitisobtained.Themultiple-phaseoptimalcontrolproblemarisingfromtheaforementionedlow-thrustorbitaltransferwitheclipsingisthereforegivenasfollows.Determinethetrajectory(p(L),f(L),g(L),h(L),k(L),m(L),t(L)),thecontrol(ur(L),u(L),uh(L)),andtheeventconstraintparametersL(r),(r=1,...,R)]TJ /F4 11.955 Tf 12.06 0 Td[(1),thatminimizethecostfunctional J=tf(6{1) 111

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subjecttothedynamicconstraintsofEqs.( 4{9 )and( 4{10 ),theinitialconditionsofEq.( 4{21 ),theterminalconditionsofEq.( 4{23 ),thepathconstraintofEq.( 4{24 ),theparameterconstraintsofEq.( 4{32 ),andtheeventconstraintsofEq.( 4{33 )-( 4{38 ).(Again,itisnotedthat=1=86400istheconversionfactorfromunitsofsecondstounitsofdays.) 6.2InitialGuessGenerationAnimportantaspectoftheapproachpresentedhereforsolvingorbittransferswitheclipsingistheinitialguessgenerationmethoddeveloped.Therststepoftheinitialguessgenerationmethodistosolveforanominaltrajectory,whereanominaltrajectoryistheoptimaltrajectoryforthenon-eclipsingcasethatwasexaminedinSection 5 .Therefore,thenominaltrajectoryisobtainedbysolvingthefollowingsingle-phaseoptimalcontrolproblem.Determinethetrajectory(p(L),f(L),g(L),h(L),k(L),m(L),t(L))andthecontrolinputs(ur(L),u(L),uh(L))thatminimizethecostfunctionalofEq.( 5{1 )subjecttothedynamicconstraintsofEqs.( 4{9 )and( 4{10 ),theinitialconditionsofEq.( 4{21 ),theterminalconditionsofEq.( 4{23 ),andthepathconstraintofEq.( 4{24 ).ThenextstepistoanalyzethenominaltrajectorytodeterminewherethespacecraftrstenterstheEarth'sshadow,whichwillvarydependingonthedateandtimeoflaunchdeparture.Thenominaltrajectoryisinitiallyanalyzedtolocatetherstshadowentrypointwhichwilloccurwhenr^s<0andjjjj=(Section 4.3.1 ).Thetrajectoryuptotheshadowentrypointthenbecomestheinitialguessfortherstphaseofthemultiple-phaseoptimalcontrolproblem.Startingattherstshadowentrypoint,theequationsofmotionarethenpropagatedforwardintimeuntilashadowexitpointisobtained,thatis,theequationsofmotionsarepropagatedwhilejjjj
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pointuntilashadowexitpointisobtained.Asbefore,theshadowexitpointdeterminestheinitialconditionsforthenextsingle-phaseoptimalcontrolproblem.Thisinitialguessgenerationprocessisrepeateduntilnoshadowregionisfoundinthenominaltrajectory.Forexample,consideranorbittransferwhereaspacecrafttravelsthroughtheEarth'sshadowonlyonetime.Analysisofthenominaltrajectorywouldleadtoatwophaseproblem.Therstphasewillsolveforthetrajectoryfromtheinitialconditiontotheentrypointoftheshadowregionandthesecondphasewillsolveforthetrajectoryfromtheexitpointoftheshadowregiontothedesiredterminalcondition.Asummaryoftheinitialguessgenerationmethodemployedinthisresearchappearsbelow.ThephasenumberisdenotedbynPandthephasenumberincreasesby1ineachloopofthegenerationmethod.ThemethodwillterminateinStep4whenthenominaltrajectorynolongercontainsashadowregion.Itisnotedthatiftheinitialnominaltrajectorybeginsinashadowregion,theequationsofmotionarepropagatedrststartinginStep5toobtaintheshadowexitpoint,whichthenbecomestheinitialconditionfortheorbittransfer. Step1:SetnP=1. Step2:Solvesingle-phaseoptimalcontrolproblemofEqs.( 5{1 ),( 4{9 ),( 4{10 ),( 4{21 ),( 4{23 ),and( 4{24 ). Step3:Analyzethenominaltrajectorytodeterminetherstinstancewherer^s<0andjjjj=,whichcorrespondstoashadowentrypoint. Step4:Ifashadowentrypointdoesnotexist,savethenominaltrajectoryastheinitialguessforphasenP,thenquit.Ifashadowentrypointdoesexists,savethenominaltrajectoryuptotheshadowentrypointastheinitialguessforphasenP. Step5:Propagatetheequationsofmotionwhilejjjj
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6.3ResultsandDiscussionTodemonstratetheapproach,theGTOtoGEO,LEOtoGEO,andSSTOtoGEOtransfersassummarizedinTable 5-1 fromSection 5 wereconsideredusingthespeciedmissioncharacteristicsshowninTable 6-1 .TheGTOtoGEOtransferwastakendirectlyfromRef.[ 47 ]inordertodemonstratethecapabilityoftheapproachdevelopedinthisresearchagainstresultsthathavealreadyappearedintheopenliterature.InaneorttobeconsistentwithRef.[ 47 ],theorbittransferwassolvedusingtheJ2gravityperturbationonlyandadeparturedateofMarch22,2000at0:00UT.TheLEOtoGEOandSSTOtoGEOtransfersweresolvedwithJ2throughJ4gravityperturbations.TheLEOtoGEOtransferwassolvedwithaWinterSolstice(December21at15:03UT)departurefortheyear2025.ThefollowingdeparturedateswereconsideredfortheSSTOtoGEOtransfer:VernalEquinox(March20at3:50UT),SummerSolstice(June20at21:44UT),AutumnalEquinox(September22at13:31UT),andWinterSolstice(December21at10:02UT)fortheyear2020.SimilartoSection 5 ,thekeyfeaturesofthetransfersareexaminedandthebehavioroftheoptimalcontrolcomponentsisanalyzedusingEqs.( 5{4 ),( 5{5 ),and( 5{6 ). 6.3.1OptimalGTOtoGEOTransferwithEclipsingFigure 6-1 showstheoptimalGTOtoGEOtrajectoryinECICartesiancoordinates(x,y,z).Themultiple-phaseoptimalcontrolproblemcontains101phases.TheEarth-shadowregionsappearnearapoapsisatthestartofthetransferandslowlyshiftawayfromapoapsistowardsthedescendingnode.Sincetheorbitalvelocityisthesmallestnearapoapsis,theamountoftimespentintheEarth'sshadowisquitesignicantatjustovereightsdays.Moreover,shadowregionsdonotexistafter59days.Theoptimalsolutionhasanaltimeof121.22days,analmassof1027.77kg,andcompletesapproximately165orbitalrevolutions.Comparedtotheoptimalsolutionwithouteclipsing,whichalsoservesastheinitialnominaltrajectoryfortheinitialguessgenerationmethoddescribedinSection 6.2 ,thenon-eclipsingtrajectoryhasanaltimeof110.58days,analmassof1031.36kg,and150.63orbitalrevolutions.Inotherwords,whenitisnecessarytotakeeclipseregionsintoaccount,the 114

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transfertakesanadditional11daysandusesapproximatelyfourmorekilogramsoffuel.Returningfocusbacktothetransferwitheclipsing,thetimehistoriesofa,e,andiareshowninFigs. 6-2A 6-2C ,respectively.ItisseenfromFig. 6-2A thatthesemi-majoraxisincreasesrapidlyforthersthalfofthetransfer,thensteadilyincreasestothedesiredterminalvalue.FromFig. 6-2B ,theeccentricityisshowntodecreaseataslowerrateinthersthalfofthetransfer,thenrapidlydecreasethroughouttherestofthetransfer.Finally,itisseenfromFig. 6-2C thattheinclinationdecreasesatanapproximatelylinearlyratethroughouttheentiretransfer.Theoptimaltransfertimeof121.22daysobtainedusingtheapproachdevelopedisincloseproximitytothetransfertimeof118.36daysobtainedinRef.[ 47 ].Now,whilethetransfertimeobtainedinthisstudyis2.4%largerthanthetransfertimeobtainedinRef.[ 47 ],thediscrepancyisduetothefactthattheapproachofthisdissertationdierssignicantlyfromtheapproachusedinRef.[ 47 ].Specically,theapproachofRef.[ 47 ]employsorbitalaveragingoftheequationsofmotion.Furthermore,Ref.[ 47 ]averagestheJ2gravityperturbationovereachorbitalrevolutionandemploysacylindricalEarth-shadowmodel.Becausetheequationsofmotionarereplacedbytheirorbitalrevolutionaverages,thesolutionobtainedinRef.[ 47 ]islessaccuratethanthesolutionobtainedusingthedirectcollocationmethodofthisstudy.Infact,thesolutionobtainedinthisdissertationsatisesthedynamicconstraintsandisfeasible.Moreover,averagingthezonalharmonicJ2hasasignicanteectonboththelongitudeoftheascendingnodeandtheargumentofperiapsis,bothofwhichdescribetheorientationoftheorbit.AveragingJ2subsequentlyinuenceswhereanEarth-shadowregionwillappearinanorbitalrevolution.Finally,thecylindricalshadowmodelemployedinRef.[ 47 ]willresultinsmallershadowregionswhencomparedagainsttheconicalprojectionmodelemployedinthisstudy.Furtherinsightintothestructureoftheoptimaltransferisobtainedbyexaminingtheoptimalcontrolu=(ur,u,uh)alongdierentsegmentsofthesolution.Figure 6-3 showsthatfoursegmentsidentifythekeyfeaturesoftheoptimalcontrol:(1)therstfeworbital 115

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revolutionsofthetransfer,(2)theregionwhereur0,(3)theregionwhereuhbecomes)]TJ /F4 11.955 Tf 9.3 0 Td[(1,and(4)thenalorbitalrevolutionsofthetransfer.First,Fig. 6-4A showstheoptimalcontrolintherstfeworbitalrevolutionsofthetransfer.Inthissegmentofthetransfer,urispositivenear==2andnegativenear=3=2whileapositiveudominatesthethrustdirectionalongallexceptjustpriorto=.Theneteectofuranduisthentoincreasethesemi-majoraxisandtodecreasetheeccentricity.Next,itisseenfromFig. 6-4B thatdi=dt(Eq.( 5{6 ))ismostnegativeimmediatelybeforetoapoapsis.ThisresultisconsistentwithFig. 6-4C wheretheminimumvalueofdi=dtcoincideswiththeminimumvalueofcos(+!)uhpriorto=.Becausetheshadowregionappearsnearapoapsis,theinclinationchangemustbeperformedelsewhere.Inthissegmentofthetransfer,theinclinationchangeisperformedmostecientlyimmediatelypriortoapoapsis.Theoptimalcontrolinthesegmentofthetransferwhereur0isshowninFig. 6-5 .Foreveryorbitalrevolutionbeyondwhereurbecomeszero(thatis,allvaluesbeyond=(2)=90.5),urchangesfromitsbehaviorintherstfeworbitalrevolutionssuchthaturisnownegativenear==2andpositivenear=3=2.Sinceur0andthethrustdirectionliespredominatelyinthepositiveudirection(exceptjustpriorto=),theoverallresultistoincreasethesemi-majoraxisandtodecreasetheeccentricity.Similartotherstfeworbitalrevolutions,di=dtismostnegativepriortoapoapsis.Evenastheshadowregionshiftsawayfromapoapsistowardsthedescendingnode,uhecientlydecreasestheinclinationbydominatingthethrustdirectionjustbefore=.Next,Fig. 6-6A showstheoptimalcontrolinthesegmentofthetransferwhereuhbecomes)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Atthisparticularpointinthetransfer,theEarth-shadowregionsceasetoexist(near=99.6)andurisnownegativenear==2andpositivenear=3=2.Moreover,unolongerdominatesthethrustdirection,reachingitsmostpositivevalueat=5=4andagraduallydecreasingvalueatperiapsis.Thecombinedeectofuranduisthentoincreasethesemi-majoraxisandtodecreasetheeccentricity.InFig. 6-6B itisseenthatdi=dt 116

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isnegativeatperiapsisandapoapsis,whileFig. 6-6C showsthatcos(+!)uhisnegativewhen0and3=4.Becausetheorbitalvelocityismuchsmalleratapoapsisthanitisatperiapsis,theinclinationchangeobservednearapoapsisislargerthanthechangeobservednearperiapsis.Finally,Fig. 6-7A showstheoptimalcontrolduringthelastfeworbitalrevolutionsofthetransfer.Shadowregionsdonotexistinthissegmentofthetransfer.Inthenalfeworbitalrevolutions,urisnegativenear==2andpositivenear=3=2,whileuispositivenear=0andisnegativenear=.Thrustinginthismannerincreasesperiapsisanddecreasesapoapsisand,thus,theoveralleectofuranduistoincreasethesemi-majoraxisandtodecreasetheeccentricity.FromFig. 6-7B itisseenthatdi=dtisaminimumnear=3=4and=7=4,whileFig. 6-7C showsthatcos(+!)uhisnegativeat=3=4and=7=4.Becausetheorbitisnearlycircularattheendofthetransfer,therateatwhichtheinclinationdecreasesisessentiallythesameat=3=4and=7=4. 6.3.2OptimalLEOtoGEOTransferwithEclipsingTheoptimalLEOtoGEOtrajectoryinECICartesiancoordinates(x,y,z)isshowninFig. 6-8 andconsistsof220phases.Theoptimalsolutionhasanalmassof569.74kg,adurationof33.69days,and223.88orbitalrevolutions.Thespacecraftdoesnotencounteranyshadowregionsafter29.51daysandspendsatotalof5.52daysofthetransferinshadow.Thetimehistoriesofa,e,andiareshowninFig. 6-9 .FromFig. 6-9 ,itisevidentthatthesemi-majoraxisincreasesataslowerrateatthebeginningofthetransfer,beforeincreasingmorerapidlyattheendofthetransfer.Furthermore,theeccentricityshowninFig. 6-9B steadilyincreasesthroughoutthersthalfofthetransferreachingamaximumvaluenear0.38beforesteadilydecreasingfortheremainderofthetransfer.Itisinterestingtonote,thatforanon-eclipsingLEOtoGEOtransfer(Fig. 5-6A ),theeccentricitystaysrelativelyclosetozero,whereasforthetransferwitheclipsingtheeccentricityincreasessignicantly.Finally,theinclinationinFig. 6-9C decreasesatanapproximatelylinearrate. 117

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ThecomponentsofthecontrolareshowninFig. 6-10 .Recallingfromthenon-eclipsingLEOtoGEOtransferstudyinSection 5.4.2 ,theradialcomponent,ur,isapproximatelyzerothroughouttheentiretransfer.FortheeclipsingLEOtoGEOtransfer,however,urisactive.Inordertounderstandthestructureoftheoptimaltransfer,itishelpfultotakeacloserlookatthecontrolcomponentsinaselectregionofthetransfersuchas=[67,72]whichisapproximatelyvedaysintothetransfer.Figure 6-11A showstheoptimalcontrolduringthesegmentofthetransferwhen=[67,72].Inthissegmentofthetransfer,theshadowregionsappearnearthedescendingnodeat3=2.Theradialcomponent,urispositivenear=3=2.Recallingthatapositiveurbetween=[,2],asitrelatestoEq.( 5{4 ),willincreaseapoapsisanddecreaseperiapsis.ThisisconsistentwithFig. 6-11B whereitisevidentthatthelargestincreaseintheeccentricityinfactoccursnear=3=2.Furthermore,thetangentialcomponent,u,ispositiveformostvaluesofwithitspeakpositivevalueoccurringjustbefore=3=2.RecallingfromEq.( 5{4 )thatapositiveuwillincreasethesemi-majoraxisanddecreasetheeccentricity,itcanbeseeninFig. 6-11B thatthesemi-majoraxisisincreasingwhenuispositiveandthattheeccentricityundergoesaslightdecrease.Despitetheslightdecreaseintheeccentricityduetoapositiveu,theoverallneteectofuranduistoincreasethesemi-majoraxisandtheeccentricity.Sincethetransferisincreasingtheeccentricity,thelargeinclinationchangerequiredispossibledespitetheshadowregions.Asseenbeforeintheothertransfers,thelargestdecreaseintheinclinationwilloccurnearapoapsis,whichisconrmedbyexaminingFig. 6-11B . 6.3.3OptimalSSTOtoGEOTransferswithEclipsingTheresultsfortheselecteddeparturedatesareshowninTable 6-2 .Whiletheminimumtimesobtainedweresimilarforthreeofthedates,theSummerSolsticedepartureyieldedthebestresultwithatransfertimeof72.60daysandhadthelargestpayloadof1122.68kg.Moreover,theleastdesirabledeparturedatewastheVernalEquinoxwherealmostthreeadditionaldayswererequiredtocompletethetransfer.Thepayloaddelivered,however,wasnearlythesameforalldatesconsidered.Forcomparison,theoptimalsolutionwithouteclipsing 118

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required72.52days,hadanalmassof1122.22kg,andcompleted60.21orbitalrevolutions.Forthisparticulartransfer,thereislittledierencebetweenthesolutionsforthenon-eclipsingandtheeclipsingtransfersforthreeofthedeparturedateschosen.TheoptimalSSTOtoGEOtrajectoriesareshowninECICartesiancoordinates(x,y,z)inFigs. 6-12 , 6-13 , 6-14 ,and 6-15 fortheVernalEquinox,SummerSolstice,AutumnalEquinox,andWinterSolsticedepartures,respectively.Foreachofthesefourtransfers,theshadowregionsappearatthebeginningofthetransferinkeylocationsthroughouttheorbit:(1)theVernalEquinoxtransferhasshadowregionsnearapoapsis,(2)theSummerSolsticetransferhasshadowregionsnearthedescendingnode,(3)theAutumnalEquinoxtransferhasshadowregionsnearperiapsis,and(4)theWinterSolsticetransferhasshadowregionsneartheascendingnode.Inadditiontoshadowregionsthatappearinthebeginningofthetransfer,theSummerSolsticeandWinterSolsticetransfershaveadditionalshadowregionsthatappearinthelastfeworbitalrevolutionsofthetransfers.ThelocationsoftheshadowregionsareconsistentwiththefactthatspacecraftinGEOwilltravelthroughtheEarth'sshadowinanorbitalrevolutionthatoccursbetweentheendofFebruaryandthemiddleofAprilorbetweenthebeginningofSeptemberandthemiddleofOctober.BecausetheSummerSolsticetransfertakes72.60days,thespacecraftwillapproachGEOnearthebeginningofSeptemberandsubsequentlytravelsthroughtheEarth'sshadowagainasitnearsitsterminalorbit.Figure 6-16 showsthesemi-majoraxistimehistoriesforallofthedeparturedatesconsidered.FortheVernalEquinox,AutumnalEquinox,andWinterSolsticedates,thesemi-majoraxisincreasesslightlyduringtherstfewdaysofthetransferandthensteadilydecreasesthroughouttherestofthetransfer.IntheSummerSolsticetransfer,however,thesemi-majoraxisdecreasesthroughouttheentiretransfer.Foralldeparturedatesconsidered,theeccentricityandinclinationbothsteadilydecreasethroughoutthetransferasshowninFig. 6-17 fortheVernalEquinoxdeparture.Thecontrolcomponentsu=(ur,u,uh)fortheVernalEquinoxdepartureareshowninFig. 6-18 .Whiletheoverallbehaviorofthecontrolcomponentsisfairlysimilarbetweenallofthedeparturedatesconsidered,furtherinsight 119

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intothestructureoftheoptimaltransfersisgainedbyexaminingthebehaviorofthecontrolcomponentsduringtherstfeworbitalrevolutionswheretheshadowregionsarepresent.Thenalfeworbitalrevolutionsarealsoexamined.ThecontrolcomponentsduringtherstfeworbitalrevolutionsforthedeparturedatesconsideredareshowninFig. 6-19 .Itisseenthattheradialcomponentofthecontrolintheinitialpartofthetransferissimilarforallofthedeparturedatesandbecomesnegativenear==2andpositivenear=3=2.Ontheotherhand,thetangentialcomponent,u,andthenormalcomponent,uh,donotexhibitsimilarbehaviorbetweenalldeparturedates.ForalldeparturedatesotherthantheVernalEquinox,uisnegativenear=0andpositivenear=,whileuhisnegativenear=0andpositivenear=.Thecyclicbehaviorofuraisesperiapsisandapoapsiswhen=[=2,3=2],andlowersperiapsisandapoapsiswhen=[3=2,5=2](includinganglewrap).BeginningwiththeSummerSolsticetransfer,showninFig. 6-19B ,theeectofuristolowerapoapsisandraiseperiapsiswhen2[0,].Becausetheshadowregionappearsnear=3=2,however,urcannotaseectivelyincreaseapoapsisanddecreaseperiapsisbetween=[,2].Inaddition,ucannotincreasebothperiapsisandapoapsis.Thus,theoverallneteectofuranduistodecreaseboththesemi-majoraxisandtheeccentricity.Next,fortheAutumnalEquinoxtransfershowninFig. 6-19C ,theshadowregionsappearnearperiapsis.Becausethespacecraftcannotthrustnear=0,ucannoteectivelylowerperiapsisandapoapsis.Theneteectofurandu,therefore,istoincreasethesemi-majoraxisandtodecreasetheeccentricity.ExaminingtheWinterSolsticetransfershowninFig. 6-19D ,theshadowregionisinitiallynear==2andthustheapoapsisdecreaseandperiapsisincreasewhen2[0,]islessthantheperiapsisdecreaseorapoapsisincreasewouldbeifthrustwereavailable.Furthermore,ucannotlowerbothperiapsisandapoapsis.Asaresult,theneteectofuranduistoincreasethesemi-majoraxisandtodecreasetheeccentricity.RecallingthatuhexhibitssimilarbehaviorfortheSummerSolstice,AutumnalEquinox,andWinterSolsticetransfers,theminimumvalueofdi=dtcorrespondstotheminimumvalueofcos(+!)uhat=.Thustheinclinationdecreasesmost 120

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rapidlynearapoapsis,wheretheorbitalvelocityofthespacecraftisthesmallest.Finally,fortheVernalEquinoxtransfershowninFig. 6-19A ,thebehaviorofuanduhisdierentfromthebehaviorobservedfortheotherdeparturedatesbecausetheshadowregionsappearnearapoapsis.FortheVernalEquinox,uismostnegativenear==4andmostpositivenear=5=4.EventhoughthebehaviorofuisdierentfortheVernalEquinoxtransferduetothelocationoftheshadowregionsnearapoapsis,ustilllowersbothperiapsisandapoapsiswhen2[3=2,5=2]andraisesbothperiapsisandapoapsiswhen2[=2,3=2].Therefore,thecombinedeectofuranduistoincreasethesemi-majoraxisandtodecreasetheeccentricity.Furthermore,thenormalcomponent,uh,ispositivenear=3=4andnegativenear=3=2.Dierentfromtheotherdeparturedates,theVernalEquinoxtransfercannotperformtheinclinationchangenearapoapsisduringtherstfeworbitalrevolutionsoftheVernalEquinoxtransfer.ExaminingFig. 6-20A ,di=dtismostnegativepriortoapoapsis,whichisconsistentwithFig. 6-20B wheretheminimumvalueofdi=dtcoincideswiththeminimumvalueofcos(+!)uhjustbeforeto=.Thecontrolcomponentsduringthenalfeworbitalrevolutionsofthetransferexhibitsimilarbehaviorforallofthedeparturedates.ThecontrolcomponentsfortheVernalEquinoxdepartureareshowninFig. 6-21 .Attheendofthetransfer,urisnegativenear==2andpositivenear=3=2,whileuisnegativenear=0andpositivenear=.Thrustinginthismannerincreasesperiapsisanddecreasesapoapsisandthustheneteectofuranduistodecreaseboththesemi-majoraxisandtheeccentricity.Lastly,uhisnegativenear=0andpositivenear=.Atthispointinthetransfer,theorbitisnearlycircularandtherateatwhichtheinclinationdecreasesinessentiallythesameatperiapsisandapoapsis. Table6-1. Missioncharacteristics. Transferm0,kgIsp,sP,kW,% GTOtoGEO12001800555LEOtoGEO100010001085SSTOtoGEO120033001065 121

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Figure6-1. TrajectoryforGTOtoGEOtransferwitheclipsing. Table6-2. NumericalresultsforSSTOtoGEOtransferswithvariousdeparturedates. NumberofTimeinFinalFinalFinalTrueDepartureDatePhasesShadow(hr)Time(d)Mass(kg)Longitude(rev) VernalEquinox1651.4775.421121.4262.22SummerSolstice2512.2172.601122.6859.31AutumnalEquinox3115.0673.491121.8659.23WinterSolstice1912.0173.061122.1859.24 122

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A B CFigure6-2. OrbitalelementtimehistoryforGTOtoGEOtransferwitheclipsing.A)avs.t.B)evs.t.C)ivs.t. 123

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A B CFigure6-3. ControltimehistoryforGTOtoGEOtransferwitheclipsing.A)urvs.t.B)uvs.t.C)uhvs.t. 124

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A B CFigure6-4. GTOtoGEOtransferwitheclipsingduringtherstfeworbitalrevolutions.A)uvs.=(2).B)ivs.=(2).C)uhandcos(+!)vs.=(2). Figure6-5. uvs.=(2)duringregionofGTOtoGEOtransferwitheclipsingwhenur0. 125

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A B CFigure6-6. SegmentofGTOtoGEOtransferwitheclipsingwhenuhbecomes-1duringtheGTOtoGEOtransferwitheclipsing.A)uvs.=(2).B)ivs.=(2).C)uhandcos(+!)vs.=(2). 126

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A B CFigure6-7. GTOtoGEOtransferwitheclipsingduringthenalfeworbitalrevolutions.A)uvs.=(2).B)ivs.=(2).C)uhandcos(+!)vs.=(2). 127

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Figure6-8. TrajectoryforLEOtoGEOtransferwitheclipsing. 128

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A B CFigure6-9. OrbitalelementhistoryforLEOtoGEOtransferwitheclipsing.A)avs.t.B)evs.t.C)ivs.t. 129

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A B CFigure6-10. ControltimehistoryforLEOtoGEOtransferwitheclipsing.A)urvs.t.B)uvs.t.C)uhvs.t. 130

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A BFigure6-11. LEOtoGEOtransferwitheclipsingwhen=[67,72].A)uvs.=(2).B)a,e,andivs.=(2). 131

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Figure6-12. TrajectoryforSSTOtoGEOtransferwithVernalEquinoxdeparture. Figure6-13. TrajectoryforSSTOtoGEOtransferwithSummerSolsticedeparture. 132

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Figure6-14. TrajectoryforSSTOtoGEOtransferwithAutumnalEquinoxdeparture. Figure6-15. TrajectoryforSSTOtoGEOtransferwithWinterSolsticedeparture. 133

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A B C DFigure6-16. Semi-majoraxistimehistoryforSSTOtoGEOtransferwithvariousdepartures.A)VernalEquinox.B)SummerSolstice.C)AutumnalEquinox.D)WinterSolstice. 134

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A BFigure6-17. EccentricityandinclinationtimehistoryforSSTOtoGEOtransferwithVernalEquinoxdeparture.A)evs.t.B)ivs.t. 135

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A B CFigure6-18. ControltimehistoryforSSTOtoGEOtransferwithVernalEquinoxdeparture.A)urvs.t.B)uvs.t.C)uhvs.t. 136

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A B C DFigure6-19. uvs.=(2)forSSTOtoGEOtransferduringtherstfeworbitalrevolutions.A)VernalEquinox.B)SummerSolstice.C)AutumnalEquinox.D)WinterSolstice. 137

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A BFigure6-20. SSTOtoGEOtransferwithVernalEquinoxdepartureduringtherstfeworbitalrevolutions.A)ivs.=(2).B)uhandcos(+!)vs.=(2). Figure6-21. uvs.=(2)forSSTOtoGEOtransferduringthenalfeworbitalrevolutionswithVernalEquinoxdeparture. 138

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CHAPTER7CONCLUSIONSInthisdissertation,theproblemofdetermininghigh-accuracylow-thrustminimum-timeEarth-orbittransferswereconsidered.Twoorbittransferstudieswereconducted.Therststudyexaminednon-eclipsingorbittransfersandthesecondstudyexaminedeclipsingorbittransfers.Bothstudiesemployedavariable-orderLGRorthogonalcollocationmethodtosolvethetrajectoryoptimizationproblems.Thesolutionsdeterminedinbothorbittransferstudiesaresomeofthersttrajectoriestobeobtainedwithoutusingorbitalaveraging. 7.1SummaryofNon-EclipsingOrbitTransferResearchInthenon-eclipsingstudy,thespacecraftwasassumedtohavepowerthroughouttheentiretransfer.Thelow-thrustorbittransferproblemwasformulatedasasingle-phaseoptimalcontrolproblem.Aninitialguessgenerationprocedurewasdevisedwhereasequenceofoptimalcontrolsub-problemsweresolved,wherethegoalofeachsub-problemwastodeterminethestateandthecontrolcomponentsthatminimizedthemeansquarerelativedierencebetweentheinitialorbitandtheterminalconditions.AsearchmethodwasalsodevelopedtohelptheNLPsolverdeterminethebestlocallyoptimalsolution.AnumericaloptimizationstudywasthenconductedtodetermineoptimaltrajectoriesandcontrolcomponentsforthreeEarth-orbittransfersusingarangeofinitialthrustaccelerationsandconstantspecicimpulsesvalues.Thekeyfeaturesofthesolutionswereidentiedandrelationshipswereobtainedthatrelatetheoptimaltransfertimeandtheoptimalnumberoftruelongitudecyclesasafunctionoftheinitialthrustaccelerationandspecicimpulse.Finally,apost-optimalityanalysiswasperformedthatveriestheoptimalityofthesolutionsthatwereobtained. 7.2SummaryofEclipsingOrbitTransferResearchIntheeclipsingstudy,thespacecraftwasassumedtohavepoweronlywhenthespacecrafthadlineofsighttotheSun.Thelow-thrustorbittransferproblemwasformulatedasamultiple-phaseoptimalcontrolproblem.Aninitialguessmethodwasdevelopedinwhich 139

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theguesswasconstructedbysolvingaseriesofsingle-phaseoptimalcontrolproblems.Byanalyzingtheresultingtrajectoriesthatwereobtainedbysolvingthesingle-phaseoptimalcontrolproblem,theapproximateshadowentranceandexitlocationsweredetermined.Anintelligentguesswasthenconstructedforthemultiple-phaseoptimalcontrolproblemwitheclipsingconstraints.Basedonthegeometryoftheshadowregion,eventconstraintswereconstructedandenforcedthatrelatetheorbitattheterminusofaneclipsetotheorbitatthestartofaneclipsewhicheliminatedtheneedtoincludecoastphasesintheoptimizationproblem.ThreeEarth-orbittransferswithvariousdeparturedateswerechosentodemonstratetheapproachdevelopedinthisresearchandthekeyfeaturesoftheoptimaltrajectorieswereanalyzed. 7.3FutureWork 7.3.1Third-BodyGravitationalPerturbationsandSolarRadiationPressureTofurtherimprovetheaccuracyofthetrajectoriesobtainedfortheEarth-orbittransfersconsidered,theeectofsolargravitationalperturbations,theeectoflunargravitationalperturbations,andtheeectofsolarradiationpressure(SRP)mustbeincluded.Includinglunisolarperturbationsisimportantformanylow-thrustEarth-orbittransfers.Asthespacecrafttravelstoorbitswithhigheraltitude,thegravitationalperturbationsduetotheSunandtheMoonbecomemoresignicantthantheeectoftheEarth'soblateness.OftentheeectoflunargravitationalperturbationsisgreaterbecausetheMoonisclosetotheEarth.IncludingtheeectofSRPisalsoimportantbecauseitisthelargestnon-gravitationalperturbationthatcanhaveasignicantimpactonthedynamicsofaspacecraftinorbit.IncludingtheeectsofSRPisdiculttomodelformanyreasons.Mostmodelsimplementaconstantsolarux,however,solaruxisknowntovarythroughouttheyearandduringirregularperiodsofsolaractivity.Furthermore,anaccurateSRPmodelrequiresprecisecomputationofthecross-sectionalareasofthespacecraft.Spacecraftsurfaceswillalsodegradeovertimeinorbitandthereectivepropertieswilluctuate.Therefore,anaccuratemodelofsatellitebehavior 140

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underperturbingforcesfromSRPmustincludeanalgorithmtodescribetheabsorptionandreectionoftherepresentativesolarray. 7.3.2OptimalityofEclipsingOrbitTransfersInthisdissertation,asearchmethodwascreatedtodeterminethebestlocallyoptimalsolutionforthenon-eclipsingorbittransferstudy.Apost-optimalityanalysiswasalsoperformedtoverifythatthelocallyoptimalsolutionwasincloseproximitytotheglobalsolution.Intheeclipsingorbittransferstudy,however,therewasnoeortmadetodetermineifthesolutionswereincloseproximitytooptimal.Futureimprovementscouldbemadetotheinitialguessgenerationmethod,whichmightyieldabetterinitialguesswithlesstimespentinshadowandsubsequentlyleadtoasolutionwithashortertransfertime.Itmayalsobeofinteresttoperformapost-optimalityanalysis. 141

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APPENDIXRELATIONSHIPBETWEENCOSTATEINMODIFIEDEQUINOCTIALELEMENTSANDCLASSICALORBITALELEMENTSInthisAppendixwederiveexpressionsforthecomponentsofthecostateoftheoptimalcontrolproblemgivenSection 5.1 intermsofthecomponentsofthecostateintermsofthemodiedequinoctialelements.First,theaugmentedHamiltonian,H,oftheminimum-timeoptimalcontrolproblemisgivenintermsofthedierentialequationsinmodiedequinoctialelementsas H=pGp+fGf+gGg+hGh+kGk+mGm+tGt)]TJ /F8 11.955 Tf 11.96 0 Td[((u2r+u2+u2h)]TJ /F4 11.955 Tf 11.96 0 Td[(1),(A{1)where(p,f,g,h,k,m,t)isthecostateassociatedwiththedierentialequationsofEqs.( 4{9 )and( 4{10 )andistheLagrangemultiplierassociatedwiththepathconstraintofEq.( 4{24 ).TheHamiltoniancanbeexpressedequivalentlyintermsoftheclassicalorbitalelementsas H=aGa+eGe+iGi+G+!G!+mGm+tGt,(A{2)where(Ga,Ge,Gi,G,G!)denetheright-handsidesofthosecomponentsoftheequationsofmotiongiveninEqs.( 4{9 )and( 4{10 )thatcorrespondtothedynamicsfortheorbitalelementsa,e,i,,!,thatis, da dL=Ga,de dL=Ge,di dL=Gi,d dL=G,d! dL=G!.(A{3)Becausethecomponentsofthecostatemandtarethesameusingeithermodiedequinoctialelementsororbitalelementsandthecontrolisthesameinbothformulations, 142

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theHamiltoniangivenineither( A{1 )or( A{1 )canbereplacedwiththereducedHamiltonian, Hr=pGp+fGf+gGg+hGh+kGk,=aGa+eGe+iGi+G+!G!.(A{4)Next,therelationshipbetweenthemodiedequinoctialelementsandtheclassicalorbitalelementsaregivenas a=a(p,f,g)=p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(f2)]TJ /F3 11.955 Tf 11.96 0 Td[(g2e=e(f,g)=p f2+g2i=i(h,k)=tan)]TJ /F6 7.97 Tf 6.59 0 Td[(12p h2+k2 1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2)]TJ /F3 11.955 Tf 11.95 0 Td[(h2=(h,k)=tan)]TJ /F6 7.97 Tf 6.59 0 Td[(1k h!=!(f,g,h,k)=tan)]TJ /F6 7.97 Tf 6.59 0 Td[(1gh)]TJ /F3 11.955 Tf 11.96 0 Td[(fk fh+gk.(A{5)Theexpressionsforda=dL,de=dL,di=dL,d=dL,andd!=dLarethengivenintermsofmodiedequinoctialelementsas da dL=@a @pdp dL+@a @fdf dL+@a @gdg dL=@a @pGp+@a @fGf+@a @gGg,de dL=@e @fdf dL+@e @gdg dL=@e @fGf+@e @gGg,di dL=@i @hdh dL+@i @kdk dL=@i @hGh+@i @kGk,d dL=@ @hdh dL+@ @kdk dL=@ @hGh+@ @kGk,d! dL=@! @fdf dL+@! @gdg dL+@! @hdh dL+@! @kdk dL=@! @fGf+@! @gGg+@! @hGh+@! @kGk.(A{6)Substituting( A{6 )into( A{4 ),thereducedHamiltoniancanbeexpressedas Hr=a@a @pGp+@a @fGf+@a @gGg+e@e @fGf+@e @gGg+i@ @hGh+@ @kGk+@ @hGh+@ @kGk+!@! @fGf+@! @gGg+@! @hGh+@! @kGk(A{7) 143

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andrearrangedtoyield Hr=a@a @pGp+a@a @f+e@e @f+!@! @fGf+a@a @g+e@e @g+!@! @gGg+i@i @h+@ @h+!@! @hGh+i@i @k+@ @k+!@! @kGk=pGp+fGf+gGg+hGh+kGk.(A{8)Equatingtermsin( A{8 )leadstothefollowingsystemofvelinearequationsthatrelate(a,e,i,,!)to(p,f,g,h,k): 266666666666664pfghk377777777777775=266666666666664@a @p0000@a @f@e @f00@! @f@a @g@e @g00@! @g00@i @h@ @h@! @h00@i @k@ @k@! @k377777777777775266666666666664aei!377777777777775(A{9)Assumingthatthesystemmatrix266666666666664@a @p0000@a @f@e @f00@! @f@a @g@e @g00@! @g00@i @h@ @h@! @h00@i @k@ @k@! @k377777777777775isinvertible,Eq.( A{9 )canbesolvedtoobtain(a,e,i,,!). 144

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[87] U.N.ObservatoryandH.N.A.Oce,TheAstronomicalAlmanacfortheYear2013.GETADDRESS:UnitedKingdomHydrographicOce,2012. [88] V.A.Chobotov,OrbitalMechanics.Washington,D.C.:AAIA,1991. [89] D.D.Wackerly,W.Mendenhall,andR.L.Scheaer,MathematicalStatisticswithApplications,7thed.London:ThomsonBrooks/Cole,2008. 151

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BIOGRAPHICALSKETCHKathrynGrahamreceivedherB.S.andM.S.inmechanicalengineeringfromtheUniversityofFloridainMay2008andDecember2010,respectively.ContinuinghereducationattheUniversityofFlorida,shereceivedherPh.D.inaerospaceengineeringinDecember2015.Shespentthesummerof2011asaSpaceScholarworkingintheSpaceVehiclesDirectorateattheKirtlandAirForceResearchLaboratoryinAlbuquerque,NM.ShewasalsoarecipientoftheJohnV.BreakwellAwardfortraveltothe2013AmericanAstronauticalSocietyAstrodynamicsSpecialistConferenceheldinHiltonHead,SC.Herresearchinterestsincludetrajectoryoptimization,optimalcontroltheory,andorbitalmechanics. 152