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Minimum-Fuel Finite Thrust Low-Earth Orbit Aeroassisted Orbital Transfer of Small Spacecraft

Permanent Link: http://ufdc.ufl.edu/UFE0043797/00001

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Title: Minimum-Fuel Finite Thrust Low-Earth Orbit Aeroassisted Orbital Transfer of Small Spacecraft
Physical Description: 1 online resource (42 p.)
Language: english
Creator: Senses, Begum
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: aeroassisted -- finite -- fuel -- leo -- minimum -- noncoplanar -- orbital -- small -- spacecraft -- thrust -- transfer
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this study, minimum fuel solutions to finite thrust noncoplanar aeroassisted orbital transfers of a small spacecraft between two low-Earth orbits are found using General Pseudospectral Optimal Control Software (GPOPS) which applies an hp-adaptive pseudospectral method. Optimal trajectories, and final mass ratios of the small spacecraft that are obtained subject to various inclination changes, heating rate constraints, and number of atmospheric passes are compared. It is observed that, fuel consumption is proportional to the inclination change, on the other hand, it is inversely proportional to the heating rate constraint. Furthermore, for the cases where the heating rate is not constrained, the number of atmospheric passes does not affect the fuel consumption. For the cases where the heating rate is constrained, increasing number of atmospheric passes, however, decreases fuel consumption.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Begum Senses.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Rao, Anil.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043797:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043797/00001

Material Information

Title: Minimum-Fuel Finite Thrust Low-Earth Orbit Aeroassisted Orbital Transfer of Small Spacecraft
Physical Description: 1 online resource (42 p.)
Language: english
Creator: Senses, Begum
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: aeroassisted -- finite -- fuel -- leo -- minimum -- noncoplanar -- orbital -- small -- spacecraft -- thrust -- transfer
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this study, minimum fuel solutions to finite thrust noncoplanar aeroassisted orbital transfers of a small spacecraft between two low-Earth orbits are found using General Pseudospectral Optimal Control Software (GPOPS) which applies an hp-adaptive pseudospectral method. Optimal trajectories, and final mass ratios of the small spacecraft that are obtained subject to various inclination changes, heating rate constraints, and number of atmospheric passes are compared. It is observed that, fuel consumption is proportional to the inclination change, on the other hand, it is inversely proportional to the heating rate constraint. Furthermore, for the cases where the heating rate is not constrained, the number of atmospheric passes does not affect the fuel consumption. For the cases where the heating rate is constrained, increasing number of atmospheric passes, however, decreases fuel consumption.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Begum Senses.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Rao, Anil.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043797:00001


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MINIMUM-FUELFINITETHRUSTLOW-EARTHORBITAEROASSISTEDO RBITAL TRANSFEROFSMALLSPACECRAFT By BEGUMSENSES ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2011

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c r 2011BegumSenses 2

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Idedicatemythesistomylovelyfamily:BerrakSenses,Aytu lSenses,BulentSenses, AyselBayar,M.HikmetBayar. 3

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ACKNOWLEDGMENTS Firstofall,Iwanttothankmyfamilyforalwayssupportingm e.Evenwhenwehave 10000kmbetweenus,Ialwaysfeelliketheyarenearme.Theya lwayshelpmetofind mywaywhenIamlost.Theytaughtmetobepeacefulandhappy.M ostimportantly, theyneverletmelosehopeandgiveupmydreams. IalsowanttothankDr.Raoforhispatience.WhenIstartedmy master'sdegree, Iwastryingtoadapttoanewcountryandanewmajor.WithoutD r.Rao,allofthese changeshavebeensopainful.Hewastheinstructorofthetwo classeswhichIlearned much.Besidesbeingaexcellentinstructor,healwaysadvis edmewell.Hewasthe onewhocalmedmedownwhenIpanic.Healwayslistenedtomean dansweredmy questionsaboutclassesbutalsoaboutlife. Iwouldnotbeabletopursuemymaster'sdegreeintheUnitedS tates,without thenancialsupportoftheTurkishgovernmentandTheScien ticandTechnological ResearchCouncilofTurkey.Ihope,eventuallythatIcanbeo neoftheengineerswho helpdevelopthespaceindustryinTurkey. Andnally,Iwanttothanktomycutelittlesisterforherpat ience.Wecouldnot spendalotoftimetogether.Ihavebeenawayfromhomesinces hewastwoyearsold. Sheneverstoppedlovingandmissingme.Iknowshelovesmewh ereverIamand whateverIamdoing.Mysweetheart,myspoiledlittlesister ,Iwillalwaysloveyou,too. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTIONTONUMERICALMETHODS .................. 9 2ASURVEYOFAEROASSISTEDORBITALTRANSFERS ........... 14 3MINIMUMFUELLEOTOLEONONCOPLANARAEROASSISTEDORBITAL TRANSFEROFASMALLSPACECRAFTWITHFINITE-BURN ......... 18 3.1Dynamics .................................... 18 3.1.1DeningControlsfortheExo-AtmosphericPhasesandR eformulation oftheDynamics ............................. 21 3.1.2DeterminingtheOptimalValueoftheThrustMagnitude ...... 22 3.1.3DeningControlfortheAtmosphericFlightPhaseandR eformulation oftheDynamics ............................. 25 3.2ConstraintsontheMotionoftheVehicle ................... 26 3.2.1TheInitialandTerminalConditions .................. 26 3.2.2InteriorPointConstraints ........................ 27 3.2.3VehiclePathConstraints ........................ 28 3.3FitnessFunction ................................ 30 3.4Results ..................................... 30 3.4.1EffectofInclinationChange ...................... 30 3.4.2EffectofHeatingRateConstraint ................... 32 3.4.3EffectofNumberofAtmosphericPasses ............... 37 4DISCUSSION ..................................... 39 REFERENCES ....................................... 40 BIOGRAPHICALSKETCH ................................ 42 5

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LISTOFTABLES Table page 3-1Physicalconstantsofthehighlift-to-dragratiospace craftandtheexponential atmosphericmodel .................................. 21 6

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LISTOFFIGURES Figure page 1-1Collocationmethodsclassdiagram ......................... 11 3-1Coordinatesystemsthatareusedinthederivationofthe dynamics ....... 19 3-2Altitude,vs.Time,SpeedandFlightPathAngleforVario usFinalInclinations ( i f ) ........................................... 32 3-3Time,vs.AngleofAttack,BankAngle,andHeatingRatefo rVariousFinal Inclinations( i f ) .................................... 33 3-4Altitude,vs.Time,Speed,andFlightPathAngleforVari ousMax.Allowable HeatingRates .................................... 35 3-5Time,vs.AngleofAttack,BankAngle,andHeatingRatefo rVariousMax. AllowableHeatingRates ............................... 36 3-6FinalMassRatio,vs.FinalInclinationsforVariousMax .AllowableHeating Rates ......................................... 37 3-7FinalMassRatio, m ( t f ) = m 0 ,vs.NumberofAtmosphericPasses,Max.Allowable HeatingRates,andFinalInclinations ........................ 38 7

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience MINIMUM-FUELFINITETHRUSTLOW-EARTHORBITAEROASSISTEDO RBITAL TRANSFEROFSMALLSPACECRAFT By BegumSenses December2011 Chair:AnilV.RaoMajor:AerospaceEngineering Inthisstudy,minimumfuelsolutionstonitethrustnoncop lanaraeroassistedorbital transfersofasmallspacecraftbetweentwolow-Earthorbit sarefoundusingGeneral PseudospectralOptimalControlSoftware(GPOPS)whichapp liesanhp-adaptive pseudospectralmethod.Optimaltrajectories,andnalmas sratiosofthesmall spacecraftthatareobtainedsubjecttovariousinclinatio nchanges,heatingrate constraints,andnumberofatmosphericpassesarecompared .Itisobservedthat, fuelconsumptionisproportionaltotheinclinationchange ,ontheotherhand,itis inverselyproportionaltotheheatingrateconstraint.Fur thermore,forthecaseswhere theheatingrateisnotconstrained,thenumberofatmospher icpassesdoesnotaffect thefuelconsumption.Forthecaseswheretheheatingrateis constrained,increasing numberofatmosphericpasses,however,decreasesfuelcons umption. 8

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CHAPTER1 INTRODUCTIONTONUMERICALMETHODS Theobjectiveofanoptimalcontrolproblemistondthestat eandcontrolthat optimizesaperformancecriteriasubjecttodynamicconstr aints,pathconstraints,initial andterminalconditions.Solutionstooptimalcontrolprob lemscanbefoundeither analyticallyornumerically.Determininganoptimaltraje ctoryanalyticallyis,however, impossibleformostoptimalcontrolproblems.Therefore,o ptimalcontrolproblems aresolvedbynumericalmethods.Numericalmethodsforsolv ingoptimalcontrol problemsfallintotwocategories:indirectanddirectmeth ods.Inanindirectmethodthe calculusofvariationsisusedtoderivetherst-orderopti malityconditions,resultingin Hamiltonianboundaryvalueproblem.TheHamiltonianvalue problemisthensolved todeterminecandidateoptimalsolutionscalledextremals .Examplesofcommonly usedindirectmethodsareindirectshooting,indirectmult ipleshootingandindirect collocationmethods[ 15 ].Althoughindirectnumericalmethodshavethepotentialt o produceaccuratesolutions,thesemethodshavesomedisadv antages.Firstly,evenfor asimpleproblem,obtainingtherst-orderoptimalitycond itionsandthensolvingthe Hamiltonianboundaryvalueproblemmaybeextremelydifcu lt.Iftheformulationofa problemchangesevenslightly,therst-orderoptimalityc onditionsmustbere-derived. Secondly,theradiiofconvergenceofindirectmethodsares mall.Consequently,an indirectmethodrequiresaverygoodinitialguess[ 3 ].Whileitisoftenstraightforwardto determineaguessfortheinitialstate,determiningagoodi nitialguessforacostateis difcultbecausethecostatedoesnothaveaphysicalinterp retation.Inaddition,inorder touseanindirectmethoditisnecessarytohaveknowledgeof theswitchingstructureof theinequalitypathconstraints. Thesecondcaseofnumericalmethodsforoptimalcontrolare directmethods.in adirectmethod,thecontinuous-timeoptimalcontrolprobl em(CTOCP)istranscribed intoanonlinearprogrammingproblem(NLP)viaastateand/o rcontrolparameterization. 9

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Althoughmanydirectnumericalmethodsarelessaccurateth anindirectnumerical methods,thesemethodshavesomeadvantages.Inadirectnum ericalmethod,the rst-ordernecessaryconditionsarenotderived.Inadditi on,theradiiofconvergenceof adirectmethodisoftenlargerthanthatofanindirectmetho d,anditisnotnecessaryto provideaguessforthenon-intuitivecostate. Anexampleofadirectnumericalmethodisthedirectshootin gmethod.Thedirect shootingmethodtranscribesaCTOCPintoanNLPviacontrolp arameterization.The solutionofanNLPgivesoptimalparametervaluesoftheappr oximatedcontrol.The directshootingmethodisusefulforonproblemswheretheso lutioncanbeaccurately approximatedusingasmallnumberofparameters. Anotherexampleofawellknowndirectmethodisthedirectmu ltiple-shooting method.Similartodirectshooting,thedirectmultiple-sh ootingisalsoacontrol parameterization.Thedifferencebetweenthedirectshoot ingandthedirectmultiple-shooting methodisthenumberoftimeintervalsandthenumberofparam eters.Whilein adirectshootingmethodtheintegrationisperformedovera singleinterval,ina multiple-shootingmethodtheintegrationisperformedove rsubintervals.Thevalue ofthestateatthestartofeverysubintervalisaparameteri ntheNLP[ 15 ].Although addingsubintervalsintroduceadditionalparameterstoth eNLP,multiple-shootingisless sensitivetoinitialguessesthanthestandardshootingmet hod. Anotherexampleofadirectnumericalmethodisadirectcoll ocationmethod. DirectcollocationmethodsconvertaCTOCPtoanNLPviaasta teandcontrol parameterization.Asopposedtodirectshootinganddirect multiple-shootingmethods, directcollocationmethodsdonotrequireknowledgeswitch ingstructureoftheinequality constraints.Otheradvantagesanddisadvantagesofadirec tcollocationmethodwillbe discussedforeachofdirectcollocationmethod.Directcol locationmethodsfallintothree categories:h-methods,p-methods,andhp-methods. 10

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COLLOCATION METHODS DirectCollocationMethods IndirectCollocationMethods LocalCollocationMethods GlobalCollocationMethods Runge-KuttaMethods OrthogonalCollocationMethods Pseudospectral Methods(OrthogonalGlobalCollocationMethods) Gauss-LobattoPseudospectralMethods -LegendrePseudospectralMethod -JacobiPseudospectralMethod Legendre-GaussPseudospectralMethods -GaussPseudospectralMethod Legendre-GaussRadauPseudospectralMethods -RadauPseudospectralMethod Figure1-1.Collocationmethodsclassdiagram Inanh-methodthestateisapproximatedusingthesamelow-d egreepolynomial. Thesemethodsalsodividethetimeintervalintosubinterva lsandaddconstraints thatenforcecontinuityinthestateateachsubintervalint erface.Convergenceofthe approximationisthenachievedbyincreasingthenumberofs ubintervals.Although h-methodshaveslowconvergencerates,usuallyNLPsareext remelysparse.Unlike anh-method,inap-methodthestateisapproximatedusingag lobalpolynomial. Convergenceisthenachievedbyincreasingthedegreeofthe polynomial. Akeyaspectofobtaininganaccuratesolutionusingap-meth odisthechoiceof collocationpoints.Collocationpointsofap-methodareof tenchosenastherootsof anorthogonalpolynomial.Threemostpopularexamplesofps eudospectralmethods aretheGausspseudospectralmethod(GPM),theRadaupseudo spectralmethod (RPM),andtheLobattopseudospectralmethod(LPM).Theset sofcollocation 11

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pointsoftheGPM,RPMandLPMarerespectivelytheLegendreGauss(LG), Legendre-Gauss-Radau,andLegendre-Gauss-Lobatto(LGL) points.Whilenoneof theendpointsoftheintervalt2 [ 1 1 ] isincludedintheLGpoints,theLGLpoints containbothoftheendpoints.Additionally,theLGRpoints onlycontainoneofthe endpoints.LG,LGRandLGLpointsaredenedastherootsofli nearcombinationsof Legendrepolynomials,andderivativesofLegendrepolynom ials. n th degreeLegendre polynomialsaredenedas P n (t)= 1 2 n n d n dtn [(t2 1 ) n ] (1–1) Therootsof n th degreeLegendrepolynomialsgivesnLGpoints.Alsoweights aregiven intermsofti andderivativeof n th degreeLegendrepolynomials. w i = 2 1 t2 i [ P (ti )] 2 ( i = 1 ,..., n ) (1–2) If n th and ( n 1 ) th degreeLegendrepolynomialsaredenotedby P n and P n 1 respectively, thentherootsof P N 1 (t)+ P N (t) givenLGRpoints[ 1 ].Theweightsaregivenas w 1 = 2 n 2 w i = 1 ( 1 ti )[ P n 1 (ti )] 2 ( i = 2 ,..., n ) (1–3) Therootsof P N 1 (t) ,-1and1pointsgivenLGLpoints.Theweightsare[ 14 ] w 1 = w + 1 = 2 n ( n 1 ) w i = 2 n ( n 1 )[ P n 1 (t)] 2 ( i = 2 ,..., n 1 ) (1–4) Toperformapseudospectralmethod,theconceptofpolynomi alinterpolationand numericalquadraturemustbeknown.Inapseudospectralmet hod,thebasisfunctionfor thepolynomialinterpolationistheLagrangepolynomials. TheLagrangepolynomialsare 12

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givenas L i (t)= N j = s j 6 = ittj t i t j s i N (1–5) sequalsonefortheippedLGRandLGLcollocationpointswhi chhavecollocatedinitial pointsandsequalszeroforLGandLGRpoints.Thesecondimpo rtantconceptisthe numericalquadrature.Numericalquadratureapproximates anintegralnumerically.The weightedsumoftheintegrandatquadraturepointsgivesana pproximatedorforsome casesexactvaluetotheintegral. Z b a f ( t ) d t N i = 1 w i f ( t i ) (1–6) where w i istheweightfunctionandNisthenumberofquadraturepoint s.Inpseudospectral methods,thehighlyaccurateGaussianquadraturerulesare used.Npointapproximation withtheLG,LGRandLGLquadraturerulesgivetheexactsolut ionofanintegralifthe polynomialdegreeoftheintegrandisatmost2N-1,2N-2,and 2N-3respectively.The weightfunctionsfortheLG,LGRandLGLquadraturerulesare givenastheEqs.( 1–2 ), ( 1–3 ),( 1–4 )[ 8 ]. Theadvantageofapproximatingthestatewithglobalpolyno mialsatorthogonal collocationpointsisthespectralconvergencerate[ 10 ].Whenap-method,however, isemployedonaproblemwhosesolutionisnon-smooth,conve rgenceratecanbe extremelyslow[ 5 ].Moreover,thedenseNLPofapseudospectralmethodandfor some problemsrequirementofanextremelyhighorderpolynomial toachieveanaccurate approximationmaycausecomputationaldeciencies. Inthethirdcollocationmethod,hp-method,thedegreeofth eapproximation, thenumberandthelocationofsubintervalsisvariedtoachi eveconvergenceofthe approximation.Therefore,itbenetsfromboththeadvanta gesofanh-methodanda p-method. 13

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CHAPTER2 ASURVEYOFAEROASSISTEDORBITALTRANSFERS Inrecentyears,smallspacecrafthavebecomeincreasingly popularinspace missionsduetotheirpotentialtoreducetheoverallcostof spacemissions.One waytoreducemissioncostistoreducetheamountoffuelrequ iredtocompletethe mission.Reducingfuelcanbeaccomplishedbyoptimizingth emaneuverrequiredin orbit.Orbitaltransferswhereallofthemaneuversrequire stheuseoffuel,suchas Hohmanntransfersrequireextremelylargefuelconsumptio n.Analternativetoan all-propulsivemaneuverisatransferwheresomeoftheorbi tchangeisperformed usingatmosphericforce.Suchmaneuverallcalledaeroassi stedorbitaltransfers.Since aeroassistedtransferwasrstpublished[ 11 ],numerousstudieshavebeenconducted onaeroassistedorbitaltransfers.Inthissectionasurvey ofaeroassistedorbitaltransfers isgiven.Moredetailedsurveysonearlyworkonaeroassiste dorbitaltransferscanbe foundinreferences[ 12 ]and[ 19 ].Aeroassitedorbitaltransfers(AOT)decreasesfuel consumptionduringvariousmissionswithaltitude,and/or inclinationchangesinthe neighborhoodofanatmosphere.Thesemissionsfallintothr eecategories:synergetic planechanges,orbitaltransfers,andplanetarymissions. Synergeticplanechange maneuversusebothpropulsiveandaerodynamicforcesandas pacevehiclestartsan synergeticplanechangemaneuverusuallyfromalow-Eartho rbit(LEO)orbit.Aeroglide andaerocruisemaneuversarethetwotypesofsynergeticpla nechangemaneuvers. Anaeroglidemaneuverincludespropulsiveforceonlyatexo -atmosphericightphases andanaerocruisemaneuverincludespropulsiveforceatbot hexo-atmosphericand atmosphericightphases.Duringanaerocruisemaneuver,t hrustatatmospheric ightphasecancelsthedragforce.Orbitaltransfersaresi milartosynergeticplane changemaneuvers.Thedifferencebetweenthesemaneuversi stheinitialorbitofthe spacevehiclewhichisusuallyahigh-Earthorbitforanorbi taltransfer.Aerodynamic forcescanalsobeusedduringplanetarymissions.Thesemis sionscanbeachieved 14

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eitherbyanaerocapturemaneuverorbyamultiple-passaero brakingmaneuvers.An aerocapturemaneuvertransfersaspacevehiclefromahyper bolicapproachinitialorbit toitsnalorbitaroundaplanet.Duringthismaneuver,atmo sphericpassesdecrease thevelocityofthespacevehicleandthisleadsthespaceveh icletobecapturedby thegravityeldoftheplanet.Moreoveraerobrakingmultipassmaneuverscombine rocketpropulsionwithhighaltitudeatmosphericpassesto circularizeanellipticalinitial orbit.Therocketpropulsionduringtheatmosphericightp haseadjuststhealtitude ofthespacevehiclebecauseifthespacevehicledoesnotgod eepenoughintothe atmosphere,thevelocitydoesnotdecreaseenough.Ontheot herhand,goingtoodeep intotheatmospherecausesenormousaerodynamicheating.T rajectoriesduringall ofthesethreemissionscanbeoptimizedaccordingtodiffer entperformancecriteria suchas,energyloss,inclinationchange,andfuelconsumpt ion.Theminimizationof theenergylosswasstudiedastheperformancecriterioninr eferences[ 9 ],and[ 20 ].In reference[ 9 ]theminimumenergylosssolutionstoanimpulsiveaeroglid emaneuver betweentwononcoplanarcircularorbitswerefound.Whilet herstimpulsede-orbited thespacevehicle,thesecondimpulselifteditfromtheedge ofsensibleatmosphere toitsnalaltitude,andthethirdimpulsecircularizedits nalorbit.Asaresult,itwas foundtobethat,maximizingtheatmosphereexitvelocitymi nimizestheenergyloss duringtheatmosphericightphase.Inreference[ 20 ]theauthorsfoundminimumenergy solutionstoLEOtoLEOnoncoplanaraeroassistedorbitaltr ansferofahighlifttodrag ratiospacevehiclesubjecttovariousheatingrateandheat ingloadconstraintsusing directmultipleshootingmethod.Thisvehicleperformedan aerocruise,aeroglide,and aerobrakemaneuver.Theauthorsalsofoundmaximuminclina tionchangesolutionsfor thesamemaneuverssubjecttosameconstraints.Consequent ly,itwasobservedthat anaerobrakemaneuverhasbetterperformanceforbothperfo rmancecriteriathanthe othermaneuvers.References[ 18 ]and[ 16 ]alsousedthemaximizationoftheinclination changeastheperformancecriterion.Inreference[ 18 ],theauthorsfoundoptimal 15

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solutionstononcoplanarmultiple-passaeroassistedorbi taltransfersfortwomaneuvers. Duringtherstmaneuver,thespacevehiclede-orbited,dec reaseditsaltitudeand directlywentintotheatmosphere.Duringthesecondmaneuv erthespacevehicledid notgodirectlyintotheatmosphere.Itroundedseveraltime saroundtheplanetand changeditsorbitalplanebeforeitenteredtheatmosphere, andthenitcompletedthe maneuverwithmultipleatmosphericpasses.Thesetwomaneu verswerecomparedand itwasshownthatalthoughthemaximuminclinationchangeva luesarealmostthesame forbothmaneuvers,thetotalighttimesdiffer.Thesecond maneuveraccomplishes thesameamountofinclinationchangewithalongerightdur ationwhichreduces theheatingloadonthespacevehicle.Inreference[ 16 ]maximuminclinationchange solutionstogeostationaryEarthOrbit(GEO)toLEOnoncopl anaraeroassistedorbital transferswithanimpulsiveaeroglidemaneuverforvarious heatingrateconstraints areobtainedbysoftwareSOCS[ 2 ](SparseOptimalControlSoftware)whichapplies adirectcollocationmethod.Inthisresearchitwasobserve dthatfortheheatingrate isnotconstrained,increasingthenumberofatmosphericpa sses,decreasesthe magnitudeofimpulsivethrustmaneuver( 4 V )slightly.Ontheotherhand,forthecases wheretheheatingrateisconstrained,thenumberofatmosph ericpassesaffects 4 V signicantly.Theeffectofthenumberofatmosphericpasse son 4 V decreases,while thenumberofatmosphericpassesincreases.Theauthorsals ofoundminimumfuel solutionssubjecttosomeheatingrateconstraints.Theysh owedthatdecreasingthe heatingrateconstraintdecreasestheeffectofthenumbero fatmosphericpasses on 4 V .Furthermore,increasingthenumberofatmosphericpasses decreasesthe fuelconsumptionregardlessofthevalueofheatingratecon straints.Reference[ 6 ] consideredaLEOtoLEOnoncoplanaraeroassistedorbitaltr ansferofahighL/Dratio spacecraftwhichcompletedanimpulsiveaeroglidemaneuve r.First,theHohmann transferswerecomparedtoaeroassistedorbitaltransfers ,anditwasshownthat aeroassistedorbitaltransfersaremorefuelefcientthan theHohmanntransfers,ifthe 16

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inclinationchangeismorethan15deg.Secondly,theeffect ofnumberofatmospheric passesonfuelconsumptionwasinvestigated,anditwasfoun dthataeroassistedorbital transferswithtwoatmosphericpassesgivethemostfuelef cienttrajectories.This resultcontraststotheresultthatwasobtainedinreferenc e[ 16 ]whichconsidereda muchmoremassivespacecraft.Inreference[ 13 ]minimum-fueltrajectorytoimpulsive thrustcoplanarhighEarthorbit(HEO)toLEOaeroassistedo rbittransferwasfound. Thisreferencegaveimportantresultsabouttheatmospheri centranceandtheexitight pathangles.Firstly,itwasshownthatIncreasingtheight pathangleattheatmospheric entranceincreasesthelowestaltitudethatthespacevehic leexperiencesandtherefore decreasesthemaximumheatingratethatthespacecraftexpe riences.Secondly,the ightpathanglenearzeroattheatmosphericexitgivesmore fuelefcienttrajectories. Asseenfromtheliteraturereview,therearenumerousresea rchonaeroassisted orbitaltransfers.Althought,thesestudiesinvestigated aeroassistedorbitaltransfers fordifferenttypesofinitialorbits,maneuvers,initialm asses,andtheyapplieddifferent numericalmethodstondoptimaltrajectories,allofthese studieshaveacommonpoint: impulsivethrust.Although,theimpulsivepropulsionassu mptioncanbegoodenough,if themaximumthrustmagnitudeislargeandthethrustduratio nsofavehicleareshorter thancoastingdurations,inpracticalapplications,thrus tstaysatturnonpositionfor aniteamountoftime,andthethrustmagnitudeisconstrain edbytherocketlimits. Therefore,nitethrustmodelsthrustinpracticalapplica tionsmorepreciselythanthe impulsivethrust. 17

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CHAPTER3 MINIMUMFUELLEOTOLEONONCOPLANARAEROASSISTEDORBITAL TRANSFEROFASMALLSPACECRAFTWITHFINITE-BURN 3.1Dynamics Inthisstudy,minimum-fuelnitethrustLEOtoLEOnoncopla naraeroassisted orbitaltransfersofasmallspacecraftforvariousinclina tionchangessubjecttovarious heatingrateconstraintsareinvestigated.Allofthenonco planaraeroassistedorbital transfersisachievedusingthreedifferenttypesofphases .Thesesphasetypesare givenas;exo-atmosphericburn,atmosphericightwithout burn,exo-atmosphericcoast. Thespacecraftonlythrustsduringtheexo-atmosphericig htphases.InthisSection theequationsofthemotionduringanaeroglidemaneuverare given.Itisassumed thatEarthisnon-rotating,thatthespacecraftismodeleda pointmass,andthatthe atmospheremodelisexponential. Therstphaseofthenoncoplanaraeroassistedorbitaltran sferistheexo-atmospheric ightburnphase.Dynamicequationsofthespacecraftdurin ganexo-atmosphericburn phasearegivenas r = v sing(3–1) q= v cosgcosy r cosf(3–2) f= v cosgsiny r (3–3) v = T cose m m r 2 sing(3–4) g= 1 v T sinecoss m m r 2 v 2 r cosg (3–5) y= 1 v T sinesins m cosg v 2 r cosgcosytanf (3–6) m = T g 0 I sp (3–7) In( 3–1 )-( 3–7 ) r denotesthegeocentricradius, m denotesthemassofthesmall spacecraft,andqdenotesthelongitude,fdenotesthelatitude, v denotesthespeed. 18

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gdenotestheightpathangle,yistheheadingangle.Thelongitude,latitude,ight pathangle,andheadingangleareshowningure 3.1 .Moreover, T denotesthethrust magnitude,edenotstheanglebetweenthevelocityandthethrust,sisthebankangle, and I sp istheenginespecicimpulse Figure3-1.Coordinatesystemsthatareusedinthederivati onofthedynamics Duringunpoweredexo-atmosphericight,theonlyexternal forcesthataffectthe spacecraftareliftanddrag.Eqs.( 3–8 )-( 3–13 ),givethedynamicequationsofthesmall spacecraftduringanatmosphericightphase. r = v sing(3–8) q= v cosgcosy r cosf(3–9) f= v cosgsiny r (3–10) v = D m m r 2 sing(3–11) g= 1 v L coss m m r 2 v 2 r cosg (3–12) y= 1 v L sins m cosg v 2 r cosgcosytanf (3–13) 19

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where D isthedragforce, L istheliftforce,andmistheEarthgravitationalparameter. Thedragandliftforcesaregivenas D = 1 2rv 2 AC D (3–14) L = 1 2rv 2 AC L (3–15)ristheatmosphericdensitywhichismodeledasr=r0 exp ( bh ) (3–16) wherer0 istheatmosphericdensityatsealevel, A isthevehiclereferencearea, C D is thedragcoefcient,and C L istheliftcoefcient.bistheinverseofthedensityscale heightand h = r R e isthealtitude.Theliftanddragcoefcientsaremodeledus inga dragpolaras C L = C Laa(3–17) C D = C D 0 + KC L 2 (3–18) Intheseequations, C Laistheliftslope,aistheangleofattack, C D 0 isthezeroliftdrag coefcient,and K isthedragpolarconstant.Thephysicalconstantsofthespa cecraft andtheatmosphericmodelisgiveninTable 3-1 Thethirdphaseofthemaneuveristheexo-atmosphericcoast phase.Inthis phasethereisnoexternalforcethataffectsthespacecraft .Ifthetermsthatinclude thethrustmagnitude( T )areeliminatedfromEqs.( 3–1 )-( 3–7 ),thedynamicsduringthe exo-atmosphericcoastphaseareobtained.Thefourthphase isagainanexo-atmospheric burnphase.TheequationsarealreadygivenasEqs.( 3–1 )-( 3–7 ). 20

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Table3-1.Physicalconstantsofthehighlift-to-dragrati ospacecraftandtheexponential atmosphericmodel Constant Value r0 (kg/ m 3 ) 1.225 b(1/m) 1/7200 A ref ( m 2 ) 1 K 1.4 C La 0.5699 m 0 (kg) 818 I SP (s) 310 T (Newton) 2500 h atm ( km ) 110 R e (m) 6378145 m( m 3 = s 2 ) 3.986012x 10 14 3.1.1DeningControlsfortheExo-AtmosphericPhasesandR eformulationof theDynamics Duringtheexo-atmosphericburnphasethecontrolischosen asthedirectionofthe thrust.Thecontrolcomponentsaregiveninequation( 3–19 ) u 1 = coseu 2 = sinecossu 3 = sinesins(3–19) 21

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Thereformulatedexo-atmosphericightdynamicsaretheng ivenasfollows: r = v sing(3–20) q= v cosgcosy r cosf(3–21) f= v cosgsiny r (3–22) v = Tu 1 m m r 2 sing(3–23) g= 1 v Tu 2 m m r 2 v 2 r cosg (3–24) y= 1 v Tu 3 m cosg v 2 r cosgcosytanf (3–25) m = T g 0 I sp (3–26) 3.1.2DeterminingtheOptimalValueoftheThrustMagnitude Inthesection,thethrustmagnitudeisassumedasthefourth controlcomponent duringtheexo-atmosphericburnphase.Theoptimalvalueof thethrustisfound usingthecalculusofvariations,andPontryagin'sminimum principal.OnlytheEqs. ( 3–23 ),( 3–24 ),and( 3–25 )change.Thereformulateddynamicequationsduring exo-atmosphericburnphasearegivenbytheEqs.( 3–30 )-( 3–32 ). r = v sing(3–27) q= v cosgcosy r cosf(3–28) f= v cosgsiny r (3–29) v = u 4 u 1 m m r 2 sing(3–30) g= 1 v u 4 u 2 m m r 2 v 2 r cosg (3–31) y= 1 v u 4 u 3 m cosg v 2 r cosgcosytanf (3–32) m = u 4 g 0 I sp (3–33) 22

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Tondtheoptimalsolutionbyusingthecalculusofvariatio ns,therststepisto determinetheHamiltonian.Thecostfunctionalofthenonco planaroptimalcontrol problemischosenasthemaximizationofthenalmass ( m ( t f )) ofthesmallspacecraft. J = m ( t f ) (3–34) ThecostfunctionalthatisgivenastheEq.( 3–34 )isamayercost.Themayercostcan beconvertedtoaLagrangecost.TheLagrangecostisfoundto beas J = Z t f t 0 mdt + m ( t 0 ) where m ( t 0 ) denotestheinitialmassofthesmallspacecraft.Now,dene thestateand costatevectors: x = f ( x ( t ) u ( t ))=[ r qf v gy m ] T (3–35) p =[ p 1 p 2 p 3 p 4 p 5 p 6 p 7 ] T (3–36) where p isthecostateand x isthestatedynamics.Additionally,thepathconstraint C ( x ( t ) u ( t )) isgivenas C ( x ( t ) u ( t ))= 1 u 2 1 + u 2 2 + u 2 3 Thentheaugmentedcostfunctionalisfoundas J a = m ( t 0 ) nTx( x ( t 0 ) t 0 x ( t f ) t f )+ Z t f t 0 [ m p T ( x f ( x ( t ) u ( t ))) zT C ( x ( t ) u ( t ))] dt wherex( x ( t 0 ) t 0 x ( t f ) t f ) isthevectorofboundaryconditions, C ( x ( t ) u ( t )) isthevector ofpathconstraints,nandzaretheLagrangemultipliersassociatedwiththeboundary 23

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conditionsandpathconstraintsrespectively.ThentheHam iltonianisfoundas H = m + p T f ( x ( t ) u ( t )) m z( 1 u 2 1 + u 2 2 + u 2 3 )= m + p 1 ( v sing)+ p 2 v cosgcosy r cosf + p 3 v cosgsiny r + p 4 u 4 u 1 m m r 2 sing + p 6 v u 4 u 3 m cosg v 2 r cosgcosytanf + p 5 v u 4 u 2 m m r 2 v 2 r cosg p 7 u 4 g 0 I sp z( 1 u 2 1 + u 2 2 + u 2 3 ) (3–37) Theoptimalcontrolisfoundusingtherstorderoptimality conditionforcontrol.H u 1 = 2zu 1 + p 4 u 4 m = 0 u 1 = p 4 u 4 2zm (3–38)H u 2 = 2zu 2 + p 5 u 4 mv = 0 u 2 = p 5 u 4 2zmv (3–39)H u 3 = 2zu 3 + p 6 u 4 mv cosg= 0 u 3 = p 6 u 4 2zmv cosg(3–40)H T = p 4 u 1 m + p 5 u 2 mv + p 6 u 3 mv cosg p 7 g 0 I sp = 0 (3–41) where u 1 u 2 u 3 aretheoptimalvaluesof u 1 u 2 ,and u 3 respectively.Asitisseenfrom Eq.( 3–37 ) u 4 islinearintheHamiltonian.Therefore,Eq.( 3–41 )doesnotgiveany informationabouttheoptimalvalueof u 4 .Todeterminetheinformationabouttheoptimal u 4 ( u 4 ),thePontryagin'smaximumprincipalisapplied. H ( x u p t ) H ( x u p t ) (3–42) p 4 u 4 u 1 m + p 5 u 4 u 2 m v + p 6 u 4 u 3 m v cosg p 7 u 4 g 0 I sp +z ( u 2 1 + u 2 2 + u 2 3 ) p 4 u 4 u 1 m + p 5 u 4 u 2 m v + p 6 u 4 u 3 m v cosg p 7 u 4 g 0 I sp +z ( u 2 1 + u 2 2 + u 2 3 ) (3–43) where x u ,and p denotestheoptimalstate,controlandcostaterespectivel y. Equation( 3–43 )issimpliedas ( u 4 u 4 )[ p 4 u 1 m + p 5 u 2 m v + p 6 u 3 m v cosg p 7 g 0 I sp ] 0 (3–44) 24

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Fromequation( 3–44 ) u 4 isfoundas u 4 = 8>>>><>>>>: 0 if p 4 u 1 m + p 5 u 2 m v + p 6 u 3 m v cosg p 7 g 0 I sp < 0 T max if p 4 u 1 m + p 5 u 2 m v + p 6 u 3 m v cosg p 7 g 0 I sp > 0 Undetermined if p 4 u 1 m + p 5 u 2 m v + p 6 u 3 m v cosg p 7 g 0 I sp = 0 (3–45) S = p 4 u 1 m + p 5 u 2 m v + p 6 u 3 m v cosg p 7 g 0 I sp Siscalledtheswitchingfunction.Asseeninequation( 3–45 ),iftheswitching functionislessthanzero,thentheoptimalthrustisfoundt obeatitsminimumvalue, zero.Ontheotherhand,iftheswitchingfunctionislargert hanzero,theoptimalthrustis foundtobeatitsmaximumvalue, T max .Theportionofthetrajectorythatcorresponds tothesituationwhere u 4 isundeterminediscalledthesingulararc.Theoptimalcont rol canbedeterminedusinghigherorderoptimalityconditions .Inthisstudyoptimalcontrol problemdoesnothaveasingulararc.Therefore,thesolutio ntothesingulararcproblem isbeyondthescopeofthisresearch.Allinall,inthissecti onitisshownthattheoptimal thrustmagnitudetakesonlytwovalueszeroor T max .Ifthespacecraftisthrusting, theoptimalthrustmagnitudeis T max and,obviously,ifitisnotthrustingthethrust magnitudeiszero.Therefore,addingthethrustmagnitudea sthefourthcomponent duringexo-atmosphericburnphasebecomescomputationall yinefcient. 3.1.3DeningControlfortheAtmosphericFlightPhaseandR eformulationofthe Dynamics Angleofattack,(a),andbankangle,(s)arethecontrolcomponentsduring atmosphericightphase.Inthisresearch,insteadofusing angleofattack,andbank angleasthecontrolcomponents, u 1 ,and u 2 areusedasthecontrolcomponentswhich aredenedas u 1 = C L sinsu 2 = C L coss(3–46) 25

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Reformulationoftheatmosphericightdynamicsaregivena sEqs.( 3–47 )-( 3–52 ). r = v sing(3–47) q= v cosgcosy r cosf(3–48) f= v cosgsiny r (3–49) v = D m m r 2 sing(3–50) g= 1 v rv 2 A 2 m u 2 m r 2 v 2 r cosg (3–51) y= 1 v rv 2 A 2 m cosgu 1 v 2 r cosgcosytanf (3–52) Thebankanglesandtheangleofattackacanbefoundintermsof u 1 and u 2 .The relationshipisgivenas 3.2ConstraintsontheMotionoftheVehicle Thespacecraftstartsandcompletestheaeroglidemaneuver atnoncoplanar circularlow-Earthorbits.Thisconstrainstheinitialvel ocity,initialightpathangle, terminalvelocity,andterminalightpathangle.Moreover ,itisassumedthatthe initialinclinationofthecircularlow-Earthorbitiszero andtheterminalinclinationis i f .Moreover,theightpathanglegisconstrainedattheentranceandtheexitofthe atmosphere.Thealtitudeandtheangleofattackareconstra inedduringtheatmospheric ight.Inthissection,alloftheseconstraintsaregiven.3.2.1TheInitialandTerminalConditions Theinitialandterminalorbitsareassumedcircularandthe altitudesoftheselow earthorbitsarechosenas h 0 = h f = 185 2 km.Thevelocityofthesmallspacecraftatthe initialandterminalorbitsisfoundas v ( t 0 )= v ( t f )= r m h 0 + R e = 7 793 km = s where R e istheradiusofearth.Theinitialandterminalvalueofigh tpathanglegisalsoconstrainedtobezero.Additionally,theinclinati onsoftheinitialandterminal 26

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circularorbitsarechosentobezeroand i f respectively.Theinclinationsofthecircular orbitsarefoundas i = cos 1 ( cos (f) cos (y)) Anotherconstraintisontheinitialmassofthespacecraft. Itischosentobe818kg.All oftheinitialandconstraintsaregivenasfollows InitialConditions TerminalConditions t 0 = 0 t f free r ( t 0 )= h 0 + R e = 6563345 m r ( t f )= h f + R e =6563345m q( t 0 )= 0 radian q( t f ) free f( t 0 )= 0 radian f( t f ) free v ( t 0 )= 7 793 km/s v ( t f )= 7 793 km/s g( t 0 )= 0 radian g( t f )= 0 radian y( t 0 )= 0 radian y( t f ) free m ( t 0 )= 818 kg m ( t f ) free i ( t 0 )= 0 radian i ( t f ) = cos 1 ( cos (f( t f )) cos (y( t f ))) radian 3.2.2InteriorPointConstraints Thesmallspacecraftenterstheatmosphereat t atm 0 andat t atm 0 .Thealtitudeofthe spacecraftiscomputedas R e + h atm h atm whichistheboundaryofsensibleatmosphere whichequals110km.Thesmallspacecraftexitstheatmosphe reat t atm f .Similartothe entrancealtitude,theexitaltitude r ( t atm f ) iscomputedas R e + h atm .Also,theightpath angleisconstrainedattheatmosphericentranceandtheatm osphericexit.Allofthe interiorpointconstraintsaregivenas 27

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AtmosphericEntranceInteriorPointConstraint AtmosphericExitInteriorPointConstraint r ( t atm 0 )= R e + h atm r ( t atm f )= R e + h atm g( t atm 0 ) 0 g( t atm f ) 0 3.2.3VehiclePathConstraints Duringtheatmosphericightphasethealtitudeofthesmall spacecraftisconstrained. Asaforementioned,theboundaryofthesensibleatmosphere is110km.Thereforethe altitudeofthespacecraftmustbebetween0and110km. 0 h h atm (3–53) Theequations( 3–17 )and( 3–18 )areonlyvalidiftheangleofattackislessthana criticalangle.Thiscriticalangleiscalledthestallinga ngle[ 17 ].Inthisresearchthe stallingangleis 40 o .Stallingangleisdenotedbyamax 0 aamax Asitisseenfromtheequation( 3–17 ),theangleofattackaislinearintheliftcoefcient C L .Therefore,constraining C L alsoconstrainsa. 0 C L C Lmax (3–54) where C Lmax correspondstothevalueoftheliftcoefcientwhenainequation( 3–17 )is equaltothestallingangle.Itisfoundtobeapproximately0 .4.Theconstraintonthelift coefcient( C L )alsoconstrainsthecontrolcomponents u 1 and u 2 C Lmax ( u 1 u 2 ) C Lmax 0 u 1 2 + u 2 2 C 2 Lmax (3–55) ThestagnationpointheatingrateiscomputedusingtheChap manequation[ 7 ] 28

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Q = Q constant (r=r0 ) 0 5 ( v = v c ) 3 15 (3–56) InEq.( 3–56 ), Q constant isconstant( Q constant =19987W/cm 2 ),and v c isequalto p m= R e Theheatingrateconstraintisgivenasfollows 0 Q Q (3–57) where Q isthemaximumallowableheatingrate.Thenaturallogarith moftheheating rateconstraintiscomputedas log0 log Q log Q log Q log Q log Q = log ( Q constant (r=r0 ) 0 5 ( v = v c ) 3 15 ) = log Q constant + 0 5log (r=r0 )+ 3 15log ( v = v c ) Fromtheequation( 3–16 ) log (r r0 ) equalsto bh ,consequentlythenaturallogarithmof heatingrateisconstrainedas log Q constant 0 5bh + 3 15log ( v = v c ) log Q (3–58) Theonlypathconstraintduringtheexo-atmosphericighti sgivenas u 2 1 + u 2 2 + u 2 3 = 1 (3–59) Toconclude,Eq.( 3–59 )givesthepathconstraintduringtheexo-atmosphericigh t phaseandEqs.( 3–53 ),( 3–54 ),( 3–55 ),( 3–58 )givethepathconstraintsduringthe atmosphericightphase. 29

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3.3FitnessFunction Theobjectiveoftheoptimalcontrolproblemisminimizingt hefuelconsumption. Thisobjectivecanalsobeachievedbymaximizingthenalma ssofthesmallspacecraft. J = m ( t f ) (3–60) 3.4Results TheoptimalcontrolproblemthatisdenedinChapter 3 isscaledusingcanonical units.TheunitoflengthischosentobetheradiusofEarth ( R e ) ,theunitoftimeis chosentobe p R e =m.RecallthatmistheEarthgravitationalparameter.Additionally,the unitofmassischosentobetheinitialmassofthesmallspace vehicle.Solutionsare obtainedbyusingGPOPSwhichisahpadaptivepsuedospectra lmethodoptimal controlsoftwareusesSNOPTastheNLPsolverandINTLABasth eautomatic differentiator.InthisstudyMATLAB-R2010bonMacBookPro 2.53GHZCore2Duo runningMacOSX10.6.7isusedtorunGPOPSversion4.0. First,theminimumfuelnoncoplanaraeroassistedorbitalt ransferproblemis solvedforvariousinclinationchangeswithoutanyheating rateconstraintandeffectof inclinationchangeonoptimaltrajectoriesisinvestigate d.Second,variousheatingrate constraintsareaddedtotheoptimalcontrolproblemtounde rstandtheeffectofheating rateconstraintsonoptimaltrajectories.Finally,optima lsolutionsforvariousnumberof atmosphericpassesarecompared.3.4.1EffectofInclinationChange Optimaltrajectoriesofone-passaeroassistedorbitaltra nsferswithvarious inclinationchangesarefoundtodeterminetheeffectofinc linationchange.Duringeach ofthesetransfersheatingrateisnotconstrained.Therst gurethatshowstheeffect ofinclinationchangeonoptimaltrajectoriesisFig. 3-2 whichgivesaltitudevs.timefor variousinclinationchanges.Thisgureindicatesthatthe spacevehiclegoesdeeper intotheatmosphereforlargerinclinationchanges.Fig. 3-2B showsaltitude,vs.speed 30

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duringtheatmosphericightphaseforvariousinclination changes.Thespacevehicle enterstheatmosphereslowerforlargerinclinationchange s.Asloweratmospheric entranceleadstoasmallerliftforceandwithasmallerlift forceandthespacevehicle cangodeeperintotheatmosphere.Next,Fig. 3-2C givestherelationshipbetween altitudeandightpathangle.Whileightpathanglesatthe atmosphericentrance isalmostthesameforallinclinationinclinationchanges, theightpathanglesatthe atmosphericexitdiffersproportionaltotheinclinationc hange.Thelargestdifference betweenexitightpathanglesis,however,around4degwhic hisasmalldegreeand ignorable.Todeterminetheeffectofinclinationchangeon optimalcontrolduringthe atmosphericightphase,Figs. 3-3A ,and 3-3B areplotted.Fig. 3-3A providesangleof attack,vs.timeforvariousinclinationchanges.Forallin clinationchanges,rstlythe vehiclestartsdecreasingitsangleofattackattheatmosph ericentrance.Secondly,the vehicleincreasesitsangleofattackafteralmost500secon d,ifthevehicleisperforming aninclinationchangemorethan15deg.Asaforementioned,t hevehiclegoesdeeper intotheatmospheretoachievelargerinclinationchanges, andthevehicleneedsbigger liftforcetoreachtheedgeofsensibleatmosphere.Therefo re,forlargerinclination changes,theangleofattackincreasesafterthevehicleexp eriencesitslowestaltitude. Secondcontrolcomponentduringtheatmosphericightphas eisthebankangle.Bank angleaffectsdirectionoftheliftforce.Toestablishthee ffectofinclinationchangeon bankangleFig. 3-3B isgiven.Itisobservedthat,thespacecraftstartsincreas ingits bankangleattheatmosphericentrancewhichincreasesposi tieliftforce.Positivelift isachievedatnegativedegreesbecauseforallinclination changesthespacevehicle entersandleavestheatmosphereatsouthhemisphere.Fig. 3-3C showsthatthespace vehicleexperiencesthemaximumheatingratewhenitexperi encesitslowestaltitude. Theheatingrateisfoundtobeinverselyproportionaltothe altitudebecauseatlower altitudesthespacevehicleexperiencesdenseratmosphere 31

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0 500 1000 1500 2000 20 40 60 80 100 120 140 160 180 200 Time(s)Altitude(km) i f =15deg i f =25deg i f =35deg i f =45deg AAltitudevs.Time 6 6.5 7 7.5 8 8.5 9 30 40 50 60 70 80 90 100 110 Speed(km/s)Altitude(km) i f =15deg i f =25deg i f =35deg i f =45deg BAltitudevs.Speed -4 -2 0 2 4 6 8 30 40 50 60 70 80 90 100 110 FlightPathAngle(deg)Altitude(km) i f =15deg i f =25deg i f =35deg i f =45deg CAltitudevs.FlightPathAngle Figure3-2.Altitude,vs.Time,SpeedandFlightPathAnglef orVariousFinalInclinations ( i f ) 3.4.2EffectofHeatingRateConstraint Optimaltrajectoriesofone-passaeroassistedorbitaltra nsferswith30deg inclinationchangearefoundsubjecttovariousheatingrat econstraintstodetermine theeffectofheatingrateconstraintonoptimaltrajectori esandnalmassratios.The heatingrateconstraintaffectsthelowestaltitudethatth espacevehicleexperiences 32

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200 400 600 800 1000 1200 10 15 20 25 30 Time(s)AngleofAttack(deg) i f =15deg i f =25deg i f =35deg i f =45deg AAngleofAttackvs.Time 200 400 600 800 1000 1200 -160 -140 -120 -100 -80 -60 -40 Time(s)BankAngle(deg) i f =15deg i f =25deg i f =35deg i f =45deg BBankAnglevs.Speed 200 400 600 800 1000 1200 0 500 1000 1500 2000 Time(s)HeatingRate(W/cm 2 ) i f =15deg i f =25deg i f =35deg i f =45deg CHeatingRatevs.Time Figure3-3.Time,vs.AngleofAttack,BankAngle,andHeatin gRateforVariousFinal Inclinations( i f ) (Fig. 3-4 ).Thelowestaltitudeincreasesfortighterheatingrateco nstraints.Moreover, forthecases,wheretheheatingrateconstraintislessthan orequalto400W/cm 2 thespacevehicleascendsforalmost400secatthebeginning ofoptimaltrajectories. Thisincreasesthespeedofthespacevehicleattheatmosphe ricentrancewhichcan 33

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beobservedinFig. 3-4B .Increasingentrancespeedleadsincreasestheliftforcet hat preventsthespacevehiclegoingdeepintotheatmosphere.N ext,Fig. 3-4C shows altitude,vs.ightpathangle.Inthisgureequilibriumgl idesegmentsisobservedfor thecaseswheretheheatingrateisconstrainedDuringthese segmentsthealtitude changes,buttheightpathanglealmostremainsthesame.Fi gs. 3-5A ,and 3-5B providetheeffectofheatingrateconstraintontheoptimal controlduringatmospheric ightphase.AsitisobservedfromFig. 3-5A ,forthecaseswheretheheatingrateisnot constrainedanditisconstrainedwith600W/cm 2 thevehiclestartstheatmospheric ightphasewithanangleofattackaround20deg.Ontheother hand,formore tightenedheatingrateconstraintstheangleofattackisfo undtobeatitsmaximum allowablevaluewhichisaround40deg.Moreover,theangleo fattackremainsatthis angleforaperiodoftimewhichisinverselyproportionalto themaximumallowable heatingrate.Thedifferencebetweenangleofattacksatthe atmosphericentrance isrelatedtotheexperiencedlowestaltitude.Forthecases wheretheheatingrateis notconstrainedoritisconstrainedwith600W/cm 2 ,thespacevehiclegoesdeepinto theatmosphereandexperiencesdenseratmospherewhichlea dstolargerliftforce. Thereforethevehicledoesnotneedtoentertheatmospherew ithalargeangleof attacktoexperiencealargeliftforce.Fig. 3-5B showsbankangle,vs.time.Forthe caseswheretheheatingrateisconstrained,bankangleincr easesalmostlinearlyfor aperiodoftime.Thisperiodcoincidewiththeperiodofequi libriumglide.Afterthe equilibriumglidesegment,bankangleincreasesdrastical lyandthereforepositivelift forceincreasesdrasticallythatleadstoascendingofvehi cle.Figure 3-5C givesheating ratesofthespacecraft.Iftheheatingrateisconstrainedt hespacevehiclereaches itsmaximumallowableheatingrateandremainsatthisheati ngrateforaperiodof timewhichbecomeslongerforlowerheatingrateconstraint s.Thisincreasesthetotal atmosphericighttime,thereforeachievingexpectedincl inationchangecanbeachieved withoutgoingdeepintotheatmosphere. 34

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0 500 1000 1500 2000 2500 3000 20 40 60 80 100 120 140 160 180 200 Time(s)Altitude(km) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 AAltitudevs.Time 6.5 7 7.5 8 8.5 9 9.5 30 40 50 60 70 80 90 100 110 Speed(km/s)Altitude(km) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 BAltitudevs.Speed -4 -2 0 2 4 6 30 40 50 60 70 80 90 100 110 FlightPathAngle(deg)Altitude(km) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 CAltitudevs.FlightPathAngle Figure3-4.Altitude,vs.Time,Speed,andFlightPathAngle forVariousMax.Allowable HeatingRates Fig. 3-6 givesnalmassratiosforvariousinclinationchanges,and heating rateconstraints.Thisgurealsogivesnalmassratiosfor nitethrustnoncoplanar orbitaltransferwithoutanyatmosphericpasses.Firstly, itisobservedthatnitethrust noncoplanaraeroassistedorbitaltransfersaremorefuele fcientthannitethrust noncoplanarorbitaltransferswithoutanyatmosphericpas sesregardlessofinclination 35

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0 500 1000 1500 2000 5 10 15 20 25 30 35 40 45 Time(s)AngleofAttack(deg) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 AAngleofAttackvs.Time 0 500 1000 1500 2000 -150 -100 -50 0 Time(s)BankAngle(deg) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 BBankAnglevs.Speed 0 500 1000 1500 2000 0 500 1000 1500 2000 Time(s)HeatingRate(W/cm 2 ) Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 CHeatingRatevs.Time Figure3-5.Time,vs.AngleofAttack,BankAngle,andHeatin gRateforVariousMax. AllowableHeatingRates changevalues.Secondly,fuelconsumptionofthesmallspac evehicleisfoundtobe proportionaltotheinclinationchange,andinverselyprop ortionaltotheheatingrate constraint. 36

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20 25 30 35 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 InclinationChange(deg)FinalMassRatio m 0 =m f WithoutAeroassist Q = 1 Q =600W/cm 2 Q =400W/cm 2 Q =200W/cm 2 Figure3-6.FinalMassRatio,vs.FinalInclinationsforVar iousMax.AllowableHeating Rates 3.4.3EffectofNumberofAtmosphericPasses Figs. 3-7A 3-7D comparenalmassratiosofanitethrustsmallspacevehicl efor variousinclinationchangeswhichperformsmulti-passnon coplanaraeroassistedorbital transferssubjecttoheatingrateconstraints.Fig. 3-7A showsthatiftheheatingrate isnotconstrainednumberofatmosphericpassesdoesnotaff ectthenalmassratio. Otherwise,iftheheatingrateisconstrained,increasingn umberofatmosphericpasses decreasesfuelconsumption.(Figs. 3-7B 3-7D ). 37

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1 2 3 0.4 0.45 0.5 0.55 0.6 0.65 NumberofAtmosphericPassesFinalMassRatiom 0 /m f i f =20deg i f =30deg i f =40deg AMax.AllowableHeatingRate=Inf 1 2 3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 NumberofAtmosphericPassesFinalMassRatio m 0 =m f i f =20deg i f =30deg i f =40deg BMax.AllowableHeatingRate=600W/cm 2 1 2 3 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 NumberofAtmosphericPassesFinalMassRatio m 0 =m f i f =20deg i f =30deg i f =40deg CMax.AllowableHeatingRate=400W/cm 2 1 1.5 2 2.5 3 0.2 0.3 0.4 0.5 0.6 0.7 NumberofAtmosphericPassesFinalMassRatio m 0 =m f i f =20deg i f =30deg i f =40deg DMax.AllowableHeatingRate=200W/cm 2 Figure3-7.FinalMassRatio, m ( t f ) = m 0 ,vs.NumberofAtmosphericPasses,Max. AllowableHeatingRates,andFinalInclinations 38

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CHAPTER4 DISCUSSION Thegoalofthisstudywastodetermineminimumfuelsolution stonitethrustLEO toLEOnoncoplanaraeroassistedorbitaltransfersofasmal lspacecraftforvarious inclinationchangessubjecttovariousheatingrateconstr aintsandunderstandtheeffect ofinclinationchange,heatingrateconstraint,andnumber ofatmosphericpassesonfuel consumption.Resultsofthisstudysuggestthatwhilefuelc onsumptionisproportional toinclinationchanges,itisinverselyproportionaltothe heatingrateconstraints. Furthermore,numberofatmosphericpassesdoesnotaffectf uelconsumption forthecaseswhereheatingrateisnotconstrained.Otherwi se,iftheheatingrate isconstrained,increasingthenumberofatmosphericpasse sdecreasesthefuel consumption. Inreference[ 16 ],minimumfuelimpulsivethrustnoncoplanarGeostationar y EarthOrbit(GEO)toLEOorbitaltransfersofmassivespacec raftswerestudiedand itwasobservedthatincreasingthenumberofatmosphericpa ssesdecreasesfuel consumptionofmassivespacecraftregardlessofheatingra teconstraint.Thisresult coincideswithmyobservationsonlyforthecaseswhereheat ingrateisconstrained. Ontheotherhand,myobservationsforthecasesheatingrate isnotconstrained coincidewiththeresultsinreference[ 6 ].Inthisreference,theauthorsinvestigated minimumfuelsolutionstoimpulsivethrustnoncoplanaraer oassistedorbitaltransfers ofsmallspacecraft,andoptimalsolutionsarefoundtobein sensitivetothenumberof atmosphericpassesregardlessoftheheatingrateconstrai nt. Finally,forfuturework,numberofatmosphericpassesmayb eincreasedand maximumnumberofatmosphericpassesthatstopsdecreasing thefuelconsumption maybefoundforvariousinclinationchanges,andheatingra teconstraints. 39

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REFERENCES [1] Abramowitz,M.andStegun,I.A. Handbookofmathematicalfunctionswith formulas,graphs,andmathematicaltables .Doverpublications,1964. [2] Betts,J.T.andHuffman,W.P.“Sparseoptimalcontrolsoftw areSOCS.” MathematicsandEngineeringAnalysisTechnicalDocumentMEA-L R-085,Boeing InformationandSupportServices,TheBoeingCompany,POBo x 3707(1997): 98124–2207. [3] Bryson,A.E.andHo,Y.C. Appliedoptimalcontrol .AmericanInstituteof AeronauticsandAstronautics,1979. [4] Darby,ChristopherLiu. Hp-pseudospectralmethodforsolvingcontinuous-time nonlinearoptimalcontrolproblems .Ph.D.thesis,UniversityofFlorida,Gainesville, Florida,2011. [5] Darby,C.L.,Hager,W.W.,andRao,A.V.“Anhp-adaptivepseu dospectralmethod forsolvingoptimalcontrolproblems.” OptimalControlApplicationsandMethods (2010). [6] Darby,C.L.andRao,A.V.“Minimum-FuelLow-EarthOrbitAer oassistedOrbital TransferofSmallSpacecraft.” JournalofSpacecraftandRockets 48(2011).4: 618–628. [7] Detra,R.W.,Kemp,N.H.,andRiddell,F.R.“AddendumtoHeat TransferofSatellite VehiclesRe-enteringtheAtmosphere.” JetPropulsion 27(1957):1256–1257. [8] Gautschi,Walter. NumericalAnalysis:AnIntroduction .Birkhauser,1997. [9] Hull,DG,Giltner,JM,Speyer,JL,andMapar,J.“Minimumene rgy-lossguidancefor aeroassistedorbitalplanechange.” JournalofGuidance,Control,andDynamics 8 (1985).4:487–493. [10] Hussaini,M.Y.andZang,T.A.“Spectralmethodsinuiddyna mics.” Annualreview ofuidmechanics 19(1987).1:339–367. [11] London,H.S.“Changeofsatelliteorbitplanebyaerodynami cmaneuvering.” JournaloftheAerospaceSciences 29(1962).3:323–332. [12] Mease,KD.“Optimizationofaeroassistedorbitaltransfer -Currentstatus.” Journalof theAstronauticalSciences 36(1988):7–33. [13] Mease,KDandVinh,N.X.“Minimum-fuelaeroassistedcoplan arorbittransferusing lift-modulation.” JournalofGuidanceControlDynamics 8(1985):134–141. [14] Michels,HH.“AbscissasandweightcoefcientsforLobatto quadrature.” MathematicsofComputation 17(1963).83:237–244. 40

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[15] Rao,A.V.“Asurveyofnumericalmethodsforoptimalcontrol .” Advancesinthe AstronauticalSciences 135(2009).1:497–528. [16] Rao,A.V.,Tang,S.,andHallman,W.P.“Numericaloptimizat ionstudyof multiple-passaeroassistedorbitaltransfer.” OptimalControlApplicationsand Methods 23(2002).4:215–238. [17] Vinh,N.X.,Busemann,A.,andCulp,R.D. Hypersonicandplanetaryentryight mechanics .UniversityofMichiganPress,1980. [18] Vinh,N.X.andMa,D.M.“Optimalmultiple-passaeroassiste dplanechange.” Acta Astronautica 21(1990).11-12:749–758. [19] Walberg,G.D.“Asurveyofaeroassistedorbittransfer.” JournalofSpacecraftand Rockets(ISSN0022-4650) 22(1985):3–18. [20] Zimmermann,F.andCalise,A.J.“Numericaloptimizationst udyofaeroassisted orbitaltransfer.” JGUIDCONTROLDYN 21(1998).1:127–133. 41

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BIOGRAPHICALSKETCH BegumSenseswasborninAnkara,Turkey.Shegraduatedfromt heIstanbul TechnicalUniversity,PhysicsEngineering.Shewasawarde da”FellowshipforMaster ofSciencedegreeinAerospaceEngineering”byTheScienti candTechnological ResearchCouncilofTurkeyin2009.SheisamemberofVehicle Dynamicand OptimizationLaboratoryandworksunderthesupervisionof Dr.AnilV.Rao.Her recentresearchinterestisminimum-fuelnitethrustlowEarthorbitaeroassisted orbitaltransferofsmallspacecraft.Shealsoworkedonhyb ridoptimizationmethodsfor interplanetarytrajectoriesanddisplaytechnologiesina erialvehicles.Shereceivedher MasterofSciencedegreeattheUniversityofFlorida,Depar tmentofMechanicaland AerospaceEngineeringinthefallof2011. 42