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Spatio-Temporal Atmospheric Density Forecasting for Drag-Based Propellant-less Spacecraft Maneuvering

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Title:
Spatio-Temporal Atmospheric Density Forecasting for Drag-Based Propellant-less Spacecraft Maneuvering Theory and Mission Design
Creator:
Guglielmo, David C
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (94 p.)

Thesis/Dissertation Information

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

Subjects

Subjects / Keywords:
Aircraft maneuvers ( jstor )
Analytical forecasting ( jstor )
Atmospheric density ( jstor )
Atmospheric models ( jstor )
Atmospherics ( jstor )
Forecasting techniques ( jstor )
Sails ( jstor )
Spacecraft ( jstor )
Spatial resolution ( jstor )
Trajectories ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
drag -- maneuver -- spacecraft
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:
Spacecraft maneuver in Low Earth Orbit for a variety of reasons. They may perform in-orbit maintenance, assume a formation, or perform a rendezvous. Using atmospheric drag as a control force, it is possible to create rendezvous maneuvers between two spacecraft without the use of propellant, given accurate forecasting of atmospheric density. By forecasting atmospheric density, it is possible to predict the effects of drag on the trajectory of a spacecraft. This work presents a general method to predict the density over the upcoming orbits of a satellite. Atmospheric density is estimated through the use of a density calibrator, and then forecasted. Using this forecast aids in designing more accurate guidance trajectories, which can be tracked by a drag-actuated controller. Forecasting multiple trajectories allows the creation of a spatial density map, from which the forecasted density can be retrieved. The effect of improved density forecasting on maneuvering is shown through the creation of a drag-based rendezvous maneuver. A guidance relative trajectory for two spacecraft is created and tracked using a Lyapunov-based controller. Using three forecasted trajectories and interpolation, the spatial density map was created. Guidances created using this spatial density map were shown to be more realistic and more easily tracked than guidances created using a single forecasted trajectory. Geomagnetic activity is also shown to have a major effect on the efficacy of the methods. Increased geomagnetic activity results in increased atmospheric density, which provides a larger control input and larger effect from increasing the spatial resolution. Finally, the design of a 3U CubeSat small spacecraft is presented. Using the algorithms presented in this work, this spacecraft will perform a rendezvous maneuver. The spacecraft is equipped with the first repeatedly-retractable drag sail onboard a CubeSat, which will be used as the actuator for the drag-based maneuvers. ( 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: BEVILACQUA,RICCARDO.
Local:
Co-adviser: FITZ-COY,NORMAN G.
Statement of Responsibility:
by David C Guglielmo.

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Source Institution:
UFRGP
Rights Management:
Copyright Guglielmo, David C. 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.
Classification:
LD1780 2015 ( lcc )

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SPATIO-TEMPORALATMOSPHERICDENSITYFORECASTINGFORDRAG-BASEDPROPELLANT-LESSSPACECRAFTMANEUVERING:THEORYANDMISSIONDESIGNByDAVIDGUGLIELMOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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2015DavidGuglielmo

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ACKNOWLEDGMENTSIwouldliketothankDr.RiccardoBevilacquaforallhissupportandhelpinorganizingmyresearchandprovidingmewithnewopportunities.Hehassentmeacrossthecountryandoverseas,broughtmetotheUniversityofFlorida,andsecuredfundingformyresearch.Hisexpertiseandexperiencehavebeeninvaluabletome.Dr.DavidPerezhasactedasasecondadvisortome,helpingwithmyresearch,ndingpreviouswork,andhelpingmewithdifcultconcepts.HehasalsoactedasaguidefromthebeginningofmytimeworkingintheADAMUSLab.Additionally,IwouldliketothankeveryoneelseIhaveworkedwithintheADAMUSLab.Icouldn'taskforabetterresearchenvironment.ThisresearchwassupportedbytheUnitedStatesOfceofNavalResearch(ONR)undertheYoungInvestigatorProgram(Awardno.N00014-15-1-2087),andwassupportedbytheUniversityofFloridaResearchandEngineeringEducationFacility(UFREEF)FellowshipinSummer2015.TheUniversityofFloridahasalsogenerouslygiventheADAMUSLabspaceandstart-upfundingtocontinueourworkhere. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 ProblemBackground ................................. 12 ExtendingtheStateoftheArt ............................ 12 AtmosphericDensityModeling ........................ 13 AddingSpatialResolutiontoAtmosphericDensityForecasting ...... 14 AtmosphericModelsUsedinThisWork ...................... 15 SolarandGeomagneticIndices ........................... 16 ImplementationOnboardaSpacecraft ....................... 17 TablesandFigures .................................. 17 2THEORETICALBACKGROUND .......................... 23 AtmosphericDensityVariations ........................... 23 ReferenceFrames .................................. 24 EquationsofMotion ................................. 24 FastFourierTransforms ............................... 26 TablesandFigures .................................. 27 3FORECASTINGATMOSPHERICDENSITY .................... 29 OverviewofForecastingProcedure ........................ 29 ForecastingAlgorithm ................................ 29 DensityCalibrator ............................... 30 DensityForecasting .............................. 32 TablesandFigures .................................. 35 4ADDINGSPATIALRESOLUTION ......................... 38 UsingSpatialResolutiontoCreateaRendezvousManeuver .......... 38 CreatingandTrackingtheGuidance ........................ 40 EndConditionsforGuidanceandTracker ..................... 40 TablesandFigures .................................. 42 4

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5RESULTS ....................................... 43 Calibrator ....................................... 43 DensityForecasting ................................. 43 ValidatingMethodofGuidanceandTracking ................... 44 EffectofGeomagneticActivity ........................... 45 EffectofAtmosphericWinds ............................ 46 TablesandFigures .................................. 48 6IMPLEMENTATIONONBOARDASPACECRAFT ................. 71 MissionOverview .................................. 71 PowerSystem .................................... 73 Location,Pointing,andRadio ............................ 74 MissionPhases .................................... 75 TablesandFigures .................................. 76 7CONCLUSION .................................... 82 8SUGGESTIONSFORFUTUREWORK ...................... 84 IndexForecasting .................................. 84 LimitingtheMotionoftheDragSailSubsystem .................. 84 REFERENCES ....................................... 86 BIOGRAPHICALSKETCH ................................ 93 5

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LISTOFTABLES Table page 2-1CommonSolarandGeomagneticIndices ..................... 27 5-1InitialConditionsforRendezvous .......................... 48 5-2MetricsforCalibratorFits .............................. 49 5-3SpacecraftParameters ............................... 49 5-4WeightsandScalingUsedinCalibrator ...................... 49 5-5PerformanceMetricsforHighKpCases ...................... 68 5-6PerformanceMetricsforLowKpCases ...................... 68 5-7CasesofVaryingWinds,Summarized ....................... 69 6-1PowerSystemEstimates .............................. 79 6-2PADDLESSuccessLevels ............................. 80 6

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LISTOFFIGURES Figure page 1-1ExampleofDifferentialDragManeuvering ..................... 18 1-2ComparisonofJB2008toHASDMDensity .................... 19 1-3JB2008InputsandOutputs ............................. 19 1-4ComparisonofNRLMSISE-00toJacchiaData .................. 20 1-5NRLMSISE-00InputsandOutputs ......................... 20 1-6ComparisonofDTM-2013DensitytoMeasuredAtmosphericDensityfromVariousSpacecraftMissions ............................ 21 1-7DTM-2013InputsandOutputs ........................... 22 2-1ECI(XYZ)andLVLH(xyz)Frames ......................... 28 2-2FFTSignalDecomposition ............................. 28 3-1DensityPriortoManeuverEpoch .......................... 35 3-2FrequenciesExtractedfromTimeWindows .................... 36 3-3ForecastingofFrequencyComponents ...................... 36 3-4ForecastingofMagnitudeComponents ...................... 37 3-5DensityDriftOver8Windows ............................ 37 4-1AddingSpatialResolution .............................. 42 4-2ForecastingMultipleTrajectoriesAddsSpatialResolution ............ 42 4-3TrackingtheChaserintheTargetLVLHFrame .................. 42 5-1HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MinimumDrag . 50 5-2HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MediumDrag .. 50 5-3HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MaximumDrag . 51 5-4LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MinimumDrag .. 51 5-5LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MediumDrag .. 52 5-6LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MaximumDrag . 52 5-7HighKpCasePrimaryFrequenciesforTarget,MinimumDrag .......... 53 5-8HighKpCasePrimaryFrequenciesforTarget,MediumDrag .......... 53 7

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5-9HighKpCasePrimaryFrequenciesforTarget,MaximumDrag ......... 54 5-10LowKpCasePrimaryFrequenciesforTarget,MinimumDrag .......... 54 5-11LowKpCasePrimaryFrequenciesforTarget,MediumDrag ........... 55 5-12LowKpCasePrimaryFrequenciesforTarget,MaximumDrag .......... 55 5-13HighKpCasePrimaryMagnitudesforTarget,MinimumDrag .......... 56 5-14HighKpCasePrimaryMagnitudesforTarget,MediumDrag ........... 56 5-15HighKpCasePrimaryMagnitudesforTarget,MaximumDrag .......... 57 5-16LowKpCasePrimaryMagnitudesforTarget,MinimumDrag .......... 57 5-17LowKpCasePrimaryMagnitudesforTarget,MediumDrag ........... 58 5-18LowKpCasePrimaryMagnitudesforTarget,MaximumDrag .......... 58 5-19HighKpCaseDensityDrift ............................. 59 5-20LowKpCaseDensityDrift .............................. 59 5-21HighKpCaseForecastingforTarget ........................ 60 5-22HighKpCaseForecastingforChaser ....................... 61 5-23LowKpCaseForecastingforTarget ........................ 62 5-24LowKpCaseForecastingforChaser ........................ 63 5-25GuidancesCreatedUsingSimulatedPerfectForecasts ............. 64 5-26LyapunovFunctionUsingSimulatedPerfectForecasts .............. 64 5-27HighKpCaseGuidanceComparison ........................ 65 5-28FinalPhaseofHighKpRendezvousDetail .................... 65 5-29LowKpCaseGuidanceComparison ........................ 66 5-30FinalPhaseofLowKpRendezvousDetail ..................... 66 5-31HighKpCaseLyapunovFunction .......................... 67 5-32LowKpCaseLyapunovFunction .......................... 67 5-33HighKpCaseControlChanges ........................... 68 5-34LowKpCaseControlChanges ........................... 69 5-35LVLH^zDisplacementBetweenSpacecraft,HighKp(Figurecourtesyofauthor) 70 8

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5-36LVLH^zDisplacementBetweenSpacecraft,LowKp(Figurecourtesyofauthor) 70 6-1PADDLESwithAllHardwareFullyDeployed .................... 77 6-2PADDLESCubeSatBusComponents ....................... 77 6-3PumpkinGPSRM-1SharedRadioandGPSBoard ................ 78 6-4BlueCanyonTechnologies(BCT)XACTAttitudeDeterminationandControlSystem(ADACS) ................................... 78 6-5PADDLESPowerSystem .............................. 79 6-6PADDLESControlSystem .............................. 80 6-7DragSailandLocationonRearofPADDLES ................... 80 6-8DragSailExplodedView .............................. 81 6-9PADDLESHardwareStates ............................. 81 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySPATIO-TEMPORALATMOSPHERICDENSITYFORECASTINGFORDRAG-BASEDPROPELLANT-LESSSPACECRAFTMANEUVERING:THEORYANDMISSIONDESIGNByDavidGuglielmoDecember2015Chair:RiccardoBevilacquaMajor:AerospaceEngineeringSpacecraftmaneuverinLowEarthOrbitforavarietyofreasons.Theymayperformin-orbitmaintenance,assumeaformation,orperformarendezvous.Usingatmosphericdragasacontrolforce,itispossibletocreaterendezvousmaneuversbetweentwospacecraftwithouttheuseofpropellant,givenaccurateforecastingofatmosphericdensity.Byforecastingatmosphericdensity,itispossibletopredicttheeffectsofdragonthetrajectoryofaspacecraft.Thisworkpresentsageneralmethodtopredictthedensityovertheupcomingorbitsofasatellite.Atmosphericdensityisestimatedthroughtheuseofadensitycalibrator,andthenforecasted.Usingthisforecastaidsindesigningmoreaccurateguidancetrajectories,whichcanbetrackedbyadrag-actuatedcontroller.Forecastingmultipletrajectoriesallowsthecreationofaspatialdensitymap,fromwhichtheforecasteddensitycanberetrieved.Theeffectofimproveddensityforecastingonmaneuveringisshownthroughthecreationofadrag-basedrendezvousmaneuver.AguidancerelativetrajectoryfortwospacecraftiscreatedandtrackedusingaLyapunov-basedcontroller.Usingthreeforecastedtrajectoriesandinterpolation,thespatialdensitymapwascreated.Guidancescreatedusingthisspatialdensitymapwereshowntobemorerealisticandmoreeasilytrackedthanguidancescreatedusingasingleforecastedtrajectory. 10

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Geomagneticactivityisalsoshowntohaveamajoreffectontheefcacyofthemethods.Increasedgeomagneticactivityresultsinincreasedatmosphericdensity,whichprovidesalargercontrolinputandlargereffectfromincreasingthespatialresolution.Finally,thedesignofa3UCubeSatsmallspacecraftispresented.Usingthealgorithmspresentedinthiswork,thisspacecraftwillperformarendezvousmaneuver.Thespacecraftisequippedwiththerstrepeatedly-retractabledragsailonboardaCubeSat,whichwillbeusedastheactuatorforthedrag-basedmaneuvers. 11

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CHAPTER1INTRODUCTION ProblemBackgroundManyspacecraftareplacedinLowEarthOrbit(LEO),approximately160-1000kmaltitude.SpacecraftcanmaneuverinLEOforstation-keeping,in-orbitmaintenance,repair,docking,andformationight.Formationightcanbedenedasmultiplespacecraftyinginadenedpattern.Formationightcanbeusedforobservationmissions(suchasPRISMA[ 1 ]),densitymeasurements(suchasGRACE,CHAMP,orGOCE),orothermissions.Traditionally,relativemaneuveringisperformedusingonboardthrusters[ 2 ]orelectricpropulsion[ 1 ].Alternatively,relativemaneuveringcanbeperformedbyvaryingtheaerodynamicdragonspacecraft,knownasdifferentialdragtechniques.Differentialdraghasthedualbenetsoffuel-freemaneuveringandlackofthrusterplumeimpingementonnearbyspacecraft.Figure 1-1 showsanexampleofrelativemaneuveringwithdifferentialdrag.Spacecraft1(SC1)maintainsitsorbit.Spacecraft1(SC2)increasesitsdragareatoreduceorbitalenergy,placingitinahigherangularvelocityorbit.SC2thenapproachesSC1frombelow,atwhichpointSC1canincreaseitsdragareatomaintainitsorbitrelativetoSC2. ExtendingtheStateoftheArtSeveralprimarymethodsareusedtomaneuverspacecraftinLEO.Thrustershavebeenusedinpreviousworktomaneuverusingpropellantinaxedorarbitrarydirection[ 3 ].ElectricpropulsionwasrecentlyusedontwoBoeing702SPsatelliteslaunchedwithaFalcon9rocket[ 4 ].Theseelectricpropulsionsystemsareusedaslow-thrustengines[ 5 ].Solarsailscanalsobeenusedtomaneuverspacecraftinorbit(inadditiontointerplanetarytravel[ 6 ]).Solarsailsrelyonpressurefromthesolarwindtoaddenergy 12

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totheorbit.Somepreviousmissionsinclude[ 7 – 23 ].Finally,electrodynamictethersarearelativelynewtechnology,usingtheLorentzforcetoeitherpropelthespacecraftorconvertorbitalenergyintospacecraftelectricalpower.CalPolyhasdevelopedtheCP6CubeSat,whichusesatetherforpropulsion[ 24 ].Allofthesemethodshavesuccessfullybeenusedasde-orbitdevicesaswell.TheuseofdifferentialdragtocreatearendezvousmaneuverwasoriginallyproposedbyLeonardetal.[ 25 ],usingrotatingpanelstovarythecross-sectionalarea.PerezandBevilacquaalsousedrotatingpanelswithaLyapunovcontrolstrategy[ 26 ],inadditiontoVirgiliandRoberts'useofrotatingpanelsintheDsatQB50mission[ 27 ].PreviousworkbyPerezetal.demonstratedtheabilitytocreatearendezvoussuccessfullywithdifferentialdrag[ 26 ].Othershavesuccessfullyuseddifferentialdragtechniquesaswell[ 27 – 39 ].Theuseofdifferentialdragseparatesspacecraftthroughorbitaldecay.ShufordandKumarbothexaminedtheseparationpossiblewithdrag[ 31 , 33 , 34 ]andfoundthatdifferentialdragcanbeusedtoaltertherelativepositionofspacecraft.ClydeSpaceLtd.manufacturestheAerodynamicEndOfLifeDeOrbitSystem(AEOLDOS)[ 40 ],intendedtousedragtoremovesmallspacecraftfromorbit.Toomanymaneuverswilleventuallyde-orbitaspacecraft,butbyusingdragonasmallerscale,differentialdragtradesaltitudeformaneuvers,decayingtheorbitslightlywitheachmaneuver. AtmosphericDensityModelingPreviousdifferentialdragworkhasgenerallyassumedconstantatmosphericdensitythroughoutthemaneuver,whichdoesnotrepresenttheatmosphereaccurately.Someworkhasalsobeenperformedincharacterizingtheatmospheretoprovideamoreaccuratedensityestimation.BowmanandMoe[ 41 ]determinedthedragcoefcientofsatellitesusingmeasuredtemperatureanddensityinLEO,usingittodeterminetheorbitaldecayforthosesamesatellites,comparingittotheJacchia70model[ 42 ].Bruinsmaetal.detailtheprocessofdeterminingtheatmosphericdensityinorbitfrom 13

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themotionoftheCHAMPmission[ 43 ],whichwasalsoperformedbySuttonetal.[ 44 ].CHAMPdatawasagainusedtopredicttheatmosphericdensityencounteredinorbitbyStastnyetal.[ 45 ].Ratherthanmodeltheatmosphere,onlythefutureorbitsofthesatelliteofinterest(CHAMP)wereconsidered.ThisimproveduponpreviousworkbyWright[ 46 ].Dell'ElceandKerschenpresentedanimprovedmodelforatmosphericdensity,whilepresentinganoptimalcontrolmethodforadifferentialdrag-basedrendezvousmaneuver[ 37 ].AdditionalworkbyPerezinvolvedforecastingatmosphericdensityderivedfromtheatmosphericmodelNRLMSISE-00forvariousnumbersoforbitsforspacecraftinLEO[ 47 ].Finally,modelingworkbyPerezpresentedadensitycalibratortoestimateatmosphericdensityfromacombinationofdensityestimatedfromthreeexistingatmosphericmodels[ 48 ].Atmosphericdensityuncertaintylimitstheaccuracyofdensityforecastingandmodeling.Inthedifferentialdrag-basedmaneuveringworkbyPerez[ 49 ],constantdensitywasassumed,whichisunrealistic.TheforecastingworkbyPerez[ 47 ]islimitedtoasingleforecastedtrajectory,whichbecomesunrealisticifthespacecraftdoesnotfollowthisorbit.BuildingontheworkbyPerez[ 47 , 49 ],thisresearchaddsspatialresolutiontoatmosphericdensityforecasting,wherespatialresolutionisdenedasknowledgeoftheatmosphericdensityatdifferentaltitudesforfuturetimes.Under-forecastingatmosphericdensityimpliesthatspacecraftorbitswilldecaymorethanexpected,whileover-forecastingimpliestoo-conservativeestimatesoforbitaldecayandmaylimitthenumberordurationofmaneuversplanned.Additionally,ifspacecraftleavetheforecastedorbit,theforecasteddensitywillnolongerbeaccurate.Addingspatialresolutiontoatmosphericdensityforecastingcompensatesforspacecraftleavingaforecastedtrajectory.Inthiswork,spatialresolutionisusedwithanexistingdifferentialdrag-basedrelativemaneuveringalgorithm[ 49 ]tocreatearendezvousmaneuver.Itwillbeshownthataddingspatialresolutionimprovesdifferentialdrag-basedmaneuvering. 14

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AddingSpatialResolutiontoAtmosphericDensityForecastingAdensitycalibratorispresentedtoestimateatmosphericdensityfromexistingatmosphericdensitymodels.Usingthecalibrateddensityasainput,theatmosphericdensityisforecastedforfuturetimes.Thisprocessisrepeatedtodevelopaseriesofforecasteddensitytimeseries,whicharetheninterpolatedtoprovideamoreaccuratedensityestimate.Theimprovementinatmosphericdensityforecastingisthenveriedwiththecreationofarendezvousmaneuver. AtmosphericModelsUsedinThisWorkVarioustypesofatmosphericmodelshavebeendeveloped,manyofwhicharesummarizedbyVallado[ 50 ].Atmosphericmodelsmaybeempirical,relyingonpreviouslymeasureddata,orphysics-based,andcalculatethestatesofsomeoralloftheatmosphere.Physics-basedmodelscanbemoreaccuratebutatthecostofvastlyincreasedcomputationalcomplexity.Threeatmosphericmodelsareusedinthiswork,allempirical.Therst,JB2008(Jacchia-Bowman2008)isderivedfromtheJacchiadiffusionequations[ 51 ].JB2008incorporatesUSAF(UnitedStatesAirForce)densitydatafrom1997-2007,takenfromsatelliteswithorbitsbetween175kmand1000km,HASDM(HighAccuracySatelliteDragModel[ 52 ])datafrom2001-2005,CHAMP(CHAllengingMinisatellitePayload[ 53 ])datafrom2001-2005,andGRACE(GravityRecoveryandClimateExperiment[ 54 ])datafrom2001-2005.TheF10.7,F10,S10,M10,Y10,andDstindicesareused;detailsoftheseindicescanbefoundinTable 2-1 .AcomparisonofthedensityratiobetweenJB2008andHASDM,thepreviousatmosphericdensitystandard,asafunctionofthegeomagneticactivity,takenbetween2001and2007isshowninFigure 1-2 ,takenfrom[ 55 ].F10BinFigure 1-2 indicatesthe81-daycenteredaverageoftheF10.7solaruxindex.InputsandoutputsfortheJB2008modelareshowninFigure 1-3 .Thesecondmodel,NRLMSISE-00,isalsobasedontheJacchiaequations,butusesalargerdatabasethantheJB2008model.Additionaldatafromincoherentscatter 15

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radar[ 56 ]andmassspectrometerswasusedtoimprovetheaccuracy.TheF10.7indexisusedwithits81-daycenteredaverageF10.7A,inadditiontotheApindex,anindicatoroftheglobalgeomagneticactivity;botharefoundinTable 2-1 .Themodelalsoincludestheoptiontouseatime-historyoftheApindextomorecloselymodeltheprecedingconditions.NRLMSISE-00iscomparedtotheJacchia70model[ 42 ],astheHASDMmodeldidnotyetexistatthetimeofcreationofNRLMSISE-00.Figure 1-4 showsthecomparisonbetweenNRLMSISE-00,thepreviousversionMSIS-90,Jacchia70,andtheoriginalJacchiadataforaltitudesbetween600kmand900km.Thenaturallogoftheratiosbetweenthedensityisshown;anaturallogofzeroindicatesthedensityisidenticaltotheJacchia-70modeldensity.Theverticalbarsontheplotrepresentthe1rangeofJacchiadatausedforthatlevelofgeomagneticactivity,withthecontainednumberindicatingthenumberofpointsused.BothversionsofMSISshowadependenceofatmosphericdensityonthesolarradioux.Figure 1-4 isbasedonaguretakenfrom[ 56 ].TheNRLMSISE-00modelinputsandoutputsareshowninFigure 1-5 .Thenalmodelused,DTM-2013,isbothmoreaccurateandlessbiasedthanJB2008andNRLMSISE-00,accordingtoBruinsma[ 57 ].JB2008andNRLMSISE-00areusedtoapproximateDTM-2013,representingtheapproximationofmorein-orbitdensitydatafromlessaccurateapproximationsofatmosphericdensity.AsproposedbyDeWitetal.[ 58 ],theF30index,scaledtotheunitsofF10.7,isusedwithDTM-2013inplaceoftheF10.7index,astheF30indexismorerepresentativeoftheUVsolaruxthanF10.7.TheKmindexisintendedtobeusedwiththemodel,butitcanalsobeusedwiththeKpindex.DTM-2013isagaincomparedtoCHAMP,GRACE,andothermissionsinFigure 1-6 ,takenfrom[ 57 ].ThisgureinsteadshowsthemeandensityratioofthemeasureddensitycomparedtotheDTM-2013densityforeachyearfrom1992.TheinputsandoutputsforDTM-2013areshowninFigure 1-7 . 16

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SolarandGeomagneticIndicesProxiesforsolarandgeomagneticactivityarerepresentedinvariousindices.AfulldiscussionoftheeffectofgeomagneticactivityonatmosphericdensitycanbefoundinChapter 2 .CommonsolarandgeomagneticindicesaredescribedinTable 2-1 .AdditionalindicesareoutlinedbyTobiskaetal.in[ 59 ].GeomagneticinthistablereferstoindicesquantifyingtheEarth'smagneticeld,andSolarFluxreferstoindicesquantifyingsolarelectromagneticradiation. ImplementationOnboardaSpacecraftMissionandhardwaredesignofarealspacecraftispresentedinthiswork,whichwilluseforecastedatmosphericdensitytoplanmaneuversrelativetoamovingpoint,or“virtualspacecraft”.Theforecastingalgorithmpresentedinthisworkwouldbemodiedtoallowforestimationofrealdensitydata,ratherthanDTM-2013density.Usingtheforecastingalgorithmallowsforaccurateestimationofatmosphericdensity,andimprovesuponpreviousmethodsbyrequiringlessdataforforecastingthanpreviouswork[ 47 ].Additionally,theforecastingmethodcanbeadaptedtoruninreal-timeonboardaspacecraft. 17

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TablesandFigures Figure1-1. ExampleofDifferentialDragManeuvering(ImagecourtesyADAMUSLab) 18

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. . . HASDMDensity . JB2008 . JB2006 . J70MOD . NRLMSIS Figure1-2. ComparisonofJB2008toHASDMDensity(AdaptedFromSource:Bowmanetal.[ 55 ]) . . JB2008 . Location . Time . Indices . Location . Time . Density . Sunlocation . Temperature Figure1-3. JB2008InputsandOutputs(FigureCourtesyofAuthor) 19

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. . . MSISE-90 . NRLMSISE-00 Figure1-4. ComparisonofNRLMSISE-00toJacchiaData(AdaptedFromSource:Piconeetal.[ 56 ]) . . NRLMSISE-00 . Location . Time . Indices . Location . Time . Indices . LocalSolarTime . Density . Temperature Figure1-5. NRLMSISE-00InputsandOutputs(FigureCourtesyofAuthor) 20

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Figure1-6. ComparisonofDTM-2013DensitytoMeasuredAtmosphericDensityfromVariousSpacecraftMissions(Source:Bruinsmaetal.[ 57 ]) 21

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. . DTM-2013 . Location . Time . Density . ExosphericTemperature . LocalTemperature . MeanMolecularMass . TemperatureGradientat120km . SpeciesConcentrations Figure1-7. DTM-2013InputsandOutputs(FigureCourtesyofAuthor) 22

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CHAPTER2THEORETICALBACKGROUND AtmosphericDensityVariationsAtmosphericdragreducesorbitalenergy,whichresultsinorbitaldecayandeventualatmosphericre-entry.Althoughaltitudeisthemaindriverofdensity,withageneraltrendtowardlowerdensityathigheraltitudes,atmosphericdensityalsodependsongeomagneticactivity.Geomagneticactivityisprimarilydrivenbytheresultsofsolarux.Thesolaruxvarieswithtwomaincycles;thesolarcyclehasaperiodofapproximately11years,andtherotationofthesunhasaperiodofapproximately27days[ 60 ].Additionally,theEarth'sday-nightcycleduetoitsrotationaffectsthesolaruxreachingthehemispheresofEarth.Variationsinsolarandgeomagneticactivityarerepresentedbygeomagneticandsolarindices.Awell-knownindexistheF10.7index,whichrepresentsthesolaruxatawavelengthof10.7cm.SolaruxdisruptstheEarthsmagneticeld;theKpindexisaproxyforthisdisruption.Geomagneticactivityresultsinincreasedatmosphericdensityintheregionsofhighactivity.Additionalgeomagneticdisruptionscauseanincreaseinatmosphericdensity.AsseeninWalterscheid[ 61 ],geomagneticactivityaddsenergytoairmolecules.Theadditionalenergycauseslightermolecularmassairmoleculestorise,whicharereplacedbyheavierairmolecules,increasingtheatmosphericdensityintheregionofhighactivity.Theuseofatmosphericdensitymodelsgivesamoreaccuraterepresentationofthedensity.Inthiswork,JB2008andNRLMSISE-00representmeasurementsfromonboarddensitysensors,whileDTM-2013representsthein-orbitdensity.Asthesemodelsareallempirical,allthreerequireatime-historyofgeomagneticindicesasinputs.Therefore,thesemodelscanonlybeuseddirectlypriortothemaneuverepoch, 23

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sincetheatmosphericindicesarenotyetavailableforfuturetimes.MoredetailsabouttheatmosphericindicescanbefoundinTable 2-1 . ReferenceFramesTworeferenceframesareusedinthisworktoassistincalculationandvisualization.TheEarth-CenteredInertial(ECI)framerepresentsastationary,non-rotatingframecenteredintheEarth,representedbytheXYZaxesinFigure 2-1 .^XpointsinthedirectionofthemeanvernalequinoxatJ2000.0(January1st,2000,at12:00:00.000TerrestrialTime),^ZpointstowardtheNorthpole,and^Y=^Z^X.TheECIframeisnottrulyinertialbutisassumedtobeinertialforshort-termmaneuvers.ItisusedfororbitalpropagationsincetheorbitisconstantintheECIframeintheabsenceofperturbations.AlsoshownistheLocal-Vertical-Local-Horizontalframe(LVLH),representedbythexyzaxesinFigure 2-1 .Theseaxesarexedtothebodyofthespacecraft.The^xaxispointsradiallyawayfromtheEarth,the^zaxispointsinthedirectionoftheangularmomentumofthespacecraftorbit(normaltotheorbitalplane),and^y=^z^x.TheLVLHframeisusedwhenvisualizingthemotionofonespacecraftrelativetoanother. EquationsofMotionThreemainforcesactonspacecraftinLEO.Aftergravity,thelargestforceactingonspacecraftresultsfromatmosphericdrag.Additionally,theJ2perturbationaffectsthemotionoforbitalplanes.Two-bodygravityrepresentsthesimplestpossiblerepresentationoftheforcesinvolvedinorbit.TheaccelerationduetogravityisproportionaltothemassoftheEarth,aswellastheinversecubeofthedistancefromEarth'scenter.ag=)]TJ /F8 11.955 Tf 12.19 8.08 Td[( r3r=398600.4418km3 s2whereagrepresentstheaccelerationduetogravity,rrepresentstherelativepositionintheECIframeofaspacecraftfromthecenteroftheEarth,isthestandardgravitional 24

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parameterofEarth[ 62 ],andrrepresentsthescalarpositionofaspacecraft,orthemagnitudeofr.Theinversecuberelationbetweenthepositionandgravitationalaccelerationimpliesarapiddrop-offofgravitywithincreasingorbitalaltitude.Thetwobodymodelissimplertocalculate,butdoesnotaccuratelycaptureorbitalperturbationsduetotheoblatenessoftheEarth.Earth'sgravitationaleldcontainsharmonics,dominatedbywhatiscommonlycalledtheJ2perturbation.Theeffectofthisperturbationontheaccelerationofaspacecraftcanbefoundasfollows:aJ2=)]TJ /F6 11.955 Tf 15.72 8.09 Td[(1 2r5J2Re22666643)]TJ /F6 11.955 Tf 11.96 0 Td[(15Z2 r23)]TJ /F6 11.955 Tf 11.96 0 Td[(15Z2 r26Z+3)]TJ /F6 11.955 Tf 11.96 0 Td[(15Z2 r2377775r (2)J2=10826310)]TJ /F11 7.97 Tf 6.59 0 Td[(8 (2)Re=6378km (2)whereZrepresentsthe^ZcomponentofthepositionofaspacecraftintheECIframeandRerepresentsthemeanradiusoftheEarth[ 62 ].J2isthecoefcientofthersttermofthezonalharmonicsandisunitless[ 62 ].Equation 2 characterizesthemotionofaspacecraftintheabsenceofdrag,whichisunrealistic.Thedragforceisproportionaltoatmosphericdensity,anddrivenbythegeometryofaspacecraft.Thedragforceisproportionaltothesquareofthevelocityofaspacecraftthroughamedium.ThedragaccelerationintheECIframecanbecalculatedasfollows:aD,ECI=)]TJ /F6 11.955 Tf 15.47 8.09 Td[(1 2mACDv2v v (2)ThedragaccelerationisdenotedaD,andthesubscriptindicatesthisistruefortheECIframe.Arepresentsthecross-windareaofaspacecraft,CDisthedragcoefcient,vrepresentstheECIvelocityvector,andvrepresentsthespeed.Thedragaccelerationactsdirectlyagainstthevelocityofthespacecraft. 25

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Addingallaccelerationstogetherproducesthetotalaccelerationonthespacecraft.a=ag+aJ2+aD,ECI (2)a=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(r r3)]TJ /F6 11.955 Tf 18.37 8.09 Td[(1 2r5J2Re22666643)]TJ /F6 11.955 Tf 11.96 0 Td[(15Z2 r23)]TJ /F6 11.955 Tf 11.96 0 Td[(15Z2 r26Z+3)]TJ /F6 11.955 Tf 11.95 0 Td[(15Z2 r2377775r)]TJ /F6 11.955 Tf 18.14 8.09 Td[(1 2mACDv2v v (2) FastFourierTransformsFastFourierTransformswillbeusedinthisresearchtoconverttime-domainsignalsintothefrequencydomain.Figure 2-2 showstheadditionofthreesignalsinbothdomains.Byplottingtheamplitudevsfrequencyinthefrequencydomain,apeakindicatestheamplitudeofthatfrequencycomponent.Sincethedensityisrepresentedasatime-series,adiscrete-timefouriertransformisused,whichisintendedforadiscretesetofpointsratherthanacontinuoussignal.Eqn. 2 showsthetransform.Y(k)=NXj=1W(j)]TJ /F11 7.97 Tf 6.59 0 Td[(1)(k)]TJ /F11 7.97 Tf 6.59 0 Td[(1)n (2)Wn=exp )]TJ /F15 8.136 Tf 8.47 0 Td[(2i = N (2)whereYrepresentsthemagnitudeoftheoutputsignalforfrequencyk,irepresentstheimaginaryunit,jindicatestheindexofthepointintheinputtime-seriesoflengthN.TheMATLABimplementationisusedinthisresearch. 26

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TablesandFigures Table2-1.CommonSolarandGeomagneticIndices NameIndexTypeMeasuredQuantity&Source KpGeomagneticPlanetary3-HourRangeIndex.Quantizesthe3-hourrangeofthemostactivehorizontaleldcompo-nent.Averagedvaluesfrom11observatories.[ 63 ]KmGeomagneticPlanetary3-HourRangeIndexcalculatedbytheInstitutfurGeophysikderGottingenUniversitat,F.R.Germany.[ 64 ]apGeomagneticLogarithmicscalingofKpindex.[ 63 ]ApGeomagneticDailyaverageof8apindices.[ 63 ]AAGeomagneticLogarithmicscalingofk-indicesfromantipodalobservatories.Quantizesthegloballevelofgeomag-neticactivity.[ 63 ]DstGeomagneticDisturbanceamplitude-stormtime.TracksvariationsintheelectriccurrentsseveralEarthradiiabovetheequator.[ 63 ]MgIIGeomagneticMgIICore-to-WingIndex.RatioofthehandklinesofthesolarMgIIemissiontothebackgroundMgIIemission.[ 65 ]E10.7SolarFluxExtremeUltraviolet(EUV)index,10.7cmwave-length[ 59 ]F10.7SolarFluxSolarRadioux,10.7cmwavelength[ 59 ]F10.7aSolarFlux81-dayaverageofF10.7[ 59 ]S10.7SolarFluxSolarUVux,10.7cmwavelength[ 59 ]F30SolarFluxSolarUVux,30cmwavelength[ 57 ]BartelNumberSolarFluxPositionin27-daysolarcycle[ 63 ] 27

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Figure2-1. ECI(XYZ)andLVLH(xyz)Frames(Source:Curtis[ 62 ]) Figure2-2. FFTSignalDecomposition.ReprintedwithpermissionfromtheSparseFourierTransformProject,MIT http://groups.csail.mit.edu/netmit/sFFT/algorithm.html (October19,2015) 28

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CHAPTER3FORECASTINGATMOSPHERICDENSITYThegoalofthischapteristodescribetheprocessofforecastingatmosphericdensity.Atmosphericdragisproportionaltoatmosphericdensity.Forecastingatmosphericdragallowsamoreaccuratepredictionoftheexpectedtrajectoryofasatellite.Thisworkelaboratesonanovelapproachtoforecastingatmosphericdensity. OverviewofForecastingProcedureFigure 3-1 illustratesthedensitymeasuredbyasatelliteinagenericorbit.Theoverallbehaviorofthedensitycanbemodeledasaquasi-periodictime-serieswithadrift.Thesinusoidalbehaviorisaresultoftheorbitalmotionofthespacecraftandtheday/nightcycle,describedinChapter 2 .Thedensitydriftcomesfromorbitaldecayandgeomagneticactivity(alsodescribedinChapter 2 ).Theexhibitedbehaviorsuggeststhatforasinglesatellite,thedensityencounteredalongorbitscanbecharacterizedbyaniteFourierSeriessuperimposedonseculartermswhichcharacterizethedrift.Characterizationoftheperiodicanddriftcomponentsofthedensityisperformedseparately.FFTsareusedtoextracttheperiodiccomponentsofthedensitytimeseries.TheFFTneedstobesuppliedwithpseudo-measurements,whicharegeneratedwiththeuseofapresenteddensitycalibrator.Propagationoftheorbitofaspacecraftinreversetimeisusedtosupplythedatasetfordevelopmentofthecalibrator.Densitydriftforecastingisaccomplishedwithextrapolationofafunctiont.InpreviousworkbyPerezetal.[ 47 ]atmosphericdensitywasforecastedalongtheorbitofaspacecraftusingayearofpreviousdata,whichcaptureddailyandseasonalvariations.Thisworkalgorithmisintendedtoruninreal-time,andsousesdaysratherthanmonthsofinputdatatoforecast;arequirementofayearofdataisnotpracticalforin-orbitforecasting. 29

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ForecastingAlgorithm DensityCalibratorAsmentioned,thedatasetforthedensitycalibratorisgeneratedbypropagatingthestateofaspacecraftbackwardforNdays.AteachofTtimesteps,thedensityfromJB2008,NRLMSISE-00,andDTM-2013isrecorded,aswellasthetime,ECIpositionandECIvelocity.ARunge-Kutta(4,5)solver(implementedintheMATLABode45function)isusedtopropagatethestateofthespacecraft.Eachtime-seriesofdensitypointsissplitintothettingandtestingset,bothofwhicharehalfthelengthofthefullset.ThegoalofttingthedensitycalibratoristominimizetheerrorbetweenDTM-2013densityandthecalibrateddensity.Aweightedfunctionofeachdensitywithabiasisusedtorepresentthecalibrateddensity.JB,S=JB)]TJ ET q .478 w 190.71 -297.15 m 196.74 -297.15 l S Q BT /F8 11.955 Tf 190.71 -303.97 Td[(JB JB,MSIS,S=MSIS)]TJ ET q .478 w 319.95 -297.15 m 325.99 -297.15 l S Q BT /F8 11.955 Tf 319.95 -303.97 Td[(MSIS MSIS (3)wherekrepresentstheindexofeachpointinthetdataset.ThedensityforJB2008andNRLMSISE-00isrepresentedbyJBandMSIS,respectively,where JBand MSISrepresentthemeanofeach.ThesubscriptSindicatesthedataisscaledaccordingtothemeanandstandarddeviationofthetdataset.Scalingthedataaccordingtothemeanandstandarddeviationallowsforimprovednite-precisioncomputationalaccuracy.ThegoalofttingthedensitycalibratoristominimizetheerrorbetweenthecalibrateddensityandDTM-2013density.minimizeMSEMSE=1 TTXk=1(^DTM(k))]TJ /F8 11.955 Tf 11.95 0 Td[(DTM(k))2 (3)^DTM=w1JB,S,t+w2MSIS,S,t+b (3)Thecalibrateddensityisrepresentedby^DTM.Theweightsofthecalibrator,w1andw2,andthebiasb,areadjustedtominimizethefunction. 30

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Afterttingthecalibrator,itistestedwiththetestdatasettoensureitcangeneralizewithanewdataset.Theerrorisfoundinasimilarwayaswiththetset.MSE=1 TTXk=1(^DTM(k))]TJ /F8 11.955 Tf 11.96 0 Td[(DTM(k))2 (3)^DTM=w1JB,S,test+w2MSIS,S,test+b (3)whereknowrepresentstheindexofeachpointinthetestingdataset.Theweightsw1andw2andthebiasbareretainedfromthetdataset.ThescaleddensityJB,SandJB,Sarecalculatedbyscalingtothemeanandstandarddeviationofthetdataset.Calibratorperformanceisevaluatedwiththemean-squared-error(MSE)ofthetestdataset.Ifthisexceedsathreshold,thecalibratoristoospecictobeusedwithadatasetotherthanthetdataset.Previousiterationsofthedensitycalibratorhadlargeerrorsoverthetdataset,andcouldnotbeused.Initially,anArticialNeuralNetwork(ANN)wasusedforthecalibrator[ 66 ].AnANNusesaweightedfunctiontocreateanoutputforasetofinputs.Traininginputsandtargets(desiredtrainingoutputs)areusedtoadjusttheweightsofthefunctionandalteritsoutputtobetterapproximatethefunction.JB2008andNRLMSISE-00densitywereusedastheinputdensity,andDTM-2013densitywasthedesiredoutput.Thiswasdroppedduetothelackofrangeinthetrainingdata.Sinceallofthetrainingdatawasfromasingletrajectory,itwaseasytodevelopaANNthatwouldoutputonlythatdata,knownasmemorizingorovertraining.Partwaythroughthedevelopmentofthemodel,itwasdiscoveredthatbecauseatmosphericdensityisstronglycorrelatedwiththeatmosphericindicesKpandF10.7,aneuralnetworkcalibratorcouldnotbetrainedusingatmosphericdensitydataalone.Inanefforttocorrelatethedensitywithgeomagneticactivityratherthantime,theindicesmentionedabovewereaddedtothecalibratorANNasadditionalinputs.However,thisdidnotsuccessfullyproduceageneralcalibratorANN;thecalibratorwasstillmemorizing. 31

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ThedensitycalibratoriscapableofestimatingtheDTM-2013densityasafunctionofJB2008andNRLMSISE-00density.BecauseJB2008andNRLMSISE-00bothrelyonmeasuredatmosphericindices,neithercanbeusedforfuturetimes,limitingtheuseofthedensitycalibratortopasttimesaswell.Thecalibrateddensitymustbeforecasttoobtainadensityestimateafterthestart(epoch)ofamaneuver. DensityForecastingThecalibratorcanbeusedtoprovidedensityestimates,whicharethenusedtopropagateaspacecraftbackwardintime.Usingthedensityresultingdensitytime-seriesfrombeforetheepoch,thedensitytime-seriesaftertheepochcanbeforecasted.Thistime-seriesisde-trended(thelinearcomponentissubtracted)andsplitintoMtimewindows.Thisdoesnotaffecttheperiodiccomponentsbutmakesthemeasiertovisualize.AFFTisusedtoextracttheperiodiccomponentsfromeachtimewindow.TheFFTisrepeatedforeachtimewindowandthemainJlargestamplitudecomponentsarerecordedforeach.Figure 3-1 showstheresultingdensityfromreversetimepropagationofaspacecraft.Thedensitytime-seriesissplitintoeighttimewindowsinthiscase.Figure 3-2 showsthefrequenciesresultingfromeachofthetimewindows.Thefrequenciesdonotvarysignicantlyinthiscase,butthemagnitudesofeachfrequencyaredifferentforeachwindow.TheJlargestamplitudefrequenciesarelinearlyextrapolatedtondthefrequenciesandmagnitudesexpectedafterthemaneuverepoch.Thisforecastswhatperiodiccomponentsareexpected,andtheweightingofeachone.TheseforecastsareshowninFigures 3-3 and 3-4 .Thephaseofthesignalcannotbeforecastedaseasilyastheamplitudeandfrequency.Thefrequencycomponentsvaryfromthebeginningtotheendofeachtimewindow.TheFFTcanreturnthephaseofasignal,butthephasewillbedifferentatthebeginningandendofatimewindowduetothetimeoffsetandfrequencyshift.Foreachcomponent,thephaseoftheforecastedcomponentisadjustedsuchthatthephasebeforeandaftertheepochmostcloselymatches,minimizingdiscontinuities. 32

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Thedensitydriftisforecastedseparatelybyminimizingtheerrorbetweenafunctionandtheoveralldensitytime-series.Afterforecastingthecomponents,thesignalisreconstructedforfuturetimes.Thisreconstructedsignalrepresentsthecompletedforecastfortheatmosphericdensity.Figure 3-5 showsthettingofafunctiontothedensitypriortotheepoch,anditsextrapolationtoestimatethedriftaftertheepoch.Forecastingatmosphericdensityprovedtobedifculttoaccomplish,andmoreiterationswereneededthanforthecalibrator.Initially,forecastingwasattemptedwithanANNaswell[ 66 ],aswasperformedbyDr.Perez[ 67 ]forasingletrajectory.Twodaysofthedensitytime-serieswereusedasinputstotrainthepredictorANN,withthenexttwodaysasthetargets.Shiftingforwardbytwodays,thenexttwodayswereusedasvalidationinputsandtargets.ValidationinANNsinvolvesusingasecondsetofinputsandtargets.Whenthetestingerrorstartstoincreaseonthevalidationset,thentrainingisstopped.Finally,shiftingforwardanothertwodays,thetestingsetisusedtochecktheforecastingperformanceoftheANNonadifferentdataset.Unfortunately,muchliketheANNcalibrator,thelimitedrangeofthedataresultedineitherunderttingofthedataormemorization.Therewasnotenoughvariationinthedatatodevelopageneralforecastingmethod,andsothisideawasdropped.Aswiththedensitycalibrator,toremovethecorrelationbetweendensityandtime,andreplaceitwithacorrelationbetweendensityandgeomagneticactivity,thesameatmosphericindiceswereaddedtothepredictorANN.Atthetime,thecalibratorANNwithindiceshadnotyetbeenruledout.Thetraining,validation,andtestingdatasetswerethesameasthosedescribedabove.Also,aswiththedensitycalibrator,addingtheindiceswasnotsufcienttopreventovertraining.AnothereffectnoticedduringtrainingoftheANNswasrelatedtothearrangementofthedata.ForANNs,agivensetofinputswillalwaysproducethesameoutput(s).Eachofthepastdensitytime-seriesendsattheepochdensity,buteachofthefuturedensitytime-seriesbeginsatthesamepoint,andendsatdifferentdensitypoints.Since 33

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eachdensitypointinthefuturetime-seriesisforecastedusingthedensitytwodaysbeforeitasaninput,allofthefuturedensitytime-serieswouldbedriventothesamepointsinceallwouldhavethesameinputatthattime.ThisrulesouttheuseofasingleANNinthemethodsdescribedabove.Therefore,thenextattemptwastoreversethedirectionofthepastdensitytime-series.Inthisway,alltime-series,pastandfuture,wouldstartattheepochdensity.However,thiswasunsuccessful,sincetheinputdatastilldidnotbehaveliketheoutputdata.Forexample,inthecaseofthemaximumdragtime-series,thedensitydecreasedmorethanintheothertwocases;trainingtheANNinthemannerdescribedresultedinaforecastoflowerdensityforthiscase,whentheforecastshouldresultinahigherdensity.Thiswasdroppedasaresult.Additionally,whiletestingthepreviousmethod,amodicationwasmadetothesingle-ANNmethod.Sinceoneproblemwasthattherewasasmallrangeofdatatousefortesting,anattemptwasmadetousealargerdataset.Approximatelysixdaysofthedensitytime-serieswereusedasinputsfortrainingandvalidation,andsixdays(withfourdaysoverlapping)wereusedasthetargets.Thisdidnotprovetoworkanybetterthantheoriginalmethod,however,anditwasdropped.Sinceonereasonthepreviousmethodsdidnotworkwastheendpointproblemdescribedabove,anattemptwasalsomadetousethreeANNs,onepertime-series.TheseANNsweresharedbetweenthetargetandchaserspacecraft.However,thismethodstillledtoovertraining,wasexcessivelycomputationallyintensive(ontheorderofdaystotrain),preventingitsuseonboardaspacecraft,andwasnotrealistic,asitrequiredadifferentANNforeachcase.KernelmethodswerethenusedtoreplaceANNstoforecastthedensity.Kernelmethodsareanalternateforecastingalgorithm,whichusethekerneltrick[ 68 ].Usingthismethod,thedatasetistransformedintoahigher-dimensionalspace,whichmakesseparationofthedataeasier.Thesamedatasetsareusedforthekernelmethodsas 34

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fortheANNmethods.Thekernelmethodsstillsufferedtheproblemofmemorization,althoughtoalesserextentthantheANNs.Onelastattempttousethekernelmethodswastheadditionoftheindicestothekernelforecastingasasecondinput.Again,thiswasdonetoimprovethecorrelationbetweentheatmosphericdensityandtheatmosphericindices.However,thisdidnotimprovethegeneralizationcapabilityofthekernelmethods.Acombinationcalibratorandpredictorwasconsidered,butruledout.Sinceadifferentforecastingmethodwouldhavebeenrequiredforeachcase,thisimpliesdifferentcalibrationoneachcaseaswell.Sinceonlyonepasttime-seriesofpointswasavailable(representingtheminimum-dragcase),onlyonecalibratorcouldbetted,makingthismethodimpossible. TablesandFigures Figure3-1. DensityPriortoManeuverEpoch(Figurecourtesyofauthor) 35

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Figure3-2. FrequenciesExtractedfromTimeWindows(Figurecourtesyofauthor) Figure3-3. ForecastingofFrequencyComponents(Figurecourtesyofauthor) 36

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Figure3-4. ForecastingofMagnitudeComponents(Figurecourtesyofauthor) Figure3-5. DensityDriftOver8Windows(Figurecourtesyofauthor) 37

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CHAPTER4ADDINGSPATIALRESOLUTIONThepreviouslydescribedalgorithmforecastsasingledensitytimeseriesforasinglespacecraft.Ifthespacecraftdeviatesfromthisorbit,thedensityforecastwillbeincorrect.Forecastingadditionaltime-seriesatdifferentaltitudesaddsspatialresolutiontotheforecast.Figure 4-1 showstheadditionofspatialresolutiontoatmosphericdensityforecasting.Thespacecraftisnolongerrequiredtofollowanexpectedtrajectoryexactlytohaveanaccuratedensityestimate.Instead,thedensityisinterpolatedbasedontheproximityofthespacecrafttoeachforecastedtrajectory.Thespacecraftusedinthisresearchcanvarytheircrosswindarea,varyingtheiratmosphericdrag.Bybackpropagatingthespacecraftstatewiththepreviouslydescribedpropagator,usingthepreviouslydescribeddensitycalibratorfordensityestimation,threedensitytimeseriesarerecordedbeforetheepoch-oneforthemaximumdrag,minimumdrag,andonefortheaveragedrag.Figure 4-2 showstheforecastingprocesstocreatemultipletrajectories.ThealgorithmdescribedinChapter 3 isrepeatedforthreetrajectoriesperspacecrafttocreatethedensityeld.Thesetime-seriesareusedwiththepreviouslydescribedmethodtoforecastcorrespondingdensitytime-seriesaftertheepoch.Theserepresentthetrajectoriesforecastedfortheminimum,medium,andmaximumdragcases.Tolocatethepositionsofeachdensitypoint,thestateofthespacecraftisthenpropagatedforwardwiththepreviouslydescribedpropagatorwitheachforecastedtime-seriesandthecorrespondingcrosswindarea.Thispairseachdensitypointwithalocation,sothatthesepointscanthenbeinterpolatedateachtimesteptoprovideamoreaccurateestimationofthedensityatthattimestep. UsingSpatialResolutiontoCreateaRendezvousManeuverUsingspatialresolutionwithdensityforecastinggivesamoreaccurateestimateoftheatmosphericdensityforecast.Theimprovementsinatmosphericdensityforecasting 38

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willbevalidatedbyusingspatialresolutionwiththeexecutionofarendezvousmaneuver.Thegoaloftherendezvousistodrivetwospacecraft,denotedthetargetandchaser,toarelativepositionandvelocityofzero.Aguidancerelativetrajectoryiscreatedrst,indicatingthedesiredpathofthechaserrelativetothetarget.Theinterpolateddensityforecastisusedtoprovidedensityestimation.Thecontrolinputtothespacecraftisafunctionoftheerror.Fortheguidance,thegoalistoreachzerorelativepositionandvelocity,sotheerroristherelativepositionandvelocity,denotederguidancebelow.Thedesiredrelativepositioninthetarget-centeredLVLHframeisdenotedbyxdandyd,indicatingthedifferenceinthe^xand^yLVLHdirections,respectively.Therelativevelocitycomponentsaredenotedas_xdand_yd.erguidance=266666664xdyd_xd_yd377777775)]TJ /F12 11.955 Tf 11.95 49.14 Td[(2666666640000377777775 (4)ertracker=266666664xdyd_xd_yd377777775)]TJ /F12 11.955 Tf 11.95 49.13 Td[(266666664xguidanceyguidance_xguidance_yguidance377777775 (4)Next,theguidanceistracked.Thegoaloftrackingistofollowtheguidancetrajectoryascloselyaspossible,sincetheguidancewasintendedtoprovidethedesiredpath.Theerrorisdenedastherelativepositionandvelocityoftheguidancesubtractedfromthetrackingrelativepositionandvelocity,denotedasertrackerabove.Therelativepositionandvelocityoftheguidanceisdenotedwiththesubscriptguidance.ApreviouslydevelopedLyapunov-basedcontrollerisusedtoadjustthecrosswindareaofthetargetandchaserbasedontheLVLHerror[ 49 ].Thecontrolinputsaturates 39

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at+1,)]TJ /F6 11.955 Tf 9.29 0 Td[(1,or0.Thiscorrespondstothetargetspacecraftmaximizingdragandthechaserminimizingdrag,thereverse,andbothspacecraftminimizingdrag,respectively.Thecontrolisupdatedevery10minutesfortheguidance,and1minutefortracking.Thereducedcontrolintervalforthetrackingcomparedtotheguidanceallowsthetrackertoaccuratelytracktheguidanceandminimizedeviations.Thecontrollawisasfollows:u=)]TJ /F2 11.955 Tf 9.3 0 Td[(sign(erT P B ) (4)whereurepresentsthepreviouslymentionedcontrolinput,errepresentstheerrorvectorused,and P and B areconstantgainmatrices. CreatingandTrackingtheGuidanceTocreatetheguidance,thestateofthetargetandchaserspacecraftarerstpropagated,startingfromtheinitialconditions,withminimumdragfor10minutes.Aftereverytimestepof1minute,thedensityestimateforeachspacecraftisupdatedbyinterpolatingthedensityeldforthattimestep.Ateach10minutecontrolupdate,thecontrolinputtothespacecraftisupdatedaccordingtothecontrollawabove.Thisisrepeateduntiltherendezvousiscompletedortheforecasteddensityisexhausted.TheequationsofmotiondescribedinChapter 2 areusedtopropagateboththetargetandchaserspacecraft.Figure 4-3 showsthetrackingofthechaserspacecraftfromthetargetspacecraft.Trackingoftheguidanceisaccomplishedinasimilarmanner.Thespacecraftareimmediatelypropagated,usingthesamecontrollawasdescribedabove,butwithDTM-2013useddirectlytoprovidedensityinformation,representingpropagationinorbit.Asmentionedabove,thecontrolanddensityarebothupdatedeveryminutewhentrackingtheguidance.TheequationsofmotiondescribedinChapter 2 areagainusedtopropagateboththetargetandchaserspacecraft.Thetrackingisrununtileitherthetrackertrajectoryhasreachedzeroortheforecasttimeintervalisreached. 40

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EndConditionsforGuidanceandTrackerForasuccessfulmaneuver,boththeguidanceandtrackerwillreachzerorelativepositionandvelocity.However,duetodensityuncertaintywhencreatingtheguidance,theguidanceandtrackerwillnotnecessarilyreachtherendezvousatthesameelapsedtime.Theguidancemayreachzerowithoutthetrackerreachingzeroatall.Thisindicatesthatthedensityforecastdidnotaccuratelyrepresentthereal-worlddynamicsorconditions.Theguidancemayalsohaveassumedalargercontrolforce(fromatmosphericdrag)thanwhatwasactuallyavailable.Occasionally,theguidancewillnotreachzero,butthetrackerwill.Thiscanresultfromacombinationofpoorforecastingandpoortracking,asaresultofthepoorforecasting.Aguidancethatdoesnotreachzerowouldnotbeused,butitispossibleforthetrackertoreachzerowhenfollowingsuchatrajectory.Thisshouldnotbeconsideredasuccessfulmaneuver.Additionally,itispossibleforneithertheguidanceortrackertoreachzero.Thisisindicativethatthedensityforecastwasinaccurate,orthatthecontrolforcefromatmosphericdragwastoolowtoreachtherendezvousintime. 41

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TablesandFigures . . Spacecraftstartingposition . Spacecraftactualtrajectory . Minimumdragtrajectory . Mediumdragtrajectory . Maximumdragtrajectory Figure4-1. AddingSpatialResolution(Figurecourtesyofauthor) . . Threetrajectoriesareforecastedforeachcase . Maximumdragtrajectory . Mediumdragtrajectory . Minimumdragtrajectory . Minimumdragtrajectory . Mediumdragtrajectory . Maximumdragtrajectory Figure4-2. ForecastingMultipleTrajectoriesAddsSpatialResolution(Figurecourtesyofauthor) . . Chaserspacecraft . Targetspacecraft . Y . X . Z Figure4-3. TrackingtheChaserintheTargetLVLHFrame(Figurecourtesyofauthor) 42

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CHAPTER5RESULTSThissectionshowstheresultsobtainedfromapplyingthemethodsdescribedintheprevioussections.TheperformanceofthecalibratorisveriedwhenestimatingDTMdensityfromJB2008andNRLMSISE-00density.ForecastingofthecalibrateddensityiscomparedtoactualDTMdensity.Thisforecasteddensityisthenusedtoprovidethedensityinformationforarendezvousmaneuver.Resultsfortherendezvousmaneuverarecomparedforasingleforecasteddensitytime-seriesandforaninterpolateddensityeld.Theeffectofatmosphericindices,whichaffecttheatmosphericdensity,isshownontheefcacyofthemaneuver.Theeffectofatmosphericwindsisalsoshown.TheepochconditionsforallcasesaredetailedbelowinTable 5-1 . CalibratorThetargetspacecraftispropagatedbackwardfor8daysusingmaximumdragtoproducethecalibratordataset.ThedensityusedtopropagatewastakenfromDTM-2013.Usinga1minutetimestep,thisresultsin11520densitypointsforeachofthethreemodels.Splittingthisintwo,thisresultsin5760pointseachforthetandtestingsets.TheperformancemetricsanddatausedforthelineartcalibratordescribedinChapter 3 areshowninTable 5-2 ,forboththetandtestdata.ThehighPearsoncoefcientandlowerrorindicatethatthecalibratorcanaccuratelyestimateDTM-2013densityforboththetandtestdataset.ThePearsoncoefcientrcanbecalculatedasfollows:r=P5760k=1DTM(k)^DTM(k) P5760k=1(DTM(k))2P5760k=1(^DTM(k))2 (5)whereDTMand^DTMrepresenttheactualandcalibratedDTM-2013densityforthetortestset.Theindexkisusedtoindicatethetimestepused.Asmentionedpreviously,themeanandstandarddeviationarecalculatedbasedonthetdataset. 43

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DensityForecastingAsdescribedinChapter 4 ,thestateofthetargetandchaserspacecraftispropagatedinreversetimewithminimum,medium,andmaximumdrag.EachdensitytimeseriesisthenusedwiththeforecastingmethodsdescribedinChapter 3 .Eighttimewindowswereused,ofonedayeach.TheresultingamplitudespectraforeachcaseareshowninFigures 5-1 through 5-6 .Fromtheamplitudespectra,itcanbeseenthatthefrequencycomponentsdonotchangesignicantlyfromonetimewindowtoanother.However,theamplitudesofthecomponentsaredifferentineachtimewindow.ThisisconrmedwithFigures 5-7 through 5-18 ,showingtheevolutionofthefrequencyandmagnitudecomponentsforthetargetspacecraftforeachtimewindow.Aspreviouslydescribed,thedensitydriftisforecastedseparately.Figures 5-19 and 5-20 showthedensitytfortheHighKpandLowKpcases,respectively.Byreconstructingtheforecastedsignalcomponents,theatmosphericdensitycanbeforecasted.DTM-2013wasusedtopropagatethestateofeachspacecraftforwardfortwodays,representingaperfectcalibratorandforecast.Figures 5-21 through 5-24 showthecomparisonbetweentheforecasteddensityforeachcaseandtheDTMdensityforthatcase. ValidatingMethodofGuidanceandTrackingThecalibrationandforecastingarenotperfect.Sincetheoverallmethodreliesontheaccuracyofthecalibratorandtheforecast,priortousingthefullalgorithm,therststepistovalidatethatthemethodworks.Thegoalistoshowthattheadditionofspatialresolutiontoatmosphericdensityforecastingcanbeusedtocreateguidancesthatcanbemoreeasilytracked,whencomparedtoguidancescreatedwithasingleforecastedtrajectory.Theguidancesresultingfromtheuseofasingleforecastedtrajectoryrepresentthepreviousstateoftheart,whichisimprovedwiththeuseofinterpolateddensity.These 44

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guidancesareshowninFigure 5-25 .Theblackguidancerepresentstheguidancecreatedwhenusinginterpolateddensity,allothersresultfromusingasingleforecastedtrajectory.Trackingoftheseguidancesisimprovedwhenusinginterpolateddensity.Figure 5-26 showstheLyapunovfunctionresultingfromtrackingoftheguidances.ThemeanLyapunovfunctionisreducedwhenusingtheinterpolateddensity.TheLypaunovfunctionisdenedasfollows:VL=err P er (5)whereerrepresentsthetrackingerror.VLisideallydriventozeroastherendezvousiscompleted. EffectofGeomagneticActivityAsmentionedpreviously,thegeomagneticdisturbanceimpactstheatmosphericdensity,affectinghowmuchspatialresolutionisobtainedfromagivendifferenceincrosswindarea.Twocaseswereusedtoshowtheeffectofgeomagneticactivity,denotedHighandLowKp.Bothcasesusedthepreviouslymentionedpropagator,calibrator,forecastingmethod,andcontroller.ThehighKpcasehadanepochofJanuary18th,2005,00:00:00.000UTC,andthelowKpcasehadanepochofJanuary27th,2005,00:00:00.000UTC.ThesecasesrepresentsustainedhighandlowKp,respectively.TheadditionalgeomagneticactivityinthehighKpcaseresultedinadditionalseparationbetweentheforecastedtrajectories,asexpected.Figures 5-21 through 5-24 ,inadditiontoshowingthedensityforecastcomparison,alsoshowthedensityresultingfrominterpolationoftheforecasteddensityeldinblue.Theseinterpolateddensitypointsfallwithinthedensityeld.Examiningtheguidancesthatresultfromtheforecasteddensityrevealsthedifferencesintheguidancebehaviorunderdifferentgeomagneticactivity.Theguidances 45

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aresmaller(lessdisplacementfromthetarget)inthehighKpcaseduetotheadditionalcontrolforceavailablefromtheincreasedatmosphericdensity.TheguidancesareclusteredmoretightlyinthelowKpcasebecausetheforecasteddensitiesandtrajectoriesareclosertogetherinthelowKpcase,sointerpolationdoesnotsignicantlyalterthedensity.BecauseaddingspatialresolutionimprovesthedensityforecastinginthehighKpcase,butnotthelowKpcase,theLyapunovfunctionisreducedinthehighKpcasebutnotthelowKpcase,whenaddingspatialresolutiontotheforecasting.AnadditionalmetricoftheeffectsofimprovingtheforecastisseeninFigures 5-33 and 5-34 .Thenumberofcontrolchangesindicatesthenumberoftimesthecontrolinputisupdated.Alownumberofcontrolchangescaneitherreectthatthenaturaldynamicsofthesystemarebeingfollowed,orthatthecontrolinputisinsufcient.ForthelowKpcase,thelownumberofcontrolchangesincombinationwiththehigherrorindicatesthatthecontrolinputisinsufcienttocreatearapidrendezvous. EffectofAtmosphericWindsTherelativemaneuveringalgorithmdescribedinChapter 3 isonlyintendedtoaccountforinplanemaneuvers.Outofplanemotionhasbeendisregarded,sinceatmosphericdragcanonlyproduceaccelerationsalongthedirectionofthevelocity.Thissimplifyingassumptionisvalidsincetheout-of-planeorbitalperturbationsoneachspacecraftareapproximatelythesameduetobothspacecraftsharinganorbitalplane.Earthrotatesaboutthe^ZintheECIframe,resultinginmovementoftheatmosphereunderthespacecraft.Co-rotationofEarth'satmosphereresultsinrelativevelocityoftheatmospheretospacecraft,potentiallyaffectingtheorbitalplaneduetodrag.Sinceboththetargetandchasedspacecraftfollowsimilarorbits,bothspacecraftshouldexperiencesimilarorbitperturbations,withoneslightlydelayed.Basedonconversationofmomentum,co-rotationoftheatmosphereat414kmaltitudecanbeanywherefrom0-1xthetangentialvelocityoftheEarth'ssurface(assumingeitherno 46

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rotationortotalrotation,respectively).Equation 5 showsthecalculationoftherelativevelocityoftheatmosphereasafunctionoftheco-rotationfractionKC.vrel=v)]TJ /F6 11.955 Tf 11.95 -.16 Td[((KC!Earthr) (5)wherevrelisthevelocityofthespacecraftrelativetotheatmosphere,!EarthistheangularvelocityoftheEarth,ristheECIpositionofaspacecraft,andvistheECIvelocityofthespacecraft.Completeatmosphericco-rotationwasaddedtoboththehighandlowkpcasesdescribedpreviously,andthedisplacementintheLVLH^zdirectionwastrackedthroughoutthemaneuver.Figures 5-35 and 5-36 showtheresulting^zdisplacementresultingfromthealgorithmsdescribedinChapter 3 ,usingspatialresolutionforthedensity.Whentrackingtheguidance,thehighKpcasehasamaximum^zdisplacementofapproximately25m,andthelowKpcasehasamaximumdisplacementofapproximately30m.Attheendofthemaneuver,thesearereduced;the^zdisplacementreachesamaximumofapproximately15minthehighKpcase,andamaximumof10minthelowKpcase.Thesecanbeassumednegligiblewhencomparedtothemuchlargerrendezvousdistancesinvolved.Inadditiontotheco-rotationoftheatmosphere,horizontalwindscanalsodisruptorbits.BasedonthewindmodelHWM07[ 69 ],atmosphericwindsintherangeoforbitsreachedbythespacecraftduringtherendezvousareexpectedtoreachnomorethan50m s,anddecreaseinvelocitywithdecreasingaltitude.Asthespacecraftaretravelingapproximately7000m s,thehorizontalwindscanbeconsiderednegligible.However,duetotheco-rotationdescribedabove,fasterwindsmaybepresent.Varyingthewindalongthedirectionofthevelocitychangesthedragforcebyvaryingthevelocityofthespacecraftthroughthemedium.Headwindsincreasetheatmosphericdrag,whiletailwindsdecreaseit.Aconstantvelocitywindwillsometimes 47

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aidthemaneuverandsometimeshinderit.Insteadofvaryingthewind,theareawasscaledby0.85,1,or1.15foreachspacecraft,tosimulatetheeffectofhavingalowerorhighercontrolforceavailable.Thisrepresentsaconstantwindinthedirectionthatwillmosthinderoraidthemaneuver.UsingthehighKpconditionsdescribedpreviously,9dragcongurationsweredeveloped.Table 5-7 detailsthecongurationsforeach.ThehighKpmaneuverwasusedtostudytheeffectofatmosphericwinds.Case5inTable 5-7 isaduplicateofthepreviouslyshownhighKpmaneuver.All9caseswereabletoreachtherendezvous.Apparentwindsmadetherendezvousmoredifcultbutnotimpossible.Thesewindsrepresentfarmoretargetedandrapidlymovingwindsthanwilleverbeencountered,indicatingatmosphericwindswillnotbeaproblem. TablesandFigures Table5-1. InitialConditionsforRendezvous TargetChaserTargetChaser CaseHighKpLowKpStartDateJan5th,2005Jan18th,2005StartTime00:00:00.000UTCFrequencyComponents10TimeWindows8PastDays8ECIXPosition,km-1330-1329-1330-1329ECIYPosition,km-3732-3733-3732-3733ECIZPosition,km5499549754995497ECIXVelocity, km = s 0.3110.3100.3110.310ECIYVelocity, km = s 6.3046.3046.3046.304ECIZVelocity, km = s 4.3544.3564.3544.356 48

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Table5-2. MetricsforCalibratorFits PearsonCoefcientrMeanSquaredError(MSE) FitDataSet0.9161.16810)]TJ /F11 7.97 Tf 6.59 0 Td[(8 kg2 = m6 TestDataSet0.9306.41110)]TJ /F11 7.97 Tf 6.59 0 Td[(8 kg2 = m6 Table5-3. SpacecraftParameters ParameterValue Mass10kgMinimumArea3.40910)]TJ /F11 7.97 Tf 6.58 0 Td[(7km2MediumArea1.590910)]TJ /F11 7.97 Tf 6.59 0 Td[(6km2MaximumArea2.840910)]TJ /F11 7.97 Tf 6.59 0 Td[(6km2DragCoefcient2(unitless) Table5-4. WeightsandScalingUsedinCalibrator ParameterValue w19.4610)]TJ /F11 7.97 Tf 6.59 0 Td[(5w12.8110)]TJ /F11 7.97 Tf 6.59 0 Td[(4b1.0410)]TJ /F11 7.97 Tf 6.59 0 Td[(3kg m3 JB9.58010)]TJ /F11 7.97 Tf 6.59 0 Td[(4JB3.59410)]TJ /F11 7.97 Tf 6.59 0 Td[(4 MSIS1.14110)]TJ /F11 7.97 Tf 6.59 0 Td[(3MSIS3.42910)]TJ /F11 7.97 Tf 6.59 0 Td[(4 49

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Figure5-1. HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MinimumDrag(FigureCourtesyofAuthor) Figure5-2. HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MediumDrag(FigureCourtesyofAuthor) 50

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Figure5-3. HighKpCaseTargetAmplitudeSpectra,ForecastedDensity,MaximumDrag(FigureCourtesyofAuthor) Figure5-4. LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MinimumDrag(FigureCourtesyofAuthor) 51

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Figure5-5. LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MediumDrag(FigureCourtesyofAuthor) Figure5-6. LowKpCaseTargetAmplitudeSpectra,ForecastedDensity,MaximumDrag(FigureCourtesyofAuthor) 52

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Figure5-7. HighKpCasePrimaryFrequenciesforTarget,MinimumDrag(FigureCourtesyofAuthor) Figure5-8. HighKpCasePrimaryFrequenciesforTarget,MediumDrag(FigureCourtesyofAuthor) 53

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Figure5-9. HighKpCasePrimaryFrequenciesforTarget,MaximumDrag(FigureCourtesyofAuthor) Figure5-10. LowKpCasePrimaryFrequenciesforTarget,MinimumDrag(FigureCourtesyofAuthor) 54

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Figure5-11. LowKpCasePrimaryFrequenciesforTarget,MediumDrag(FigureCourtesyofAuthor) Figure5-12. LowKpCasePrimaryFrequenciesforTarget,MaximumDrag(FigureCourtesyofAuthor) 55

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Figure5-13. HighKpCasePrimaryMagnitudesforTarget,MinimumDrag(FigureCourtesyofAuthor) Figure5-14. HighKpCasePrimaryMagnitudesforTarget,MediumDrag(FigureCourtesyofAuthor) 56

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Figure5-15. HighKpCasePrimaryMagnitudesforTarget,MaximumDrag(FigureCourtesyofAuthor) Figure5-16. LowKpCasePrimaryMagnitudesforTarget,MinimumDrag(FigureCourtesyofAuthor) 57

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Figure5-17. LowKpCasePrimaryMagnitudesforTarget,MediumDrag(FigureCourtesyofAuthor) Figure5-18. LowKpCasePrimaryMagnitudesforTarget,MaximumDrag(FigureCourtesyofAuthor) 58

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Figure5-19. HighKpCaseDensityDrift(FigureCourtesyofAuthor) Figure5-20. LowKpCaseDensityDrift(FigureCourtesyofAuthor) 59

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Figure5-21. HighKpCaseForecastingforTarget(FigureCourtesyofAuthor) 60

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Figure5-22. HighKpCaseForecastingforChaser(FigureCourtesyofAuthor) 61

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Figure5-23. LowKpCaseForecastingforTarget(FigureCourtesyofAuthor) 62

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Figure5-24. LowKpCaseForecastingforChaser(FigureCourtesyofAuthor) 63

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Figure5-25. GuidancesCreatedUsingSimulatedPerfectForecasts(FigureCourtesyofAuthor) Figure5-26. LyapunovFunctionUsingSimulatedPerfectForecasts(FigureCourtesyofAuthor) 64

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Figure5-27. HighKpCaseGuidanceComparison(FigureCourtesyofAuthor) Figure5-28. FinalPhaseofHighKpRendezvousDetail(FigureCourtesyofAuthor) 65

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Figure5-29. LowKpCaseGuidanceComparison(FigureCourtesyofAuthor) Figure5-30. FinalPhaseofLowKpRendezvousDetail(FigureCourtesyofAuthor) 66

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Figure5-31. HighKpCaseLyapunovFunction(FigureCourtesyofAuthor) Figure5-32. LowKpCaseLyapunovFunction(FigureCourtesyofAuthor) 67

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Table5-5. PerformanceMetricsforHighKpCases MinDragMedDragMaxDragInterpolated MeanLyapunovFunction18238258951747616136GuidanceControlChanges405407427417TrackerControlChanges389407395401 Table5-6. PerformanceMetricsforLowKpCases MinDragMedDragMaxDragInterpolated MeanLyapunovFunction6416912346898041GuidanceControlChanges35379TrackerControlChanges133999195 Figure5-33. HighKpCaseControlChanges(FigureCourtesyofAuthor) 68

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Figure5-34. LowKpCaseControlChanges(FigureCourtesyofAuthor) Table5-7. CasesofVaryingWinds,Summarized 123456789 TargetArea0.85x01.15x0.85x01.15x0.85x01.15xChaserArea0.85x0.85x0.85x0001.15x1.15x1.15xMeanLyapunovFunction,unitless111311412929706445821613524755614882475521585GuidanceControlChanges417417417417417417417417417TrackerControlChanges409389357351401385345385421 69

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Figure5-35. LVLH^zDisplacementBetweenSpacecraft,HighKp(Figurecourtesyofauthor) Figure5-36. LVLH^zDisplacementBetweenSpacecraft,LowKp(Figurecourtesyofauthor) 70

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CHAPTER6IMPLEMENTATIONONBOARDASPACECRAFTThealgorithmsdescribedintheprevioussectionsareintendedforeventualapplicationonboardanactualspacecraft.Propellant-lessAtmosphericDifferentialDragLowEarthOrbitSpacecraft(PADDLES)isa3UCubeSat,whichcanbedenedasaclassofsatelliteintendedforuniversityresearch.PADDLESisshowninFigure 6-1 . MissionOverviewPADDLESwillbelaunchedfromtheInternationalSpaceStation.Duetotheuncertaintyonlaunchprovider,PADDLEScanbedeployedanywherefromapproximately300to500kmaltitude.Thisisprimarilydrivenbymissiontimeandmaneuvertime;themissionisexpectedtolast2-3months,andmaneuversmaytakedaysbutshouldnottakeweeks.Theorbitaleccentricityalsoshouldbeminimized,whichwillmaximizetheperigeealtitude,maximizingthemissiontime.Withanapogeeof420kmandaperigeeof416km,theISSorbiteccentricityisapproximately0.0342,whichisacceptableforthePADDLESmission.PADDLESisanticipatedtobelaunchedfromtheNanoRacksCubesatDeployer(NRCSD)[ 70 ],locatedintheISSKibomodule.ThisisanalternativetothemorecommonlyusedPoly-PicosatelliteOrbitalDeployer(P-POD)[ 71 ],withlessstringentrestrictions,makingiteasiertodesign.TheNanoRacksdeployercansupportuptoa6.5U(1Udenedasstandardized10cmx10cmx10cmunit)spacecraft,orcombinationsthatadduptoamaximumof6.5U.Spacersareaddedtotakeupanyextralength.PADDLEScontainsadragsailtovarythecrosswindarea,varyingtheatmosphericdrag,whichwillbeusedtoactuatethepreviouslydescribedmaneuvers.Thecontrolalgorithmdescribedinprevioussectionsismodiedtoreectavailabledata.ThePADDLESmissioninvolvesasinglespacecraft,soallmaneuveringisperformedrelative 71

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toamovingpoint,orvirtualspacecraft.Additionally,thecalibratorwillestimaterealatmosphericdensity,andsowouldneedtoberebuiltwithrealdata.Tomeasurethein-orbitatmosphericdata,PADDLESisequippedwithaNRL-developedmassspectrometer.TheramfaceofPADDLEScontainsamassspectrometer,referredtoasRAM-Sensor(RAMS).ThissensorisNRL-developedandfunded.Itwillindirectlymeasureatmosphericdensitywhileinorbit.Oncethedensityisforecastedusingthemodiedcontrolalgorithm,PADDLESwillusearepeatedly-retractabledragsailtomaneuver.TheactuatorforthemaneuverstobeundertakenbyPADDLESisthedragsailsystem.DevelopmentbeganatRensselaerPolytechnicInstitute(RPI)andcontinuedatUniversityofFlorida(UF).Thedragsailconvertsrotarymotionofthecenterofthesailtolinearmotionofthecorners,allowingthesailtoopenandcloseasdesired.Figure 6-7 showsthemountinglocationontherearofPADDLES.Basedonanexistingorigamipattern[ 72 ],thesailusesaspiralfoldthatwrapsaroundthecenter.Drivenfromthecenter,rotatingthecenterdeploysorretractsthesail.Themotionisconstrainedbywireloopsinthecorners,whichturntherotarymotionintolinearmotion.Sinceallfourcornersmoveradially,thesailincreasesinareaasthecornerstraveloutward,untilthesailbecomesnearlyat.Thewireloopsusedtoconstrainthemotionrideoncollapsibleboomsbasedonmeasuringtape.Tosavespace,theseboomspackatforlaunchanddeployment.Additionally,iftheboomsbecomedeected,thedesignassuresthattheywilltendtosnapbackintoplace.SincethesailishiddenfromthewindbytheCubeSatchassiswhenfullyretracted,theminimumareais0.01m2,andthemaximumareaisapproximately0.15m2.Thisisdifculttoestimateaccuratelyduetotheexibilityofthesail.Figure 6-8 showsallthemajorhardwareinthedragsailsubsystem.Thedragsailandmassspectrometerbothrequirespecicconditionstofunctionproperly.TheremainderofthehardwareonboardPADDLESsupportsthemission. 72

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Asspacecrafttendtohavemanyfailurepoints,andaredependentonothersystemsworking[ 16 ],PADDLEShasvaryingsuccesslevelsdenotedinTable 6-2 .ProvidingmultiplesuccesslevelsallowsPADDLEStobesuccessfulevenifthird-partysystemsdonotwork.ThemaincontrolsystemofPADDLESistheCubeSatbus.APumpkin,Inc.ightmotherboardisusedtocontroloallothersystems.TheCubeSatbuscontainsastack-throughconnectorlinkingallthemajorboardsinPADDLES.Figure 6-2 showsthearrangementofeachboard.Differingouterdimensionsimpliesthatwhiletheorderisarbitraryintermsofsignaling,thiscongurationistheonlyarrangementthatclearsallhardwareontheoutsideofthestack. PowerSystemThepowersystemisintegraltooperationofPADDLES,andmustbedesignedtoproduceanaveragepowerabovewhatPADDLESrequires,overthelifeofthemission.UsingaSTKanalysis,thiswasdeterminedtobeapproximately4.4Woverthelifeofthemission.However,sincethepowerrequirementswillvaryandsolarpowerwillvaryaswell,PADDLEScontainsanonboardbatteryandpowerhandlingsystem.ThesolarpanelslocatedneartheramfaceofPADDLESareusedtoproducepower,whichisthenroutedthroughtheXUEPSboard[ 73 ]andstoredinthebatterywhennotinuse.ThisarrangementisshowninFigure 6-5 .ThepowersystemonboardPADDLESshouldbeabletopowerthespacecraftatfullpowerforacompleteorbit.Itisnotpossibletohavemorethanonecompleteorbitindarkness(duetoEarth'sorbit),sothisissufcienttodeterminetheminimumbatterycapacity.PADDLEScandrawamaximumof10.15W,andanorbitlastsapproximately90minutes,implyingaminimumbatterycapacityof15.22W-hrisrequired.Toprovideanextramargin,aPumpkin,Inc.40W-hrbatteryisused.Tables 6-1 and 6 indicatetheassumptionsinvolvedindeterminingthepowerrequirements.Theseassumptionsreecttheconservativeapproachtopowersystemselectiondescribedabove. 73

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PowerwillbesuppliedbytwoPumpkin,Inc.solarpanels.PADDLESwillpartiallysuntracktomaximizesolarpower.Sincethelongaxisofthespacecraftmustremainalignedwiththevelocity,PADDLESrotatesaboutthelongaxisuntilthepowergenerationrateismaximized. Location,Pointing,andRadioThesharedGlobalPositioningSystem(GPS)andradioboardisprovidedbyPumpkin.ThisboardusesaNovAtelOEM615GPSreceiverwithCoordinatingCommitteeforMultilateralExportControls(COCOM)altitudeandvelocitylimitsremoved[ 74 ].ThisallowstheGPStobeusedinorbit.ATaoglasAP.25E.07.0054AGPSantennaisusedontheanti-nadirfaceofthespacecraft.ThisisplacedsoastocapturethemaximumnumberofGPSsatellites,asallGPSsatelliteswillbeinahigherorbitthanPADDLES,andthismaximizesthechancesofreachingfourormoresatellites.PointingradiallyawayfromtheEarthminimizesthechancesofblockingtheGPSsignalwiththechassis.TheradioreceiverisalsosharedwiththeGPSboardtosavespaceinthechassis.Theradiocommunicatesovertheamateurradioband.TheGPSRM-1,thepreviousmodel,isshowninFigure 6-3 ,duetoInternationalTrafcinArms(ITAR)restrictions.TheBlueCanyonTechnologiesXACTAttitudeDeterminationandControlSystemisusedtodeterminetheattitudeofPADDLES[ 75 ].Usingaside-mountedstartrackershowninFigure 6-4 andthelocationreportedbytheGPSsystem,theXACTisabletousevisualstardatatodeterminewhichwayPADDLESispointed.InthecasethatPADDLESispointedintheincorrectdirection,theXACTisequippedwiththreereactionwheelsthatcanbespunupordowntovarytheattitude.Thesehavethepotentialtosaturate,orreachthemaximumpossiblespeed.Threemagnetorquersarealsoinstalled,whichcanrejectreactionwheelangularmomentumtotheEarth'smagneticeld,allowingthewheelstode-saturatewhenneeded.As 74

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mentionedintheprevioussection,PADDLESwillpartiallysuntrack;thismotionandstabilizationwillbeprovidedbytheXACT.ThemassspectrometeronboardPADDLESrequiresclosealignmenttothevelocityvectortoensureaccuratemeasurementoftheparticleow.BasedoncommunicationwithNRL(UnitedStatesNavalResearchLaboratory),themassspectrometermustbepointedwith3degreesoftheramfaceofthespacecraft.Additionally,thisfaceshouldbealignedascloselyaspossiblewiththevelocityvector,asmisalignmentswillaffectthecross-windarea.ThestartrackeronboardPADDLES,usedtodeterminetheattitude,alsorequirespointingatthesky.Thisrequiresrotationabouttheramaxis,whichdoesnotchangethecross-windarea. MissionPhasesWhileintheNanoRacksdeployer,PADDLESwillinitiallybeinthe“Off”softwarestateshowninFigure 6-9 .Inthisstate,thehardwareistotallyoff.RemovingtheRemoveBeforeFlight(RBF)pinwhileintheNanoRacksdeployerbringPADDLEStothe“Startup”state.Themotherboardwillinitializethehardware,softwareinterruptsandsoftwaretasks.The“Diagnostic”stateisreachedwhenthehardwareisreadytouse.SincePADDLESisstillintheNanoRacksdeployeratthispoint,communicationwiththemotherboardispossible.ReleasingtheseparationswitchesonejectionandreleasefromtheNCRSDtransitionsPADDLEStothe“Waiting”State.The“Waiting”stateservestwopurposes.AccordingtotheNanoRacksspec[ 70 ],PADDLEScannotdeployfor30minutesafterreleasefromthedeployertopreventdamagetobothPADDLESandthelauncher.“Waiting”propagatesthecurrenthardwareandsoftwarecongurationfor30minutes.IntheeventthatPADDLESdoesnotreleasefromtheNanoRacksdeployerproperly,thisallowstheoperator30minutestore-depresstheseparationswitches(returningPADDLEStothe“Diagnostic”state)or 75

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reinserttheRBFpin(returningPADDLEStothe“Off”state).Alternately,aslongastheseparationswitchesarepressedatleastonceeveryhalfhour,loadingPADDLESintotheNanoRacksdeployercantakeaslongasnecessary.WhenPADDLEShasbeendeployedfromthedeployerintoLEO,itwillreachthe“Deployment”stateafter30minuteshaveelapsed.Atthispoint,allhardwareisretracted.Burnwiresrstreleasethesolarpanels,whichalsoallowstheboomstodeploy.Anadditionalburnwireisusedtoreleasetheantennas.Deploymentoftheextrnalhardwareallowsaclearviewbytheonboardstartracker,sode-tumblingwilltakeplaceatthistime.Thesailalsoisunobstructedatthispoint,andcanopenandcloseasdeterminedbytheonboardcontroller.ThisbringsPADDLEStothe“Running”state,inwhichitwillremainformuchofthemission,duringwhichallthehardwareisoperational.Atmosphericdensitymeasurementsdrivetheopeningandclosingofthesailwhileinorbit.Thepreviousstatesarenolongerreachableatthispoint.PADDLESgenerallyrunsataslightpowerdecitduetothelimitationsonsuntracking.Whenthebatteryislow,PADDLESentersthe“Charging”state.Thesailisclosedtomaintaintheorbit,thenthecontrollerandsailmotorandcircuitareshutoff.TheGPSisturnedoffaswell.Instead,positionandvelocityestimateswillbepropagatedusingtheonboardInertialMeasurementUnit.TheEPSboardwillmonitorthechargingstateofPADDLESsothatthemotherboardwillreturnPADDLEStothe“Running”statewhenthebatteryreachessufcientcharge.Whenthemissioniscomplete,orPADDLESnolongerhassufcientaltitudetoeffectanymaneuvers,PADDLESwillenterthe“De-Orbit”statewhencontrolledtodosobythegroundstation.PADDLESwillfullyopenthedragsailwhilecontinuingtomakeatmosphericmeasurements,andwillcontinuesendingradiosignalsaslongaspossible.Thehardwarewillburnupasitenterstheatmosphere.Thisstateisreachablefromboththe“Running”and“Charging”states. 76

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TablesandFigures Figure6-1. PADDLESwithAllHardwareFullyDeployed(PhotoCourtesyofAuthor) Figure6-2. PADDLESCubeSatBusComponents(ImageCourtesyofAuthor) 77

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Figure6-3. PumpkinGPSRM-1SharedRadioandGPSBoard(Source:Pumpkin[ 76 ]) Figure6-4. BlueCanyonTechnologies(BCT)XACTAttitudeDeterminationandControlSystem(ADACS)(Source:BlueCanyonTech.[ 75 ]) 78

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SystemAveragePowerAssumptions DragSail0.013WOnfor5%oftheorbit,5changesperhourRadio0.0092W90minuteorbittimeusedMotherboardandPPM0.25WAlwaysonGPS0.535W50%dutycycleEPS0.1WAlwaysonXACT2WAlwaysonRAMS1.5WAlwaysonTotal4.408W Table6-1. PowerSystemEstimates SystemPeakSteady-StateAssumptionsDataSource SailMotor0.27W0WFaulhaberdataRadioModule5W0WStackdataunavailableAstroDevdataMotherboard&PPM0.25W0.05WMotherboardpowersPPMPumpkindataGPS1.03W0.04WStackdataunavailablePumpkindataEPS0.1W0.1WClydeSpacedataADACS2W0.5WBCTdataRAMS1.5W1.5WEstimateU.S.NRLTotal10.15W2.19W . . Panel1A . Panel1B . Panel2A . Panel2B . EPSBoard . Battery . OtherSystems Figure6-5. PADDLESPowerSystem(FigureCourtesyofAuthor) 79

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. . CSKMBandPPM . BCTXACT . DragSail . Power . GPS . Deployment . Radio . RAMS . Deployer . SeparationSwitches . BurnWires Figure6-6. PADDLESControlSystem(FigureCourtesyofAuthor) Figure6-7. DragSailandLocationonRearofPADDLES(ImageCourtesyofAuthor) Table6-2. PADDLESSuccessLevels SuccessLevelResult MinimumSuccessPADDLESdeploysandsuccessfultwo-waycommunicationisestablishedAdditionalSuccessAboveplusallsensorsfunctionproperlyAdditionalSuccessAboveplusthesailopensandclosesasdesiredMaximumSuccessAboveplusrendezvousmaneuversaresuccessfullycompleted 80

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Figure6-8. DragSailExplodedView(ImageCourtesyofAuthor) . . De-Orbit . Charging . SpecialCases . De-OrbitCommand . Running . Running . BatteryOK . BatteryLow . De-OrbitCommand . Deployment . Waiting . OrbitInsertion . Diagnostic . Startup . Off . InNCRSD . PullRBFPin . Eject . Press . Separation . Switch . ReplaceRBFPin . ReplaceRBFPin Figure6-9. PADDLESHardwareStates(FigureCourtesyofAuthor) 81

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CHAPTER7CONCLUSIONThelineartcalibratorhasbeenshowntosuccessfullyestimateDTM-2013densityfromJB2008andNRLMSISE-00densityforthesametimeandlocation.ThisallowsittobeusedasaproxyforDTM-2013,whichrepresentstheobservedatmosphericdensity.Usingthismethodtoestimatedensityforpasttimes,thedensityforfuturetimesisforecastedusingthemethodsdescribedinthiswork.Thisforecasteddensityisthenusedtocreatethreeforecastedtrajectoriesforthetargetandthechaserspacecraft,addingspatialresolutiontothedensityforecast.Implementationofdensityforecastingmethodsusingspatialresolutionhasbeenshowntoimprovethetrackingofdifferentialdrag-basedguidances.Theguidancescreatedusingasingleforecastedtrajectorytoprovidethedensityinformationwerecomparedtotheguidancecreatedusingspatialresolutionandinterpolatingbetweenthreeforecastedtrajectories.TheLyapunovfunction,indicatingtrackingerror,waslowestwhentrackingtheguidancecreatedusingspatialresolutioninthehighKpcase,indicatingthatthisguidancewaseasiesttotrack.Conversely,addingspatialresolutioninthelowKpcasepreventedthecreationofaconsistentrelativeguidance,whichdidnotreducetheLyapunovfunctionwhentrackingtheguidancecreatedusingspatialresolution.Thisisduetothesimilarityoftheforecastedtrajectories.Interpolatingbetweenthreesimilartrajectoriesresultsinaninterpolateddensitythatdoesnotvarysignicantlyfromanyindividualtrajectory.Fromthis,itcanbeconcludedthataddingspatialresolutionismorebenecialduringperiodsofhighgeomagneticactivity,duetotheincreaseinseparationbetweenforecastedtrajectoriesinthepresenceofhigheratmosphericdensity.ImplementationofthisalgorithminaspacecraftisperformedusingPADDLES,a3UCubeSat.PADDLESwillactuatedifferentialdrag-basedrendezvousmaneuverswithrespecttoamovingpoint,or“virtualspacecraft”.AfteritslaunchfromtheInternational 82

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SpaceStation,theonboarddragsailwilldeployandretracttocreatethecontrolforcenecessarytomaneuverinorbit.Themissionwilllastseveralmonths,whichissufcientlylongtoactuatemanymaneuversofseveraldayseach.ThesemaneuverswillberepeateduntilPADDLESnolongerhassufcientaltitudeandde-orbits. 83

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CHAPTER8SUGGESTIONSFORFUTUREWORK IndexForecastingAspreviouslymentioned,thesolarandgeomagneticindicesareonlyavailableinthepast.Sincemanyatmosphericmodelsdependonsolarandgeomagneticindices,thisprecludestheiruseinthefuture.ThisworkcouldbegreatlysimpliedbyusinganANNtopredictthesolarandgeomagneticindicesintothefuture,allowingthedirectuseofJB2008andNRLMSISE-00inthefuture.Togetherwiththecalibrator,thiswouldallowdirectcalibrationtoDTM-2013inthefuture,whichwouldthenbeusedtocreatethedensityelds,whichwouldbeusedasnormal.DTM-2013cannotbeuseddirectlyinthefuturebecauseitistakenasthetruthmodelandthereforeassumedtobeunavailableforgeneratingtheguidance.Someworkhasalreadybeendoneinpredictingsolarandgeomagneticindices.Gleisneret.al[ 77 ]usedANNstoforecastsolarwindspeed,protonnumberdensity,andthesouthwardcomponentoftheplanetarymagneticeld.Dimitrievet.al[ 78 ]usedANNstoforecastF10.7,sunspotnumber,andthemeansolarmagneticeldvalue.Similarly,Huanget.al[ 79 ]usedsupportvectormachinestoforecastonlyF10.7.Additionally,someworkhasbeendoneincorrelatingindicestootherindices,whichwouldallowthepredictionofthedesiredindicesifotherindicescouldbepredictedandacorrelationfound.TemerinandLi[ 80 ]presentapredictionmethodforDstfromthesolarwind.MacPhersonet.al[ 81 ]detailacorrelationbetweentheF10.7indexandsunspotnumber.Finally,theSOLAR2000modelbyTobiskaet.al[ 82 ]modelsthesolaroutputtopredicttheindicesonEarth. LimitingtheMotionoftheDragSailSubsystemThecontrolsystemforthedragsailsubsystemisstillunderdevelopment.Becausethemotormustrotate1.25turnstofullyopenthesail,thismakesitverydifculttodesignalimitswitch.Someoptionsthatarebeingexploredinclude: 84

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• Gearingthelimitswitch.Thiswouldconverttherotationonthecentercolumnintoasmallerrotationelsewhere,allowingtheuseofanencoderasalimitswitch. • Combiningatoggleandmomentaryswitch.Thestateofthetoggleswitchindicateswhetherthecentercolumnisaboveorbelow360degreesofrotation. • Mountingthecontrolhardwarebelowthesail. • Addingascrewthreadtothecentercolumnandmountingthelimitswitchesvertically. 85

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[73] CubeSatPower, http://www.clyde-space.com/documents/2818 ,ClydeSpace,lastAccessed1October2015. [74] J.Graham-Cumming,“GAGA-1:CoComlimitforGPS,” http://blog.jgc.org/2010/11/gaga-1-cocom-limit-for-gps.html ,November2010,lastAccessed1October2015. [75] BCTXACTHighPerformanceAttitudeControlforCubeSats, http://bluecanyontech.com/wp-content/uploads/2015/05/XACT-Data-Sheet 1.0.pdf ,BlueCanyonTechnologies,lastAccessed1October2015. [76] A.E.Kalman,“GPSRM-1GPSReceiverModuleHardwareRevision:C,” http://www.cubesatkit.com/docs/datasheet/DS CSK GPSRM 1 710-00908-C.pdf ,December2013. [77] H.GleisnerandH.Lundstedt,“ANeuralNetwork-BasedLocalModelforPredictionofGeomagneticDisturbances,”JournalofGeophysicalResearch,vol.106,no.A5,pp.8425,2001 http://dx.doi.org/10.1029/2000ja900142 . [78] A.Dmitriev,Y.Minaeva,Y.Orlov,M.Riazantseva,andI.Veselovsky,“SolarActivityForecastingon1999-2000byMeansofArticialNeuralNetworks,”inEuropeanGeophysicalSociety(EGS)XXIVGeneralAssembly,1999. [79] C.Huang,D.-D.Liu,andJ.-S.Wang,“ForecastDailyIndicesofSolarActivity,F10.7,UsingSupportVectorRegressionMethod,”ResearchinAstronomyandAstrophysics,vol.9,no.6,pp.694,2009 http://dx.doi.org/10.1088/1674-4527/9/6/008 . [80] M.TemerinandX.Li,“ANewModelforthePredictionofDstontheBasisoftheSolarWind,”JournalofGeophysicalResearch,vol.107,no.A12,2002 http://dx.doi.org/10.1029/2001ja007532 . [81] K.MacPherson,A.Conway,andJ.Brown,“PredictionofSolarandGeomagneticActivityDataUsingNeuralNetworks,”JournalofGeophysicalResearch,vol.100,no.A11,pp.21735744,November1995 http://dx.doi.org/10.1029/95ja02283 . [82] W.K.Tobiska,T.Woods,F.Eparvier,R.Viereck,L.Floyd,D.Bouwer,G.Rottman,andO.White,“TheSOLAR2000EmpiricalSolarIrradianceModelandForecastTool,”JournalofAtmosphericandSolar-TerrestrialPhysics,no.62,pp.1233,2000 http://dx.doi.org/10.1016/s1364-6826(00)00070-5 . 92

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BIOGRAPHICALSKETCHDavidGuglielmoearnedhisB.S.andM.S.degreesinMechanicalEngineeringfromRensselaerPolytechnicinstitutein2010and2012,respectively,wherehewontheChangetheWorldChallengebothinfall2011andfall2012.In2013,hejoinedtheADvancedAutonomousMultipleSpacecraft(ADAMUS)LaboratoryatRensselaerPolytechnicInstitute.Priorto2013,Dr.GuglielmoworkedintheCenterforAutomationTechnologiesandSystems(CATS)atRensselaerPolytechnicInstitute,withDr.DanielWalczyk.In2014,hemovedwithDr.RiccardoBevilacquatotheUniversityofFlorida.Whileastudent,Dr.Guglielmowasselectedforthe2015MathematicalModelingandOptimizationInstituteinShalimar,Florida,jointlysponsoredbytheAirForceOfceofScienticResearch(AFOSR)andtheMunitionsDirectorate(RW)oftheAirForceResearchLaboratory(AFRL).Dr.Guglielmoalsoplacedsecondinthe2015UniversityofFloridaMechanicalandAerospaceEngineeringTeachingCompetition.HewasselectedtopresentatboththeUniversityofFloridaControlSystemsSymposiumandGraduateStudentResearchDay.Dr.GuglielmowasawardedhisPh.D.inAerospaceEngineeringinDecember2015.Dr.GuglielmoisamemberoftheAmericanInstituteofAeronauticsandAstronautics(AIAA)andtheAmericanSocietyofMechanicalEngineers(ASME),andholdsaTechnician-classamateurradiolicense.Dr.GuglielmoisSTK10certied. 93

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In2013,Dr.GuglielmopresentedOrigami-basedDragSailforCubeSatPropellant-freeManeuveringinthe5thNanosatelliteSymposiuminTokyo,Japan.Additionally,in2014,hepresentedPropellant-lessAtmosphericDifferentialDragLowEarthOrbitSpacecraft(PADDLES)MissioninTheCommerceofSmallSatellitesConference,inLogan,Utah.Dr.Guglielmoco-authoredSpatialResolutioninDensityPredictionforDifferentialDragManeuveringGuidance,presentedattheAAS/AIAAAstrodynamicsSpecialistConferenceinBoulder,CO.Dr.GuglielmoalsopreviouslypresentedhisresearchattheASMEFuelCellConferencein2011and2012. 94