Spacecraft Attitude Resource Sharing

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Title:
Spacecraft Attitude Resource Sharing
Physical Description:
1 online resource (199 p.)
Language:
english
Creator:
Johnson, Shawn C
Publisher:
University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Fitz-Coy, Norman G
Committee Members:
Dixon, Warren E
Barooah, Prabir
Mcnair, Janise Y
Lacy, Seth

Subjects

Subjects / Keywords:
attitude -- data -- disaggregated -- fractionated -- fusion -- resource -- satellite -- sharing -- spacecraft -- tasking
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Disaggregated spacecraft systems distribute the functionality of traditional monolithic spacecraft across several platforms. This dissertation addresses some of the technical challenges associated with a disaggregated attitude determination system (ADS), where the disaggregated ADS is defined by a chief spacecraft, with relative and inertial attitude sensors, and multiple deputy spacecraft, with only inertial attitude sensors or no sensors at all. A tracking controller and reference signal kinematics are developed for the chief to track the deputies and measure their relative attitude states. These measurements are used in the formulation of a disaggregated implementation of the extended Kalman filter (EKF) for inertial attitude estimation. The disaggregated EKF produces multiple inertial attitude state estimates of the chief and deputies. Euclidean state and attitude state data fusion are reviewed and used to develop a generalized attitude data fusion law based on the attitude error vector. To avoid numerical complexities associated with constrained attitude parameterizations, minimal vectorial attitude parameterizations are investigated for data fusion. The accuracy of the vectorial parameterizations is assessed, and it is shown that the classical Rodrigues parameters are a sufficient parameterization for minimal attitude data fusion, but that higher-order Rodrigues parameters can be used to improve the accuracy. The minimal parameterizations are applied to data fusion laws using the shadow parameters of the vectorial parameterizations, as well as a local error representation.  A greedy tasking algorithm is also developed. This algorithm makes use of the current covariance information of the deputies to task the chief to track deputies and measure their relative states. Characterizations of the size of the state-error covariance matrix are investigated in simulation based on the trace, matrix norms, and differential entropy. Simulation results show that for measurement sharing, greedy tasking can improve upon the accuracy of the chief and deputies using a fixed tasking strategy. Furthermore, Monte Carlo results show that the greedy tasking algorithm is mostly insensitive to the covariance metric.   This manuscript concludes with numerical simulations of two attitude resource sharing scenarios: (i) between a chief and a single deputy and (ii) between a chief and two deputies. These simulations demonstrate that attitude resource sharing can be used to increase the attitude knowledge of a deputy equipped with low performance attitude sensors and also improve the accuracy of the attitude knowledge of all spacecraft (including the chief) when all spacecraft are equipped with similar sensors.
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.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Shawn C Johnson.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Fitz-Coy, Norman G.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-02-28

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lcc - LD1780 2013
System ID:
UFE0045813:00001


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SPACECRAFTATTITUDERESOURCESHARING By SHAWNC.JOHNSON ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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2013ShawnC.Johnson 2

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Inlovingmemoryofmymother,CaroleCushmanJohnson 3

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ACKNOWLEDGMENTS Imetmyadvisorandcommitteechair,Dr.NormanG.Fitz-Coy,rstastheprofessor ofmyundergraduateAstrodynamicscourse.Atthattime,heimpresseduponmethe importanceoffundamentalsandtheperilsoftakingshortcuts.Throughoutmydoctoral studiesIhavenotonlylearnedfromhisexperience,butalsofromhissuperbintegrity andcharacter.ThesearetraitsthatIwillaimtosharewiththosethatImentorandIam verythankfulforhistirelesssupport.Ialsowouldliketothank,Drs.PrabirBarooah, WarrenDixon,SethLacy,andJaniseMcNair,forsupportingandfosteringmyresearch ideasandservingasmembersonmysupervisorycommittee.Iamparticularlygrateful toDr.SethLacyforprovidingthecatalysttothisresearch,asmymentorattheAirForce ResearchLaboratory,andtheplethoraofdiscussionsalongtheway. Inaddition,therearemanyindividualsthathavegivenmestrengththroughoutthis endeavor.IwouldliketothankthepastandpresentmembersoftheSpaceSystems Groupfortheircamaraderie,aswellasmanyengagingresearchdiscussionsoverthe years.Ofthisgroup,IamparticularlythankfultoDr.JosueMu nozforprovidingme motivationsinceourrstyearincollege.Iamalsothankfultomyfriendsandfamilyfor theirloveandencouragement.Particularly,Iamindebtedtomyparentsforhelpingme navigatelife'smanychoices,providingguidancewhenaskedupon,butneverattempting tocontrolmypaththeGNCoflife.TomyPapa;theoriginalGatorinourfamily;thank youforinstillinginmeapassionandenthusiasmforlearning.Tomygrandmother,thank youforbeinganexampleofpositivity,patience,andlove;andforputtingthatTangram setinmyhandswhenIwasthree.Iamgratefultomysiblingsforteachingmemore thanIcouldeverlearninaclassroom,Iwillalwayslookuptoyou.Finally,Iwouldliketo thankmywifeforherenduringloveandsupport.EverydayIamhumbledbyhercaring natureandabilitytobringoutthebestinothers. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................9 LISTOFFIGURES.....................................10 ABSTRACT.........................................13 CHAPTER 1INTRODUCTION...................................15 1.1ResourceSharing...............................18 1.2RelevantMissions...............................22 1.2.1RendezvousandDockingMissions..................22 1.2.2Space-basedInterferometry......................24 1.2.3SatelliteNetworks...........................25 1.2.4SystemF6................................25 1.3ProblemStatements..............................26 1.3.1Problem1:InertialAttitudeEstimationinaDisaggregatedSystem28 1.3.2Problem2:AttitudeDataFusion...................29 1.3.3Problem3:SensorTasking......................30 1.4ThesisStatements...............................31 1.4.1Thesis1:OnInertialAttitudeEstimationforDisaggregatedSystems31 1.4.2Thesis2:OnMinimalAttitudeParameterizationDataFusion....31 1.4.3Thesis3:OnSensorTaskingforAttitudeResourceDistribution..31 1.5DissertationOrganization...........................32 2SYSTEMDYNAMICSANDCONTROL.......................34 2.1GeneralNomenclature.............................34 2.2RigidBodyKinematics.............................34 2.3CoordinateFrames...............................35 2.3.1Earth-CenteredInertialECI.....................36 2.3.2Earth-CenteredEarth-FixedECF..................36 2.3.3HillLVLH................................36 2.3.4PerifocalPQW.............................36 2.3.5BodyFrameB.............................37 2.4AttitudeParameterizationsandKinematics.................37 2.4.1AttitudeMatrix.............................39 2.4.2EulerAngles..............................42 2.4.3Axis-Angle................................43 2.4.4EulerSymmetricParametersUnitQuaternions..........45 2.4.5ClassicRodriguesParametersCRPs................46 5

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2.4.6ModiedRodriguesParametersMRPs...............46 2.4.7StereographicOrientationProjectionsSOPs............47 2.4.8GeneralizedCayleyTransformation..................49 2.4.9VectorialAttitudeParameterizations.................50 2.5SummaryofAttitudeParameterizations...................53 2.6AttitudeDynamics...............................55 2.7AttitudeTrackingControl............................56 2.7.1DesiredCoordinateFrames......................56 2.7.1.1RelativespacecraftsharingframeSH..........57 2.7.1.2GroundtargetframeT...................57 2.7.2DesiredQuaternionKinematics....................59 2.7.2.1Relativespacecraftsharingframekinematics.......59 2.7.2.2Groundtargetframekinematics..............60 2.7.3Clohessy-Wiltshire-HillCWHEquations...............61 2.7.4AttitudeTrackingControllerDerivation................63 2.8Summary....................................65 3DISAGGREGATEDATTITUDEESTIMATION...................67 3.1GeneralNomenclature.............................67 3.2InertialAttitudeSensors............................68 3.2.1SunSensors..............................69 3.2.2Magnetometers.............................69 3.2.3StarTrackers..............................71 3.2.4RateGyroscope............................73 3.3DeterministicAttitudeDetermination.....................74 3.3.1q-algorithm...............................76 3.3.2OtherAttitudeDeterminationAlgorithms...............78 3.4KalmanFilterReview..............................79 3.5ExtendedKalmanFilter............................84 3.6ExtendedKalmanFilterforInertialAttitudeEstimation...........85 3.6.1SensitivityMatrixDerivation......................86 3.6.2DiscretePropagationDerivation....................88 3.6.3ExampleInertialAttitudeEKFwithaMagnetometerandSunSensors forLowPrecisionPointing.......................89 3.6.4ExampleInertialAttitudeEKFwithaStarTrackerforHighPrecision Pointing.................................91 3.7RelativeAttitudeSensors...........................94 3.8DeterministicRelativeAttitudeDetermination................97 3.9InertialAttitudeEstimationinaDisaggregatedSystem...........98 3.9.1SensitivityMatrixDerivation......................98 3.9.2DiscretePropagationDerivation....................101 3.9.3NotesonDisaggregatedAttitudeEstimation.............104 3.10Summary....................................104 6

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4ATTITUDEDATAFUSION..............................105 4.1DataFusionOverview.............................105 4.2DataFusionReview..............................106 4.3DataFusionandtheErrorState.......................108 4.3.1EuclideanFusionErrorState.....................108 4.3.2AttitudeFusionErrorState.......................110 4.4UnknownCross-CorrelationDataFusionAlgorithms............112 4.4.1CovarianceIntersection........................113 4.4.2EllipsoidIntersection..........................115 4.4.3SummaryofDataFusionLawsforAttitudeStates..........117 4.5ReviewoftheUnitQuaternionforDataFusion...............118 4.6VectorialAttitudeParameterization......................119 4.7ParameterizedAttitudeErrorVector.....................120 4.7.1First-OrderModelofAttitudeErrorVector..............121 4.7.2DataFusionJacobian.........................122 4.7.3AttitudeErrorVectorApproximationAccuracy............123 4.7.4UnitQuaternionTransformation....................128 4.8MinimalAttitudeParameterizationDataFusion...............128 4.8.1ShadowSwitchingDataFusionProcess...............129 4.8.2Example:FRPCovarianceIntersectionwithShadowSwitching..131 4.9VectorialAttitudeDataFusionwiththeLocalErrorApproach.......131 4.10CommentsonAttitudeDataFusion......................134 4.10.1ANoteonMRPSingularitiesandShadowSwitching........135 4.10.2ANoteontheMinimizationCriterionfortheCIAttitudeFusionLaw136 4.10.3ANoteonCovarianceIntersectionandEllipsoidalIntersection...137 4.10.4ANoteonAttitudeDataFusionwithAppendedStateVectors...138 4.11TwoStarTrackerDataFusionExample...................138 4.12Summary....................................141 5STOCHASTICGREEDYSENSORTASKING...................143 5.1SensorTaskingOverview...........................143 5.2TaskingProblemStatement..........................145 5.3TaskingSolutionMethodology.........................146 5.4ResourceSharingAssumptiononDynamicsTimeConstants.......147 5.5Tasking.....................................148 5.5.1BaselineTasking............................148 5.5.2GreedyTasking.............................149 5.6PerformanceMetrics..............................151 5.6.1Trace...................................152 5.6.2MatrixNorms..............................152 5.6.3DifferentialEntropy...........................154 5.7Simulations...................................155 5.7.1PerfectRelativeAttitudeSensorAssumption.............156 5.7.2SimulationInitialization.........................156 7

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5.7.3Single-runResults...........................158 5.7.4MonteCarloAnalysis..........................162 5.7.5ANoteontheTracePerformanceMetric...............165 5.8Summary....................................165 6SPACECRAFTATTITUDERESOURCESHARINGSIMULATIONS.......167 6.1SimulationInitializations............................167 6.2TwoSpacecraftAttitudeResourceSharing.................168 6.2.1TwoSpacecraftwithStarTrackersScenario.............169 6.2.2TwoSpacecraftwithCoarseDeputySensorsScenario.......170 6.3ThreeSpacecraftAttitudeResourceSharing................172 6.3.1ThreeSpacecraftwithStarTrackersScenario............174 6.3.2ThreeSpacecraftwithCoarseDeputy-1SensorsScenario.....175 6.4TaskingAlgorithmComparisonforAttitudeEstimateResourceSharing withDataFusion................................178 6.5ObservationsonSpacecraftAttitudeEstimateResourceSharing.....181 6.6Summary....................................184 7Conclusions......................................185 7.1RevisitingtheResearchQuestionsandThesisStatements........185 7.2FutureWork...................................186 REFERENCES.......................................188 BIOGRAPHICALSKETCH................................199 8

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LISTOFTABLES Table page 1-1Resourcesharingtechnologiesforthespacecraftsubsystems.........21 2-1Severalgeneratingfunctionsandthevectorialparameterizationof SO ...52 2-2InversekinematicsJacobianforseveralvectorialattitudeparameterizations..54 2-3Propertiescomparisonofseveralattitudeparameterizations...........54 3-1AssumptionsforapplicationoftheKalmanlter..................80 3-2DiscreteKalmanlter................................84 3-3ListofmodicationstotheKalmanlterfortheextendedKalmanlter.....85 3-4InertialattitudeextendedKalmanlter.......................90 3-5Stateinitializationsforcoarsesensorinertialattitudesimulations........90 3-6Timeparametersforcoarsesensorinertialattitudesimulations.........90 3-7Stateestimateinitializationsandsensorcharacteristicsforcoarsesensorinertial attitudesimulations..................................91 3-8Stateinitializationsforneprecisioninertialattitudesimulations.........93 3-9Stateestimateinitializationsforneprecisioninertialattitudesimulations...93 3-10ExtendedKalmanlterforinertialattitudeestimationinadisaggregatedsystem103 4-1Resourcesharingdatafusionandsystemarchitecturetaxonomy........106 4-2Transformationsforvectorialattitudeparameterizationsandtheunitquaternion128 5-1Stateinitializationsfortaskingsimulations.....................157 5-2Controlandtimeparametersfortaskingsimulations...............157 5-3Sensorcharacteristicsfortaskingsimulations...................158 5-4Stateestimateinitializationsfortaskingsimulations................158 5-5AggregatedTaskingPerformanceComparisonRelativeSensor10 FOV RMSErrorarcsec.................................161 5-6Taskingperformancecomparisonrelativesensorwith60 FOVRMSerror arcsec........................................162 6-1Sensorcharacteristicsusedinattitudeestimateresourcesharingsimulations.168 9

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LISTOFFIGURES Figure page 1-1Fractionatedelementstechnologyreadinesslevel................19 1-2Potentialbenetofthereductioninmassofasmallsatellite...........20 1-3IllustrationoftheOrbitalExpressrendezvousanddockingmission.......24 1-4Resourcesharingincommunicationsnetworks..................25 1-5IllustrationofSystemF6cluster-levelfaulttolerance...............26 1-6Fundamentalfunctionsfortheattituderesourcesharingprocess........28 1-7Attituderesource-sharingarchitecture.......................29 2-1Commoncoordinateframes.............................37 2-2Relativeattitudecompositions............................40 2-3Stereographicprojections..............................48 2-4Commonstereographicprojections.........................49 2-5Trackingcoordinateframes.............................58 3-1Descriptionofatypicalstartracker.........................71 3-2Descriptionofthegeometryoffocal-planemeasurementsusingavirtualCCD74 3-3MagneticeldandsunvectorcollinearityforanexampleinertialattitudeEKF withamagnetometerandsunsensors.......................91 3-4TruebodyangularvelocityforanexampleinertialattitudeEKFwithamagnetometer andsunsensors...................................92 3-5QuaternionerrorforanexampleinertialattitudeEKFwithamagnetometer andsunsensors...................................92 3-6AvailablestarvectorsforanexampleinertialattitudeEKFwithastartracker..93 3-7QuaternionerrorforanexampleinertialattitudeEKFwithastartracker....94 3-8NotionaldepictionofaVISNAVmeasurement...................95 3-9DescriptionoftheVISNAVsensorgeometry....................96 4-1Stargraphrepresentationforattituderesourcesharing..............106 4-2Notionalcovarianceintersectionuncertaintyellipsecomparison.........113 10

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4-3Notionalellipsoidalintersectionuncertaintyellipsecomparison.........115 4-4Linearityofseveralordersofthehigher-orderRodriguesparameters......124 4-5AttitudeerrorvectorapproximationaccuracyfortheMRPsandFRPs......126 4-6AttitudeerrorvectorapproximationaccuracyfortheMRPsandFRPsusing thelocalerrorrepresentation............................133 4-7AttitudeerrorvectorapproximationaccuracywiththeCRPsusingthelocal errorrepresentation.................................134 4-8Justicationfordatafusionatthequaternionunitsphereusingtheshadow parameterizations..................................136 4-9ExampletwostartrackerdatafusionusingtheFRPsandshadowparameters.139 4-10Pointingerrorcomparisonoftwostartrackerswithandwithoutdatafusion...140 4-11ComparisonofstartrackerdatafusionwithMRPshadowparameterandlocal errorrepresentation.................................140 5-1Stochastictaskinggreedycontrolproblem.....................146 5-2Resourcesharingprocess..............................147 5-3RelativepositionintheHillframe..........................159 5-4RMSerrortaskingperformancecomparisonrelativesensor10 FOV.....160 5-5RMSerrortaskingperformancecomparisonrelativesensor60 FOV.....163 5-6MonteCarloattitudeerrortrajectoriesfora10 FOVrelativeattitudesensor usingthetracemetric................................164 5-7MonteCarlosampledatacomparisonoftheRMSerror.............165 6-1Angularvelocityresultingfromtorque-freemotionforthedeputiesinallsimulated scenarios.......................................169 6-2Localandfusedquaternionerrorforthetwospacecraftstartrackerscenario.170 6-3Chiefattitudecontrolforthetwospacecraftstartrackerscenario........171 6-4Covarianceintersectionweightsforthetwospacecraftstartrackerscenario..171 6-5Twospacecraftlocalandfusedquaternionerrorwithcoarsedeputyattitude sensing........................................172 6-6Chiefattitudecontrolforthetwospacecraftcoarsedeputyattitudesensing scenario........................................173 11

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6-7Covarianceintersectionweightsforthetwospacecraftcoarsedeputyattitude sensingscenario...................................173 6-8Effectsobservedattheonsetofdatafusionwithpreciseandcoarsesensors.174 6-9Localandfusedquaternionerrorforthethreespacecraftstartrackerscenario176 6-10Chiefattitudecontrolforthethreespacecraftstartrackerscenario.......177 6-11Covarianceintersectionweightsforthethreespacecraftstartrackerscenario.177 6-12Localandfusedquaternionerrorforthethreespacecraftscenariowithcoarse Deputy-1sensing...................................178 6-13ChiefattitudecontrolforthethreespacecraftscenariowithcoarseDeputy-1 sensing........................................179 6-14Covarianceintersectionweightsforthethreespacecraftscenariowithcoarse Deputy-1sensing...................................179 6-15Affectoftaskingalgorithmchoiceontheerrorboundsofathreespacecraft scenariowithneprecisionsensing........................181 6-16Affectoftaskingalgorithmchoiceontheerrorboundsofathreespacecraft scenariowithcoarseDeputy-2attitudesensing..................182 6-17EffectoftaskingontheChiefboresightattitudeaccuracy............183 6-18ChiefattitudecontrolforthethreespacecraftscenariowithcoarseDeputy-1 sensingusingRound-robintasking.........................183 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SPACECRAFTATTITUDERESOURCESHARING By ShawnC.Johnson August2013 Chair:NormanG.Fitz-Coy Major:AerospaceEngineering Disaggregatedspacecraftsystemsdistributethefunctionalityoftraditional monolithicspacecraftacrossseveralplatforms.Thisdissertationaddressessomeof thetechnicalchallengesassociatedwithadisaggregatedattitudedeterminationsystem ADS,wherethedisaggregatedADSisdenedbyachiefspacecraft,withrelative andinertialattitudesensors,andmultipledeputyspacecraft,withonlyinertialattitude sensorsornosensorsatall. Atrackingcontrollerandreferencesignalkinematicsaredevelopedforthechiefto trackthedeputiesandmeasuretheirrelativeattitudestates.Thesemeasurementsare usedintheformulationofadisaggregatedimplementationoftheextendedKalmanlter EKFforinertialattitudeestimation.ThedisaggregatedEKFproducesmultipleinertial attitudestateestimatesofthechiefanddeputies.Euclideanstateandattitudestatedata fusionarereviewedandusedtodevelopageneralizedattitudedatafusionlawbased ontheattitudeerrorvector.Toavoidnumericalcomplexitiesassociatedwithconstrained attitudeparameterizations,minimalvectorialattitudeparameterizationsareinvestigated fordatafusion.Theaccuracyofthevectorialparameterizationsisassessed,anditis shownthattheclassicalRodriguesparametersareasufcientparameterizationfor minimalattitudedatafusion,butthathigher-orderRodriguesparameterscanbeused toimprovetheaccuracy.Theminimalparameterizationsareappliedtodatafusionlaws 13

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usingtheshadowparametersofthevectorialparameterizations,aswellasalocalerror representation. Agreedytaskingalgorithmisalsodeveloped.Thisalgorithmmakesuseofthe currentcovarianceinformationofthedeputiestotaskthechieftotrackdeputiesand measuretheirrelativestates.Characterizationsofthesizeofthestate-errorcovariance matrixareinvestigatedinsimulationbasedonthetrace,matrixnorms,anddifferential entropy.Simulationresultsshowthatformeasurementsharing,greedytaskingcan improveupontheaccuracyofthechiefanddeputiesusingaxedtaskingstrategy. Furthermore,MonteCarloresultsshowthatthegreedytaskingalgorithmismostly insensitivetothecovariancemetric. Thismanuscriptconcludeswithnumericalsimulationsoftwoattituderesource sharingscenarios:ibetweenachiefandasingledeputyandiibetweenachiefand twodeputies.Thesesimulationsdemonstratethatattituderesourcesharingcanbeused toincreasetheattitudeknowledgeofadeputyequippedwithlowperformanceattitude sensorsandalsoimprovetheaccuracyoftheattitudeknowledgeofallspacecraft includingthechiefwhenallspacecraftareequippedwithsimilarsensors. 14

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CHAPTER1 INTRODUCTION Spacecraftdesignisprimarilydrivenbythetime,cost,andrisktolaunchmass toorbitandthecostofdevelopingsophisticatedspacecraftandpayloadsthatmust surviveandoperatewithintheharshorbitalenvironment.In2002,thiscostaveraged between$5000/lbto$10000/lbforlaunchtolowEarthorbitLEO[1].Duetothese factors,spacecraftcancostupwardsofseveralbillionsofdollarsandtakeyearsto develop.Asaresult,traditionalmonolithicspacecraftaredesignedtoberiskaverse. Forthepurposeofthisdiscussion,riskisdenedastheproductoftheprobabilityofa failureoccurringandthelossinvalueifafailureoccurs.Sincethecostofmonolithic spacecraftisimplicitlyhigh,theyareconservativelydesignedinordertominimizethe probabilityoffailure.Reducingtheprobabilityoffailureisaccomplishedbyincorporating redundanciesandover-engineeringatthesystemandsubsystemlevel.Additionally, thecostandtimenecessarytodeveloptraditionalspacecraftcanbeprohibitivefor responsivemissions,wheretoobjectiveistorespondtoachangingmissiondemand asfastaspossible.However,recentresearchanddevelopmentofsmallspacecraft technologyanddisaggregatedspacearchitecturesareaddressingthechallenges oftimelyandcost-effectivemissiondevelopment.Thesesmallerspacecraftcannot replaceallofthecapabilityofmonolithicspacecraft,buttheydohavethepotentialto complementtheirlargercounterpartsandprovidenewcapability. Smallspacecraftdenitionsvarybasedonsize,weight,andpowerSWaPaswell ascost[2].Inthecontextofthispaper,smallspacecraftareclassiedbasedonmass. Massisagoodsurrogatedescriptionbecauseitinherentlydrivesthespacecraftform factor,availablepower,andcost.Smallspacecrafttypicallyvaryfrommini-satellites < 500kgdowntopico-satellites < 1kg.However,recentresearchisfurtherpushing massboundariesintothefemto-satelliteregime < 100gsometimesreferredtoasa satellite-on-a-chip[3],aswellasallthewaydowntospacecraftwithdimensionsas 15

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neasinterplanetarydustandmassontheorderofmilligrams[4].Allclassesofsmall spacecraftarebeinginvestigatedfortheirpotentialforshorterdevelopmenttimesand improvedcapabilityovertraditionalspacecraftforcertainmissiontypescommonly, distributedsensing. Complementingsmallspacecrafttechnologyistheadventofdisaggregated spacecraftarchitectures.Thesearchitecturesmakeuseofseveralsmallerspacecraftto produceacapabilitytraditionallyprovidedbyamonolithicspacecraft.Thedisaggregated spacecraftisanevolutionofthefractionatedspacecraft,whichspatiallydistributes someorallofthespacecraftbuselementsthatarephysicallyconnectedwithina traditionalmonolithicspacecraftinordertoachieveanewcapability.Thedifferencein disaggregationandfractionationissubtle,inthattheindividualnodesinadisaggregated architecturecontainallofthespacecraftbuselements,butinsteaddistributethe spacecrafttoachieveacapability.Whereas,thefractionatedspacecraftspatially distributesatraditionalspacecraft'sbuselements.Althoughthisdifferenceexists,many ofthebenetsoffractionatedspacecraftareequivalenttothosefordisaggregated spacecraft.Itisimportanttonotethatthedisaggregatedconceptisnewerthanthe fractionatedconcept,andthusthereislessinformationavailablethanforfractionated systems.Thefractionatedliteratureisreviewedbelow,butisintroducedwiththe intentiontomotivateadisaggregatedapproach.Assuch,fractionatedanddisaggregated spacecraftwillbeusedinterchangeablyfortheremainderofthiswork. Therstbenettobestudiedforfractionatedspacearchitectures,andthus disaggregatedarchitectures,wasexibility[5].Flexibilityisdenedastheability ofthesystemtobemodiedtodojobsnotoriginallyincludedintherequirements denitionandtheconclusionofthatstudywasthatifexibilityisvaluedhighenough, thenafractionatedsystemmaybepreferredoveratraditionalmonolithicsystem. Otherpropertiesthatdemonstratethebenetsoffractionatedarchitecturesare responsiveness,maintainability,andscalability.Responsivenessistheabilityto 16

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meetchangingrequirementsorconditionsquickly,whichcorrespondsintheframework toashort-ormedium-termadaptationtoanychangeinrequirementsorconditions, maintainabilityistheabilityofasystemtobekeptinanappropriateoperating condition,andscalabilityistheabilityofasystemtomaintainitsperformance andfunction,andretainallitsdesiredpropertieswhenitsscaleisincreasedgreatly withouthavingacorrespondingincreaseinthesystem'scomplexity[6].However,the conclusionofMathieu'sthesiswasthatindustryhadnoincentivetopursuefractionated spacearchitectures.Hence,therstcustomerwouldhavetobethegovernment. Subsequently,in2007theDefenseAdvancedResearchProjectsAgencyDARPA beganinvestigatingfractionatedspacearchitectures.AtDARPA,Brown[7,8]used avalue-centricdesignmethodologytoquantitativelydemonstratethebenetsof fractionatedsystemsforrobustnesstosubsystemfailures,andresponsivenessand adaptabilitytoevolvingmissionrequirements.However,todate,zerofractionated spacecraftmissionshaveownandseveralchallengesexistinthedevelopment ofthenecessarycapabilitybeforetherstmissionscanbeown.Morerecently, disaggregatedspacecrafthavebeenunderinvestigationbytheUnitedStatesAirForce withallofthesepotentialbenetsinmindforthespace-acquisitionprocess[9]. Disaggregatedspacecraftsystems,likefractionatedspacecraft,relyonresource sharing,inter-satellitenetworkprotocolsandcommunication,clusterying,and distributedcomputing[10].Inthiscontext,aresourceisdenedasanyelementor capabilitythatcanbetransferredfromonespacecrafttoanother.Thepossiblesharable resourcescorresponddirectlywiththespacecraft'ssubsystemsandpayload.The relevantsubsystemsinclude,powergenerationanddistribution,attitudedetermination andcontrol,orbitdeterminationandcontrol,communications,thermalmanagement, andcommandanddatahandling.Missionspecicpayloadresourceexamplesinclude, observationalsensorsandhigh-bandwidthdownlinkcommunications.Resourcesharing istheactoftransferringresourcesbetweenspacecraftandisthefocusofthisresearch. 17

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1.1ResourceSharing Resourcesharingcreatesseveralnewchallengeswithrespecttotraditional self-containedspacesystems.Specically,monolithicspaceassetscontainsubsystems thatarephysicallyconnectedandthatcanperformallnecessaryfunctionsdirectly on-board.Therefore,anoptimizedsetofhardwareandsoftwareareon-boardtomeet themissionrequirements.Forexample,amissionthatneedsprecisionpointingwillbe metwithaspacecraftdesignthathasanon-boardsensorsuitetodetermineattitude andactuatorsforcontrollingattitude.However,whenresourcesharingisrequired, eitherbyhardwareandsoftwarefailureorbydesign,thespacecraftmuststillbeable togeneratethenecessaryresources,exchangethoseresourcesandcoordinatethe actionsofthedisaggregatedcapabilitytorespondtomissionrequirements.These functionalrequirementsmotivatetheneedforhardwarethatcanfacilitatethesharingof resourcesandalgorithmsthatcanefcientlycoordinatethespacecrafttoallocateand distributetheresourcestoaccomplishtherequiredtasking. Guoconductedasurveyofthecurrentstate-of-the-artinfractionatedspace resourcesharingforseveralofthebuselements[11].Thissurveyfocusedonarchitectures, networking,wirelesscommunication,distributedcomputing,andwirelesspowertransfer. Thefractionatedarchitectureformissionplanningandsystemcapabilityhasbeen analyzedusingvalue-centricdesigntools[8]aswellasagent-basedsystemstheory [12].ThetechnologyreadinesslevelTRLformanyofthebuselementsareshownin Figure1-1A.Space-basedwirelesscommunicationandnetworkinghavethehighest TRL,whichisintuitiveasthesetwoelementsarethemostfundamentaltofractionated spacecraft,becausetheyarerequiredforeveryothercapability.Intra-spacecraft wirelesscommunication,takingplacebetweencomponentswithinaspacecraft,has beendemonstratedonorbitwiththeSpaceShuttleandInternationalSpaceStation atbandwidthsupto900MHz.Inter-spacecraftwirelesscommunication,takingplace betweenspacecraftinanetwork,hasbeendemonstratedwithradiofrequencyRFas 18

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wellasLASER.LASERhastheadvantageofbeinglowerpowerandhighbandwidth, andin2008wasusedontheTerraSAR-Xspacecrafttotransmitatadata-rateof 5.6Gbps.Thecaseforwirelesspowertransferforfractionatedspacesystemswas presentedbyTurner[13].Sincethen,thetechnologyforwirelesspowertransferhas beenunderdevelopmentandtheefciencyofthesesystems,specicallythrough microwavebeaming,isprovingtheviabilityofpowerresourcesharing[14].However, wirelesspowertransferhasonlybeendemonstratedontheground. A B Figure1-1.Fractionatedelementstechnologyreadinesslevel.ATRLradargram adaptedfrom[11].BNASAtechnologyreadinesslevel[15]. SomespacecraftbuselementswereabsentfromGuo'ssurvey,astheyareless mature.Forexample,actuationisalsoaviableresourcetoshare.Inthepastdecade, activemagneticcontrolhasbeenunderinvestigationforformationyingmissionsusing Coulombforces[16]andactivemagnets[17,18],andhashasalsobeenstudiedinthe contextofresourcesharing[19].Thistechnologyisstillatthetheoreticallevel. Thelastbuselementofinterest,andthefocusofthisresearch,isthesharingof sensormeasurementsfordistributedattitudedeterminationandcontrolsystems.O'Neill performedhighlevelanalysisontheeffectoffractionationonpointing-intensivemissions 19

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toshowthatdecouplingpointing-intensivesubsystemsfromtheothersubsystemscould reducemassandcost[20].Thisreductioninmasscanalsopotentiallyimprovethe attitudemaneuverabilityofaspacecraftasshowninFigure1-2.Thisgureshowsan examplespacecraftwithaxedactuatorthatistypicalinsizeforanano-satellite.Asthe sensor-to-vehiclemassfractionisreduced,theslewratecanpotentiallybeincreased whileconsumingthesamepower.Duetophysicalconstraintsimposedbytheoptical benchofmanyattitudesensors,themassfractionofthesesensorsisgreaterforsmaller spacecraftthanfortraditionalmonolithicspacecraft.Thatis,foramonolithicspacecrafta startrackerisrelativelysmall,howeverforasmallsatellite,suchasa1UCubeSat,astar trackercantakeupconsiderablemassandvolume[21].Asimilarincreaseinslewing performancecouldbeseenbyincreasingthesizeoftheactuators,butattheexpense ofincreasedinertia.Thepointremainsthattheremaybebenetfromanattitudecontrol perspective.However,thetheoryandimplementationforresourcesharingarestillata lowTRL.Asummaryoftheresourcesharingelementswithkeytechnologiesisprovided inTable1-1. Figure1-2.Potentialbenetofthereductioninmassofasmallsatellite Terminologyforattitudeisnowintroduced.Thetermsinertial,absolute, relative,external,andinternalareoftenassociatedwithattitude.Inthecontext ofthisresearch, inertialattitude referstotheorientationofaspacecraftwithrespectto 20

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Table1-1.Resourcesharingtechnologiesforthespacecraftsubsystems SubsystemTechnologiesFunctions Communicationswireless-link,networkingtransmittingandreceivingdata ElectronicPowerSystemmicrowavebeamingtransmittingandreceivingpower AttitudeDeterminationrelativeattitudesensorsattitudeestimationanddatafusion Propulsioncoulombcontrolorbitalmaneuvering CommandandDataHandlingparallelcomputing,client-serverinformationprocessing aninertialreference.Sensorsthatprovideinformationaboutinertialattitudearereferred toasinertialattitudesensors,inwhichexamplesincludesunsensors,magnetometers, andstartrackers. Relativeattitude referstotheorientationofaspacecraftwith respecttoanotherspacecraftornon-inertialreference.Similarly,sensorsthatprovide informationaboutrelativeattitudearereferredtoasrelativeattitudesensors,inwhich examplesincluderadiofrequency,vision-basedsensors,andlasercommunication.One lastsensorthatdeservesattentionistherategyroscope,whichisaninertialsensorthat measurestheangularvelocityinbodycoordinates. Resourcescanalsobedifferentiatedbytheirniteness.Aniteresourceisdened asanyresourcethatwhentransferredischangedonboththesharingandreceiving spacecraft.Examplesofniteresourcesincludepower,communicationsdownlink,and positionandattitudeactuation.Studieshaveexaminedniteresourcesinfractionated spacecraftfromanagent-basedperspective[22]andappliedgametheory[12]tothe resourceallocationproblem.Ontheotherhand,anon-niteresourceisdenedasany resourcethatwhentransferredtoareceivingspacecraftremainsunchangedonthe sharingspacecraft.Examplesofnon-niteresourcesincludesensormeasurementsor otherdata.Itisimportanttonotethatthetransferofnon-niteresourcesmayrequire theutilizationofniteresources,suchaspowerorfuel.Anillustrativeexampleis resourcesharingofsensormeasurements.Dependingonthehardwareutilized,sensor measurementsmayrequirechangesinattitudeorpositionrelyingonpowerand/or propellantaswellasinter-satellitecommunications,whichconsumepower. Attitudesensormeasurementsareanon-niteresourcethatarenecessaryto determineandcontrolthespacecraft'sattitude.Ithasbeenshownthatindisaggregated 21

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systemsthereisaconuenceofnavigation,communications,andcontrolforposition control[23].Thissameconuencenaturallyextendstodisaggregatedattitude determinationsystemsADSs.Additionally,Blackmoreprovedthatobservabilityin adistributedattitudedeterminationsystemrequiresarelativesensorpathfromthe spacecraftbacktoanabsolutesensor[24].Therefore,inordertoshareinertialattitude, relativeattitudesensorsarenecessarytocreatetherelativestatepathfromaninertial attitudesensor.Missionswithrelevancytorelativestatesensingandresourcesharing aredetailedinthefollowingsection. 1.2RelevantMissions Therearetwopredominanthardwaretechnologiesthatmustexisttoenableon-orbit attituderesourcesharing.Therstisaninter-satellitecommunicationslink.This communicationslinkenablescoordinationanddata-sharingwithinthenetwork.The secondenablinghardwareisrelativestatesensors.Relativestatesensorshaveheritage inmissionsrangingfromrendezvousanddockingtospace-basedinterferometry,and althoughthesemissiontypesdonotrequireresourcesharing,theyprovidevaluable insightintotheresourcesharingmechanism.Relevantmissionsforrelativestate sensingtechnologiesinightdemonstration,aswellasproposedfractionatedsystems, arediscussedinSubsections1.2.11.2.4 1.2.1RendezvousandDockingMissions RendezvousanddockingRNDinvolvesthefarandcloseapproachfromone spacecrafttoanother,aswellasthephysicallinkingofthetwospacecraft.RND missionsrelyheavilyonthedeterminationofrelativeattitudeandpositionstatesfor theapproachtrajectoryanddockingpose[25].ThesensorsutilizedforRNDmissions areonlyapplicabletoproximityoperations.Forexample,atlargerdistancesradio frequencyRFsensorsaretoonoisytobeutilizedforhighaccuracyattitudeknowledge. Similarly,vision-basedsensorssufferatlargedistances,becausetheincidentlightwill coalesceandtheobjectwillappearasasinglepoint. 22

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Earlysystems,suchastheRussian Kurs ,utilizedRFsignalsinatarget-chaser congurationtoperformRND[26].Morerecentmissionshaveusedvisiblespectrum signalsmeasurementsforrelativestatesensing.The SpacecraftfortheUniversalModicationofOrbits/Front-endRoboticsEnablingNear-termDemonstration SUMO/FREND wasatechnologydemonstrationspacecraftdesignedtoserviceor modifytheorbitsofotherspacecraft.The FREND packageutilizedcomputervision andposeestimationtechniquesforautonomousRND[27]. XSS-10 [28]and XSS11 [29]werespacecraftdesignedtoperformresidentspaceobjectRSOinspection andrendezvousmaneuversusingrelativenavigationsensors. XSS-10 utilizedthe visiblecamerasystemVCSforrelativestatesensing. XSS-11 combinedanactive LIDARsensorwithavisiblecameraandstartrackersystem.The DART spacecraftwas designedtousetheadvancedvideoguidancesensorAVGSforRND.TheAVGSwas designedtotrackretro-reectorducialsonanotherspacecraftforrelativestatesensing, butsuffereda`'soft-collision`'thatresultedinadepletionofallofitspropellantandwas unabletodemonstratethemissionobjectives[30]. OrbitalExpress wasthefollow-onto DART andconsistedoftwospacecraft,ASTROandNextSat.Thespacecraftutilized theAVGSandsuccessfullydemonstratedRNDmaneuvers[26].The Autonomous TransferVehicleATV wasaEuropeanSpaceAgencyESAdesignedspacecraftthat usedvideometerandtelegoniometerasrelativestatesensorstodockandresupplythe InternationalSpaceStationISS[31].The formationautonomyspacecraftwiththrust, relnav,attitude,andcrosslinkFASTRAC usedrelativeglobalpositionsignalGPS measurementsforrelativenavigation[32]. Severalspacecrafthavedemonstratedrelativeattitudeandpositionmeasurements throughavarietyofsensorsincloseproximity.Thesemissionshavehelpedpave thewayfordisaggregatedattitudedeterminationsystems,becausetheyprovidethe requisiterelativestatesensingpath.However,thesetechnologiesmustbematuredor extendedtoprovidethenecessaryrelativestateprecisiontobesuitableforresource 23

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sharingmissions.Itisassumedthatthistechnologywillcontinuetomatureand ultimatelycomplementthisresearchforpracticalimplementation. Figure1-3.IllustrationoftheOrbitalExpressrendezvousanddockingmission[33] 1.2.2Space-basedInterferometry The StarLight missionisaproximityformationyinginterferometrymissionthat makesuseoftheAutonomousFormationFlyingAFFsensor[34].Similartothe RussianKursproximitysensor,theAFFutilizesRFsignalstomeasurerelativestates tomoderateprecision.AlasermetrologysystemtakesoverfortheAFFtoyieldthe precisionnecessaryforinterferometry. Theproposed LaserInterferometerSpaceAntennaLISA missionusesa distributedspacecraftplatformtocreatealargeapertureantennaforsensinggravitational wavesatlowfrequencies[35].Specically,threeidenticalspacecraftforma5million km equilateraltriangleformationwithstringentrequirementsonrelativepositionand attitude.Duetothelargeastronomicaldistancesinvolved,LISArequiresinter-satellite communicationwiththeDeepSpaceNetworkDSNandpreciserelativestate knowledgeandcontrolbetweentheinterferometerplatforms.Forthismission,star trackersarenotaccurateenoughtomaintainrelativeattitudetotherequiredprecision. LISAutilizesquadlaserdetectorsfordirectionsensingtotheotherspacecraftinthe 24

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formation.Thislaserisusedasatrackinginputforthecontrollerinordertomaintain preciserelativeattitude. 1.2.3SatelliteNetworks Existingsatellitenetworkshavealreadydemonstratedinter-satellitenetworking andcommunication.Thesecapabilitiesarethemostfundamentaltodisaggregated spacecraftarchitectures,asallotherdisaggregatedcapabilitiesrelyonthem.Two examplesofresourcesharingincommunicationsnetworksarethe TrackingandData RelaySatelliteSystemTDRSS [36]showninFig.1-4Aand Iridiumconstellation [37]showninFig.1-4B.Thesesystemsweredesignedtorelaydataviainter-link communicationstootherspacecraftandthegroundandhaveestablishedheritagein wirelesslyconnectedsatelliteconstellationsandbasicdatasharing. A B Figure1-4.Resourcesharingincommunicationsnetworks.AIllustrationoftheTDRSS constellation[38].BIllustrationoftheIridiumconstellation[39] 1.2.4SystemF6 AfractionatedmissionthatwaspursuedbytheDefenseAdvancedResearch ProjectsAgencyDARPAisSystemF6Future,Fast,Flexible,Fractionated,Free-Flying Spacecraft.Thevalue-caseforfractionatedspacearchitectureswasmadebyBrown in2006[10].Subsequently,in2007,DARPAbeganpursuingfractionatedspace architecturesthroughSystemF6.SystemF6isaninstantiationofthefractionated spacecraftarchitectureparadigmshiftproposedbyBrown,withthegoaltoprovide 25

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evidenceofthevalueoffractionationthroughightdemonstrationofmultiplefractionated capabilities.Ofparticularinteresttothisresearch,istheideaofafractionatedattitude determinationsystem.Fractionatedattitudehasbeensuggestedasamechanism forincreasedrobustnesswithoutrequiringsignicantadditioninhardware.Shown inFigure1-5,fractionatedattitudewasidentiedasademomissionforSystemF6 faulttoleranceattheclusterlevel,whereifastartrackerfailsinthenetworkitmaybe replacedthroughresourcesharing[40].AspartoftheDARPASystemF6project, Figure1-5.IllustrationofSystemF6cluster-levelfaulttolerance[40] OrbitalSciencesCorporationdevelopedPleiades[41],whichwasapilotstudyfora notionalfractionatedspacecraftsystem.Thesystemarchitectureforthismissionwas aLEOEarth-observationsciencemissionwithdistributedimagers.Thesystemwas fractionatedatmultiplelevelsincludingtheimager,communicationsdownlink,data storage,anddataprocessing. 1.3ProblemStatements Resourcesharinghasbeenidentiedasacriticalcapabilitynecessaryfor disaggregatedspacecraftsystems.Atahighlevel,thisisalocalizationproblem,as studiedinthesimultaneouslocalizationandmappingliterature[42],andmorerecently 26

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usingrelativeattitudelocalization[43,44].Theobjectiveofanyattitudedetermination systemistoestimatetheattitudeofthespacecraftasaccuratelyaspossible.Asa result,anaturalattitude-basedobjectiveforthedisaggregatedsystemistominimizethe uncertaintyintheattitudeestimation,summedacrossthenetwork.Thatis, min n X i =1 Z T 0 k e i k dt where e i istheattitudestateerrorforthe i th spacecraft, n isthetotalnumberof spacecraft,and T isthetime-horizonofinterest. TheresourcesharingprocessissummarizedinFigure1-6,wherethecontrol ofinformationissought.Specically,informationisgeneratedthroughsensingand incorporatedtothestateknowledgethroughestimation.Allinformationsourcesare thencombinedtoimproveknowledgeacrossthedisaggregatedsystem.Finally,thenext informationsourceischosen.Thisprocessreliesonfourkeyfunctions,including: 1.Inter-satellitecommunication 2.Estimationofinertialattitudeusingrelativeattitudesensormeasurements 3.Fusionofmultipleinformationsources 4.Decisionalgorithmsfordistributionofattituderesources Ofthesefourchallenges,inter-satellitecommunicationisassumedtoexist.Thelast threeareaddressedinthisworksee:Section1.4. Twodenitionsfortheattituderesourceareexploredinthisresearch: 1.Attitudemeasurementsoriginatingfromthechief'ssensors,whichareshared directlywiththedeputies 2.Processedattitudeestimates,whicharesharedbetweenthechiefanddeputies Whenprocessedestimatesareshared,datafusionisthetoolusedforcombining multiplesourcesofinformation. ConsiderthegeneralspacecraftnetworkshowninFigure1-7,wherethechief representsanodewitharelativeattitudesensorandthedeputiesarenodesinthe 27

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Figure1-6.Fundamentalfunctionsfortheattituderesourcesharingprocess networkwithoutrelativeattitudesensing,andpotentiallywithoutattitudesensing, altogether.Thisnetworkcaneitherdescribeaformation-yingsystem,wherethe relativepositionandattitudeofthespacecraftaretobecontrolled,oraconstellation, wherethespacecrafthavesomecommonobjective,butdonotrelyonmaintainingtheir relativestates.Thechallengesinestimation,datafusion,andsensortaskingforattitude resourcesharingarenowfocusedintothreeproblemstatementswithrelatedresearch questions. 1.3.1Problem1:InertialAttitudeEstimationinaDisaggregatedSystem Relativeattitudesensorsareusedtodirectlyestimatetherelativeattitudebetween achiefanditsdeputies[45,46].Theinertialattitudeofadeputycanbeproducedby localinertialsensors.Alternatively,thecompositionofrelativeattitudeofthedeputy withrespecttothechiefandinertialattitudeofthechiefcanbeusedtogeneratenew knowledgeaboutthedeputiesinertialattitude.Takingthisintoconsideration,arelevant researchquestionis,howcaninertialattitudebeestimatedinadisaggregatedsystem throughtheuseofrelativeandinertialattitudesensors? 28

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Figure1-7.Attituderesource-sharingarchitecture 1.3.2Problem2:AttitudeDataFusion Informationtransferbetweenthechiefanddeputiesisaccomplishedviathe wirelesslinkinFigure1-7.Attituderesourcesharingrequiresthatattitudemeasurements orestimatebesharedbetweenspacecraftthroughthislink.Inthecaseofshared attitudeestimates,multiplesourcesofthesameinformationaregenerated,asrelative andinertialattitudecanbecomposedtodescribetheinertialattitudeofanother spacecraft.Datafusionseeksanoptimalcombinationofinformationtoproducea bestestimatewheninformationdependencyisuncertain,asisthecasewithcovariance intersection[47,48],ordependencyisknown,asinthecaseofcorrelatedtrackfusion [49].Togenerateafusedestimate,datafusionalgorithmsoptimizethecombinationof multi-sensorinformationbyminimizingacostfunction,whichisaproblemdependent 29

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measureoftheerrorbetweenthestateestimatesandthefusedstate.Fortypical applications,thefusionerrorisameasureoftheEuclideandistance.Howeverattitude belongstothespaceofspecialorthogonalmatrices, SO ,whichdescribestherelative orientationbetweentwocoordinatesystems,andthusaEuclideandistanceerroris inappropriateforattitude.Therefore,theproblemoffusingmultipleattitudeinformation sourcesrequiresanalternativedenitionofthefusionerror.Tworesearchquestionsthat areaddressedinthisresearchare: 1.Howcanexistingdatafusionalgorithmsbeextendedtoattitudestates? 2.Whatisafastandaccurateattitudeparameterizationfordatafusion? 1.3.3Problem3:SensorTasking Thethirdobjectiveofthisresearchistodevelopalgorithmsfortaskingsensorsto allocateanddistributeattituderesourceswithinaspacecraftnetwork.Theprimaryfocus ofthisobjectiveistoinvestigatenetworksconsistingoffractionatedADSdistributed acrossachiefspacecraftandmultipledeputies.Toachievethebestperformance,for aparticularcostfunction,thesensorwillideallybetaskedfortheinnitehorizon.The resultingtaskingisthesolutiontoastochasticoptimalcontrolproblem. AwellknownstochasticoptimalcontrolproblemistheLinearQuadraticGaussian LQGcontroller.TheLQGcontrollerhasaclosed-formsolutionduetotheproperties ofseparationandcertaintyequivalence.Theseparationpropertystatesthatthe optimalcontrollerandestimatoraresolvedseparately[50,51].Thatis,thecriterion fordeterminingthecontrolisindependentoftheestimationandviceversa.Certainty equivalencestatesthatanoptimaldeterministiccontrollerisequivalenttothesame controllawreplacedbyoptimalestimatedstates[52].Forexample,underveryspecic assumptions,theLinearQuadraticGaussianLQGcontrollawsolvesthedeterministic LinearQuadraticRegulatorLQRandoptimalLinearQuadraticEstimatorLQE independentlyviatheircorrespondingRiccatiequations[53].However,fordynamic attitudesensortasking,wheretheobjectiveistominimizethepointingerrorinthe 30

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system,thecontrolleractionsarebasedonsensingobjectives.Thisleadstothedual effectwherecontrolnotonlyaffectsthestatesbutalsotheuncertaintyinthestates.This istheconverseoftypicalfeedbackcontrollawswheresensormeasurementsarebased oncontrollerobjectivesthatonlyaffectthestate.Controlforsensingisnontrivialand resultsinanabsenceofthecertaintyequivalencepropertyandalackinseparationof estimationandcontrol. Duetoalackinseparationandcontrol,optimalsolutionsarecomputationally expensiveanddonotscalewellwithlargernetworkscurseofdimensionality. Therefore,theresearchquestionforsensortaskingis,howcanasensorbetasked tominimizethetotalattitudeerrorinadisaggregatedattitudedeterminationsystem, whilealsobeingcomputationallyinexpensive,andscalingwellwithlargernetworks? 1.4ThesisStatements ToaddresstheresearchquestionspresentedinSection1.3,thefollowingdescribes thethesisstatementsforestimation,datafusion,andsensortaskingasappliedto attituderesourcesharing. 1.4.1Thesis1:OnInertialAttitudeEstimationforDisaggregatedSystems Giveninter-satellitecommunication,anextendedKalmanlterforestimatinginertial attitudefromrelativeandinertialattitudesensorsinadisaggregatedsystemcanbe utilizedwhilethelinearizationremainsvalid. 1.4.2Thesis2:OnMinimalAttitudeParameterizationDataFusion ExistingdatafusionmethodsonEuclideanstatespacescanbeextendedto theattitudestatespacesof SO usingtheparameterizationindependentattitude errorvector.Properchoiceofaminimalattitudeparameterizationcanleadtoan unconstrained,global,andnonsingulardatafusionprocess. 1.4.3Thesis3:OnSensorTaskingforAttitudeResourceDistribution Greedysensortaskingisrobustforattituderesourcedistributionindisaggregated spacecraftnetworks,asitavoidstheplanningprocessandtheissueswithseparationin 31

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estimationandcontrol.Theprimaryfactorstoconsiderforthegreedydecisionalgorithm aretherelativestateswithconstraintsandtheirrespectiveuncertainties. 1.5DissertationOrganization Thisdissertationisdividedintosevenchaptersandproceedswiththefollowing organization. Chapter2introducesthefundamentalspacecraftattitudeparameterizationsand kinematics.Thenattitudedynamicsandcontrolareintroducedinthecontextofthe trackingproblemtodistributeresources.Thischapterlaysthegroundworkforthe remainderofthedissertation. Chapter3summarizesexistingattitudesensorsandattitudedeterminationand estimationresultsforinertialandrelativeattitude.Thischapterconcludeswiththe developmentofaninertialattitudeestimatorforthechieftoestimatethedeputies inertialattitudestatesviatherelativeattitudesensor.Usingtheseresults,thechiefand deputiesproducemultipleestimatesofeachspacecraft'sinertialattitude.Thisleads tothenextchapteronefcientmethodsofcombiningmultiplesourcesofinformation datafusion. Chapter4reviewsexistingattitudedatafusiontechniquesandprovidesageneral frameworkforextendingthetheoryofdatafusionforEuclideanstatespacesto attitudestateson SO .Minimalattitudeparameterizations,namelythevectorialsets introducedinChapter2,aredevelopedtoprovideaglobalnonsingularunconstrained parameterizationforattitudedatafusion.Theaccuracyoftheminimalparameterizations fordatafusionarecompared.Thischapterconcludeswithanotionaltwo-star-tracker datafusionproblemtoprovidefurtherevidencefortherstthesisstatementonminimal attitudeparameterizationsfordatafusion. Chapter5introducestheproblemofsensortaskingforattituderesourcesharing. Agreedytaskingalgorithmforthecapturingofrelativestatemeasurementsfrom thechieftothedeputiesisdevelopedthataddressestheissuesofuncertainty 32

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minimization,computationaltractability,andarchitecturescalability.Thecontentofthis chaptersupportsthesecondthesisstatementonsensortaskingforattituderesource sharing.Thischapterconcludeswithafour-spacecraftsensortaskingexampletoverify theapproach. Chapter6presentsanexampleresourcesharingarchitecture,whichcombines theresultsofChapter4andChapter5.Particularly,twoandthreespacecraftformation yingnetworksaresimulated.Resourcesharingisinvestigatedforcaseswithcoarse andneprecisioninertialattitudesensorswithahighprecisionrelativelink.Datafusion isbasedonthefourth-orderRodriguesparametersandthesensortaskingalgorithmare applied.Resultsareprovidedthatindicatethevalidityoftheapproachandprovidethe nalevidenceforthethesisofthisdissertation. Chapter7drawsconclusionsfromtheprecedingchaptersandmakesrecommendations forfutureresearch. 33

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CHAPTER2 SYSTEMDYNAMICSANDCONTROL Attituderesourcesharinginadisaggregatedspacecraftsystemisbasedonthe rotationalmotionofeachspacecraftandalsoontheparameterizationusedtodescribe thatmotion.Resourcesharingalsoreliesonthecompositionofrelativeandinertial attitudemeasurements.Therefore,anantecedenttoresourcesharingandsensor taskingisafundamentalknowledgeofinertialandrelativeattitudedynamicsand attitudeparameterizations. 2.1GeneralNomenclature Scalars,vectors,andmatriceson R n areubiquitouslyusedthroughoutthiswork. Thenomenclatureforthesequantitiesisprovidedforclaritytothereader.Scalarsare writtenaslowercaseletterse.g., a .Vectorsandcolumnmatrixrepresentationsof vectorswillbewritteninlowercaseboldfonte.g., a .Generalmatricesarewrittenin upper-caseletterse.g., A .Finally,thequaternionsarewrittenasthelowercase, q 2.2RigidBodyKinematics Themotionofasinglespacecraftcanbetreatedasarigidbody,wherethemutual distancebetweentheparticlesmakingupthespacecraftareinvariablewithanite mass.Asinglerigidspacecrafthassixdegrees-of-freedomthreefortranslation andthreeforrotation.Notethatnon-rigidmotion,bymeansofexibleorarticulating members,createadditionaldegrees-of-freedom,butwillnotbeinvestigatedinthis research.FromChasle'stheorem,themostgeneraldisplacementofarigidbodyisthe translationalongalinecoupledwitharotationaboutthatline[54].Itisthuspertinentto denewhatismeantbytranslationandrotation. Translationalmotionoccurswhenthedisplacementofallparticlesintherigidbody followparallelpaths.Whereas,rotationalmotionoccurswhenthereisanonparallelpath followedbysomeparticlesintherigidbody.Therefore,atminimum,sixcoordinates 34

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arenecessarytodescribethemotionofaspacecraft,thatis,threecoordinatesforthe translationandthreecoordinatesfortherotation. Forattitudeapplicationstherearetwoconceptswhichhavemanydenitionsin theliterature,andthusacleardenitionisnecessaryinthecontextofthisdiscussion thatoftheframeofreference,andthecoordinateframe.Theframeofreference iswherealldistancesandtimearemeasuredfortheapplicationofNewtonianand Euleriandynamics.Assuch,theframeofreferenceisnon-acceleratingandsaidto beinertial.Inpractice,deninganexactframeofreferencewithphysicalsignicance formeasurementsisimpossible.Therefore,theframeofreferenceischosenthatis approximatelyinertialwithrespecttothetimehorizonofinterestandthedistances expected,suchastomaintainthenon-accelerationrequirement.Theseconddenition isinregardtothecoordinateframe.Acoordinateframeisasetofbasisvectorsdened byaplaneandadirectioninthatplane,wheretheseconddirectionisorthogonaltothe planeandthethirddirectionfollowsfromtheright-handrule;forminganorthonormal set. Thedenitionofattitudeisbasedintherepresentationofunitvectorsdenedby coordinateframes.Attitudeinvolvesthedescriptionoftherotationaldegrees-of-freedom ofarigidbodyandisdenedasthe relativeorientationbetweentwocoordinatefames, oneofwhichistypicallyxedtotherigidbody .Whendealingwithspacecraft,ofprimary concernistheorientationofacoordinateframexedtotherigidspacecraftwithrespect tosomeothercoordinateframe.Severalofthesecoordinateframesaredescribedinthe followingsection. 2.3CoordinateFrames Asattitudedescribestherelativeorientationbetweencoordinateframes,itis importanttounderstandthedenitionoftheframesthataretypicallyencountered inspacecraftapplications.Startingwithaninertialcoordinateframewherethe fundamentallawsofdynamicshold,thecoordinateframeswillbedenedinthe 35

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sequenceoriginatingwiththeinertialframeofreferenceandendingwiththebody, whosemotionisofinterest. 2.3.1Earth-CenteredInertialECI ForEarthorbitingapplications,acommoninertialframeistheECIframe.Theframe F I ,showninFigure2-1A,isdenedbytheorthonormalbasisvectors f X I Y I Z I g Thesedirectionsareassumedtobexed.However,duetothenutationoftheEarth, adirectionspeciedtoaninertialpointinspacesuchasastarwillaccelerateasthe originatingpointofthedirectionisxedtotheEarth.Torectifythisinvalidationofthe assumption,theJ2000frameconsidersthereferencedirectionsataspecicinstancein time.Specically,theJ2000frameisdenedbytheequatorialplane,inwhich X I liesin theplaneandpointsinthedirectionfromthecenteroftheEarthtotherstpointofAries onJanuary1,2000. 2.3.2Earth-CenteredEarth-FixedECF TheECFframeisxedtotheEarthandisusedtodescribetherotationofthe EarthwithrespecttotheECIframe,giventhattheEarthrotateswithanangularvelocity E = F = Z F ,where =7.292115 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 rad = s [55].TheECFframe F F ,shownin Figure2-1A,isdenedbytheorthonormalbasisvectors f X F Y F Z F g ,wherethe X F lies intheequatorialplaneismeasuredbytheEarthhourangleEHArelativeto X I 2.3.3HillLVLH TheHillframe F H ,showninFigure2-1B,isanexampleofalocal-vertical-local-horizontal LVLHframe.Itisthebasistypicallyusedforrelativetranslationalmotionofmultiple spacecraft.Itisdenedbytheorthonormalbasisvectors, X H = r k r k Z H = h k h k and Y H = Z H X H ,where r isthepositionvectorfromthecenteroftheearthtothecenterof massofthespacecraftand h istheangularmomentumvectoroftheorbit. 2.3.4PerifocalPQW ThePerifocalframe F P ,showninFigure2-1B,iscommonlyusedataninertial frameofreferenceforEarthorbitalapplications.Ithashasoriginattheoccupiedfocus 36

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oftheconicsectiondescribingtheorbitalmotionandisdenedbytheorthonormalbasis vectors P = e k e k W = h k h k ,and Q = W P ,where e istheeccentricityvector. 2.3.5BodyFrameB TheBodyframe F B ,isdenedbyorthonormalbasisvectors, f X B Y B Z B g .This frameisxedtothespacecraftandusedtodescribetheattitudeofthespacecraftwith respecttotheothercoordinateframesdescribedinthissection.Typically,theframe willoriginateatthecenterofmassofthespacecraft,whichisconvenientforattitude dynamics.Fortheremainderofthediscussiontofollow,astartrackerisassumedtobe alignedwith Z B andarelativeattitudesensorisalignedwith X B A B Figure2-1.Commoncoordinateframes.AInertialandEarth-xedframes.BHilland perifocalframes. 2.4AttitudeParameterizationsandKinematics Attitudecanbedenedasthemappingofvectorsinonebasistoanother,suchthat x 0 = R x where x 0 isacolumnmatrixoftheavectorrepresentedin F E 0 coordinates, x isthe columnmatrixofthatsamevectorrepresentedin F E coordinates,and R 2 SO R 3 3 mapsvectorsfrom F E to F E 0 .Theparameterizationof SO isaclassical problem.Sinceattitudedescribesthethreedegrees-of-freedomofthespacecraft's 37

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rotationalmotion,thereareaminimumofthreeparametersthatarenecessary todescribethespacecraft'sattitude.However,ingeneral,theattitudematrixthat describesthemappingbetweentwobasisisnon-Euclidean,belongingtothegroupof specialorthogonalmatricesdenotedby SO .Thereareanumberofnon-Euclidean representationsthatparameterizetheattitude.ThroughuseofBrouwer'stheoremon invarianceofdomain,Stuelpnagelshowedthatallminimalattitudeparameterizations containsingularities[56].Infact,itwasshownthatittakes,atminimum,veparameters to1-to-1globallyparameterize SO .Asolutiontotheexistenceofsingularitiesis touseredundantrepresentationsorensurethatthattheminimalparameterization singularityisoutsideoftheattituderegionofinterest. Sincethechoiceofattitudeparametersisnon-unique,manyattitudeparameterizations havebeendeveloped,eachwithspecicadvantagesanddisadvantages.Anoftencited surveyoftheattitudeparameterizationsbyShusteriscontainedinReference[57],which wascurrentupto1993.AsurveyofthosecontainedinShuster'spaper,aswellasa descriptionofmorerecentattitudeparameterizationsfoundbystereographicprojections, higherorderCayleytransformsisdescribedinReference[58].Finally,ageneralization ofaxis-anglederivedparameterized,termedthevectorialattitudeparameterizations, asdescribedinReference[59].Abriefsummaryoftheresultsfromtheliteratureis providedinSections2.4.12.4.8. Therearemanyimportantcharacteristicstoconsiderwhenchoosingaparameterization, suchasthepresenceofsingularities,numberofparametersandimposedconstraints, computationoftranscendentalfunctions,andkinematiclinearity.Ideally,aparameterization shouldbeglobally-singularityfree,minimalinnumberofparametersandconstraints, andlinearwithouttranscendentalfunctions.However,sinceallminimalparameterizations aresingular,eithersingularitieswillbeencounteredwithaniterotation,orredundant parametersmustbeutilized.Thechoiceofparameterizationiscriticalfortheeffectiveness ofattitudecontrol,estimation,anddatafusionalgorithms,asarediscussedinthiswork. 38

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Thefollowingsubsectionsreviewseveralattitudeparameterizationwiththeir rst-orderkinematicsandcompositionlaws.Additionally,thesecond-orderkinematics areprovidedfortheunitquaternionparameterizations,whichisnecessaryforthe attitudetrackingcontrollawthatenablestheonespacecrafttotrackanother. 2.4.1AttitudeMatrix Thefoundationofattitudeoriginateswithavector, x ,expressedintwodifferent coordinateframes.Eachcoordinateframeisdescribedbytheorthonormalbasis vectors,suchthat F E = e 1 e 2 e 3 and F E 0 = e 0 1 e 0 2 e 0 3 .Thevector, x ,representedin eachbasisis x = x 1 e 1 + x 2 e 2 + x 3 e 3 = x 0 1 e 0 1 + x 0 2 e 0 2 + x 0 3 e 0 3 Therefore,inordertotransformfrombasis F E to F E 0 ,thedirectioncosinesdenedby thecomponents R ij = e 0 i e j parameterizethematrix, R .Thecolumnmatrixrepresentationofthevector x in F E 0 ,as denotedby x 0 iscomputedthrough x 0 = R x Thedirectioncosinematrixisequivalentlytermedtheattitudematrixwhendiscussing thegeneralproperorthogonalmatrix[57].Mathematically,thismatrixisanorthogonal transformation,belongingtothegroup SO .Thistransformationispreservingofthe vector'slengthandorientation,thatis, RR T = I 3 3 where I 3 3 istheidentitymatrixofappropriatedimension.Basedonthisdention, R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = R T .Itisalsoproper,satisfyingdet R =+1.Therearenineparametersnecessaryto describetheattitudematrix,andhence, R containssixconstraints.Asaresult,there isalargeredundancyin R ,whichcomesatthecostofhighercomputationalburden. 39

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However,theredundancyin R hastheadvantageofformingagloballyuniqueand singularity-freeattitudeparameterizationthatcanbeusedforcontinuoussingularity-free controllaws[60]. Thecompositionoftwosequentialrotations R 1 followedby R 2 is, R = R 2 R 1 Rotationsdonotcommute.Asaconvention, inertialattitude istermedtodescribethe orientationbetweenarigidbodyandaninertialcoordinatesystem.Whereas, relative attitude istermedtodescribetheorientationbetweentwonon-inertialcoordinate frames.Therelativeattitudebetweentworigidbodiescanbedescribedthroughthe aboveattitudecompositionlawasshowninFigure2-2. Therotations A and B representtheinertialattitudeof F A and F B withrespect totheinertialreference, F I ,respectively.Observingthedirectionoftherotations,the relativeattitudefrom F B to F A ,isdenotedby C andisequivalentto A T B ,asseenin Figure2-2A.Whereas,Figure2-2Bdepictstherelativeattitudefrom F A to F B ,and isequivalentto C T = B T A .Therefore,therelativeattitudeisdenedthroughattitude compositions. A B Figure2-2.Relativeattitudecompositions.A F B to F A .B F A to F B 40

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Anotherimportantfact,isthatinnitesimalrotationsarecharacterizedbythematrix R = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.147 Td [([ n ] where n istheaxisofrotation, isaninnitesimalrotationaboutthataxis.Notethat the[ n ]notationrepresentstheskew-symmetricmatrixequivalentofthecross-product operator,suchthat [ n ]= 2 6 6 6 6 4 0 )]TJ/F61 11.9552 Tf 9.298 0 Td [(n 3 n 2 n 3 0 )]TJ/F61 11.9552 Tf 9.299 0 Td [(n 1 )]TJ/F61 11.9552 Tf 9.299 0 Td [(n 2 n 1 0 3 7 7 7 7 5 ThisfactwillbeexploitedforthedevelopmentsinChapter4ondatafusion.Using thedenitionofthederivativeandmakinguseofthecompositionofrotationsandthe denitionofaninnitesimalrotation,theskew-symmetricmatrixthatisparameterized bytheangularvelocityvectoroftherotatedframewithrespecttotheoriginalframein rotatedcoordinatesis [ ] = R R T Alternatively,givenknowledgeoftheangularvelocity,theattitudematrixkinematicsare givenby R = )]TJ/F20 11.9552 Tf 11.291 -0.146 Td [([ ] R whichcanbeintegratedtoyieldtheattitudeovertime.Duetonumericaltruncation,six constraintsmustbeimposedtoensure R 2 SO Thesecondderivativeoftheattitudematrixis R = )]TJ/F20 11.9552 Tf 11.291 -0.131 Td [( [ ] )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ ][ ] R oralternativelytheskew-symmetricmatrixparameterizedbytheangularaccelerationis [ ] = )]TJ/F20 11.9552 Tf 11.929 2.385 Td [( RR T + [ ][ ] 41

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Thesecondderivativesaredevelopedfortheattitudematrixandtheunitquaternionin anticipationofthethetrackingcontrollerdescribedSection2.7. 2.4.2EulerAngles Amaximumofthreeanglesisnecessarytodescribetheattitudebetweentwo arbitrarycoordinateframes, F E and F E 0 .TheEuleranglesaretheanglesthroughwhich threeconsecutiverotationsaremadeinsequencethatformthecompositerotation describingtheattitudebetweenthetwobases.ConsiderEulerangles, f 1 2 3 g ,where thefundamentalrotationsaboutdirections f 1 2 3 g aredescribedby R 1 1 = 2 6 6 6 6 4 100 0cos 1 sin 1 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin 1 cos 1 3 7 7 7 7 5 R 2 2 = 2 6 6 6 6 4 cos 2 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin 2 010 sin 2 0cos 2 3 7 7 7 7 5 R 3 3 = 2 6 6 6 6 4 cos 3 sin 3 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin 3 cos 3 0 001 3 7 7 7 7 5 Euleranglesequencesarenon-uniqueandcanbecategorizedassymmetricand asymmetric.Symmetricsequences,suchas3-1-3,havesingularitiesat 2 = n n = 1,2,3,....Whereas,theasymmetricsequences,suchas3-2-1,havesingularitiesat 2 = n = 2 n =1,3,5,.... Theattitudematrixparameterizedbythe3-2-1sequenceofEuleranglesis R 321 1 2 3 = 2 6 6 6 6 4 c 1 c 2 s 1 c 2 )]TJ/F61 11.9552 Tf 9.299 0 Td [(s 2 c 1 s 2 s 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(s 1 c 3 s 1 s 2 s 3 + c 1 c 3 c 3 s 3 c 1 s 2 c 3 + s 1 s 3 s 1 s 2 c 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(c 1 s 3 c 2 c 3 3 7 7 7 7 5 where c cos and s sin 42

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Onceagain,consideringthe3-2-1sequenceofEulerangles,thekinematicsare 2 6 6 6 6 4 1 2 3 3 7 7 7 7 5 = 1 c 2 2 6 6 6 6 4 0 s 3 c 3 0 c 2 c 3 )]TJ/F61 11.9552 Tf 9.299 0 Td [(c 2 s 3 c 2 s 2 s 3 s 2 c 3 3 7 7 7 7 5 withinversekinematicsgivenby = 2 6 6 6 6 4 )]TJ/F61 11.9552 Tf 9.298 0 Td [(s 2 01 c 2 s 3 c 3 0 c 2 c 3 )]TJ/F61 11.9552 Tf 9.299 0 Td [(s 3 0 3 7 7 7 7 5 2 6 6 6 6 4 1 2 3 3 7 7 7 7 5 Clearly,thesingularitymanifestsinthekinematicsat 2 = n = 2 n =1,3,5,.... Thisfactholdsforallasymmetricsequences.Althoughitwasnotexplicitlyshown,all symmetricsequenceshavekinematicsingularitiesat 2 = n n =1,3,5,.... Euleranglesformaminimalattitudeparameterizationandareamenableto rapidcomputationwiththecaveatthattranscendentalfunctions,intheformof sinesandcosines,mustbecomputed.However,theyalsosufferfromsingularities. SingularitiescanbeavoidedbyswitchingbetweensymmetricandasymmetricEuler anglesequences,butotherminimalparameterizationsexistthatcanalsoavoid singularities,butwithoutthetranscendentalfunctionsrequiredbytheEulerangles. 2.4.3Axis-Angle Euler'srotationtheoremstates,arotationaboutapointisalwaysequivalentto arotationaboutalinethroughthepoint[54].Followingfromthistheoremandthe previousdevelopmentoftheattitudematrix,theattitudematrixcanbeparameterized byanaxisandangledecomposition.Givenaunitvector, n ,directedalongtheaxisof rotation,andanangle throughwhichthebodyisrotated,theattitudematrixcanbe parameterizedbyanaxisandtheangleofrotationaboutthataxis,whichisexpressed 43

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throughtheEuler-Rodriguesformulaas R n =cos I 3 3 + )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos nn T +sin [ n ]. Theaxis-anglerepresentationisa4-parametersetthatrequiresthecomputationof transcendentalfunctions.Additionally,thisrepresentationhasa2-to-1correspondence withtheelementsoftheattitudematrixandisthusnon-unique;thatis, R n = R )]TJ/F101 11.9552 Tf 9.299 0 Td [(n )]TJ/F23 11.9552 Tf 9.299 0 Td [( .UsingEq.2,thekinematicsfortheaxis-anglerepresentationare n = 1 2 I 3 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(cot 2 [ n ] [ n ] = n T Theinversekinematicsaregivenby = n +sin n )]TJ/F20 11.9552 Tf 11.955 -0.131 Td [( 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos [ n ] n Thereisasingularityat =0and = 2 .However,allsingularitieslieatthenull rotationstate,whichcanbeaccountedforintherepresentation.Thisfactdoespose problemsforstabilizingcontrollers,asthe cotangent functionisill-conditionednear thesingularitypoints,whicharethetypicaldesiredequilibriaofastabilizingcontroller. FINDREFERENCE Thesecondorderkinematicsare n = 1 2 [ n ] + 1 4 csc 2 2 )]TJ/F101 11.9552 Tf 5.48 -9.684 Td [(nn T )]TJ/F101 11.9552 Tf 11.955 0 Td [(n T n I 3 3 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 cot 2 )]TJ/F101 11.9552 Tf 5.48 -9.684 Td [(nn T )]TJ/F101 11.9552 Tf 11.956 0 Td [(n T n I 3 3 = n T + n T andthesecondorderinversekinematicsare = n + [ 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos I 3 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(sin [ n ]] n + [ sin I 3 3 )]TJ/F20 11.9552 Tf 11.956 -0.131 Td [( 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos [ n ]] n 44

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2.4.4EulerSymmetricParametersUnitQuaternions Anextensiontotheaxis-anglerepresentationismadebydening q sin 2 n and q 4 cos 2 .Thesenewparametersmakeupafour-parametersetwhichhaveequivalent algebratoHamilton'squaternion.Theunitquaternion, q =[ q T q 4 ] T isaonceredundant parameterizationthatmustsatisfytheconstraint q T q + q 2 4 =1. Theunitquaternionparameterizestheattitudematrixas R q = q 2 4 )]TJ/F101 11.9552 Tf 11.955 0 Td [(q T q I 3 3 +2 qq T )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 q 4 [ q ]. Thekinematicsoftheunitquaternionarebilinearsuchthat q = 1 2 q = 1 2 q where q = 2 6 4 q 4 I 3 3 + [ q ] )]TJ/F101 11.9552 Tf 9.298 0 Td [(q T 3 7 5 = 2 6 4 )]TJ/F20 11.9552 Tf 11.291 -0.146 Td [([ ] )]TJ/F49 11.9552 Tf 9.299 0 Td [(! T 0 3 7 5 Theinversekinematicsarenonlinearandgivenby =2 T q q Theunitquaternioneliminatesthetranscendentalfunctionsfoundintheaxis-angle representationandresultsinasingularity-freeandbilinearkinematicdescription.As aresulttheunitquaternionistheparameterizationofchoiceformanymodernattitude estimationandcontrolalgorithms.However,theunitquaternionispreservingofthe 2-to-1natureoftheaxis-anglerepresentationandisthussubjecttoissuessuchas wind-upinfeedbackcontrollers.Therefore,caremustbetakeninapplicationoftheunit quaternion[60]. 45

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2.4.5ClassicRodriguesParametersCRPs TheparametersoftheEulersymmetricparameterscanbecombinedtoformthe classicRodriguesparametersCRPs,alsoknownastheGibbsvector,whichare denedby q q 4 =tan 2 n Thiscombiningofparametersisanalogoustoaprojectionofthequaternion4-sphere ontoa3-dimensionalhyperplaneasdiscussedinSubsection2.4.7.Theattitudematrix parameterizedbytheCRPsis R = 1 1+ T f )]TJ/F49 11.9552 Tf 11.955 0 Td [( T I 3 3 +2 T )]TJ/F20 11.9552 Tf 11.956 0 Td [(2[ ] g ThekinematicsfortheCRPsaregivenby = 1 2 )]TJ/F61 11.9552 Tf 5.479 -9.684 Td [(I 3 3 + [ ] + T andtheinversekinematicsby = 2 1+ T I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ ] TheCRPsareanunconstrainedminimalparameterizationthatavoidthecomputation oftranscendentalfunctions,butalsointroduceasingularityat = .Inaddition,the CRPsare1-to-1for 2 )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ .However,thisnonsingularrangeisquitelimitingfor unconstrainedrotationsandthereisnoescapingthesingularitiesduetotheir1-to-1 nature. 2.4.6ModiedRodriguesParametersMRPs Formanyapplications,asingularityat = ,aspresentwiththeCRPs,may notbeacceptable.ThemodiedRodriguesparametersMRPsshiftthissingularityto = 2 ,bydening q 1+ q 4 =tan 4 n 46

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TheattitudematrixexpressedwiththeMRPsis R = I 3 3 + 8[ ] 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(4 )]TJ/F49 11.9552 Tf 11.955 0 Td [( T [ ] + T 2 ThekinematicsfortheMRPSaregivenby = 1 4 )]TJ/F20 11.9552 Tf 10.461 -9.684 Td [(1 )]TJ/F49 11.9552 Tf 11.956 0 Td [( T I 3 3 +2 [ ] +2 T andinversekinematicsare = 4 1+ T 2 )]TJ/F20 11.9552 Tf 10.46 -9.684 Td [(1 )]TJ/F49 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 [ ] +2 T ThemathematicalsingularitiespresentintheMRPscanbeovercomebyswitching totheshadowset,denedby s )]TJ/F101 11.9552 Tf 9.299 0 Td [(q 1 )]TJ/F61 11.9552 Tf 11.956 0 Td [(q 4 whenasingularityisnear.Theshadowsetresultsfromthe2-to-1non-uniquenessof theMRPrepresentation.Thatis,theMRPsandtheirshadowsetrepresentthesame attitude,butencountersingularitiesatdifferentorientations,wheretheMRPisvalidfor 2 )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 andtheshadowsetfor 2 0,+4 2.4.7StereographicOrientationProjectionsSOPs Stereographicprojectionsareageometrictechniqueforprojectingaunitsphere ontoaplane.Aprojectionpointischosenonthesurfaceoftheunitsphereaswellas aprojectionplane.Allpointsontheunitspherearethenmappedtothisplane.This ideahasbeenextendedtodevelopnewattitudeparameterizationsbyprojectionofthe 4-dimensionalspherethatisformedbytheconstraintsurfaceoftheunitquaternion, ontothe3-dimensionalhyperplane.ThisconceptwasrstusedbyMarandiand ModitodeveloptheMRPs[61]asseeninFigure2-4B.Schaubextendedthisworkto developthesymmetricstereographicorientationparametersSSOPsandasymmetric 47

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stereographicorientationparametersASOPs[62].TheSSOPsaredenedby = q q 4 )]TJ/F61 11.9552 Tf 11.955 0 Td [(a wheretheconstant, a ,dictatesthelocationofthesingularityandisdenedby a =cos s 2 TheASOPsdonothavesuchacompactform,asthegeometricsingularitydependson boththeaxisandangle. A B Figure2-3.Stereographicprojectionsadaptedfrom[62].ASymmetricstereographic projections.BAsymmetricstereographicprojections. TheCRPsandMRPsaresubsetsofSSOPshavingaprojectionpointalongthe q 4 axis.ItisclearfromFigure2-4AthattheCRPsaresingularat = ,wherethe projectionofpoint q =[0,0,0, )]TJ/F20 11.9552 Tf 9.298 0 Td [(1] T onthethree-sphererepresentingthequaternion unityconstraintisatinnityontheCRPhyperplane.Similarly,fromFigure2-4B,itis clearthattheMRPissingularat = 2 .Otherparameterscanbedevelopedusing stereographicprojections 48

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A B Figure2-4.Commonstereographicprojectionsadaptedfrom[58].AStereographic projectionofCRPs.BStereographicprojectionofMRPs. 2.4.8GeneralizedCayleyTransformation TheCayleytransformprovidesamappingfromthegroupofskew-symmetric matrices, so ,tothespecialorthogonalmatrices, SO .Astheattitudematrix belongsto SO ,thismappinghassignicanceforattituderepresentations.TheCayley transformisdenedas R Q = I 3 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(Q I 3 3 + Q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = I 3 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(Q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 I 3 3 + Q where Q isaskew-symmetricmatrix.Byletting Q =[ ],theCayleytransform generatestheCRPs.FromthisobservationandthedevelopmentoftheMRPs,Tsiotras usedcomplexanalysiswithconformalmappingstomotivatetheextensionoftheCayley transformtohigherorderswiththegeneralizedCayleytransform,giveninEq.2 [63].Thesehigherordertransformswereshowntoparameterizetheattitudematrix andexpandthesingularity-freeregionofapplicabilityoftheCRPandMRPattitude parameterizations. R Q = I 3 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(Q n I 3 3 + Q )]TJ/F61 7.9701 Tf 6.587 0 Td [(n 49

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Therefore,thesecondorderCayleytransform n =2with Q =[ ]yieldstheMRPs. Anexamplehigher-orderparameterizationwasdevelopedwiththe4 th orderCayley transform n =4using Q =[ ],where q 1+ q 4 p 2+ q 4 =tan 8 n Thisfourthorderparameterizationplacesthesingularitiesat 4 .However, Cayley-basedparameterizationsoforder n 3havedisadvantagesinthekinematical description,wherethedifferentialequationisnolongerdenedbyasecond-order polynomialexpression,asisthecasewiththeCRPsandMRPs. 2.4.9VectorialAttitudeParameterizations Recently,thefamilyofminimalattitudeparameterizationsknownasthevectorial attitudeparameterizations[59]wereintroducedandsubsequentlycharacterizedwith respecttoanonlinearityandsingularityindex[64].Thisgeneralizationofminimal attitudeparameterizationwasusedtogenerateafamilyofattitudecontrollawstoshape thetransientperformanceofthenonlinearattitudedynamics[65].Withtheexceptionof theEulerangles,alloftheminimalparameterizationsreviewedinthissectionbelongto alargerclassofattitudeparametersknownasthevectorialattitudeparameterizations. ThevectorialattitudeparameterizationsfollowfromEuler'srotationtheoremandhave theform, r = r n whichmaptheaxisofrotation, n ,andangleofrotation, ,throughthegenerating function, r ,toconstructafamilyofminimalattitudeparameterizations.Twoimportant familiesofvectorialattitudeparameterizationsarethesineandtangentfamilies,dened as r s m = k s sin 2 m 50

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and r t m = k t tan 2 m respectively,where k s and k t arescalingcoefcientsand m istheorder.Allorders oftheRodriguesparametersbelongtothetangentfamily,suchthatthehigher-order RodriguesparametersaredenedbyEq.2when k t =1.Whereasthehigher-order sineparametersHOSPsaredenedbyEq.2when k s =1.However,anarbitrary numberofvectorialparameterizationsexistofwhich,examplesbasedonhigher-order azimuthalprojectionsweredenedinReference[64]. Asdescribedin[64],thegeneratingfunction r willproduceattitudeparameterizations belongingtotwosingularityclassesbasedonwhetherthegeneratingfunctionislocally bi-Lipschitzcontinuouswithin 2 [ )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ ] .ClassIparameterizationsarenotlocally bi-Lipschitzcontinuouswithin 2 [ )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ ] andthushaveunavoidablekinematic singularitieswithin 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 ,+2 ] .Theseparameterizationsarethemostprohibitive forcontrol,estimation,anddatafusionwhenunconstrainedmotionisencountered. ClassicexamplesofClassIparameterizationsaretherst-ordersineandtangent vectorialparameterizations,whichfor k s k t =1arereferredtoastheorthographic parametersOPsandclassicalRodriguesparametersCRPs,respectively.Class IIparameterizationsarelocallybi-Lipschitzcontinuouswithin 2 [ )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ ] and thushaveavoidablesingularitiesforallrotationanglesthroughuseoftheshadow parameterizations.Shadowparameterizationsareconstructedthroughtheprojections basedonthetwo-to-onenatureofthequaternion.Theuseoftheshadowsproduces aglobally-singularityfreeparameterization.ExamplesofClassIIparameterizations arethesecond-ordersineandtangentparameterizations,whichfor k s k t =1are knownastheLambertparametersLPsandmodiedRodriguesparametersMRPs, respectively.Oneotherparameterizationofnoteistherotationvector,alsoknownas 51

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theequidistantorientationparametersEOPs,whichisaparameterizationdened 8 2 R )-222(f 0 g i.e.,allpointsonthereallineexceptzero. AsummaryofseveralvectorialattitudeparameterizationsisgiveninTable2-1,with denitionsforthegeneratingfunctionandassociatedparameterizationoftheattitude matrixandsingularityclass.Inthistable,thefourth-orderRodriguesparametersFRPs areanexampleofthegeneralHORPs. Table2-1.Severalgeneratingfunctionsandthevectorialparameterizationof SO Name r r R r EOPs cos I 3 3 )]TJ/F20 6.9738 Tf 11.158 4.445 Td [(sin [ ] + cos )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 2 T CRPs tan 2 1 1+ 2 \000 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 I 3 3 )]TJ/F20 9.9626 Tf 9.962 0 Td [(2 [ ] +2 T MRPs tan 4 1 1+ 2 2 \000 1 )]TJ/F20 9.9626 Tf 9.963 0 Td [(6 2 + 4 I 3 3 )]TJ/F20 9.9626 Tf 9.962 0 Td [(4 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 [ ] +8 T FRPs tan 8 nh 1+ 2 4 )]TJ/F20 6.9738 Tf 6.227 0 Td [(32 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 2 2 i I 3 3 )]TJ/F20 6.9738 Tf 6.227 0 Td [(8 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 1 )]TJ/F20 6.9738 Tf 6.227 0 Td [(6 2 + 4 [ ] +32 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 2 T o 1+ 2 4 HORPs m tan 2 m I 3 3 )]TJ/F20 9.9626 Tf 9.963 -0.122 Td [([ m ] m I 3 3 + [ m ] )]TJ/F61 6.9738 Tf 6.227 0 Td [(m m > 2 OPs sin 2 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F20 9.9626 Tf 9.962 0 Td [(2 2 I 3 3 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 1 2 [ ] +2 T LPs sin 4 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F20 9.9626 Tf 9.962 0 Td [(8 2 +8 4 I 3 3 )]TJ/F20 9.9626 Tf 9.962 0 Td [(4 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 2 )]TJ/F20 9.9626 Tf 10.793 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 1 2 [ ] +8 )]TJ/F20 9.9626 Tf 4.567 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 T Descriptionoftheshadowparameterizationsresultingfromstereographic projectionsoftheunitquaternionontothetangentplanehavebeendiscussedindetail in[62,63].Theseresultsweregeneralizedforthevectorialattitudeparameterizationby Tanygin[64],inwhichtheleftshadowparametersaredenedby r +2 andtheright shadowsby r )]TJ/F20 11.9552 Tf 12.104 0 Td [(2 .Furthermore,usingtheseshadows,aglobalnonsingularattitude representation, r isdenedby r = 8 > > > > < > > > > : r +2 n 2 ,2 ] r n 2 [ )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ ] r )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 n 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 )]TJ/F23 11.9552 Tf 9.298 0 Td [( Itisimportanttonotethatalthoughthehigher-orderRodriguesparametersand higher-ordersineparameterswillhavefurtherbifurcationsleadingtomanyshadow sets,onlytheshadowsdescribedabovearenecessaryastheyspan 2 [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 ] whichisallthatisnecessaryduetotheperiodicinvarianceofattitude. 52

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Takingthetime-derivativeofEq.2yields r = r n + r n where r = @ r =@ .Applyingtheknownrelationshipsfortheaxis-anglerepresentation time-derivatives, and n [57],toEq.2,leadstotheforwardkinematicsform r = G r where G r istheforwardkinematicalJacobian,whichisingeneralanonlinear functionof r .AtableofseveralforwardkinematicalJacobianmatricesareprovided inReference[64].However,fordatafusion,theinversekinematicsJacobianisof interest,whichrelatestheangularvelocitytothetime-derivativeoftheparameterization through = H r r where H r = G )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 r istheinversekinematicalJacobian.Thekinematicsand inversekinematicsforvariousordersoftheRodriguesparametershavealreadybeen summarizedinSections2.4.5-2.4.8. Table2-2providesasummaryoftheinversekinematicalJacobianforallofthe vectorialattitudeparameterizationsdescribedinTable2-1.Thistablewillbeextensively usedinthedevelopmentsinChapter4,whereproperchoiceofattitudeparameterization fordatafusionisbasedonthelinearityoftheinversekinematicsJacobian. 2.5SummaryofAttitudeParameterizations Table2-3summarizesthepropertiesoftheattitudeparameterizationspresented inSection2.4.Thepropertiesofinterestarewhethertheparameterizationisglobal, unique,constrained,nonlinear,orcontainssingularities.Globalreferstoitsability todescribeallattitudesandrelatedkinematicconditionsandisdirectlytiedtothe presenceofsingularities.Uniquenessreferstowhethertheparameterizationis1-to-1 53

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Table2-2.InversekinematicsJacobianforseveralvectorialattitudeparameterizations Name r H r EOPs sin I 3 3 )]TJ/F20 6.9738 Tf 11.158 4.542 Td [( 1 )]TJ/F20 6.9738 Tf 6.227 0 Td [(cos 2 [ ] + 1 2 h 1 )]TJ/F20 6.9738 Tf 11.158 4.444 Td [(sin i T CRPs 2 1+ 2 I 3 3 )]TJ/F20 9.9626 Tf 9.963 -0.122 Td [([ ] MRPs 4 1+ 2 2 )]TJ/F20 9.9626 Tf 8.717 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 I 3 3 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 [ ] +2 T FRPs 8 1+ 2 4 )]TJ/F20 9.9626 Tf 8.717 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 )]TJ/F20 9.9626 Tf 10.793 -8.07 Td [(1 )]TJ/F20 9.9626 Tf 9.963 0 Td [(6 2 + 4 I 3 3 )]TJ/F20 9.9626 Tf 9.963 0 Td [(4 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 [ ] +2 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(5 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 2 + 4 T HORPs m 2 m 1+ 2 I 3 3 )]TJ/F20 6.9738 Tf 20.951 4.444 Td [(2 2 2 m 1+ 2 [ m ] + h 2 m 2 m 1+ 2 m )]TJ/F20 6.9738 Tf 22.585 4.444 Td [(2 3 m 1+ 2 i m T m OPs 2 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 1 2 h )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 I 3 3 )]TJ/F29 9.9626 Tf 9.963 8.07 Td [()]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 1 2 [ ] + T i LPs 4 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 1 2 h )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 2 )]TJ/F20 9.9626 Tf 10.793 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 I 3 3 )]TJ/F20 9.9626 Tf 9.963 0 Td [(2 )]TJ/F20 9.9626 Tf 4.566 -8.07 Td [(1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 3 2 [ ] + )]TJ/F20 9.9626 Tf 4.567 -8.07 Td [(3 )]TJ/F20 9.9626 Tf 9.962 0 Td [(2 2 T i withattitudematrix.Constraintsrefertotheredundancyintheparameterizationwith respecttotherotationaldegrees-of-freedom.Nonlinearitydescribestheformofthe kinematicaldifferentialequations,whereTransc.referstotranscendentalfunctionsand Poly.referstopolynomialsofatleastordertwo.Finally,singularityreferstothetype ofsingularity,whereGeo.referstothesingularitybeingofgeometricorigin,inwhich notallattitudescanbedescribedbytheparameterization,Kin.referstokinematic singularities,inwhichtheparameterizationorangularvelocityescapestoinnityinnite timeatcertainattitudes,andNullstatereferstosingularitieswherethenullstatedoes notexistfortheparameterization. Table2-3.Propertiescomparisonofseveralattitudeparameterizations NameSymbolGlobalUniqueConst.NonlinearitySingularity AttitudeMatrix R YesYes6LinearNonsingular EulerAngles 1 2 3 NoNo0Transc.Kin. Axis-Angle n NoNo1Transc.Nullstate Quaternions q YesNo1LinearNonsingular EOPs NoNo1Transc.Nullstate CRPs NoYes0Poly.Geo./Kin. MRPs NoNo0Poly.Geo./Kin. FRPs NoNo0Poly.Geo./Kin. HORPs m NoNo0Poly.Geo./Kin. OPs NoYes0Poly.Geo./Kin. LPs NoNo0Poly.Geo./Kin. HOSPs m NoNo0Poly.Geo./Kin. Theattitudematrixistheonlyuniqueglobalnonsingulardescriptionofattitude. However,nineparametersarenecessaryinitsdescription.Theunitquaternionreduces 54

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thenumberofparametersdowntofourandisstillgloballynonsingularwithlinear kinematics.However,itstillrequiresasingleconstraintandisnon-unique.Theunit quaternionpropertiesaresufcientformanyapplicationsinattitudeestimationand controlasdescribedinSection2.7andChapter3,respectively.However,inChapter4, minimalparameterizationswillbeutilizedforfastandefcientattitudedatafusion algorithmdevelopment,duetotheirunconstrainednature. 2.6AttitudeDynamics Focusisnowturnedtowardsattitudedynamicsandcontrol.Therotationalmotion ofrigidspacecraftisgovernedbyEuler'sSecondLawofMotion.Thislawstatesthat thetime-rate-of-changeoftheangularmomentumofarigidbody, H ,aboutthecenterof massoraninertialpoint,pointc,isequivalenttothenetexternaltorque, ,actingon thebody.Thatis, H c = Ingeneral,theexternaltorqueinorbitwillconsistofthecontrolinput,aswellas disturbancetorquesoriginatingfromenvironmentalfactorsandnon-environmental factors.Examplesofenvironmentalfactorssolarpressure,aerodynamicdrag,magnetic interactions,andinertia-inducedgravitygradients.Anexamplenon-environmentalfactor ismassejection[66]. Equation2isavectorexpression.Whenthevectorsarerepresentedinbody coordinates,theattitudedynamicsequationsare J = )]TJ/F20 11.9552 Tf 11.955 0 Td [([ ] J where J istheconstantinertiamatrix, istheangularvelocityofthebodyrelativetothe inertialframe,and[ ] J isreferredtoasthegyroscopictorque. Givenknowledgeofthespacecraftinertiaaswellastheenvironmentaleffects,the attitudedynamicscanbeusedasaplantmodelforthedesignofacontrolinputtoaffect 55

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theattitudemotion.Designingacontrolinputtotrackanotherspacecraftforattitude resourcesharingisthefocusofSection2.7. 2.7AttitudeTrackingControl Uptothispointinthediscussion,attitudeparameterizationshavebeenreviewed thatdescribetherelativeorientationofacoordinateframexedtoaspacecraftwith respecttoanotherframeofinterest.Additionally,attitudedynamicshavebeenreviewed thatdescribetheevolutionofthespacecraft'srotationalmotion.Utilizingtheseresults, thenextsectionisconcernedwithcontrollingtherotationalmotionofaspacecraftto accomplishsomemissionspecicobjective.Specically,attitudecontrolisnecessary forattituderesourcesharing,asitallowsonespacecrafttopointasensoratanother spacecrafttocapturemeasurementsabouttheobservedspacecraft'sstates.In thediscussionthatfollows,coordinateframesaredevelopedtodescribeadesired orientationforresourcesharingandanonlinearcontrollawisdevelopedtoensurethat thedesiredsignalistracked. 2.7.1DesiredCoordinateFrames Anattitudetrackingcontrollerrequiresknowledgeofthekinematicsofareference quaterniontrajectorywithderivativesuptosecondorder.Thefollowingsectiondenes severalimportantreferencevectorsandtheirassociatednaturalcoordinatesystem. Thereferencevectorsarethenusedtoderiveanassociatedcoordinateframe.Finally, thekinematicrelationshipsforeachofthesecoordinateframesaredeveloped.Unless otherwisespecied,itisassumedthattheprimarypointingdirectionisalignedwiththe X B and,ifnecessary,asecondarypointingdirectionisalignedwiththe Z B .Additionally, unlessotherwisespecied,allderivatives,denotedby areassumedtobeinertial. Thisassumptionisvalidinmostcases,asmostreferencevectorsareknownininertial coordinatesandthereforecanbedifferentiateddirectlywithouttheneedtointroduce furthercoordinatetransformations. 56

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2.7.1.1RelativespacecraftsharingframeSH Theprimaryobjectiveofattituderesourcesharingistransformofattitudeinformation betweentwospacecraft.Therefore,theobjectiveistotrackanattitudetrajectorythat pointsthechief'srelativeattitudesensorataneighboringdeputytocapturerelative attitudeinformation.Thispointingdirectionisdenedbytherelativepositionvector. Inthecontextofthissection,therelativepositionvectorisdenotedby ,whichisa standardconvention.However,itshouldnotbeconfusedwiththeCRPsinthebroader discussionsinotherchapters.Touniquelydenethebasistriad,asecondarypointing vectorwillremaingeneralandisdenotedby .Projectionsensuretheorthogonality ofthisbasis.Forexample,theprimarygoalistosenseanotherspacecraft,butthe secondaryobjectivemaybetohaveanorthogonalfaceofthespacecraftthatis sun-pointing.Inthatcase,if isdenedasthepositionvectorfromthebodytothe sun, r B = ,thenthe Z sh willbeasclosetosun-pointingwhilepursuingtheprimary objectivewhichisdenedbythe X sh pointingalongtherelativepositionvector.This denitionalsomaintainstheorthogonalityofthebasesvectors.Therefore,thesharing frame,showninFigure2-5A,isdenedbytheorthonormalbasisvectors, f X sh Y sh Z sh g where X sh = k k Y sh = X sh k X sh k Z sh = X sh Y sh 2.7.1.2GroundtargetframeT Thegroundtargetframe F T ,showninFigure2-5B,isdenedbytheorthonormal basisvectors, f X t Y t Z t g .ACartesiancoordinatesystemisattachedtoTandaligned withthebasisvectorsdenedby Z t = k k Y t = Z t k Z t k X t = Y t Z t where isthepositionvectorfromthecenterofmassofthechieftothegroundtarget point.Ifthemissionofthespacecraftistotrackgroundtargetsbutperiodicallyshare 57

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measurementswithanotherspacecraftyinginformation,thenchoosing = ,ensures thechief'sattitudeisasneartothesecondarysharingobjectiveaspossible,while satisfyingtheprimaryobjective. RecallfromEq.2thatthetargetquaternionisdenedbytherelativetarget vector andamission-dependentvectorthatclosesthetriad.Bychoosingthe mission-dependentvectortobetherelativepositionvectorfromchieftodeputy, thespacecraftwilltrackthetargetwhilealsominimizingtheslewtoshare.Since is measuredininertialcoordinates,thematrix R t =[ X t Y t Z t ] T denestheattitudematrixfromtheinertialcoordinatestothetargetcoordinates.The processoutlinedinReference[57]isusedforconvertingtheattitudematrixinEq.2 tothequaternionrelatingthetargetframetotheinertialframe.Therefore,thetarget quaternion, q t ,isknown. Thereferenceattitudetrajectoryisdenedbythebasisvectors Z t = T 1 = 2 Y t = [ Z t ] [ Z t ] T [ Z t ] 1 = 2 X t = [ Y t ] Z t A B Figure2-5.Trackingcoordinateframes.ABodyandsharingframes.BBodyandtarget frames. 58

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2.7.2DesiredQuaternionKinematics Theattitudematrixwasconstructedfromasetoftime-varyingbasisvectors.By differentiatingthesevectors,thederivativesoftheattitudematrixcanbeconstructed. Then,thederivativesoftheattitudeparameterizationofinterestcanbebackedoutand usedinthecontrollersynthesis. 2.7.2.1Relativespacecraftsharingframekinematics Itassumedthatallunitvectorsandtheirderivativesareknownininertialcoordinates. Ifthisisnotthenaturaldescription,thentransformationsmustbesubstitutedintothese expressions.Therefore,thesharingframebasisvectorsare X sh = T 1 = 2 Y sh = [ ] X sh [ ] X sh T [ ] X sh 1 = 2 Z sh = [ X sh ] Y sh Sincetheunitvectorsarerepresentedininertialcoordinatesanddenebydesired coordinateframe,theynaturallyformtheattitudematrixdenedby R sh =[ X sh Y sh Z sh ] T whichtransformsvectorsrepresentedininertialcoordinatestosharingframecoordinates. Takingthetime-derivativeofthebasisvectorsyields X sh = T )]TJ/F49 11.9552 Tf 11.955 0 Td [( T T X sh Y sh = \002 X sh + [ ] X sh [ ] X sh T )]TJ/F20 11.9552 Tf 11.955 -0.132 Td [( [ ] X sh \002 X sh + [ ] X sh T [ ] X sh T [ ] X sh # Y sh Z sh = X sh Y sh + [ X sh ] Y sh 59

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whichcanbeusedtoproducethederivativeoftheattitudematrix.Takingthesecond derivativeoftheunitvectorsleadsto X sh = T )]TJ/F49 11.9552 Tf 11.956 0 Td [( T )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 T I 3 x 3 T X sh + T )]TJ/F49 11.9552 Tf 11.955 0 Td [( T T X sh Y sh = 1 [ ] X sh T [ ] X sh nh \002 X sh + [ ] X sh [ ] X sh T )]TJ/F20 11.9552 Tf 11.955 -0.131 Td [( [ ] X sh \002 X sh + [ ] X sh T )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 \002 X sh + [ ] X sh T [ ] X sh I 3 x 3 i Y sh + h \002 X sh +2 X sh + [ ] X sh [ ] X sh T )]TJ/F20 11.9552 Tf 11.955 -0.132 Td [( [ ] X sh \002 X sh +2 X sh + [ ] X sh T i Y sh o Z sh = X sh Y sh +2 X sh Y sh + [ X sh ] Y sh whichcanbeusedtoconstructthesecondderivativeoftheattitudematrix. 2.7.2.2Groundtargetframekinematics Note:Iftherelativepositionvectorismeasureddirectly,itwillbeknowninbody coordinates.Inthiscase,therelativepositionvectormustbetransformedtoinertial coordinatesforthisanalysis.Alternatively,ifdifferencesininertialpositionareutilized, noadditionalmodicationisnecessary.Also,thetargetvector, ,isderivedfroma giventargetpositionininertialcoordinatesandtheabsolutepositionofthespacecraft, knowninertially.Therefore,thetargetvectoranditsderivativesareallknownininertial coordinates.Usingthisknowledge,theinertialderivativesareformulatedas Z t = T )]TJ/F49 11.9552 Tf 11.955 0 Td [( T T Z t Y t = Z t + [ Z t ] [ Z t ] T )]TJ/F20 11.9552 Tf 11.955 0 Td [( [ Z t ] Z t + [ Z t ] T [ Z t ] T [ Z t ] # Y t X t = Y t Z t + [ Y t ] Z t andthederivativeoftheattitudematrixis R t =[ X t Y t Z t ] T 60

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Thesecondderivativeofthebasesvectorsis Z t = T )]TJ/F49 11.9552 Tf 11.955 0 Td [( T )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 T I 3 x 3 T Z t + T )]TJ/F49 11.9552 Tf 11.955 0 Td [( T T Z t Y t = 1 [ Z t ] T [ Z t ] Z t + [ Z t ] [ Z t ] T )]TJ/F20 11.9552 Tf 11.955 0 Td [( [ Z t ] Z t + [ Z t ] T )]TJ/F20 11.9552 Tf 13.15 0 Td [(2 Z t + [ Z t ] T [ Z t ] I 3 x 3 Y t + Z t +2 Z t + [ Z t ] [ Z t ] T )]TJ/F29 11.9552 Tf 13.151 9.684 Td [()]TJ/F20 11.9552 Tf 5.479 -9.83 Td [([ Z t ] Z t +2 Z t + [ Z t ] T Y t X t = Y t Z t +2 Y t Z t + [ Y t ] Z t andthesecondderivativeoftheattitudematrixis R t =[ X t Y t Z t ] T Recallthatthesecondderivativeof canbecomputedfromtheClohessy-Wiltshire-Hill CWHequationsandtransformedtoinertialcoordinates.AbriefreviewoftheCWH equationsisgiveninSection2.7.3. Usingattitudematrixanditsderivatives,thecorrespondingangularvelocityand derivativeofangularvelocityforthetarget,requiredbythecontroller,canbecomputed as [ d ] = )]TJ/F20 11.9552 Tf 11.929 2.385 Td [( R t R T t [ d ] = )]TJ/F20 11.9552 Tf 11.929 2.385 Td [( R t R T t )]TJ/F20 11.9552 Tf 14.585 2.385 Td [( R t R T t 2.7.3Clohessy-Wiltshire-HillCWHEquations Sincethetrackingsignalforresourcedistributionutilizesinformationabout therelativepositionvector,thissectionreviewsthedescriptionofrelativeposition. Determiningtherelativemotionanditsderivativeswhenbothsatelliteshaveknowledge oftheirinertialpositionandderivativesresultsfromadifferencingintheirrespective states.However,ifrelativepositionismeasuredonboardthespacecraft,thenthe descriptionofmotiontakesplaceinanon-inertialcoordinatesystem.Underthese 61

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conditions,therearemanyrelativemotionformulations,whichmakevaryingassumptions. ThemostcommonrelativemotionequationsaretheClohessy-Wiltshire-HillCWH equations[67,68].TheCWHequationsassumethattheonespacecraftisboundedto acircularorbitandthatallotherspacecraftareincloseproximitywithrespecttotheir distancefromtheEarth.Otherrelativemotionformulationsexiststhatrelaxthecircular orbitrequirements.ThemoststudiedistheTschauner-Hempelequations[68,69].That formulation,howevermakesuseofthetrue-anomalydomaintoobtainclosed-form solutions.Inpractice,thetrueanomalymuststillbemappedtothetimedomainsolving Kepler'sEquation,whichhasnoclosed-formsolution[70].Theassumptionsforthe CWHequationstoholdwillalwaysbevalidfortheinterestofthisresearch,butthe circularassumptioncanberelaxedtomakeuseoftheresultsforgeneralelliptical motionofthechief. Giventhechieffollowsacircularorbitandthedeputiesareincloseproximitytothe chief,theCWHequationsdescribesthemotionoftherelativepositionvectorinLVLH coordinates.TheCWHequationsareshowninstate-spaceform,inEq.2, 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x y z x y z 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000100 000010 000001 3 n 2 0002 n 0 000 )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 n 00 00 )]TJ/F61 11.9552 Tf 9.298 0 Td [(n 2 000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x y z x y z 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 62

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ThestatetransitionmatrixSTMoftheCWHequationshasananalyticform,duetoits lineartime-invariantstructure,andisexpressedas t t 0 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 )]TJ/F20 11.9552 Tf 11.956 0 Td [(3 c nt 00 s nt = n 2 )]TJ/F61 11.9552 Tf 11.955 0 Td [(c nt = n 0 )]TJ/F20 11.9552 Tf 9.298 0 Td [(6 nt )]TJ/F61 11.9552 Tf 11.955 0 Td [(s nt 10 )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 )]TJ/F61 11.9552 Tf 11.955 0 Td [(c nt = n 4 s nt = n )]TJ/F20 11.9552 Tf 11.955 0 Td [(3 t 0 00 c nt 00 s nt = n 3 ns nt 00 c nt 2 s nt 0 )]TJ/F20 11.9552 Tf 9.299 0 Td [(6 n )]TJ/F61 11.9552 Tf 11.955 0 Td [(c nt 00 )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 s nt )]TJ/F20 11.9552 Tf 9.298 0 Td [(3+4 c nt 0 00 )]TJ/F61 11.9552 Tf 9.299 0 Td [(ns nt 00 c nt 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 where n isthemeanmotionofthechieforbit,and c nt cos nt and s nt sin nt .The statetransitionmatrixrepresentationoftheCWHequationsis x t = t t 0 x 0 = t )]TJ/F61 11.9552 Tf 11.955 0 Td [(t 0 x 0 giventheinitialstate x 0 =[ x 0 y 0 z 0 x 0 y 0 z 0 ] T IfusingtheCWHequations,therelativepositionvectorexpressedinLVLH coordinatesmustbetransformedtoinertialcoordinatesanddifferentiatedaccordingly. Therefore,theattitudematrixfromLVLHtoinertialcoordinatesanditsderivativeswill enterintothedesiredkinematicalrepresentation. Aftertransformingtherelativepositionanditsderivativesintoinertialcoordinates, substitutionintothedesiredattitudekinematicexpressionsyieldsatrajectorythatcanbe trackedtopointarelativesensoralongtherelativepositionvector. 2.7.4AttitudeTrackingControllerDerivation Theprimaryobjectiveofthechiefistopointitsrelativeattitudesensoralongthe relativepositionvectorjoiningitselfwiththedeputy.Thisobjectiveissatisedwhen thebodyframeisalignedwiththedesiredframewithzerorelativeangularvelocity. Thereforethecontrolobjectiveistodrivetheerrorquaternion, q e =[ q T e q 4 e ] T ,denedby Eq.2,totheidentityquaternion [ 0,0,0,1 ] T andtheerrorangularvelocity,denedby 63

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Eq.2,tozero. q e = q q )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 d = q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 d q )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 d q e = )]TJ/F61 11.9552 Tf 11.956 0 Td [(R B = D d Theerrorstatesrequire q d ,whichisthequaternionusedtorepresentadesired orientation, d ,whichisthedesiredangularvelocity,and R B = D ,whichistheattitude matrixfromdesiredtobodycoordinates. Theattitudetrackingcontrollermustgenerateastabilizingcontrolinputthatdrives theerrorquaternionanderrorangularvelocity,asymptoticallytotheidentityquaternion andzero,respectively.Givenastate x ,Lyapunov'ssecondmethod,alsoknownas thedirectmethod,statesthattheexistenceofaLyapunovcandidatefunctionLCF satisfyingthefollowingconditions 1. V 0 =0 2.lim k x k!1 V x = 1 3. V x > 0, 8 x )-222(f 0 g 4. V x < 0 8 x )-222(f 0 g issufcienttoproveglobalasymptoticstabilityGAStotheequilibriumstate,provided onlyoneequilibriumstateexists.Duetothe2-to-1natureofthequaternions,MRPs, andothernon-uniqueattitudeparameterizations,theattitudetrackingfeedback controllerusingtheserepresentationscanonlybelocallyasymptoticallystableLAS. Thatis,therearetwoequilibriaforthesystem.Additionally,attitudestatesarebounded andthustheLyapunovfunctioncannotberadiallyunbounded.However,duetothe persistenceofexcitation,thetrajectorywillneverremaintrappedintheundesired 64

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equilibriumstateandallotherattitudeinitialconditionswillasymptoticallyapproachthe desiredequilibrium. ALCFisconstructedinEq.2thatisbothpositivedeniteandradially unbounded.Thechoiceof K asascalarmultipleofinertiamatrixfollowsfromthe discussioninReference[71]. V = 1 2 T e K )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 J e + )]TJ/F61 11.9552 Tf 11.955 0 Td [(q 4 2 e 8 K > 0 BytakingtheinertialderivativeofVinEq.2,theLyapunovderivative, V = T e K )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 )]TJ/F20 11.9552 Tf 11.956 -0.147 Td [([ ] J )]TJ/F61 11.9552 Tf 11.955 0 Td [(J R B = D d )]TJ/F61 11.9552 Tf 11.955 0 Td [(JR B = D d + Kq 4 e q e canbeshowntobenegativedenitewiththecontrolinput = [ ] J + J R B = D d + JR B = D d )]TJ/F61 11.9552 Tf 11.955 0 Td [(Kq 4 e q e )]TJ/F61 11.9552 Tf 11.955 0 Td [(C e 8 C > 0. Therefore,thecontrollerinEq.2ensuresthatallofthesufcientconditionsfor Lyapunovstability[72]aremetandthesystemisasymptoticallystableexceptforthe specialcasewhen e = 0 and q 4 e =1.Althoughthespecialcaseisunlikelydueto perturbations,causingapersistenceofexcitation,thequaternionfeedbackcontrolinput isnon-uniqueandprovisionsshouldbemadewhenusingthiscontrollerifthatcaseis expected.Also,notethatthiscontrollawrequiresthatthedesiredquaternionistwice differentiable. 2.8Summary Therequisitetheoryforattitudeparameterizations,kinematics,dynamics,and controlweredevelopedinthischapter.Theattitudeparameterizationswerecompared basedontheiradvantagesanddisadvantageswithrespecttoseveralimportant properties.Specically,theattitudematrixwasshowntobewithoutanydeciency inuniquenessorsingularity,butattheexpenseofhighredundancyandnumberof constraints.Thequaternionishighlyregardedduetoitslinearityandlackofsingularity 65

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andsingleredundancy.However,cautionmustbegiventoitsnon-uniquenessand pronenesstowind-upinattitudecontrol,aswellasenforcementofitsconstraint.Finally, thevectorialattitudeparameterizationswerereviewed,astheyareunconstrainedand canbemadetobegloballynonsingularthroughuseoftheirshadowsets. Attituderesourcesharingrequiresthechieftopointarelativeattitudesensorat itsdeputiestomeasuretheirrelativestates.Inordertoaccomplishthisobjective,the second-orderkinematicsofthedesiredattitudewereformulatedtogenerateareference trajectoryfortrackingcontrol.Anonlineartrackingcontrollerwasderivedthrough Lyapunov'sdirectmethodthatmadeuseoftheunitquaternionparameterizationand attitudematrix.Theseresultsarecriticalforsharingattitudeinformationbetweenthe chiefanddeputies. 66

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CHAPTER3 DISAGGREGATEDATTITUDEESTIMATION Thegoalofinertialattitudeestimationinadisaggregatedsystemisforeach spacecraftinthenetworktoestimatetheentirestateofthenetworkusingonlylocal information.Inthedisaggregatedestimationscheme,thechiefmakesuseoftherelative attitudesensortocapturevectormeasurementstothedeputyofinterest.Inaddition, thechiefestimatesitsinertialattitude.Thischapterdevelopsthetheoryforthechiefto estimateadeputy'sinertialattitude.Inaddition,thetheoryispresentedforthedeputy toestimateitsinertialattitude,butusinganaugmentedstatevectortoaccountforthe otherspacecraftinthenetwork.Thus,thechiefanddeputiesusethesamestatevector andtheattitudeestimationforthedisaggregatedsystemyieldsmultipleestimatesof theinertialstatesofthechiefandeachdeputy,alongwiththeircross-correlations.This methodisbenecialwhenallinformationiscombinedthroughdatafusion. ThischapterrstreviewsbasicresultsinestimationandKalmanltering.The remainderofthechapterisdividedintotwomainsections.Intherstofthesesections, inertialattitudesensorsandestimationaresurveyedandreviewed.Inthesecondof thesesections,theextendedKalmanlterequationsforthedisaggregatedattitude estimationschemearederived. 3.1GeneralNomenclature Inthischapter,compactnotationisneededtodescribethemeasurementquantities andthestateestimates.Particularly,unitvectorsdescribingadirectionaretheprimary measurementsnecessaryforattitudesystems,wherethedirectionisspeciedby alinesegmentconnectingtwopoints.Theframeinwhichthosepointsareknown isimportantaswellasthebasisinwhichthatvectorisrepresented.Thenotation forvectormeasurementsis A a B = A i j k ,where a isthevector,thetilderepresentsa measuredquantity,theleftsuperscriptdenotestheframethat a isrepresented,and therightsubscriptstatesthatthe i th -vectororiginatesfromapointxedinin F A and 67

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endsinapointxedin F B attimeindex k .Forparameterizedattitudestates,the notationis x )]TJ/F61 7.9701 Tf -1.123 -8.72 Td [(B = A j k ,where x istheattitudestate,thehatdesignatesthequantityisastate estimate,therightsuperscriptdenotesthepre-orpost-updatestatus,andtheright subscriptstatesthattheattitudeisfrom F B to F A attimeindex k .Thetimeindexis oftendropped,butincludedwhenrelevanttothediscussion.Fortheattitudematrix,the notationis R B = A ,wheretherightsubscriptstatesthat R transformsvectorsrepresented in F A to F B .Finally,thefollowingequivalencywillbeusedextensively, R B = A = R x )]TJ/F61 7.9701 Tf -1.124 -8.721 Td [(B = A 3.2InertialAttitudeSensors Attitudesensorsaredevicesthatmeasurequantitiesusedtodetermineattitude. Inertialattitudesensorsmeasurevectorsrepresentedinthebodyframe,whichwith mathematicalmodelsofthevectorsrepresentedintheinertialframearerelatedthrough theinertialattitudematrix.Mostattitudesensorsmeasureanglesthatareusedto determineadirection,andthusthesesensorsproduceunitvectormeasurements. However,therearesensorsthatmeasurefullvectorinformation,thatis,magnitudeand direction.Alsonote,withoutlossofgenerality,thesensorframeandbodyframeare assumedtobealigned. Inthefollowingsection,measurementmodels,inertialreferencemodels,and errorsourcesaredescribedforthreetypicalinertialattitudesensors.Sunsensorsand magnetometersaretypicallysmall,lowpower,sensorsusedforlowprecisionattitude measurements.Thesesensorswillbereferredtoascoarseattitudesensors.Itis importanttonotethathigheraccuracyversionsofthesesensorsexist,buttypicallyat theexpenseoflargeSWaP.Startrackersareneprecisionattitudesensors,whichare typicallyhigherinSWaPthanthecoarsesensors.Therefore,startrackersareideally suitedforattituderesourcesharinginprecisionpointingapplications. 68

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3.2.1SunSensors Letthedirectionfromthespacecrafttothesunbedenotedby s ,andreferredto asthesunvector.Sunsensorsmeasurethesunvectorwithrespecttothebodyframe, s B .Givenamathematicalmodeloftheinertiallyrepresentedsunvector, s I ,theinertial attitudematrixrelatesthemeasuredandmodeledsunvectorthrough B s = R B = I I s Asunsensormeasuresthesunvectorwithameasurementmodelgivenby B s = R B = I I s + s where s iswhitezero-meanGaussiannoise.Therandomnoiseinasunsensoris aresultofseveralfactorssuchas,EarthandLunaralbedo,reections,surfacearea deviations,andtemporalvariationsinthesun'selectromagneticradiationoutput[73,74]. TheinertialmodelforthesunvectorisgivenasafunctionoftheJuliandate,which speciestheeclipticlongitudeofthesun, e andtheobliquityoftheecliptic, [55].The inertialsunvectorisgivenby I s = 2 6 6 6 6 4 cos e cos sin e sin sin e 3 7 7 7 7 5 3.2.2Magnetometers Letthemagneticeldvectorataparticularpositionbedenotedby m .Unlikethe sunvectordirection,whichisaunitvector,themeasuredmagneticeldvectorisatrue vectorwithmagnitudeanddirectionandmustbenormalized,whichyields B m .Given amathematicalmodelofthenormalizedinertiallyrepresentedmagneticeldvector, I m ,theinertialattitudematrixrelatesthemeasuredandmodeledmagneticeldvector 69

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through B m = R B = I I m Athree-axismagnetometerTAMmeasuresthemagneticeldvectorwitha measurementmodelgivenby B m = R B = I I m + m where m iswhitezero-meanGaussiannoise.TherandomnoiseintheTAMisaresult ofinternalmagneticandelectricelds.Inaddition,errorsthatexistintheinertialmodel canbeincludedinthemeasurednoise.Thepresenceofinternaleldstypicallylimits theuseofmagnetometerstolowerorbits,asthestrengthoftheeldfollowsaninverse cubelawwithdistancefromthesource.Systematicerrorsourcessuchassensor misalignmentcanexist,butcanbeaccountedforthroughcalibration. Severalinertialmagneticmodelsexists,rangingfromalow-ordermagneticdipole modeltohigh-ordersphericalharmonicsmodels.OneexampleistheWorldMagnetic ModelWMM[75].TheWMMisa12 th -ordersphericalharmonicsmodelthatincludes coreandsurfaceeffectsaswellaslinearsecularvariations.Themagneticdipolemodel isgivenby I m = R 3 H 0 r 3 3 )]TJ/F101 11.9552 Tf 5.479 -9.683 Td [(d T r r r 2 )]TJ/F101 11.9552 Tf 11.955 0 Td [(d # where R istheradiusoftheEarth, H 0 isthemagneticeldintensitycomputedfrom therstordercoefcientsoftheWMMatthedateandtimeofinterest, r istheorbital positionvectorininertialcoordinates,and d isthedirectionofthedipoleaxisininertial coordinates.Themagneticdipolemodeisusedastheinertialreferenceforallmagnetic eldvectorsimulationsinthiswork. 70

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3.2.3StarTrackers Startrackers,likesunsensors,measuredirectionstoinertiallyknownpoints. However,measurementsofstarsdifferfromthesuninthattheycanbeapproximated asinertiallyxedpointsinspace.Thus,theinertialrepresentationstardirectionsare independentofthespacecraftposition. Startrackersarehighlyaccurateinertialattitudemeasuringdevices,capableof sub-arcsecaccuracies.Unlikesunsensorsandmagnetometers,startrackerscan measuremorethanonereferencevector.Startrackerstakefocal-planemeasurements withaphotovoltaicsensor,andmapthosemeasurementstotheinertiallocation containedinanon-boardstarcatalog.AnexamplestartrackerisshowninFigure3-1A. Figure3-1Bshowsapin-holemodelforastartrackercapturingline-of-sightmeasurements. Themeasurementmodelforthestartrackeris A B Figure3-1.Descriptionofatypicalstartracker.AStartrackerhardware[76].BStar trackermeasurement. I b i = R B = I I b i + s where B b i isthemeasureddirectionofthe i th -starinthebodyframe, R B = I istheattitude matrixmappinginertialtobodycoordinates, I b i isthecatalogeddirectionofthe i th -star 71

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intheinertialframe,and s iswhitezero-meanGaussiannoisethatcorruptsthe measurement.Focalplanemeasurementsaremodeledthroughthevectorformofthe collinearityequations,suchthat B b i = 1 q f 2 + x 2 i + y 2 i 2 6 6 6 6 4 )]TJ/F61 11.9552 Tf 9.298 0 Td [(x i )]TJ/F61 11.9552 Tf 9.298 0 Td [(y i f 3 7 7 7 7 5 I b i = 2 6 6 6 6 4 b x i b y i b z i 3 7 7 7 7 5 where f isthefocallengthofthesensorand x i y i arethecoordinatesofameasurement onthefocalplane,and b x i b y i b z i aretheinertialcoordinatesofastaridentiedinthe catalog.Duetothenarroweld-of-viewofatypicalstartracker,attitudeaccuracyabout theboresightdirectionistypicallyanorderofmagnitudelargerthanthetransverse directions[77]. Startrackerstypicallyoperateintwomodes.Therstmodeistheattitude acquisitionorlost-in-spacemode,wherestarsareassociatedthroughpattern recognitionalgorithmswithrespecttoastoredstarcatalogofknownstars,which yields b I i .Thesecondmodetracksthealreadyidentiedstarsandincreasesaccuracy bylteringphenomenonsuchasstarstreakingusingangularvelocityinformation.For thepurposesofthisdiscussion,itisassumedthatthestarshavealreadybeenidentied andprocessed.Thevirtualstartrackercharge-coupleddeviceCCDmodeldescribed inReferences[78,79]isusedinthisresearchandshowninFigure3-2.Thegeometry isreviewedtoprovideacompletepictureofthevectormeasurementsoriginatingfroma startracker. ThevectorsidentifyingthecornersoftheCCDasdirectedfromthepinholeare denotedby f S 1 ,..., S 4 g .Thesecornervectorsrepresentedinrightascensionand 72

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declinationcoordinatesare S T 1 = 2 6 6 6 6 4 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin # cos # sin cos # cos 3 7 7 7 7 5 S T 2 = 2 6 6 6 6 4 sin # cos # sin cos # cos 3 7 7 7 7 5 S T 3 = 2 6 6 6 6 4 sin # )]TJ/F20 11.9552 Tf 11.291 0 Td [(cos # sin cos # cos 3 7 7 7 7 5 S T 4 = 2 6 6 6 6 4 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin # )]TJ/F20 11.9552 Tf 11.291 0 Td [(cos # sin cos # cos 3 7 7 7 7 5 where # istherightascensionand isthedeclination.Thesecoordinatesarerelatedto theeld-of-viewFOV,denotedby # x and # y ,as # = # x 2 ,cos 2 = cos # x +cos # y cos # x +1 ThenormaldirectionstotheCCDare n 12 = S 1 S 2 k S 1 S 2 k n 23 = S 2 S 3 k S 2 S 3 k n 34 = S 3 S 4 k S 3 S 4 k n 41 = S 4 S 1 k S 4 S 1 k Astarvector, S k ,ismeasuredbythesensorifallofthefollowingconditionsare satised: S T k n 12 < 0, S T k n 23 < 0, S T k n 34 < 0, S T k n 41 < 0. Therefore,thevirtualCCDissuppliedwithidentiedstarsfrompatternrecognitionand trackingalgorithmstodetermineifthestarisconsistentwiththeCCDgeometry.Given consistency,ameasurementiscaptured,whichissubjecttoerrorssourcessuchas misalignments,sensornoise,andambientlight. 3.2.4RateGyroscope Therategyroscopeisadeviceusedtomeasureangularratesofarigidbodywith respecttoaninertialframe.Gyroscopesareoftenreferredtoasinertialsensors, astheymakeuseoftheinertiapropertiesofthedevice.However,gyroscopesdo notdirectlyproduceinertialattitudeinformation.Instead,theyintegrateangularrate measurementsandmakeuseofareferenceinitialconditiontoyieldinertialattitude. Thismakesthegyroscopeapowerfultoolforpropagatinginertialattitudewithhigh 73

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Figure3-2.Descriptionofthegeometryoffocal-planemeasurementsusingavirtual CCD accuracy.Farrenkopf'sgyroscopemodelisassumed[80],whichisgivenby = )]TJ/F49 11.9552 Tf 11.955 0 Td [( )]TJ/F49 11.9552 Tf 11.955 0 Td [( v = u where istheratebias, v iszero-meanGaussianwhitenoisecorruption, u isa zero-meanwhiteGuassianrandomvariablethatrepresentsrandomwalkinthedrift rate.Thecovariancestatisticsoftherandomvariablesforwhitenoisecorruptionandthe randomwalkaregivenby, 2 v I 3 3 and 2 u I 3 3 ,respectively. 3.3DeterministicAttitudeDetermination GivenvectormeasurementsoriginatingfromthesensorsdescribedinSection3.2, thegoalistodeterminethespacecraftattituderelativetoaninertialframe.Attitude determinationisrelevantinthecontextofresourcesharingbecauseitallowsthe spacecrafttoinitializeitsattitudeestimator.AdoptingtheterminologyofWertz[73], attitudedeterminationiseitherdeterministicoroptimal.Thedeterministicsolution isobtainedwhentwonon-parallelvectorsaremeasuredinonebasisandarealso 74

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knowninanother.TheTRIADalgorithm[81]isadeterministicattitudedetermination method,whichconstructsthreeorthonormalvectorsusingthefamiliarGramm-Schmidt orthogonalizationprocessknownfromlinearalgebra.Optimalattitudedetermination occurswhenthesystemisoverdeterminedandisbasedinoptimizationtheoryand usethestatisticalpropertiesofthesensors.Ansurveyofseveralexistingattitude determinationalgorithmsisprovidedinReference[82]. Deterministicattitudedeterminationisdenedwhentherearetheminimumnumber ofvectormeasurementsnecessarytocomputetheinertialattitude.Black'sTRIAD algorithmsolvesthedeterministicattitudedeterminationproblem,giventwononparallel unitvectors, B b 1 and B b 2 measuredinthebodyandthecorrespondingunitvectors, I b 1 and I b 2 ,knownintheinertialframe.Theattitudematrixrelatesthebodyandinertial representationsthrough B b 1 = R B = I I b 1 B b 2 = R B = I I b 2 Thesevectorsspanaplaneandoverdeterminethesystem.Therefore,theTRIAD methodextractstheminimuminformationfromtothetwovectorstoconstructthree orthonormalvectorsusingtheGram-Schmidtorthogonalizationprocesssothat, B b 0 1 = B b 1 B b 0 2 = [ B b 1 ] B b 2 k [ B b 1 ] B b 2 k B b 0 3 =[ B b 0 1 ] B b 0 2 I b 0 1 = I b 1 I b 0 2 = [ I b 1 ] I b 2 k [ I b 1 ] I b 2 k I b 0 3 =[ I b 0 1 ] I b 0 2 Usingthenewlydenedprimevectors,theattitudefromtheTRIADalgorithmisgivenby R B = I = 3 X i =1 B b 0 i I b 0 i T Determiningtheattitudeofaspacecraftfrommorethantheminimumnumberof vectormeasurementsisknownasoptimalattitudedetermination[73].Solvingfor R B = I fromvectorsrepresentedintwobaseswasoriginallyposedbyWahba[83]asabatch 75

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leastsquaresproblemwithcostfunctiongivenby J R B = I = n X j =1 k B b j )]TJ/F61 11.9552 Tf 11.955 0 Td [(R B = I I b j k 2 InthedecadesthatfollowedthepublishingofWahba'sproblem,manysolutionshave beendeveloped.Therstmajorbreakthroughcamewiththeparameterizationofthe attitudematrixwiththequaternion. 3.3.1q-algorithm Davenport'sq-algorithm[84]providesasolutiontotheWahbaproblemby parameterizingtheattitudematrixwiththeEulersymmetricparametersandreducingthe minimizationproblemtotheeigenvalue-eigenvectorproblem.ByrewritingtheWahba problemas J R B = I = n X j =1 B b j )]TJ/F61 11.9552 Tf 11.956 0 Td [(R B = I I b j T B b j )]TJ/F61 11.9552 Tf 11.955 0 Td [(R B = I I b j Expandingtheright-handsideofthecostfunctionshowsthattheonlytermdependent on R B = I is )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 P n j =1 B b j T R B = I I b j .Therefore,the R B = I thatminimizes J R B = I isfoundby maximizingthegainfunction,sothat argmax R B = I g R B = I = n X j =1 B b j T R B = I I b j Thisproblemcanbeshowntobeequivalentto argmax R B = I g R B = I = tr R B = I VW T where W = h B b 1 B b n i and V = h I b 1 I b n i ,duetothepropertyofthetrace operator.A3 x 3matrix, B ,knownastheattitudeprolematrix,canthenbedened suchthat B WV T .Themajorbreakthroughintheq-algorithmwastheuseofthe parameterizationoftheattitudematrixwiththeEulersymmetricparameters.Using Eq.2,thereducedgainfunctionis g q = q 2 4 )]TJ/F101 11.9552 Tf 11.955 0 Td [(q T q + q T S q +2 q 4 z T q 76

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where tr B T S B + B T ,and[ z ]= B T )]TJ/F61 11.9552 Tf 11.955 0 Td [(B Thisleadstothequadraticform g q = q T Kq where K 2 6 4 S )]TJ/F23 11.9552 Tf 11.955 0 Td [( I 3 3 z z T 3 7 5 Since K isasymmetricmatrix,ithasallrealeigenvaluesandisdiagonalizable. Additionally,alleigenvaluescanbeshowntosumtozero. Duetothequaternionunityconstraint,thisproblemcanbefurtherposedasa constrainedoptimizationproblem,where maxg q = q T Kq subjecttoq T q )]TJ/F20 11.9552 Tf 11.955 0 Td [(1=0. UsingthemethodofLagrangemultipliers,anaugmentedgainfunctionisgivenby Kq = q whichreducestotheclassiceigenvalue-eigenvectorproblem.Substituting Kq into Eq.3,yieldsthemostimportantresult g q = Therefore,theoptimalattitudequaternioneigenvectorisassociatedwiththemaximum eigenvalueof K ,suchthat Kq opt = max q opt Davenportusedthepowermethod[85]todeterminethemaximumeigenvalue,thus yieldingtheoptimalquaternion.Othermethodsforsolvingforthemaximumeigenvalue 77

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exists,aswellasotherparameterizationsoftheattitudematrix,whichhaveyieldedfast andnearlyexactsolutionstotheWahbaproblem. 3.3.2OtherAttitudeDeterminationAlgorithms Newerdevelopmentsinattitudedetermination,sincethebreakthroughofthe q-Davenportmethod,differmainlyintheparameterizationoftheattitudematrixandthe associatedapproximationsornumericalmethodsusedtosolveWahba'sproblemwith thatparameterization.Shuster'sQUaternionESTimatorQUESTalgorithm[81,86] wastherstfastapproximatesolutiontotheWahbaproblempresentedinEq.3by introducingtheGibbsvectorparameterizationandusing max =1asaninitialguess foraNewton-Raphsonsolvertoapproximatelydeterminethemaximumeigenvalueof theattitudematrix.Thismethodprovedtobehighlyaccurateapproximationthatrapidly convergestotheoptimalsolution. Othernotableattitudedeterminationtechniqueshavebeendevelopedthatspanthe parameterizationsofattitude.Thequaternion-basedmethodsfollowasimilarapproach totheq-algorithmbututilizeotherformulationsforsolvingtheeigenvectorproblem. SingularValueDecompositionSVD[87]providesanexactsolutiontotheeigenvector problembutiscomputationallyexpensive.Otherquaternionmethodsinclude,Filter QUEST[88],REQUEST[89],EstimatoroftheOptimalQuaternionESOQ[90] andESOQ2[91],whichprovideclosed-formsolutionstotheoptimalattitude.A directattitudematrixmethodwasdevelopedinFastOptimalAttitudeMatrixFOAM [92]thatbypassestheparameterizationtothequaternionaltogether.EULER-2and EULER-n[93]parameterizetheattitudematrixinWahba'sproblemusingtheaxis-angle representation.Inthepastdecade,MRPshavebecomemorepopularasanattitude representationandasolutiontoWahba'sproblemwasfoundbyparameterizationwith theMRPsbyModiedRodriguesAttitudeDeterminationMRAD[94].Anelegant solutionwasrecentlyfound,calledOptimalLinearAttitudeEstimatorOLAE[95],which usedtheCayleytransformtotransformWahba'sproblemintoaleastsquaresproblem 78

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withouthavingtosolvetheeigenvectorproblem.Mostrecently,theCayleytransform wasappliedtoconstructahigher-ordermethodcalledtheCayleyAttitudeTechnique [96]. Therearemanyattitudedeterminationtechniquestochoosefromforresource sharingapplications.Mostofwhichseekanefcientsolutiontoaneigenvalue-vector problem.However,themainpurposeofattitudedeterminationinthiscontextisto initializetheattitudeestimator,whichisthefocusofthenextsection. 3.4KalmanFilterReview Althoughattitudedeterminationtechniquesarecapableofdeterminingtheattitude directlyfromvectormeasurements,lteringthemeasurementscanleadtosignicant improvementsinaccuracy[97].AttitudeestimationtypicallyreliesonKalmanltering andthusabriefreviewofimportantpropertiesoftheKalmanlterarediscussed. Allstateinformationismeasuredorderivedfrommeasurementsobtainedwith asensororfromasystemmodel.Measurementsareneverperfectandaresubject tonoise.Additionally,allmodelsofasystemaresimplicationsofphysicalreality, whereunmodeledeffectsaretermedperturbations.Asaresult,stateinformationis neverpreciselyknownfrommeasurementsnorfrommodels.Stateestimationisthe processofdeterminingsystemstatesfromnoisymeasurementsandperturbedmodels. Anestimateissimplyanapproximationofthestategivenimperfectknowledge.An estimatorisamappingthattakesinmeasurementsandperturbedsystemdynamicsand outputsanestimateofthesystemstate.Themostappliedsequentialestimatoristhe Kalmanlter,whichisarecursiveoptimallterandone-steppredictor. Attituderesourcesharingisintimatelyconnectedtotheuncertaintyintheattitude statesofthechiefanddeputies.Uncertaintyarisesduetoimperfectknowledgeofa systemstateorprocess.Probabilitytheoryservesasthebasisofhandlinguncertainty inthisresearch.Therearemanymethodsofdeterminingattitudebasedonvector 79

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measurements.However,givenknowledgeoftheuncertaintyinthesystemandsensor measurements,betterperformancecanbeachievedthroughltering. BeforepresentingtheKalmanlterequations,itisnecessarytointroduceseveral importantpropertiesofthesystemnoise.Theoriginofsensornoiseissimilartothat ofperturbationsinasystemdynamicsmodel,wherenoiseresultsfromunmodeled effectsinthesensor.Noisecaneitherbecorrelatedintimeoruncorrelatedintime, wherewhitenoiseisuncorrelatedintimeandcolorednoiseiscorrelatedintime.In additiontocorrelation,thestatisticsofnoisecanbedescribedbymanyprobability densityfunctions.AllofthenoisetermsinthisdiscussionwillbemodeledasGaussian, asnecessaryfortheoptimalityoftheKalmanlter.Theprobabilitydensityfunctionfor aGaussianrandomvariableisfullydescribedbyitsmeanandcovariance.Another propertyofinterestisthelinearityofthestateevolutionofthesystem.TheKalman lterisonlyoptimalforlinearsystems,butthisconictswiththerealitythatmost physicalsystemsexhibitnonlinearity.Table3-1providesasummaryoftheassumptions necessaryfortheoptimalKalmanlter. Table3-1.AssumptionsforapplicationoftheKalmanlter CharacteristicAssumption ObservabilityThesystemstatesofinterestmustbeobservable SystemDynamicsLinear NoiseGaussianzero-meanprobabilitydensityfunctions Whitenoisesequencesindependenceintime Independenceofprocessandmeasurementnoise GiventhatalloftheassumptionsinTable3-1aretrue,theKalmanlteristhe recursivestateestimatorandone-steppredictor,whichconvergestotheoptimalstate estimatesunderallmeaningfulmetricse.g.,meansquarederror,mode,median,or varianceandachievestheCramer-Raolowerbound[9799]. Considerthecontinuouslinearsystemsdynamicsgivenby x t = F t x t + G t w t 80

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where x t isthestateofthesystem, F t isthestatematrix, G t istheprocessnoise inputmatrix,and w t istheprocessnoise.Thestatisticsoftheprocessnoisearegiven by E [ w t ] = 0 E w t w t T = Q t t )]TJ/F23 11.9552 Tf 11.955 0 Td [( where t )]TJ/F23 11.9552 Tf 12.071 0 Td [( istheDiracdeltafunctionand E [ ] istheexpectationoperator.Thelinear measurementmodelisgivenby y t = H t x t + v t where y t arethemeasurements, H t istheobservationmatrix,and v t isthewhite zero-meansensornoisewithstatisticsgivenby E [ v t ] = 0 E v t v t T = V t t )]TJ/F23 11.9552 Tf 11.955 0 Td [( Since F t and G t aretime-varyingfunctions,thestatetransitionmatrixSTMis thesolutiontothefollowingdifferentialequation @ @ t t t 0 = F t t t 0 t 0 t 0 = I n n Fortimeinvariantsystems,theSTMisgivenby t t 0 = e F t )]TJ/F61 7.9701 Tf 6.587 0 Td [(t 0 Fortheremainderofdiscussions,itisassumedthatthetimestepissufcientlysmall fortheSTMtobeaccuratelyapproximatedbythematrixexponential.Additionally,the modelaboveisforcontinuouslinearsystems,howeverinthisresearch, allsystems willbeassumedtobediscreteasimplementedon-boardaspacecraftwhereanatural discretizationoccurswiththeregularsamplingoftherategyroscope 81

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Thecontinuousdynamicscanbediscretizedtoyield x k +1 = k x k + k w k where k isthediscretestatetransitionmatrixand k isthediscreteprocessnoise matrix.Similarly,thediscretemeasurementmodelisgivenby y k = H k x k + v k where v k 0 R k .Thediscretestatetransitionmatrixisgivenby k = e F t k t where t = t k +1 )]TJ/F61 11.9552 Tf 11.955 0 Td [(t k However,thediscreteprocessnoisecovariancematrix, Q ,requiresthedifcult computationoftheintegralgivenby Q = Z t k +1 t k e F t k +1 )]TJ/F24 7.9701 Tf 6.586 0 Td [( G Q G T e F T t k +1 )]TJ/F24 7.9701 Tf 6.587 0 Td [( d Agooddiscussiononthediscretizedprocessnoisecovariancematrixisfoundin Reference[97],wherethemainnumericalresultcomesfromReference[100].Abrief solutionisprovided. First,theaugmentedmatrix, A ,isformedby A = 2 6 4 )]TJ/F61 11.9552 Tf 9.298 0 Td [(F t k G t k Q t k G t k 0 F T t k 3 7 5 t ItwasshowninReference[100]thattakingmatrixexponentialof A producesthe discretestatetransitionmatrixandprocesscovariancematrixexactlyfortimeinvariant 82

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systems,suchthat e A = 2 6 4 B 11 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 k Q 0 n n T k 3 7 5 where B 11 isnotusedanyfurther.Fortime-varyingsystems,thissolutionisaccurate forsufcientlysmalltimesteps.However,forsmallenoughtimestepstheSTMcanbe approximatedwiththefollowingrst-ordermodelsforthestate-transitionmatrixgivenby = I n n )]TJ/F61 11.9552 Tf 11.955 0 Td [(F t k t andtheprocesscovariancematrixcomputedby Q k = G t k Q t k G T t k t ArigorousderivationofthediscreteKalmanltercanbefoundinReferences[98, 99].ThediscreteKalmanltercontainsseveralsteps.Therststepistheinitialization ofthelter,wheretheinitialstatesandcovarianceareassumedbasedonknowledge oftheproblemofinterest.Nextthegainiscomputed.TheKalmangainisthedriving forcebehindtheoptimalityofthelterandbalancesuncertaintyinthedynamicsprocess andthemeasurementsystem.Nextthestatesandcovarianceareupdatedusingthe computedgainwhichmodiestheinnovationtocorrectthepredictedstatefromtheprior information.Finally,thestatesandcovariancearepropagatedone-stepprediction tothenexttimesteptobeupdatedbyfuturemeasurements.Thenalresultsofthe KalmanlteraresummarizedinTable3-2.Note,thestatecovariancematrixisdenoted by P anddenedby P = E xx T 83

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Table3-2.DiscreteKalmanlter Initialize x t 0 = x 0 P t 0 = P 0 Gain K k = P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.916 -7.463 Td [(k H k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k + V )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 Update P + k = I n )]TJ/F61 9.9626 Tf 9.963 0 Td [(K k H k x )]TJ/F61 6.9738 Tf -0.917 -7.462 Td [(k P )]TJ/F61 6.9738 Tf -0.697 -7.462 Td [(k x + k = x )]TJ/F61 6.9738 Tf -0.916 -7.463 Td [(k + K k y k )]TJ/F101 9.9626 Tf 9.963 0 Td [(h k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k Propagate P )]TJ/F61 6.9738 Tf -0.698 -7.462 Td [(k +1 = k P + k T k + Q k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k +1 = k x + k TheKalmanlterisanimportanttoolforlinearsystems,buttheattitudesystem dynamicsarenonlinear.Therefore,theKalmanltercannotbedirectlyapplied. HandlingthisnonlinearityisthedomainoftheextendedversionoftheKalmanlter. 3.5ExtendedKalmanFilter Considerthecontinuousnonlinearsystemthatisafneintheprocessnoisesuch that, x t = f x t + g t w t where f x t isanonlinearfunctionofthestatesand g t isanonlinearprocessnoise function.Themeasurementmodelisgivenby y t = h x t + v t where h x t isanonlinearfunctionofthestates. Fornonlinearsystems,suchastheevolutionofattitudeforarigidbody,theKalman lterdoesnotproducetheoptimalestimate.However,therearemodicationstothe Kalmanlterfornonlinearsystems,suchastheextendedKalmanlterEKF[101,102] andtheunscentedKalmanlter[48],aswellasMonteCarlo-basedmethods[103], thatseekapseduo-optimalestimate.TheextendedKalmanlterwasrstapplied 84

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bySchmidttotheproblemofinertialnavigationinthe1960s[101],andsatisesthe estimationrequirementsforthefollowingdevelopments. SomeaspectsofthesystemnonlinearityaremaintainedintheextendedKalman lter,whereasotheraspectsrequirealinearizationaboutthecurrentstateestimate nottobeconfusedwithalinearizationaboutthetruestate.Asummaryofthe modicationstotheKalmanlterisprovidedinTable3-3. Table3-3.ListofmodicationstotheKalmanlterfortheextendedKalmanlter StepModication GainSensitivitymatrixisformedfromalinearizationofthemeasurementfunctionabout thecurrentstateestimate UpdateThenonlinearmeasurementfunctionisusedintheinnovation PredictionOne-steppredictorutilizesthenonlinearsystemdynamics PropagationCovariancepropagationutilizestheJacobianofthenonlinearsystemdynamicsabout thecurrentstateestimate TheextendedKalmanlterfollowsthesamestepsasdescribedfortheKalman lter,butwiththemodicationsinTable3-3usingthelinearizationsofthestatematrix andobservationmatrixgivenby F t = @ @ x t f x t j x t )]TJ/F20 11.9552 Tf 145.885 -0.299 Td [( and H x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(k = @ h @ x j x )]TJ/F61 5.9776 Tf -0.733 -6.584 Td [(k 3.6ExtendedKalmanFilterforInertialAttitudeEstimation Attitudeestimationhasalonghistory,inwhichmanyofthemajorresultswere surveyedinReference[104].Likethedifferencesseeninattitudedetermination, thestructuraldifferencesinattitudeestimationcenteraroundthechoiceofattitude parameterization.Theprimarychallengescreatedbyattitudestatesareassociatedwith themultiplicativenatureoferror,attitudesingularities,andenforcementofconstraints. Theunitquaternionisapopularparameterizationforattitudeestimationduetoitsmany favorablepropertiesrefertoTable2-3.However,whencarryingallcomponentsofthe 85

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quaternion,theunityconstraintcausesthestateerrorcovariancematrixtobesingular. Thisisexpected,asthestatesarenolongerindependent.Ifthestateestimateand thetruestatearesufcientlyclose,thentheattitudeerrorstatecanbereducedtothe 3-stateattitudeerrorvector. InReference[104]itwasalsoshownthatwithoutlossofgenerality,theattitude EKFisassumedtomakeuseofarategyroscope,whichservesasasurrogatefora systemdynamicsmodel.Farrenkopf'sgyroscopemodelisassumed,whichincludesa ratebias,whitenoise,andrandomwalkinthedriftrate.Thestatesofinterestforinertial attitudeestimationaretheunitquaternion, q B = I ,relatingtheattitudefromthebodyto inertialframeandthegyrobias, .Inthissection,thesubscriptsonthequaternionwill bedropped,suchthattheattitudestateissimply q .Therefore,theerrorstateutilizedin theextendedKalmanlteraretheattitudeerrorvector, ,andthebiaserror, TheEKFrequiresaninitializationofthequaternionestimate, q ,aswellasthe gyroscopebias, ,andthestateerrorcovariance, P .Theinitialquaternionestimate shouldbeascloseaspossibletothetruequaternion,duetothelinearization,andcan beobtainedfromanyoftheattitudedeterminationtechniquesdescribedinSection3.3. Anaccurateinitialestimatemaintainsthelinearityoftheapproximationandensuresthat thelterwillnotdivergeduetononlinearities.Thegyroscopebiascanbemeasured onthegroundpriortolaunch,butwilltypicallybealteredduringlaunchandrequire updating.Thederivationforthesensitivitymatrixanddiscretepropagationarereviewed inthenextsection. 3.6.1SensitivityMatrixDerivation Vectormeasurementstaketheformof B b i = R q I b i + B i 86

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wheretheestimatedmeasurementisgivenby B b )]TJ/F61 7.9701 Tf -0.406 -8.501 Td [(i = R q )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [( I b i Thesensitivityisgivenbythedifferencebetweentheunknowntruthandtheestimated measurement,suchthat B b = B b )]TJ/F61 7.9701 Tf 11.955 7.928 Td [(B b )]TJ/F20 11.9552 Tf -52.674 -31.833 Td [(= R q I b i )]TJ/F61 11.9552 Tf 11.955 0 Td [(R q )]TJ/F20 11.9552 Tf 7.085 -4.937 Td [( I b i FromEq.2,thecompositionrelatingthetruequaternionparameterizedattitudematrix andtheestimatedstateattitudematrixyields R q = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.147 Td [([ ] R q )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [(. Thenalformofthesensitivityequationforasinglemeasurementis B b = h R q )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [( I b i i wherethecomponentsoftheattitudeerroraregivenby =[ e 1 e 2 e 3 ] T .Therefore,the sensitivitymatrixfor N vectormeasurementsofthecombinedquaternionandbiasstate vectoris H k x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(k = 2 6 6 6 6 4 h R q )]TJ/F20 11.9552 Tf 7.085 -4.338 Td [( I b 1 i 0 3 3 . h R q )]TJ/F20 11.9552 Tf 7.084 -4.338 Td [( I b N i 0 3 3 3 7 7 7 7 5 Thesevectormeasurementshavenosensitivitywithrespecttothebias.However,the biasisobservableasaresultofthecouplingbetweenthequaternionratesandthe angularvelocity. 87

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3.6.2DiscretePropagationDerivation Thelaststepondiscretepropagationwarrantsfurtherdescription.Usingapower seriesapproximation,thekinematicquantitiesarepropagatedby + k = k )]TJ/F20 11.9552 Tf 14.576 2.05 Td [( + k q )]TJ/F61 7.9701 Tf -0.646 -8.501 Td [(k +1 = + k q + k where + k = 2 6 4 cos 1 2 k + k k t I 3 3 )]TJ/F29 11.9552 Tf 11.955 9.684 Td [( + k + k )]TJ/F20 11.9552 Tf 13.59 2.05 Td [( + T k cos 1 2 k + k k t 3 7 5 and + k = sin )]TJ/F20 7.9701 Tf 6.675 -4.976 Td [(1 2 k + k k t + k k + k k Thecovarianceispropagatedby P )]TJ/F61 7.9701 Tf -0.836 -8.502 Td [(k +1 = k P + k T k + G k Q k G T k where G k = 2 6 4 )]TJ/F61 11.9552 Tf 9.299 0 Td [(I 3 3 0 3 3 0 3 3 I 3 3 3 7 5 = 2 6 4 11 12 21 22 3 7 5 88

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and 11 = I 3 3 )]TJ/F20 11.9552 Tf 11.956 -0.147 Td [([ ] f sin k k t g k k + [ ][ ] f 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(cos k k t g k k 2 12 = [ ] f 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos k k t g k k 2 )]TJ/F61 11.9552 Tf 11.955 0 Td [(I 3 3 t )]TJ/F20 11.9552 Tf 11.956 -0.146 Td [([ ][ ] fk k t )]TJ/F20 11.9552 Tf 11.955 0 Td [(sin k k t g k k 3 21 =0 3 3 22 = I 3 3 and Q k = 2 6 4 )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [( 2 v t + 1 3 2 u t 3 I 3 3 )]TJ/F29 11.9552 Tf 11.291 9.684 Td [()]TJ/F20 7.9701 Tf 6.675 -4.977 Td [(1 2 2 u t 2 I 3 3 )]TJ/F29 11.9552 Tf 11.291 9.683 Td [()]TJ/F20 7.9701 Tf 6.675 -4.976 Td [(1 2 2 u t 2 I 3 3 )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [( 2 u t I 3 3 3 7 5 AsummaryofthestepsfortheattitudeEKFprocessisprovidedinTable3-4. 3.6.3ExampleInertialAttitudeEKFwithaMagnetometerandSunSensorsfor LowPrecisionPointing Anexamplecoarsesunsensorandmagnetometer-basedextendedKalmanlter wasmodeledandsimulatedusingtheinitialstateinformationgiveninTable3-5,the timeparametersinTable3-6andtheinitialstateestimateinformationinTable3-7. ThespacecraftisplacedinalowEarthorbitatapproximately400 km altitude,where themagneticeldisstrong,butalsomoreuncertain,wherethemagnetometerhasa standarddeviationof1 .Thesunsensorsareassumedtoavoideclipsethroughoutthe simulationandhaveastandarddeviationof0.1 ThesunvectormodelanddipolemodeldescribedinSection3.2serveasthe knowninertialmodelsforthemeasuredvectors.Sunsensorsandamagnetometer producetwovectors,whichistheminimumnecessaryforthree-axisattitudedetermination. Theorbitalperiodforthisorbitisapproximately90minutes.Figure3-3showsthat duetotheperiodicityoftheorbitalmotion,thesunvectorandmagneticeldvector periodicallybecomenearlycollineartwiceperorbit,duetothesymmetryofthemagnetic eld.Whenthemeasurementsarenearthecollinearconguration,theerrorincreases 89

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Table3-4.InertialattitudeextendedKalmanlter Initialize x t 0 = x 0 = q T 0 T 0 T P t 0 = P 0 q t 0 = q 0 t 0 = 0 Gain K k = P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.916 -7.463 Td [(k H k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.917 -7.463 Td [(k + V )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 H k x )]TJ/F61 6.9738 Tf -0.917 -7.462 Td [(k = 2 6 6 6 4 h R q )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [( I b 1 i 0 3 3 . h R q )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [( I b N i 0 3 3 3 7 7 7 5 Update P + k = I 3 3 )]TJ/F61 9.9626 Tf 9.963 0 Td [(K k H k x )]TJ/F61 6.9738 Tf -0.916 -7.463 Td [(k P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k x + k = + T k + T k T = K k y k )]TJ/F101 9.9626 Tf 9.962 0 Td [(h k x )]TJ/F61 6.9738 Tf -0.916 -7.462 Td [(k y k = 2 6 4 B b 1 B b N 3 7 5 h k x )]TJ/F61 6.9738 Tf -0.916 -7.462 Td [(k = 2 6 4 R q )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [( I b 1 R q )]TJ/F20 9.9626 Tf 6.725 -3.616 Td [( I b N 3 7 5 q + k = q )]TJ/F61 6.9738 Tf -0.484 -7.463 Td [(k + 1 2 q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(k + k + k = )]TJ/F61 6.9738 Tf -0.526 -7.463 Td [(k + + k Propagate P )]TJ/F61 6.9738 Tf -0.698 -7.463 Td [(k +1 = k P + k T k + G k Q k G T k q )]TJ/F61 6.9738 Tf -0.483 -7.462 Td [(k +1 = + k q + k + k = k )]TJ/F20 9.9626 Tf 12.215 1.708 Td [( + k Table3-5.Stateinitializationsforcoarsesensorinertialattitudesimulations ParameterValueUnits Inertia: J diag 300,100,200 kg m 2 InitialStates: r 0 [ 6778.10.01620.0080 ] T m v 0 [ 0.00006.64113.8342 ] T m/s q 0 [ 0100 ] T 0 [ )]TJ/F20 9.9626 Tf 7.748 0 Td [(0.0030.0020.004 ] T rad/s Table3-6.Timeparametersforcoarsesensorinertialattitudesimulations ParameterValueUnits Time: t up 1 s t prop 0.1 s T 225 min 90

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Table3-7.Stateestimateinitializationsandsensorcharacteristicsforcoarsesensor inertialattitudesimulations ParameterValueUnits StateEstimate: q 0 [ 0100 ] T 0 10 deg = hr v 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 u 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 asseeninFigure3-5.Additionally,duetotheangularvelocityofthespacecraftshown inFigure3-4,inthiscaseresultingfromtorque-freemotion,theerrorineachaxis exhibitsoscillatorybehavioratthefrequencyoftheangularvelocity,asinformation iscontinuouslypassingbetweenthebodyaxes,giventheslowlyevolvingvector measurementswiththeorbitalmotion.Inthesesimulations,theattitudeisestimated tosub-degreeaccuracyasevidencedbythe3boundsinFigure3-5. Figure3-3.MagneticeldandsunvectorcollinearityforanexampleinertialattitudeEKF withamagnetometerandsunsensors 3.6.4ExampleInertialAttitudeEKFwithaStarTrackerforHighPrecision Pointing AnexamplestartrackerinertialattitudeEKFwasmodeledandsimulatedin MATLAB[105].Thesesimulationsmakeuseofthestartrackermodeldiscussedin Section3.2.ThestarcatalogaccompanyingReference[97,106]wasusedwithastar magnitudethresholdofsix.InthesesimulationstheFOVofthestartrackeris6 ,thus thereismuchlessinformationavailableforrotationsalongtheboresight,andless opportunitytoobservestarsfarawayfromtheboresightduetothenarrowFOV.A 91

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Figure3-4.TruebodyangularvelocityforanexampleinertialattitudeEKFwitha magnetometerandsunsensors Figure3-5.QuaternionerrorforanexampleinertialattitudeEKFwithamagnetometer andsunsensors maximumofsixstarswereassumedtobeabletobeprocessedwhenmeasurements weretaken.Thestartrackermodelassumesthatthestarassociationproblemhas alreadybeensolved.Theinitialpositionandvelocitystatesareslightlydeviatedfrom thepreviousexampleandthespacecraftstatesaresummarizedinTable3-8.Initial stateestimatesaregiveninTable3-9.ThesametimeparameterslistedinTable3-6 wereusedinthesesimulations.Ingeneral,theaccuracyofastartrackerwillimprove asthenumberofstarsidentiedincreases.Figure3-6depictsthenumberofstars availableoverthesimulation,whereitisassumedthatthemaximumnumberofstars 92

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Table3-8.Stateinitializationsforneprecisioninertialattitudesimulations ParameterValueUnits Inertia: J diag 300,100,200 kg m 2 InitialStates: r 0 [ 6778.100 ] T km v 0 [ 06.64123.8343 ] T km/s q 0 [ 1000 ] T 0 [ 0.0010.0010.001 ] T rad/s Table3-9.Stateestimateinitializationsforneprecisioninertialattitudesimulations ParameterValueUnits StateEstimate: q 0 [ 1000 ] T 0 0.1 deg = hr v p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(7 rad = s 1 = 2 u p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(10 rad = s 3 = 2 is6.FromFigure3-7,theerrorexhibitsspikeswhenthenumberofstarsisreduced. Anotherwellknownfactthatisobserved,isthatmeasurementscontainlessattitude informationaboutthesensorboresightthanthetransverseaxes.Thatis,aline-of-sight measurementparallelwiththeboresightdirectionprovidesnoinformationabout therotationaboutthataxis.Thestartrackeriscapableofproducinghighlyaccurate arc-secondlevelaccuracy,whichisordersofmagnitudebetterthanthesunsensorsand magnetometer-basedattitudeestimationshowninFigure3-5.Arc-secondswillserveas theprimarymeasureofattitudeerrorinthiswork,where1 =3600 arc )]TJ/F61 11.9552 Tf 11.955 0 Td [(second Figure3-6.AvailablestarvectorsforanexampleinertialattitudeEKFwithastartracker 93

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Figure3-7.QuaternionerrorforanexampleinertialattitudeEKFwithastartracker 3.7RelativeAttitudeSensors Recallthatattitudedescribestheorientationfromoneframewithrespecttoanother. Relativeattitudeisconcernedwiththeorientationofaframexedtoaspacecraftwith respecttoaframexedtoanotherspacecraftorothernon-inertialframe.Therefore, relativeattitudesensorsprovidethecapabilitytomeasurequantitiesthatcanbe relatedbacktorelativeattitude.Allrelativeattitudesensorsmeasurequantitiesthat providedirectioninformation.Examplesofrelativeattitudesensorsthatprovideasingle directionarerelativeglobalpositioningsystemGPSsignals[107,108]andvarious electromagneticsensingdevicessuchasradiofrequency[26,109].Anexampleof arelativeattitudesensorcapableofmeasuringmultipledirectionsisvision-based navigationVISNAV.ThisVISNAVsensorservesastheprimaryrelativeattitudesensor inthisresearchandisdiscussedinmoredetailbelow. Multi-directionrelativeattitudesensorsareanalogoustostartrackersforinertial sensing.VISNAVsensorsmeasurethedirectiontomultipleducialpointsonanobject ofinteresttodeterminerelativeattitudeandpositionasshowninFigure3-8.Thechief usesdetectionopticsandafocal-planedetectorthataredesignedtodifferentiate receivedsignalsfromthedeputy'sopticalducialstodeterminemultipleline-of-sight 94

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vectors.AnexamplesystemofthisnatureisdescribedinReference[110],which requiresthespacecrafttobeincloseproximity.Duetothecloseproximityrequirement, relativeattitudeandrelativepositionarecoupled. Figure3-8.NotionaldepictionofaVISNAVmeasurement VISNAVutilizessimilarhardwaretoastartracker.However,differentiatingducials isnotapatternrecognitionproblemlikethestartracker,butinsteaditisaproblemof physicallydifferentiatingtheducialbytheemittedsignals.Giventhisdifferentiation, directionstotheducialsarecapturedwithafocal-planedetectorandmodeledusing thevirtualCCDdescribedinSection3.2.Sincetheduciallocationsareknownin deputycoordinates,knowledgeoftherelativepositionprovidesenoughinformationfor theline-of-sightvectorstobedeterminedasshowninFigure3-9.Themeasurement modelforVISNAVsensoris C b D = C i = R C = D D b D = C i + vis where C b D = C i isthemeasureddirectionofthe i th -ducialinthechiefframe, R C = D is theattitudematrixmappingdeputytochiefcoordinates, D b D = C i isthedirectionofthe i th -ducialinthedeputyframe,and vis iswhitezero-meanGaussiannoisecorrupting themeasurement.Focalplanemeasurementsaremodeledthroughthevectorformof 95

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Figure3-9.DescriptionoftheVISNAVsensorgeometry thecollinearityequations,suchthat C b D = C i = 1 q f 2 + x 2 i + y 2 i 2 6 6 6 6 4 )]TJ/F61 11.9552 Tf 9.299 0 Td [(x i )]TJ/F61 11.9552 Tf 9.299 0 Td [(y i f 3 7 7 7 7 5 D b D = C i = 1 p X i )]TJ/F61 7.9701 Tf 11.955 3.454 Td [(D x i 2 + Y i )]TJ/F61 7.9701 Tf 11.955 3.454 Td [(D y i 2 + Z i )]TJ/F61 7.9701 Tf 11.955 3.454 Td [(D z i 2 6 6 6 6 4 X i )]TJ/F61 7.9701 Tf 11.955 4.339 Td [(D x i Y i )]TJ/F61 7.9701 Tf 11.955 4.339 Td [(D y i Z i )]TJ/F61 7.9701 Tf 11.955 4.339 Td [(D z i 3 7 7 7 7 5 where X i Y i Z i aretheknownduciallocationsindeputycoordinates, x i y i arethe focalplanecoordinatesoftheducialmeasurement, D i istherelativepositionvector betweenthechiefanddeputy,and f isthefocallengthoftheVISNAVsensor. The VISNAVsensorwillserveastherelativeattitudesensorfortheremainderofthiswork. AlimitationoftheVISNAVsensoristhatitonlyprovidesprecisemeasurements incloseproximity.Asthedistancebetweenthesensorandducialsincreases,the ducialapproachacoalescedpointsource,whichcontainsnoattitudeinformation. Therefore,utilizationofthissensorislimitedtoclose-proximityapplications,suchas 96

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distributing-sensingformation-yingmissions.Itisalsoassumedthatadaptiveoptics areimplementedtoalterthefocallengthoftheVISNAVsensorandtheintensityoflight emittedfromtheducialsasafunctionofrange. 3.8DeterministicRelativeAttitudeDetermination Inthegeneralcase,whererelativepositionisunknown,severalmethodshave beendevisedtodeterminerelativepositionandattitudesimultaneouslyfromthe VISNAVmeasurements.TherstapproachusesaGaussianLeastSquaresDifferential CorrectionGLSDCprocess[110]toiterativelydeterminerelativepositionandattitude. Thismethodsuffersfromsensitivitytoinitialconditionerrorsandislesscomputationally thanothermethods.Thenextapproachbypassesattitudedeterminationandutilizes lteringtodirectlyconvergeuponthestates[111].However,althoughthismethodis robusttoerrorsininitialconditions,itrequiresconvergencebeforeasuitablerelative attitudeestimateisobtained. Twomethodsforrelativeattitudeandpositiondeterminationwerepresentedin Reference[112].Therstmethod,LinearAlgebraResectionApproachLARA,creates ahomogeneoussystemofequationsandndsthesolutiontotherelativeattitude andpositionproblemasthelefteigenvectorassociatedwiththeminimumsingular valueofthehomogeneousequations.Throughanull-spaceargument,thiscase imposesarestrictionthatatminimum,6ducialsmustbevisible.Thisisprohibitivein application.Thesecondapproach,FirstAttitudeFreeApproachFAFA,recaststhe combinedrelativepositionandattitudeproblemintoanonlinearsystemofequations parameterizedsolelybytherelativeposition.Theresultingequationisan8 th -order polynomialwithrootsyieldingtherelativeposition.Afterisolatingtherealsolutionsof thispolynomial,therelativeattitudeissolvedforusinganyofthemanyattitude-only determinationtechniques,suchasQUESTorOLAE.Althoughthistechniqueis fairlyrobust,itstillrequiresnumericaliteration.Thenalapproachutilizesposition triangulationtodeterminedistancestotheducials[113].Thesedistancesthenprovide 97

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enoughinformationtodetermineattitudeindependentlyofposition.Finally,theattitude estimateisusedtodeterminetherelativeposition. Contrarytothegeneralcaseofrelativeattitudeandpositiondetermination,if relativepositionisalreadyknown,thentheWahbaproblemcanbesolvedtodetermine relativeattitude.Thiswillbethecasethatisassumedinthiswork,asthefocusison attituderesourcesharingandnotonthegeneralcaseoftranslationsandrotations. 3.9InertialAttitudeEstimationinaDisaggregatedSystem Fortheattitudeestimatorfordisaggregatedspacecraft,thegoalistohaveeach spacecraftinthenetworkmaintainanestimateoftheinertialattitudeofallspacecraft inthenetwork,butonlyusinglocalmeasurements.Thechiefiscapableofmeasuring itsinertialattitudeaswellastherelativeattitudetothedeputies.Thedeputiesare onlycapableofcapturingvectormeasurementstoestimatetheirinertialattitude.To accomplishthis,thechiefutilizesaninertialsensor,suchasastartracker,andarelative sensor,suchasVISNAV. Inthecaseofasinglechiefand n deputies,thestatevectoris x = q T C q T D 1 ,..., q T D n T C T D 1 ,..., T D n T andtheerrorstateis x = T C T D 1 ,..., T D n T C T D 1 ,..., T D n T ThedifferencewiththeEKFfordisaggregatedspacecraftwithrespecttothestandard inertialattitudeEKFisinthederivationofthesensitivitymatrixandthestateand covariancepropagation. 3.9.1SensitivityMatrixDerivation Considerthecaseofthechiefmeasuringtherelativeattitudestatesofthe i th -deputy anditslocalstates.Itisassumedthattherelativepositionvectorbetweenthechief and i th -deputyisknownandanyuncertaintythereinisaddedasprocessnoiseinthe 98

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dynamicsmodel.Therefore,the j th -vectorindeputycoordinatesismappedtochief coordinatesthroughtherelativeattitudematrix,suchthat C b D i = C j = R q D i = C D b D i = C j Alternatively,insteadofmappingtheknowndeputyrepresentedvectorsthroughthe relativeattitudematrix,theattitudecompositionlawproducestheequivalentforms C b D i = C j = R q C R T q D i D i b D i = C j and C b )]TJ/F61 7.9701 Tf -0.406 -8.72 Td [(D i = C j = R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.579 -8.501 Td [(D i D i b D i = C j Thesensitivityequationisonceagainthedifferencebetweenthetrueandestimated measurement,suchthat C b = C b D i = C j )]TJ/F61 7.9701 Tf 11.955 7.928 Td [(C b )]TJ/F61 7.9701 Tf -0.406 -8.72 Td [(D i = C j = R q C R T q D i D i b D i = C j )]TJ/F61 11.9552 Tf 11.955 0 Td [(R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.502 Td [(D i D i b D i = C j FromEq.2,thecompositionrelatingthetruequaternionparameterizedattitudematrix andtheestimatedstateattitudematrixyields R q C = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.147 Td [([ C ] R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C and R q D i = )]TJ/F61 11.9552 Tf 5.48 -9.683 Td [(I 3 3 )]TJ/F29 11.9552 Tf 11.955 9.683 Td [( D i R q )]TJ/F61 7.9701 Tf -0.58 -8.502 Td [(D i 99

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forthechiefanddeputy,respectively.Therefore, C b = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.147 Td [([ C ] R q )]TJ/F61 7.9701 Tf -0.579 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i )]TJ/F61 11.9552 Tf 5.48 -9.684 Td [(I 3 3 + D i D i b D i = C j )]TJ/F61 11.9552 Tf 11.955 0 Td [(R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C A T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i b D i = C j = R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i D i b D i = C j )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ C ] R q )]TJ/F61 7.9701 Tf -0.579 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i b D i = C j )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ C ] R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i D i b D i = C j Neglectingthehigher-ordertermsinEq.3resultsintherstordermodelofthe sensitivityequation, C b = h R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.579 -8.501 Td [(D i D i b D i = C j i C )]TJ/F61 11.9552 Tf 11.955 0 Td [(R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i h D i b D i = C j i D i ThesensitivitymatrixforasingleVISNAVmeasurementon-boardthechiefis H C x )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [(= h R q )]TJ/F61 7.9701 Tf -0.58 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.579 -8.502 Td [(D i D i b D i = C j i )]TJ/F61 11.9552 Tf 9.298 0 Td [(R q )]TJ/F61 7.9701 Tf -0.579 -8.617 Td [(C R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i h D i b D i = C j i 0 3 3 0 3 3 ThesensitivitymatrixforasingleVISNAVmeasurementon-boardthechieffollows directlyfromthetwo-spacecraftderivation,exceptthatthecolumndeputysensitivity correspondswiththemeasuredspacecraft.Supposethe i th -deputyismeasuredbythe VISNAVsensor,thenthesensitivitymatrixis H C x )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [(= h h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C j i 0 3 3 i )]TJ/F20 6.9738 Tf 6.226 0 Td [(1 )]TJ/F61 9.9626 Tf 7.749 0 Td [(R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i h D i b D i = C j i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.226 0 Td [(3 i i Therefore,for N starand M ducialmeasurements,thechiefsensitivitymatrixis H C x )]TJ/F20 11.9552 Tf 7.085 -4.936 Td [(= 2 6 6 6 6 6 6 6 6 6 6 6 4 h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C C b 1 i 0 3 3 i )]TJ/F20 6.9738 Tf 6.226 0 Td [(1 0 3 3 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i . . . h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C C b N i 0 3 3 i )]TJ/F20 6.9738 Tf 6.226 0 Td [(1 0 3 3 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C ,1 i 0 3 3 i )]TJ/F20 6.9738 Tf 6.226 0 Td [(1 )]TJ/F61 9.9626 Tf 7.749 0 Td [(R q )]TJ/F61 6.9738 Tf -0.484 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i h D i b D i = C ,1 i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.226 0 Td [(3 i . . . h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C M i 0 3 3 i )]TJ/F20 6.9738 Tf 6.226 0 Td [(1 )]TJ/F61 9.9626 Tf 7.748 0 Td [(R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i h D i b D i = C M i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i 3 7 7 7 7 7 7 7 7 7 7 7 5 Thissensitivitymatrixisnolargerinrowsthanthetwo-spacecraftcase.However, therecanbemanymorecolumns,wherethenumberofcolumnsgrowsatarate 100

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of6+6 n +1,where n isonceagainthenumberofdeputies.Mostimportantly, duetolinearityandsuperposition,Murrell'salgorithmcanbeusedtoprocesseach measurementsequentially. Similarly,the i th deputysensitivitymatrixwith L inertialvectormeasurementsis H D i x )]TJ/F20 11.9552 Tf 7.084 -4.936 Td [(= 2 6 6 6 6 4 0 3 3 0 3 3 i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 h R q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i b 1 i 0 3 3+6 n )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 i . . . 0 3 3 0 3 3 i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 h R q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D i D i b L i 0 3 3+6 n )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 i 3 7 7 7 7 5 Asthedeputycanonlyobtainlocalinertialmeasurements,thesensitivitymatrixhas columnsofzerosforthechiefstateandbiases. 3.9.2DiscretePropagationDerivation Considerthecontinuous-timeerrorstatedynamicstobeusedinadisaggregated estimationschemeforthe n -deputycase,suchthat x dc = F da x dc + G da w dc where F da = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F20 11.9552 Tf 11.291 -0.147 Td [([ c ] 0 3 3 )]TJ/F61 11.9552 Tf 9.298 0 Td [(I 3 3 0 3 3 0 3 3 0 3 3 0 3 3 )]TJ/F20 11.9552 Tf 11.291 -0.147 Td [([ d ,1 ] 0 3 3 )]TJ/F61 11.9552 Tf 9.298 0 Td [(I 3 3 0 3 3 0 3 3 . . . . 0 3 3 0 3 3 0 3 3 )]TJ/F20 11.9552 Tf 11.291 -0.146 Td [([ d n ] 0 3 3 )]TJ/F61 11.9552 Tf 9.299 0 Td [(I 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 . . . . 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 101

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G da = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F61 11.9552 Tf 9.299 0 Td [(I 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 )]TJ/F61 11.9552 Tf 9.299 0 Td [(I 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 I 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 3 I 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Thecontinuous-timeprocessnoisecovariancematrixisgivenby Q da =diag )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [( 2 C v I 3 3 2 D 1 v I 3 3 ,..., 2 D n v I 3 3 2 C u I 3 3 2 D 1 u I 3 3 ,..., 2 D n u I 3 3 Thediscretesolutionsforthestatetransitionmatrixandprocesscovariancematrixare alsofoundthroughEq.3andEq.3,respectively.Finally,eachofthequaternion andbiasstatesarepropagatedthroughEq.3andEq.3,respectively. Thechiefanddeputyeachindividuallyestimatetheentirestateofthetwo spacecraftnetwork.Thechiefhasthepotentialforfullobservabilityofthesystem states.Whereas,thedeputyonlyhasobservabilityoveritsownstatesi.e.,thedeputy quaternionandbias.Forinstance,withoutcombininginformationfromthechief estimates,thedeputiesknowledgeofthechiefinertialattitudewillalwaysdiverge. Table3-10summarizestheEKFtobeusedinthedisaggregatedsystemforattitude resourcesharing.However,forthedeputytomakeuseoftheknowledgeofitsstates fromthechief,datafusionisrequired.Abenetofthisdatafusion,isthatthechiefalso standstogaininperformancethroughthedatafusionprocess.Resultsaredeveloped inChapter4.Ifthetimestepissmallenough,thenthecomputationalcomplexityof thepropagationonlyincreaseswiththeorderofmatrixmultiplication.However,ifthe small-time-stepapproximationdoesnothold,thenthematrixexponentialmustbe computed,whichiscomputationallyexpensiveforlargestatesizes.Therefore,this approachisonlyforsmallernetworksofspacecraftrequiringresourcesharing. 102

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Table3-10.ExtendedKalmanlterforinertialattitudeestimationinadisaggregated system Initialize x t 0 = x 0 = q C j T 0 q D 1 j T 0 ,..., q D n j T 0 C j T 0 D 1 j T 0 ,..., D n j T 0 T P t 0 = P 0 Gain K k = P )]TJ/F61 6.9738 Tf -0.697 -7.462 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.917 -7.462 Td [(k H k x )]TJ/F61 6.9738 Tf -0.917 -7.462 Td [(k P )]TJ/F61 6.9738 Tf -0.698 -7.462 Td [(k H T k x )]TJ/F61 6.9738 Tf -0.917 -7.462 Td [(k + R )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 Chief: H C x )]TJ/F20 9.9626 Tf 6.725 -3.616 Td [(= = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 h R q )]TJ/F61 6.9738 Tf -0.483 -7.563 Td [(C C b 1 i 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 0 3 3 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i . . . h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C C b N i 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 0 3 3 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C ,1 i 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 )]TJ/F61 9.9626 Tf 7.749 0 Td [(R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.484 -7.463 Td [(D i h D i b D i = C ,1 i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i . . . h R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.484 -7.463 Td [(D i D i b D i = C M i 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 )]TJ/F61 9.9626 Tf 7.749 0 Td [(R q )]TJ/F61 6.9738 Tf -0.484 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i h D i b D i = C M i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Deputies: H D i x )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [(= 2 6 6 6 4 0 3 3 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 h R q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b 1 i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i . . . 0 3 3 0 3 3 i )]TJ/F20 6.9738 Tf 6.227 0 Td [(1 h R q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b L i 0 3 3+6 n )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 i 3 7 7 7 5 Update P + k = I 6 n 6 n )]TJ/F61 9.9626 Tf 9.962 0 Td [(K k H k x )]TJ/F61 6.9738 Tf -0.916 -7.463 Td [(k P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k x + = + T C + T D 1 ,..., + T D n + T C + T D 1 ,..., + T D n T = K k y k )]TJ/F61 9.9626 Tf 9.963 0 Td [(h k x )]TJ/F20 9.9626 Tf 6.724 -3.616 Td [( Chief: y C = 2 6 6 6 6 6 6 6 6 6 4 C b 1 C b N C b D i = C ,1 C b D i = C M 3 7 7 7 7 7 7 7 7 7 5 h C x )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [(= 2 6 6 6 6 6 6 6 6 6 4 R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C C b 1 R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C C b N R q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C ,1 R q )]TJ/F61 6.9738 Tf -0.484 -7.564 Td [(C R T q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b D i = C M 3 7 7 7 7 7 7 7 7 7 5 q + C j k = q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C j k + 1 2 q )]TJ/F61 6.9738 Tf -0.483 -7.564 Td [(C j k + C j k + C j k = )]TJ/F61 6.9738 Tf -0.526 -7.564 Td [(C j k + + C j k Deputies: y D i = 2 6 4 D i b 1 D i b L 3 7 5 h D i x )]TJ/F20 9.9626 Tf 6.725 -3.615 Td [(= 2 6 6 4 R q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b 1 R q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i D i b L 3 7 7 5 q + D i j k = q )]TJ/F61 6.9738 Tf -0.483 -7.463 Td [(D i j k + 1 2 q )]TJ/F61 6.9738 Tf -0.484 -7.463 Td [(D i j k + D i j k + D i j k = )]TJ/F61 6.9738 Tf -0.525 -7.463 Td [(D i j k + + D i j k Propagate P )]TJ/F61 6.9738 Tf -0.697 -7.463 Td [(k +1 = k P + k T k + Q k Chief:Deputies: q )]TJ/F61 6.9738 Tf -0.483 -7.563 Td [(C j k +1 = + C j k q + C j k q )]TJ/F61 6.9738 Tf -0.484 -7.462 Td [(D i j k +1 = + D i j k q + D i j k + C j k = C j k )]TJ/F20 9.9626 Tf 12.214 1.708 Td [( + C j k + D i j k = D i j k )]TJ/F20 9.9626 Tf 12.215 1.708 Td [( + D i j k 103

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3.9.3NotesonDisaggregatedAttitudeEstimation Thedisaggregatedapproachrequiresthatthegyromeasurementsbesharedover awirelesslinkattheregularsamplingintervalfordiscretepropagation.However,in Reference[104]itwasnotedthatalthoughthelargestnumericalburdenintheattitude Kalmanlteroccurswiththecomputationofthetransitionmatrixandthecontribution oftheprocessnoisetothestatecovariancematrix,thiscomputationcantakeplace atmuchlargertimescalesthanthestatepropagation.Therefore,eachdeputycan locallypropagatetheirstatesatthefrequencyoftheirlocalgyroupdateswithoutbeing burdenedsoheavilybytheprocessnoisecomputations. Theapproachtakeninthissectionshouldonlybeusedforsmallerformation yingnetworks,wherecommunicationisavailablewhenrequired.Modicationswillbe necessarytotheformulationiflargernetworksandirregularcommunicationisexpected. 3.10Summary Inertialandrelativeattitudedeterminationandestimationweresurveyed.Inertial attitudeisdeterminedbysolvingtheWahbaproblem.Therearemanysolutionsto thisproblem,suchasthepopularQUESTalgorithm.However,othermethodssuchas ESOQ-2andOLAEhavebeenshowntoprovidefastercomputationalsolutions.Dueto thenonlinearnatureofattitudeestimation,theExtendedKalmanFilterforquaternion kinematicswassurveyed.Next,relativeattitudedeterminationwassurveyed.Duetothe couplednatureofrelativeattitudeandposition,relativeattitudedeterminationismore complexthaninertialattitudedetermination.However,closed-formsolutionsdoexist. Inthelastpart,theEKFfordisaggregatedspacecraftwasdeveloped.Totheauthor's knowledge,thisistherstderivationandapplicationofrelativeattitudesensorsused todirectlyestimatetheinertialstatesofmultiplespacecraft.Theseequationshave applicationtospacecraftseekingimprovedinertialattitudeestimatesthroughattitude resourcesharing.ThemultipleestimatesoriginatingfromtheEKFforthedisaggregated systemaretheinputsfordatafusionalgorithms. 104

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CHAPTER4 ATTITUDEDATAFUSION Datafusionhasbeendevelopedtooptimallycombinemultiplestateestimates indisaggregatedsystems,wherethestateshavetypicallybeenEuclideaninerror. Morerecently,resultshavebeendevelopedforattitudedatafusion,whichaccount forthemultiplicativenatureofattitudecompositions.Thischapterreviewsexisting resultsindatafusionforEuclideanandattitudestates.Contributionsaremadeinthe parameterizationofattitudefordatafusionandparticularlyinthecharacterizationof errorcreatedbyusingaminimalattitudeparameterizationforfastattitudedatafusion. Thischapterisdividedintothreeparts.Intherstpartitisshownthatmany existingdatafusionalgorithmscanberepresentedintermsofthefusionerror,which istheerrorbetweentheindividualstateestimatesandthefusedstate.Basedonthe fusionerrorrepresentation,itisshownthatexistingEuclideanstatedatafusionlaws canbegeneralizedtoattitudestatesthroughthelossfunctionperspectivewiththe attitudeerrorvectorservingasthefusionerrorstate.Thesecondpartderivesaform oftheattitudeerrorvectorusingthevectorialattitudeparameterizationsanddiscusses thebenetsoftheseminimalattitudeparameterizationsfordatafusion.Thethirdpart investigatesthevectorialshadowparameterizationsandlocalerrorrepresentationsfor fastandaccurateattitudedatafusion. 4.1DataFusionOverview Inthediscussionsthatfollow,resourcesharingisassumedtoberepresentedwith astargraphwherethechiefspacecraftisconnectedtoeachofthedeputyspacecraft, butthedeputiesareonlylinkedtothechief.ThiscongurationisshowninFigure4-1. Thistypeofconnectivityisjustiedbythefactthatbetterinertialattitudeestimationis soughtthroughresourcesharing,andthustheonlyadditionalknowledgeprovidedto thedeputiesisgarneredthroughtherelativeattitudelinkonthechiefandtheimproved knowledgeofthechiefsgarneredfromthedeputies.Therecouldbemultiplechiefs,but 105

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thiscaseisnotconsidered,althoughitonlyrequiresaminorextensiontotheEKFfor disaggregatedsystemsformulationpresentedinChapter3. Figure4-1.Stargraphrepresentationforattituderesourcesharing Givenastargraphnetwork,thereareseveralimportantcharacteristicsthat dictatethealgorithmsandperformanceofresourcesharing.Asummaryofimportant characteristicsfordatafusionisgivenin[114].Ataxonomyforfusioncharacteristics fortheresourcesharingproblemisshowninTable4-1.Thisisnotmeanttobean exhaustivelist,asaspectssuchasimperfect,conicting,andspuriousdatawillnotbe considered.However,thesepropertiescanbeextendedtothepresentdiscussion. Table4-1.Resourcesharingdatafusionandsystemarchitecturetaxonomy CharacteristicDescriptionOptions ProcessingFrameworklocationofdatafusion Centralized,Decentralized DataCorrelationstateestimatedependency Correlated,Uncorrelated Connectivitygraphstructureofnetwork Full,Star Temporalstructurevariationinthegraphedges Dynamic,Static Directivityone-wayortwo-waycommunication Symmetric,Asymmetric Dataprocessingformofinformationsources Pre-processed,Rawdata Ofthesecharacteristics,thedatafusionproblemconsideredinthisresearchis thedecentralizedstargraphwithcorrelatedstates,staticgraphedges,symmetric communication,andpre-processedinformation.Itshouldbenoted,thatalthoughthe communicationsnetworkisstatic,therelativeattitudeconnectivityistimevaryingand basedonthetasking. 4.2DataFusionReview Datafusionisaproblemthathasarisenduetointerestinmulti-sensorsystems, eitheron-boardasingleplatformordistributedacrossanetworkofplatforms.Data 106

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fusionexploitstheknowndependencyintheinformationsources,asisthecasewith Kalmanltering[98],orseeksanoptimalcombinationofinformationtoproduceabest estimatewheninformationdependencyisuncertain,asisthecasewithcovariance intersection[47,48].Togenerateafusedestimate,datafusionalgorithmsoptimizethe combinationofmulti-sensorinformationbyminimizingacostfunction,whichissome measureoftheerrorbetweenthestateestimatesandthefusedstate.Fornon-attitude applicationswithunconstrainedstates,thefusionerrorisameasureoftheEuclidean distance.However,attitudebelongstothespaceofspecialorthogonalmatrices, SO andthusameasureoferrorthroughtheEuclideandistanceisnotappropriate. Theauthorhasobservedthatthepropertiesofthedatafusionalgorithmare inheritedfromtheattitudeparameterization.Recentadvancesinattitudedatafusion algorithmshavemadeuseoftheunitquaternion[46,115,116]andthemodied RodriguesparametersMRPs[117,118]toparameterizetheattitudeerrorvector. However,bothoftheseparameterizationsposedifcultiesinthedatafusionsolution. Theunitquaternionparameterizationisconstrained,whichmustbehandledinthe datafusionoptimizationexplicitlythroughtheuseofLagrangemultipliers.Solving theconstrainedoptimizationproblemaddsadditionalnumericalcomplexityfor low-dimensionalsystemsandbecomesburdensomeforhigh-dimensionalsystems. Alternatively,theMRPparameterization,whichisfromalargerclassofminimalattitude parameterizationsknownasthevectorialattitudeparameterizations,isunconstrained buthassingularitiesat 2 requiresasingularityavoidancealgorithm.Asingularity avoidancemethodthattransformstheglobalMRPdatafusionproblemtoalocal representation,whereasolutionisensuredtobesingularityfreehasbeenproposed in[118].Afterthedatafusionoccurs,thesolutionisthentransformedbacktothe globalMRPparameterization,whereitisthenfurthertransformedbacktotheoriginal parameterizationutilizedintheattitudeestimationorcontrolalgorithms.Toimprove uponthispriorartinattitudedatafusion,recentresultsintheminimalvectorialattitude 107

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parameterizationareappliedtotheparameterizationoftheattitudeerrorvector.Due totheperiodicinvarianceofattitude,thedatafusionparameterizationonlyrequires singularitiesnotbepresentfor 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 ,+2 ] .Thevectorialparameterizationsare knowntoproduceparameterswithtwodistinctsingularityclasses.Ofthesetwoclasses, onepermitsashadowparameterization,whichareusedtoproduceanunconstrained, globallynonsingularparameterizationoftheattitudeerrorvector.Additionally,itisshown thattheparameterizationdeterminestheaccuracyofthedatafusionprocess. 4.3DataFusionandtheErrorState Oneclassicdatafusionproblemisthatoflinearsystemswithtwoindependent Gaussianinformationsourcestypically,adynamicprocessmodelandameasurement model.Theoptimalweightedcombinationoftheindependentdynamicsprocessand measurementmodelsisgeneratedbytheKalmanlter.However,whentheinformation sourcesareknowntobedependentorthecorrelationisunknown,theKalmanlter cannolongerbedirectlyapplied[119].Thisclassofproblemisencounteredin multi-sensorsystemsandavarietyofdatafusionalgorithmshavebeencreatedto generateconsistentestimatesthatareoptimalinthesenseofsomerelevantcost function.Thesolutiontobothofthesedatafusionproblemsareafunctionofthe Euclideandistancebetweenthestateestimatesandthefusedstate. 4.3.1EuclideanFusionErrorState Considerthesetofpairs, x i P i ,thatdescribesthemeanandcovarianceof multiplesourcesofasystemstate.Theobjectiveofanydatafusionsystemisto optimallycombinethestateestimates, x i togenerateanewfusedstate, x f ,suchthat x f = f x x 1 ,..., x n withapproximatecovarianceofthefusedstategivenby P f = f P P 1 ,..., P n 108

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Here,thefunction, f x ,isthestatefusionlawand f P isthecovarianceapproximationlaw. Thecovarianceapproximationmustbeconsistenttoensurethatitneverunderestimates theuncertaintyinthesystem,whichcouldleadtodivergence.Therefore,thegoalof datafusionistooptimallyandconsistentlycombinemultiplesourcesofinformation.In termsofthesecondmoment,consistencyissatisedwhen P f )]TJ/F20 11.9552 Tf 14.371 2.385 Td [( P f 0, whichdesignatesthedifferenceisapositivesemi-denitematrix,and P f istheactual covarianceofthefusedstateand P f isanapproximationforthefusedstatecovariance thatisconsistentwithanycrosscorrelationbetweenthestateestimates.Clearly,a consistentfusedstatecovarianceexistsifthecross-correlationisknown.Furthermore, considertheclassofdatafusionlawsthatcanberewrittenintermsofthefusionerror, suchthat 0 = f x x 1 ,..., x n where x i = x )]TJ/F20 11.9552 Tf 13.965 0.108 Td [( x i 2 R n and x isanypossiblevalueofthefusedstate.Forthisclassof problems,thefusionlawisrecastastheminimizationofalossfunction, J ,suchthat x f =argmin x J x 1 ,..., x n Therst-orderoptimalityconditionforthisoptimizationproblemis 0 = dJ x 1 ,..., x n d x Fromthechainruleofdifferentiation,theoptimalityconditionis 0 = n X i =1 @ x T i @ x @ J x 1 ,..., x n @ x i However,thedatafusionJacobianformedfromthepartialderivativesofthefusion errorstatewithrespecttothefusionstateistheidentitymatrix,andthustheoptimality 109

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conditionis 0 = n X i =1 @ J x 1 ,..., x n @ x i For x 2 R n ,theoriginalfusionlawisrecapturedfromthisminimization.However,for statesthatparameterize SO ,thefusionlawisalteredbasedonthederivativeofthe errorstatewithrespecttotheattitudeparameterization. 4.3.2AttitudeFusionErrorState Inordertoapplydatafusionalgorithmstostatesparameterizedin SO ,the attitudeerrorstatemustbewellunderstood.Ageneraldiscussionofattitudeerror wasdiscussedinChapter2andisnowspecializedinthecontextofdatafusion.The attitudefusionerrorstateisrepresentativeoftherotationfromtheindividualattitude stateestimatestothefusedattitudestate.FromEulersrotationtheoremwhichstates arotationaboutapointisalwaysequivalenttoarotationaboutalinethroughthepoint [54], SO canbeparameterizedbyanaxisandtheangleofrotationaboutthataxis, whichisexpressedthroughtheEuler-RodriguesformulagiveninEq.2. Theattitudeerrorvectorisderivedfromtheattitudecompositionlawasdescribedin Eq.2.Forageneralcompositionofattitudes, R 0 = RR wheretheoriginalattitude, R ,istransformedby R ,toproducethenalattitude R 0 Since f R R R 0 g2 SO and RR T = I 3 3 ,then R = R 0 R T .2 Ofinterest,isavectorformofthecompositionlaw.Fortransformationsassociated withsmallangulardisplacements,theEuler-Rodriguesformulabecomes R = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ n ] + O )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(k n k 2 110

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When issufcientlysmall,higherordertermscanbeneglectedand R = I 3 3 )]TJ/F20 11.9552 Tf 11.955 -0.147 Td [([ ] Therefore,theattitudeerrorvectoris [ ] = I 3 3 )]TJ/F61 11.9552 Tf 11.955 0 Td [(R 0 R T Theattitudeerrorvectorcanthenbedenedthroughtheaxisandangleby = n Clearly,theattitudeerrorvectorisparameterizationindependentasitisafunctionofthe compositionof RR 0 2 SO .Therefore,thechoiceofparameterizationoftheattitude errorvectorwilldenethesuitabilityfordatafusion. Let s 2 R 3 beanyminimalattitudeparameterizationof R .Now,applyingtheattitude errorvectortothefusionerrorstateformulatedinSection4.3,yields s f =argmin s J 1 ,..., n Thelossfunctionapproachtoattitudedatafusionwasrstintroducedtoextend applicationofthecovarianceintersectionlawtoattitudestates[115].Thelossfunction perspectiveforEuclideanstatesproducesthesamefusedstateastheoriginal fusionlaw.However,forminimalattitudeparameterizationsthatarefollowfromthe Euler-Rodriguesformula, n and )]TJ/F23 11.9552 Tf 9.299 0 Td [( )]TJ/F101 11.9552 Tf 9.298 0 Td [(n representthesameattitude.Differentiating thequadraticlossfunctionensuresthatthefusedattitudeisinvarianttothisnon-unique choice. FollowingthederivationoftheEuclideanstates,therst-orderoptimalitycondition forattitudelossfunctionminimizationis 0 = dJ 1 ,..., n d s 111

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Fromthechainruleofdifferentiation,theoptimalityconditionis 0 = n X i =1 @ T i @ s @ J 1 ,..., n @ i UnlikethecaseforEuclideanstates,thedatafusionJacobianwillnot,ingeneral,bethe identitymatrix.Therefore,boththedatafusionJacobianandrepresentationoftheerror statemodifythedatafusionlawwithrespecttotheoriginalformulationforEuclidean states.Thesemodicationsareparameterizationdependentandhaveanimpactonthe characteristicsandaccuracyoftheresultingattitudefusionlaw.Thatis,theaccuracyof theattitudeerrorvectorapproximationwillinuencetheerrorinthefusionlaw. Insummary,thissectiondemonstratedthatthelossfunctionapproachasoriginally developedforthecovarianceintersectionalgorithmwithquaternionparameterization [115]andlaterappliedtotheMRPparameterization[117,118],appliestoanydata fusionlawandattitudeparameterization,giventhatthefusionlawcanbewrittenin termsofthefusionerror,asdescribedinEq.4.Section4.4demonstrateshowtypical datafusionalgorithmscanberewrittenintermsofthefusionerrorandthenadaptedto attitudeerrorstatesutilizinganyparameterizationoftheattitudeerrorvector. 4.4UnknownCross-CorrelationDataFusionAlgorithms Theproblemofdatafusionwithunknowncorrelationiscommonindecentralized multi-sensorsystemswhereinformationissharedacrossthenetwork.Twoalgorithms fordatafusionwithunknowncross-correlationwillbeextendedtoattitudeproblemsvia thelossfunctionandthefusionerror.TherstisthecovarianceintersectionCIfusion law,whichhasalreadybeenwrittenintermsofthelossfunctionoftheattitudeerror vectorin[115]andrelatedtotheworkonquaternionaveraging[120].Thesecondisthe ellipsoidintersectionEIfusionlaw,whichoffersalessconservativeapproachtodata fusionwithrespecttothecovarianceintersectionalgorithm,whereconsistencyhasbeen demonstratedinsimulation[121123]. 112

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4.4.1CovarianceIntersection Considerthesetofstateestimatepairs x i P i ,whichdescribetherstandsecond momentsoftherandomvariable, x ,withanarbitraryprobabilitydensityfunction.The covarianceintersectionCIfusionlawyieldsafusedestimate,whichisaconvex combinationofthestateestimates.Figure4-2representstheanotionalexampleofthe covarianceintersectionresultsforatwo-dimensionalsystem.Figure4-2Ashowsthe fusedstatecovarianceellipseforthe1boundonthestateerrorcenteredatzerofor theoriginalstateestimatesandcovarianceintersectionsolution,denotedby E 0, P 1 E 0, P 2 and E 0, P CI ,respectively.withminimumcostaswellasasuboptimalchoiceofweightings. Asexpected,thefusedstatecovarianceellipseliesoutsideoftheoriginalstateestimate covarianceellipses.Figure4-2BdemonstrateshowtheCIfusedstateisacompromise betweentheoriginalstateestimates. A B Figure4-2.NotionalcovarianceintersectionuncertaintyellipsecomparisonA Covarianceintersectioncenteredatzero.BCovarianceintersection centeredatmean. TheCIdatafusionmethodisgeneralfor n sensorsandisgivenbythefusionlaw x CI = P CI n X i =1 i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i x i 113

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andcovarianceapproximationlaw P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 CI = n X i =1 i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i Theweightings, i ,mustsumtoone,whereanyconvexcombinationsatisesthe covarianceintersectionfusionlaw,asdemonstratedbythedottedlineinFig.4-2A. Alternatively,thedashedlineinFig.4-2Arepresentsamoreappropriatechoice ofweightingsasevidencedbythesmallersizeincomparisontothearbitraryCI estimate.Inthiscase,theweightingsareoptimallydeterminedwithrespecttoa minimizationcriterion,typicallythetraceordeterminantoftheinformationformof thefusedcovariance,toproduceafusedresultthatisconsistentwithallpossible cross-correlationsbetweenthestateestimates. Thestatefusionlawisrewrittenintermsofthefusionerroras 0 = n X i =1 i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i x i whichistheresultofminimizing J CI x 1 ,..., x n = n X i =1 i x T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i x i Adaptingthislossfunctiontoattitudeyields, J CI 1 ,..., n = n X i =1 i T i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i i withthecorrespondingoptimalitycondition, 0 = n X i =1 i @ T i @ s P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i i ThechoiceoftheunitquaternionorMRPs,ashasalreadybeenappliedtoCI,only changetheparameterizationJacobianandattitudeerrorvectorformdescribedin Eq.4.ItisimportanttonotethatEq.4isgeneralforanyattitudeparameterization appliedtotheCIfusionlaw.ThisfactisutilizedintheexamplepresentedinSection4.8.2. 114

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4.4.2EllipsoidIntersection UnliketheCIfusionlaw,whichpresumesaformforthefusionlawwithnoregard tothemutualinformationread:correlation,theEllipsoidalintersectionEIfusionlaw explicitlycharacterizesmutualinformationbeforederivingthefusedstate.Inthisform, thestateestimatesaredecomposedintoexclusiveandmutualinformation,suchthat theunknownmutualinformationcovarianceisdeterminedasthemaximumpossible correlationbelongingtotheenclosureoftheunionofthestateestimatecovariances,as showninFig.4-3.Therefore,byconstruction,theapproximatefusedcovariancematrix fortheEIfusionlawwillalwaysbelessthanorequaltothesizeofthatderivedfromthe CIfusionlaw. A B Figure4-3.NotionalellipsoidalintersectionuncertaintyellipsecomparisonAEllipsoidal intersectioncenteredatzero.BEllipsoidalintersectioncenteredat mean. ThederivationoftheEIfusionlawfollowsfromthediscussionabove,whichyields x EI = P EI 2 X i =1 P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i x i )]TJ/F41 11.9552 Tf 11.956 0 Td [()]TJ/F27 7.9701 Tf 6.323 4.936 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 EI = )]TJ/F61 11.9552 Tf 5.48 -9.684 Td [(P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 + P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F41 11.9552 Tf 11.955 0 Td [()]TJ/F27 7.9701 Tf 6.324 4.936 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 where and )]TJ/F20 11.9552 Tf 9.646 0 Td [(arethemutualmeanandcovariance,respectively.Sincethemutual meanandcovarianceareunknown,consistencyisassuredifthelargestpossible 115

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correlationisconsidered.ThisistheessenceoftheEIfusionlaw.Themaximummutual informationcovarianceisfoundthroughthefollowingconvexoptimizationproblem, )]TJ/F20 11.9552 Tf 9.645 0 Td [(=argmax 2 R n n n X i =1 q subjectto E 0, P 1 [E 0, P 2 E 0, where E 0, P i arethecovarianceellipsoidsofthestateestimatesand E 0, isthecovariance ellipsoidfortheunknownmutualinformationbetweenthestateestimatesandthemutual mean.Thisisanalyticallyfoundbysolvingforthejointdiagonalizationof P 1 and P 2 toproduce P 1 = I n n and P 2 = D 2 throughtheeigen-decompositionon D )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(1 2 1 S )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 1 P 1 + P 2 S 1 D )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(1 2 1 .Clearlythistransformationon P 1 yields I n n .Theeigen-decompositionofthe transformationon P 2 is D )]TJ/F20 5.9776 Tf 7.782 3.259 Td [(1 2 1 S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 P 2 S 1 D )]TJ/F20 5.9776 Tf 7.782 3.259 Td [(1 2 1 = S 2 D 2 S T 2 Thus,themaximummutualinformationisfoundanalyticallythrough )]TJ/F20 11.9552 Tf 9.644 0 Td [(= S 1 D 1 2 1 S 2 D )]TJ/F61 11.9552 Tf 5.106 1.794 Td [(S T 2 D )]TJ/F20 5.9776 Tf 7.782 3.259 Td [(1 2 1 S T 1 where D )]TJ/F20 11.9552 Tf 8.427 1.793 Td [(= 8 > < > : max [ D 2 ] ij ,1,if i = j 0,if i 6 = j Themutualmeanispotentiallysingularbyconstruction,butcanbeavoidedwiththe followingapproximation = )]TJ/F61 11.9552 Tf 5.48 -9.684 Td [(P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 EI )]TJ/F41 11.9552 Tf 11.956 0 Td [()]TJ/F27 7.9701 Tf 6.323 4.936 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 +2 I n n )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 )]TJ/F61 11.9552 Tf 10.461 -9.684 Td [(P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 EI )]TJ/F61 11.9552 Tf 11.955 0 Td [(P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 2 + I n n x 1 + )]TJ/F61 11.9552 Tf 5.479 -9.684 Td [(P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 EI )]TJ/F61 11.9552 Tf 11.955 0 Td [(P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 1 + I n n x 2 where =0ifanyeigenvalueof P j ,resultingfromthejointdiagonalizationof P i and P j yielding P i = I ,hasmagnitudeslightlylargerthanone,orelse = ,where issome smallperturbation[123]. 116

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GiventheEIfusionlawandfollowingthedevelopmentoftheerrorstateloss function,theEIfusionstateisfoundbyminimizingthefollowinglossfunction J EI x 1 x 2 x = 1 2 2 X i =1 x T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i x i )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 x T )]TJ/F27 7.9701 Tf 6.324 4.936 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 x where x i x EI )]TJ/F101 11.9552 Tf 12.632 0 Td [(x i and x x EI )]TJ/F49 11.9552 Tf 12.632 0 Td [( .AswasthecasewiththeCIfusionlaw,it isimportanttoviewthesystemfromthisperspective,asattitudecompositionsarenot purelyadditive.Forattitudestates,thelossfunctionbecomes J EI 1 2 = 1 2 2 X i =1 T i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i i )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 T )]TJ/F27 7.9701 Tf 6.323 4.936 Td [()]TJ/F20 7.9701 Tf 6.587 0 Td [(1 TheoptimalityconditionfortheEIfusionlawis 0 = 2 X i =1 @ T i @ s P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i i )]TJ/F29 11.9552 Tf 11.955 20.443 Td [( @ T @ s )]TJ/F27 7.9701 Tf 6.324 4.936 Td [()]TJ/F20 7.9701 Tf 6.586 0 Td [(1 Therefore,theEIfusionlawforattitudestatescanbedeterminedbyrstndingthe maximummutualinformationthroughEq.4andEq.4,andthenmakinguseofthe optimalityconditioninEq.4toalgebraicallysolvefortheEIoptimalfusedstate. 4.4.3SummaryofDataFusionLawsforAttitudeStates ModicationofexistingdatafusionlawsthatwereoriginallyderivedforEuclidean stateshasdemonstratedtheapplicabilityofthesesamefusionlawsforstatesthat parameterize SO .TheCIandEIfusionlaws,wereshowntohavetheappropriate formtobeviewedfromthelossfunctionperspective.Next,Section4.5willreviewthe quaternionparameterizationfordatafusionandthechallengesassociatedwithits redundancy.Then,thevectorialparameterizationoftheattitudeerrorvectorandthe generationofthevectorialparameterizationJacobianarediscussedinSection4.6in regardtotheirapplicationtodatafusion. 117

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4.5ReviewoftheUnitQuaternionforDataFusion TheEuler-Rodriguessymmetricparametersaredescribedbytheunitquaternion, q = q T v q 4 T ,anddenedbytheaxisandangleofrotationthrough q v =sin 2 n q 4 =cos 2 where q v and q 4 arethevectorandscalarportionofthequaternion,respectively.The unitquaternionparameterizationforattitudeisgivenby R q = )]TJ/F61 11.9552 Tf 5.48 -9.684 Td [(q 2 4 )]TJ/F101 11.9552 Tf 11.955 0 Td [(q T v q v I 3 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 q 4 [ q v ] +2 q v q T v Theunitquaternionisapopularattitudeparameterizationduetoitbeingglobally-singularity freewiththebilinearkinematics q = 1 2 q = 1 2 q where q = 2 6 4 q 4 I 3 3 + [ q v ] )]TJ/F101 11.9552 Tf 9.298 0 Td [(q T v 3 7 5 = 2 6 4 )]TJ/F20 11.9552 Tf 11.291 -0.147 Td [([ ] )]TJ/F49 11.9552 Tf 9.298 0 Td [(! T 0 3 7 5 However,amajordrawbackoftheunitquaternionfordatafusionisthatitrequires theconstraint, q 2 4 + q T v q v =1, besatised.Theattitudematrixparameterizedbytheunitquaternionandassociated quaternionkinematicsmaintaintheconstraintbyconstructionandareonlysubjectto numericalprecisionerrors,whichcanbemanagedthroughbrute-forcenormalization. 118

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RecallthegeneraloptimizationproblemforattitudedatafusioninEq.4.When isformulatedwithaconstrainedattitudeparameterization,theoptimizationproblem mustbemodiedtosatisfytheequalityconstraints.Thenewoptimizationproblemis givenby minimize J 1 ,..., n subjectto g s )]TJ/F61 11.9552 Tf 11.955 0 Td [(c =0, where g istheconstraintfunctionand c istheconstraintvalue.Thisoptimization problemcanbesolvedthroughthemethodofLagrangemultipliers[124].Various algorithmsforsolvingtheconstrainedoptimizationproblemwiththemethodofLagrange multipliershavebeenimplementedforthequaterniondatafusionproblem[115,116]. However,theadditionoftheunityconstraintaddsalgorithmicandnumericalcomplexity withrespecttotheunconstrainedoptimizationproblem.Thismotivatestheuseof minimalattitudeparameterizationsforattitudedatafusion. 4.6VectorialAttitudeParameterization Theparameterizationoftheattitudeerrorvectordictatesthecharacteristicsof theattitudedatafusionproblemasaconsequenceofEq.4.Theunitquaternion parameterizationrequiresthatasingleconstraintbesatised,whichgenerates additionalcomplexitywithrespecttoanunconstrainedoptimizationproblem.Itiswell knownthatallminimalattitudeparameterizationsareunconstrained,butalsosingularfor particularattitudes.Singularitiescanmanifestthrougheitherofgeometricorkinematic origin.Geometricsingularitiesarearesultofthedescriptionofattitude,inwhichcertain orientationscausetheparameterstoincreasetoinnity.Kinematicsingularitiesarea resultofthemotionofattitude,inwhichsmallchangesinangularvelocitycorrespondto aescapeoftheparameterizationderivativetoinnityinnitetime,andviceversa.Both ofthesesingularitytypescancauseissuesfordatafusion. 119

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Forsystemswithunconstrainedrotationalmotion,thepresenceofkinematical singularitiesposesdifcultiesforattitudecontrol,estimation,anddatafusion.Due tothesingularitiespresentinallminimalattitudeparameterizations,theconstrained unitquaternionparameterizationperformswellforattitudecontrol[125127]and estimation[103,104].Inthisresearchitisassumedthattheunitquaternionisused forcontrolandestimation.Alternatively,itisknownthatcertainminimalattitude parameterizations,suchastheMRPsandhigherorderRodriguesparametersHORPs, canalsobeusedtoproduceglobalsingularity-freekinematicsbymakinguseof theshadowparameterizations[62,63].Furthermore,datafusiononlyrequiresthe attitudeparameterizationbefreeofkinematicalsingularitiesfortherangeofrotations, 2 [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 ] ,duetotheperiodicinvarianceofattitude.However,datafusionneed notutilizethesameparameterizationasthatwhichisutilizedintheestimationand controlalgorithms,asthequaternionsproduceaconstrainedoptimizationproblemfor datafusion.Tocircumventthisproblem,vectorialparameterizationsoftheattitudeerror vectorareinvestigatedtogeneratefastandaccurateattitudedatafusionlaws. 4.7ParameterizedAttitudeErrorVector Thevectorialattitudeparameterizationsandtheirkinematicsdiscussedin Section4.6arenowusedtoderivetheattitudeerrorvectors.Theattitudecomposition lawprovidesameasureofchangeinattitudefromonecongurationtoanother. Parameterizingtheattitudecompositionlawwiththegeneralvectorialparameterization yields R = R )]TJ/F101 11.9552 Tf 5.48 -9.684 Td [(r 0 R T r Therefore,giventwoattitudes,representedby r 0 and r ,theattitudeerroron SO is exactlygivenbyEq.4.Althoughthisexpressionisexact,thegoalistodevelopa parameterizedattitudeerrorrepresentationasrequiredbythedatafusionoptimality conditioninEq.4.Whentheprincipalrotationsaresmall,theattitudeerrorvector 120

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canberepresentedaccuratelybyalinearmodelforEq.4.Smallrotationsarea validassumption,asattitudeestimationisassumedtoproduceestimatessatisfyingthe small-angleassumptionevenwithcoarseattitudesensors. 4.7.1First-OrderModelofAttitudeErrorVector RecallthattheattitudeerrorvectordenedinEq.4isparameterization independent.Oneapproachtoformulatingtheattitudeerrorvectoristhroughsequential compositionson SO whenparameterizedthroughthegeneralattitude s on R 3 .To simplifyexpressions,higher-ordertermsareneglectedwhentheattitudeerrorvectoris assumedtobesmall.Alternatively,considerthegeneralnonlinearkinematicsgivenin Eq.2,whicharederivedthroughinnitesimalrotations.ATaylorseriesexpansion withrespecttotimeyields r t = r t 0 + r t 0 t + 1 2! r t 0 t 2 + ForsmalltimestepsandusingEq.2,alinearmodelis r t = r t 0 + G r t 0 t 0 t Ifoverthetimestep t ,thecondition, t 0 = t 0 t ,islocallysatised,thenthe Taylorseriesapproximationisgivenby r t = r t 0 + G r t 0 t 0 Therefore,thetruncatedapproximationfortheattitudeerrorvectoris t 0 = G )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 r t 0 r t )]TJ/F101 11.9552 Tf 11.955 0 Td [(r t 0 Makinguseoftheinversekinematicformandremovingthetimeindex,theapproximate modelis = H r )]TJ/F101 11.9552 Tf 5.48 -9.684 Td [(r 0 )]TJ/F101 11.9552 Tf 11.956 0 Td [(r 121

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Thisapproximationwillserveastherepresentationoftheattitudeerrorvectorfordata fusion. 4.7.2DataFusionJacobian Theinversekinematicsarenowshowntobeintimatelytiedtoattitudedatafusion. AnexactdenitionoftheparameterizationJacobianisrelatedtotheangularvelocity, whereonerelevantdenitionoftheangularvelocityis = d dt wherethecorrespondingdifferentialrelationshipis d = dt Similarly,theparameterizationtime-derivativeisdenedby r = d r dt whichhasthedifferentialrelationship d r = r dt Itshouldbenotedthattheseareexactdifferentialrelationships.Therefore,theforward andinversekinematics,asrepresentedthroughdifferentials,are d r = G r d and d = H r d r respectively,andthus, d d r = H r 122

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Asaresult,theJacobianrequiredbythedatafusionoptimalityconditioninEq.4,is simplythetransposeofitsinversekinematicalJacobian.Thatis, d T d r = H T r TheinversekinematicJacobianissingularforallminimalvectorialattitudeparameterizations. Whatisimportant,isthatfortheClassIIparameterizationswhich,recallingfrom Chapter2,arebestowedwithashadowset,thekinematicalsingularitiescanbe avoided. 4.7.3AttitudeErrorVectorApproximationAccuracy Choiceofparameterizationoftheattitudeerrorvectorshouldavoidgeometric singularitiesassociatedwithconversionbetweentheunitquaternionsandkinematic singularitiesassociatedwithchangesintheparameterization.InReference[64]itwas shownthattheaccuracyofpropagationforthevectorialattitudeparameterizations isrelatedtothekinematicalconditionnumber, ,whichrelateserrorsinangular velocitytoerrorsinthetime-derivativeoftheparameterization.Furthermore,itwas shownthattheMRPshavethelowestkinematicalconditionnumberwhenswitching betweentheshadowparameterizationandthusarethemostaccuratevectorialattitude parameterizationforpropagation. ThegeneralchangeinattitudewasdenedinEq.4.Fordatafusion,theattitude errorvectordenedforsmallrotationsisprovidedforgeneralcompositionsinEq.4. Forthevectorialparameterizations,therst-orderapproximationoftheattitudeerror vectorwasderivedinEq.4.Theaccuracyoftherst-orderapproximationofthe attitudeerrorvectorischaracterizedbythedifferenceinthetruevalueoftheattitude errormatrixwithrespecttotheattitudematrixparameterizedbytheapproximateattitude errorvector,overtherange 2 [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 ] .Anappropriatemeasureinthedifferenceis 123

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theEuclideandistance, J ,suchthattheerrorisdenedby J = k R r r 0 )]TJ/F61 11.9552 Tf 11.955 0 Td [(R r k f where kk f representstheFrobeniusmatrixnorm.Thequantity r isrelatedto thoughascaling.Forinstance,fortheMRPs,thescalingisfour,whereasfortheFRPs thisscalingiseight.Thechoiceofattitudeparameterizationisbasedontheobjectiveto minimizethisadditiveerror,howeverotherobjectivefunctionssuchasthemultiplicative eigenangleerrorexhibitthesametrends.Asaresult,parameterizationsthatexhibit linearbehaviorfor 2 [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 ] areidealfordatafusion.Considerthecaseof theclassicalRodriguesparametersCRPs, ,andrespectivehigherordersofthe Rodriguesparameters,suchasthethird-orderRodriguesparametersTRPsand fourth-orderRodriguesparametersFRPs, .Figure4-4Ademonstratesthatthehigher orderRodriguesparametersincreaseinlinearitywiththeorderoftheparameterization. Thisfactoflinearityfollowsdirectlyfromtherepeatedhalfangleformulationofthe higher-orderRodriguesparameters,wherethetangentfunctionisnearlylinearforsmall angles.Inthelimit,thehigher-orderRodriguesparametersrepresenttheprincipal rotationvector,whichisanexactlylinearfunctionof ,bydenition.Tocomparethe A B Figure4-4.Linearityofseveralordersofthehigher-orderRodriguesparametersA Linearmodelcomparison.BLinearmodelresidual. 124

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linearityoftheparameterizations,aleastsquaresminimizationwasperformedtot arst-ordermodelusingdatapointsoriginatingfromhalfofthenonsingularrangeof theset.Aresidualisdenedasthedifferencebetweentheparameterizationandthe rst-ordermodel.FromFig.4-4B,theresidualshowsthateachorderoftheRodrigues parameterbeginstoexhibitdecaytosingularityattheprincipalrotationassociated withhalfofthesingularityangle.Forexample,theCRPsbegintodivergeat = 2,the MRPsat ,andsoon.However,theshadowparameterizationsofferrelieffromthese nonlinearitiesfor m 2.Bymakinguseoftheswitchingsurfacecos = 2=0i.e., q 4 =0 toswitchbetweentheshadowparameterizations,thevectorialattitudeparameterization willexhibitthemostlinearrelationshippossibleforthatset.Thisisduetothesymmetry oftheparameterabout =0,suchthattheaforementionedswitchingsurfaceensures thesymmetricsetisfurthestfromsingularity.Thisimpliesthattheprincipalrotation remainsassmallaspossibleforthatparameterization.Figure4-5showstheeffects ofprincipalrotationangleandattitudeerrorangleonthelinearapproximationwhen switchingbetweentheshadowparameterizationsfortheattitudeerrorvectorforthe modiedRodriguesparametersandthefourth-orderRodriguesparameters.TheMRPs representthestate-of-the-artforunconstrainedattitudedatafusion.Inthisgraph,the attitudeerrorangleisdepictedbyapositiveconstantprincipalrotationangleoffset, suchthat =tan 4 n 0 =tan + 4 n and =tan 8 n 0 =tan + 8 n 125

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Theexacterrorcompositionon SO R r ,iscomputedthroughEq.4withthe parameterizedattitudematrixfoundinTable2-1.Therst-orderapproximationofthe compositioniscomputedthroughEq.4usingtheparameterizedinversekinematical JacobianinTable2-2.Note,theordinateinFig.4-5isscaledlogarithmically,duetothe investigationofmultipleordersofmagnitudefortheattitudeerrorangleinradians.The curvesfor 2 [ )]TJ/F23 11.9552 Tf 9.299 0 Td [( ,+ ] aregeneratedusingthelistedparametersandtheirshadows areusedfor 2 )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 )]TJ/F23 11.9552 Tf 9.298 0 Td [( [ + ,+2 .TherstobservationfromFig.4-5isthatsince Figure4-5.AttitudeerrorvectorapproximationaccuracyfortheMRPsandFRPs boththeMRPsandFRPspermitashadowparameterization,bothsetshaveerrors thatasymptoticallyapproachzeroat = f 0, 2 g .Thisisexpected,astheshadow parameterizationiswelldenedwhentheoriginalsetisill-conditioned. Thenextobservationisthattheswitchingcondition, q 4 =0,providesaveryrobust lawfordescriptionoftheattitudeerrorvector.Thisfactisillustratedbythepeakinerror observedat = .Sincethepeakoccursattheintersectionoftheparameterization withitsshadowset,theerrorisminimalwiththisswitchingcondition.Thatis,anyother switchingpointwouldresultinalargerpeakerroratthisintersection.However,asthe orderoftheattitudeerrorincreases,theswitchinglawdescribedby q 4 =0issuboptimal, 126

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asillustratedbythejumpdiscontinuityintheerrorat = for = f 1 g radians. ThegraylineinthemagniedcircledregioninFig.4-5demonstratesthatatlarge angleerrors,theswitchingconditioncouldbeadjustedtoreducetheerrorinthe approximation.However,largeerrorsinvalidatethelinearizationsmallangleassumption usedintheformulationofstandarddatafusionandstateestimationalgorithms.Inthese cases,modicationstotheswitchinglawwillimprovetheerror,butmoreimportantly thedatafusionandestimationalgorithmsmustbereformulatedtoaccountforthe nonlinearityintheattitudedynamicsprocess.Thefactremainsthatforsmallerrors,the switchinglaw q 4 =0isveryaccurateandrobustandthuspreferredfordatafusionunder typicalconditions. ThenalandparticularlypowerfulobservationisthattheFRPsproducean attitudeerrorvectorwhichisnearlyanorderofmagnitudemoreaccuratethanthe MRPsatthepeakerrorassociatedwith .BoththeMRPsandFRPsaresubject tosmallpolynomialerrorsourcesasindicatedbytheirrespectiveattitudematrix parameterizations.Thehigher-orderpolynomialfortheFRPsapproximatesalinear functionin asevidencedbyFig.4-4.Thisimprovementinaccuracyisconsistentfor allordersofattitudeerrorseethelegendinFig.4-5.Itisimportanttonotethatas theattitudeerrorgrows,thelinearapproximationerrorbeginstoapproachtheorderof theattitudeerroritself.Whereas,forsmallattitudeerrors,thelinearapproximation isseveralordersofmagnitudesmaller,andonlyhasasmallaffectonthefusion process.However,fromanaccuracystandpoint,forallcasestheFRPsareapreferable parameterizationfordatafusionincomparisontotheMRPs.Itisexpectedthatasthe orderofhalvinginthetangentincreasesforthehigher-orderRodriguesparameters, thatthelinearapproximationwillbemoreaccuratefor 2 [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 ,+2 ] .However,there maybeincreasingnumericalcomplexityastheorderincreases.Inthelimit,theCayley transformproducestherotationvector,whichiscompletelylinearin .However,the kinematicsareill-conditionedat =0.Theseobservationsareillustrativeofthebenet 127

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ofusingthehigher-orderRodriguesparametersfortheattitudeerrorvector,withthe caveatthatatrade-offdoesexistwithrespecttonumericalcomplexity. 4.7.4UnitQuaternionTransformation Section4.7.3showedthatminimalattitudeparameterizationsaresuitedfordata fusionastheyareunconstrainedandcanbemadetobehighlylinearfor 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 ,+2 ] However,theunitquaternionstillhaslargeutilityforcontrolandestimation.This sectionprovidesthedetailsforconvertingtoandfromtheunitquaternion.Particularly, ifthestateestimatesoriginatefromtheunitquaternion,thenwhenusingthevectorial parameterizationfordatafusion,itisnecessarytotransformbackandforthbetween thevectorialattitudeparameterizationsandtheunitquaternion.Table4-2providesa summarytheforwardandinversetransformations. Table4-2.Transformationsforvectorialattitudeparameterizationsandtheunit quaternion Name rr q q v r q 4 r EOPs 2tan )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 q v q 4 sin 2 cos 2 CRPs q v q 4 1 1+ 2 1 2 1 1+ 2 1 2 MRPs q v 1+ q 4 2 1+ 2 1 )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 1+ 2 FRPs q v 1+ q 4 + p 1+ q 4 4 1 )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 1+ 2 2 1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 2 + 4 1+ 2 2 OPs q v )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 1 2 LPs 2 1+ q 4 1 2 q v 2 )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 1 2 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 2 4.8MinimalAttitudeParameterizationDataFusion ClassIIvectorialattitudeparameterizationsareminimal,unconstrained,and nonsingularwhenpairedwiththeirshadowparameterizations.Thesepropertiesmake themidealforparameterizationoftheattitudefusionerrorvector.Inthispart,ageneral approachtoattitudedatafusionusingtheshadowparameterizationswitchinglawis presented.TheseresultsarethendiscussedwithrespecttotheCIandEIfusionlaws introducedearlierinSections4.4. 128

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4.8.1ShadowSwitchingDataFusionProcess Itisassumedthatthedatafusionprocessbeginswithquaternionestimatesand thatClassIIvectorialparameterizationsareutilizedforthefusionprocesstoavoid solvingfortheconstrainedoptimization.SincetheClassIIparametershaveshadows, denotedby r S ,singularityavoidancecanthusbeaccomplishedbyswitchingtothe shadowsetusingaswitchinglaw. InsightintotheswitchinglawcanbegainedfromanunderstandingofMRP-based attitudecontrol.TheMRPswitchingconditionforattitudecontrolistypicallychosen tobe T =1[62].Thisconditionisequivalenttoswitchingattheunitspherewhere = )]TJ/F49 11.9552 Tf 9.299 0 Td [( S .Theimplicationofthisswitchinglawisthattheattitudeparameterizationis alwaysboundedbyunitmagnitude,whichisdesirablefortuningtheresultingcontroller. However,ingeneraltheswitchingsurfacetakestheform T = c where c isanarbitraryrealconstant.Justlikethecaseforcontrollaws,thefusion lawforthevectorialattitudeparameterizationsisbasedonthedesiretoboundthe normof r ,whichresultsinananalogousswitchingconditiontothatofMRP-based attitudecontrol.However,theboundednessinthiscaseensuresthatdatafusionwillbe numericallystable,suchthatsmallchangesintheparameterizationareindeedreective ofsmallanglechangesintheattitude. Assumingthattheparameterizationforestimationandcontrolistheunitquaternion, thedatafusionswitchingconditionfortheMRPsinEq.4isequivalenttothe switchingsurfaceontheunitquaternion, q 4 = c q where c q isanarbitraryrealconstant.SwitchingattheunitspherefortheMRP,givenby c =1,isequivalenttothequaternionswitchingsurfaceat c q =0.Therefore,thechoice 129

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forvectorialattitudeparameterization, r ,ofthefusionattitudeerrorvectorisgivenby r = 8 > < > : r ,if q 4 0 r S ,if q 4 < 0 whichisunconstrainedandgloballynonsingular.Additionally,thischoiceof c q ensures thedatafusionwilloccuratthemaximumdistancefromsingularity,wherethedistance isdenedthroughattitudecomposition.Theimplicationsarethatthisswitchingcondition willproducethemostaccuraterepresentationfordatafusion,astheparameterization willremainthemostlinearasdemonstratedinFig.4-5. Fromtheabovediscussion,asummaryofthevectorialparameterizationsfusion lawwithshadowparameterizationswitchingisprovidedinthefollowingalgorithm: Algorithm1providesanunconstrainedglobalnonsingulardatafusionprocesswhich Algorithm1 Vectorialattitudeparameterizationdatafusionwithshadowswitching 1: procedure G ENERAL F USION P ROCESS q i 2:Constructthesetofquaternionestimates, q i ,tobefusedandensurethatquaternionvectorportionspointinthesamedirection 3:Transformthequaternionstateestimatestothevectorialparameterizationbyarbitrarilytestingthe rstquaternionestimateinthesetusingthefollowingswitchingcondition: 4: if q 4, i 0 then 5:transformallquaternionestimatestotheirrespectivevectorialestimate, r usingthe r q column inTable4-2 6: elseif q 4, i < 0 then 7:transformallquaternionestimatestotheirrespectiveshadowvectorialestimate, r S ,using Eq.2 8: endif 9:SolveforthefusedstatefromtheoptimalityconditioninEq.4,whichfortheCIandEIalgorithmsarecomputedthroughEq.4andEq.4,respectively. 10:Transformthefusedstatebacktotheunitquaternionparameterizationusingthe q r and q 4 r columnsinTable4-2 11: endprocedure canbeappliedtotheunknowncorrelationCIandEIalgorithms.Infact,thisprocessis generaltoanyattitudedatafusionlawsatisfyingthelossfunctionforminEq.4.An exampleofthisprocessutilizingtheFRPsappliedtotheCIfusionlawisdiscussedin thenextsection. 130

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4.8.2Example:FRPCovarianceIntersectionwithShadowSwitching Recalltheformofthelossfunctionforthecovarianceintersectionalgorithm, Eq.4andtheattitudeerrorvectorparameterizedbytheFRPsgiveninTable2-2. Usingthesedenitions,theFRPparameterizedcovarianceintersectionlossfunctionis J CI CI = n X i =1 i CI )]TJ/F20 11.9552 Tf 14.266 0 Td [( i T H T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i H i CI )]TJ/F20 11.9552 Tf 14.265 0 Td [( i FollowingfromEq.4,theCIFRPoptimalityconditionis 0= n X i =1 i H T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i H i CI )]TJ/F20 11.9552 Tf 14.265 0 Td [( i T Solvingthelinearsystemofequationsfor CI ,yieldsthefusedattitudegivenby CI = n X i =1 i H T i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i H i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n X i =1 i H T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i H i i Therequiredinverseisfora3 3matrixandisguaranteedtoexist.Existencefollows since P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i existsbyconstruction,thesimilaritytransform, H T i P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i H i ,producesthe sameeigenvaluesas P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i ,andthematrixtoinvertresultsfromaconvexcombinationof H T i P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i H i with P n i =1 i =1. 4.9VectorialAttitudeDataFusionwiththeLocalErrorApproach Analternativetotheshadowswitchingmethodisthelocalerrorapproach developedinReference[118].Thismethodhasthebenetsofbuilt-insingularity avoidanceaswellassuperbkinematicconditioning.Thelocalerroristheresultof computingtheerrorquaternionwithrespecttoareferencequaternion.Followingthe notationofReference[118],thederivationisforthelocalerrorissummarized.Letthe quaternionestimatebedenotedby q i ,thereferencequaternionas q ,andthefused quaternionas q f .Thefusedquaternioncanberelatedtothequaternionestimatesbya smallquaternionerror, q ,suchthat q f = q i q i 131

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Analternativedenitionofthefusedquaternionistousedeviationsfromthereference trajectory, q ,suchthat q f = q q Thereferencetrajectoryandquaternionestimatearerelatedby q = q i q i andthus q i = q q )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i where q i isonceagainsmall.CombiningthedenitionsofEq.4,Eq.48,and Eq.4,theerrorsarerelatedby q = q i q )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 i andthroughattitudecompositions q i = q q i NowdeningthevectorialparameterizationofthequantitiesinEq.420,therst-order modeloftheattitudeerrorvectorusingthelocalerrorapproachis i = H r q i r q )]TJ/F101 11.9552 Tf 11.955 0 Td [(r q i where r q i isthevectorialparameterizationof q i and r q isthevectorialparameterization os q .Algorithm2summarizesthedatafusionmethodusingthelocalerrorapproach. Figure4-6depictstheapproximationerrorcenteredabouttheidentitystateoutto 5 = 180 rad .TheFRPsstillproduceanorderofmagnitudeimprovementovertheMRPs usingthelocalerrorapproach.However,evenat5 = 180 rad error,theapproximation errorismultipleordersofmagnitudelowerthanthefusionerror.Therefore,thelocal 132

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Algorithm2 Vectorialattitudeparameterizationlocalerrordatafusionalgorithm 1: procedure G ENERAL L OCAL E RROR D ATA F USION P ROCESS q i 2:Constructthesetofquaternionestimates, q i ,tobefusedandensurethatquaternionvectorportionspointinthesamedirection 3:Constructthesetoflocalquaternionerrorstateestimateswithrespecttoareferencequaternion usingEq.4 4:Transformthelocalquaternionerrorstateestimatestothevectorialparameterization 5:SolveforthefusedstatefromtheoptimalityconditioninEq.4,whichforCIandEIalgorithms arecomputedthroughEq.4andEq.4,respectively 6:Transformthefusedlocalerrorstatetothelocalquaternionerrorusingthe q v r and q 4 r columns inTable4-2 7:Transformthefusedlocalquaternionerrorstatebacktoanabsolutefusedquaternionusingthe inverseofthereferencequaternion 8: endprocedure errorapproachisapproximatelyequalnomattertheminimalparameterization.Based onthisobservation,itcanbearguedthatinfacttheCRPsshouldbeutilizedasthebest parameterizationforminimaldatafusion,astheyaremorecomputationallyefcientthan theMRPsandFRPs.ThelocalerrorrepresentationensuresthattheCRPswillstayfar awayfromtheirsingularityat Figure4-6.AttitudeerrorvectorapproximationaccuracyfortheMRPsandFRPsusing thelocalerrorrepresentation Takingintoaccounttherobustnessofthelocal-errorapproachtotheattitude parameterization,Figure4-7demonstratesthatalthoughtheCRPsareanorder 133

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ofmagnitudeworsethantheMRPsandtwoordersofmagnitudeworsethanthe FRPs,duetothelocalerrorrepresentation,theyarestillmorethanadequatefor precisionattitudeapplicationsandaddonlynegligibleerrorforcoarseattitude estimation.However,thechoiceiswiththedesignerofthetobalancetheaccuracy oftheparameterizationvsthecomputationalexpense. Figure4-7.AttitudeerrorvectorapproximationaccuracywiththeCRPsusingthelocal errorrepresentation Intuitively,thereferencequaternionshouldbechosentominimizethedistance withrespecttotheidentityquaternion.Thischoiceensuresthatwhicheverminimal parameterizationischosenwillbethehighestaccuracyforthatparameterization. ThisleadsdirectlytothequaternionaveragingsolutionmentionedinReference[118]. However,usingthelocalerrorapproachmakesthesolutionveryinsensitivetothe choiceofthereferencetrajectory,andasaresultthereferencetrajectorycanbechosen tobeanyofthestateestimateswithoutanoticeablelossinperformance. 4.10CommentsonAttitudeDataFusion Throughoutthisresearch,severalnewobservationsweremadeinthestudyof attitudedatafusion.Severalnotesrelatedtoattitudeparameterizationsandfusionlaws areprovidedinthefollowingsubsections. 134

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4.10.1ANoteonMRPSingularitiesandShadowSwitching InReference[117],singularityavoidancewasdescribedforaglobalMRPapproach usingasimilarswitchingconditionaswasdescribedinthissection.Additionally,a localMRPapproachwaspresentedandfurtherexpoundeduponinReference[118], whichdidnotrequireuseoftheshadowparameterizations.InReference[117],both ofthesemethodswereshowntobemorenumericallyefcientthanexistingquaternion datafusionlaws.Additionally,theglobalswitchinglawwasdemonstratedtobemore numericallyefcientthanthelocalMRPsingularityavoidancemethod.Thisefciency differenceisaresultofthelocalerrorapproachnecessitatingadditionaltransformations toandfromtheoriginalattitudeparameterizationaswellasthepotentialneedfor quaternionaveragingtoensurethefusionerrorremainssmall. AlthoughtheglobalMRPswitchingwasshowntobemoreefcientinReference[117], itisstatedinReference[118]thatdatafusionneartheunitspherewillcauseissueswith aglobalMRPdatafusionroutineutilizingshadowswitching.Inparticular,itisstated thatadditivecorrectionstothefusedattitudemayresultinanerroneousswitchingof theMRPtoitsshadowparameterization.Althoughitistrueforallconditionsotherthan ontheunitsphere,thatwhentheMRPisinsidetheunitspheretheshadowMRPis outsidetheunitsphere,itisnottruethattheMRPalwaysremainsinsideoftheunit sphere.Bycontinuity,theMRPwillbegreaterthanoneforall > andunboundedat thesingularityexistingat 2 .MRPcontinuitynearthequaternionunitsphereisshown inFig.4-8,wherethearrowspointinthedirectionofchangewithincreasing ,such thatas q movesto q 0 movesto 0 .Theequivalentantipodalattitudeof )]TJ/F101 11.9552 Tf 9.298 0 Td [(q movesto )]TJ/F101 11.9552 Tf 9.298 0 Td [(q 0 ,as S movesto S 0 .BoththeMRPandtheshadowmovecontinuouslythroughthe perimeteroftheunitspherealongtheprojectionplane.Thus,whentheMRPislocated ontheunitsphereat q i =+1thentheshadowMRPisalsoontheunitsphere,butat q i = )]TJ/F20 11.9552 Tf 9.299 0 Td [(1,wheretheconverseofthisstatementisalsotrue.Therefore,assumingthatall ofthevectorportionsofthequaternionspointinthesamedirection,theMRPcouldonly 135

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erroneouslybefusedasashadowMRPifthestateestimateandthefusedstatehavea fusionerrorof2 .Thisisaclearviolationofthesmallangleassumptionthatwasused intheconstructionoftheMRPcompositionlawandattitudeerrorvectordescription.As aresult,theMRPswitchinglawprovidesagloballynonsingularapproachtodatafusion. ThesesameargumentsholdfortheotherClassIIvectorialattitudeparameterizations. Figure4-8.Justicationfordatafusionatthequaternionunitsphereusingtheshadow parameterizations 4.10.2ANoteontheMinimizationCriterionfortheCIAttitudeFusionLaw TheCIfusionlawisequivalenttoaone-dimensionalminimizationproblem,where theminimizationcriterionistypicallychosentobethetraceordeterminantofthe informationformoftheapproximatefusedcovariancematrix.Thisminimizationcriterion isthusproblemdependent.Thefocusofthisstudyisforattitudedatafusion,andthus theperformancemetricforattitudeestimationanddatafusionisthepointingaccuracy. Onedenitionforthepointingaccuracyis J pointing = p e 2 where e istheerroroftheestimatewithrespecttothetruestate.Insimulation,the performanceofthedatafusionlawcanbeevaluatedwithrespecttoatruthmodelto 136

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determinethetrueerrorinthesystem.However,thestateerrorcannotbedirectly characterizedforphysicalsystemsunlessthestateisknownexactly,whichinthatcase precludestheneedfordatafusionandestimation.Therefore,agoodsurrogateforthe pointingaccuracyofunbiasedestimatesisthestate-errorcovariancematrix,which correspondstouncertaintyboundsontheerrorofthestateestimatewithrespecttothe unknowntruth.Fromthisalternativedenition,thepointingaccuracyisgivenby J pointing = p tr P CI Sincethesquarerootismonotonic,thetraceisanequivalentminimizationcriterionto thesquarerootofthetrace.Assuch,thegoalofdatafusionshouldbetominimize thetraceofthefusedcovariancematrix.Thus,thereisanequivalencewiththe application-specicdatafusionperformanceandtheminimizationcriterionofthe covarianceintersectionalgorithm.InReference[115],thetracewasshownempirically tobetheoptimalCIminimizationcriterionforattitudeapplications.However,theCI minimizationcriterionfollowsdirectlyfromthefactthattheevaluationcriterionfor attitudeisthepointingaccuracy,whichwasjustshowntobeequivalenttominimizing thetraceofthestate-errorcovariancematrix.Thisinformalproofdemonstratesthatfor attitudeapplications,oneshouldalwaysminimizethetracefortheCIfusionlaw. 4.10.3ANoteonCovarianceIntersectionandEllipsoidalIntersection Onthesurface,itwouldappearthatadrawbackforutilizingtheEIfusionlawis thatitisonlyapplicabletothefusionoftwostates,whereastheCIfusionlawcanbe computedforanynumberofstateestimates.However,sequentialfusionwiththeEI fusionlawcanbeaccomplished,wherethestatesareiterativelyfusedtogetherin sequence.ThissamemethodologyhasbeenimplementedfortheCIfusionlawand shownnumericalandoptimalityimprovements[128]. 137

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4.10.4ANoteonAttitudeDataFusionwithAppendedStateVectors Formostattitudeestimationapplications,otherparametersaretypicallyjointly estimatedalongsidetheattitude.Theresultsdevelopedinthischapterdemonstrate animprovedapproachtoparameterizingtheattitudeerrorvectorforfusingattitude information.However,alloftheseresultsareequallyvalidforcaseswithappendedstate vectorsthatareinclusiveofEuclideanstates,suchasbiasesandmisalignments. 4.11TwoStarTrackerDataFusionExample Considerthecaseoftwostartrackersoutttedononespacecraft.Therststar trackerisassumedtobealignedwiththespacecraftbodyz-axisandthesecond startrackerisalignedwiththebodyx-axis.Bothstartrackersareequalincapability. Eachstartrackerprocessestheirlocalmeasurementsanddevelopsanestimatefor thespacecraft'sattitude.Thesimulationparametersforthestartrackerexamplein Section3.6.4areusedinthesesimulationsfortheinitialstateandstateestimate parameters.Figure4-9providestheerrorresultsforthestartrackersimulations.The errorfortherststartrackerisshowninFigure4-9A,whereitisveriedthatrotations abouttheboresight,paralleltothebodyz-axis,havethelargesterror.Similarly,for thesecondstartracker,Figure4-9Bexhibitsthesamebehavior,buttheboresightis paralleltothebodyx-axis.UsingtheFRPdatafusionlaw,Figure4-9Cdemonstrates thebenetsofdatafusionwiththestateestimatesareofsimilaraccuracy.Inthiscase, thefusedattitudecombinestheexcellenterrorperformanceofthex-axisusingtherst star-trackerandthez-axisusingthesecondstartracker. TheperformanceimprovementsarefurthershowninFigure4-10,whichshowsa comparisoninpointingerrorwhenusingeitherofthestartrackersindividuallyversusthe fusedpointingerror.Thefusedstateestimateismuchlesssensitivetosensoreffects originatingfromeithersensor,astypicallytheothersensorwillprovideanyinformation thatisunavailableataparticularinstanceintime. 138

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A B C Figure4-9.ExampletwostartrackerdatafusionusingtheFRPsandshadow parameters.AStartracker1errorwithoutdatafusion.BStartracker2 errorwithoutdatafusion.CFusedstartrackererror. 139

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Figure4-10.Pointingerrorcomparisonoftwostartrackerswithandwithoutdatafusion Figure4-11AdemonstratestheapproximationerrorthatexitswiththeMRPsduring theconvergingphaseofthelter.Duringconvergence,theerrorisstillhighandthe approximationerroractuallycausesthefusedestimatetobelargerinerrorthaneither oftheindividualstateestimates.However,whenusingthelocalerrorrepresentationas isdoneinFigure4-11B,theeffectoftheapproximationismitigatedandthefusedstate exhibitsthebenetsofthedatafusionprocess. A B Figure4-11.ComparisonofstartrackerdatafusionwithMRPshadowparameterand localerrorrepresentation.AMRPfusionwithshadowswitchingBMRP fusionwithlocalerrorrepresentation 140

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4.12Summary AgeneralframeworkforextendingEuclideanstatedatafusionlawstoattitude stateswaspresentedinthisChapter.Thevectorialattitudeparameterizationsservedas aframeworkforgeneratingminimalunconstrainedattitudedatafusionlaws.First-order approximationsoftheattitudeerrorvectorwereshowntobemoreaccuratefor higher-orderRodriguesparameterswithrespecttothemodiedRodriguesparameters andclassicalRodriguesparameters.ItisinterestingtonotethatwhentheHORPswere developed,Tsiotrasstated,Itstillremains,however,todeterminetheapplicabilityof thesehigherorderparametersinrealisticattitudeproblems[63].Thischapterhas shownthattheHORPsareahighlyaccurateparameterizationforminimalattitudedata fusion. Giventhedescriptionoftheattitudeerrorvectoraccuracy,twomethodswere investigatedforattitudedatafusion.Therstmethodmakesuseoftheshadowsetfor theClassIIvectorialparameterizationstoprovideanunconstrainedglobalnonsingular parameterizationfordatafusion.However,itisnotedthattheattitudeerrorvector accuracyishighestwhencenteredaroundthenullattitudestate.Therefore,thelocal errorrepresentationwasadoptedtoimprovetheaccuracyfordatafusion.Basedonthe resultsofthelocal-error,forinnitesimalrotations,datafusionisinvarianttotheminimal attitudeparameterization,asallsetsareinnitesimallylinear.Evenforrelativelylarge attitudeerrorsseveraldegrees,theattitudeerrorvectorlinearapproximationerroris ordersofmagnitudelessthanthefusionerror.Therefore,theCRPsaretheparameter ofchoiceforfastattitudedatafusion.Ifasmallimprovementinaccuracyisdesired fractionsofapercent,thenhigher-orderparameterizationscanbeused. RecallingtherstthesisstatementondatafusioninSection1.4,thischapter presentedfactsweretoextendexistingdatafusionmethodsonEuclideanstatespaces canbeextendedtotheattitudestatespaces.Itwasalsoseenthatbychoicethe combinationofchoosinganappropriateminimalattitudeparameterization,alongwith 141

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thelocalerrorrepresentation,leadstoafamilyoffastunconstrainedglobalnonsingular attitudedatafusionlaws. 142

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CHAPTER5 STOCHASTICGREEDYSENSORTASKING Thischapterisconcernedwithhowtomosteffectivelytaskthechieftocapture andsharerelativeattitudemeasurementswithrespecttothedeputiestominimizethe attitudeerrorofthedeputies.Firstthesensortaskingproblemisintroduced,whichis followedbytheintroductionofxedandgreedytaskingalgorithms.Metricsarethen investigatedforthegreedytaskingapproach.Simulationsareperformedtothexed taskingalgorithmwiththegreedytaskingalgorithm.Inaddition,severalmetricsare compared,andMonteCarloanalysisisusedtoprovideastatisticalcomparisonofthe taskingmetrics. 5.1SensorTaskingOverview Inadeterministicsetting,ataskingproblemtypicallyinvolvesthesolutionto atravelingsalesmanproblem,whereusingtheanalogy,asalesmanmustvisita prescribedsetofhousesintheshortestpath.Therearemanysolutionstothisproblem, oneofthemostwellknownandclassicapproachesbeingtheHungarianAlgorithm [129].Thetravelingsalesmanproblemsuffersfromthecurseofdimensionalityandoften requiresheuristicsandproblemdependentalgorithmmodicationsforlargernetworks. Unlikethetravelingsalesmanproblem,thesensortaskingproblemtakesona modiedstructure,whereeachtargetmayneedtobevisitedmorethanonceand theedgeweightingsarebasedontheperformancemetricofinterest,whichmaybe stochasticinnature.Therefore,sensortaskingrequiresamodiedsolutionapproach. Inthedesiredsolution,thesensorfollowsapathofleastresistancetodistribute measurementsorinformationtoasetofdeputies,whileminimizingfuelandtime. However,theobjectivesofuncertaintyminimizationandtraditionalcontrolobjective minimizationsareconfounded.Intheeventthattheobjectiveistominimizethepointing errorinthesystem,thecontrolleractionsarebasedonsensingobjectives.Ifthe objectiveistominimizefuelandtime,thecontrolisonlyasgoodasthecertaintyinthe 143

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stateestimates.Therefore,thecontrolandtaskingobjectivesareinseparable.Optimal solutionstoproblemswiththisstructurearecomputationallyexpensiveandsufferfrom thecurseofdimensionality. Othershaveinvestigatedsimilarproblems.Aninformation-basedapproachto sensortaskingfordecentralizedcontrolofvehiclesconstrainedtoplanarmotionand taskedwithinformationgatheringwasdevelopedinReference[130].Theperformance objectiveofthesensortaskingproblemwasevaluatedbyinformationmeasures.One questionthatisaddressedinthischapteris,areinformation-theoreticmeasures appropriateforattitudesensortasking?TheworkinReference[130]wasconcerned withsystemswhereinformationchangescontinuouslywiththedynamicstatesof anetworkofvehicles,suchthatacontinuouscontrolinputonthevehiclecausesa continuousinformationchange.Sensortaskingforattituderesourcesharingisalso concernedwiththeproblemofinformationgathering.However,intheframework understudy,informationisonlygainedwhentherelativeattitudesensoriscapturing measurements,whichcanrequiresignicanttimestepstoslewtothesharing orientation.Therefore,forasmalllook-aheadtime,relativeattitudemaynotbeavailable duringtheslew,andthusnonewinformationisgathered. Optimalsensortaskingusinginformationmetricshasalsobeenappliedto space-basedsensorsfortheplanartranslationalmotionproblem[131].Taskingwas basedonconstraintswithregardtorangeandeld-of-view,andappliedtothespace situationalawareness.Thetaskingsolutioninvolvedsolvingadiscrete-timelinear programmingproblem.However,thisdecision-makingdidnotconsiderthesensor dynamicsandcouplingbetweenthesensorstatesandtheobservations.Anotherplanar exampleusedinformationmetricsandappliedthemtoamyopicread:greedytasking forspacesituationalawarenessSSA[132,133].Finally,aninformationreceding horizonapproachtosuboptimalsensortaskingwasdevelopedanddemonstratedfor one-dimensionalharmonicoscillators[134,135].Stochasticoptimalcontroltheorywas 144

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usedtoposethetaskingproblemasapartiallyobservableMarkovdecisionprocess POMDP,inwhichstochasticsimulationwasusedalongwithastochasticrelaxationof thedeterministictaskingcontrol.However,themethodofReference[134,135]relieson theassumptionthatinformationgainsoccurwithinthehorizonofthecontrol. Itwaspreviouslyreportedthattheattitudeuncertaintycouldbeusedformulti-objective attituderesourcesharinginatwo-spacecraftscenario[136],wherethespacecraft motionhassixdegrees-of-freedom.Thefullsixdegrees-of-freedomcasecauses additionalcomplexitywithrespecttotheplanarsolutionsdescribedintheaforementioned references.ThischapterextendstheresultsofReference[136]toimprovepointing performanceforageneralnetworkofspacecraftinfull6-DOFmotionbysharingattitude measurementsbasedongreedytaskingalgorithms.Forthedisaggregatedsystem consideredinthiswork,thechiefdecisionisbasedonthedeputyuncertainties. 5.2TaskingProblemStatement Considerachiefspacecraft,denotedby C ,whichiscapableofmeasuringthe relativeattitudeofadeputy,denotedby D ,butonlywhenwithintheFOVoftherelative attitudesensor.Inthecasewhenthedeputieslackinertialsensing,theinformation fromthechiefisthesolesourceofinertialattitudeknowledge.Whenthedeputieshave inertialsensors,thesharedresourcesmaybeusedtoimprovethelocalstateestimate throughdatafusionasdemonstratedinsection4.11.Allspacecraftareequippedwith rategyrosforimprovedprecisionattitudepropagation. Thestructureofthesensortaskingproblemresultsintwocoupledcontrolproblems, whicharecascadedinaninner-outerloopframework.Theinnercontrolloopisa discretetaskingcontrol.Theoutercontrolloopisacontinuouscontrolinputtocommand theactuatorstotrackadeputytomeasurerelativeattitude.Thisisacontrolforsensing problem.Ifoptimalityissoughtintheinnitehorizonorsomenitehorizon,the effectofthetwocontrolinputsisobfuscatedbythecouplingofestimationandcontrol. Estimationandcontrolareinseparableduetothecascadingofthecontrolloopsand 145

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theirrespectivecontrolobjectives.Theproblemisthentodesignthetaskingcontroland dynamicscontrolundertheseconditions. Inadditiontothechallengeofseparation,optimalsensortaskingforcontinuous dynamicalsystemswithasensingobjectivesufferfromthecurseofdimensionality,as thenumberofpathsinthedynamicprogrammingproblemgrowsunboundedwiththe numberoftimestepsandnodesinthesensornetwork. 5.3TaskingSolutionMethodology Inordertomitigatetheissueswithlackofseparabilityinestimationandcontrol andthecurseofdimensionality,agreedysensortaskingapproachispursuedinthis chapter.Greedyapproachesarealsoreferredtoasmyopic,asthelogicbehindthese approachesisnear-sightedandonlyconsidersinformationimmediatelyavailablefor makingdecisions[137].Greedylogiconlyrequiresthecurrentstateofthesystemand thusseparatesestimationandcontrolforthetaskingproblem,wherethediscreteinner taskingcontrolloopfeedstheouterdynamicscontrolloopwiththetask.Theouter controlloopresultsintrackingproblemwithcontrolinputdesignedtosatisfytheinner controllooptask.Figure5-1showsthecontrolloopinblockdiagramformwiththe greedyapproach.GiventheGreedytasking,areal-timefeedbackcontrollawcanbe Figure5-1.Stochastictaskinggreedycontrolproblem usedforguidingthechiefspacecraftfromonetasktoanotherandtheestimatorsforall spacecraftcanbeupdatedandpropagatedrecursively.Finally,certaintyequivalence 146

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holdsanddeterministiccontrollawscanbeusedinsteadofstochasticallysynthesized controllers. Figure5-2demonstratestheproposedmethodforattituderesourcesharing inadisaggregatedattitudedeterminationsystem.Therststepistoconstructthe instantaneousnetworkofspacecraftthatarewithinthesensorconstraints.Next,the currentstatesofeachspacecraftareupdatedbasedontheiravailablemeasurements. Thedeputiesthencommunicatetheircurrentstateuncertaintybacktothechieffor decision-making.Basedontheprovidedinformation,thesensorgeneratesatasking queueandchoosesthehighestprioritytaskinthequeuetopursueforresource sharing.Inordertosharemeasurements,thechiefspacecraftmustreorienttopoint theboresightoftherelativeattitudesensoralongtherelativepositionvector,inwhich thedesiredquaternionkinematicsforresourcesharingwerederivedinSection2.7.2. Oncethechiefspacecraftisreorientedtothesharingconguration,measurementsare capturedandattituderesourcesaresharedoverthecommunicationlinkbetweenthe deputyandchief.Thisprocessisrepeatedthroughoutthemission. Figure5-2.Resourcesharingprocess 5.4ResourceSharingAssumptiononDynamicsTimeConstants Thetimeconstantforlterdivergenceisassumedtobemuchlargerthanthetime constantfortheattitudecontroller.Thesetimeconstantsaredirectlyrelatedtothegyro precisionandactuatorsizing.Thisassumptionensuresthatadeputywithoutinertial sensingwillnotdivergebetweenthetimeittakesthechieftoslewtopointtherelative 147

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attitudesensoratthedeputy.Ifthedeputyhasaninertialsensor,thisassumptionis unnecessary. 5.5Tasking Sensormanagementsolutionsareclassiedasxed,greedy,oroptimal.Fixed strategiespredeneataskingorder,thusensuringregularresourcesharingforall deputies.However,axed-strategycannotadapttounexpectedchangesinsystem conditionspotentiallyleadingtopoorsystemperformance.Optimaltaskingisat theotherextreme,whereaperformanceindexisoptimizedoveratimehorizonto generateascheduleforthesensortoprovidemeasurementstothedeputies.However, duetotheinnercontrollooprequiredtoreorientthechiefspacecraftpursueatask, estimationandcontrolareinseparable.Therefore,optimaltaskingiscomputationally challengingandnotpursuedinthisresearch.Inbetweenthesetwotaskingextremes istheGreedystrategy,whichmakesuseofthecurrentknowledgeofthesystem naturallyseparatingestimationandcontrol.Thepotentialadvantageoverthexed strategyisthatperformanceisconsidered,andthesystemcanadapttochangesbased onthecurrentinformation. 5.5.1BaselineTasking Thissectiondescribesanexamplexed-taskingalgorithmthatisutilizedasa baseline.ThestrategyisreferredtoastheRound-robinstrategyandissummarizedin Algorithm3.Round-robintaskingisinitializedbythesetofdeputies, D ,thechiefnode, C ,andthecommunicationrange, D .Inthisow,rstthesetofavailabledeputiesinthe network, D D ,aredeterminedbasedonthedeputiesinrangeofcommunicationwiththe chief,wherethecolumnmatrixofcommunicationrangeforthenetworkisdesignated by D andthecurrentrangeofthe j th )]TJ/F61 11.9552 Tf 12.148 0 Td [(deputy is .Next,thecurrentstatesareupdated basedontheircurrentinformationavailabletothechiefandeachdeputy.Forthisxed policy,thechiefsequentiallycyclesthroughtheavailabledeputies.Itisassumedthatin ordertoproceedwiththenexttask,thesensormustrstsatisfythecurrenttask.This 148

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requiresthatthesensorgeneratesatrackingtrajectorybasedonknownrelativestate kinematicsandtracksthetrajectoryuntiltherelativesensorisalignedwiththerelative positionvectorandwithintherangeconstraint, D .Whentheseconditionsaremet, thechiefmeasurestherelativeattitudeofthetaskeddeputyandprovidesitssensor measurementstothedeputyforintegrationintothedeputyestimator. Algorithm3 Fixed-cycleresourcesharing 1: procedure R OUND -R OBIN T ASKING P 0 D C D 2: Constructtaskingset, D D = f D j j j < D j g 8 j 3: Choosetargetcyclically, D = j +1, j n 1, j > n ,where D 2 D D 4: Generatetrackingtrajectory q i ,for C tosharewiththetaskedspacecraft, D 5: Tracktrajectory 6: if TrackingError relativesensorconstraint then 7: Update x )]TJ/F61 7.9701 Tf -1.099 -8.501 Td [(i j k to x + i j k and P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(i j k to P + i j k 8: Fusestates,ifnecessary 9: Propagatestates, x )]TJ/F61 7.9701 Tf -1.099 -8.501 Td [(i j k +1 and P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(i j k +1 10: Goto Constructtaskingset 11: else 12: Fusestates,ifnecessary 13: Propagatestates, x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(i j k +1 and P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(i j k +1 14: Goto Tracktrajectory 15: endif 16: endprocedure 5.5.2GreedyTasking AnalternativetotheRound-robinstrategy,andthefocusofthischapter,isto maximizeaperformancemetricbasedonthecurrentstateuncertaintyofthedeputies. Forattitude-intensivemissions,theuncertaintyofinterestisthepointingprecision. Therefore,thegoalisminimizetheattitudestate-errorcovariance.ForlinearGaussian systems,theprobabilitydistributiondescribingthestateerrorisfullycapturedbyits meanandcovariance.Additionally,forunbiasedestimators,themeanstateerroris zero.Therefore,acompletedescriptionofthestateerroruncertaintyisembeddedinthe state-errorcovariancematrix. 149

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ThegreedyalgorithmutilizesthesameinitializationastheRound-robinalgorithm andisequivalenttothexed-strategyuntilthetaskingdecisionismade.Tomakea taskingdecision,thechiefgathersinformationonthedeputyuncertainties,whichis assumedtobefullydescribedbythestateerrorcovariancematrix, P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(i j k .Thechief thengeneratesataskingpriority.Followingthechoiceofatask,thechieftracksthe taskeddeputyuntilthedeputyuntilarelativeattitudemeasurementisavailable.With theavailabilityofameasurement,thedeputyincorporatesthisnewinformationintoits stateestimate.ThegreedytaskingisdescribedinAlgorithm4.Thisalgorithmstates Algorithm4 Uncertainty-basedresourcesharing 1: procedure G REEDY T ASKING P 0 D C D 2: Constructtaskingset, D D = f D j j j < D j g 3: Fromattitudeestimatorcomputestatemeanandcovariance, x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(i j k P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(i j k 4: Computeuncertainty-basedmetric, J j = f P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k 5: Choosenexttargetsuchthat, D =argmax j 2 N J j ,where D 2 D D 6: Generatetrackingtrajectory q i ,for C tosharewiththetaskedspacecraft, D 7: Tracktrajectory 8: if TrackingError relativesensorconstraint then 9: Update x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(i j k to x + i j k and P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(i j k to P + i j k 10: Fusestates,ifnecessary 11: Propagatestates, x )]TJ/F61 7.9701 Tf -1.1 -8.502 Td [(i j k +1 and P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(i j k +1 12: Goto Constructtaskingset 13: else 14: Fusestates,ifnecessary 15: Propagatestates, x )]TJ/F61 7.9701 Tf -1.1 -8.501 Td [(i j k +1 and P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(i j k +1 16: Goto Tracktrajectory 17: endif 18: endprocedure thatthechiefwillshareresourceswiththemostuncertaindeputyandreorienttoprovide itwithmeasurements.Oncetheinformationcontainedinthemeasurementisshared andanupdateoccurs,theuncertaintyinthatdeputywillbereduced.Iftheuncertainty issufcientlyreduced,thechiefwillbetaskedwithadifferentdeputy.Therefore,the sensortaskingadaptsbasedontheinformationgainedfromacquiringmeasurements andtheinformationlostduetocovariancepropagation. 150

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Thealgorithmasitstands,placesequalweightingontheuncertaintyofeach spacecraft.However,ifonespacecraftrequireshigherprecisionattitudeknowledge, anyofthesealgorithmscanbemodiedtoincorporateaconvexweightingrelationship inthetaskingdecision.Thegreedytaskingalgorithmdoesnotspecifyaperformance metric.Section5.6overviewsseveralperformancemetricsthatcharacterizethestate uncertainty. 5.6PerformanceMetrics Thereareseveralpropertiesofinterestwhentaskingthechieftocapturerelative attitudemeasurementsofthedeputies.Ideally,ataskingalgorithmwilldrivetheerror inthesystemtoitsminimumandconsumeaslittlefuelandtimeaspossible.However, lackofseparationandcontrolmakedecisionsbasedonfuelconsumptionandtime difculttopursue.Therefore,onlyinstantaneousstatesareavailable.Particularly,state estimationprovidesameasureoftheinstantaneousuncertaintyinthesystem.Alogical strategythen,isforthechieftoprovidemeasurementstothedeputywiththehighest needatthepresenttime.Deningthehighestneedbasedontheuncertaintymotivates acharacterizationofstateuncertaintysize. Thecovariancematrixisameasureofthevarianceorspreadinuncertainty describedbytheprobabilitydensityfunctionofarandomvariable.Sincetherandom variableofinterestistheerrorstateofthesystem,thecovariancematrixwillprovide acomparativemeasureofuncertaintyforalldeputyattitudeestimates;astheyareall centeredatzeroassumedtobeunbiasedwiththeirrespectivevariances.Thehigher thevariance,thelesscertainthedeputystateknowledge.Thisimpliesthatdeputies needforadditionalinformationisgreatest.Therefore,highervariancescorrespondto lowerprecisionintheattitudeknowledge. Thegoalistochooseametricthatbestcapturestheuncertaintyofeachspacecraft, sothatthechiefspacecraftwillprioritizeitstaskingtoreducethepointingerrorin themaximumuncertaintydeputy .Inthissection,thesizeofthecovariancematrix 151

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isconsidered.Thereareseveralpossibilitiesthatexistforcharacterizingthesizeof theuncertaintydescribedbyaprobabilitydensityfunction.Normsprovideameasure ofdistanceforvectorspaces,whichisspecicallyapplicabletothevectorspaceof matricesdenedover R m n .Thissectiondescribesseveralmetricsthatareafunctionof thecovariancematrix.Thesemetricspopulatetheuncertaintyfunction, f P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(j j k ,intthe greedytaskingdescribedinAlgorithm4. Considertwoelements, A and B ,whicharemembersofaset.Ametricdenesthe distance, d ,betweenthosetwoelements.Moreformally,ametricsatisesthefollowing properties[138]: 1.Positivity: d A B 0,and d A B =0 A = B 2.Symmetry: d A B = d B A 3.Triangleinequality: d A B + d A C d B C 5.6.1Trace Thetraceofasymmetricpositivedenitematrix A ,denotedas tr A ,isthesum ofthediagonalelementsof A andsatisesallconditionsofametric.When A isthe state-errorcovariancematrix, P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(j j k ,thetracesumsthesmallerroranglecovariancein allthreerotationaldegreesoffreedom.Therefore f P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(j j k = tr P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(j j k providesameasureoftheinstantaneoustotalattitudeuncertaintyofaspacecraft. 5.6.2MatrixNorms Alternatively,matrixnormsprovideameasureofdistance,whereallnormsona vectorspacearemetrics.Normsaredenotedby kk .Consideran n m matrix, A ,with thefollowingproperties[85,139]: 1.Positivity: k A k 0for A 6 =0and k A k =0 A =0 2.Scalarhomogeneity: k A kj jk A k 152

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3.Triangleinequality: k A + B kk A k + k B k Additionally,thenormofamatrixwillassumethepropertyofsub-multiplicativity;thatis, k AB kk A kk B k Normsarenon-unique,butareequivalentinconvergencefornitedimensionalvector spaces.Althoughequivalentinconvergence,eachnormprovidesadifferentmeasure ofsizeandthuscanprovidedifferentmeasuresofuncertainty.Threenormswillbe explored. The1-normisthemaximumabsolutevaluecolumnsumofamatrix.Appliedtoa state-errorcovariancematrix,the1-normprovidesameasureofthemaximumtotalerror contributedbyasingledirection. f P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(j j k = k P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k k 1 =max j n X i =1 j a ij j Duetosymmetryofthestate-errorcovariancematrix,themaximumabsolutevaluerow sum,knownasthe 1 -norm,isequaltothe1-normandcanbeusedinterchangeablyin thisapplication. The2-normisthesquarerootofthemaximumeigenvalueofthematrix, P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j T k P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k ; thatis f P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k = k P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(j j k k 2 = q max P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j T k P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k Thisnormprovidesanothermeasureofthemaximumerrordirection,butduetothe productrequiredbythedeterminantincorporatesamultiplicativeerroreffect. Unlikethe1-and2-norms,theFrobeniusnorm, f P )]TJ/F61 7.9701 Tf -0.836 -8.502 Td [(j j k = k P )]TJ/F61 7.9701 Tf -0.837 -8.502 Td [(j j k k F = q tr P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j T k providesameasureofEuclideandistanceforalltermsinthestate-errorcovariance matrixrelativetozero.Therefore,alltermsareweightedequallywhendeterminingthe 153

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sizeofthematrix.Fortheresourcesharingproblem,thisnormstatesthateachdirection andcrosscorrelationtermisequallyimportant,andthusismeasureofthetotalerrorin thesystem.However,whenrelatingtheFrobeniusnormtopointing,weightisgivento thecross-correlationterms,althoughtheydonotdirectlyimpacttheattitudeaccuracy. 5.6.3DifferentialEntropy Inattitudeestimation,informationiscontinuouslygainedandlostinacycle correspondingtomeasurementupdatesandpropagation,respectively.Givenknowledge oftheunderlyingprobabilitydistributiondescribingtherandomvariableforthestate error,animportantquestionishowmuchinformationiscontainedinthatrandom variable?Informationtheory,originallydevelopedtounderstandcommunicationlink efciency,isndingmanyusesinmodernengineeringapplications.Inthiscontext, informationtheoryisutilizedtogenerateametricforthestate-errorcovariancematrix size. InformationtheorywasintroducedbyShannon[140]asamathematicalformulation forcommunicationlinkefciency.ShannonformulateddatatransmissionasaMarkov processanddevelopedthequantityentropy, H ,todescribehowmuchinformationis generatedorlostbythatprocess.Entropywasconstructedwiththefollowingproperties [141]: 1. H max isthemaximumentropy,whichoccurswiththesureevent 2. H =0occurswhenalleventsareequallypossiblei.e.,uniformdensity 3. H x y H x + H y i.e.,thetriangleinequalityholds Consideradiscreterandomvariable, X ,withprobabilitymassfunction, p x .The entropysatisestheabovepropertiesandisdenedas H X = )]TJ/F29 11.9552 Tf 11.291 11.357 Td [(X x p x log p x Shannonentropydescribesthecompactnessofarandomvariable. 154

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Forcontinuousrandomvariables,theShannonentropyisreferredtoasdifferential entropy[141]andisdenedas H X = )]TJ/F29 11.9552 Tf 11.291 16.272 Td [(Z X f x logf x dx ForGaussianrandomvariables,thedifferentialentropyis f P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k = H P )]TJ/F61 7.9701 Tf -0.836 -8.501 Td [(j j k = 1 2 log h e n det P )]TJ/F61 7.9701 Tf -0.837 -8.501 Td [(j j k i where det denotesthedeterminant.Differentialentropyprovidesameasureofthe volumeofthesmallestsetcontainingmostoftheprobabilitycontainedinaprobability densityfunction.Therefore,theuniformdistributionwillhaveamaximumentropy, becauseallvaluesinthesupportsetcontainthesameinformation.Whereas,the sureeventhasminimumentropy.Foralldistributionsbetweenthosetwobounds, entropyprovidesameasureofspreadintheprobabilitydensityfunction.Asaresult,the differentialentropy,asafunctionofthestate-errorcovariancematrix,providesanatural measureofuncertaintyalbeit,itisnottechnicallyametric. 5.7Simulations Totestthetaskingalgorithmsandperformancemetrics,asimulationframework wasconstructedusingtheMATLABsoftware[105].Modelswereconstructedfor theorbitalandattitudemotionofmultiplespacecraftyinginformation.Specically, simulationswereconductedonanetworkconsistingoffourspacecraft,yingwith unconstrained6-DOFmotion.Thisnetworkiscomprisedofasinglechiefspacecraftand threedeputies.Thedeputiesareprovidedwithvaryingdelityrategyrostodemonstrate theutilityofthedevelopedalgorithms,suchthatdeputy-1isprovidedwiththeworst gyro,withincreasingprecisionuptodeputy-3havingagyrothatisequivalentto thechiefspacecraft.Algorithm3wasusedasbaselineforcomparisontotheGreedy approachdescribedinAlgorithm4. 155

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5.7.1PerfectRelativeAttitudeSensorAssumption Inordertoisolatetheaffectsofthesensortaskingalgorithmsandlimitthenumber ofvariableparameters,itisassumedforthesimulationsthatfollow,thattherelative attitudeknowledgeisperfect.Thissimplication,eliminatestheeffectsofestimation anddatafusioninadisaggregatedsystem,andisolatesthetaskingperformance.Using thisassumption,thesharedstartrackermeasurementsaretransformedusingtheexact relativeattitudematrix,suchthat D i b j = R T q C = D C b j + R T q C = D C i j =1,2,..., N Inthissetting,measurementsareshareddirectlytothedeputiesandutilizedasifthey originatedon-boardthedeputies.Itshouldbeemphasizedthatthisassumptionisonly inplacetoavoidconfoundingeffectswithrelativeattitudeanddatafusion.Alsonote, thisassumptionwillberelaxedinthefullresourcesharingsimulationspresentedin Chapter6. 5.7.2SimulationInitialization Usingthesimulationtestbed,thefour-spacecraftscenariowasinvestigated,with inertialmotionofthechiefandrelativemotionofthedeputieslistedinTable5-1. Table5-2liststhecontrolparametersfortheLyapunovcontrollerderivedin Section2.7.4andspecicallytopopulatethegainsintheresultingcontrollerin Eq.2.Thesaturationlimitisindicativeofthelimitexpectedforthesizeofthe spacecraft.Simulationswereperformedfor225minutestoensurethattheestimators convergedandperformancewasbasedonthesteady-statebehavior.Thegyrosoperate atasamplingfrequencyof10Hzandthestartrackeronthechiefissampledevery second. Inordertodemonstratetheperformanceofthealgorithmsdiscussed,each spacecraftwasprovidedwithadifferentgyroperformancespecication,asthegyro performancespecicationscontroltheinformationlossprocessintheattitudeestimators 156

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Table5-1.Stateinitializationsfortaskingsimulations Spacecraft: J s J 1 J 2 J 3 diag 300,100,200 kg m 2 ChiefInitialStates: r C j 0 [ 6778.100 ] T km v C j 0 [ 06.64123.8343 ] T km/s q C j 0 [ 1000 ] T C j 0 [ 0.0010.0010.001 ] T rad/s Deputy-1InitialStates: D 1 j 0 [ 5 )]TJ/F20 9.9626 Tf 7.749 0 Td [(1015 ] T m D 1 j 0 [ )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.0100 )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.01130 ] T m/s q D 1 j 0 [ 0100 ] T C j 0 [ )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.0030.0020.004 ] T rad/s Deputy-2InitialStates: D 2 j 0 [ )]TJ/F20 9.9626 Tf 7.749 0 Td [(10525 ] T m D 2 j 0 [ )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.02000.02260.0100 ] T m/s q D 2 j 0 [ 0010 ] T D 2 j 0 [ 0.0010.003 )]TJ/F20 9.9626 Tf 7.748 0 Td [(0.003 ] T rad/s Deputy-3InitialStates: D 3 j 0 [ )]TJ/F20 9.9626 Tf 7.749 0 Td [(15 )]TJ/F20 9.9626 Tf 7.748 0 Td [(1030 ] T m D 3 j 0 [ 0.01000.0339 )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.0200 ] T m/s q D 3 j 0 [ 0001 ] T D 3 j 0 [ 0.002 )]TJ/F20 9.9626 Tf 7.749 0 Td [(0.0010.003 ] T rad/s Table5-2.Controlandtimeparametersfortaskingsimulations ParameterValueUnits Time: t up 1 s t prop 0.1 s T 225 min Controller: u max 5 Nm K diag,6.667,13.333 C diag,46.667,93.333 presentedinChapter3.Thechiefspacecraftpropagateswithastate-of-the-artgyro. Deputy-3utilizesanequivalentgyrotothechief,whereasdeputy-2propagateswith alowerperformancegyroanddeputy-1propagateswiththemostbiasedandnoisy sensor.Table5-3providesdetailsonthespeciccharacteristicsofthesensorsand Table5-4detailsthestateestimateinitialization.Itshouldbenotedthatthechiefis assumedtoslewatarateslowenoughfortheaccuracyofthesensortohold,even duringthetaskingmaneuvers.Additionally,theeld-of-viewontheperfectrelative 157

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attitudesensorissimulatedwithbothasmallandlargeeld-of-viewintheresulting simulations. Table5-3.Sensorcharacteristicsfortaskingsimulations ParameterValueUnits ChiefStarTracker: FOV star 6 deg star 1.67 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 deg ChiefVISNAV: FOV vis Varying deg vis 1.67 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(3 deg ChiefGyro: C v p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(7 rad = s 1 = 2 C u p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(10 rad = s 3 = 2 Deputy-1Gyro: D 1 v 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 D 1 u 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 Deputy-2Gyro: D 2 v 5 p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(7 rad = s 1 = 2 D 2 u 5 p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(10 rad = s 3 = 2 Deputy-3Gyro: D 3 v p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(7 rad = s 1 = 2 D 3 u p 10 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(10 rad = s 3 = 2 Table5-4.Stateestimateinitializationsfortaskingsimulations ParameterValueUnits ChiefEstimate: q C j 0 [ 1000 ] T C j 0 0.1 deg = hr Deputy-1Estimate: q D 1 j 0 [ 0100 ] T D 1 j 0 10 deg = hr Deputy-2Estimate: q D 2 j 0 [ 0010 ] T D 2 j 0 5 deg = hr Deputy-3Estimate: q D 3 j 0 [ 0001 ] T D 3 j 0 1 deg = hr 5.7.3Single-runResults Therelativemotionofthethreedeputieswithrespecttothechieffollowsfrom theCWHequationsandinshowninFigure5-3.Athree-dimensionalviewisshownin Figure5-3A,followedbyorthographicprojectionsinFigure5-3B-5-3D.Thesegures demonstratethattherelativemotionisboundedandthatthespacecraftareyingin proximityforutilizationofaVISNAV-likerelativeattitudesensor.Theboundednessalso impliesthatthecommunicationlinkispersistent. 158

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A B C D Figure5-3.RelativepositionintheHillframe.AThree-dimensionalBX-Yprojection. CX-Zprojection.DY-Zprojection Figure5-4showsanattitudeprecisioncomparisonoftheRound-robinstrategyto theuncertainty-basedgreedyalgorithmswithvaryingperformancemetricforasingle 225minutesimulation.Alsorecallthatonesecondofarcisequal1600 th ofadegree. Eachbarrepresentstheroot-mean-squareRMSerroroverthesimulationduration, whichwascomputedforeachalgorithmandtheirassociatedmetrics.Thebarsare dividedinthepointingerrorcontributedbythesmallrotationsabouteachbodyaxis.The RMSwascomputedforthetimeperiodbeginningaftereachestimatorhasconverged inthiscase,chosenas10min.Inthesesimulationsa10 -FOVsensorisassumed. 159

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Theweightedtracemetricgreedyallowsthesystemdesignertogivemoreprioritytoa specicdeputy.Intheweightedcase,Deputy-1wasassumedtorequiremoreprecision thandeputy-2anddeputy-3withweighting W 1 =0.5, W 2 =0.25,and W 3 =0.25.That is,deputy-1wasweightedtwiceasmuchastheotherspacecraft.FromFigure5-4,the A B C D Figure5-4.RMSerrortaskingperformancecomparisonrelativesensor10 FOV.A Chief.BDeputy-1.CDeputy-2.DDeputy-3. chiefattitudeerrorisanorderofmagnitudegreaterthanthedeputies.Thisisadirect resultofthesparsityofmeasurementsavailabletothedeputiesincomparisontothe chief;thatis,thechiefmeasurementsaredividedamongstthedeputies.Furthermore, theslewbetweentasksresultsinnoresourcesharinginthattimeperiod. Figure5-4Ashowsthatthechiefisdominatedbyerrorrotationsabouttheboresight directionoftherelativeattitudesensor.Thisresultsfromthefactthatdirection measurementsprovidenoinformationaboutrotationsaboutavectorparalleltothat direction.Thissamebehaviorisnotobservedinthebargraphsdescribingthedeputies 160

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inFigures5-4B-5-4D.Thisisaresultoftherelativeattitudemappingbetweenthechief andthedeputies,wheretheboresightdirectiononthechiefismappedtoatime-varying directiononthedeputy.Thedeputiesmotionareperiodicduetothetorque-freemotion assumedintheirsimulation.Thisperiodicityresultsintheobserveddistributionof uncertainty. Whencomparingthetaskingperformance,thechiefismostlyinsensitivetothe taskingalgorithmandperformancemetric.However,therearesomeindirectdynamic affectsfromthevaryingattitudetrajectories,whichresultfromthetaskingalgorithms. Theattitudetrajectorydirectlyimpactsthestareldthatisobservabletothestartracker atanypointintime,andthustheaccuracyofestimatesderivedfromthestartracker. Anotherobservationisthattheround-robintaskingalgorithmyieldslowperformance forDeputy-1andDeputy-2buthighperformanceforDeputy-3.Recall,thatDeputy-3 containsthebestgyrospecications,andthuslosesinformationatamuchlower ratethanDeputy-1andDeputy-2.Usingaround-robinapproach,Deputy-3obtains measurementsatanequalratetotheotherdeputiesalthoughitlosesinformationat alowerrate.However,moremeasurementsforDeputy-3reducedthemeasurements availabletoDeputy-1andDeputy-2.Table5-5tabulatestheaggregatedresultsofthe RMSpointingperformance.ImprovementinDeputy-3comesatthecostofreduced performanceoftheotherdeputieswhenutilizinganuncertainty-basedgreedyapproach. Fromtheseresults,thetraceandweightedtracemetricyieldthelowestaggregated uncertainty.However,theseresultsareforonesimulationandforonlyoneeld-of-view. Table5-5.AggregatedTaskingPerformanceComparisonRelativeSensor10 FOV RMSErrorarcsec Round-robinTraceWeightedTraceShannonTwo-normFrobenius Chief0.470.530.510.500.510.49 Deputy-112.288.555.998.368.578.53 Deputy-211.307.019.288.568.138.25 Deputy-36.005.615.805.664.857.50 Sum30.0521.6921.5823.0822.0524.77 161

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Inadditiontothesimulationofa10 FOVVISNAVsensor,simulationswerealso performedfora60 FOVVISNAVsensor.AlargerFOVsensorresultsinlessslewing bythechieftoprovidemeasurements.Figure5-5providesacomparison.Thegeneral behaviorofthiscaseisconsistentwiththesmallereld-of-viewsensor.However,there aresomedifferences.Specically,thebesttaskingmetricforeachdeputyhaschanged. Additionally,Table5-6showstheperformancemetricproducingthehighestaggregate pointingperformancehaschanged.Itisimportanttonotethatthesesimulationsassume perfectrelativeattitudemeasurementswhenthedeputyiswithintheeld-of-view oftherelativeattitudesensoronthechief.Additionally,measurementsareshared insteadofstateestimatethroughdatafusion.Therefore,thepointingperformance observationsfromthemetricareconfoundedwiththedynamiceffectsfromtheslewing andthebenetsofdatafusionarenotobserved.However,basedontheassumptions, asignicantimprovementwasobservedintheaggregateaccuracywhenusingthe greedyalgorithmincomparisontotheround-robinstrategy.AMonteCarloanalysisis performedinSection5.7.4tocomparetheaffectsofthetaskingmetric,statistically. Table5-6.Taskingperformancecomparisonrelativesensorwith60 FOVRMSerror arcsec Round-robinTraceWeightedTraceShannonTwo-normFrobenius Chief0.830.800.980.650.770.76 Deputy-17.174.593.634.843.865.16 Deputy-24.563.544.403.254.014.34 Deputy-31.653.083.992.763.192.71 Sum14.2112.0113.0011.5011.8312.97 5.7.4MonteCarloAnalysis AMonteCarloanalysiswasperformedtocomparetheuncertaintymetric.One hundredsamplesweredrawnuniformlyfromtheunderlyingprobabilitydistributions. Figure5-6showsthetimehistoryoftheattitudeerrorofthechiefanddeputiesfor allMonteCarlosamples.Therstobservationisthateachofthedeputiesexhibits oscillatorybehaviorintheirerror.Theseoscillationswereobservedtooccuratthe frequencyoftheangularvelocityofthespacecraft;thatis,usingtheparametersin 162

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A B C D Figure5-5.RMSerrortaskingperformancecomparisonrelativeeensor60 FOV.A Chief.BDeputy-1.CDeputy-2.DDeputy-3. Table5-1,thenormoftheangularvelocityofDeputy-1throughDeputy-3,is5.39 E )]TJ/F20 11.9552 Tf 12.296 0 Td [(3 rad/s,4.36 E )]TJ/F20 11.9552 Tf 12.756 0 Td [(3rad/s,and3.74 E )]TJ/F20 11.9552 Tf 12.756 0 Td [(3rad/s,respectively.Sincetheattitudemotion ofthedeputiesistorque-free,angularmomentumisconserved,andthenormofthe angularvelocityisconstantthroughoutthesimulation.Anotherobservationisthatthe 3boundsdoinfactboundtheerrorwithapproximately99.7%condence;thatis, veryfewerrorsfalloutsideofthe3bounds.Thisveriesthattheltersareconsistent, evenwhenmeasurementsaresparse,suchasthecaseforDeputy-3inFigure5-6D. Finally,itcanbeseenthatthetaskingexhibitsonlyasmallvariabilitywithrespecttothe MonteCarlosampling.Thisvariabilityisbetterelucidatedinthebox-and-whiskerplot showninFigure5-7.Thebox-and-whiskersplotvisualizesthemedian,25-thpercentile, 75-thpercentile,andminimumandmaximumvaluesoftheMonteCarlosampled data.OverlayedonthischartaretheRMSerrorsamplepointsfromtheMonteCarlo 163

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A B C D Figure5-6.MonteCarloattitudeerrortrajectoriesfora10 FOVrelativeattitudesensor usingthetracemetric.AChief.BDeputy-1.CDeputy-2.DDeputy-3. sampling,whicharejitteredforvisualization.Themeanandstandarddeviationarealso visualizedbytheredlineandtopandbottomoftheblueregion,respectively.Based onthisdata,theone-normmetrichadthelowestsamplemeanandvarianceinthe RMSerror.Additionally,the10 -FOVdatawasmorespreadoutthanthe60 -FOVdata. Therelativemotionisprimaryfactorthatwasnotremovedintheexperiments,butnot accountedforinthetasking.Thedifferenceinperformancebetweenthe10 -FOVdata and60 -FOVdataisaresultoftherelativetranslationalmotionofthechiefwithrespect 164

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Figure5-7.MonteCarlosampledatacomparisonoftheRMSerror tothedeputies.Therefore,modicationstothegreedytaskingalgorithmtoaccountfor relativetranslationalmotionmayimprovetheperformance. 5.7.5ANoteontheTracePerformanceMetric AswasthecaseinthenoteondatafusionseeSection4.10.2,withoutthe effectsofrelativemotion,thetracemetricshouldprovidetheoptimalmetricfor uncertainty-basedtaskinginattitudeproblems,aspointingisdenedthroughthe squarerootofthetraceofthestate-errorcovariancematrix.Itispostulatedthatifthe relativemotiondynamiceffectswerenotpresent,thiswouldhavebeenobservedinthe simulations.However,relativemotiondoesfactorintosensortaskingperformancefor theattituderesourcesharingproblem.Althoughtheone-normproducedbetterresults inthesimulationspresented,theimprovementsweremarginaloverthetracemetricand maybeattributedtotherelativemotion. 5.8Summary TwonewmethodswerepresentedinthisChapter.Therstmethodutilized information-guidedgreedysensortaskingtoovercometheissuewithseparation 165

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inestimationandcontrol.Thesensor-taskingproblemforadisaggregatedattitude determinationsystemwasintroduced.Optimalsolutionstothisproblemwereshown tosufferfromcouplinginestimationandcontrol.Asasolutiontothisproblem,greedy taskingalgorithmsweredeveloped.Todrivethegreedyalgorithms,metricsbased onthestate-errorcovariancematrixwereintroduced.Specically,thematrixtrace, matrixnorms,anddifferentialentropywereusedasameasureoftheuncertaintyin eachspacecraft.Thesensorwasthentaskedwithprovidingmeasurementstothemost uncertainspacecraftusingAlgorithm4.WiththeGreedytaskingthesystempointing errorwasreducedconsiderablyoverthebaselineRound-robinstrategyofAlgorithm3. MonteCarlosimulationswereperformedtodeterminethestatisticalsignicancein performancebetweenthedifferenttaskingmetrics.Resultsshowedthatforaparticular eld-of-view,allofthetaskingmetricsgeneratedverysimilarresults,butthatthe one-normwasmarginallybetterthantheothersfortheparticularscenariossimulated. Themainresultofthischapteristhatperformanceinattitudesensortaskingproblemsis confoundingbetweentheeffectsofrelativemotion. 166

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CHAPTER6 SPACECRAFTATTITUDERESOURCESHARINGSIMULATIONS Inthischapter,simulationsarepresentedforseveralattituderesourcesharing scenarios.Thesescenariosarechosentoberepresentativeofmissionsthatcould benetfromattituderesourcesharing.Simulationsaredividedintothreeparts.Inthe rstpart,resourcesharinginatwospacecraftnetworkisconsidered.Inthiscase,there isnotaskingdecisionrequiredandthechiefisassumedtoalwaystrackthedeputy. Resourcesharingwithtwospacecraftdemonstratestheeffectofdatafusiononthe attitudeaccuracyofthechiefanddeputy.Inthesecondpart,resourcesharingbetween threespacecraftisconsidered.Inthiscase,thechieftracksthetwodeputiesfollowing asequenceresultingfromthetaskingalgorithm.Resultsinthispartdemonstratethe combinedeffectofdatafusionandsensortasking.Inthelastpart,thegreedyand round-robintaskingalgorithmsarecomparedforthethreespacecraftresourcesharing scenarios. 6.1SimulationInitializations Forthetwoandthreespacecraftscenarios,theinitialstatesandstateestimatesare initializedwiththeparametersinTable5-1andTable5-4,respectively.Thesimulations areperformedfor225minutes,whichequatestoapproximately2.5orbitsinthe assumed400kmorbitofthechief.Thedeputiesyinboundedorbitsaroundthe chiefasshowninFigure5-3.Table6-1liststhesensorcharacteristicsfortheattitude sensorsandrategyroscopes. Forallsimulations,attitudedatafusionusestheFRPparameterizationwithlocal errorrepresentationasappliedtothecovarianceintersectionfusionlaw,describedin Section4.9.Thersttwopartsassumegreedysensortaskingwiththetracemetric andthelastpartprovidesacomparisonofresultswiththeround-robinstrategy.The followingquantitieswillbepresentedtocomparetheperformanceandbehaviorofthe resourcesharingsimulations: 167

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Table6-1.Sensorcharacteristicsusedinattitudeestimateresourcesharingsimulations ParameterValueUnits StarTracker: star 1.67 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(3 deg FOV star 6 deg VISNAV: vis 1.67 10 )]TJ/F20 6.9738 Tf 6.226 0 Td [(3 deg FOV vis 10 deg Magnetometer: mag 1 deg SunSensor: sun 0.1 deg ChiefGyro: C v p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 C u p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 Deputy-1Gyro: D 1 v 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 D 1 u 10 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 Deputy-2Gyro: D 2 v 5 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 D 2 u 5 p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 Deputy-3Gyro: D 3 v p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(7 rad = s 1 = 2 D 3 u p 10 10 )]TJ/F20 6.9738 Tf 6.227 0 Td [(10 rad = s 3 = 2 1.Thesmallanglecomponentsoftheattitudeerrorvector, f e 1 e 2 e 3 g ,with3boundsforthechiefanddeputieswithandwithoutdatafusion 2.CommandedtrackingtorqueforthechiefcontrollerfromEq.2andgainsfrom Table5-2 3.Angularvelocityofthechiefanddeputies 4.Covarianceintersectionweights Forallcases,theangularvelocityofthetwo-deputiesresultfromtorque-free motion,andareprovidedinFigure6-1. 6.2TwoSpacecraftAttitudeResourceSharing Withasinglechief-deputypair,resourcesaresharedatthecommunicationrate oncethechiefhasacquiredthesharingattitudewhiletrackingthedesiredcontrolsignal describedinSection2.7.1.Thiscasedemonstratesattituderesourcesharingwith datafusionwithouttasking.ThedisaggregatedestimationschemeandEKFequations providedinTable3-10areimplementedforthechief-deputypair.Inthisscenario, simulationsareconductedforthecasewhenthedeputyhaslocalaccesstocoarse 168

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A B Figure6-1.Angularvelocityresultingfromtorque-freemotionforthedeputiesinall simulatedscenarios.ADeputy-1.BDeputy-2. attitudeusingsunsensorsandamagnetometer,andthecasewhenishaslocalaccess toastartracker.ThechiefhaslocalaccesstoastartrackerandVISNAVsensorwith specicationslistedinTable6-1. 6.2.1TwoSpacecraftwithStarTrackersScenario TheattitudeerrortrajectoryforbothspacecraftisshowninFigure6-2.Comparing theresultsofthechieferrorinFigure6-2AwiththedeputyerrorinFigure6-2B,shows thatwithoutdatafusionthedeputyhasasteadystateerrornearlytwiceaslargeas thechief.Thisisaconsequenceofthedeputyusinganoisiergyroscopethanthe chief.However,theincorporationofattituderesourcesharingreducestheestimate errorintheboresightaxisofeachspacecraft.Thisisanalogoustothetwostartracker casesimulatedattheendofChapter4.Sincethechiefalwaystracksthedeputy, thecontroltorqueandangularvelocityoperateatnominaltrackinglevelsasseen inFigure6-3A-6-3B.Specically,itcanbeseenthattheperiodofthesesignalsare 169

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consistentwiththechief'sorbitalmotion,whichrecallwasapproximately90minutes. Sincebothspacecrafthavethesamestartracker,butdifferentgyros,Figure6-4shows thatingeneraltheweightingsareequal.However,thereisaslightpreferenceforthe chiefinformation. A B Figure6-2.Localandfusedquaternionerrorforthetwospacecraftstartrackerscenario. AChief.BDeputy. 6.2.2TwoSpacecraftwithCoarseDeputySensorsScenario Figure6-5showstheattitudeerrorinthechiefanddeputyforthecasewhendata fusionisused,aswellaswhenattitudeonlylocaldataisused.Sincethechiefhas signicantlybettersensingthanthedeputy,Figure6-5Ashowsthatforcoarsedeputy sensing,thechief'sattitudeerrorisinvarianttoattitudeestimateresourcesharing,but thedeputyerrorisimprovedbyordersofmagnitude,asseeninFigure6-5B.Likethe twospacecraftwithstartrackersscenario,thechieftracksthedeputyfortheentiretyof thesimulation.Asaresult,Figure6-6showsthatthecontrolinputandangularvelocity areverysmall.Inaddition,thedeputyneverusesitslocalstateestimateafterthe 170

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A B Figure6-3.Chiefattitudecontrolforthetwospacecraftstartrackerscenario.AControl torque.BAngularvelocity. Figure6-4.Covarianceintersectionweightsforthetwospacecraftstartrackerscenario sharingcongurationisacquiredbythechief,becausetheinformationfromthechief isalwayssuperiortoinformationlocaltothedeputy.ThisfactisevidencedbytheCI weightingsinFigure6-7. Inadditiontothelong-termbehaviorseenintheabovegures,Figure6-8shows aplotofatransienteffectintheattitudeerrorattheonsetofdatafusion.Attheinstant 171

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A B Figure6-5.Twospacecraftlocalandfusedquaternionerrorwithcoarsedeputyattitude sensing.AChief.BDeputy. therstrelativeattitudemeasurementisavailable,theestimateofthedeputyonthe chiefisverypoor.However,vectormeasurementsoriginatingfromtherelativeattitude sensorarehighlyaccurate.Therefore,applicationofthesensitivitymatrixinEq.3at therstmeasurementhasconsiderableerrorduetotheerrorin R T q )]TJ/F61 7.9701 Tf -0.58 -8.501 Td [(D .Therefore,the Kalmangaincomputationisbasedonthesensitivitymatrixwithrst-ordererrors.This causestheerrortobrieygrowandthendecayafterrelativeattitudemeasurements beginloweringtheerrorinthedeputyinertialattitude,suchthatthesensitivitymatrix onlyexhibitssecond-ordererrors. 6.3ThreeSpacecraftAttitudeResourceSharing Withmorethanonechief-deputypair,thechiefmustdecidewhichspacecraft requiresanimprovedstateestimatebycapturingrelativeattitudemeasurements.These simulationsimplementattitudeestimateresourcesharingthroughdatafusionand sensortasking.Thesimulationsinthissectioncombinemanyofthedevelopments 172

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A B Figure6-6.Chiefattitudecontrolforthetwospacecraftcoarsedeputyattitudesensing scenario.AControltorque.BAngularvelocity. Figure6-7.Covarianceintersectionweightsforthetwospacecraftcoarsedeputy attitudesensingscenario presentedinthisdissertation.Specically,theEKFforadisaggregatedADSon threespacecraftisimplementedbasedonthegeneraln-spacecraftextendedKalman lteringequationsfordisaggregatedADSprovidedinSection3.9.TheFRPlocalerror covarianceintersectionalgorithmisonceagainused.Fortasking,thegreedytasking 173

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Figure6-8.Effectsobservedattheonsetofdatafusionwithpreciseandcoarsesensors algorithmderivedforattitudemeasurementsharinginChapter5.Withthisframework, simulationsareconductedforthecasewhenDeputy-1hascoarseinertialattitude sensors,andthecasewhenallspacecrafthavestartrackers.ThechiefandDeputy-2 usestartrackersinbothcases. 6.3.1ThreeSpacecraftwithStarTrackersScenario Thiscaseisanalogousinstructuretothetwospacecraftwithstartrackers scenario,exceptnowthechiefmusttrackbothdeputiesbasedonataskingalgorithm. AjuxtapositionoftheattitudeerrorforthethreespacecraftisshowninFigure6-9. Figure6-9Ashowsthatdatafusionforthechiefonlyreducestheerrorintheboresight axiswhenthechiefhasslowlyevolvingdynamics,whicharerepresentedbythespikes in e 3 forthelocalchiefattitudeerror.Theslowlyevolvingangularvelocityofthechief 174

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isdepictedinFigure6-10B.RecallfromFigure6-1thatDeputy-1andDeputy-2have comparablyslowevolvingattitudestates.Duetotheseslowlyevolvingstates,thelocal estimateerrorrotationsabouttheboresightaxisasconsiderablylargerthanforthe transverseaxes.However,attituderesourcesharingsignicantlyimprovestheerrors abouttheboresightdirectionofthestartrackers.Errorsaboutthetransverseaxes arealsoreducedforbothdeputies,astheirsteadystatevaluesarehigherthanthe informationprovidedbythechief,duetodegradedgyrospecicationswithrespectto thechief.Figure6-11showsthattheinformationfromallspacecraftareusedfordata fusiontoimprovetheirstateestimates.Therefore,resourcesharingbenetedallthree spacecraftinthenetwork,butespeciallythedeputies,whichhadlowerqualitylocal informationavailablewithoutresourcesharing. 6.3.2ThreeSpacecraftwithCoarseDeputy-1SensorsScenario Inthelastcase,thechiefandDeputy-2areequippedwithstartrackers,and Deputy-1isequippedwithcoarseinertialattitudesensors.Figure6-12providesa juxtapositionoftheerrorineachspacecraft.Muchofthebehaviorexhibitedinthethree spacecraftwithstartrackersscenarioappliestothiscase.However,itisobserved inFigure6-12A,thatdatafusionhasmoreofanaffectonthechiefthaninthethree startrackercase.Althoughthereisanimprovementduetodatafusion,thechief performanceisdegradedwithrespecttothethreestartrackercase.Expectedly, theerrorshowninFigure6-12BforthelocalestimateofDeputy-1ismuchgreater thanforthethreestartrackercase.Inthissamegure,resourcesharingsignicantly reducestheerrortocomparablelevelswiththechiefandDeputy-2.Similarbehavior isshowninFigure6-12CforDeputy-2.SinceDeputy-1hassuchpoorlocalsensing, inclusiveofapoorgyro,thechiefisalwaystaskedwithDeputy-1,whichresultsinthe lownominalcontroltorqueandangularvelocityseeninFigure6-13.Infact,theseare theexactprolesseeninthetwospacecraftscenarios,sinceDeputy-1iscommonto thesimulations.Furthermore,Figure6-14showsthatoutsideofearlytransients,the 175

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A B C Figure6-9.Localandfusedquaternionerrorforthethreespacecraftstartracker scenario.AChief.BDeputy-1.CDeputy-2 covarianceintersectionalgorithmneverusedDeputy-1information,insteadrelyingona combinationofinformationfromthechiefandDeputy-2togeneratethebestresults. 176

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A B Figure6-10.Chiefattitudecontrolforthethreespacecraftstartrackerscenario.A Controltorque.BAngularvelocity. Figure6-11.Covarianceintersectionweightsforthethreespacecraftstartracker scenario 177

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A B C Figure6-12.Localandfusedquaternionerrorforthethreespacecraftscenariowith coarseDeputy-1sensing.AChief.BDeputy-1.CDeputy-2 6.4TaskingAlgorithmComparisonforAttitudeEstimateResourceSharingwith DataFusion Attituderesourcesharingwithdatafusiondoesnotexhibitthesamebehavior asmeasurementsharing.Thissectioncomparestheperformanceofthethree 178

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A B Figure6-13.Chiefattitudecontrolforthethreespacecraftscenariowithcoarse Deputy-1sensing.AControltorque.BAngularvelocity. Figure6-14.Covarianceintersectionweightsforthethreespacecraftscenariowith coarseDeputy-1sensing spacecraftscenariosusingthegreedyandRound-robintaskingalgorithms.Since 179

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thetwospacecraftscenariosweretreatedastrackingproblems,theydonottwithinthis discussion. Asmentionedinthesimulationdescriptions,thethreespacecraftscenarios simulatedinSection6.3madeuseofthegreedytaskingalgorithmwiththetrace metric.Inthissection,resultsforthesamethreespacecraftscenariosareprovided, butwiththeRound-robintaskingstrategy.Forthethreestartrackercase,acomparison ofthe3bounderrorforthelocalandfusedstateestimatesisshowninFigure6-15. BycomparingFigure6-15AandFigure6-15B,itisseenthattheresourcesharing improvementsaremostlyinsensitivetothetaskingalgorithm.Therearesomespikes thatarepresentwiththegreedytaskingerrorofthechiefthatarenotpresentinthe resultsfortheRound-robinstrategy.Thecauseofthesespikesaremoreevidentinthe caseoftwostartrackersandacoarseattitudesensor. Thetaskingalgorithmimpactstheattitudeerrorthemostwhentheresulting motionofthechiefvariesfromrapidslewingtoslowslewing.Thisisexactlythecase encounteredforthethreespacecraftscenariowiththechiefandDeputy-2equipped withstartrackersandDeputy-1equippedwithcoarseattitudesensors.SinceDeputy-1 hascoarsesensorsandalowerprecisiongyro,Figure6-13Bshowedthatgreedy taskingresultsinanearstationarychief,whomisonlytaskedwithtrackingDeputy-1 forrelativeattitudesensing.Alternatively,theRound-robintaskingstrategycauses thechieftorapidlyslewandtrackthedeputies,alternatingbetweenthetwoaftera relativeattitudemeasurementiscaptured.Figure6-17showstheangularvelocityand errorforthechiefusingthegreedyandRound-robintaskingalgorithms.Forperiods oflowangularvelocity,theboresightaxiserrorgrowsuntilthechiefbeginstoslewata higherrate.Whenthechiefmotionisnearlystationary,theattitudeerrorintheboresight directionachievesahighersteady-stateerror.Therefore,greedytasking,whichonly accountsforthestateuncertaintyofthedeputies,doesnotproperlyaccountoneofthe primaryfactorsaffectingthesystemperformance.Theresultingcontroltorqueforthe 180

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A B Figure6-15.Affectoftaskingalgorithmchoiceontheerrorboundsofathreespacecraft scenariowithneprecisionsensing.AGreedytasking.BRound-robin tasking. Round-robinstrategyisgiveninFigure6-18.Thecontroltorquehasgonefromalow nominaltrackingsignaltoaggressiveslewmaneuvering.Clearly,inthiscase,thereisa trade-offbetweentheattitudeaccuracyandthecontroltorque. 6.5ObservationsonSpacecraftAttitudeEstimateResourceSharing Resourcesharinginanetworkofspacecraftwithsimilarattitudesensorcapability improvestheperformanceofallspacecraftwhentherelativeattitudesensorisof similarperformancetotheinertialsensors.Thisimprovementisadirectresultof datafusion,whichoptimallycombinesmultipleinformationsourcestoproduceabest estimate.Inthiscase,thespacecraftarelesssensitivetothetaskingalgorithm.In thesecases,greedyandRound-robintaskingwillcausethespacecrafttoswitch constantlyamongstthespacecraftandconsumeconsiderabletorqueaswasseeninthe 181

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A B Figure6-16.Affectoftaskingalgorithmchoiceontheerrorboundsofathreespacecraft scenariowithcoarseDeputy-2attitudesensing.AGreedytasking.B Round-robintasking. examples.However,thechiefcouldinsteadpursueanon-sharingobjectiveinadditionto pursuingrelativeattitudemeasurementsperiodically.Thetrade-offsforthechiefwould bebetweenattitudeaccuracy,controltorque,andtheamountoftimepursuingother non-sharingobjectives. Whenthedeputiesareprovidedwithdissimilarsensingcapability,thetasking algorithmplaysalargerroleinthepointingperformanceofthespacecraftinthenetwork andthedeputiesrelymoreonthesharedresource.Inthesecases,itwasseenthatthe chiefmotionwascoupledwithitsuncertainty,whichpropagatedtotheuncertaintyof theotherspacecraftthroughdatafusion.Inthesecases,greedytaskingreducesthe amountofslewingbymaintainingaxonthedeputywiththehighestuncertainty,but 182

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Figure6-17.EffectoftaskingontheChiefboresightattitudeaccuracy Figure6-18.Chiefattitudecontrolforthethreespacecraftscenariowithcoarse Deputy-1sensingusingRound-robintasking attheexpenseofsomeerrorimprovementsexperiencedbymoreaggressiveslewing. Thecouplingbetweenattitudeerrorandslewingshouldbeinvestigatedfurtherinfuture studies. 183

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6.6Summary Thischapterpresentedsimulationsontwoandthreespacecraftarchitectures, whereattitudestateestimatesweresharedbetweenthechiefanddeputiesusingdata fusion.Fromaperformanceperspective,sharedstateestimatesusingdatafusionare preferredtosharedstatemeasurements,wheredatafusionresultsinbetterutilizationof alltheinformationavailabletothechiefanddeputies.Whenthedeputiesareequipped withinertialattitudesensorscomparabletothechief,allspacecraftbenetfromattitude resourcesharing.However,thischapterconcludedwiththeobservationthatwhenone deputyhaslessaccurateattitudesensors,slewingmaneuversbythechiefcanaffectthe attitudeaccuracyperformanceofallspacecraft.Sincethechief'smotionisconfounded withthetaskingalgorithm,thetaskingalgorithmwillneedtoaccountforthedynamics explicitly.Onenalremarkonthesimulationsisthattheresultsfromthischapterveried themanyofthebenetsofattituderesourcesharing.Particularly,spacecraftequipped withlow-gradeattitudesensorscanseelargeattitudeimprovementsandspacecraft withhighaccuracysensorscanseeperformancegreaterthanthespecicationsofthat individualsensorallow. 184

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CHAPTER7 CONCLUSIONS Anewparadigmofspacecraftarchitecturesisbeinginvestigatedtoprovide improvementsinmissionexibility,responsiveness,andthreatmanagement.This newparadigmisreferredtoasthedisaggregatedspacecraftsystem,wherethe functionalitiesofatraditionalmonolithicspaceassetaredisaggregatedacrossseveral smallerplatformsconsistingofachiefandmultipledeputies.Thisresearchaddressed someofthetechnicalchallengesassociatedwiththedisaggregationoftheattitude determinationsystem. 7.1RevisitingtheResearchQuestionsandThesisStatements InSection1.3.1theproblemwasposedastohowtoperformdatafusionfor attitudestatesandwhatattitudeparameterizationshouldbeused.Itwashypothesized inSection1.4.1thattheattitudeerrorvectorcouldextendknowndatafusionlaws toattitudestatesandthatappropriateparameterizationofthecanproducean unconstrainedglobalnonsingularattitudedatafusionlaw.Thedevelopmentsin Chapter4veriedthesehypothesis.Inaddition,itwasshownthattheaccuracyof attitudedatafusionisbasedonthechosenattitudeparameterization.Theclassical RodriguesparametersCRPswereshowntobeafastandsufcientlyaccurate parameterization,whereasthehigher-orderRodriguesparameterswereshownto improvetheaccuracybutattheexpenseofadditionalcomputations.Inmostinstance, theCRPsarethepreferredparameterizationfordatafusion. InSection1.3.2theproblemoftaskingthechieftosharerelativeandinertial attitudemeasurementswasposedtominimizetheattitudeestimationerrorofthe deputies.ItwashypothesizedinSection1.4.2thatgreedysensortasking,asafunction ofdeputycovarianceinformation,couldbeusedtodistributeresourcesandimprovethe attitudeaccuracyofthedeputies.ThedevelopmentsinChapter5veriedthehypothesis forthecasewhenmeasurementsaresharedfromthechieftothedeputies.Specically, 185

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itwasshownthatgreedytaskingcouldbeusedasacomputationallyinexpensive methodtoimprovetheaccuracyofadisaggregatedADS,incomparisontoxed Round-robintasking.Intheresultinggreedytaskingalgorithm,metricsbasedonthe deputycovarianceinformationwerecomputedbythedeputiesandcommunicatedtothe chief.Thechiefusedthecovariancemetricinthetaskingdecisiontosharerelativeand inertialattitudemeasurements.MonteCarlosimulationsshowedthatthegreedytasking algorithmwasmostlyinsensitivetothecovariancemetricform.InChapter6itwas observedthatgreedytaskingbasedsolelyonthedeputycovarianceinformationwas inefcientforsharedattitudestateestimates.Inthiscase,itwasseenthatacoupling existsbetweenthechiefdynamicsandstateuncertainty.Specically,theattitudemotion causesachangeinchiefuncertaintythatpropagatestothedeputiesviadatafusion.As aresult,thebaselineRound-robinstrategyimprovedtheaccuracyofthedeputiesby exploitingthisdynamiceffect. 7.2FutureWork Thisresearchmadecontributionsintheareasofinertialattitudeestimation,attitude datafusion,andsensortaskingfordisaggregatedattitudedeterminationsystems.The resultsfromthisresearchhavealsospawnednewresearchdirectionsineachofthese areas. Intheareaofdisaggregatedinertialattitudeestimation,theauthorrecommends explorationofthefollowingtopics: Investigatedisaggregatedattitudeestimationthatdoesnotrequireeachspacecraft maintainestimatesoftheentirenetwork Investigateintermittentcommunicationforinformationtransferbetweenthechief anddeputies Determinethefactorsinuencingtheaccuracyofcurrentrelativeattitudesensors toyieldmeasurementsaspreciseasastartracker Investigateotherrelativeattitudesensorssuchasdirectionmeasurements obtainedthroughlasercommunicationssystems,whicharenotlimitedtotheclose proximityrequirementoftherelativeattitudesensorassumedinthismanuscript 186

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Intheareaofattitudedatafusion,theauthorrecommendsexplorationofthe followingtopics: Implementtheattitudeerrorstateinanellipsoidalintersectionalgorithmand comparetheresultswiththecovarianceintersectionalgorithm Furtherinvestigatetheaffectsofchiefattitudemotiononchiefanddeputy uncertainty,whichresultfromthetrackingcontroller Investigatetheaffectsofrelativetranslationalmotionbetweenthechiefand deputiesondatafusionandstateuncertainty Intheareaofsensortaskingindisaggregatedattitudedeterminationsystems,the authorrecommendsexplorationofthefollowingtopics: Investigatetheaffectsofchieftranslationalandrotationdynamicsonthegreedy taskingalgorithm Developthestatemodelsandcostfunctionstoderiveoptimalsensortasking algorithmsforattituderesourcesharing Developsensortaskingalgorithmsthataccountforfuelconsumptioninadditionto uncertainty 187

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BIOGRAPHICALSKETCH ShawnChristopherJohnsonwasborninDeLand,Florida,in1983.Growingup neartheSpaceCoast,hegazeduptotheskytoviewmanySpaceShuttlelaunches anddevelopedanexcitementforallthingsspace.Withthedesiretoonedaybecome anastronomer,inhighschoolhespentthesummerof2000attheUniversityofFlorida performingresearchinnear-infraredastronomyaspartoftheStudentScienceTraining ProgramundertheadvisementofDr.LaurenJones.Realizingthatheenjoyedamore hands-onapproachtospaceexploration,heoptedtostudyaerospaceengineering attheUniversityofFlorida,wherehereceivedhisB.S.inthefallof2005,alongwith minorsinbiomechanicsandbusinessadministration.Shawncontinuedhisstudiesat theGeorgiaInstituteofTechnology,whereheearnedanM.S.inaerospaceengineering inthespringof2008asagraduateresearcherintheAerospaceSystemsDesign LaboratoryundertheadvisementofDr.DimitriMavris.Thereheresearchedadvanced designmethodologiesandsystemsengineering.Realizingthathispassionstill remainedwithspacesystems,hereturnedtotheUniversityofFloridatoenterthe doctoralprogramasamemberoftheSpaceSystemsGroupundertheadvisement ofDr.NormanG.Fitz-Coy.Alongwithconductingtheresearchdocumentedinthis dissertation,hehashadthepleasureofworkingontheUniversityofFloridaCubeSat, SwampSat. Throughouthistenureinacademia,Shawnalsohadthepleasureofbeinga teachingassistantattheUniversityofFloridaforgraduateandundergraduatedynamics andcontrols.Outsideoftheclassroomandlab,heinternedwithRegeneration Technologies,Inc.now,RTIBiologics,Inc.inthesummerandfallof2005,withthe GeorgiaTechResearchInstituteinthesummerof2009,andwasaSpaceScholarat theAirForceResearchLaboratoryforthesummerof2011and2012.Upongraduation, ShawnwilljoinNASA'sJetPropulsionLaboratory. 199