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Techniques for Improved Performance in Cmg Based Attitude Control Systems

Permanent Link: http://ufdc.ufl.edu/UFE0045267/00001

Material Information

Title: Techniques for Improved Performance in Cmg Based Attitude Control Systems
Physical Description: 1 online resource (199 p.)
Language: english
Creator: Nagabhushan, Vivek
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: acs -- alignment -- attitude -- cmg -- estimation -- imbalance -- jitter -- misalignment -- precision -- spacecraft
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Control moment gyroscopes (CMG) are desired as actuators for spacecraft attitude control due to certain properties which include rapid retargeting, precision pointing, and low specific mass (mass per unit torque output). Ambiguity in some of the on-orbit parameters such as actuator alignment, and mechanism imperfections lead to deteriorated performance of the attitude control system. Novel techniques are developed to address some of these issues. Imprecise knowledge of the gimbal orientation of a CMG can lead to spacecraft pointing errors. Knowledge of spacecraft angular acceleration is shown to be beneficial in the design of techniques to estimate on-orbit parameters. A method to estimate the spacecraft angular acceleration using linear acceleration measurements is developed. The method uses six uniaxial accelerometers and a Kalman filter to obtain bias-free estimates of the angular acceleration, along with smoothed angular velocity estimates. Using the estimates of angular acceleration, a technique to estimate the on-orbit gimbal orientation is developed. Measurements of spacecraft angular velocity, and angular acceleration along with measurements of angular velocity and acceleration of the CMG flywheel are used in a linear least squares construct to estimate the unknown orientation. Three least squares solution variants are discussed, and their performances are compared. High-fidelity simulations utilizing data from real hardware are presented. Imbalance in the flywheel of a CMG leads to high frequency attitude disturbance called jitter. A three-flywheel system is developed to reduce the magnitude of jitter emitted by the CMG. The dynamics of jitter due to rotor imbalance is investigated, and a modification to the CMG flywheel system which involves the replacement of the single flywheel by a three-flywheel system is proposed. This method overcomes the need for precision balancing of the flywheels, provides limited redundancy against flywheel failure, and provides a long term jitter management solution. The dynamics of the three-flywheel system are developed and elaborate simulations are performed to verify the validity of the method. The effect of single/multiple flywheel failure in the three-flywheel system is investigated. The power and mass characteristics are analyzed and compared with those of the single-flywheel system.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Vivek Nagabhushan.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Fitz-Coy, Norman G.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045267:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045267/00001

Material Information

Title: Techniques for Improved Performance in Cmg Based Attitude Control Systems
Physical Description: 1 online resource (199 p.)
Language: english
Creator: Nagabhushan, Vivek
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: acs -- alignment -- attitude -- cmg -- estimation -- imbalance -- jitter -- misalignment -- precision -- spacecraft
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Control moment gyroscopes (CMG) are desired as actuators for spacecraft attitude control due to certain properties which include rapid retargeting, precision pointing, and low specific mass (mass per unit torque output). Ambiguity in some of the on-orbit parameters such as actuator alignment, and mechanism imperfections lead to deteriorated performance of the attitude control system. Novel techniques are developed to address some of these issues. Imprecise knowledge of the gimbal orientation of a CMG can lead to spacecraft pointing errors. Knowledge of spacecraft angular acceleration is shown to be beneficial in the design of techniques to estimate on-orbit parameters. A method to estimate the spacecraft angular acceleration using linear acceleration measurements is developed. The method uses six uniaxial accelerometers and a Kalman filter to obtain bias-free estimates of the angular acceleration, along with smoothed angular velocity estimates. Using the estimates of angular acceleration, a technique to estimate the on-orbit gimbal orientation is developed. Measurements of spacecraft angular velocity, and angular acceleration along with measurements of angular velocity and acceleration of the CMG flywheel are used in a linear least squares construct to estimate the unknown orientation. Three least squares solution variants are discussed, and their performances are compared. High-fidelity simulations utilizing data from real hardware are presented. Imbalance in the flywheel of a CMG leads to high frequency attitude disturbance called jitter. A three-flywheel system is developed to reduce the magnitude of jitter emitted by the CMG. The dynamics of jitter due to rotor imbalance is investigated, and a modification to the CMG flywheel system which involves the replacement of the single flywheel by a three-flywheel system is proposed. This method overcomes the need for precision balancing of the flywheels, provides limited redundancy against flywheel failure, and provides a long term jitter management solution. The dynamics of the three-flywheel system are developed and elaborate simulations are performed to verify the validity of the method. The effect of single/multiple flywheel failure in the three-flywheel system is investigated. The power and mass characteristics are analyzed and compared with those of the single-flywheel system.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Vivek Nagabhushan.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045267:00001


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TECHNIQUESFORIMPROVEDPERFORMANCEINCMGBASEDATTITUDE CONTROLSYSTEMS By VIVEKNAGABHUSHAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c 2013VivekNagabhushan 2

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ToRaaji,Amma,Anna,andCutie 3

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ACKNOWLEDGMENTS IamindebtedtomyadvisorDr.NormanFitz-Coywhohasnotonlytrainedmeas astudentbuthastaughtmebyexample,severalvaluablelessonsonbeingdisciplined anddedicated.Hehasalwaysinstilledcondenceinmeandlistened,critiqued,and encouragedeveryresearchidea.Henevertoldmewhatwaswrong,butmadesure thatIrealizedthesamebyencouragingmetodeliberateonthematter.Suchexercises havebeenthemostvaluablepartofmyeducationattheuniversity.Ithankhimagain forbeinginuentialinbecomingthepersonIamtoday.Ithankmycommitteemembers Dr.JohnConklin,Dr.WarrenDixon,andDr.GuidoMuellerfortheiradviceandfor alltheinterestingdiscussionsthathelpedmeimprovethematerialpresentedinthis dissertation.Ialsothankthemforbeingresponsiveinprovidingfeedbackonmy dissertation. IamevergratefultomywifeMeghana,myparentsArundathiandNagabhushan, andmysisterNamithawithoutwhoselove,support,andpatience,Iwouldnothavebeen abletocomethisfar.Ithankmyparentsfortheirrelentlesstrustandcondenceinme. Ithankmywifeforinuencingmeineverywalkoflife,forherunconditionallove,for helpingmedeveloppatienceandcomposure,andforalwaysbeingbymyside. Igivemyspecialthankstoallmycolleagues,pastandpresent,Dr.ShawnAllgeier, Dr.SharanAsundi,DanteBuckley,KatieCason,Dr.TakashiHiramatsu,Shawn Johnson,Dr.FrederickLeve,Dr.JosueMu noz,KunalPatankar,BungoShiotaniatthe spacesystemsgroupfortheirvaluablediscussionsandsupport.Ialsothankthemfor helpingmedevelopteamskillswhileworkingonSwampSat.Iwouldalsoliketothank theadvancedtechnologyresearchandengineeringcenterASTRECanditsmembers fortheirsupportinconductingsomeoftheresearchpresentedinthisdissertation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 ABSTRACT.........................................12 CHAPTER 1INTRODUCTION...................................14 1.1ControlMomentGyroscopeCMGGimbalMisalignment.........17 1.2RotorImbalanceandAttitudeJitter......................18 1.3ContributionsandOutline...........................20 1.3.1EstimationofSpacecraftAngularAcceleration............20 1.3.2On-OrbitEstimationofCMGGimbalOrientation...........21 1.3.3On-OrbitJitterControlUsingaThree-FlywheelSystem.......21 2SPACECRAFTATTITUDEDYNAMICSANDCONTROL.............23 2.1AttitudeRepresentation............................23 2.1.1FrameofReferenceandCoordinateBases.............24 2.1.2DirectionCosineMatrixDCM....................25 2.1.3Axis-AngleRepresentation.......................26 2.1.4EulerSymmetricParametersorUnitQuaternion..........28 2.1.5EulerAngles..............................29 2.2AttitudeKinematics...............................29 2.2.1DifferentialEquationforaDCM....................30 2.2.2DifferentialEquationforAxis-AngleRepresentation.........31 2.2.3DifferentialEquationforUnitQuaternions..............31 2.3AttitudeDynamics...............................32 2.3.1MassanditsMoments.........................32 2.3.2AngularMomentum...........................34 2.3.3EquationsofMotionforaRigidBody.................36 2.3.4EquationsofMotionforaRigidBodywithaMomentumDevice..38 2.3.5EnvironmentalTorques.........................39 2.3.5.1Gravitationaltorque.....................39 2.3.5.2Aerodynamictorque.....................40 2.3.5.3Radiationtorque.......................41 2.3.5.4Magnetictorque.......................41 2.4AttitudeControl.................................42 2.4.1ActiveAttitudeControl.........................43 2.4.2AttitudeControlErrors.........................44 5

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2.4.3AttitudeActuators............................45 2.4.3.1Externaltorqueactuators..................45 2.4.3.2Momentumactuators....................46 3CONTROLMOMENTGYROSCOPES.......................50 3.1ControlMomentGyroscopeVariants.....................51 3.1.1Double-GimbalControlMomentGyroscope.............51 3.1.2VariableSpeedControlMomentGyroscope.............52 3.1.3Scissored-PairControlMomentGyroscope.............53 3.1.4AdaptiveSkewControlMomentGyroscope.............54 3.2Single-GimbalControlMomentGyroscope.................54 3.2.1CongurationsofMultipleCMGs...................56 3.2.1.1SingularitiesinCMGs....................57 3.2.2MechanicalConstructionofaTypicalCMG..............58 3.2.2.1MechanismimperfectionsinCMGs............59 3.2.3DynamicsofaSpacecraftwithaCMG................60 4ESTIMATIONOFSPACECRAFTANGULARACCELERATION.........69 4.1Background...................................69 4.2Methodology..................................71 4.2.1AngularAccelerationUsingUniaxialAccelerometers........72 4.2.2AngularAccelerationUsingTriaxialAccelerometers.........74 4.3EstimationofAngularAcceleration......................76 4.3.1MeasurementModelforAngularAcceleration............77 4.3.2EstimationofBias-FreeAngularAcceleration............82 4.4SimulationsandResults............................85 5ON-ORBITESTIMATIONOFCMGGIMBALORIENTATION...........96 5.1Background...................................96 5.2Methodology..................................100 5.2.1SpacecraftEquationsofMotion....................100 5.2.2LeastSquaresSolution........................104 5.3Measurements,Sensors,andImplementation................110 5.3.1MeasurementofFlywheelStates...................111 5.3.2MeasurementofSpacecraftStates..................112 5.3.2.1Spacecraftangularvelocity.................113 5.3.2.2Spacecraftangularacceleration..............114 5.3.3CorrelatingFlywheelandSpacecraftMeasurements........114 5.4SimulationsandResults............................115 5.4.1Case1:LowSignaltoNoiseRatioandSmallMisalignment....116 5.4.2Case2:HighSignaltoNoiseRatioandSmallMisalignment....117 5.4.3Case3:HighSignaltoNoiseRatioandLargeMisalignment....118 5.4.4Case4:Truncated A Matrix......................118 6

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6ON-ORBITJITTERCONTROLUSINGATHREE-FLYWHEELSYSTEM....130 6.1ExistingJitterMitigationMethods.......................130 6.2FlywheelImbalanceandSpacecraftJitter..................132 6.3JitterMitigationUsingaThree-FlywheelSystem..............137 6.4SimulationsandResults............................143 6.4.1Single-FlywheelSystem........................143 6.4.2IdealSystem..............................144 6.4.3High-FidelitySimulations........................145 6.4.4EffectofFlywheelFailure.......................148 6.4.4.1Failureofywheel F 1 ....................148 6.4.4.2Failureofywheel F 2 or F 3 .................149 6.4.4.3Failureofywheel F 1 ,andeither F 2 or F 3 .........149 6.4.4.4Failureofywheel F 2 and F 3 ................150 6.4.5PowerandMassTrades........................150 6.4.6IdentifyingtheFlywheelPhaseAngle.................152 7CONCLUSIONSANDFUTUREWORK......................173 APPENDIX ANOTATIONS.....................................176 BFLYWHEELIMBALANCE..............................179 B.1StaticandDynamicImbalanceinFlywheels.................179 B.2BalancingQuality................................184 REFERENCES.......................................188 BIOGRAPHICALSKETCH................................199 7

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LISTOFTABLES Table page 4-1SimulationParametersforAngularAccelerationEstimation...........88 5-1SimulationParametersforGimbalAngleEstimation...............119 5-2ResultsSummaryCase1.............................120 5-3ResultsSummaryCase2.............................120 5-4ResultsSummaryCase3.............................120 5-5ResultsSummaryCase4.............................120 6-1ImbalanceDataofCommercialMomentumActuators..............154 6-2Single-FlywheelSystemSimulationParameters..................154 6-3IdealThree-FlywheelSystemSimulationParameters...............155 6-4Three-FlywheelSystemCommonSimulationParametersHi-Fidelity.....156 6-5G0.4Three-FlywheelSystemSimulationParametersHi-Fidelity........157 6-6G1Three-FlywheelSystemSimulationParametersHi-Fidelity.........158 6-7G2.5Three-FlywheelSystemSimulationParametersHi-Fidelity........159 6-8G6.3Three-FlywheelSystemSimulationParametersHi-Fidelity........160 6-9MassandPowerParametersofSingle-FlywheelSystem.............160 6-10MassComparisonBetweenSingle-andThree-FlywheelSystems.......161 6-11PowerComparisonBetweenSingle-andThree-FlywheelSystems.......161 8

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LISTOFFIGURES Figure page 2-1AxisAngleRepresentation.............................48 2-2RigidBody......................................48 2-3ActiveAttitudeControlBlockDiagram.......................49 2-4AttitudeDeterminationandControlSystemErrors................49 3-1Single-GimbalControlMomentGyroscope....................65 3-2Double-GimbalControlMomentGyroscope....................65 3-3ScissorPairControlMomentGyroscopeSchematic...............66 3-4AdaptiveSkewControlMomentGyroscope....................66 3-5ControlMomentGyroscopeCMGCongurations................66 3-6IntegratedMiniaturePyramidalArrangementofCMGs IMPAC .........67 3-7ExplodedViewofthe IMPAC ............................67 3-8RigidSpacecraftwithCMG.............................68 4-1SpacecraftwithUniaxialAccelerometersandGyroscope............89 4-2SpacecraftwithTriaxialAccelerometersandGyroscope.............89 4-3SpacecraftInertialSensors.............................90 4-4TrueSpacecraftAngularVelocityandAngularAcceleration...........91 4-5AngularVelocityMeasurements...........................92 4-6FilteredAngularVelocityEstimates.........................92 4-7EstimatedAngularAccelerationBias........................93 4-8EstimatedAngularAcceleration...........................93 4-9ErrorinMeasuredAngularVelocity.........................94 4-10ErrorinFilteredAngularVelocityEstimates....................94 4-11ErrorinAngularAccelerationEstimates......................95 4-12DifferentiationofAngularVelocity..........................95 5-1SchematicRepresentationofaSingleCMGinanArbitrarySpacecraft.....121 9

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5-2Cross-CorrelationPlotBetween and .....................121 5-3SimulinkModelforEmulatingSensorMeasurements...............122 5-4FlywheelMeasurementsCase1.........................123 5-5SpacecraftMeasurementsCase1........................124 5-6Monte-CarloAnalysisResultsCase1......................125 5-7FlywheelMeasurementsCase2.........................126 5-8SpacecraftMeasurementsCase2........................127 5-9Monte-CarloAnalysisResultsCase2......................128 5-10SpacecraftMeasurementsCase3........................129 6-1SchematicRepresentationofSpacecraftwithUnbalancedFlywheel......162 6-2SchematicoftheThree-FlywheelSystem.....................162 6-3AngularVelocityofSpacecraftwithSingleUnbalancedFlywheel........163 6-4FastFourierTransformofSpacecraftAngularVelocity..............164 6-5SimulationofanIdealThree-FlywheelSystem..................165 6-6FlywheelAngularVelocitiesinaHigh-FidelityThree-FlywheelSystem.....166 6-7SpacecraftAngularVelocityinaHigh-FidelityG0.4Three-FlywheelSystem..166 6-8SpacecraftAngularVelocityinaHigh-FidelityG1Three-FlywheelSystem...167 6-9SpacecraftAngularVelocityinaHigh-FidelityG2.5Three-FlywheelSystem..167 6-10SpacecraftAngularVelocityinaHigh-FidelityG6.3Three-FlywheelSystem..168 6-11ComparisonofResidualJitteratVariousBalancingQualities..........168 6-12SimulationofaThree-FlywheelSystemwithaFailedFlywheel F 1 ........169 6-13SimulationofaThree-FlywheelSystemwithaFailedFlywheel F 2 or F 3 ....169 6-14SimulationofaThree-FlywheelSystemwithFailedFlywheels F 1 and F 2 ....170 6-15SimulationofaThree-FlywheelSystemwithFailedFlywheels F 2 and F 3 ....170 6-16ComparisonofFlywheelFailureModesinaThree-FlywheelSystem......171 6-17IndicativeCongurationsofSingle-andThree-FlywheelSystems........171 6-18PowerComparisonBetweenSingle-andThree-FlywheelSystems.......172 10

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B-1HomogeneousFlywheel...............................185 B-2FlywheelwithOnlyStaticImbalance........................185 B-3FlywheelwithOnlyDynamicImbalance......................186 B-4BalancedFlywheelSystem.............................186 B-5BalancingQualityasperISO1940/1........................187 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy TECHNIQUESFORIMPROVEDPERFORMANCEINCMGBASEDATTITUDE CONTROLSYSTEMS By VivekNagabhushan May2013 Chair:NormanG.Fitz-Coy Major:AerospaceEngineering ControlmomentgyroscopesCMGaredesiredasactuatorsforspacecraft attitudecontrolduetocertainpropertieswhichincluderapidretargeting,precision pointing,andlowspecicmassmassperunittorqueoutput.Ambiguityinsomeofthe on-orbitparameterssuchasactuatoralignmentandmechanismimperfectionssuchas imbalanceinywheelsleadtodeterioratedperformanceoftheattitudecontrolsystem. Noveltechniquesaredevelopedtoaddresstheseissues. ImpreciseknowledgeofthegimbalorientationofaCMGcanleadtospacecraft pointingerrors.Knowledgeofspacecraftangularaccelerationisshowntobebenecial inthedesignoftechniquestoestimateon-orbitparameters.Amethodtoestimatethe spacecraftangularaccelerationusinglinearaccelerationmeasurementsisdeveloped. ThemethodusessixuniaxialaccelerometersandaKalmanltertoobtainbias-free estimatesoftheangularaccelerationalongwithsmoothedangularvelocityestimates. Usingtheestimatesofangularacceleration,atechniquetoestimatetheon-orbitgimbal orientationisdeveloped.Measurementsofspacecraftangularvelocityandangular accelerationalongwithmeasurementsofangularvelocityandaccelerationofthe CMGywheelareusedinalinearleastsquaresconstructtoestimatetheunknown orientation.Threeleastsquaressolutionvariantsarediscussed,andtheirperformances arecompared.High-delitysimulationsutilizingdatafromcommerciallyavailable hardwarearepresented. 12

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ImbalanceintheywheelofaCMGleadstohighfrequencyattitudedisturbance calledjitter.Athree-ywheelsystemisdevelopedtoreducethemagnitudeofjitter emittedbytheCMG.Thedynamicsofjitterduetorotorimbalanceisinvestigated andamodicationtotheCMGywheelsystemwhichinvolvesthereplacement ofthesingleywheelbyathree-ywheelsystemisproposed.Themethodresults inasystemthatproduceslowerjittercomparedtothesingle-ywheelsystem, provideslimitedredundancyagainstywheelfailure,andconstitutesalongterm jittermanagementsolution.Thedynamicsofthethree-ywheelsystemaredeveloped andelaboratesimulationsareperformedtoverifythevalidityofthemethod.Theeffect ofsingle/multipleywheelfailureinthethree-ywheelsystemisinvestigated.Thepower andmasscharacteristicsareanalyzedandcomparedwiththoseofthesingle-ywheel system. 13

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CHAPTER1 INTRODUCTION Attitudeisdenedastheorientationofanobjectusuallyaspacecraftrelative toanobservationreference.Thestudyofspacecraftattitudeinvolvesunderstanding theevolutionofrotationalmotionofthespacecraftinorbitattitudedynamicsand developingcontrolhardwareandalgorithmstoachievethedesiredstateattitude determinationandcontrol.Attitudecontrolincludesattainmentofthedesiredattitudeas wellasthedesiredangularvelocity.Attitudecontrolisessentialforalmostallspacecraft forasuccessfulmissionandsomeoftheapplicationsareasfollows.Attitudecontrol mayberequiredfordirectingasensorsuchasacameraonaspacecrafttoanobject ofinterestonEarth,termedgeospatialimaginge.g., WorldView spacecraft,orienting thespacecraft'ssolararraytofacetheSun,observationofcelestialbodiese.g., Hubblespacetelescope ,directingacommunicationantennatoagroundstationon theEarthe.g., Iridium satellites,orientingthrusterstoperformstationkeepingGPS satellitesandotherorbitalmaneuverse.g.,STSorbiter,etc.Wertz[1]classies attitudecontrolintotwotypesviz. attitudestabilization and attitudemaneuvercontrol Attitudestabilizationinvolvesmaintainingaparticularattitudeforalongperiodoftime whereasattitudemaneuveringinvolvesreorientationfromoneattitudetoanotherusing attitudecontrolhardware.Basedonthemethodused,attitudecontrolcanbefurther classiedinto passive and active .Inpassiveattitudecontrol,externalenvironmental forcese.g.,gravitygradient,solarpressureareusedtoattainadesiredattitude, whereasactiveattitudecontrolusesactuatorse.g.,reactionwheelsandcontrol momentgyroscopes,feedbacksensorse.g.,starsensor,magnetometer,gyroscope, etc.,andsomecontrolstrategytoperformreorientingmaneuvers.Typically,attitude maneuvercontrolimpliescompletethree-axisrotationalcontrolofthespacecraft. Theattitudeactuatorsalongwiththecontrolhardwareandsoftwareconstitutethe attitudecontrolsystemACS.TheACSisakeycontributortotheoverallmission 14

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successandmoreoften,acriticalone.Attitudecontrolhasevolvedsincethelaunch oftherstspacecraft,the Sputnik ,in1957,whichhadnoattitudecontrol. Explorer-1 thatwaslaunchedshortlyafterhadsingleaxisattitudestabilizationbutlackedany activecontrol.Today,almostallmajorspacecrafthaveactivethree-axisattitudecontrol thatallowthemtopointtoarequireddirectioninspace.Dependingonthemission requirements,theACSofsomespacecraftaremoreprecisethantheothers.Thereare spacecrafttodaythatrequirepointingaccuraciesuptofractionsof arcseconds suchas HubbleSpaceTelescope [2]andsomethatrequirelittleafewdegreestonopointing e.g., O/OREOS [3]and GeneSat [4].Thedeviationbetweenthedesiredattitude andactualattitudeisknownasattitudeerror.Severalparameterizationsofattitude andattitudeerrorexist[5].Inadditiontostringentpointingrequirements,spacecraft suchasthe Hubblespacetelescope and GravityProbeB [6]requireanultra-quietlow vibrationspacecraftplatform.Preciseattitudecontrolischallengingduetointernaland externaldisturbancesonthespacecraftandlimitationsimposedbyattitudesensorsand actuators.Internaldisturbancesincludejitterfromangularmomentumcontroldevices e.g.,reactionwheels,controlmomentgyroscopes,etc.andgimbaleddevicese.g., gimbaledantennaandsolararraydrives,propellentslosh,andmovementofother rotaryinternalcomponentssuchascryogeniccoolerpumps.Externaldisturbances, primarilyenvironmentaleffectssuchasatmosphericdrag,gravitygradientandmagnetic torquesimparttorqueonthespacecrafttherebyaffectingthespacecraftattitude. Inefciencyofcontrolalgorithmsalsocontributesignicantlytoattitudeerror.Inthis dissertation,decienciesinattitudeactuators,specicallythoseassociatedwiththe controlmomentgyroscopeCMGthatmayleadtoattitudeerrorsareconsidered. Controlmomentgyroscopeisatypeofmomentumexchangedevicethatcontrols theattitudeofthespacecraftthroughmomentumredistribution.Theyaredesired fortheirattractivefeatureswhichincluderapidretargetingandprecisionpointing capabilities,lowspecicpowerpowerperunittorque,andlowspecicmassmass 15

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perusingtorque.TheyCMGshavebeenresearchedforutilityinspacecraftattitude controlsincethe1970sSkylabin1973[79],USmilitaryobservationsatellites Key-Hole:KH-11andKH-12in1976,andtheMIRspacestationin1986[10].They werealsosuccessfullyimplementedontheInternationalSpaceStation[11,12]. TherstreportedcommercialuseoftheCMGwasontherecentEarth observationspacecraft Worldview-1 [13].AgenericCMGconsistsofaspinning rotorthatisgimbaledaboutoneormoreaxes.Momentumexchangebetweenthe spacecraftandtheCMGisachievedbyreorientingthespinningrotoraboutthe gimbaledaxis.Asopposedtoothermomentumexchangeactuatorssuchasreaction wheelsandmomentumwheels,CMGsarespecialinachievingthenewmomentum statesfaster,andwithbetterprecision[10,1416].SeveralvariantsoftheCMGexist basedonthethenumberofactuateddegreesoffreedome.g.,single-gimbalCMG, double-gimbalCMG,variable-speedCMG.Thesevariantsarediscussedbrieylater inthedissertation.Themostpopularvariant,thesingle-gimbalCMGSGCMGis mainlyconsideredinthisdissertation.Aswithanyothercomplexandprecisecontrol mechanism,attitudecontrolusingCMGsisfacedwithchallengesposedbymechanism imperfectionsandhardwarelimitations.Aparticularchallengeisthatoftheambiguity intheon-orbitorientationorangularpositionofthegimbals.Theuncertaintymayresult frommisalignmentofthegimbalsorfailureofthegimbalangularpositionsensorduring launch.Theangularpositionofthegimbalrequiredforcontrolisthereforeunknown andmayleadtopointingerrors.Anotherchallengethatmayleadtopoorpointing performanceistheimbalanceinCMGrotors.Imbalanceiscausedbynon-homogeneity oftherotormaterials,manufacturingerrors,andlimitationsofbalancingmachines. Thisimperfectioncanleadtoattitudejitterinspacecraftandaffectstheperformanceof instrumentsandsensorsthatrequirestablepointing[1722].Thequestfordeveloping smallerspacecrafte.g.,micro-,nano-,andpico-satellites[2325]hasbeenfueling thedevelopmentofsmallerandhighperformanceattitudeactuators[10,2630].The 16

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miniaturizationofattitudeactuatorspresentadditionalchallenges[27,31,32]whichmay increasetheoccurrenceandeffectsofgimbalmisalignmentandrotorimbalance. 1.1ControlMomentGyroscopeGimbalMisalignment Accurateknowledgeofalignmentororientationofmomentumactuatorsspecically, theirindividualmomentumvectorsrelativetothespacecraftiscriticaltoachieve precisionpointingofthespacecraft.Thedifferencebetweentheactualon-orbitand pre-launchcalibratedorientationsoftheactuatoristermedactuatormisalignment. Thesemisalignmentsmaybesmallinmagnitudesub-degreeandcanoccurdueto severalreasonsincludingassemblyerrors,measurementlimitations,launchshockand vibration,andon-orbiteffectse.g.,differentialthermalexpansion.Smallmisalignments occurinxedactuatorssuchasreactionwheelsandmomentumwheels,andan approximatealignmentoftheactuatorisknownapriori.IngimbaledactuatorsCMGs, rotarysensorfailureand/orgimbalrestraintfailureduringlaunchcanleadtototal ambiguitylargegimbalmisalignmentintheorientationofthegimbal.Anaccurate knowledgeofthegimbalorientationisrequiredtopreciselycontroltheCMGoutput torqueinordertoperformprecisionpointing.Usually,someformofabsoluteangular feedbackdevicee.g.,encoder,resolver,etc.isincorporatedwithinthetheCMG toprovidemeasurementsofthegimbalangle[33].Forreasonsindicatedearlier, theremaybemisalignmentbetweenthefeedbackdeviceandtheCMGgimbalthat introducesanerrorinthemeasurement.Inextremecases,CMGsmayevenhave failedencoderse.g.,GOESandMarsOdyssey.Encoderfailurecanbecausedby radiationdamagetotheelectronics,breakageofencoderopticalwheelorchangein itsopticalproperties.Inothercasesanincrementalencoder/resolvermayhavebeen usedinplaceofanabsoluteone[24,27]astheyarecheaperandhigherresolution encoderscanbepackagedintoarelativelysmallerspace[33].Anincrementalencoder providesinformationaboutthechangeinthegimbalorientationasopposedtothe absoluteorientation.Insuchcases,theabsoluteorientationofthegimbalisrequired 17

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tobedeterminedoncesothatitcanthenbepropagatedutilizinginformationfromthe incrementalencoder.Amethodtoestimatetheon-orbitorientationoftheCMGgimbal mayprovetobeaneffectivesolutionfortheabovementionedproblems.Further,the abilitytoestimateactuatororientationandlargemisalignmentsonorbitmayvoidthe needforgroundcalibrationoftheseactuators.Thismayleadtopotentialsavingsincost andtime,asdesiredbyresponsivespacemissions. Estimationofactuatoralignmenthasreceivedlesserattentionasopposedto estimationofattitudesensoralignment[3438]andattitudeestimationingeneral [3946].Majorityoftheactuatoralignmentestimationmethodspublishedthusfar usetheexpressionfortheinertialangularmomentumofthespacecraftandassume ittobeaconstantoveracertainperiodoftime[4750].Thisapproachrequires accurateknowledgeofthespacecraftattitudeandunderthepresenceofexternal disturbancetorques,theassumptionofconstantangularmomentumisviolated. Anotherpublication[51]usesthesimpliedattitudeequationsofmotioninsteadofthe expressionforangularmomentum.However,thismethodneglectsthenon-linearterms intheequationsofmotionassumingthemtobesmall,anddeterminestheangular accelerationofthespacecraftbydifferentiatingangularvelocity.Thisdissertation developsamethodtoestimatethegimbalorientationofCMGsusingthecomplete attitudeequationsofmotioninclusiveofthenon-linearterm.Further,amethodto estimatetheangularaccelerationofthespacecraftusinglinearaccelerometersis developedwhichisshowntobemoreaccurateincomparisontodifferentiatingangular velocitymeasurements. 1.2RotorImbalanceandAttitudeJitter Avibration-freespacecraftenvironmentisdesiredbycertaintypesofpayloadswith precisionpointingrequirementssuchasastronomicaltelescopes,Earthobservation andremotesensingsystems,opticalcommunicationsystemsandotherdirectional communicationantennae.On-boardvibrationsaffectthepointingstabilityofthe 18

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spacecraftresultinginattitudeoscillationscalledjitter.Vibrationsalsocausetranslational effectswhicharelesssevereinaffectingtheperformanceofsuchpayloads.Only rotational/attitudeeffectsarethereforeconsideredhere.Obtainingaquietlow-vibration environmentisoftenachallengeforspacecraftdesignersduetothepresenceof variousdisturbancesourcesonthespacecraft.Amajorandcommonsourceofvibration istheimbalanceofrotorsinmomentumactuators.Momentumactuatorsconsistof rotorsspinningatvariablee.g.,reactionwheelsoratconstante.g.,CMGsangular speeds.ThepresenceofstaticanddynamicimbalancesseeAppendixBinspinning rotorscausesanoscillatingtorquewhichresultsinjitter.Traditionally,therotorsare preciselybalancedtoreducethemagnitudeofimbalance.Theprecisionbalancing processistimeconsumingandexpensive,andtheresidualimbalancecanstillresultin signicantjitter.Further,theimbalanceintherotorsincreaseduringtheirlifetimedue toseveraleffectse.g.,thermaldistortionoftherotors,hoopstrain,etc.,resultingina gradualdeteriorationfromtheirbeginning-of-lifeBOLvalue.Thiseffectisoftenseenin compositeywheelswhichsufferfromimbalancegrowthduetointer-laminaraws[52]. Severalmethodsinvolvingpassiveand/oractiveisolatorshavebeendevisedand implementedtoattenuatevibrationsandreduceitsimpactontheperformanceofthe payloads[5360].Thesemethodshoweverdonotreducethevibrationsemittedbythe source,buteitherisolatethesourceofvibrationfromtherestofthespacecraft,orisolate thepayloadalonewhichissensitivetovibrations.Insomecases,boththepayloadand thesourcesofvibrationareisolated.Therehavehoweverbeennoeffortstoreduce themagnitudeoftheemittedjitteritself,apartfromprecisionbalancingtheywheels. Thejittermitigation/controlmethoddescribedinthisdissertationreducesthevibrations emittedfromtheCMGbyreplacingthesingleywheelrotorbyasetofthreeywheels. Byimplementingthismethod,theywheelsystemachievesstaticanddynamicbalance levelsthatareotherwisedifculttoachieveusingtraditionalbalancingmethods.The methodalsoprovidesacertainlevelofadaptabilitytochangesinywheelimbalance, 19

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andhence,overthelifetimeoftheactuator,deliversasmallerdeviationfromtheBOL jitter. 1.3ContributionsandOutline Thecontributionofthisdissertationfocusesonmethodstoimprovetheperformance ofattitudeactuators,specicallyCMGs.First,amethodtoestimatetheangular accelerationofthespacecraftusinglinearaccelerometersisdeveloped.Theangular accelerationsodeterminedisusedtodevelopamethodtoestimatetheunknowngimbal orientationofanSGCMG.Finally,atechniquetoreducethejitteroutputofaCMGusing athree-ywheelsystemisdeveloped. 1.3.1EstimationofSpacecraftAngularAcceleration Knowledgeofspacecraftangularaccelerationmaybenetestimationofspacecraft dynamicparameterssuchasmasspropertiesandactuatoralignments.Asopposedto differentiationofangularvelocitymeasurementswhichampliessensornoise,amethod todeterminebias-freeangularaccelerationusingmeasurementsoflinearacceleration atpre-determinedpointsinthespacecraftisdeveloped.Multiplelinearaccelerometers areselectivelysituatedinthespacecraftforobtainingthesemeasurements.Aspecic congurationofsixuniaxialaccelerometersthatmaybefeasibleforimplementation onaspacecraftisdeveloped.Acongurationwiththreetriaxialaccelerometersis alsopresentedforcomparisonpurposes.Further,theeffectofsensorbiasandnoise ontheangularaccelerationestimatesisdiscussedandcompared.Anapproximate linearmodelfortheangularaccelerationmeasurementsisdeveloped.Thelinear measurementmodelisthenusedtoidentifytheeffectivebiasintheangularacceleration estimatesinaKalmanlterconstruct.Inadditiontoidentifyingthebias,thelteralso producessmoothedangularvelocityestimates.Simulationsareperformedusingdata fromcommerciallyavailablesensors. 20

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1.3.2On-OrbitEstimationofCMGGimbalOrientation Oneofthebenetsofangularaccelerationmeasurementsviz.itsuseinthe estimationofspacecraftdynamicparametersisappliedtodevelopamethodtoestimate theunknowngimbalorientationofSGCMGsusingon-orbitmeasurements.Euler's equationsofmotionareusedasthebasisfortheestimationtechnique.Measurements ofspacecraftangularvelocity,andangularaccelerationalongwithmeasurements ofangularvelocityandaccelerationoftheSGCMGywheelareutilized.Itisshown that,byextractingtheelementsofthedirectioncosinematrixDCMthatrepresent therelativeorientationofthegimbalinthespacecraft,thenon-linearequations ofmotioncanbeexpressedinalinearform.Linearleastsquarestechniquesare thendevelopedtoestimatetheunknownorientation.Threedifferentleastsquares solutionmethodologiesarediscussed,andtheirperformancesarecompared.Bias-free spacecraftangularaccelerationandlteredangularvelocityaredeterminedusing linearaccelerometermeasurementsasindicatedintheprevioussection.Themethodis applicableindependentofthemagnitudeofgimbalalignment.High-delitysimulations utilizingdatafromcommerciallyavailablehardwarearepresented.Althoughtheconcept associatedwiththemethodhasbeenpresentedspecictothealignmentofCMG gimbals,itcanbeextendedforalignmentestimationofothermomentumactuators. 1.3.3On-OrbitJitterControlUsingaThree-FlywheelSystem SpacecraftjitterduetoywheelimbalanceinaCMGisconsidered.Amodication totheywheelsystemthatprovidesalongtermjittermanagementsolutionisproposed. ThemodicationinvolvesreplacingtheywheelinaCMGbyathree-ywheelsystem. Themethodperformson-orbitbalancingoftheywheelsystembyadjustingtherelative imbalancebetweentheywheels.Althoughonlyattitudeeffectsareconsidered, thethree-ywheelapproachalsoreducesthemagnitudeoftranslationaljitter.This methodovercomestheneedforhighbalancequalityofywheelsandprovidessome redundancyagainstywheelfailure.Thedynamicsofthethree-ywheelsystemare 21

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developedandtheproceduretoperformon-orbitbalancingisdescribed.Elaborate simulationsareperformedtoverifythevalidityofthemethod.Theperformancesofthe proposed,andthesingle-ywheelsystemarecompared.Theeffectofsingle/multiple ywheelfailureinthethree-ywheelsystemisinvestigated.Thepowerandmass characteristicsareanalyzedandcomparedwiththoseofthesingle-ywheelsystem. Commentsonthedesignofthethree-ywheelsystemaremadetoindicateits practicality.Thetechniquecanalsobeappliedtoothermomentumactuatorsidentically. Thejittermitigationmethoddevelopedreducesthemagnitudeofjitteremittedby theCMGasopposedtothecurrentmethodsthatisolatethejitterfromtheaffected spacecraftareas.However,themethodcanbeusedinconjunctionwithisolation mechanismstofurtherimprovethejitterenvironment. Theremainderofthedissertationisorganizedasfollows.Anintroductiontovarious aspectsofspacecraftattitudedynamicsandcontrolincludingattituderepresentation, attitudekinematics,developmentofequationsofmotion,on-orbitdisturbances andattitudecontrolhardwarearediscussedinChapter2,furnishingsometheory thatmayberequiredforthechaptersthatfollow.TheconceptofaCMG,types ofCMGs,mechanicalconstructionofatypicalCMG,andhardwarelimitationsare discussedinChapter3.AdetailedanalysisoftheSGCMGdynamicsisalsoprovided. Developmentofamethodtoestimatetheangularaccelerationofaspacecraftusing linearaccelerometersisdescribedinChapter4.Adiscussionofactuatormisalignment estimationandthedevelopmentofatechniquetoperformon-orbitestimationofgimbal orientationofanSGCMGisprovidedinChapter5.Adiscussionofywheelimbalance, attitudejitterandamethodtoreducethemagnitudeofjitterproducedbytheCMGis providedinChapter6.Theconceptsofstaticanddynamicimbalanceinaywheel whichmayprovideadditionalinsightintothedevelopmentsinChapter6arediscussed inAppendixB.ConcludingremarksareprovidedinChapter7.Descriptionofthe notationsusedthroughoutthedissertationisprovidedinAppendixA. 22

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CHAPTER2 SPACECRAFTATTITUDEDYNAMICSANDCONTROL Theprinciplesthatgoverntherotationalmotionofarigidspacecraftinorbitand methodstocontrolthespacecraft'sattitudearediscussedinthischapter[1,61].The denitionofattitude,itsrepresentations,andattitudekinematicsarealsodiscussed. Theattitudedynamicsofarigidbodyisstudiedfollowedbythedevelopmentofattitude equationsofmotionfor1.arigidbodyand,2.arigidbodywithanarbitrarymomentum actuator,fromrstprinciples.Abriefdescriptionoftheenvironmentalorexternaltorques thataffectthemotionofthespacecrafte.g.,gravitygradient,magnetic,aerodynamic andradiationtorquesisprovided.Finally,attitudecontrol,attitudecontrolerrors,and typesofattitudecontrolarebrieydiscussed.BasedonK onig'stheorem[62,63],ifthe centerofmass c.m. ofabodyisxedrelativetoalltheotherpointsinthebodyrigid body,thenthedynamicsofthetherotationalattitudemotionisdecoupledfromthe translationalrectilinearmotion.Thisenablesustostudytheattitudedynamicsofa rigidspacecraftwithoutregardtothetranslationaleffects.Anarbitrarychangeinthe congurationofabodyisthereforeadisplacementofthe c.m. andarotationaboutthis point.Further,sincefromanattitudeperspective,the c.m. ofthespacecraftbehaves similartoaninertialorxedpoint,thespacecraftpropertiessuchasinertiadyadicand angularmomentumareusuallyexpressedaboutthispoint. 2.1AttitudeRepresentation Allrepresentationsofattitudearerelative.Arigidbodyhasthreedegreesof rotationalfreedominthethreedimensionalDEuclideanspace,andaminimumof threeparametersarethereforeneededtorepresenttherelativeattitude.Thereare severalparametersetalternativestorepresenttheattitudeofarigidbody.Someofthe parametersetsthatarerelevantandsufcienttodescribetheworkpresentedinthis dissertationarediscussedhere.Adetailedsurveyofattituderepresentationscanbe foundin[5]. 23

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2.1.1FrameofReferenceandCoordinateBases Asystemofreferenceforspaceandtimeiscalledaframeofreference.Aframe ofreferenceisanon-physicalquantity,andonlynotional.InNewtonianmechanics,a singleframeofreferenceinwhichthemeasuresofspaceDEuclideanspaceand timeareinvariantisusedtodescribemotion. Ameasurementsystemisrequiredtoexpressthephysicalquantitiesobserved inaframeofreference.Asetofbasisvectors,calledacoordinatebasisorcoordinate system 1 isusedforthispurpose.Alogicalchoiceofsuchasetforrepresentationin3D Euclideanspaceisthe dextral right-handed orthonormal basiswhichconsistsofthree unitvectorse.g., X Y Z normaltoeachotherinaright-handedsense.Acoordinate basisisdenotedbythescriptedletter C ,withasubscriptthatdenesthefeatureofthe coordinatebasis.Thecomponentsofthephysicalquantityexpressedusingacoordinate basisarecalledcoordinates.Forinstance,avector a canbeexpressedusingthe h X Y Z i coordinatebasisas a = a x X + a y Y + a z Z oralternativelyasacolumnmatrix a = a x a y a z T .Thecoordinates a x a y a y arethecomponentsofthevector a alongtheirrespectivebasisdirections.Axed,ornon-acceleratingcoordinatebasis iscalledaninertialcoordinatebasisoraninertialcoordinatesystemdenotedby C I Themeasuresinaninertialcoordinatebasisareabsolute.Acoordinatebasisthatonly translateswithaconstantvelocityisalsoinertial.Inthisdissertation,thecoordinate basisattachedtothespacecraftbodyiscalledthebodycoordinatebasis,denotedby C B .Considertworigidbodieswithacoordinatebasis C j attachedtoeachrigidbody. Theorientationattitudeofthesecondrigidbodyrelativetotherstisdenedasthe orientationofthecoordinatebasisvectorsof C 2 asexpressedinthecoordinatebasis 1 Thesearesometimesreferredtoasreferenceframesbuttoavoidconfusion betweenframeofreferenceandreferenceframethisdissertationavoidsusingthe termreferenceframe. 24

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C 1 .Determiningtheattitudeofarigidbodythereforeinvolvesexaminingtheprojections ofitsbodyxedbasisvectorsontothebasisvectorsofacoordinatebasistowhichthe attitudeisreferenced.Thereareninesuchprojectionsandnotallareindependent. 2.1.2DirectionCosineMatrix Asdiscussedpreviously,theattitudeofarigidbodyisdeterminedbycomputing theprojectionsofthebody-xedcoordinatebasisvectorsontoacoordinatebasisof inetrest.Thereareninesuchprojectionsandarereferredtoasthedirectioncosines.A 3 3matrixconsistingoftheseprojectionsiscalledadirectioncosinematrixDCM. Considertwoarbitrarycoordinatebases C A and C B ,representedbytheunitvectors h X A Y A Z A i and h X B Y B Z B i ,respectively.Now,eachbasisvectorof C B canalsobe expressedin C A asalinearcombinationofthebasisvectorsof C A asgivenby X B = c 11 X A + c 12 Y A + c 13 Z A Y B = c 21 X A + c 22 Y A + c 23 Z A Z B = c 31 X A + c 32 Y A + c 33 Z A where,thecoefcientsarethedirectioncosinesbetweenthecorrespondingbasis vectors.Now,consideranarbitraryvector v ,whichcanbeexpressedin C A and C B as A v = x a X A + y a Y A + z a Z A = [ x a y a z a ] T B v = x b X B + y b Y B + z b Z B = [ x b y b z b ] T ItcanbeshownthatusingtherelationsinEq.2,thecoordinatesofthevector v in C B canbeexpressedintermsofcoordinatesin C A inacompactformasgivenby B v = R BA A v where,thematrixofcoefcients R BA givenby R BA = 0 B B B B @ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 1 C C C C A 25

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iscalledtheDCMorcoordinatetransformationmatrixthattransformsvectorrepresentations in C A tothatof C B .AlthoughtheDCMdoesnotcausephysicalrotationofthevectorit operateson,itissometimesreferredtoasarotationmatrixasitrepresentstherelative angulardisplacementorrotationbetweenthetwocoordinatessystemsorrigidbodies involved.TheDCMhasthefollowingproperties[61] 1.TheDCM R isanorthonormalmatrixi.e.,theinnerproductbetweenanytworows oranytwocolumnsiszero,andtheinnerproductofaroworcolumnwithitselfis equalto1 2.Thedeterminantofaproperright-handed R isequalto+1 3.TheinverseoftheDCMequalsitstransposei.e., R T R = R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R = 1 4.Allnineelementsof R arenotindependentandareboundbytheconstraintsof orthonormality 5.TheDCMhasonerealeigenvalue, 1 =1andtwocomplexeigenvaluesin conjugatepairs 2 = e )]TJ/F59 7.9701 Tf 6.587 0 Td [(j 3 = e )]TJ/F59 7.9701 Tf 6.586 0 Td [(j .Thetraceofthematrix,whichisequaltothe sumoftheeigenvaluesisthereforegivenby Tr R =1+2cos 6.TheproductoftwoDCMsisalsoaDCMandsuccessivetransformationsare pre-multiplied.Forinstance,acompositetransformationinvolvingatransformation from C A to C B followedbyatransformationfrom C B to C C isrepresentedby R CA = R CB R BA 2.1.3Axis-AngleRepresentation Eulerdiscoveredthatthegeneraldisplacementofarigidbodyaboutapointis arotationaboutanaxispassingthroughthatpoint[6163].Wecanthusrepresent arigidbodyrotationoritsattitudeusingtherotationaxis a ,andtherotationangle seeFigure2-1asaparameterset a .TheaxisofrotationiscalledtheEuler axisandtheangleofrotationisknownastheEulerangledifferentfromEulerangle sequenceinSection2.1.5.Notethattheaxisanglerepresentationhasfourparameters a = a 1 a 2 a 3 T and ,andoneconstraintgivenby k a k =1.ThecorrespondingDCM 26

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representationoftheattitudecanbeobtainedintermsoftheEuleraxisandangleusing therelationgivenby R a =cos 1 + )]TJ/F20 11.9552 Tf 11.955 0 Td [(cos a a T )]TJ/F20 11.9552 Tf 11.955 0 Td [(sin a Thenotation R a representsanangulardisplacementof betweentwoarbitrary coordinatesystems,abouttheaxis a .Sincetheaxisremainsunchangedduringthe rotation,itisevidentthattheEuleraxisistheeigenaxisoftheDCMassociatedwith therealeigenvalue, 1 =1.Whentherotationaxisisoneofthecoordinatebasis vectors,therotationiscalledaprincipalrotation.Principalrotationsareveryuseful forillustratingrigidbodyrotations.UsingEq.2,theDCMsassociatedwiththethree principalrotations,aboutthe X Y ,and Z axes,respectivelyaregivenby R X = 0 B B B B @ 100 0cos sin 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin cos 1 C C C C A R Y = 0 B B B B @ cos 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin 010 sin 0cos 1 C C C C A R Z = 0 B B B B @ cos sin 0 )]TJ/F20 11.9552 Tf 11.291 0 Td [(sin cos 0 001 1 C C C C A GivenaDCM,theEulerangleandaxiscanbedeterminedusingEq.2and a = 1 2sin R T )]TJ/F58 11.9552 Tf 11.955 0 Td [(R whenTr R isnotequalto3or-1.WhenTr R =3,therigidbodywouldhave undergonezeronetdisplacementandhencetheaxisofrotationcannotbedetermined. 27

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WhenTr R = )]TJ/F20 11.9552 Tf 9.299 0 Td [(1,therotationangle = ,andhencetherotationaxiscannotbe determinedusingEq.2.Adetaileddiscussioncanbefoundin[61]. 2.1.4EulerSymmetricParametersorUnitQuaternion Afrequentlyusedsetofparameterswhichavoidstheindeterminacyseeninthe axis-anglerepresentationaretheEulersymmetricparameters.Thisparameterset isidenticaltoaunitquaternionandfollowsthesamemathematicalstructure.The termquaternioninusedinthisdissertationtorefertoEulersymmetricparameters. QuaternionwasatermcoinedbyHamilton[64]forfour-parameterhypercomplex numbers.Aquaternioniscomposedofavectorpart q andascalarpart q 4 andis denotedby q = q + q 4 .Thecomplexconjugateofaquaternionisdenotedby q y = q )]TJ/F59 11.9552 Tf 11.957 0 Td [(q 4 Thescalarandthevectorpartsofaquaternionaregivenby q =sin 2 a = 2 6 6 6 6 4 q 1 q 2 q 3 3 7 7 7 7 5 q 4 =cos 2 where, a istheaxisofrotation,and istheangleofrotationidenticaltotherepresentation inSection2.1.3.Thenormoftheunitquaternionisgivenby q q 2 1 + q 2 2 + q 2 3 + q 2 4 =1 Theequivalenttransformationmatrixcanbedeterminedfromthequaternionrepresentation using R = q 2 4 )]TJ/F20 11.9552 Tf 11.955 -0.131 Td [( q T q 1 +2 q q T )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 q 4 [ q ] Withtwosuccessiverotations, q followedby p ,thecompositequaternion, r ,representing thecompositerotationisgivenby r = p 4 q 4 )]TJ/F20 11.9552 Tf 11.955 -0.131 Td [( p T q + )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(p 4 q + q 4 p )]TJ/F20 11.9552 Tf 11.955 -0.146 Td [([ p ] q 28

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Moreonquaternionalgebracanbefoundin[64]and[5]. 2.1.5EulerAngles Anyarbitraryrigidbodyrotationcanberepresentedasacompositionofthree successiverotationsoftherigidbodyaboutitsbodyxedcoordinateaxes.The rotations,asdescribedinSection2.1.5arecalledprincipalrotations,andtheangles throughwhichtheprincipalrotationsareperformedarecalledtheEulerangles.The sequenceinwhichtherotationsareperformediscalledtheEulersequence.There aretwelvesuchsequencescomprisingofsymmetricandasymmetricsequences.An exampleofasymmetricEulersequenceisthe3-1-3sequencecommonlyusedinspace ightmechanicswhichinvolvesarotationaboutthethird Z axisfollowedbyarotation abouttherst X axisandagainarotationaboutthe Z axis.The3-1-3rotationthrough angles ,and ,respectivelyisrepresentedby R 313 = R Z R X R Z AnexampleofanasymmetricEulersequenceisthe3-2-1yaw-pitch-rollsequence commonlyusedinaircraftmechanics.Thisinvolvesarotationaboutthethird Z axis followedbyarotationaboutthesecond Y axisandnallyarotationabouttherst X axis.The3-2-1rotationthroughangles ,and ,respectivelyisrepresentedby R 321 = R X R Y R Z OtherattituderepresentationssuchasRodriguezparameters,ModiedRodriguez parameters,Cayley-Kleinparameters,etc.arenotusedinthisdissertationandhence notdiscussedhere.Thesamecanbefoundin[5]. 2.2AttitudeKinematics Attitudekinematicsdealswiththegeometryofrotationalmotionwithoutconsideration totheforcesaffectingmotion.Rotationalmotionofarigidbodyiscompletelydescribed byitsattitudeandangularvelocity.Theangularvelocityofarigidbodyisdenedasthe 29

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rateofchangeofitsbodyxedcoordinatebasisvectorsrelativetoanothercoordinate basisofinterest.Sincethebasisvectorsareofunitmagnitude,theratechangeisa resultoftherotationoftherigidbody.Theangularvelocityisconsideredtobeabsolute whenitisdenedrelativetoaninertialcoordinatebasis.Theangularvelocity,unlike attitudeisatruevectorandhencefollowsthelawofvectoraddition.Denotingthe angularvelocityofarigidbody A orabodyxedbasis C A relativetoanotherrigidbody B by BA ,andsoon,wecanwrite CA = CB + BA Usingthedenitionofangularvelocity,theratechangeofavector v asobservedina coordinatebasis C B isgivenby v = @ v @ t + BA v where,thepartialderivativerepresentstheratechangeasobservedinacoordinate basis C A inwhichthevector v onlychangesinmagnitude. Itisevidentthattheattitudeofarigidbodywithnon-zeroangularvelocityis timedependent.Theevolutionofattitudecanbedeterminedbyintegratingattitude differentialequationsinvolvingtheangularvelocity.Thedifferentialequationsforvarious attituderepresentationsdiscussedinSection2.1aregivenbelow.Thederivationof theseequationscanbefoundin[61]. 2.2.1DifferentialEquationforaDCM ThedifferentialequationforaDCMthatcanbeusedforattitudepropagationis givenby R = )]TJ/F20 11.9552 Tf 11.291 -0.146 Td [([ ] R IntegratingEq.2yieldsthetimehistoryofattitudeofarigidbodywhentheinitial attitudeinknown.Whileintegrating,theorthonormalitycondition R T R = 1 mustbe continuouslyexaminedtoavoidaccumulationofnumericalerrors. 30

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2.2.2DifferentialEquationforAxis-AngleRepresentation ItwasshownearlierthataDCMcanbeexpressedintermsoftherotationaxisand angle a .UsingEq.2andEq.2,itcanbeshownthatthepropagationofthe axisandangleparametersisgivenbythedifferentialequationas a = 1 2 a )]TJ/F20 11.9552 Tf 11.955 0 Td [(cot 1 2 a a = a T Thetimehistoryofattitudeintermsof a and canbedeterminedbyintegrating Eq.2.Whileintegrating,theconstraint a T a =1shouldbecontinuallyexamined. 2.2.3DifferentialEquationforUnitQuaternions Recognizingtherelationbetweentheaxis-angleparameters a andtheunit quaternion q q 4 asgivenbyEq.2,wecanuseEq.2todeterminethedifferential equationfortheunitquaternionas q = 1 2 )]TJ/F20 11.9552 Tf 5.479 -9.83 Td [([ q ] + q 4 1 q 4 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 q T whichcanbecompactlywrittenasgivenby q = 1 2 Q where,the4 3matrix Q isgivenby Q = 0 B @ [ q ] + q 4 1 )]TJ/F20 11.9552 Tf 11.291 -0.132 Td [( q T 1 C A Thetimehistoryofattitudeintermsoftheunitquaternioncanbedeterminedby integratingEq.2.Whileintegrating,theconstraintcondition q T q =1mustbe continuallyexamined.Alternatively,anotherformofEq.2generallyusedforattitude 31

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propagationisgivenby q = 1 2 [ ] q where,the4 4matrix isgivenby = 0 B @ )]TJ/F20 11.9552 Tf 11.291 -0.146 Td [([ ] )]TJ/F20 11.9552 Tf 11.291 -0.131 Td [( T 0 1 C A ThedifferentialequationforattitudeintermsofEuleranglesequencesisnotdiscussed hereasitwillnotbeusedinanypartofthisdissertation.Thesamecanbefoundin[61]. 2.3AttitudeDynamics Therepresentationofattitudeandthegeometryofrotationalmotionwerediscussed inSections2.1and2.2,respectively.Inthissection,weshalldiscussthedevelopment ofequationsofmotionthatgoverntheevolutionofrotationalmotionofarigidbody R undertheinuenceofexternalforcesandtorques.Theresearchpresentedinthis dissertationdoesnotrequirethediscussionofexiblebodydynamicsandishence omitted.Further,theattitudemotionisconsideredindependentofthetranslational dynamics.VectorialmechanicsortheNewton-Eulerformulationisusedinderivingthe equationsofmotion.Theexistenceofaninertialpointandaninertialcoordinatebasis isassumed.Quantitiesexpressedrelativetotheinertialpointand/orcoordinatebasis aretermedinertialorabsolutequantities.Forinstance,theabsolutelinearvelocityof apointisitsvelocityrelativetotheinertialpoint,andtheabsoluteangularvelocityofa rigidbodyisitsangularvelocityrelativetotheinertialcoordinatebasis. 2.3.1MassanditsMoments Thetotalmass, m ofarigidbodyisameasureoftheamountofmattercontained withinthevolumethatdenestherigidbody.Considerahomogeneousrigidbodywith constantmass,inwhichcasethemassandvolumeofarigidbodyarerelatedbya 32

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simpleexpressiongivenby m = Z R dV = Z R dm where,theuniformdensityoftherigidbodyisdenotedbytheconstant ,and dV isa differentialvolumeelement.Consideranarbitraryrigidbodyofmass m asshownin Figure2-2.Let O beapointofinterestintherigidbody.Thepositionofthedifferential masselement dm from O isrepresentedbythepositionvector r c + r ,where r c isthe positionofthe c.m.C oftherigidbodyrelativeto O ,and r isthepositionofadifferential mass dm from C .Therstmomentofmassaboutthepoint O denotedby c o isdened as c o Z R )]TJ/F65 11.9552 Tf 5.479 -9.683 Td [(r c + r dm Itcanbeshownthattherstmomentofmassofarigidbodyaboutits c.m. isequalto zero,andhencewecanwriteEq.2as c o = m r c + Z R r dm = m r c Thepositionofthe c.m. ofarigidbodyrelativetoanarbitrarypointisthereforegivenby r c = c o m Forarigidbodywithconstantmass,thelocationofthecenterofmassrelativetothe otherpointsintherigidbodydoesnotchange.Thesecondmomentofmassofarigid bodyaboutanarbitrarypoint O ,alsoknownastheinertiadyadicisdenedas J o Z R r c + r r c + r 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( r c + r r c + r dm whichcanberewrittenasthepopularparallelaxistheorem Huygens-Steiner theorem givenby J o = I c + m )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r c r c 1 )]TJ/F65 11.9552 Tf 11.955 0 Td [(r c r c 33

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where,thecentroidalinertiadyadicoftherigidbodyisgivenby I c = Z R r r 1 )]TJ/F65 11.9552 Tf 11.956 0 Td [(rr dm Theinertiadyadicisameasureofthemassdistributioninabody.Theinertiadyadic representedusingacoordinatebasisisgivenbytheinertiamatrix J .Theinertiamatrix issymmetricandpositivedenite[61].Theeigenvectorsof J areknownastheprincipal axesofinertiaandtheeigenvaluesaretheprincipalmomentsofinertia J 1 J 2 J 3 Theeigenvectorsformasetoforthonormalbasisandcanhencebeusedtodene acoordinatebasis, C P .Theinertiadyadicwhenexpressedinthisbasisisadiagonal matrixwiththeprincipalmomentsofinertiaalongitsdiagonalasgivenby P J o =diag J 1 J 2 J 3 Whentheeigenvectorsofaninertiamatrixexpressedinanarbitrarycoordinatebasis C A arestackednexttoeachother,theresulting3 3matrixistheDCMthatrepresents thetransformationfromthecurrentcoordinatebasis C A totheprincipalcoordinatebasis C P .Thecoordinatetransformationthattransformstheinertiamatrixexpressedina basis C A toabasis C B isgivenby B J o = R BA A J o R BA T Inanarbitraryinertiamatrix,theoff-diagonalcomponentsarecalledtheproductsof inertia.Variousotherpropertiesoftheinertiamatrixcanbefoundin[61]. 2.3.2AngularMomentum Angularmomentumofarigidbodyaboutanarbitrarypoint O isdenedasthe rstmomentoflinearmomentumabout O .Theabsoluteangularmomentumofarigid bodyorasystemofrigidbodiesisalwaysconservedunlessacteduponbyexternal torques.Theexpressionfortheangularmomentumofarigidbodyisdevelopedbelow. 34

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TheabsolutelinearvelocityofapointonarigidbodyshowninFigure2-2isgivenby v = v o + )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(r c + r where v o istheabsolutevelocityofthepoint O and istheabsoluteangularvelocityof therigidbody.Thetotallinearmomentumoftherigidbodyisdenedas p Z R v dm andbyusingEq.2,wecanwrite p = m v o + m )]TJ/F46 11.9552 Tf 5.48 -9.684 Td [(! r c + Z R r dm p = m v o + m )]TJ/F46 11.9552 Tf 5.48 -9.683 Td [(! r c However,ifweconsiderthe c.m. oftherigidbodyinsteadofthepoint O ,thetotallinear momentumisgivenby p = m v c where, v c istheabsolutevelocityofthe c.m. Theabsoluteangularmomentumofthe rigidbodyaboutthepoint O isdenedas h o Z R )]TJ/F65 11.9552 Tf 5.479 -9.683 Td [(r c + r v dm UsingtheexpressionforthelinearvelocityinEq.2,wecanwriteEq.2as h o = m )]TJ/F65 11.9552 Tf 5.48 -9.683 Td [(r c v o )]TJ/F65 11.9552 Tf 11.955 0 Td [(v o Z R r dm + Z R r c + r r c + r 1 )]TJ/F20 11.9552 Tf 11.955 -0.132 Td [( r c + r r c + r dm UsingthedenitionsfortherstandsecondmomentsofmassgivenbyEq.2,and Eq.2,respectivelytheexpressionfortheangularmomentumcanbewrittenas h o = )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(c o v o + J o where,theterm h o isknownasthetotalorabsoluteangularmomentumasitincludes theeffectofmotionofpoint O [61,65].Itmaybeadvantageoustoconsiderthe c.m. of 35

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therigidbodyasthepointaboutwhichtheangularmomentumiscomputedasthe velocityofthe c.m. isonlythetranslationalvelocityoftherigidbodyandisnotaffected byitsrotation.Thus,ifwechoosethispointtobethecenterofmassoraninertialpoint, thersttermofthesuminEq.2vanishes.Further,theinertiadyadicisnowgivenby I c .Therefore,theequationoftheangularmomentumofarigidbodyrelativetoits c.m. is givenby h c = I c However,inthecaseofaspacecraft,theremaybeseveralcomponentsthatmove relativetothespacecraftstructure,andsomepointotherthanthe c.m. maybe convenienttocomputetheangularmomentum. 2.3.3EquationsofMotionforaRigidBody Euler'slawsandtheexpressionfortheratechangeofangularmomentumareused toderivetheattitudeequationsofmotionofasinglerigidbody.Theprinciplesusedin thisbasicexampleareusedindevelopingtheequationsofmotionformorecomplex systemsinthesubsequentchapters. Euler'srstlawstatesthat thetotalexternalforceactingonarigidbodyequalsthe massoftherigidbodytimestheaccelerationofitscenterofmass asgivenby m v c = n X i =1 F i Euler'ssecondlawstatesthat thetotalexternalmomenttorqueappliedtoarigid bodyaboutanarbitrarypointintherigidbodyisequaltotheratechangeofangular momentumaboutthesamepoint,ifthechosenpointisaninertialpointorthecenterof mass .Theratechangeofangularmomentumofarigidbodyaboutits c.m. istherefore givenby h c = I c + I c = ext c 36

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whereseeFigure2-2, ext c isthetotalexternaltorqueontherigidbodyaboutits c.m. givenby ext c = n X i =1 r i F i Theexternaltorquesonaspacecraftincludetheenvironmentaldisturbancetorques seeSection2.3.5andattitudecontroltorquesproducedbythrusters.Equation2, knownasEuler'sequationscompletelyrepresenttheattitudeequationsofmotionfora rigidbody.However,whenthepointaboutwhichthedynamicsareevaluatedisnotthe c.m. oraninertialpoint,additionaltermsareinvolvedandtheequationsofmotionare derivedasfollows.Theabsoluteangularmomentumoftherigidbodyaboutthepoint O givenbyEq.2canberewrittenas h o = m )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r c v o + m )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(r c )]TJ/F46 11.9552 Tf 5.479 -9.684 Td [(! r c + h c = m )]TJ/F65 11.9552 Tf 5.479 -9.683 Td [(r c )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(v o + r c + h c Notingthat v o + r c isthevelocityofthe c.m. v c ,wecanwrite h o = m )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r c v c + h c Thetimeratechangeof h o isthereforegivenby h o = m )]TJ/F58 11.9552 Tf 6.412 -9.517 Td [( r c v c + m )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r c v c + h c UsingEuler'srstlawgivenbyEq.2,Eq.2,andEq.2wecanwrite h o = m )]TJ/F58 11.9552 Tf 6.412 -9.517 Td [( r c v c + n X i =1 )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r c + r i F i Finally,usingEq.2andEq.2,theratechangeoftheabsoluteangularmomentum aboutanarbitrarypoint O intherigidbodyisgivenby h o + v o p = ext o 37

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where, ext o isthetotalexternaltorqueontherigidbodyabout O .Whenthepoint O isaninertialpointorthecenterofmass,itcanbeseenthattheequationreducesto Euler'sequationsgivenbyEq.2.Therearenoanalyticalsolutionstothegeneral formofEuler'sequationsforacompositionofmultiplerigidbodiessuchasaspacecraft. However,closedformsolutionscanbefoundforasinglerigidbodywithzeroexternal torque[61].Numericalintegrationoftheequationsofmotionishenceusedinthis dissertationtodeterminetheevolutionofspacecraftstates. 2.3.4EquationsofMotionforaRigidBodywithaMomentumDevice Whenamomentumdevicesuchasareactionwheelisembeddedinarigidbody rigidspacecraft,thetotalangularmomentumisequaltothesumoftheangular momentumoftherigidbodyandtheangularmomentumofthemomentumdevicesas givenby h c = J c + X h i p i where J c istheinertiadyadicofthesystemconsistingoftherigidbodyandthe momentumdevicesaboutthesystem's c.m. ,and h i istheangularmomentumof the i th momentumdeviceaboutapoint p i .Thepoint p i issuchthatitisxedintherigid bodyandliesontherotationaxisofthemomentumdevice.Theratechangeofangular momentumofasuchassystem,usingEuler'slawisgivenby h c = J c + X h i p i + J c + X h i p i = ext c where h i p i isevaluatedrelativetotherigidbody.Usually,thestatesofthemomentum devicee.g.,angularvelocityandaccelerationofaywheelaremeasuredina coordinatebasisxedtothedevice.InsuchcasesaDCM R BA i thattransforms representationsfromacoordinatebasis C A i xedinthe i th momentumdevicetothe bodyxedcoordinatebasis C B isrequiredforthematrixrepresentationoftheequations 38

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ofmotionasgivenby 2 B h c = J c + X R BA i h i p i + [ ] J c + X R BA i h i p i = B ext c Thecoordinatizedequationsofmotionareusedfornumericalanalysisandsimulation purposes. 2.3.5EnvironmentalTorques Theattitudemotionofarigidbodyisaffectedbyexternaltorquesasrepresented bytherighthandsideofEq.2.Thespacecraftexperiencesthesetorquesfromthe surroundingenvironment.Althoughthemagnitudeofthesetorquesseemnegligible, theycausesignicantdriftofthespacecraftattitudeoveralongperiodoftime.These externaltorquesthatariseduetotheinteractionofthespacecraftwithitssurrounding environmentarediscussedinthissection. 2.3.5.1Gravitationaltorque Non-sphericalwithrespecttomassandinertiaspacecraftexperienceatorquedue tothenon-uniformityofthelocalgravitationaleldoverthevolumeofthespacecraft. Theresultisanon-coincidencebetweenthespacecraft'scenterofmassandcenterof gravity.Newton'slawofgravitationstatesthat thegravitationalforcebetweentwopoint massesinspaceisproportionaltoproductofthemassesandinverselyproportional tothesquareofthedistanceseparatingthetwopoints .Sinceeverypointonthe spacecraftisatadifferentdistancefromthegravitationalsourcese.g.,Earth,Moon, etc.,thelocationofthespacecraft'scenterofgravityisdifferentfromthelocationof itscenterofmass.Theapproximaterstordergravitationaltorqueonthespacecraft aboutits c.m. duetotheEarth'sgravitationaleld[1,61]isgivenby g c = 3 r 3 n )]TJ/F65 11.9552 Tf 5.48 -9.683 Td [(I c n 2 RefertothenoteinAppendixAforthenotationschemeusedinthisrepresentation. 39

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where,thecentroidalinertiaofthespacecraftisdenotedby I c and n isthenadir directionfromthe c.m. ofthespacecraft.ThegravitationalparameteroftheEarth =3.986 10 14 Nm 2 = kg and r isthedistancebetweenthe c.m. oftheEarthandthe c.m. ofthespacecraft.InderivingEq.2,ithasbeenassumedthatthespacecraft isasinglerigidbodywithitsdimensionsmuchsmallercomparedtoitsdistancefrom theEarth,andtheEarthissphericalandsymmetricwithrespecttomass.Itcanbe notedthatthegravitationaltorquevanisheswhenanyoneoftheprincipalaxesofthe spacecraftalignswiththenadirdirection. 2.3.5.2Aerodynamictorque Themechanicsofaerodynamicforceexperiencedbyspacecraftinorbitissimilar tothatobservedinatmosphericight.However,thedrasticallylowatmospheric densityrequiresadifferentapproachtomodelingtheseforces.Eveninthelowest orbitspossiblewithoutrapiddecayduetoatmosphericdrag,themeanfreepathof theparticlesisgreaterthan1 km .Afreemolecularowmodelnointeractionwith surroundingparticlescanthereforebeassumed.Theforceexertedbytheparticleon thespacecraftisthroughmolecularmomentumtransfer[61]andadiffusereection mechanismwheretheimpactingparticlegetsembeddedinthespacecraftand transfersallitsmomentumisassumed.Theresultingforceisinthedirectionofthe relativevelocitybetweenthespacecraftandthelocalatmospheredenotedby v rel .The aerodynamicforceresultsinatorqueonthespacecraftwhenthevectorfromthe c.m. of thespacecrafttoeffectivecenterofpressure r p isnotparalleltotherelativevelocity vector v rel .Theaerodynamicforceonthe i th faceofthespacecraftisgivenby f aero i = 1 2 )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [( a A i v 2 rel C d v rel where a istheatmosphericdensity, A i theareaofthe i th facenormalto v rel ,and C d is thedragcoefcient.Thetorqueduetotheaerodynamicforceonthe i th faceaboutthe 40

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centerofmassofthespacecraftisgivenby aero i c = r pi f i where r pi isthedistancefromthespacecraft c.m. tothecenterofpressureofthe i th face ofthespacecraft.Thetotalaerodynamictorqueisgivenby aero c = X i aero i c 2.3.5.3Radiationtorque Electromagneticradiationexertspressureonthesurfaceofspacecraftwhichresults inaforceactingatthecenterofpressure.Theradiationpressurecanbeexplainedin termsofitscorpuscularnature[61].Themechanicsofradiationtorqueissimilartothat ofaerodynamictorqueandisgivenbyananalogousexpression.Theexpressionsfor theforceexertedonthespacecraftduetotheradiationpressure p onthe i th faceis givenby f rad i = 1 2 pA i C d s where,thedirectionofincidentradiationisdenotedby s .Theresultingtorqueaboutthe spacecraft c.m. isgivenby rad i c = r pi f rad i Thetotaltorqueexertedonthespacecraftduetoradiationpressureisgivenby rad c = X i rad i c ForEarthorbitingspacecraft,themajorsourceofradiationistheSunandthemean Solarpressure p =4.5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 N = m 2 2.3.5.4Magnetictorque Aspacecraftcontainsseveralsourcesofmagneticeldsuchasmotorsand electroniccircuits.Moreoftenthannot,thereisnetresidualmagneticmoment M dueto 41

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thesesources.Atorquegivenby mag c = M B isexertedonthespacecraftwhenthismagneticmomentinteractswiththemagneticux B oftheenvironment[61]. Thetotalexternaltorqueexertedonthespacecraft,aboutits c.m. isequaltothe sumoftorquesinEqs.2,2,2,and2givenby ext c = g c + aero c + rad c + mag c 2.4AttitudeControl Attitudecontrolisdenedastheprocessofachievingandmaintainingdesired orientationsofthespacecraftinspace.Spacecraftcontainspecicsubsystemscalled attitudedeterminationandcontrolsystemsADCSthatperformthisfunction.The ADCSofaspacecraftcomprisesofvarioussensors,actuatorsandcontrolalgorithms. Twotypesofattitudecontrolexistviz.attitudestabilizationandattitudemaneuver control[1].Attitudestabilizationistheprocessofmaintainingtheexistingattitudefor acertainperiod,whereasattitudemaneuvercontrolinvolvesreorientationbetween attitudesatadesiredrate.Basedonthemethodused,attitudecontrolcanbeclassied intopassiveandactive.Passiveattitudecontrolcanonlyprovidestabilizationand doesnotinvolvefeedbackfromattitudesensors.Theydonotactivelyapplycorrective torquesonthespacecrafttoachieveadesiredattitude.Passiveattitudecontrolsystems usespin-stabilizationorenvironmentaltorquessuchasgravitygradienttoachievea particularpredeterminedorientation.Forinstance,aspacecraftusinggravitygradient basedattitudestabilizationwillattainanattitudesuchthatitsminoraxisofinertiais alignedwiththegravitationalaccelerationvector.Thisrelativeorientationcannotbe changedwithoutanactiveattitudecontrolsystem.Spin-stabilizedspacecraftspin ataconstantrateabouttheiraxisofmaximuminertiaorcontainawheelthatspins 42

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atconstantrate.Sincetheangularmomentumofthespacecraftisconserved,the spacecraftcontinuestopointinthesameinertialdirectionunlessaffectedbyexternal torques.Suchspacecraftalsorequirenutationdamperstosuppressthespacecraft's nutatingmotion.Spacecraftwithpassiveattitudecontrolsuchasgravitygradient andspin-stabilizationusuallyrequireanadditionalactiveattitudecontroltocounter theexternalandinternaldisturbances.Anactiveattitudecontrolsystemismore sophisticatedandmoreaccuratecomparedtoapassiveattitudecontrolsystem.In thisdissertation,weonlyconsideraspectsrelatedtoactiveattitudecontrol. 2.4.1ActiveAttitudeControl Activeattitudecontrolistypicallyautonomousandinvolvesthecontinuous applicationofcorrectivetorqueonthespacecraftbasedonanerrorsignal.Atypical activeattitudecontrolprocessisshowninFigure2-3.Thevariouscomponents, bothhardwareandprocessalgorithmsshowninFigure2-3constitutetheADCS ofthespacecraft.ThefunctionofanactiveADCSistoreorientthespacecraftto anydesiredattitudewithaparticularaccuracyandspeed.TheADCSshouldalso compensateforenvironmentaldisturbancesactingonthespacecraft.Anestimate oftheexistingorientationandangularvelocityofthespacecraftisprovidedbythe attitudedeterminationsystemADSwhichconsistsofsensorse.g.,starsensor,sun sensor,gyroscopesetc.andanattitudedeterminationscheme/algorithm.Theerror betweentheexistingattitudeandthedesiredattitudeisinputtotheattitudecontrol systemACSthatconsistsofattitudecontrolalgorithmsandactuators.Basedonthe attitudeerror,anappropriatesetofcommandsiscomputedusingtheattitudecontrol algorithmsandissuedtotheactuatorse.g.,thrustersandmomentumactuators. Theactuatorsthereforeapplythedesiredtorqueonthespacecraft.The`plant'and actuatormomentumactuatorsonlydynamicsarecoupledandcanberepresentedby Eq.2.Theterms ext c and P h i pi inEq.2constitutethecontrolinputstorques intheactiveattitudecontrolsystem.Boththesecontrolinputsalterthemomentum 43

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stateofthespacecrafttherebyinducingrotationalmotion.Theexternalcontroltorque )]TJ/F46 11.9552 Tf 5.479 -9.684 Td [( ext c canbeappliedusingactuatorssuchasthrustersandelectromagneticcoils.The internalcontroltorquerepresentedby P h i pi isappliedusingmomentumactuators suchasreactionwheels,momentumwheels,andcontrolmomentgyroscopesCMG. Whentheactuatorswhicharexedinthespacecraftchangetheirangularmomentum state,thespacecraftalsochangesitsmomentumstatesuchthatthenetinertialangular momentumisconserved.Duetohardwarelimitationsassociatedwithsensorsand actuators,efciencyofcontrolalgorithms,andexternaldisturbances,thereexistsan errorbetweenachievedanddesiredattitudes.Inthisdissertationweareconcernedwith theattitudeerrorsthatresultfromactuatordeciencies.Thevarioustypesofattitude errorsandattitudeactuatorsarediscussedbelow.Descriptionofothercomponentsof theADCSsuchasattitudecontrolalgorithmsandADSarenotrequiredfortheensuing researchdiscussionandarehenceexcluded.Thesamemaybefoundin[1]. 2.4.2AttitudeControlErrors Attitudecontrolisaprocessthatorientsaspacecraftbody-xedcoordinatebasis intoalignmentwithadesiredreferencecoordinatebasis.Duetocertainlimitations ofattitudecontrolhardwareandalgorithms,thealignmentisnotperfect.Attitude errorisdenedastherelativeorientationbetweenthetrueandthedesiredattitudes ofthespacecraft.Errorsintroducedateachstageintheattitudecontrolprocess accumulateresultinginanetattitudeerror.Sincetheattitudeerrorrepresentations arenottruevectors,theerrorsarenotadditiveanditishencedifculttorepresentthe cumulativeerror.Theindividualerrorsaredescribedusingtheaxis-anglerepresentation Section2.1.3.TheseerrorsareillustratedinFigure2-4. InaccuraciesintheADSleadtoerrorbetweentheactualattitudeandtheestimated attitude,knownasknowledgeerror.Thiserrorisrepresentedusingtheparameterset )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(a ke ke asshowninFigure2-4.IfaperfectACSweretobeassumed,thetotalattitude 44

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errorwouldbeequaltoonlytheknowledgeerror.Thisimpliesthattheaccuracyof ADCSislimitedbytheperformanceofitsattitudesensors. Attitudeactuatorlimitationsandinefciencyofcontrolalgorithmscausemisalignment betweentheestimatedattitudeandthedesiredattitude.Thiserroristermedascontrol errorandisrepresentedbytheparameterset a ce ce asshowninFigure2-4. Furthermore,externaldisturbancesandactuatorimperfectionse.g.,unbalanced rotorsleadtovariationsabouta mean attitudeofthespacecraft.Thisinstabilityofthe spacecraftattitudeinknownasstabilityerrorandisrepresentedbytheparameterset a se se inFigure2-4.Anexampleofsucherroristheattitudejittercausedbyonboard vibrations.Itshouldbenotedthatthedirectionoftheaxis a se mayvarydependingon thedirectionofdisturbance,andtheangle se isameasureofthepeak-to-peakangular erroraboutthegivenaxis.Thestabilityerroristhereforerepresentedbyanerrorconein Figure2-4. Thetotalattitudeerroristhecumulativeeffectofalltheabovementionederrors. However,iftheaxes a ke a ce ,and a se areparallel,thenthenetattitudeerroristhesum oftheangles ke ce ,and se 2.4.3AttitudeActuators Attitudeactuatorscanbeclassiedintoexternaltorqueactuatorsandmomentum exchangeactuators. 2.4.3.1Externaltorqueactuators Externaltorqueactuatorsapplyanexternaltorqueontothespacecraftthereby affectingitsattitude.Additionalangularmomentumisaddedtothespacecraftinthe process,andafterthedesiredattitudehasbeenattained,torqueneedstobeapplied againtoremovetheadditionalmomentum.Externaltorqueactuatorscanbeclassied intomassexpulsiondevicesandenvironmentaltorqueactuators. Massexpulsiondevicesorthrustersconsistofapropellentstoredonboardthe spacecraftthatisexpelledthroughanozzlewithhighvelocity.Thisimpartsaforceon 45

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thespacecraftatthepointofattachmentofthenozzlewhichinturnappliesatorque aboutthe c.m. ofthespacecraft.Thelimitationsoftheseactuatorsincludelowaccuracy, limitedlifeandhighcost.Furthermore,theforceimpartedonthespacecraftbyasingle thrusterinducesorbitalchangesandthereforetheythrustersneedtobeusedinpairs. Aminimumofsixpairsthrustersarethereforerequiredtoeffectpurethree-axis attitudemotion.Examplesofmassexpulsiondevicesincludecoldgasthrustersandion thrusters. Environmentaltorqueactuatorssuchasmagnetictorquersinteractwithenvironmental forcestoimparttorqueonthespacecraftaboutadesireddirection.Magnetictorquers produceamagneticeldthatinteractswiththeEarth'slocalmagneticeldtoproducea torqueasgiveninEq.2.Torquecannotbegeneratedalongthedirectionofthelocal magneticeld,andmagnetictorquerscannotbeusedataltitudeswherethemagnetic eldisveryweak.Attitudecontrolusingmagnetictorquersislimitedtolowaccuracyand lowslewrateapplications. 2.4.3.2Momentumactuators MomentumactuatorssuchasreactionwheelsandCMGsareelectromechanical devicesthateffectspacecraftmotionbymutualexchangeofangularmomentumwith thespacecraft.Sincemomentumactuatorsonlyconsumeelectricalpowerproduced onboardbysolarcells,theirlifeistheoreticallyunlimited. Areactionwheelisthesimplestmomentumactuatorandconsistsofaywheelthat iscontrolledbyamotor.Areactionwheelxedinaspacecraftcanincreaseordecrease speedtochangeitsangularmomentum.Thespacecraftreactstothischangesothat thenetinertialangularmomentumisconserved.Aminimumofthreereactionwheels arerequiredforindependentthree-axisattitudecontrol.Typically,asetoffourreaction wheelsisusedtoprovideredundancy. Incomparisontoareactionwheel,CMGsaremorecomplexmechanisms,anda genericCMGconsistsofaspinningywheelthatisgimbaledaboutoneormoreaxes. 46

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Thedirectionoftheywheel'sspinaxisorangularmomentumvectorisalteredby rotationofthegimbals.This,theneffectsspacecraftmotiontoconservethenetinertial angularmomentum.CMGsprovidebetteraccuraciesandfasterslewratescomparedto reactionwheels.AdetaileddiscussiononCMGsisprovidedinChapter3. Thebenetsofusingamomentumactuatorforattitudecontrolincludehigher accuraciesandfastersettlingtimes.Theirlimitationsincludehighermasscompared toexternaltorqueactuators,disturbancescausedbybearingfrictionandunbalanced ywheels,andcontrolcomplexity.Furthermore,momentumactuatorscansaturate afteracertaintimeperiodandrequireexternaltorqueactuatorstoreturnthemtotheir nominalstate.Forinstance,consideraspacecraftwithreactionwheelssubjectto environmentaltorques.Topreservetheattitudeofthespacecraft,thereactionwheels spinuptoabsorbtheangularmomentumofthespacecraft.Afteracertainperiodof time,thereactionwheelsmayreachtheirmaximumoperatingspeedandcannolonger controlthespacecraftattitude.Insuchcases,externaltorqueactuatorssuchasa thrustermustbeusedtoimparttorqueonthespacecraftandreturnthereactionwheels totheirnominaloperatingconditions.Thisoperationiscalledmomentumdesaturation ormomentumdumping.Allspacecraftusingmomentumactuatorsthereforehavesome additionalexternaltorqueactuatortoprovidedesaturationcapability. 47

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Figure2-1.AxisAngleRepresentation[61] Figure2-2.RigidBody 48

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Figure2-3.ActiveAttitudeControlBlockDiagram Figure2-4.AttitudeDeterminationandControlSystemErrors 49

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CHAPTER3 CONTROLMOMENTGYROSCOPES ControlmomentgyroscopesCMGhavebeenresearchedforutilityinspacecraft attitudecontrolsincethe1960s[10]andwereimplementedinspacecraftbeginningin the1970sSkylabin1973[79],USmilitaryobservationsatellitesKH-11andKH-12 in1976,andtheMIRspacestationin1986[10].Theyhavealsobeensuccessfully implementedontheInternationalSpaceStation[11,12].Therstreportedcommercial useoftheCMGwasontherecentEarthobservationspacecraft Worldview-1 [13]. CMGsfunctionasattitudeactuatorsbyeffectinginternalangularmomentumexchange. AgenericCMGconsistsofaspinningywheelthatisgimbaledaboutoneormore axes.TheangularmomentumoftheCMGischangedbyreorientingtheywheel spinaxisortheangularmomentumvectorbygimbalrotation.Thechangeinthe angularmomentumoftheCMGcausesthespacecrafttorotatesuchthatthenetinertial angularmomentumisconserved.Therateatwhichtheangularmomentumvectorof theywheelisreorientedisproportionaltothemagnitudeoftorqueimpartedonthe spacecraft.InsomeCMGs,theywheelspeedisalsousedasacontrolparameter tochangetheangularmomentum.Thus,CMGscaneffectbothchangeindirection andmagnitudeoftheangularmomentumvector.SeveralCMGvariantsexistbased onthemethodinwhichthechangeoftheangularmomentumvectoriseffectedsee Section3.1.Incomparison,reactionwheelschangetheirangularmomentumonly bychangingtheirmagnitude.Sincereorientingtheangularmomentumvectorofa setofactuatorstoachieveadesirednetmomentumstateisfasterthanchanging theirmagnitudetoachievethesamestateasinareactionwheel,theCMGsare capableofrapidlyreorientingthespacecraft.Further,sincethedirectionofthe angularmomentumvectoriscontrolledmorepreciselythanitsmagnitude,CMGs alsoenableprecisionpointingofspacecraft[10,15,16].CMGsexhibitlowerspecic massmassofCMGperunittorqueoutputandlowerspecicpowerpowerperunit 50

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torqueoutputconsumptioncomparedtoreactionwheelswiththesametorqueand angularmomentumcapacity[27,66,67].Fromavibrationstandpoint,sinceCMGs operatewithaconstantywheelspeed,thedisturbancecausedbyywheelimbalance ismoremanageable[26]comparedtoreactionwheelswhichhavealargervibration spectrum[17].Despitetheseadvantages,CMGshavebeenseldomusedduetotheir controlcomplexities,andaconditioncalledsingularityinwhichacongurationofCMGs cannotproducetorquealongagivendirection.AlthoughCMGsingularitiesarenot thefocusofthisdissertation,abriefdiscussiononsingularityisprovidedlaterinthis chapterforcompleteness.CMGvariantsarebrieydiscussedinSection3.1.Thebasic andmostwidelyusedvariantviz.thesingle-gimbalCMGSGCMGisconsidered inthisdissertationandisdiscussedindetailinSection3.2.PropertiesofSGCMGs, mechanicalconstructionofatypicalSGCMGanddevelopmentoftheSGCMGdynamics aredescribed.Abriefdiscussiononactuatorimperfectionsthatleadtoattitudeerrorsis alsoprovided. 3.1ControlMomentGyroscopeVariants ThebasicandmostwidelyadoptedvariantofCMGsisthesingle-gimbalcontrol momentgyroscopeorSGCMG.AnSGCMGconsistsofaywheelspinningata constantnominalspeedandisgimbaledaboutasingleaxis.Thedecienciesof anSGCMGactuatoranditscontributiontoattitudeerroristhemainfocusofthis dissertation.TheremainingvariantsoftheCMGarerstdiscussedbrieyandthe SGCMGisdealtwithindetailinSection3.2. 3.1.1Double-GimbalControlMomentGyroscope Adouble-gimbalcontrolmomentgyroscopeDGCMGconsistsofywheelspinning withconstantnominalangularvelocitythatismountedontwoorthogonalgimbalswith onegimbalnestedinsidetheother.AschematicofaDGCMGandapictureofthe DGCMGusedontheSkylabareshowninFigure3-2.ADGCMGthushastwodegrees offreedomforoutputtorquecontrol.Theratechangeofangularmomentumortorque 51

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outputofaDGCMGisgivenby d h dt = 1 h + 2 )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(J 1 1 + h where, h istheangularmomentumofthespinningywheel, J 1 istheinertiadyadic oftheinnergimbalandtheywheel,andthegimbalratesgivenby 1 = 1 g 1 and 2 = 2 g 2 arethecontrolparameters.WhenanyoneofthegimbalaxesoftheDGCMG alignswiththeangularmomentumvector,itlosesacontroldegreeoffreedomand cannotproduceanytorquealongthatdirection.Thisconditionisknownasgimbal lock.ItcanbeobservedfromFigure3-2Athataportionofthetorqueproduceddue tothemotionofonegimbalhastoberesistedbytheotherinordertoapplytorqueon tothespacecraft.ThegimbalmotorsofaDGCMGarethereforerequiredtohavea highertorquecapability,andconsequentlyconsumemorepowerrelativetotheSGCMG gimbalmotors.AsetofthreeDGCMGsweresuccessfullyemployedintheSkylab attitudecontrolsystem[8,9].MethodstocontrolDGCMGsandtoavoidgimballock havealsobeenresearched[7,68,69].IncomparisonwithanSGCMG,themechanical constructionoftheDGCMGismorecomplex. 3.1.2VariableSpeedControlMomentGyroscope VariablespeedcontrolmomentgyroscopesVSCMGrepresentanothermethod ofprovidinganadditionalcontroldegreeoffreedomoverthebasicSGCMG.AVSCMG isexactlysimilarinconstructiontoanSGCMGbuttherotorspeedinaVSCMGis notconstantandcanbevariedsimilartoareactionwheel.Thus,inadditiontothe gyroscopictorqueoutput,areactiontorqueisalsoproducedbytheangularacceleration oftheywheel/rotor.Theratechangeofangularmomentumoroutputtorqueofa VSCMGisthereforegivenby d h dt = I + g h where, istheangularaccelerationoftheywheeland I representstheinertiadyadic oftheywheel.Thegimbalangularvelocity andtheywheelangularacceleration 52

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arethereforethecontrolparametersoftheVSCMG.TheconceptofVSCMGs wasintroducedbyFordandHall[70]whichwerethencalledgimbaledmomentum wheels.Theadditionaldegreeoffreedomgainedbyywheelaccelerationprovides acapabilitythatmayhelpescapesingularcongurations.Asingularityanalysisof VSCMGscanbefoundin[70].Similartoareactionwheel,thevariablespeedofa VSCMGywheelintroducesawidespectrumofvibrationsthataffectthepointing accuracyofthespacecraft.Thetorqueoutputcapacityalongtheadditionaldegreeof freedomissignicantlylesscomparedtothegyroscopictorque.Itisthereforedesirable tominimizetheuseoftheadditionaldegreeoffreedom,andseveralpublications investigatecontrolmethodsforVSCMGsthatcanachievethisgoal[7175].Another importantutilityofVSCMGsisywheelenergystorage.Suchasystemisreferredto inliteratureasintegratedpowerandattitudecontrolsystemIPACS.Algorithmsthat intelligentlymanagepowerandmomentumdistributiontoachievepre-denedobjectives havealsobeendeveloped[7680].RealizationofanIPACSsystemrequiresadditional technologiessuchasmagneticbearingstominimizefrictionlossinthebearings[81]. 3.1.3Scissored-PairControlMomentGyroscope Ascissored-paircontrolmomentgyroscopeconsistsoftwoidenticalSGCMGs whoseywheelsspinatequalandconstantnominalangularvelocities.Thegimbalsof thetwoSGCMGsaremechanicallycoupledsuchthatthegimbalrotationofoneCMGis theoppositeoftheotherasshowninFigure3-3.Thiswillconstrainthetorqueoutputof thesystemtoliealongastraightlinesimilartoareactionwheel.Thisconceptwasrst introducedforspacecraftattitudecontrolbyCrenshaw[82].Bymechanicallycoupling thetwogimbals,therequirementoftwogimbalmotorsisreducedtoone,andisclaimed toreducepowerconsumption[83].ThreeorthogonalpairsofsuchCMGscanbeused toprovidethree-axiscontrol.However,itcanbeseenfromFigure3-3thatwhenthe momentumaxesofSGCMGsinapairarealigned,theycannotproduceanytorque output. 53

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3.1.4AdaptiveSkewControlMomentGyroscope TheinclinationofthegimbalaxisofanSGCMGinthespacecraftbody-xed coordinatebasisiscalledtheskewangleoftheSGCMG.Traditionally,theskew-angle isconstanti.e.,theSGCMGisrigidlyxedtothespacecraft.Recently,Kojima,et al.[84,85]havestudiedSGCMGswithvariableadaptiveskewangles.SuchaCMG isreferredtoasanadaptiveskewcontrolmomentgyroscopeASCMG.Apyramidal arrangementoffourASCMGsisshowninFigure3-4.SimilartotheVSCMGand DSCMGdiscussedearlier,theASCMGalsohastwodegreesoffreedomusable foroutputtorquecontrol.Simulationsperformedonapyramidalclusteroffoursuch ASCMGshaveshowntoreducesettlingtime,butthebenetoftheadditionaldegree offreedominsingularityavoidancehasnotbeendemonstrated[85].Thetorqueoutput equationoftheASCMGissimilartothatoftheDGCMGgivenbyEq.3. 3.2Single-GimbalControlMomentGyroscope ThebasicformofaCMGisthesingle-gimbalcontrolmomentgyroscopeor SGCMG 1 .TherstcommercialspacecrafttouseCMGswasthe WorldView-1 spacecraftdevelopedby DigitalGlobe [13]in2007.Considertheschematicofan idealCMGshowninFigure3-1A.Itconsistsofahomogeneousywheelspinning ataconstantnominalspeed,andattachedtoasinglerotatinggimbal.Theangular momentumvectorofthespinningywheeldenotedby h isalongthespinaxis,andis ofaconstantmagnitude.Theywheelismountedonthegimbalsuchthattheangular momentumvectorisorthogonaltotheaxisofrotationofthegimbaldenotedby g .For aCMGwithoutlimits/constraintsontherangeofgimbalmotion,theangularmomentum trajectoryforacompleterotationofthegimbaltracesacirclewithradiusequalto k h k asshowninFigure3-1B.Sincetherotorspeedremainsconstant,thechangeinthe 1 Unlessotherwisementioned,SGCMGswillbereferredtoasCMGsforthe remainderofthedissertation. 54

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angularmomentumispossibleonlybygimbalrotation.Whenachangeintheangular momentumoftheCMGiseffectedbygimbalmotion,therotorimpartsatorqueonthe gimbalsacrossitssupportsbearingsinanefforttoconservetheangularmomentum. Themagnitudeoftorqueisequaltotheproductofthemagnitudesoftheangular momentumandthegimbalspeed ,andthedirectionisdependentontheorientation ofthegimbal .Whenthegimbalisxedinaspacecraft,thistorquecausesarotation ofthespacecraftsuchthattotalinertialangularmomentumspacecraft+CMGis conserved.ItcanbeseenfromFigure3-1Bthatthechangeintheangularmomentum d h dt isperpendiculartoboththeangularmomentumdirectionandthegimbalaxis.This change,equaltothetorqueoutputoftheCMGistangenttotheangularmomentum spacecircleatthepointcoincidingwiththetipoftheangularmomentumvector. Thetorqueandtheangularmomentumthereforelieinaplanenormaltothegimbal axis.Consideracoordinatebasisformedbytheinitialangularmomentumvector h thegimbalaxis g ,andanothervectorthatcompletestherighthandedbasis.This coordinatebasisisxedwithrespecttothespacecraftandisreferredtoasthegimbal coordinatebasis.ThemeasuresoftheCMGparameterssuchasangularmomentum andtorqueoutputareexpressedinthisbasis.Thus,whenexpressedinthegimbal coordinatebasis,theratechangeofangularmomentumoftheCMGrelativetothe spacecraftortorqueoutputoftheCMGisgivenby d h dt = @ h @ = h where isequalto g .Thedependenceoftheoutputtorqueonthegimbalangle isembeddedintheterm @ h @ .SincethetorqueoutputofasingleCMGspansatwo dimensionalspace,acombinationoftwoormoreCMGsisrequiredforproducingthree dimensionaltorque. 55

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3.2.1CongurationsofMultipleCMGs ACMGcongurationisidentiedbythenumberofCMGsandtheorientationof theCMGsinaspacecraft.TheorientationofaCMGisdenedbytheinclinationof itsgimbalaxisinthespacecraftbodycoordinatebasis.Severalcongurationscanbe formedbyvaryingtheorientationofthegimbalaxes.Someofthesecongurationsare showninFigure3-5where,thebluearrowsindicatethegimbalaxes.Themostcommon congurationfoundinliteratureisthepyramidalcongurationshowninFigure3-5A. ThegimbalaxesoftheCMGsinthiscongurationarenormaltotheinclinedplanesofa pyramid.TheboxcongurationhasfourCMGswiththeirgimbalaxesalongfouredges ofaboxasshowninFigure3-5B.TherooftopcongurationhasCMGsinmultiplesof twosuchthateachpairofCMGshavetheiraxesontheinclinesofaroofasshown inFigure3-5C.Thetotalangularmomentumof n CMGsisgivenbythesumofthe individualangularmomentumoftheCMGsas h = n X i =1 h i SincetheangularmomentumofeachCMGisdependentonitsgimbalorientation,we canwritethetotalangularmomentumasgivenby h = n X i =1 h i i Asdiscussedpreviously,theangularmomentumspaceofasingleCMGisrepresented byacircleandtheangularmomentumisthesameforgimbalorientations and +2 .Thusthecongurationspaceofthegimbalanglescanberepresentedbyan n -dimensionaltorus.Theangularmomentumspace H of n CMGsobtainedbysumming theangularmomentumvectorsofallCMGsforeverypossiblegimbalorientation,can thereforebeseenasavectorvaluedmappingbetweenthe n -dimensionalconguration space D n ofthegimbalangles i tothe3DEuclideanspace. 56

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3.2.1.1SingularitiesinCMGs Adiscussiononthesingularities,althoughnotthemainfocusofthisdissertation, isprovidedforcompletenessasattitudeerrorsareinevitablewhenasetofCMGs transitionthroughasingularconguration.Considerthevectorvaluedmapfromthe gimbalanglecongurationspace D n totheangularmomentumspace H .Forany arbitrarydirection u ,thereexistsasetofgimbalanglesapointin D n suchthatthe projectionofthetotalangularmomentumalong u isamaximum.Similarlythereexists anothersetofgimbalanglesforwhichtheprojectionofthetotalangularmomentum along u isaminimum.Themaximumprojectionsoftheangularmomentumofthe n CMGsalongeverydirectioninthe3DEuclideanspaceformanenvelopeknownasthe externalmomentumenvelopeandtheminimalprojectionsformtheinternalmomentum envelope[14,8688].Now,consideracongurationofgimbalanglessuchthatthe totalangularmomentumofthe n CMGsisapointontheexternalmomentumenvelope. Therecannotbeanychangeintheangularmomentumalongtheoutwardnormalto theangularmomentumenvelopeatthispoint.Similarly,iftheangularmomentumstate wasapointontheinternalmomentumenvelope,therecannotbeanychangeinthe angularmomentumalongtheinwardnormal.Thissituationiscalledsingularity,andin thiscongurationofthegimbalangles,theCMGscannotproducetorquechangeinthe angularmomentumalongaparticulardirectioncalledthesingulardirection[14,8691]. Thegimbalanglecongurationinthissituationiscalledasingularconguration. UsingEq.3,theratechangeofthetotalangularmomentumrelativetothe spacecraftisgivenby d h dt = n X i =1 @ h i @ i d i dt whichcanbewritteninmatrixformas d h dt = J 57

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where, J isthe3 nJacobian ,and isthe n 1vectorofthegimbalrates.The Jacobian isafunctionofthegimbalangles i andrepresentsamapfromthe n 1 gimbalratevectortothe3 1outputtorquevector.The Jacobian hastobefullrank rank3inordertoproducetorquealonganyarbitrarydirectionin3D Euclidean space. Whenthegimbalanglesareinasingularconguration,therankofthe Jacobian isless than3andthereforecannotproducetorquealongthesingulardirection.Thesingular directionisorthogonaltoallthecolumnsinthe Jacobian Despitetheirseveralbenets,CMGshavebeenseldomusedinthepastdue tocontrolcomplexitiescausedbysingularities.This,hashoweverattractedalot ofresearchinthedevelopmentofmethodstoescapesingularcongurationswith minimaldisturbancetothespacecraft'sattitude.Severalmethodsincludingmechanism modications[79,6875,8285]andcontrolstrategies[66,89,9295]toovercome theeffectofsingularityhavebeendeveloped. 3.2.2MechanicalConstructionofaTypicalCMG ThecomponentsofatypicalCMGarediscussedinthissection.Thediscussion mayprovideaninsightintothesourceofvariousdecienciesintheCMGcomponents andcongurationthatmayaffectitsperformance.Thevariouscomponentsare illustratedusingtheassembledandexplodedviewsofasingleunitoftheintegrated miniaturepyramidalarrangementofCMGs IMPAC showninFigure3-6andFigure3-7, respectively.Thedesignofthe IMPAC isbasedoncertainrequirementsanddesign constraintsofaspecicmission[24].However,thecongurationandcomponentsare similartothatofagenericCMG. ACMGconsistsofaywheelthatisdesignedtomaximizeitsinertiatomassratio. Theywheelissupportedusingasetofbearingsinastructurecalledtheywheel housing.Thebearingsaredesignedtominimizefrictionandprovidesufcientsupport stiffnesstotheywheel.Theywheelhousingandotherstructuralcomponentsare manufacturedusingamaterialwithhighstiffnesstomassratio.Theywheelhousing 58

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alsocontainstheywheelmotorthatisusedtospintheywheel.Theywheelmotor istypicallyabrushlessmotor.Theywheelmotorcontrolelectronicsaresometimes accommodatedontheywheelhousing.Theywheelhousing,alongwiththeywheel, bearings,motor,andcontrolelectronicsconstitutetheywheelassembly.Theywheel assemblyisgimbaledonasetofbearingslocatedonthegimbalhousing.CMGsare designedsuchthatthechangeintheinertiaoftheywheelassemblyexpressedin thespacecraftbodycoordinatebasisduetogimbalmotionisminimum[27].Agimbal motorwhoserotaryportionisattachedtotheywheelhousingcontrolsthegimbal motion.Thegimbalassemblymaysometimescontainmechanicalgearingbetweenthe gimbalmotorandtheywheelhousing.Arotarypositionsensorsuchasanencoder isattachedtothegimbaltoprovideitsangularpositionrelativetothegimbalhousing. Sincethegimbalmotionisunconstrained,aslipringthatprovidesanuninterrupted powerandcommunicationpassagetotheywheelmotoranditselectronicsisused. Thegimbalbearings,mechanicalgears,andslipringsofferfrictionalresistanceto gimbalmotion.Theeffectofdecienciesinthecomponentsontheperformanceofthe CMGarediscussedbelow. 3.2.2.1MechanismimperfectionsinCMGs Precisionpointingofaspacecraftrequiresanattitudecontrolsysteminwhichthe torqueoutputoftheactuatorisexactlyasdesiredbytheattitudecontrolalgorithm. However,inreality,certainmechanismimperfectionsexistinattitudeactuatorsthat inducedisturbancestotheoutputtorque.Theseimperfectionsmaybeduetolimitations intheactuatormanufacturingprocess,efciencyofthecomponents,orduetoexternal forcese.g.,launchloadsthatmaycompromisetheperformanceofanotherwise accurateactuator.InaCMG,anunbalancedywheelandfrictionduetobearingsand slipringsaretwosuchimperfectionsthatleadtopointingerrors.Additionally,uncertainty insomeoftheactuatorparametersrequiredbytheattitudecontrolalgorithmmayalso 59

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introduceattitudeerror.Forinstance,theuncertaintyintheorientationofthegimbal leadstoerrorsintheactuatoroutputtorque. Flywheelimbalancecanresultfromnon-homogenousmaterials,impreciseywheel geometryanderrorsintheywheelbalancingprocess.Theimbalanceinthespinning ywheelcreatesauctuationinthedirectionofangularmomentumthatleadstoattitude jitterofthespacecraft. Flywheelandgimbalbearings,andslipringsaredesignedtoofferlowfrictional resistance.Thereishoweverasmallamountofresidualfrictionthatoffersresistance tothemotionoftheywheelandgimbals.Thefrictionaltorqueisinternal,andhence doesnotdirectlyaffecttheattitude.Nevertheless,duetheirunpredictablenature,friction causesdisturbancestosmoothattitudecontrolthatleadtopointingerrors.Frictional lossesalsoincreasethepowerconsumptionoftheattitudeactuator. AttitudecontrolusingCMGsrequireanaccurateknowledgeofthegimbal'sangular position.Misalignmentbetweenthegimbalandthemechanism'srotaryfeedbacksensor e.g.,encodermaybecausedduringassemblyandlaunch.Additionallyfaultysensors andsensorfailuremayleadtototalambiguityinthegimbal'sangularposition.Sincethe outputtorqueoftheactuatordependsonthegimbal'sangularposition,anuncertaintyin itsknowledgecancausetheactuatoroutputtobeerroneous. Attitudeerrorscausedbyywheelimbalanceandambiguityinthegimbal orientationareconsideredinthisdissertation.Aknowledgeofthedynamicsofa spacecraftwithaCMGisrequiredinordertoanalyzetheeffectsofthesedeciencies anddevelopmitigationstrategies.Thesameisdiscussedinthefollowingsection. 3.2.3DynamicsofaSpacecraftwithaCMG TheequationsofmotionofaspacecraftwithasingleCMGaredeveloped.The dynamiccharacteristics/behaviorofaspacecraftwithasingleCMGsufcesthe needsofthedevelopmentsgiveninthisdissertation.ConsideringmultipleCMGs doesnotprovideanyadditionalinformation,whileconsideringasingleCMGprovides 60

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moreclarityandperhapsabetterinsightintothedynamiceffects.Theywheelof theCMGisassumedtobehomogeneousandperfectlybalancedforthecurrent discussion.TheeffectofanunbalancedywheelisdiscussedindetailinChapter6. Further,thespinaxisoftheywheelandthegimbalaxisareassumedtobeperfectly orthogonal.Consideraspacecraftwithits c.m. at C asshowninFigure3-8.Let C B bethespacecraftbodyxedcoordinatebasisandlet betheangularvelocityofthe spacecraft.AsingleCMGsimilartotheoneshowninFigure3-1Aislocatedinthe spacecraft.Considerapoint G whichislocatedattheintersectionoftheywheelspin axisandthegimbalaxis.Thepositionof G relativeto O isdenotedby r g .Acoordinate basis C G calledthegimbalcoordinatebasisisattachedtothespacecraftsuchthat Z G isalongthegimbalaxis.Asecondcoordinatebasis C F calledtheywheelcoordinate basisisattachedtothegimbalsuchthat X F isalongthespinaxisoftheywheeland Z F isalongthegimbalaxis.Theangularvelocityoftheywheelrelativetothegimbal about X F isdenotedby .Let O bethe c.m. oftheywheelwithmass m w .Theposition of O relativeto G isdenotedby r o .Itshouldbenotedthat r o isalongthetheywheel spinaxisandisthereforeperpendiculartothegimbalaxis.Theangulardisplacement betweentheywheelcoordinatebasisandthegimbalcoordinatebasisaboutthe Z G isdenotedby .Theangle iscalledthegimbalangleorthegimbalorientation.The angularmomentumofthespacecraftstructureexcludingonlytheCMGrotaryparts about C isgivenby h s c = I s c Now,considerasmalldifferentialmass dm oftheywheellocatedatadistance r from O .Thevelocityofthedifferentialmassrelativeto C isgivenby v = r g + )]TJ/F46 11.9552 Tf 5.48 -9.684 Td [(! + r o + )]TJ/F46 11.9552 Tf 5.479 -9.684 Td [(! + + r where, = Z G istheangularvelocityofthegimbalrelativetothespacecraft,and = X F istheangularvelocityoftheywheelrelativetothegimbal.Theangular 61

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momentumoftheCMGabout C isgivenby h cmg c = Z )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g + r o + r v dm whichcanberewrittenas h cmg c = I w o + I w o + I w o + I g g + r g m w )]TJ/F58 11.9552 Tf 7.603 -7.742 Td [( r o + m w \000 r g + r o )]TJ/F65 11.9552 Tf 5.48 -9.683 Td [(r g + r o 1 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F65 11.9552 Tf 5.48 -9.683 Td [(r g + r o )]TJ/F65 11.9552 Tf 12.952 -9.683 Td [(r g + r o where,theinertiadyadicoftheywheel I w o andtheinertiadyadicofthegimbal I g g are givenby I w o = Z )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r r 1 )]TJ/F65 11.9552 Tf 11.955 0 Td [(rr dm I g g = m w )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r o r o 1 )]TJ/F65 11.9552 Tf 11.955 0 Td [(r o r o InderivingEq.3,theywheelhousingandotherelementssuchastheslipring attachedtothegimbalseeFigure3-7areassumedtobemassless.Includingthe massoftheseelementswouldonlychangethevalueof I g g and J s c ,anddoesnotchange thestructureoftheequation.Thetotalangularmomentumofthespacecraftaboutits c.m. isthereforegivenbythesumofEq.3andEq.3as h c = J s c + I w o + I w o + I g g + r g m w )]TJ/F58 11.9552 Tf 7.603 -7.741 Td [( r o where,theinertiadyadic J s c isgivenby J s c = I s c + I w o + m w \000 r g + r o )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(r g + r o 1 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F65 11.9552 Tf 5.48 -9.683 Td [(r g + r o )]TJ/F65 11.9552 Tf 12.952 -9.683 Td [(r g + r o ItshouldbenotedthattheinertiadyadicinEq.3istimevaryingasitdependsonthe timevaryingvector r o .Consideringtheassumptionsofabalancedywheelandthe orthogonalityoftheywheelandgimbalaxes,thetimeratechangeofthespacecraft 62

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angularmomentumabout C isgivenby h c = J s c + J s c + I g g + )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(I w o + I g g + I w o + r g m w )]TJ/F58 11.9552 Tf 7.603 -7.742 Td [( r o + h c + I w o + )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g m w )]TJ/F58 11.9552 Tf 7.603 -7.742 Td [( r o Thetimeratechangeoftheinertiadyadic J s c isgivenby J s c = )]TJ/F58 11.9552 Tf 7.603 -7.742 Td [( I w o )]TJ/F65 11.9552 Tf 11.955 0 Td [(I w o + r o )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g + r o 1 )]TJ/F58 11.9552 Tf 12.888 0.167 Td [( r o )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(r g + r o + )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g + r o r o 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g + r o r o andthetimerateofchangeoftheinertiadyadicisgivenby I g g I g g = I g g )]TJ/F65 11.9552 Tf 11.955 0 Td [(I g g where, r o = r o .However,inreality,aCMGisusuallydesignedsuchthatthequantity r o isminimumandthechangeintheinertiaduetogimbalmotionisnegligible[27]. Thustheinertiaoftheywheelasexpressedintheywheelcoordinatebasisisalmost equaltotheinertiaoftheywheelasexpressedinthespacecraftbodycoordinatebasis. Iftheeffectsof r o andchangeininertiaduetogimbalmotiononthedynamicsofthe spacecraftareneglected,thenEq.3canberewrittenas h c = J s c + I w o )]TJ/F58 11.9552 Tf 7.602 -7.742 Td [( + + h c + I w o where,nowthetimeinvariantinertiadyadic J s c isgivenby J s c = I s c + I w o + m w )]TJ/F65 11.9552 Tf 5.479 -9.684 Td [(r g r g 1 )]TJ/F65 11.9552 Tf 11.955 0 Td [(r g r g ThesystemshowninFigure3-8hasverotationaldegreesoffreedomandishence completelyrepresentedbyasetofveequationsofmotion.Therstthreeequations aregivenbyEq.3whichdescribesthe3Drigidbodymotionofthespacecraft.The othertwoequationspertaintothemotionoftheywheelandthemotionofthegimbal. Theseequationsaredevelopedbelow.Theangularmomentumoftheywheelaboutthe 63

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point O isgivenby h w o = I w o )]TJ/F61 11.9552 Tf 5.48 -9.684 Td [( + + Theratechangeoftheangularmomentum, h w o alongthe X F directionisequaltothe torqueattheinterfaceoftheywheelandtheywheelhousing[61].Itcanbeshown thatthisscalarequationisgivenby )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(I w o + I w o X F = w wherethetorque w includesfrictionandmotortorque.Equation3providesa differentialequationforthequantity .Theangularmomentumofthegimbalaboutthe point G isgivenby h g g = I g g )]TJ/F58 11.9552 Tf 7.602 -7.741 Td [( + + I w o )]TJ/F58 11.9552 Tf 8.238 -7.742 Td [( + Theratechangeofangularmomentum, h g g alongthegimbalaxis Z G isequaltothe torqueattheinterfaceofthegimbalandthegimbalhousing.Itcanbeshownthatthis scalarequationisgivenby h g g Z G = )]TJ/F65 11.9552 Tf 5.48 -9.684 Td [(I g g + I g g + I w o + I w o Z G = g wherethetorque g includesthetorqueduetogimbalmotor,gimbalbearingfriction,and slipringfriction.Equation3providesadifferentialequationforthequantity .The completesetofequationsrequiredtodescribetheattitudemotionofthespacecraft isgivenbyEqs.3,3,and3.Itisrecalledthattheeffectofimbalancein theywheelisnotcapturedinthedynamicequationsdevelopedaboveasitwillbe consideredexplicitlyinChapter6. Themajorcontributionsofthedissertationarediscussedinthefollowingchapters. 64

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ASchematic BAngularMomentumSpace Figure3-1.Single-GimbalControlMomentGyroscope ASchematic BSkylabDGCMG Figure3-2.Double-GimbalControlMomentGyroscope 65

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Figure3-3.ScissorPairControlMomentGyroscopeSchematic Figure3-4.AdaptiveSkewControlMomentGyroscope[85] A Pyramid B Box C Rooftop Figure3-5.ControlMomentGyroscopeCMGCongurations 66

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ACADModel BPrototype Figure3-6. IMPAC Figure3-7.ExplodedViewofthe IMPAC 67

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Figure3-8.RigidSpacecraftwithCMG 68

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CHAPTER4 ESTIMATIONOFSPACECRAFTANGULARACCELERATION Knowledgeoftheangularaccelerationofthespacecraftisrequiredforthe estimationofgimbalorientationofacontrolmomentgyroscopeCMGaspresented inthisdissertation.Thischapterdiscussesamethodtoestimatebias-freespacecraft angularaccelerationusinglinearaccelerationmeasurements. 4.1Background On-orbitestimationofspacecraftpropertiessuchasinertia,actuatoralignment, etc.isessentialforaccurateattitudecontrol.Therehasbeenaconcertedeffortto developsuchestimationtechniques[4850,96,97].Thecapabilitytoobtainangular accelerationestimatesmaycomplementsuchestimationtechniques.Considerthe systemofsecond-orderdifferentialequationsknownasEuler'sequationsthatgovern theattitudemotionofarigidbodyasgivenbyEq.2.Thenetexternaltorque ext c onthespacecraftisassumedtobeknownornegligible.Mostspacecrafttodayare equippedwithgyroscopesthatprovidemeasurementsofthespacecraftangularvelocity. Itisstraightforwardtoobservethatif,inadditiontotheangularvelocity,measurements ofangularaccelerationareavailable,onecouldventureintodeterminingsomeofthe unknownparameterssuchastheactuatoralignmentrepresentedbythetransformation matrix R BA inEq.2.Unlikethecaseoflinearacceleration,whereaccelerometers canbeusedfordirectmeasurement,therearenosensorsthatcandirectlymeasure angularaccelerationalthoughtherehavebeensomeeffortsinthepasttodevelop suchsensors[98,99].Thesemethodshavenotbeensuccessfulasthesensorcannot distinguishbetweeneffectscausedbylinearandangularaccelerations.Adirectand bruteforcemethodtodeterminetheangularaccelerationofarigidbodyspacecraftis todifferentiatetheangularvelocitymeasurements.Thisapproachhowever,amplies signalnoiseandishenceseldomused.Anothermethodtodeterminetheangular accelerationofarigidbodyisbymeasuringlinearaccelerationatvariouspoints 69

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inthebody.TherstpublicationsofthismethodwerebyCorey[100]in1962,and bySchuleretal.[101]in1967,andwereseekingsolutionsforagyro-freeinertial navigationsystem.Themethodhashencebeenpursuedinvariousdisciplines suchasbiomechanics[102,103],robotics[104],projectileguidance[105],vibration control[106],andinertialnavigationsystems[107].However,nopublicationshavebeen foundthatstudytheuseofaccelerometersspecicallyformeasurementofspacecraft angularacceleration. Determinationofangularaccelerationusinglinearaccelerometersdependson thenumber,location,andmeasurementaxisoftheaccelerometers.Weshallreferto aparticularcombinationoftheseparametersasaconguration.Themathematical structureofthesolutionchangesbasedonthecongurations.Ithasbeenshownthat theminimumnumberoflinearaccelerationmeasurementsrequiredtodetermineboth angularvelocityandaccelerationofarigidbodyisnine[108110].Chenetal.[107] determineangularaccelerationonlyusingsixlinearaccelerationmeasurementsandthe angularvelocityisfoundbyintegration.Theirmethodhoweverrequirestheinitialangular velocity ,andplacescertainrestrictionsontheplacementandorientationofthe accelerometerswhichmaybeunsuitableforimplementinginaspacecraft. Thischapterpresentsamethodtouniquelydeterminethespacecraftangular accelerationusinglinearaccelerationmeasurementsfromsixuniaxialaccelerometers. Theeffectofsensornoiseandbiasontheangularaccelerationestimatesareevaluated. Thecaseofnineaccelerationmeasurementsfromthreetriaxialaccelerometers whichhasbeenpreviouslystudied[109]isalsodiscussed,andanexplanationforthe preferenceoftheuniaxialcongurationisgiven.Angularvelocitymeasurementsfroma gyroscopeareassumedtobeavailableinthedevelopmentofthismethod.Thesolution isfurtherenhancedbyincludingaKalmanltertoidentifytheeffectivebiasinthe angularaccelerationmeasurements.Inadditiontoestimatingthebias,theKalmanlter alsoproducessmoothedangularvelocityestimates.Simulationsareperformedusing 70

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datafromcommerciallyavailablesensorstoshowtheeffectivenessandimplementability ofthesolutionmethod.Theremainderofthechapterisorganizedasfollows.First,the kinematicprinciplesofobtainingangularaccelerationfromuniaxialaccelerometersand triaxialaccelerometersarediscussedintheSection4.2.Theeffectofgyroscopeand accelerometersensornoiseandbiasonthederivedangularaccelerationestimates isthendiscussedinSection4.3.AlinearKalmanlterforestimatingthesensorbias andobtainingsmoothedangularvelocityestimatesispresented.Simulationresults thatverifythecapabilityofthemethodarepresentedinSection4.4.Theangular accelerationthusestimatedisusedinamethodtodeterminetheunknownorientationof aCMGgimbalinChapter5. 4.2Methodology Theapproachforestimatingtheangularaccelerationofarigidspacecraftusing linearaccelerometersisdescribedinthissection.Itisassumedthattheangular velocitymeasurementswhichcanbeobtainedusingsensorssuchasagyroscopeare available.Linearaccelerationisapointpropertywhileangularaccelerationisaproperty oftherigidbody.Thelinearaccelerationofapoint i onarigidbodyexpressedinthe spacecraftbodyxedcoordinatebasisisgivenby a i = a o + [ ] r i + [ ] [ ] r i where, a o istheaccelerationofanarbitraryreferencepoint O inthespacecraftand r i isthepositionofpoint i from O .Adifferenceequationwhichisindependentofthe measurement a o givenby a ij = )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(a i )]TJ/F58 11.9552 Tf 12.453 0 Td [(a j = [ ] r ij + [ ] [ ] r ij isusedtodeterminetheangularacceleration.Thepoints i and j areanytwoarbitrary pointsontherigidspacecraft,and r ij isthepositionofpoint i relativeto j .Itcanbeseen fromEq.4thatlinearaccelerometermeasurementsatmultiplepoints,together 71

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withtheangularvelocitymeasurementscanbeusedtodeterminetheunknown angularacceleration.Commerciallyavailableaccelerometersarefoundintwotypes viz.uniaxialandtriaxialaccelerometers.Uniaxialaccelerometersmeasurethelinear accelerationofapointalongasingledirection.Theoutputofalinearaccelerometerat apoint i ishenceascalarvaluedenotedby a i .Atriaxialaccelerometerconsistsofan integratedassemblyofthreeorthogonaluniaxialaccelerometers.Themeasurement directionsformasetofthreeorthogonalbasisvectorsthatconstitutetheaccelerometer coordinatebasis.Theoutputofatriaxialaccelerometeratapoint i ishencethetotal linearaccelerationatthatpoint,denotedby a i ,andexpressedinacoordinatebasis denedbythethreeorthogonalmeasurementaxes.Amethodtodeterminetheangular accelerationofaspacecraftusingsixuniaxialaccelerometersisdevelopedbelow.A methodthatusesonlytriaxialaccelerometers[109]isalsodiscussedforcomparison purposes. 4.2.1AngularAccelerationUsingUniaxialAccelerometers Itisshownthatwecancompletelydeterminetheangularaccelerationofarigid spacecraftbyaspeciccongurationofsixuniaxialaccelerometers.Consideran arbitraryrigidspacecraftwithsixuniaxialaccelerometersasshowninFigure4-1. Measurementsorobservationsareexpressedusingthespacecraftbodycoordinate basis h X Y Z i .Athree-axisgyroscopeprovidesmeasurementofthespacecraft angularvelocity.Thegyroscopeaxesareassumedtobeparalleltothebodyxed coordinatebasisforsimplicity.Asetoftwouniaxialaccelerometersareplacedalong eachofthe X Y ,and Z axes.Themeasurementdirectionofeachaccelerometeris indicatedbythesolidarrowsinFigure4-1.Theaccelerometersalongthe Z axisare locatedatpoints z 1and z 2,andaredenotedby A z 1 and A z 2 ,respectively.Thepositions of A z 1 and A z 2 from O aregivenby r z 1 and r z 2 ,respectively.Theaccelerometers A z 1 and A z 2 areorientedsuchthattheymeasureonlythex-componentoftheacceleration attheirrespectivepoints.Theaccelerometersalongthe Y axisdenotedby A y 1 and A y 2 72

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areorientedsuchthattheymeasureonlythez-componentofaccelerationat y 1and y 2, respectively.Theaccelerometersalongthe X axisdenotedby A x 1 and A x 2 areoriented suchthattheymeasureonlythey-componentofaccelerationat x 1and x 2,respectively. Thereasonforthechoiceoftheaboveorientationswillbecomeclearintheensuing discussions.Theaccelerationatpoints z 1and z 2aregivenby a z 1 = a o + [ ] r z 1 + [ ] [ ] r z 1 a z 2 = a o + [ ] r z 2 + [ ] [ ] r z 2 UsingEqs.4and4,thedifferenceintheangularaccelerationatpoints z 1 and z 2 denotedby a z 12 isgivenby a z 12 = [ ] r z 12 + [ ] [ ] r z 12 Intheaboveequation,thequantitiesthataremeasuredinclude a z 12 and ,and is thequantitythatneedstobedetermined.Groupingthemeasuredquantities a z 12 and together,thedifferenceequationiscompactlywrittenasgivenby a z 12 = [ ] r z 12 where,thenonlinearterm a z 12 thatcomprisesthegyroscopeandaccelerometer measurementsisdenedas a z 12 a z 12 )]TJ/F20 11.9552 Tf 9.886 -0.147 Td [([ ] [ ] r z 12 .Theangularaccelerationcannot bedeterminedsolelyusingEq.4asthequantity r z 12 thatmaps a z 12 to isnotfull rank .Sincetheaccelerometersat z 1 and z 2 canonlymeasurethex-componentofthe angularaccelerationdenotedby a x z 1 and a x z 2 ,respectively,wecanwritex-componentof angularaccelerationsas a x z 1 = a z 1 T X a x z 2 = a z 2 T X a x z 12 = a x z 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(a x z 2 73

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WecanthereforewritetheprojectionofEq.4alongthe X axisas a x z 12 = )]TJ/F20 11.9552 Tf 5.48 -9.831 Td [([ ] r z 12 T X + )]TJ/F20 11.9552 Tf 5.48 -9.831 Td [([ ] [ ] r z 12 T X Usingvectoridentities,wendthattheprojectionyieldsthey-componentoftheangular acceleration y inthespacecraftbodycoordinatebasisasgivenby y = T Y = a x z 12 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.83 Td [([ ] [ ] r z 12 T X k r z 12 k = a z 12 T X k r z 12 k Thespecicorientationoftheaccelerometersonthe Z axispermitsthestructureof thissolution.Theothercomponentsoftheangularaccelerationinthespacecraftbody coordinatebasisaresimilarlyfound,andthetotalangularaccelerationofthespacecraft inthebodycoordinatebasisisgivenby B = 0 B B B B @ x y z 1 C C C C A = 0 B B B B B B B B B B @ a z y 12 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.83 Td [([ ] [ ] r y 12 T Z k r y 12 k a x z 12 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F20 11.9552 Tf 5.48 -9.83 Td [([ ] [ ] r z 12 T X k r z 12 k a y x 12 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.479 -9.83 Td [([ ] [ ] r x 12 T Y k r x 12 k 1 C C C C C C C C C C A = 0 B B B B B B B B B @ )]TJ/F58 11.9552 Tf 5.977 -9.683 Td [(a y 12 T Z k r y 12 k a z 12 T X k r z 12 k a x 12 T Y k r x 12 k 1 C C C C C C C C C A ItcanbeseenfromEq.4thattheaccelerometermeasurementsarecompletelyused andnopartoftheindependentmeasurementsarediscarded.Thereforeaminimumof sixuniaxialaccelerometersarerequiredfordeterminingtheangularaccelerationofa rigidspacecraftwhentheangularvelocityisknown. Itshouldbenotedthatthecongurationofsixuniaxialaccelerometersisequivalent toacombinationofonetriaxialaccelerometerlocatedatthereferencepoint,andthree uniaxialaccelerometersalongeachofthethreeaxes.This,alternatebutequivalent congurationmaybemorefeasibletoimplementonaspacecraft. 4.2.2AngularAccelerationUsingTriaxialAccelerometers Atriaxialaccelerometerconsistsofanintegratedassemblyofthreeorthogonal uniaxialaccelerometers,andtheirmeasurementdirectionsformasetofthree 74

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orthogonalbasisvectorsthatconstitutetheaccelerometercoordinatebasis.Including anarbitrarycoordinatetransformationfromtheaccelerometercoordinatebasistothe spacecraftbodycoordinatebasisdoesnotaffectthesolutionstructure,andforsimplicity, weassumethattheseaccelerometercoordinatebasesareparalleltothespacecraft bodycoordinatebasis.Considerasetofthreenon-collineartriaxialaccelerometers placedarbitrarilyinarigidspacecraftasshowninFigure4-2.Theaccelerometers arelabeledas A 1 A 2 ,and A 3 ,andareatpositions r 1 r 2 ,and r 3 from O ,respectively. Themeasurementaxesofthegyroscopesare,again,assumedtobeparalleltothe spacecraftbodycoordinatebasis.UsingEq.4,twoindependentdifferenceequations fromthreetriaxialaccelerometerscanbegeneratedasgivenby a 12 = [ ] r 12 + [ ] [ ] r 12 a 13 = [ ] r 13 + [ ] [ ] r 13 Intheaboveequations,themeasuredquantitiesinclude a 12 a 13 ,and .Similarto Eq.4,bygroupingthemeasuredquantitiesineachequationtogether,wecanwrite a 12 = [ ] r 12 a 13 = [ ] r 13 Equation4canhencebecompactlywrittenas 0 B @ a 12 a 13 1 C A | {z } a = 0 B B B B B B B B B B B B B B @ 0 r z 12 )]TJ/F59 11.9552 Tf 9.299 0 Td [(r y 12 )]TJ/F59 11.9552 Tf 9.299 0 Td [(r z 12 0 r x 12 r y 12 )]TJ/F59 11.9552 Tf 9.299 0 Td [(r x 12 0 0 r z 13 )]TJ/F59 11.9552 Tf 9.299 0 Td [(r y 13 )]TJ/F59 11.9552 Tf 9.298 0 Td [(r z 13 0 r x 13 r y 13 )]TJ/F59 11.9552 Tf 9.298 0 Td [(r x 13 0 1 C C C C C C C C C C C C C C A | {z } D 75

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Now,thematrix D whichcontainscomponentsoftherelativedistancebetweenthe accelerometershasa rank equalto3whenthepoints A 1 A 2 ,and A 3 arenon-collinear. Thus,theangularaccelerationoftherigidspacecraftcanbefoundusingthepseudo-inverse solutionasgivenby = D T D )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 D T a OnobservingEqs.4through4,itcanbeseenthataminimumofthreetriaxial accelerometersarerequiredinordertodeterminetheangularacceleration.Withtwo triaxialaccelerometers,wewouldhavejustonedifferenceequationwhichisinsufcient todeterminetheunknown .Further,thestructureofthepseudo-inverseresultsina fusionofthesensormeasurementscontainedin a whichisjustiedbythefactthatthe ninelinearaccelerationmeasurementsthreetriaxialaccelerometersdonotrepresent theminimummeasurementsetrequiredtoestimatetheangularaccelerationofarigid body. Aswithanyothersensor,theoutputonanaccelerometerisaffectedbybiasand randomnoise.Ananalysisoftheseeffectsontheestimatesoftheangularacceleration andatechniquetoobtainbias-freeangularaccelerationisdescribedinthefollowing section. 4.3EstimationofAngularAcceleration Thestochasticproperties mean and covariance oftheaccelerometerand gyroscopemeasurementnoisearetypicallyobtainedfromthemanufacturer'sdatasheet orvialaboratoryexperiments.Thesepropertieshelpinestimatingtheperformanceof thesensors,andalsoindesigningstateestimatorsandlters.Alinearmeasurement modelbasedonthesestochasticpropertiesistypicallyusedforthispurpose.Measurement modelshavebeenstandardizedforsensorssuchasgyroscopesandlinearaccelerometers. Theangularaccelerationofthespacecraftisnotdeterminedfromalinearmeasurement, butdeterminedusinganon-linearequationinvolvingmeasurementsfromlinear accelerometersandgyroscopeasgivenbyEq.4andEq.4.Wetherefore 76

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needtodeviseameasurementmodelinordertoevaluatetheperformanceofthe estimationmethod,andtohelpinthedesignofltersthatcanimprovetheaccuracyof theestimates.Westartbydescribingthemeasurementmodelsfortheaccelerometer andthegyroscope,andthenusethesemodelsinthenon-linearkinematicequations derivedintheprevioussectiontoconstructanapproximatelinearmeasurementmodel fortheangularacceleration.Usingthislinearmeasurementmodel,aKalmanlteris designedtoidentifythebiasintheangularaccelerationestimates. 4.3.1MeasurementModelforAngularAcceleration Themeasurementmodelsoftheaccelerometerandgyroscopeareconsideredrst astheyareneededtoconstructameasurementmodelfortheangularacceleration.The outputmodelofatriaxialaccelerometerisconsideredasageneralcase,andtheoutput modelofauniaxialaccelerometerwouldbeaprojectionofthetriaxialcasealongthe measurementdirection.Fortheaccelerometeratpoint i ,therelationshipbetweenthe measuredacceleration e a i andthetrueacceleration a i ismodeledas e a i = a i + a i + a i a i = a i where, a i isthebias,and a i istherandomnoiseintheaccelerometermeasurements. Thebiasrandomwalkorinstabilityisgivenby a i inEq.4where, a i isarandom noiseprocess.Therandomnoiseprocesses a i and a i areassumedtobezero-mean, Gaussian,andwhite.Further,therandomnoisebetweenanytwocomponentsof accelerometermeasurementatapoint,andbetweenanycomponentsofmeasurement atanytwopointsareassumedtobeuncorrelated.Theseassumptions,andthe covariancesoftherandomnoisevectorsaresymbolicallyrepresentedas E f a i g = 0 E f a i )]TJ/F46 11.9552 Tf 5.978 -9.683 Td [( a i T g = 2 a 1 E f a i )]TJ/F46 11.9552 Tf 5.977 -9.683 Td [( a j T g = 0 E f a i g = 0 E f a i )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( a i T g = & 2 a 1 E f a i )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( a j T g = 0 77

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where 0 isthezerovector, 0 isthezeromatrix,and a and & a arethestandarddeviations ofeachcomponentof a i and a i ,respectively.Theangularvelocityoutputofthe gyroscopeismodeledusingthemeasurementequationgivenby e = + sv where sv isarandomnoiseprocess.Notethatthespacecraftangularvelocity measurementsareassumedtobebias-freeastheycanbeidentiedusingspacecraft attitudemeasurementsinanextendedKalmanlter.Thisbiasestimationprocesshas beenwellpublishedinliterature[39,41,43,111]andhencenotrepeatedhere.The randomnoiseprocess sv ismodeledasazero-mean,Gaussian,andwhitenoise processwithcovariance E f sv )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( sv T g = 2 sv 1 where sv isthestandarddeviationof eachcomponentoftherandomvector sv ItcanbeseenfromEq.4andEq.4thatirrespectiveofthetypeofmethod used,thedeterminationoftheangularaccelerationisbasedonthequantities a ij ,which aretheonlyquantitiesthatcontainsensormeasurements.Itisthereforenecessary torstdevelopameasurementmodelfor a ij toobtainameasurementmodelforthe spacecraftangularacceleration .UsingEqs.4and4,wecanwriteEq.4as a ij = e a ij )]TJ/F46 11.9552 Tf 12.453 0 Td [( a ij )]TJ/F46 11.9552 Tf 12.454 0 Td [( a ij + [ e ] sv r ij + sv [ e ] r ij )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( sv sv r ij where,thequantities e a ij ij ,and ij aregivenby e a ij = e a i )]TJ/F28 11.9552 Tf 12.455 1.38 Td [(e a j )]TJ/F20 11.9552 Tf 11.956 -0.147 Td [([ e ] [ e ] r ij a ij = a i )]TJ/F46 11.9552 Tf 12.453 0 Td [( a j a ij = a i )]TJ/F46 11.9552 Tf 12.453 0 Td [( a j Thecovarianceofthecompositeaccelerometernoisecanbeboundedasgivenby E f a ij )]TJ/F46 11.9552 Tf 5.978 -9.683 Td [( a ij T g 4 2 a 1 78

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ItcanbeseenfromEq.4that e a ij isanon-linearfunctionofmultiplerandomnoise processes.Further,thenonlinearrandomnoisetermsalsoinvolvetheangularvelocity. Thestatisticalpropertiesofthesenonlinearrandomnoiseprocessesarethereforetime varyinganddifculttoestimate.Thereforealinearmodelfor e a ij thatapproximatesthe nonlinearmodelisderivedasfollows.ThelastterminEq.4isverysmallcompared totheothertermsintheequationandcanhencebeignored.Considertheothertwo non-lineartermsinEq.4whichcanbeexpressedasgivenby ij = [ e ] sv r ij = k e kk sv kk r ij k sin 1 sin 2 u ij = sv [ e ] r ij = k e kk sv kk r ij k sin 1 sin 2 u where,thevectors u and u arerandomvectorsofunitmagnitude,andcovariance of 1 .Thenon-linearterms ij and ij arerandomprocesses,anditcanbeseenfrom Eq.4thattheircovarianceisdependenton k k ,andhencetimevarying.The processesarethereforenotstationary.Inordertoapproximatetheseexpressionsby alinearandstationaryform,wedeterminetheboundsontheircovariancesasfollows. Themagnitudeoftherandomvector sv givenby k sv k canbe`3 bounded'with99.7% condenceasgivenby k sv k = q x sv 2 + )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [( y sv 2 + z sv 2 k sv k q 3 sv 2 + 3 sv 2 + 3 sv 2 k sv k 5.2 sv Wedeneaparameter suchthat ij 5.2 sv k e k max k r ij k where k e k max isthemagnitudeofthemaximumspacecraftangularvelocityduringthe measurementacquisitionperiod.Wecanthereforeboundthemagnitudeoftherandom 79

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noiseprocessesinEq.4asgivenby k ij + ij k 2 ij Wenowdenearandomvariable suchthatitsstatisticsaregivenby E f ij g = 0 and E f ij )]TJ/F46 11.9552 Tf 5.977 -9.684 Td [( ij T g = 4 2 ij 9 1 OnobservingEqs.4,4,and4,wecanconcludethatthetraceofthe covariancematrixoftherandomvector isgreaterthanorequaltothetraceofthe covariancematrixofthesumoftherandomprocesses )]TJ/F46 11.9552 Tf 5.977 -9.684 Td [( ij + ij asgivenby Tr E n ij )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( ij T o Tr E n )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( ij + ij )]TJ/F46 11.9552 Tf 13.45 -9.684 Td [( ij + ij T o Thestochasticprocessesrepresentedbythesumofthenon-linearterms )]TJ/F46 11.9552 Tf 5.978 -9.684 Td [( ij + ij can thereforebereplacedwithasinglerandomvariable ij toapproximatethenon-linear expressioninEq.4toalinearmeasurementmodelasgivenby a ij = e a ij )]TJ/F46 11.9552 Tf 12.454 0 Td [( a ij )]TJ/F28 11.9552 Tf 11.956 9.683 Td [()]TJ/F46 11.9552 Tf 5.977 -9.683 Td [( a ij )]TJ/F46 11.9552 Tf 12.454 0 Td [( ij Thesumofrandomvariables, a ij )]TJ/F46 11.9552 Tf 12.454 0 Td [( ij inEq.4canberepresentedbyasingle randomvariable ij suchthat E f ij g = 0 and E f ij )]TJ/F46 11.9552 Tf 5.977 -9.684 Td [( ij T g = 4 2 a )]TJ/F20 11.9552 Tf 13.15 9.657 Td [(4 2 ij 9 1 where, )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [(4 2 a 1 isthecovarianceof a ij .Thelinearmeasurementmodelfor a ij istherefore givenby a ij = e a ij )]TJ/F46 11.9552 Tf 12.453 0 Td [( ij )]TJ/F46 11.9552 Tf 12.454 0 Td [( ij UsingEq.4,wecannowderivethemeasurementmodelfortheangularacceleration. Measurementmodelsfortheangularaccelerationdeterminedusingbothuniaxialand triaxialmethodsarecompared. 80

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Considertheexpressionfortheangularaccelerationdeterminedviauniaxial accelerometersasgivenbyEq.4.Thisvalueofangularaccelerationwouldbe accurateifexactmeasurementsfromaccelerometersandgyroscopesareavailable. Since,themeasurementsarecorruptedbysensornoiseandbias,theangular accelerationiscomputedbyreplacingthetruevaluesinEq.4bymeasuredvalues usingEq.4asgivenby e = 0 B B B B B B B B B B @ )]TJ 5.98 -8.304 Td [(e a y 12 T Z k r y 12 k )]TJ 5.98 -8.304 Td [(e a z 12 T X k r z 12 k )]TJ 5.98 -8.304 Td [(e a x 12 T Y k r x 12 k 1 C C C C C C C C C C A = 0 B B B B B B B B B @ )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(a y 12 + y 12 + y 12 T Z k r y 12 k a z 12 + z 12 + z 12 T X k r z 12 k a x 12 + x 12 + x 12 T Y k r x 12 k 1 C C C C C C C C C A ItcanbeseenfromEq.4thatthebiasandnoiseinthemeasurementslinearlyenter theexpressionforangularacceleration,andarescaledonlybythedistancebetween thetwoaccelerometers.Now,considerthecaseinvolvingtriaxialaccelerometers.Using Eq.4,themeasuredangularaccelerationisgivenby e = D T D )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 D T 0 B @ a 12 + 12 + 12 a 13 + 13 + 13 1 C A Itismoredifculttocommentontheeffectofsensorbiasandnoiseontheestimateof theangularaccelerationinthiscase.Togainmoreinsight,weconsiderthestructure ofthequantity D T D )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 D T whichconsistsoftheelementsofthevectors r 12 and r 13 .Weknowthatthevectors r 12 and r 13 lieonasingleplane.Considerthecasewhen thisplaneisthe x )]TJ/F59 11.9552 Tf 12.15 0 Td [(y planeformedbythespacecraftbodycoordinatebasisvectors X and Y andthevectors r 12 and r 13 areorthogonaltoeachother.Let r 12 = [ x 00 ] T and r 13 = [ 0 y 0 ] T .UsingEq.4,itcanbeshownthatinthiscase,theangular 81

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accelerationisgivenby e = 0 B B B B B B B B B @ )]TJ/F58 11.9552 Tf 5.977 -9.683 Td [(a 13 + 13 + 13 T Z y )]TJ/F20 11.9552 Tf 10.494 7.957 Td [( a 12 + 12 + 12 T Z x x a 12 + 12 + 12 T Y x 2 + y 2 )]TJ/F59 11.9552 Tf 13.151 8.847 Td [(y )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(a 13 + 13 + 13 T Y x 2 + y 2 1 C C C C C C C C C A OncomparingEq.4andEq.4,itcanbeseenthattheangularacceleration determinedusinguniaxialaccelerometersislessaffectedbybiasandnoiseas comparedtothatdeterminedusingonlytriaxialaccelerometers.Itisthereforepreferable touseuniaxialaccelerometerstodeterminetheangularaccelerationofthespacecraft. Thediscussionintheremainderofthechapterisbasedontheestimationofangular accelerationusinguniaxialaccelerometersasgivenbythemeasurementmodelin Eq.4. 4.3.2EstimationofBias-FreeAngularAcceleration Theeffectofsensorbiasonthedeterminationofangularaccelerationisseen inEq.4.Inmanyinstancestheinitialbiasstartupbiasoftheaccelerometersis largeandcanseverelyaffecttheaccuracyofthecomputedangularacceleration.Itis thereforenecessarytoestimatethebiasandremoveitfromthemeasurementstoobtain abetterestimateoftheangularacceleration.Equation4canberewrittenasgiven by e = 0 B B B B B B B B @ a z y 12 k r y 12 k a x z 12 k r z 12 k a y x 12 k r x 12 k 1 C C C C C C C C A | {z } + 0 B B B B B B B B @ z y 12 k r y 12 k x z 12 k r z 12 k y x 12 k r x 12 k 1 C C C C C C C C A | {z } + 0 B B B B B B B B @ z y 12 k r y 12 k x z 12 k r z 12 k y x 12 k r x 12 k 1 C C C C C C C C A | {z } 82

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UsingthenotationindicatedbytheunderbracesinEq.4,themeasurementmodelor thetruthmodelfortheangularaccelerationcanbecompactlyexpressedasgivenby = e )]TJ/F46 11.9552 Tf 12.454 0 Td [( )]TJ/F46 11.9552 Tf 12.454 0 Td [( = where, e istheangularaccelerationcomputedfromaccelerometerandgyroscope measurementsasgivenbyEq.4, isthetrueangularaccelerationofthe spacecraft,and and arethe effectivebias and effectivenoise ,respectivelyin thecomputedangularacceleration.Therandomnoise isusedtomodelthe effective biasinstability .Thecovariancesoftherandomprocesses ,and arerepresented by 2 1 ,and & 2 1 ,respectively.Itcanbeobservedthatthesecovariancescanbe computedusing a ij ,and & a .ThemeasurementmodelgiveninEq.4issimilar tothegyroscopemeasurementmodelpublishedbyFallon[112]andFarrenkopf[39] usedtoestimatethebiasingyroscopemeasurementsandtoobtainlteredestimates ofthespacecraftattitude.Ifanestimateofthebias h i canbeobtained,thenbias-free angularaccelerationcanbeestimatedas h i = e )-222(h i AlinearKalmanltertoestimatethebiasisnowdiscussed.AKalmanlterisbased onadynamicprocessmodelwhichrepresentstheevolutionofitsstatesanda correspondingsetofmeasurementsforoneormorestatesofthedynamicmodel. ThemeasurementsareusedtoupdatethestatesbasedonanoptimalKalman feedbackrule[111].TherstorderdifferentialequationsinEq.4areconsidered asthedynamicprocessmodels.Theinputtothedynamicprocessmodelisthe measurement e ,whichiscounterintuitive,butdoesnotaffecttheKalmanlterform andtheestimationprocess[111].ThemeasurementsusedforKalmanfeedbackarethe angularvelocitymeasurementsfromthegyroscopeasgivenbyEq.4.TheKalman 83

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lterisimplementedindiscretetime,andthecorrespondingdiscretetimemodelsforthe dynamics,andmeasurementaregivenby x k +1 = x k + )]TJ/F58 11.9552 Tf 10.763 0 Td [(u k + w k e y k = H k x k + sv k where,thetimedifferencebetweensuccessiveintervals k isequalto t ,and = 2 6 4 1 )]TJ/F40 11.9552 Tf 9.298 0 Td [( t 1 0 1 3 7 5 )]TJ/F20 11.9552 Tf 10.261 0 Td [(= 2 6 4 t 1 0 3 7 5 H = 1 0 x k = 2 6 4 k k 3 7 5 u k = e k w k = 2 6 4 R t 0 )]TJ/F46 11.9552 Tf 9.796 0 Td [( )]TJ/F20 11.9552 Tf 11.955 -0.132 Td [( t )]TJ/F23 11.9552 Tf 11.955 0 Td [( d R t 0 d 3 7 5 e y k = e k Thecovariancematricesoftheprocessnoise w k ,andthemeasurementnoise sv k are givenby C p = E f w k w k T g = 2 6 4 )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [( 2 t 2 + 1 4 & 2 t 4 1 )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F20 7.9701 Tf 6.675 -4.977 Td [(1 2 & 2 t 3 1 )]TJ/F28 11.9552 Tf 11.291 9.683 Td [()]TJ/F20 7.9701 Tf 6.675 -4.976 Td [(1 2 & 2 t 3 1 )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(& 2 t 2 1 3 7 5 C m = E f sv k )]TJ/F46 11.9552 Tf 5.977 -9.684 Td [( sv k T g = 2 sv 1 TheKalmanupdateequations,predictionequations,andinitialconditionsforthelter arerespectivelygivenas K k = h P )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k i H T H h P )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k i H T + C m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 h x + k i = h x )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k i + K k )]TJ 5.98 -8.471 Td [(e y k )]TJ/F58 11.9552 Tf 11.955 0 Td [(H h x )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k i h P + k i = 1 6 6 )]TJ/F58 11.9552 Tf 12.453 0 Td [(K k H h P )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k i h x )]TJ/F59 7.9701 Tf 0 -8.501 Td [(k +1 i = h x + k i + )]TJ/F58 11.9552 Tf 10.762 0 Td [(u k h P )]TJ/F59 7.9701 Tf 0 -8.502 Td [(k +1 i = h P + k i T + C p 84

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h x i t =0 = h x )]TJ/F20 7.9701 Tf 0 -8.358 Td [(0 i = 2 6 4 h i h i 3 7 5 h P i t =0 = h P )]TJ/F20 7.9701 Tf 0 -8.36 Td [(0 i = P TheoutputoftheKalmanlterincludestheestimateoftheeffectiveangularacceleration bias h i ,andsmoothedangularvelocityestimates h i .Theestimatedbias h i isremovedfromtheangularaccelerationmeasurementsasgivenbyEq.4.The Kalmanlterdiscussedaboveissimilartotheoneusedforsingle-axisspacecraft attitudeestimation[39,111],andmoredetailsonsimilarimplementationsoftheKalman ltercanbefoundinthesereferences.Bias-freeestimatesoftheangularacceleration arethusobtained.Theaddedadvantageofusingthelterisobtainingsmoothed angularvelocityestimateswhichwouldotherwisebelimitedtonoisygyroscope measurements.Simulationsarenowpresentedtoshowtheeffectivenessofthemethod. 4.4SimulationsandResults Simulationsareperformedusingdatafromcommerciallyavailablehardware, obtainedviatheirmanufacturer'sdatasheetorlaboratorymeasurements.Linear accelerometercharacteristicsarederivedfromtheQA3000inertialgradeaccelerometer fromHoneywell[113]showninFigure4-3A,andthegyroscopecharacteristicsare derivedfromtheADIS16485tacticalgradeinertialmeasurementunitIMUfromAnalog Devices[114]showninFigure4-3B.Theaccuracyoftheangularaccelerationestimates obtainedaredirectlyrelatedtotheperformanceofthesensors.Thesimulationsare performedaccordingtothefollowingsteps: 1.AspacecraftdynamicmodelsimilartoEq.2issimulatedforarbitraryinputs fromamomentumactuatortoobtainvaluesoftrueangularvelocity andtrue angularacceleration 2.Thetruelinearaccelerations a i ofthesixpointsindicatedinFigure4-1are computedusingEq.4 3.Syntheticmeasurementsoftheangularvelocity e ,andlinearaccelerations e a i arecreatedbyaddingnoiseandbiastothedataobtainedintheabovesteps.The 85

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biasandnoisecharacteristicsarebasedonthesensordataobtainedfromthe manufacturer'sdatasheet,andarelistedinTable4-1.Itshouldbenotedthatthe noisecharacteristicsareconservativeandnotrepresentativeofhighperformance sensors 4.TheangularaccelerationofthespacecraftiscomputedusingEq.4 5.TheeffectivebiasintheangularaccelerationisestimatedusingtheKalmanlter describedintheprevioussection 6.Bias-freeangularaccelerationandlteredangularvelocityestimatesarethus obtained TheparametersusedforthesimulationareshowninTable4-1.Thetruetrajectoriesof thespacecraftangularvelocityandaccelerationobtainedfromstep1ofthesimulation procedureareshowninFigure4-4.Theangularvelocitymeasurementsasobtained bythegyroscopeareshowninFigure4-5.Thesemeasurementsarebias-freeas describedearlier.ThenoiseinthemeasurementsshowninFigure4-5isclearlyseen comparedtothetrueangularvelocityinFigure4-4A.Thelteredangularvelocity, whichistherstoutputoftheKalmanlterisshowninFigure4-6.Theeffectivebias denedasgiveninEq.4isequaltothedifferenceinthebiasbetweenthetwo accelerometersinquestiondividedbythemagnitudeofthedistancebetweenthem. Forinstance,referringtoTable4-1the x )]TJ/F20 11.9552 Tf 9.298 0 Td [(componentofthetruebias x iscomputed as0.03126 rad = s 2 .Thiseffectivebiasinthecomputedangularaccelerationisthe secondoutputoftheKalmanlterandisshowninFigure4-7.Alogarithmicscalehas beenusedforthe x-axis toclearlydepicttheimprovementinthebiasestimateswith time.Thebias-freeangularaccelerationofthespacecraftdeterminedusingEq.4 isshowninFigure4-8.Theerrorbetweenthemeasuredangularvelocitygyroscope measurementsandthetrueangularvelocityisshowninFigure4-9.Thecorresponding errorinthelteredangularvelocityisshowninFigure4-10.Thetheoretical3 bounds onthelteredangularvelocityerrorarealsoshown.Thereductioninnoisesmoothing whencomparedtoFigure4-9canbeobserved.BycomparingFigures4-9and4-10,it canbeseenthatthemagnitudeofthenoiseinthelteredangularvelocityestimatesis 86

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aboutvetimeslesscomparedtotheerrorinthemeasuredangularvelocity.Theerror intheangularaccelerationestimatesofthespacecraftisshowninFigure4-11.The peakerrorintheangularaccelerationestimatesisabout 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 rad = s 2 .Theangular accelerationofthespacecraftdeterminedbydifferentiatingthemeasuredangular velocityisshowninFigure4-11,anditcanbeseenthattheestimateshavealargeerror duetonoiseamplication. Thus,aneffectivemethodtodeterminetheangularaccelerationofthespacecraft usinglinearaccelerometerswasdeveloped.Thepeakerrorintheangularacceleration estimateswasabout 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 rad = s 2 .Thiserror,however,isbasedontheperformance ofthesensors,andbetterestimatescanbeobtainedusinghighprecisionsensors. Further,theestimationofangularaccelerationaidedinobtainingsmoothedangular velocityestimates.Thestandarddeviationoftheerrorinthesmoothedangularvelocity wasvetimessmallerthanthatofdirectgyroscopemeasurements.Theknowledge ofangularaccelerationcannowbeusedintheestimationofspacecraftdynamic propertiesbasedonEuler'sattitudeequations.Onesuchutilityindeterminingthe on-orbitorientationofaCMGgimbalisdiscussedinChapter5. 87

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Table4-1.SimulationParametersforAngularAccelerationEstimation ParameterValueUnits Simulationtime200 s Solver4thorderRunge-Kutta n = a Timestep0.0002xed s Spacecraftinertia Actual J c = 2 4 0.1550 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0050 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.00500.1550 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.00050.1600 3 5 kgm 2 Dataacquisitionrate50 Hz Spacecraftinitialangular velocity = [ 0.20.10.1 ] T rad = s Gyroscopenoise sv =2.79 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 rad = s Accelerometernoise a =6.87 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 m = s 2 Accelerometerbias instability & a =2.22 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 m = s 3 Accelerometerinitialbias y x 1 =0 m = s 2 y x 2 =0.0147 m = s 2 z y 1 =0 m = s 2 z y 2 =0.0147 m = s 2 x z 1 =0 m = s 2 x z 2 =0.0147 m = s 2 Accelerometerdistances k r x 12 k =0.45 m k r y 12 k =0.45 m k r z 12 k =0.45 m Angularacceleration effectivenoise =1.192 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 rad = s 2 Angularacceleration effectivebiasinstability & =4.93 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 rad = s 3 Estimatedinitialangular velocity h i = [ 0.19880.09530.1003 ] T rad = s Estimatedinitialangular accelerationbias h i = [ 000 ] T rad = s 2 Estimatedinitialstate errorcovariance h P i = )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 1 0 0 )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 1 88

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Figure4-1.RigidBodyRepresentationofanArbitrarySpacecraftwithUniaxial AccelerometersandGyroscope Figure4-2.RigidBodyRepresentationofanArbitrarySpacecraftwithTriaxial AccelerometersandGyroscope 89

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AHoneywellQA3000 BAnalogDevices16485IMU Figure4-3.InertialSensorsforMeasurementofSpacecraftAngularVelocityand Acceleration 90

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AAngularVelocity BAngularAcceleration Figure4-4.TrueSpacecraftAngularVelocityandAngularAcceleration 91

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Figure4-5.AngularVelocityMeasurements Figure4-6.FilteredAngularVelocityEstimates 92

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Figure4-7.EstimatedAngularAccelerationBias Figure4-8.EstimatedAngularAcceleration 93

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Figure4-9.ErrorinMeasuredAngularVelocityGyroscopeNoise Figure4-10.ErrorinFilteredAngularVelocityEstimates 94

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Figure4-11.ErrorinAngularAccelerationEstimates Figure4-12.ErrorinAngularAccelerationObtainedbyDifferentiatingAngularVelocity Measurements 95

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CHAPTER5 ON-ORBITESTIMATIONOFCMGGIMBALORIENTATION Accurateknowledgeofthegimbalorientation ofacontrolmomentgyroscope CMGisrequiredforprecisionpointingcapabilitiesseeChapter3.Angularacceleration estimatesChapter6arebenecialintheestimationofactuatoralignmentssuchasthe CMGgimbalorientation.Thiscapabilityisexercisedbydevelopingamethodtoestimate theunknownorientationoftheCMGgimbalusingon-orbitmeasurements.Euler's equationsofmotionareusedasthebasisforthedevelopmentofthemethod,andthe equationsofmotionareformulatedsuchthatlinearleastsquarestechniquescanbe adoptedfordeterminingtheunknowngimbalorientation.Thesolutionmethodology presentedcanbeextendedtomisalignmentestimationofothermomentumactuators. 5.1Background Estimationofactuatoralignmenthasattractedlessattentionwhencomparedto estimationofattitudesensoralignment[3438]andattitudeestimationingeneral[39 46].Itisimportanttoreviewandunderstandtheoutcomeofthesespacecraftattitude estimationandsensoralignmentestimationcontributionsasitisstraightforwardto observethattheproblemofactuatoralignmentestimationisakintothatofattitude estimation.Anattitudeestimationproblemcanbeposedasaminimizationproblem, alsoknownastheWahbaproblem[115],whosesolutionminimizesthedifference intheobservationofvectorsintwoseparatecoordinatebasesrelatedlinearlyviaa transformationmatrixordirectioncosinematrixDCMasgivenby Given B v i = R BA A v i + i i =1,..., n Minimizef R = 1 2 n X i =1 a i k B v i )]TJ/F58 11.9552 Tf 12.453 0 Td [(R BA A v i k 2 where, i istheerrorinthe i th setofobservations.TheDCM R BA inEq.5transforms theobservationofavector v i inanarbitrarycoordinatebasis C A denotedby A v i tothe observationinanotherarbitrarycoordinatebasis C B ,denotedby B v i .Theinuence 96

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ofeachmeasurementonthemeasure f R canbecontrolledbythescalarweighting factors a i .Forinstance,Sunandstarsensorscanbeusedtoobtainmeasurements ofthedirectionofSunandstars,respectivelyinacoordinatebasisattachedtothe bodyofthespacecraft.Similarly,measurementsofthesamevectorscanbeobtained inanothercoordinatebasisofintereste.g.,ECIcoordinatebasis.Theseobservations ofthesamevectorinacoordinatebasisattachedtothespacecraftbodyandanother referencecoordinatebasisofinterestallowstheattitudeestimationproblemtobe posedasperEq.5,sothattheattitudeofthespacecraftcanbeestimatedrelative toareferenceofinterest.Severalsolutionstothisproblemhavebeenproposedand implemented[40,4246]. Now,theactuatoralignmentestimationproblemcanbeaddressedinasimilar manner,provided,theobservationsofvectorsinacoordinatebasisattachedtothe misalignedactuatorandobservationsofthesamevectorsortheireffectsinacoordinate basisattachedtothespacecraftbodyareavailable.Theexpressionforthespacecraft angularmomentumandthespacecraftattitudeequationsofmotionpermittheuseof twosuchvectorsforalignmentestimationviz.atheactuatorangularmomentumvector andbtheactuatoroutputtorquevector,respectively.Theuseofbothactuatorangular momentum[4750]andactuatoroutputtorque[51]foralignmentestimationcanbe foundinliterature.Theseapproachesarediscussedbriey.First,considerthetotal angularmomentumofarigidspacecraftaboutitscenterofmass c.m. expressedinthe spacecraftbodyxedcoordinatebasis C B asgivenby B h s c )]TJ/F58 11.9552 Tf 12.454 0 Td [(J s c )]TJ/F59 7.9701 Tf 18.308 14.944 Td [(n X i =2 B h i = R BA A h 1 where, J s c and aretheinertiamatrixaboutthe c.m. andabsoluteangularvelocity ofthespacecraft,respectively,expressedinspacecraftbodycoordinatebasis,and B h i istheangularmomentumofthe i th actuatoralsoexpressedinthespacecraftbody coordinatebasis.Theangularmomentumoftheactuatorwhosealignmentistobe 97

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determinedisexpressedinacoordinatebasis C A xedtotheactuator,andisdenoted by A h 1 .TheDCMthattransformsrepresentationsin C A to C B isrepresentedby R BA ThesimilaritybetweenEq.5andthe given observationsinEq.5canbereadily observed.Exceptforthetotalangularmomentumofthespacecraft h S andtheDCM R BA ,theremainderofthequantitiesinEq.5canbedeterminedusingvarioussensor measurements.Sincetheangularmomentumofspacecraftcannotbemeasured, thesolutionprocessfordetermining R BA assumesthespacecraftinertialangular momentumtobeconstantoveragiventimeinterval[4750].Thisassumptionallowsus toequatetheinertialangularmomentumattwoormoretimeinstantsasgivenby R t J s c t + n X i =2 B h i t + R BA A h 1 t t = R t + t J s c t + n X i =2 B h i t + R BA A h 1 t t + t where R t representsthetransformationfromthespacecraftbodyxedcoordinate basistoaninertialcoordinatebasisattime t ,orsimply,thespacecraftattitudeattime t .Giventheknowledgeofthespacecraftattitudeandothermeasurementsatvarious timeinstants,Eq.5canbeexpressedinalinearformsimilartoEq.5.Theactuator alignment R BA canthereforebeestimatedusingleastsquaressolutions[47,48], recursivesolutions[49,50],andtheseveralothersolutionstoEq.5orthegeneralized Wahbaproblem[115].Itshouldbenotedthatthismethodrequiresaccurateknowledge ofthespacecraftattitudeandunderthepresenceofexternaldisturbancetorques,the assumptionofequalityofangularmomentumatdifferentinstantsoftimeisviolated. Slightlyhigherdelitycanbeincorporatedbyincludingtheangularmomentumchange duetoknownexternaltorquese.g.,gravitygradient[47].Further,thedifference betweenthespacecraftangularmomentaattimes t and t + t hastobesignicant toprovideagoodsignaltonoiseratioSNR,andtherebyagoodestimateof R BA But,highSNRmeasurementsareoftenassociatedwithhighspacecraftangular 98

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velocitiesthataffecttheaccuracyofattitudemeasurementse.g.,starsensorsare usuallyaccurateunder4 = s [116].Now,considerEuler'sequationsofmotionofarigid spacecraftgivenby )]TJ/F28 11.9552 Tf 11.955 20.444 Td [( J s c + n X i =2 B h i + J s c + n X i =2 B h i !! = R BA A h 1 + R BA A h 1 wherethetotalexternaltorqueonthespacecraftdenotedby isassumedtobe negligible.Itcanbeseenthatinusingthisequationforestimating R BA ,theknowledge ofthespacecraftattitudeisnotrequired.But,theexpressioninEq.5isnon-linear, andhencedifculttoexpressinaWahbastyleproblemorinaformwherelinear techniquescanbeapplied.Thecouplingbetweenthespacecraftangularvelocity and theangularmomentumofthemisalignedactuator A h 1 posesdifcultiesinbeingable towritetheequationinalinearform.Duetothenon-linearnatureoftheequationand therequirementforangularacceleration measurements,Euler'sequationhasbeen mostlyavoidedasthebasisforalignmentestimation.Fosbury, etal [51]havechosento ignorethecouplingtermandproceedtoestimatethealignmentusingalinearKalman lter,whichmayproducelessaccurateresultsaswillbeshownlaterinthesimulations. Theyalsodeterminetheangularaccelerationofthespacecraftbydifferentiatingangular velocitymeasurements,whichisknowntoamplifynoise. ThischapterpresentsamethodtoestimatethegimbalorientationofCMGsbased onEuler'sequationswithoutneglectingthenonlinearcouplingterm.Itisshownthat byextractingtheelementsoftheDCM R BA ,theestimationproblemcanbesetupsuch thatthegimbalorientationcanbeestimatedusingalinearleastsquaresapproach.The methodusesmeasurementsfromspacecraftinertialsensorssuchasgyroscopesand accelerometers,andactuatorfeedbacksensors.Astrategytoestimatebias-freeangular accelerationusinglinearaccelerometermeasurementsinsteadofdifferentiatingnoisy angularvelocitymeasurementsasdescribedinChapter4isalsoused.Acomplete setofsimulationsperformedusingdatafromactualsensorhardwareispresented. 99

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Comparisonofresultsobtainedfromdifferenttypesofleastsquaressolutionisalso provided. 5.2Methodology On-orbitestimationoftheunknowngimbalorientationofaCMGusingon-board sensorsisnowdescribedindetail.Itshouldbeacknowledgedthatwecanonlynd alignments/orientationsrelativetoareferencecoordinatebasisofinterestinwhich theobservationsmeasurementscanbemadeandtheestimationaccuracyislimited bytheaccuracyofthesensors[36,117].Thereferenceinourcaseisthespacecraft bodycoordinatebasis C B relativetowhichthegimbalorientationisdetermined.Itis customarytosituateagyroscopee.g.,beropticgyroscope,MEMSgyroscope,etc. thatisxedtothespacecraftsuchthatitsorthogonalmeasurementaxesdenethis coordinatebasis. Theestimationofthegimbalorientationisperformedduringtheattitudeacquisition phasebeforecommencementofnormalmissionoperations,oneCMGatatime. Theattitudeacquisitionphaseinvolvesspacecraftdespinandothermomentum managementmaneuvers[1].Anychangeintheangularvelocityofthespacecraft causedbymaneuversrequiredfortheestimationprocesscanthusbehandledduring thisphase.WhiletheestimationisbeingperformedonaparticularCMG,theothers areinactiveanddonotcontributetothedynamics.Wethereforeconsideraspacecraft withasingleCMG,withoutlossofgenerality.Theestimationprocesscanbeperformed eitheronboardthespacecraft,oronthegroundusingtelemetrydata. 5.2.1SpacecraftEquationsofMotion ConsidertherepresentationofanarbitraryspacecraftwithasingleCMGasshown inFigure5-1.Thespacecraftbodycoordinatebasis C B givenby h X B Y B Z B i ,isused torepresentobservationsrelativetothespacecraft.Asetofbasisvectors h X F Y F Z F i representingtheywheelcoordinatebasis C F isxedtothegimbalsuchthatthevector X F isalongtheywheelspinaxis,and Z F isalongthegimbalaxis.Anothersetofbasis 100

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vectors h X G Y G Z G i representingthegimbalcoordinatebasis C G isalsoxedtothe spacecraftbutorientedsuchthatitisalignedwith C F ,whenthegimbalangle, =0. Thevalueof denedabout Z G ,rangesfrom0to2 radians .Itisthevalueof that weseektoestimateinthischapter.Theywheelisassumedtobeideallybalanced, andthepositionoftheywheel's c.m.O islocatedatadistance r o fromthespacecraft's c.m.C .TheDCMthattransformsvectorrepresentationsinthecoordinatebasis C F to representationsin C G isgivenby R GF = R Z G )]TJ/F23 11.9552 Tf 9.298 0 Td [( = 0 B B B B @ c )]TJ/F20 11.9552 Tf 9.299 0 Td [(s 0 s c 0 001 1 C C C C A where,the cosine and sine ofthegimbalanglearerepresentedbyc ands ,respectively. Theorientationofthegimbalcoordinatebasis C G relativetothespacecraftbody coordinatebasis C B isrepresentedbyanarbitrary,xed,butknownDCMgivenby R BG = 0 B B B B @ l xx l xy l xz l yx l yy l yz l zx l zy l zz 1 C C C C A ThistransformationmatrixEq.5alsoprovidestheorientationofthegimbalaxisin thespacecraftbodycoordinatebasis.Wegeneralizetherepresentationbyanarbitrary DCMtoincludeallpossiblecongurationsofCMGssuchaspyramid,rooftop,andbox congurations[66].Thus,theDCMthatrepresentsthetransformationfrom C F to C B is givenby R BF = R BG R GF FromEq.3,thetotalangularmomentumofthespacecraftaboutits c.m. isgivenby B h c = J s c + R BF )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(I w o 101

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where,thecontributionofthegimbaldynamicshavebeenexcludedasthegimbal remainsxedinthespacecraftduringtheestimationphase.Thecentroidalinertia matrixoftheywheelisdenotedby I w o ,andasdescribedinSection3.2.3,theCMGsare designedsuchthatdifferenceinthevalueof I w o asexpressedintheywheelcoordinate basisandasexpressedinthespacecraftbodycoordinatebasisisminimal.Thus,the effectofinertiachangeduetogimbalmotiononthetotalspacecraftinertiacanbe neglected.Thetotaltimeinvariantinertiamatrixofthespacecraft J s c isthereforegiven by J s c = I s c + I w o + m w )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o )]TJ/F58 11.9552 Tf 13.449 -9.684 Td [(r o T where, I s c istheinertiamatrixoftherigidspacecraftstructureabout C .Assumingthe externaltorqueactingonthespacecrafttobenegligiblecomparedtothetorqueoutput oftheCMG,theratechangeofangularmomentumofthespacecraftaboutits c.m. is givenby J s c + [ ] J s c + R BF )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(I w o + [ ] R BF )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(I w o = 0 whichrepresentstheEuler'sequationsofmotion.Thedynamicsofthespacecraftunder considerationFigure5-1arecompletelyrepresentedbyEq.5togetherwiththe initialstatesofthespacecraftandtheywheel.Theresponseofthespacecrafttoany internalchangeintheangularmomentumduetoywheelmotioncanbecomputed fromthisequation.Consequently,whenthestatesofthespacecraftandywheel areknown,certainunknownparametersintheequationcanbeidentied.Theintent ofthedevelopmentpresentedinthischapteristoidentifytheunknownvalueofthe gimbalorientation whichisembeddedinthetransformationmatrix R BF asgivenby Eq.5.Measurementsofthespacecraftangularvelocity ,thespacecraftangular acceleration ,theywheelangularvelocity ,theywheelangularacceleration andtheknowledgeoftheinertiamatrices I w o and J s c arenecessaryinordertoextract thevalueoftheunknowngimbalangleusingEq.5.Anapproximatewithupto 2%errorvalueoftheinertiamatricesisassumedtobeknown.Theinertialsensors 102

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commonlyusedonboardthespacecraftandthesensorsincorporatedinattitude actuatorsfortheircontrolcanprovidethenecessarymeasurements.Amoredetailed descriptiononobtainingthesemeasurementsispresentedlaterinthechapterunder Section5.3.Fornow,weproceedwiththediscussionofidentifyingtheunknowngimbal orientation assumingthesemeasurementsareavailable.Groupingterms,Eq.5 canbecompactlyrewrittenas s t + R BF w t + [ t ] R BF h w t = 0 wheretheexplicitdependenceofthemeasurementsontime t isindicated.Theterms s t w t ,and h w t inEq.5aregivenby s t = J s c t + [ t ] J s c t w t = I w o t h w t = I w o t ThequantityrepresentedbytherstlineinEq.5isafunctionofthespacecraft angularvelocityandacceleration,andcanbeobtainedusinginertialsensors.Similarly, thequantitiesrepresentedbythesecondandthirdlinesarefunctionsoftheywheel states,andcanbeobtainedusingvarioussensorsintheactuator.Asmentionedearlier, anideallybalancedywheelisassumedforthesakeofsimplicity,andtheinertiamatrix I w o isthereforediagonal.Bynotingthattheexpressionsfortheywheelangularvelocity andangularaccelerationrelativetothespacecraftbody,expressedin C F aregivenby F = 00 T and F = )]TJ/F20 11.9552 Tf 7.716 -7.634 Td [( 00 T ,respectively,wecanwrite F w = w 00 T F h w = h w 00 T where, w = k F w k and h w = k F h w k .Thetimedependencerepresentationhasbeen droppedforbrevity.UsingEqs.5and5,andbyextractingouttheelementsc 103

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ands from R BF ,itcanbeshownthatwecanrewriteEq.5aslineartimevarying equationasgivenby 2 6 6 6 6 4 s x s y s z 3 7 7 7 7 5 | {z } y t = 2 6 6 6 6 4 h w )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F23 11.9552 Tf 9.298 0 Td [(! y l zx + z l yx )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l xx h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! y l zy + z l yy )]TJ/F23 11.9552 Tf 11.956 0 Td [( w l xy h w x l zx )]TJ/F23 11.9552 Tf 11.955 0 Td [(! z l xx )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l yx h w )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(! x l zy )]TJ/F23 11.9552 Tf 11.955 0 Td [(! z l xy )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l yy h w )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F23 11.9552 Tf 9.298 0 Td [(! x l yx + y l xx )]TJ/F23 11.9552 Tf 11.956 0 Td [( w l zx h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! x l yy + y l xy )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l zy 3 7 7 7 7 5 | {z } A t 2 6 4 c s 3 7 5 | {z } x t where, s x isthex-componentof s inthespacecraftbodycoordinatebasisandso on.ThetermsinEq.5areeitherknownormeasuredexceptforthetwounknown parameters:c ands .Itshouldbenotedthatc ands arenotindependentandare correlated.Usingthenotationsdenotedbytheunder-braces,Eq.5iscompactly writtenas y t = A t x t Ifthegyroscopiccouplingtermbetweenthespacecraftangularvelocityandtheactuator angularmomentumgivenby [ ] R BF h w inEq.5isneglectedassuggestedin[51], thenthematrix A isgivenby A t = 2 6 6 6 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [( w l xx )]TJ/F23 11.9552 Tf 9.298 0 Td [( w l xy )]TJ/F23 11.9552 Tf 9.299 0 Td [( w l yx )]TJ/F23 11.9552 Tf 9.298 0 Td [( w l yy )]TJ/F23 11.9552 Tf 9.299 0 Td [( w l zx )]TJ/F23 11.9552 Tf 9.298 0 Td [( w l zy 3 7 7 7 7 5 Itwillbeshownlaterinthesimulationthatthisassumptionwillproducelargererrorin theestimateofthegimbalorientation.Therefore,themethoddevelopedinthischapter doesnotusethetruncated A matrix,butusesthecompleteoneasgiveninEq.5. Wehavenowexpressedthespacecraftequationsofmotioninalinearform,and canuseleastsquarestechniquestoestimatetheunknowngimbalangle 5.2.2LeastSquaresSolution Theelementsofthevector y t andthematrix A t areobtainedfromsensor measurementsandarehencecorruptedbynoise.Also,themeasurementsarein 104

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discreteformastheyareobtainedusinganalogtodigitalADconverters.Foraset of n suchmeasurements,thelinearequationEq.5canthereforebeexpressedin discreteformasgivenby e y k = e A k x k + k k =1,..., n where, tilde representsmeasuredvalues,and istherandommeasurementnoise. Itshouldbenotedthat istimevaryingandanonlinearfunctionofrandomnoise processesassociatedwiththemeasurementsofspacecraftandywheelstates.Itis thereforedifculttomakeanyconclusionsonthestatisticalpropertiese.g.,meanand covarianceof .Insomeleastsquaresproblems,thematrix A t istimevaryingbut unaffectedbynoise,andonlythemeasurementvector y t containsrandomnoise withknownstatisticalproperties.Insuchcases,wecanobtaintheoptimalestimateof x usingtheminimumvarianceleastsquaressolution[111]givenby h x k i = e A k T C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 e A k )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 e A k T C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 e y k where, C isthecovarianceoftherandomnoisevector ,givenby C = E f "" T g .Given thestatisticalpropertiesof ,Eq.5achievesthe Cram er-Rao lowerbound[111] andhenceprovidesthebest,unbiasedestimateof x ,basedonthemeasurements e y Unfortunately,asindicatedearlier,thenoisevector inourcaseisanon-linearfunction ofrandomnoisefrommultiplesensors.Statisticalpropertiesof arethereforetime varying,anddifculttodetermine.SinceEq.5involvesthecovarianceof ,itcannot beusedtodetermine h x i inourparticularcase.Therefore, brute-force leastsquares methodsthatminimizeaquadraticfunctionoftheestimationerrorwithoutregard tothestatisticalbehavioroftheerror,areconsidered.Threedifferentleastsquares solutiontechniquesarepresented.Therstapproachndstheminimum 2-norm least squaressolution[111]foreachofthe nequationsinEq.5,andthenalestimate isgivenasthesample-meanofthe nestimates.Weshallrefertothisapproachasthe 105

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independentleastsquares approachsincetheleastsquaressolutioniscomputedfor eachmeasurementsetindependently.Thesecondapproachisabatchleastsquares estimatewhichusesallofthe n -measurementssets,andthethirdisabatch,weighted leastsquaresapproach.Thethreeapproachesarediscussedbelow. Basedontheestimate h x k i atthe k th timestep,Eq.5canbewrittenas e k = e y k )]TJ/F28 11.9552 Tf 13.448 3.412 Td [(e A k h x k i k =1,..., n where e k istheestimationerrorduetouncertaintyinthe k th measurements.Consider thequadraticerrorfunctiongivenby f e k = 1 2 )]TJ/F58 11.9552 Tf 5.978 -9.683 Td [(e k T e k MinimizationoftheerrorfunctioninEq.5yieldstheleastsquaressolution[111] givenby argmin f e : h x k i = e A k T e A k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 e A k T e y k ThesolutioninEq.5constitutestheminimum 2-norm solutionwhenthepositive denitenessconditiongivenby s T e A k T e A k s > 0 issatisedwhere, s isanyarbitrary 1vector.Theleastsquaressolutionfor h x k i inEq.5canthereforebecomputedforeachofthe` n 'setofmeasurements independently.Foreachvalueof h x k i thusfound,acorrespondingvalueof h k i is computedusingthe 2-argumentarctangent or atan2 functionas h k i = atan 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(h x k i h x k i 106

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Thenalestimate, h i iscomputedasthesamplemeanofall h k i inEq.5asgiven by h i = 1 n n X k =1 h k i Thisistheindependentleastsquaressolution. Abatchleastsquaressolutioncanalsobefoundbyconsideringall` n 'measurements atonceasgivenby 0 B B B B B B B @ e 1 e 2 e n 1 C C C C C C C A | {z } e = 0 B B B B B B B @ e y 1 e y 2 e y n 1 C C C C C C C A | {z } e y 3 n )]TJ/F28 11.9552 Tf 11.291 49.136 Td [(0 B B B B B B B @ e A 1 e A 2 e A n 1 C C C C C C C A | {z } e A 3 n 2 h x i Byminimizingthequadraticerrorfunctiongivenby f e = 1 2 e T e weobtaintheminimum2-normleastsquaressolutiontoEq.5asgivenby h x i = e A 3 n 2 T e A 3 n 2 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 e A 3 n 2 T e y 3 n where,thesubscriptsinEq.5representthematrixsize.ThesolutioninEq.5is alsosubjecttothepositivedenitenessconditionsimilartoEq.5. Aparticularkindofbatchleastsquaressolution,theweightedleastsquares WLSsolution[111]allowsustoselectivelyplacehigheremphasisonlownoise measurements.TheWLSsolutionminimizesthequadraticgivenby f e = 1 2 e T We where, W isa3 n 3 npositive-deniteweighting matrix. 107

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ThesolutiontotheminimizationofEq.5istheWLSsolutiongivenby argmin f e : h x i = e A T W e A )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 e A T W e y subjecttothepositivedenitenesscondition s T e A T W e A s > 0 TheeffectivenessoftheWLSsolutiondependsonthechoiceoftheweighting parameters.Theweightingmatrixthatproducesanunbiasedestimatewithminimum varianceistheinverseoftheerrorcovariancematrixasgiveninEq.517.But,as mentionedearliersincewedonothavegoodestimatesofthecovariancematrix,a strategyforselectinganappropriate W soastoplaceemphasisonmeasurementswith highSNRisproposedasfollows.Theestimationofthegimbalorientationwasbased onEuler'sequationsgivenbyEq.5.Thisequationprovidestheaccelerationofthe spacecraftasafunctionoftheywheelaccelerationvector,whichinturnisdependent onthegimbalorientation.Measurementsetswithhigherywheelacceleration,and hencehigherspacecraftacceleration,willthereforecontain richer informationofthe gimbalorientation.Further,thesensorsusedinthemeasurementofthespacecraftand ywheelaccelerationsaremoreaccurateatlargeinputamplitudesasthesignaltonoise ratioSNRwillbehigher.Basedonthesearguments,wechooseadiagonalweighting matrixgivenby W = diag )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(w 1 x w 1 y w 1 z w 2 x w 2 y w 2 z ,..., w kx w ky w kz k =1,2,..., n suchthathigheremphasisisplacedonmeasurementsattimeinstantscorrespondingto largeywheelaccelerations.Thediagonalelementsof W inEq.5arechosenas 0 B B B B @ w kx w ky w kz 1 C C C C A = 0 B B B B @ j kx j j ky j j kz j 1 C C C C A + 0 B B B B @ w w w 1 C C C C A 108

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where, ki isthe i th componentoftheywheelangularaccelerationinthe k th measurement set,expressedintheywheelcoordinatebasis C F .Asmallvalue w > 0isaddedto theelementsoftheweightingmatrixsothat W ispositivedeniteevenwhenthe ki are zero Thethreeapproachestondingtheestimate h x i givenbyEqs.5,5,and 5requirethatmatrix e A T e A ispositivedenite.Itisnowshownthatthisisindeedtrue. Let Q = e A T e A .Weknowthat Q issymmetric,andhenceitis positivesemi-denite [118]. The eigenvalues of Q arethereforegreaterthanorequaltozero,asgivenby eig Q = i 0 i =1,2 For e A T e A tobepositivedenite,both 1 and 2 havetobegreaterthanzero.Thiscanbe trueifandonlyif e A giveninEq.5is fullrank columnrank asgivenby Q = e A T e A is p.d.iffrank e A =2 Forthematrix e A tobe fullcolumnrank ,thetwocolumnsof e A havetobelinearly independent.Toprovethis,letsusrstassumethecolumnstobelinearlydependentas givenby 2 6 6 6 6 4 h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! y l zx + z l yx )]TJ/F23 11.9552 Tf 11.956 0 Td [( w l xx h w x l zx )]TJ/F23 11.9552 Tf 11.956 0 Td [(! z l xx )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l yx h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! x l yx + y l xx )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l zx 3 7 7 7 7 5 = c 2 6 6 6 6 4 h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! y l zy + z l yy )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l xy h w )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(! x l zy )]TJ/F23 11.9552 Tf 11.955 0 Td [(! z l xy )]TJ/F23 11.9552 Tf 11.956 0 Td [( w l yy h w )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(! x l yy + y l xy )]TJ/F23 11.9552 Tf 11.955 0 Td [( w l zy 3 7 7 7 7 5 where c isanyarbitraryconstant.Onobservation,itcanbeseenthatthisassumption wouldimplytheconditionsgivenby 0 B B B B @ l xx l yx l zx 1 C C C C A = c 0 B B B B @ l xy l yy l zy 1 C C C C A 109

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Weknowthattheseconditionscannotbetrueasthetermsintheleftandtheright bracketsrepresenttherstandthesecondcolumnsofthetransformationmatrix R BG asgivenbyEq.5.Thus,weprovebycontradictionthattheequalitycondition assumedinEq.5isfalseandthematrix e A is fullcolumnrank .Since e A isfullcolumn rank,andsince Q = e A T e A issymmetric, Q ispositivedenite.Further,since W is positivedenite, e A T W e A isalsopositivedeniteasrequiredbytheWLSsolution.We nowhaveasolutionstrategyforestimatingthegimbalorientationoftheCMGusing measurementsofywheelandspacecraftangularvelocities,andangularaccelerations. Sensormeasurementsusuallyincludebiasandnoise.Thepresenceofbiasinthe measurements,especiallytheinitialorstartupbiascanhavesevereeffectsonthe accuracyofthesolution.Adiscussiononobtainingthesebias-freemeasurements isprovidedinthefollowingsection.Theresultsofallthreesolutionapproachesare comparedviasimulationsandpresentedinSection5.4. 5.3Measurements,Sensors,andImplementation Aspreviouslydiscussed,theestimationmethodisbasedonvectorobservations measurementsmadeintwoseparatecoordinatebasis.Therstsetincludes observationsoftheywheelangularvelocityandangularaccelerationrelativetothe spacecraftbody,expressedintheywheelcoordinatebasis C F .Thesecondsetincludes observationsofthespacecraftinertialangularvelocityandangularacceleration, expressedinthespacecraftbodycoordinatebasis C B .Therstphaseoftheestimation processinvolvescapturingthespacecraftresponsetoinputsfromtheywheelofthe CMGwhosegimbalalignmentistobedetermined.Anarbitrary,butpredeterminedinput accelerationcommandisprovidedtotheywheelcausingcorrespondingchangesin thespacecraftangularvelocityandacceleration.Thespacecraftandywheelangular velocitiesandaccelerationsaremeasuredsimultaneouslyusingon-boardsensors. Thebiaspresentinmeasurementsareremovedusingasequentialestimationprocess. Thesecondphase,theninvolvestheuseofthesebias-freesensormeasurements 110

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toestimatethegimbalangle asdescribedinSection5.2.Thecharacteristicsofthe sensorsandthemethodsusedtoobtaintherequiredbias-freemeasurementsarenow presented.Modelsusedtogeneratesyntheticsensordataforuseinsimulationsare alsodiscussed. 5.3.1MeasurementofFlywheelStates Severaltechniquescanbeusedtoobtaintheon-orbitangularvelocityand accelerationmeasurementsoftheywheel,andtheydependonthetypeofsensors usedintheactuators.Rotaryfeedbackdevicese.g.,Hall-effectsensors,optical encoders,andresolversaretypicallyincorporatedintomomentumactuatorstoobtain measurementsoftherotor'sangularposition[33].Theangularpositionobtained usingthe digital-like outputoftheHall-effectsensorsandencodersislessproneto high-frequencyelectricalnoise,andisusedtodeterminetheangularvelocityby nite difference .Sincethebiasinthesensoroutputpertainsonlytotheangularposition, theangularvelocitymeasurementsderivedby nitedifference wouldbebias-free.The equationusedtomodeltheangularvelocitydeterminedusingHall-effectsensorsor encodersisthereforegivenby e = + fv where,therandomnoise fv intheywheelangularvelocitymeasurementsismodeled asazero-meanGaussianwhitenoiseprocesswithacovariancegivenby E f fv )]TJ/F46 11.9552 Tf 5.48 -9.684 Td [( fv T g = 2 fv 1 Thevalueof 2 fv isderivedfromlaboratorymeasurements.Thenoiseinangularvelocity couldbeduetomeasurementerrorsorexternaldisturbanceeffectse.g.,frictiondrag andmotorcontrol. PrecisioncurrentsensorssituatedontheywheelmotorcontrollerseeFigure3-7 canmeasurethecurrentdrawnbytheywheelmotor.Thecurrentdrawnbythemotoris 111

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directlyproportionaltoitstorqueoutput[33]asgivenby i = w k t where,thetorqueconstantofthemotorisrepresentedby k t ,andexpressedin Nm = A S.I.units.FromEq.5andEq.5,theangularaccelerationoftheywheel canbedeterminedusing e = k t e i I w o X F X F = + fa where, fa istherandomnoiseintheywheelangularaccelerationmeasurements. Thebiasincurrentsensormeasurementsisequaltothecurrentdrawindicatedbythe sensorwhentheywheelangularvelocityiszero,andcanthereforebedetermined. Therandomnoise fa intheywheelangularaccelerationmeasurementsinEq.5is modeledasazero-meanGaussianwhitenoiseprocesswithacovariancegivenby E f fa )]TJ/F46 11.9552 Tf 5.479 -9.684 Td [( fa T g = 2 fa 1 Theywheelangularaccelerationcanalsobefoundbydifferentiatingtheangular velocitydeterminedfromHallsensorsandencoders.Thesensorparametersconsidered forsimulationarederivedfromtheHall-effectandcurrentsensormeasurementsofthe IMPAC CMGshowninFigure3-7. 5.3.2MeasurementofSpacecraftStates Thespacecraftstatestobemeasuredincludethespacecraftangularvelocity andangularacceleration .Thesemeasurementscanbeobtainedusinginertial sensorse.g.,gyroscopesandaccelerometerswhicharetypicallyincludedasapart ofthespacecraft'sGNCsuite.Thegyroscopeconsideredinthischapterisapart oftheADIS16485inertialmeasurementunitIMUfromAnalogDevices[114],and theaccelerometersconsideredaretheHoneywellQA-3000inertialnavigationgrade uni-axialaccelerometers[113].Thesensorsuiterequiredforthemeasurementof 112

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spacecraftangularvelocityandaccelerationincludeonetri-axialgyroscopeandsix uni-axialaccelerometers,similartoChapter4.Anindicativeplacementofthesesensors inanarbitraryspacecraftisshowninFigure4-1. 5.3.2.1Spacecraftangularvelocity Thespacecraftangularvelocityismeasuredusingthetri-axialgyroscope.The angularvelocitymeasurementsfromthegyroscopearecorruptedwithbiasandnoise, andtheoutputcanbemodeledas e = + sv + sv sv = sv where sv = sv modelsthebiasrandomwalk[39].Themeasurementnoise sv in thespacecraftangularvelocityismodeledasazero-mean,Gaussian,whitenoise processwithacovariance E f sv sv T g = 2 sv 1 .Thebiasrandomnoise sv isalso similarlymodeledwithacovarianceof E f sv sv T g = & 2 sv 1 .Thevaluesof sv and & sv areobtainedfromlaboratorymeasurementsorfromthemanufacturer'sdatasheet.In additiontothebiasrandomwalk,thesensormeasurementsarealsoaffectedbythe relativelylargeinitialbias sv .Thegyroscopebiashastobeestimatedandremoved fromthemeasurementstobeabletoobtainaccurateestimatesofthegimbalangle.The biascanbeestimatedbyusingdatafromspacecraftattitudesensorse.g.,Sunsensor, startracker,etc.,andgyroscopemeasurementsinanextendedKalmanlter.Thisbias estimationprocesshasbeenwellpublishedinliterature[39,41,43,111]andhencenot repeatedhere.Itshouldbenotedthatanymisalignmentintheattitudesensorsitself doesnotaffectthebiasestimateasthelterconvergenceisdependentonchangein attitude,andnotontheabsoluteattitude.Thegimbalangleestimationmethod,thus, isnotaffectedbyattitudesensormisalignments.Oncethebiashasbeenestimated,it canbeusedtoobtainbias-freeangularvelocitymeasurements.Theangularvelocity measurementscanthenbemodeledasgivenbyEq.4. 113

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5.3.2.2Spacecraftangularacceleration TheangularaccelerationofthespacecraftisdeterminedasdescribedinChapter4. 5.3.3CorrelatingFlywheelandSpacecraftMeasurements Acommonpracticalissueinobtainingsimultaneousmeasurementsfrommultiple sensorsistheirtimesynchronization.Asindicatedearlier,thegimbalangleestimation methoddescribedinthischapterisbaseduponmeasurementsofavectorobtained inonecoordinatebasis,anditseffectintheother.Itisthereforerequiredthatthe measurementsinbothcoordinatebasesareobtainedforsameinstantsintime.Due toseveralreasons,themeasurementsmaynotbetimesynchronized.Forexample, thespacecraftinertialsensorsgyroscopeandaccelerometerandtheywheel sensorsontheCMGhaveseparateADconverterswhichmaybeunsynchronizedin time.Theuncorrelateddatamayaffecttheaccuracyofthegimbalangleestimate. Proceedingwiththecaseasdescribedbytheaboveexample,let'sassumethe spacecraftmeasurements and aresynchronizedwitheachother,andthe ywheelmeasurements and arealsosynchronizedwitheachother.Further, let'sassumethespacecraftstatesandywheelstatesareunsynchronized,andthat theywheelmeasurementslagthespacecraftmeasurementsby t seconds.The signalswillnowhavetobesynchronizedbeforeusingthemtoestimatethegimbal angle.However,thetimelag, t isunknownandneedstobedetermined.The processofdetermining t isnowdescribed.AngularaccelerationoftheCMGywheel redistributesthespacecraftangularmomentumandhencecausescorresponding angularaccelerationofthespacecraft.Theexternaltorqueonthespacecraftis consideredtobenegligiblewhencomparedtothetorqueduetoywheelacceleration. WecanthereforeconcludebyobservingtheequationsofmotiongivenbyEq.5,that themagnitudeofywheelangularacceleration k k andthemagnitudeofspacecraft angularacceleration k k areperfectlycorrelatedwitheachotherforanygiveninput fromtheywheel.Thetimelag t betweenthemeasurementscanthereforebe 114

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determinedbycomputingthecross-correlation[119]betweenthetrajectoriesof k t k and k t k .Thecross-correlationbetweenthetrajectoriescanbecomputedusingthe xcorr commandinMatlab R .Theresultofonesuchsimulationperformedonasetof measurementswheretheywheelmeasurementslagthespacecraftmeasurementsby 200 miliseconds isshowninFigure5-2.Thecross-correlationvalueisamaximumwhen thetimeoffsetonthex-axisisequaltothelag t presentbetweenthemeasurements. Thebluelineintheplotshowsthepeakoccurringatatimeoffsetequalto200 ms whereastheredlinegeneratedusingsynchronizedmeasurementsshowspeak cross-correlationatatimeoffsetof0 ms .Thetimelag t canthusbeidentied. 5.4SimulationsandResults Simulationsbasedonaccuratelymodeledsensordatawereperformedtoevaluate theeffectivenessandimplementabilityoftheproposedestimationmethod.The sensorcharacteristicsusedinthesimulationswereobtainedeitherusinglaboratory measurementsormanufacturer'sdatasheet.Simulationsareperformedusingboth lowSNRandhighSNRmeasurementsetstoevaluatethedifferenceisestimation accuracy.Fourdifferentcaseswereconsideredforthesimulations.Therstcaseuses measurementswithlowSNRandthesecondcaseusesmeasurementswithrelatively higherSNR.Smallgimbalmisalignmentsareconsideredforthesetwocases.Athird casewithalargegimbalmisalignmentandhighSNRmeasurementsisalsopresented toshowthattheestimationaccuracyisindependentofthemagnitudeofmisalignment. Thefourthcaseusesthesamemeasurementsetandgimbalmisalignmentasthethird, buttheestimationisperformedusingthetruncated A matrixasgiveninEq.5.Each oneofthethreeleastsquarestechniquesdiscussedinSection5.2.2wereusedto estimatethegimbalangleineverycase.Thesimulationparametersthatarecommonto allfourcasesarelistedinTable5-1.Thesimulationswereperformedusing200 seconds ofemulatedsensormeasurements.TheSimulink R modelusedtogeneratethese measurementsisshowninFigure5-3.A2%differenceisassumedbetweentheinertias 115

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ofthereferencespacecraftmodelusedtogeneratedata,andthespacecrafttruthmodel usedforestimation,asseeninTable5-1.Thisaccountsforanyambiguitiesinthe knowledgeofthespacecraft'sinertia.Theaccuracyoftheresultspresentedisafunction oftheperformanceofthesensorsandtheexactnessofthedynamicmodelusedfor estimation.Ifidealnoise-freesensordataandanexactdynamicmodelwereavailable, thentheleastsquaressolutionwillnotbeanapproximation,butanexactsolution.The simulationsareperformedasperthefollowingsteps. 1.Thedynamicsofaspacecraftaresimulatedforanarbitraryywheelinputusing thereferencespacecraftdynamicmodelandreferencegimbalangle.Thetrue valuesoftheywheelstates and ,spacecraftangularvelocity ,andlinear acceleration a i ofcertainpointsinthespacecraftareobtainedfromtheoutputof thedynamicmodel 2.Theexactvaluesoftheywheelandspacecraftstatesarecorruptedusing sensorbiasandnoiseasperthesensormodels,whichthenemulateon-orbit measurements 3.Theangularaccelerationofthespacecraftisestimatedfromlinearaccelerometer dataasdescribedinChapter4.Steps1through3areaccomplishedusingthe Simulink R modelshowninFigure5-3. 4.Thebiasinspacecraftangularvelocitymeasurementsisestimatedbyanextended Kalmanlterusingspacecraftattitudedata,andthebiasinthespacecraft angularaccelerationmeasurementsisestimatedusingalinearKalmanlter asdescribedpreviously.Thebiasesarethenremovedfromtheangularvelocity andaccelerationmeasurements.Inadditiontoestimatingtheaccelerationbias, thelinearKalmanlteralsosmoothestheangularvelocitymeasurements 5.Usingbiasfreemeasurementsofthespacecraftandtheywheelstates,andan assumedcurrentbestknowledgespacecrafttruthmodel,thegimbalangleis estimatedusingtheleastsquaresmethods 6.AMonte-Carloanalysis,withdifferent seeds inSimulink'srandomnumber generator[120]isperformedtoverifytherepeatabilityofthesolutionaccuracy Theresultsfromthesimulationsarenowpresented. 5.4.1Case1:LowSignaltoNoiseRatioandSmallMisalignment Thereferencegimbalangleinthiscaseis0.1 andtheywheelinputproleis basedoncommandingtheywheeltoachieve1000rad/suntilbeingswitchedoffat35 116

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seconds.Sincethereisnoinputfromtheywheelformorethan3/4ofthesimulation time,theSNRofthemeasurementscorrespondingtothistimeperiodisverylow andcontainspoorinformationonthegimbalangle.Thebias-freeywheelangular velocityandaccelerationmeasurementsareshowninFigure5-4A,andFigure5-4B, respectively.Itcanbeseenthattheywheelacceleratestoasteadystatespeeduntil beingswitchedoffat35seconds.Theywheelthendeceleratestozerospeeddue tobearingfriction,andremainsatthatstatefortheremainderofthesimulationtime. Thespacecraftresponseangularvelocityandaccelerationtothisywheelinputis showninFigure5-5AandFigure5-5B.Theestimationerrorsofthethreemethodsfor a150trialMonte-CarloanalysisisshowninFigure5-6.Theaverageerrorofthethree estimationmethodsderivedfromtheresultsoftheMonte-Carloanalysisisgivenin Table5-2.Itcanbeseenthaterrorintheindependentleastsquaresmethodishigh, asthismethodtreatseachestimate k equally,andthemajorityoftheestimatesuse measurementswithlowSNR.Thebatchleastsquaresandtheweightedleastsquares ontheotherhandperformwellduetotheirnoise-averagingfeature.Thebatchleast squaresestimateisslightlybetterthantheweightedleastsquaresestimateasthe formeraveragesnoiseovertheentiresetofmeasurementswhereasthelatteraverages thenoiseonlyoverthefewmeasurementswithhighSNR. 5.4.2Case2:HighSignaltoNoiseRatioandSmallMisalignment Thereferencegimbalangleinthiscaseis0.1 ,buttheywheelinputproleis basedonasinusoidalangularvelocitycommandwithanamplitudeof837.76rad/s andafrequencyof0.094rad/s.Sincetheywheeliscontinuallyaccelerating,the overallSNRofthemeasurementsarerelativelyhighercomparedtothepreviouscase. Thebias-freeywheelangularvelocityandaccelerationmeasurementsareshownin Figure5-7AandFigure5-7B,respectively.Thespacecraftresponsetothisywheel inputisshowninFigure5-8AandFigure5-8B.Theestimationerrorsofthethree methodsfora150trialMonte-CarloanalysisisshowninFigure5-9.Theaverageerror 117

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ofthethreeestimationmethodsderivedfromtheresultsoftheMonte-Carloanalysisis giveninTable5-3.Itcanbeseenthaterrorintheindependentleastsquaresmethod isstillhighbut,hasimprovedfromthepreviouscaseasthemajorityoftheestimates usemeasurementswithhighSNR.Theperformancesofthebatchleastsquaresand theweightedleastsquareshavefurtherimprovedbutnow,theweightedleastsquares estimateisslightlybetterthanthebatchleastsquaresestimate.Thisisbecausethe weightedleastsquaresmethodaveragesthenoiseovermeasurementswithhighSNR whichinthiscase,constitutesmostofthemeasurements. 5.4.3Case3:HighSignaltoNoiseRatioandLargeMisalignment Thereferencegimbalangleinthiscaseis84 andemulatesasituationinwhichthe gimbalorientationiscompletelyunknown,possiblyduetoafailedencoder.Theywheel inputproleissimilartothatofcase2showninFigure5-7AandFigure5-7B.The spacecraftresponsetothisywheelinputisshowninFigure5-10AandFigure5-10B. TheestimationerrorsaregiveninTable5-4.Notethat,theerrorsreportedinthistable aretheresultsofasinglesimulationrunandnottheaverageerrorofaMonte-Carlo analysis,unlikeTables5-2,and5-3.Theintentofsimulatingthiscaseistoshowthe invarianceoftheestimationaccuracyoverthemagnitudeofmisalignment,andthiscan beveriedbycomparingthevaluesinTable5-3andTable5-4. 5.4.4Case4:TruncatedAMatrix Thesamesimulationasincase3isrepeatedbuttheestimationisperformed usingthetruncated A matrixasgivenbyEq.5toassesstheeffectofneglecting thegyroscopiccouplingtermintheequationsofmotionusedforestimation.Theother simulationparametersremainthesame.TheestimationerrorsaregiveninTable5-5.It canbeseenthat,neglectingthegyroscopiccouplingtermproducessignicantlyhigher errors. Amethodtoperformon-orbitestimationofCMGgimbalorientationbasedon Euler'sequationsofmotionwasdeveloped.Theknowledgeofangularacceleration 118

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obtainedfromthemethoddescribedinChapter4wasusedintheestimationscheme. Anaccuracyofabout1 arc-minute wasachievedusingtheweightedleastsquares solution.Theaccurateknowledgeofthegimbalorientationwillhelpinimprovingthe pointingperformanceoftheCMGactuators.Theeffectofanunbalancedywheelon thepointingaccuracyofthespacecraft,andamethodtoperformon-orbitbalancingof ywheelsispresentedinthenextchapter. Table5-1.SimulationParametersforGimbalAngleEstimation ParameterValueUnits Simulationtime200 s Solver4thorderRunge-Kutta n = a Timestep0.0002xed s SpacecraftinertiaTruel J s c = 2 4 155 )]TJ/F20 11.9552 Tf 9.299 0 Td [(5 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.298 0 Td [(5155 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.5160 3 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kgm 2 SpacecraftinertiaModel J s c = 2 4 151.9 )]TJ/F20 11.9552 Tf 9.298 0 Td [(4.9 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.298 0 Td [(4.9151.9 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.5 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.5156.8 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 kgm 2 Flywheelinertia I w o = 2 4 600 03.42250 003.4225 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 kgm 2 Gimbalframerotationmatrix R BG = 2 4 0.70710.7071 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.4082 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.57740.70710.4082 0.577300.8165 3 5 n = a Dataacquisitionrate50 Hz Spacecraftinitialangular velocity = [ 0.20.10.1 ] T rad = s Flywheelangularvelocity measurementnoise fv =1.047 rad = s Flywheelangularacceleration measurementnoise fa =0.033 rad = s 2 Gyroscopenoise sv =2.79 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 rad = s Gyroscopebiasinstability & sv =2.14 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 rad = s 2 Gyroscopeinitialbias B sv =8.73 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 [ 111 ] rad = s Accelerometernoise ia =6.87 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 m = s 2 Accelerometerbiasinstability & ia =2.22 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 m = s 3 Accelerometerinitialbiassix accelerometers ia =0.0147 [ 010101 ] m = s 2 Accelerometerdistances k r x 12 k = k r y 12 k = k r z 12 k =0.45 m WeightingParameter w =0.001 n = a 119

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Table5-2.ResultsSummaryCase1 SolutionMethodAverageError degrees IndependentLeastSquares17.3610 BatchLeastSquares-0.0928 WeightedLeastSquares-0.1250 Table5-3.ResultsSummaryCase2 SolutionMethodAverageError degrees IndependentLeastSquares-0.0684 BatchLeastSquares0.0292 WeightedLeastSquares-0.0171 Table5-4.ResultsSummaryCase3 SolutionMethodError degrees IndependentLeastSquares-0.1085 BatchLeastSquares-0.0211 WeightedLeastSquares0.0095 Table5-5.ResultsSummaryCase4 SolutionMethodError degrees IndependentLeastSquares-7.8791 BatchLeastSquares-2.4221 WeightedLeastSquares-3.2312 120

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Figure5-1.SchematicRepresentationofaSingleCMGinanArbitrarySpacecraft Figure5-2.Cross-CorrelationPlotBetween and 121

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Figure5-3.SimulinkModelforEmulatingSensorMeasurements 122

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AFlywheelAngularVelocity BFlywheelAngularAcceleration Figure5-4.FlywheelMeasurementsCase1 123

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ASpacecraftAngularVelocity BSpacecraftAngularAcceleration Figure5-5.SpacecraftMeasurementsCase1 124

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Figure5-6.Monte-CarloAnalysisResultsCase1 125

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AFlywheelAngularVelocity BFlywheelAngularAcceleration Figure5-7.FlywheelMeasurementsCase2 126

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ASpacecraftAngularVelocity BSpacecraftAngularAcceleration Figure5-8.SpacecraftMeasurementsCase2 127

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Figure5-9.Monte-CarloAnalysisResultsCase2 128

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ASpacecraftAngularVelocity BSpacecraftAngularAcceleration Figure5-10.SpacecraftMeasurementsCase3 129

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CHAPTER6 ON-ORBITJITTERCONTROLUSINGATHREE-FLYWHEELSYSTEM Ahomogeneousandperfectlybalancedywheelwasconsideredintheprevious chapters.Aywheelwithimbalanceinduceson-boardvibrationswhichcauses oscillatorytranslationalandattitudemotionofthespacecraft.Onlytheoscillatory attitudemotioncalledattitudejitterwhichaffectsthepointingstabilityofthespacecraftis considered.Athree-ywheelapproachtoreducethemagnitudeofjitteremittedbythe ywheelinacontrolmomentgyroscopeCMGisdescribedinthischapter.Themethod performson-orbitbalancingoftheywheelsystembyadjustingtherelativeimbalance betweentheywheels.Althoughonlyattitudeeffectsareconsidered,thethree-ywheel approachalsoreducesthemagnitudeoftranslationaljitter. 6.1ExistingJitterMitigationMethods Isolationofvibrationsinducedbyunbalancedrotorshasbeenstudiedextensively. Severalmethodsinvolvingpassiveand/oractiveisolatorshavebeendevisedand implementedtoattenuatevibrationsandreduceitsimpactontheperformanceof payloads[5360].Thesemethodsfunctionbyeitherisolatingthesourceofvibration fromtherestofthespacecraftorbyisolatingthepayloadalonewhichissensitive tovibrations.Insomecases,boththepayloadandthesourcesofvibrationare isolated.Passiveisolatorsareusuallysomeformofviscoelasticmateriale.g., rubberorspeciallydesignedviscousdamperswithvariablestiffnessanddamping. Theseisolatorsareplacedbetweenthevibrationsourceandthepayload.Theisolator functionsasamechanicallowpasslter,absorbingthevibrationenergyuptoacertain breakfrequencyandthusdecreasesthemagnitudeofvibrationseenbythepayload. Theyprovidereducedtransmissibilityonlybeyondthebreakfrequencyandrolloffas asecondordersystemat40dB/decade.Althoughloweringthebreakfrequencyby usingsofterisolatorsmayprovideisolationatlowerfrequencies,itreducesthestiffness ofthesupports.Thismakesthepayloadvulnerabletolargedisplacementsduring 130

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launchandcollisionwithotherspacecraftcomponents.Italsointroducesundesirable dynamicsrattleduringattitudecontrol.Passiveisolatorshavebeenusedonmany missions[5355]includingtheHubbleSpaceTelescope.Activeisolatorsinclude electromechanicalactuatorssuchasvoicecoils,magneticactuators,andpiezo-electric stacks.Multiplesuchactuatorsaresometimesusedinahexapodconguratione.g., Stewartplatform[56,57]toprovidemultidegree-of-freedomisolation.Thecontrolof theseactuatorsisbasedonfeedbackfromaccelerometers/forcesensorsmountedatthe payloadinterface,andrequireacomplexcontrolalgorithm.Theseisolatorsprovidegreat isolationatlowfrequencies,butarelimitedbythebandwidthoftheactuators/control systemsathigherfrequencies.Activeisolatorsrequirecontinuouspowertoisolate, andevensupportthepayload.Thus,activeisolatorsrequiresomesortofalaunchlock forrestraintduringlaunch.Theperformanceoftheactiveactuatorsdependsonthe efciencyofvibrationfeedbacksensorsandcontrolalgorithms.Thevariouscomponents oftheactiveisolatorsaddsignicantmasstothespacecraft.Hybridactuators[5759] thatincludeacombinationofactiveandpassiveisolatorshavebeenimplementedto provideawiderbandwidthofisolation,andasteeperrolloff.Theyhavealsobeen usedtoperformlimited,butnepointingofthepayload[57].Hybridactuatorswith adaptivedampingbasedonshapememoryalloyshavealsobeendeveloped[60]. However,hybridsystemsexperiencechallengesencounteredinbothactiveandpassive systems.Asopposedtoattenuationmechanismsjustdiscussed,faststeeringmirrors thatcompensateforjitterbydynamicallyalteringthepathoftheopticalbeamhave beenusedtomitigatejittereffectsinopticalcommunicationandimagingsatellites[121]. Anothercompensationmethodistheuseofpost-processingtechniquestorestore jitteraffectedimages[122].Thismethod,specictoimagingpayloads,isnotsufcient initselfandistypicallyusedtoaugmenttheperformanceobtainedbyphysicaljitter reductionmethodssuchasisolators. 131

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Thejittermitigationmethodsdescribedabovearemeanstoreducethe effect ofjitteronthepayload,anddonotreducethejitterproducedbythesource.This chapterintroducesamethodthatreducesthemagnitudeofjitteremittedbyywheel imbalanceinaCMG.However,themethoddevelopedinthischaptercanbeusedin conjunctionwithisolatorstofurtherattenuatethelowmagnitudejitter.Theremainderof thechapterisorganizedasfollows.Thedynamicsofaspacecraftwithanunbalanced ywheelispresentedinSection6.2.Jitterreductionusinganovelthree-ywheel systemisdescribedinSection6.3alongwiththedevelopmentofadynamicmodelfor aspacecraftwithathree-ywheelsystem.Simulationsandresultsthatcomparethe performanceofsingle-andthree-ywheelsystemsarepresentedinSection6.4.An indicativedesignofathree-ywheelsystem,powerandmasstradesarealsogiven. 6.2FlywheelImbalanceandSpacecraftJitter ConsidertheschematicofaspacecraftasshowninFigure6-1.Thespacecraft consistsofaspacecraftstructureandasingleywheelbelongingtoaCMG.Weare interestedonlyinthesourceofvibration,andnotitstransmissionthroughthespacecraft structure;arigidspacecraftishenceconsidered.Sincetheywheelimbalanceand ywheelrotationaretheonlyfactorsthatcausejitter,thegimbaloftheCMGisassumed tobexedrelativetothespacecraft.Thisassumptionallowsustostudytheeffects ofjitterindependentofthedynamicscontributedbygimbalmotion,andpermitsthe representationofthesystemsuchthattheywheelisrigidlysecuredtothespacecraft usingbearings.Itywheelcanthereforerotateaboutanaxisxedinthespacecraft body.Theywheelisassumedtohavesomeimbalance,andasingleunbalanced ywheelsingleCMGisconsideredforclarityintheensuinganalysis.Theunbalanced ywheelisrepresentedbyacombinationofahomogeneousdiscofmass m w anda smalleccentricmass m e .Thechangeinthelocationofthespacecraftcenterofmass c.m. C duetothemotionoftheeccentricmassresultsintranslationaljitter,butas statedearlier,onlyattitudejitterisconsideredinthischapter.Further,theresulting 132

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changesinthespacecraftinertiaisalsonegligible.Themotionofthespacecraft c.m. is henceignored.The c.m. ofthehomogeneousdiscisatapoint O whichliesonthe ywheelrotationaxis.Theeccentricmassisataxeddistance r e relativeto O .The angularvelocityofthespacecraft,andtheangularvelocityoftheywheelrelativeto thespacecraftarerepresentedby ,and ,respectively.Acoordinatebasis C B given by h X B Y B Z B i isattachedtothespacecraft.Anothercoordinatebasis C A givenby h X A Y A Z A i isalsoattachedtothespacecraftsuchthattheywheelrotationaxisis along X A .Thepositionoftheeccentricmassisrepresentedinacoordinatebasis C F givenby h X F Y F Z F i whichisxedtotheywheelsuchthat X F isalongtheywheel spinaxis,and Y F isalongtheprojectionof r e inthe Y F Z F plane.Itcanthereforebe observedthat Y F alignswith Y A whentheangle =0.Theangle isreferredtoas theywheelphase.ThedirectioncosinematrixDCMrepresentingthecoordinate transformationfrom C F to C A isgivenby R AF = 0 B B B B @ 100 0c )]TJ/F20 11.9552 Tf 9.299 0 Td [(s 0s c 1 C C C C A where,c ands representthe cosine and sine oftheangle ,respectively.The coordinatetransformationfrom C A to C B isarbitraryanddependsontheorientationof theCMGgimbalaxisinthespacecraft.Thistransformationisassumedtobeanidentity matrixgivenby R BA = 1 ,withoutlossofgenerality.Thecoordinatetransformationfrom C F to C B isthereforegivenby R BF = R AF .TheDCM R BF isdenotedby R forbrevity. Thetotalangularmomentumofthespacecraftabout C ,expressedinthespacecraft bodycoordinatebasisisgivenby B h c = J s c + R )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(J w o + m e r o R )]TJ/F20 11.9552 Tf 5.48 -9.83 Td [([ ] r e 133

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Theinertiamatrixofthespacecraftinclusiveoftheywheelabout C ,denotedby J s c is givenby J s c = I s c + R )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(I w o R T + m w )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o )]TJ/F58 11.9552 Tf 13.449 -9.684 Td [(r o T + m e )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o + B r e T )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(r o + B r e 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o + B r e )]TJ/F58 11.9552 Tf 13.45 -9.684 Td [(r o + B r e T andtheinertiamatrixoftheywheelabout O ,denotedby J w o isgivenby J w o = I w o + m e )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r e T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r e 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r e )]TJ/F58 11.9552 Tf 13.45 -9.684 Td [(r e T where,theinertiaoftherigidspacecraftstructure,about C isrepresentedby I s c ,and theinertiaofthehomogeneousdiscabout O isrepresentedby I w o .Indevelopingthe spacecraftattitudeequationsofmotion,weneglectthechangeintheinertiaofthe spacecraftduetothedisplacementoftheeccentricmassandalsoassumezeroexternal torqueonthespacecraft.Thus,usingEuler'ssecondlaw,thespacecraftattitudemotion isgovernedby J s c + [ ] B h c = )]TJ/F58 11.9552 Tf 11.955 0 Td [(R J F o )]TJ/F59 11.9552 Tf 11.955 0 Td [(m e r o R r e )]TJ/F58 11.9552 Tf 11.955 0 Td [(R [ ] J F o )]TJ/F59 11.9552 Tf 11.955 0 Td [(m e [ R ] r o R )]TJ/F20 11.9552 Tf 5.48 -9.83 Td [([ ] r e ItcanbeseenfromEq.6thatthefourtermsontherighthandsideoftheequation contributetotheevolutionofthespacecraftstatesastheycontinuouslyredistributethe spacecraft'sangularmomentum.Thersttwotermsresultfromywheelacceleration whichmayexcitecertainexiblemodesofthespacecraftcausingshort-termattitude jittertransients.OnobservationofEq.6,itisclearthatthiseffectcanbecaused evenwithaperfectlybalancedywheel.Thefocusofthischapter,however,isthe moresignicantsourceofjitterviz.,ywheelimbalancewhichisrepresentedbythe thirdandfourthtermsinEq.6.First,considerthefourthterminEq.6whichis aconsequenceofmasseccentricity.Sincethedirectionof B r e changescontinually duetoywheelrotation,theangularmomentumstateofthespacecraftisaffected, inducingattitudejitter.Thiseffectvanishesiftheeccentricityvector r e isequalto 0 or 134

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if r e isparalleltotherotationaxisywheelangularvelocityvector.Thus,theeffects representedbythefourthtermdependonlyonthecomponentofmasseccentricity normalto andisreferredtoas staticimbalance .Thestaticimbalanceintheywheel isusuallyexpressedin gmm andisgivenbytheproduct m e r e ,where r e = r e T Y F Alsonotethatifthespacecraftcenterofmassandtheeccentricmassarecontained inthesameplane,thenstaticimbalanceresultsinonlytranslationaljittereffects.Now, considerthethirdterminEq.6whichisaconsequenceofnon-symmetricmass distributionintheywheel.Theangularmomentumcomponent J w o changesdirection inthespacecraftbodycoordinatebasisatafrequencyequaltotheywheel'sspeed, whichimpartsanoscillatingtorqueonthespacecraftcausingattitudejitter.Thiseffect vanishesif J w o isparalleltotheangularvelocityvector,i.e.,iftheprincipalaxisof J w o coincideswiththeangularvelocityvector.Thecoincidenceoftheprincipalaxiswith theangularvelocityvectorisdependentontheexistenceofaplaneofmasssymmetry normaltotherotationaxis.Theabsenceofmasssymmetryintherotorisreferredto as dynamicimbalance .Thedynamicimbalanceisusuallyexpressedin gmm 2 andits magnitudeisgivenbytheoff-diagonalcomponentsof J w o associatedwiththedirection of .Theneteffectonthespacecraftisthesumofjitterduetostaticanddynamic imbalances.Amoredetaileddiscussiononstaticanddynamicimbalanceisprovidedin AppendixB. Themagnitudeofjitterinducedbytheywheeldependsontheamountof imbalancepresentintheywheel.Traditionally,ywheelsarebothstaticallyand dynamicallybalancedonbalancingmachinesusingthe two-planebalancing method[123, 124].Thismethodinvolvescalculatedadditionorremovalofsmallamountsofywheel massatmultiplelocationsontwoparallelplanesxedtotheywheel.Duetolimitations ofmachinesandsensorsusedinbalancing,theywheelscontainsomeamountof residualimbalanceandcannotbeperfectlybalanced.Thebeginning-of-lifeBOLstatic anddynamicbalanceachievedbysomecommerciallyavailablemomentumactuators 135

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arelistedinTable6-1.Theactuatorsarelistedintheincreasingorderoftheirimbalance values.ThelastcolumninTable6-1liststheywheelbalancequalityusedtospecify thedisturbancecharacteristicsoftheactuator.Metricstoidentifythebalancingquality ofaywheelhavebeenstandardized.ThemostcommonlyusedstandardistheISO 1940/1 BalanceQualityRequirementsforRigidRotors [125],accordingtowhichthe balancingqualityisindicatedbytheletterGfollowedbyasetofnumbers.Ahigher`G number'impliesahigherimbalanceseeAppendixB.High-speedandprecisionrotors suchasinertialreferencinggyroscopesrequireabalancingqualityofG0.4orbetter, whereasrelativelylowerprecisionrotorse.g.,jetengineturbinesandmagnetichard diskdrivesrequireabalancingqualityofG2.5.Amoredetailedapplicationlistcanbe foundin[124].ThebalancequalityasperISO1940/1foraywheelofagivenmass dependsontheresidualimbalancestaticimbalanceonlyandtheangularspeed.It hencerepresentstheeffectofimbalance,andnotjusttheimbalance.Forinstance,in Table6-1,itcanbeseenthatduetoitslowspeed,theRWA-15actuatorhasabetter balancingqualityeventhoughithaslargerimbalancevaluesascomparedtothe VRW-1actuator.Thetotalpermissibleresidualimbalanceinarotorcanbedetermined usingtheISO1940/1balancingqualitychart[125]showninFigureB-1inAppendixB. Alternatively,foragivenbalancingquality,thepermissibleresidualimbalancecanbe computedusingtherelation m e r e = 9549G m w k k wherethestaticimbalance m e r e isin gmm ,ywheelmass m w isin kg ,andywheel speedisin rpm .ItcanbeseenfromEq.6thatforthesameimbalancevalue,the balancingqualitydeteriorateswithincreaseintheywheelspeed.Itisthereforedifcult toobtainagoodbalancequalityforywheelsspinningathighspeeds.Thesame argumentisreectedinTable6-1.Itwillbeshownlaterusingsimulationsthateven thehighbalancequalityachievedbytheactuatorsasgiveninTable6-1inducejitter thatisunacceptableforcertainmissions.Further,theimbalancemaydeteriorate 136

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duetoon-orbiteffectssuchasthermaldistortion.Therefore,someformofon-orbit jittermitigationmethodhastobeemployed.Athree-ywheelsystemthatproduces signicantlylessjitterwhencomparedtothetraditionalsingle-ywheelsystemandcan dynamicallyperformon-orbitbalancingisnowpresented. 6.3JitterMitigationUsingaThree-FlywheelSystem Thethree-ywheelbasedjittermitigationmethodpresentedinthischapter reducesthejitteremittedbyunbalancedywheelsunlikeothermethodsdescribed inSection6.1thatattenuatethetransmissionofjitter.Staticanddynamicbalancingof ywheelsusingbalancingmachinesisusuallytherststeptowardsachievingfavorable jitterperformancefrommomentumactuators.However,duetouncertaintiesand imperfectionsassociatedwithbalancingsystemsandsensors,aprecisionbalanced ywheelisneverperfectandcontainsresidualimbalance.Themethodpresentedin thischapterdoesnotreducetheresidualimbalanceoftheindividualywheel,butuses twoadditional,similarlybalancedywheelstocountertheeffectofresidualimbalance. ItwasmentionedinSection6.1thattwobalancingplanesarerequiredinorderto performstaticanddynamicbalancing.Weintuitivelyextendthisideabyaugmentingthe single-ywheelsystemwithtwoadditionalywheelstoperformbalancingoperationson they.Thefollowingassumptionsaremadeinthedesignofthismethod: 1.Multipleywheelsproducedinthesamebatch,madefromthesamematerial stock,andbalancedusingthesamemachinetothesamebalancingqualitywill havenearequalstaticanddynamicimbalance 2.Thedifferenceinthestaticanddynamicimbalancebetweentheywheelsthus producedismuchsmallerthantheimbalanceoftheindividualywheels 3.Jittercausedbyywheelimbalanceisgreaterthanthatcausedbybearingrace imperfections 4.Structuraldeectionsduetohighspeedrotationhoopstrainandthermal gradientsisuniformacrossallywheels Itisemphasizedthattherstandsecondassumptionsonlyrequirethesamebalancing qualityamongtheywheelsanddo not requirehighbalancequalityoftheindividual 137

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ywheels.Thevalidityoftherstandsecondassumptionswereveriedbydiscussions withabalancingfacility[126].Thethirdassumptionisnecessaryasadamagedbearing wouldvoidthelowjitterobtainedbyabalancedsystem.Thelastassumptionimplies thattheimbalanceinalltheywheelswouldchangebythesamepercentageoftheir initialimbalancewhensubjectedtohooporthermalstresses. ConsideraspacecraftsimilartotheonepresentedinSection6.2.Butnow,we substitutethesingleywheelwiththreeywheelsthatabidebytheassumptionsabove suchthatthecombinedmassandaxialinertiaofthethree-ywheelsystemareequal tothoseofthesingleywheel.Aschematicofthethree-ywheelsystemisshown inFigure6-2.Theaxialspacingoftheywheelsisexaggeratedintheschematicfor clarity,andinanactualsystem,thethreeywheelsareintegratedcompactlytofunction togetherinavolumeslightlygreaterthanthatoccupiedbyasingle-ywheelsystem. Eachywheelisindependentlycontrolledbyitsownmotor.Similartothesingle-ywheel caseinSection6.2,thespacecraft c.m. isdenotedby C ,andthecoordinatebasis C B givenby h X B Y B Z B i isxedinthespacecraftbody.Thethreeywheelsaredenoted by F 1 F 2 ,and F 3 ,andeachywheelisrepresentedasacompositionofahomogeneous discandasmalleccentricpointmassasinthepreviouscaseofthesingle-ywheel system.Themassofthehomogeneousdiscsassociatedwithywheels F 1 F 2 and F 3 aredenotedby m w 1 m w 2 ,and m w 3 ,respectively.Thediscofywheel F 1 hastwicethe massandaxialinertiacomparedtodiscsofywheels F 2 and F 3 .Thecenterofmass ofthehomogeneousdiscsrepresentedby O 1 O 2 ,and O 3 areatdistances r o 1 r o 2 ,and r o 3 ,respectivelyfrom C .Theeccentricmassoftheywheels F 1 F 2 ,and F 3 aredenoted by m e 1 m e 2 ,and m e 3 ,respectively.Thepositionoftheeccentricmasses m e 1 m e 2 ,and m e 3 relativeto O 1 O 2 ,and O 3 aregivenby r e 1 r e 2 ,and r e 3 ,respectively.Sinceallthree ywheelsarebalancedtothesamequalityseeAppendixB,theimbalanceinywheel F 1 istwicethatofywheels F 2 and F 3 .Further,sinceequalbalancequalitiesimplyequal displacementofthe c.m. ,theeccentricityvectorsofallthreeywheelsareequaland 138

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theeccentricmassofywheel F 1 istwicethatofywheels F 2 and F 3 .Theeccentricity vectorsexpressedinacoordinatebasisattachedtothecorrespondingthreeywheels arethereforeidentical.Anarbitraryandunbiasedeccentricityvectorisassumedandis givenby r e i = r e i [ 111 ] T i =1,2,3 where, r e i isthestaticimbalanceorthemagnitudeofeccentricitynormaltotheywheel rotationaxis,andisgivenby r e i = )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r e i T Y F i Basedonthepreviousassumptionsandrationale,therelationsbetweentheeccentric massesandeccentricitymagnitudesoftheywheelsaregivenby r e 1 = r e 2 = r e 3 = r e m e 1 2 = m e 2 = m e 3 = m e Acoordinatebasis C A givenby h X A Y A Z A i isattachedtothespacecraftsuchthat X A denesthespinaxesoftheywheels.Acoordinatebasis C F 1 givenby h X F 1 Y F 1 Z F 1 i isattachedtotheywheel F 1 suchthat X F 1 isalong X A and Y F 1 isalongtheprojection of r e 1 onthe Y F 1 Z F 1 plane.Theanglebetween Y F 1 and Y A about X A isgivenby 1 Thecoordinatebasis C F 2 givenby h X F 2 Y F 2 Z F 2 i ,and C F 3 givenby h X F 3 Y F 3 Z F 3 i associatedwithywheels F 2 ,and F 3 ,respectivelyaresimilarlydened.Theywheels F 2 ,and F 3 areaxiallyseparatedfromywheel F 1 by x a asshowninFigure6-2.The positionvectors r o 2 and r o 3 arethereforegivenby r o 2 = r o 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(x a X A r o 3 = r o 1 + x a X A Thecoordinatetransformationfromthe i th ywheelcoordinatebasis C F i to C A isgivenby aDCMidenticaltoEq.6butwith replacedby i .Theangle i isthephaseofthe i th 139

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ywheel.Thecoordinatetransformationfrom C F i to C B isthereforegivenby R BF i = R BA R AF i where, R BA isanyarbitraryDCMandisassumedtobetheidentitymatrix 1 without lossofgenerality.TheDCM R BF i issimplydenotedby R i forbrevity.Thecoordinate transformationsassociatedwithywheels F 2 ,and F 3 aresimilarlydened.Theangular velocitiesoftheywheels F 1 F 2 ,and F 3 relativetothespacecraftaredenotedby 1 2 ,and 3 ,respectively.Considerthecasewhereinthephaseandangularvelocity relationsbetweentheywheelsaregivenby 2 = 3 = 1 k 1 k = k 2 k = k 3 k = Itwillbenowbeshownthat,whenthethree-ywheelsysteminFigure6-2exactly satisesEqs.6through6,6,and6,thesystemachievesbothstaticand dynamicbalance.Itcanbeshownthattheangularmomentumofarigidspacecraftwith athree-ywheelsystemexpressedinthespacecraftbodycoordinatebasis C B isgiven by B h c = J s c + 3 X i =1 R i J F i o i i + m e i r o i R i i r e i where,thespacecraftinertiamatrix J s c isgivenby J s c = I s c + 3 X i =1 R i I F i o R T i + m w i )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(r o i T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o i 1 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F58 11.9552 Tf 5.977 -9.683 Td [(r o i )]TJ/F58 11.9552 Tf 13.45 -9.683 Td [(r o i T + m ei )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(r o i + B r e i T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r o i + B r e i 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(r o i + B r e i )]TJ/F58 11.9552 Tf 13.45 -9.684 Td [(r o i + B r e i T andtheywheelinertiamatrix J F i o i isgivenby J F i o i = I F i o i + m e i )]TJ/F58 11.9552 Tf 5.977 -9.684 Td [(r e i T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r e i 1 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F58 11.9552 Tf 5.978 -9.683 Td [(r e i )]TJ/F58 11.9552 Tf 13.449 -9.683 Td [(r e i T 140

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Intheaboveexpressions, I s c istheinertiaofthespacecraftstructureabout C ,and I F i o istheinertiaofthehomogeneousdiscassociatedwiththe i th ywheelabout O i .The inertiamatrix I F i o canbeexpressedinthecoordinatebasis C F i as F i I F i o i =diag I ai I ti I ti where,theterms I ai and I ti representrespectively,theaxialandtransverseinertia ofthehomogeneousdiscofthe i th ywheel.Further,sincetheaxialinertiaofthe homogeneousdiscofywheel F 1 wasassumedtobetwicethatofthediscsassociated withywheels F 2 and F 3 ,wecanwrite 0.5 I a 1 = I a 2 = I a 3 = I a UsingEqs.6,6,and6,theangularmomentuminEq.6canberewritten as B h c = J s c + 3 X i =1 0 B B B B @ )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(I ai +2 m ei r 2 ei i s i )]TJ/F20 11.9552 Tf 11.955 0 Td [(c i )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(m ei r 2 ei i )]TJ/F20 11.9552 Tf 11.291 -0.131 Td [( c i +s i )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(m ei r 2 ei i 1 C C C C A + r o i 0 B B B B @ 0 )]TJ/F20 11.9552 Tf 11.291 -0.131 Td [( c i +s i m ei r ei i c i )]TJ/F20 11.9552 Tf 11.955 0 Td [(s i m ei r ei i 1 C C C C A where i = k i k ,andthearbitraryDCM R BA isassumedtobetheidentitymatrix 1 .It canbeinferredfromEq.6thatwitharbitraryvaluesforeachof i r e i m e i ,and i the systemisbothstaticallyanddynamicallyunbalanced.Furthermore,theassumptionsof Eqs.6,6,6,and6yield B h c = J s c + 0 B B B B @ 4 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(I a +2 m e r 2 e m e r 2 e 2s 1 +s 2 +s 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2c 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(c 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(c 3 )]TJ/F59 11.9552 Tf 9.299 0 Td [(m e r 2 e 2c 1 +c 2 +c 3 +2s 1 +s 2 +s 3 1 C C C C A + r o i 0 B B B B @ 0 )]TJ/F59 11.9552 Tf 9.298 0 Td [(m e r e 2c 1 +c 2 +c 3 +2s 1 +s 2 +s 3 m e r e 2c 1 +c 2 +c 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2s 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(s 3 1 C C C C A 141

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ThesecondandthirdtermsontherighthandsideofEq.6contributetothe dynamicimbalanceandstaticimbalance,respectivelyofthethree-ywheelsystem. OnobservationofEq.6,itisclearthatthestaticanddynamicimbalanceofthe systemdependonthephaseangles i .Further,themaximumimbalanceoccurs,when thephasesareequal,whichisaconditionanalogoustothesingle-ywheelsystem.On theotherhand,itcanbefoundthatwhenrelationbetweentheywheelphases i is givenbyEq.6,completestaticanddynamicbalanceisachieved,andtheangular momentumofthespacecraftcanthereforebewrittenas B h c = J s c +4 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(I a +2 m e r 2 e X A NotethatinEq.6,thereisnoeffectofmasseccentricities,andtheywheelangular momentumisalwaysparalleltoitsrotationaxis.Assumingzeroexternaltorqueonthe spacecraft,theratechangeofangularmomentumforthecompletelybalancedsystem Eq.6isgivenby B h c = J s c + [ ] h c = 0 Equation6representstheattitudeequationsofmotionofacompletelybalanced systemandaredevoidofanyjitter.Thus,completestaticanddynamicbalancecanbe achievedusingthethree-ywheelsystemthatperfectlysatisestheconditionsgivenby Eqs.6,6,6,and6.Theequationsofmotionofagenericthree-ywheel systemthatmaynotperfectlysatisfytheseassumptionsisgivenby J s c + [ ] B h c + 3 X i =1 R i J F i o i i + m e i r o i R i h i i r e i + 3 X i =1 R i i J F i o i i + m e i R i i r o i R i i r e i = 0 Thisequationisusedtoperformnumericalsimulationsinthenextsectiontoshowthat signicantjitterreductioncanbeachievedusingthethree-ywheelsystem. 142

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Note: Asindicatedearlier,thediscussioninthischapterandthedissertationin generalconsidersaxedmeanspeedfortheywheeltypicalofaCMGandhence thedesiredphaserelations,onceachievedarecontinuouslymaintainedasallthree ywheelspinatthesamespeed.Incaseofanactuatorsuchasareactionwheel,where theywheelaccelerationisthesourceofactuatortorqueoutput,allthreeywheelsare acceleratedcontrolledinunison.Thisensuresthattheywheelsdonotchangetheir relativephasedifference. 6.4SimulationsandResults Detailedsimulationsthatportraythecapabilityofthethree-ywheelsystemare presented.Theresultsofthesimulationsandsomepracticalimplicationssuchas ywheelfailure,andpowerandmasscharacteristicsofthethree-ywheelsystem arealsodiscussed.Thissectionisorganizedasfollows.First,simulationsofthe single-ywheelsystemshowninFigure6-1areperformedtoobtainbenchmarkjitter values.Next,abaselinesimulationisperformedonanidealthree-ywheelsystemto showthatunderidealconditions,thethree-ywheelsystemachievescompletestatic anddynamicbalance.Thisisfollowedbyasetofhigh-delitysimulationsperformed forywheelswithvariousbalancingqualities,andnon-idealconditions.Theeffectof single/multipleywheelfailuresistheninvestigatedwithanothersetofsimulations. Adiscussiononthepowerandmasscharacteristicsofthethreeywheel-systemis provided.Finally,acommentonthemeasurementofywheelphaseismade. 6.4.1Single-FlywheelSystem Thesingle-ywheelsystemissimulatedbynumericallyintegratingEq.6.The simulationparametersaregiveninTable6-2.Thespacecraftandywheelinertias arecomputedusingEq.6andEq.6,respectively.Theeccentricmass,and itspositionarechosensoastoobtainabalancingqualityofG1,whichisalsothe averagequalityoftheactuatorslistedinTable6-1.However,itshouldbenotedthat thedynamicimbalancevalueusedforthesimulationsissuperiorbetterbalanced 143

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tothoseoftheactuatorsinTable6-1.Theangularvelocityofthespacecraftobtained fromthesimulationsisshowninFigure6-3.Theangularvelocityconsistsoftwo components,viz.highfrequencyjitterandlowfrequencyprecession.Thejitteris duetotheunbalancedywheelasdiscussedearlier,andtheprecessionisdueto thegyroscopiccouplingbetweenthespacecraftangularvelocityandtheywheel angularmomentum.TheangularvelocityinFigure6-3Aisplottedfor15 seconds to showonefullcycleofprecession,beyondwhichthetrajectoryrepeatsitselfsince thereisnodampingorenergydissipationinthesystem.Thehighfrequencyjitteris seenclearlyinFigure6-3Binwhichtheangularvelocitytrajectoryisplottedforonly 0.1 seconds toshowazoomedview.AfastFouriertransformFFToftheangular velocityisshowninFigure6-4,anditcanbeseenthatthejitterfrequencyisequalto theywheelrotationspeedasexpected.ThemagnitudeofjitterasseeninFigure6-3B isabout5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 rad = s whichcorrespondstoanangularpointinginstabilityof0.19 arc-sec .Thisvalueofjittermaymeettherequirementsofcertainlow-accuracymissions, whileitisunacceptableinsome.Forinstance,theHubblespacetelescopehasan attitudestabilityrequirementof0.007 arc-sec [18]. 6.4.2IdealSystem Anideal/baselinesimulationthatdemonstratestheconceptofthethree-ywheel systemispresented.Thesystemparametersarechosensuchthattheconditionsgiven byEqs.6,6,6,and6areexactlysatised.Further,themassproperties oftheywheelsaresuchthatthesystembecomesequivalenttoasingle-ywheel systemwhenthethreeywheelsareinphase.Thesimulationparametersaregiven inTable6-3.SimulationisperformedbynumericallyintegratingEq.6,andbegins withallywheelsinphasewithrespecttoeachother,i.e., 1 = 2 = 3 ,andspinning atconstantandequalangularvelocities.Thesephaseandangularvelocityconditions resultinadynamicsystemequivalenttothesingle-ywheelsystemrepresentedby Eq.6.At10 seconds ,ywheel F 2 iscommandedatrapezoidalangularvelocity 144

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proleadditivetotheconstantangularvelocitylasting4.5 seconds suchthatit gains radians overthephaseofywheel F 1 1 .Simultaneously,ywheel F 3 isalso commandedthesameangularvelocityprolebutoppositeindirection. F 3 willtherefore lag F 1 by radians but,willbeinphasewith F 2 .Theequalandoppositemotionofthe ywheels F 2 and F 3 donotresultinanynetaccelerationonthespacecraft.However,it shouldbenotedthatwitharbitraryinitialphasesoftheywheels,itmaynotbepossible toalignthephaseswithoutimpartingsmalldisturbancestothespacecraft.Therelative phasealignmentsasgivenbyEq.6isachievedattheendofthe14.5 seconds andcompletelybalancesthethree-ywheelsystem.Thesystemdynamicsisnow representedbyEq.6whichisfreefromjitter.Sincetheangularvelocitiesofthe ywheelsareequal,therelativealignmentbetweentheywheelsremainsunchanged andthusmaintainsthedesiredphaserelations.Theresultsofthesimulationareshown inFigure6-5,whereFigure6-5Ashowsthecommandedywheelangularvelocity proles.Theangularvelocitycommandsofywheels F 2 and F 3 thatshifttherelative phasebetweentheywheelsby radians canbeseenbetween10and14.5 seconds Figure6-5Bshowsthespacecraftangularvelocity.Itcanbeseenthat,until10 seconds intothesimulation,thelevelofjittercorrespondsexactlytothesingle-ywheelcase showninFigure6-2astheimbalancesinbothsystemsareidentical.Asthephase betweentheywheelsisgraduallychangedbetween10 seconds and14.5 seconds thehighfrequencyjitteriscompletelyremovedfromthesystem.Theresidualmotionof thespacecraftisduetothecouplingbetweenthespacecraftangularvelocityandthe angularmomentumofthethree-ywheelsystem. 6.4.3High-FidelitySimulations Theabilityofanidealthree-ywheelsystemtoachievecompletestaticanddynamic balancewasdemonstratedintheprevioussimulation.Theidealsystemisbasedon theexactnessoftheconditionsgivenbyEqs.6,6,6,6,and6.Since manufacturingequipmentandmethods,materialhomogeneity,sensors,bearings,and 145

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controlsystemshaveassociatederrors,theseconditionsarenotperfectlyvalidand includesomeuncertainty.Wenowconsiderathree-ywheelsystemwithvariations intheabovementionedconditionsthatcanbeinducedbytheseuncertainties.The propertiesofahigh-delitymodelthatcapturesthesevariationsarelistedbelow. 1.Thedifferenceinstaticimbalancebetweenywheelsiswithin 2.5%ofthegiven balancingquality[126] 2.Thedifferenceindynamicimbalancebetweenywheelsiswithin 2.5%ofthe givenbalancingquality[126] 3.Thedeviationintheaxialseparation x a betweenonesetofywheelsandtheother islessthan20 m 4.Flywheelphasecontrolinaccuraciesresultindeviationsofthedesiredphase relationsby 10 Theinaccuracyconsideredismorethanthatofacommercial product[127] 5.Variationinywheelaccelerationsbetweenthethreeywheelsresultingfrom independentphase/speedcontrolareconsidered Theequationsofmotionofagenericthree-ywheelsystemrepresentedbyEq.6 issimulatedtoshowthattheuseofarealistichigh-delitythree-ywheelsystem resultsinsignicantjitterreductioncomparedtoasingle-ywheelsystem.Itwas mentionedatthebeginningofSection6.3thatthethree-ywheelsystemdoesnot requirehighbalancequalityoftheindividualywheelsandonlyrequiresthemto bebalancedtothesameapproximatequality.Separatesimulationsareperformed forfourdifferentywheelbalancingqualitiesG0.4,G1,G2.5,andG6.3toverifythe same.TheparameterscommontoallofthesesimulationsarelistedinTable6-4. Theywheelangularvelocityprolesthattriestoachievethedesiredphaserelations betweentheywheelsEq.6,andintheprocess,producesasinusoidalvariation of 10 betweentheywheelphasesisshowninFigure6-6.Thisvelocityprole isalsocommontoallthesimulations.Thesimulationparametersandresultsofa three-ywheelsystemwhoseindividualywheelsarebalancedtoaG0.4qualityare showninTable6-5andFigure6-7,respectively.Fortheotherbalancingqualities,the 146

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simulationparametersaregiveninTables6-6,6-7and6-8,andandresultsareshown inFigures6-8,6-9,and6-10.Theresultsofthesimulationsperformedforallbalancing qualitiesaresummarizedinFigure6-11,whichshowsthemagnitudesofresidual jitterofthethree-ywheelsystemandthatofacorrespondingsingleywheelsystem. Whilethemagnitudeofjitterincreasessignicantlywiththedecreaseinbalancing qualityforasingleywheelsystem,theincreaseisnotassevereforathreeywheel system.Infact,itcanbeseenfromFigure6-11thatevenwithabalancingqualityof G6.3,thethree-ywheelsystemproduceslessjitterthanasingle-ywheelsystem withaG0.4balancingquality.Themarginalincreaseintheresidualjittermagnitude ofthethree-ywheelsystemwithdecreaseinthebalancingqualityiscausedbythe 10 deviationinthephaseangles,whichamplifytheeffectoftheimbalancepresent intheywheels.Inanidealthree-ywheelsystem,theresidualjitterisalwayszero regardlessofthebalancingquality.Itcanbeinferredfromthesimulationresultsthatthe three-ywheelsystemproducesamaximumresidualjitterthatisatleast20timeslower comparedtothatproducedbyasimilarlybalancedsingle-ywheelsystem. On-orbiteffectssuchasthermaldistortionofywheelsandhoopstraintendto increasetheimbalancepresentintheywheels.Thiscausesincreasedjitterina singleywheelsystem.However,inathreeywheelsystem,sincealltheywheels aremanufacturedidenticallyfromthesamematerial,thechangeinimbalance isalmostuniformacrossallywheels.Furthermore,sinceonlythedifferencein imbalancebetweenywheels,andnottheindividualimbalanceintheywheelsaffect themagnitudeofjitterinathree-ywheelsystem,theeffectofon-orbitchangesin imbalanceisminimal. Wenowlookattheeffectoffailureofoneormoreywheelsinthethree-ywheel system. 147

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6.4.4EffectofFlywheelFailure Eachoftheywheelsinathree-ywheelsystemiscapableofbeingcontrolled independently.Whilethisprovidesacertaindegreeofredundancy,onemightconsider thesystemtohavemorepointsoffailurecomparedtothesingle-ywheelsystem. Itisthereforenecessarytoanalyzetheextentofperformancedegradation,andthe extentofcapabilityretainedbythepartiallyfailedsystem.Withthepossibilityoffailure ofsingleormultipleywheelsatanyinstant,thereareatotalofsevenfailuremodes. But,sincetheywheels F 2 and F 3 areidentical,thereareonlyfourdistinct,non-trivial modesoffailure,andtheyarediscussedbelow.Afullyfunctionaloracompletelyfailed systemisconsideredtrivial.Simulationsforallfourfailuremodesareperformedon athree-ywheelsystemwithabalancingqualityofG2.5arbitrarilyselected.The parametersusedforsimulationaresameasthoseusedforthehigh-delitysimulation ofaG2.5systemasgiveninTables6-4,and6-7.Theresultsofthesimulationforeach caseispresentedbelow. 6.4.4.1Failureofywheel F 1 Flywheel F 1 ,whichisthelargerofthethreeywheels,accountsforhalftheaxial inertiaandmassofthethree-ywheelsystem.Uponfailureofthisywheel,thesystem losesitsabilitytoproduce50%oftheactuator'storqueoutputandmomentumstorage capability.ReferringtoEq.6,itcanbeseenthatthejitterofthistwo-ywheelsystem canbeminimizedbythephaserelation 2 = 3 .Partialbalancecantherefore beachieved.ThesimulationresultofsuchasystemisshowninFigure6-12.The simulationstartswithafailedywheel F 1 .Flywheels F 2 ,and F 3 areinphaseatthis point.At10 seconds simulationtime,theseywheelsarecommandedtoachieve aphasedifferenceof radians ,whichyieldsastaticallybalancedsystem.Itcan veriedfromEq.6thatonlystaticbalanceisachievedwithsuchasystem.Dynamic imbalanceisstillpresentwhichcausessomejitterasseeninFigure6-12. 148

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6.4.4.2Failureofywheel F 2 or F 3 Flywheels F 2 and F 3 haveidenticalmassproperties,andeachywheelaccounts for25%oftheaxialinertiaofthethree-ywheelsystem.Failureofeitheroneofthe ywheelsthereforeaccountsfora25%decitintheactuator'storqueoutputand momentumstoragecapability.ReferringtoEq.6,itcanbeseenthatthejitterofthis two-ywheelsystemcanbeminimizedbythephaserelation 1 = 2 or 1 = 3 ThesimulationresultofsuchasystemisshowninFigure6-13.Thesimulationstarts withafailedywheel F 3 ,andtheywheels F 1 and F 2 areinphaseatthisinstant.At 10 seconds simulationtime, F 2 iscommandedtogain radians over F 1 ,whichonly providespartialstaticbalancetothesystem.Thisisduetothefactthatywheels F 1 and F 2 differintheirmagnitudeofstaticimbalance.ItcanbeseenfromFigure6-13 thattheresidualjitterishigherinmagnitudewhencomparedtothepreviouscase wherecompletestaticbalanceisachieved.Further,thesystemisalsodynamically unbalanced.Thus,theresidualjitterisduetobothstaticanddynamicimbalance.The changeinthe x -componentofthespacecraftangularvelocityas F 2 iscommanded togain radians over F 1 ,canbeclearlyseeninFigure6-13.Inafullyfunctional three-ywheelsystem,orinasystemwithafailedywheel F 1 ,thiseffectiscancelledout duetoopposingrotationsby F 2 and F 3 asshownearlierinFigure6-6. 6.4.4.3Failureofywheel F 1 ,andeither F 2 or F 3 Thisisaconditionofmultipleywheelfailureandtheresultingsystemisidenticalto asingle-ywheelsystem,butwithreducedtorquecapacity.Inthisparticularcase,with failureofywheels F 1 andeither F 2 or F 3 ,75%oftheaxialinertiaofthethree-ywheel systemislost.Theactuatorthereforeretainsonly25%ofitstorqueoutputand momentumstoragecapability.Thesimulationresultofsuchasystemisshownin Figure6-14,anditssimilaritywithFigure6-3Asingle-ywheelcaseisreadilyobserved. However,thefrequencyofprecessionislowerduetothelowerangularmomentumof 149

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theywheels.Evidently,asinthecaseofasingleywheelsystem,thejitterisdueto both,staticanddynamicimbalance. 6.4.4.4Failureofywheel F 2 and F 3 Thisconditionisidenticaltothepreviouscasebut,only50%oftheaxialinertiaof thethree-ywheelsystemislost.Theactuatorthereforeretainstheremaining50%ofits capacityfrom F 1 .ThesimulationresultofsuchasystemisshowninFigure6-15. AcomparisonofallfailuremodesisillustratedinFigure6-16.Thehorizontal axisindicatesthefailureconditionofthesystem.Thetwoverticalaxes,leftand right,representthejittermagnitudeandremnanttorquecapacity,respectivelyofthe failedsystem.Maximumtorqueandminimumjitterisachievedinafullyfunctional three-ywheelsystem.Thetorquecapacityoftheactuatorsandtheabilitytoperform balancingiscompromisedwithywheelfailures.Theconditionwithfailureofywheels F 2 ,and F 3 representsthecasewiththelargestjittertotorqueratio,whichtallieswiththe factthatthiscaseisidenticaltoasingle-ywheelsystem.Thethree-ywheelsystemcan thereforesustainuptotwoywheelfailures,andstillremainfunctionalbut,withreduced capacity.Partialbalancecanstillbeachievedinthecaseofsingleywheelfailures.The systemthereforeoffersasignicantadvantageoverthesingleywheelsystem,inwhich failureresultsin100%lossofactuatorcapacity. 6.4.5PowerandMassTrades Thebenetsofthethree-ywheelsystemincludejitterreductioncapabilityand redundancy.Thesebenetsareprovidedattheexpenseofadditionalmass,power, andcost.Inthissection,anestimateofthepowerconsumptionandmassofthe three-ywheelsystemispresentedandcomparedwiththatofasingle-ywheel system.Inordertomaketheseestimates,indicativedesigncongurationsofsingleandthree-ywheelsystemsareneeded.Onesuchdesigncongurationisshown inFigure6-17.Thedesignsshowninthegurearenotidealmechanicaldesigns andareonlyprovidedasanindicativereferencetoshowthevariouscomponents 150

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presentinsuchasystem.However,theindicativedesignissufcienttomakeinformed estimatesofthemassandpowerconsumption.Appropriatecomponentsareselected fromcommerciallyavailablehardware,andtheirmassandpowercharacteristicsare obtainedfromthemanufacturer'sdatasheet.Theselectionofthesecomponentswas basedonthemassandinertiaoftheywheelstheyareassociatedwith.Forinstance, thesingle-ywheelsystemconsistsofamotorwithatorquecapacitytwicethatof ywheel F 1 ,andfourtimesthatofywheels F 2 ,and F 3 ofthethree-ywheelsystem. Thisisbasedonthefactthattheywheelassociatedwiththesingleywheelsystem hasmassandaxialinertiaequaltotwicethatofywheel F 1 ,andfourtimesthatof ywheels F 2 and F 3 .Similarly,thesingle-ywheelsystemalsohasbearingsthatare bigger,andhenceprovidegreaterviscousresistanceascomparedtothebearingsof thethree-ywheelsystem.Eachywheelhasitsownencodernotshowningure oranyotherangularfeedbackdevicetoprovidefeedbacknecessaryforywheel control.Thisfeatureiscommontobothsingle-ywheelandthree-ywheelsystems. MotorcharacteristicswereobtainedfromKollmorgen[128]andbearingdatawere obtainedfromSKF[129].Theywheel,motor,encoder,andbearingdataofasingle ywheelsystemaregiveninTable6-9.Anestimateofthemassisprovidedforthe single-ywheelandthree-ywheelsystemsinTable6-10.InTable6-10,themassof eachcomponentofthethreeywheelsystemislistedasapercentageofthemass ofthecorrespondingcomponentinasingleywheelsystemtoclearlyrepresentthe increase/decreaseinmassofeachcategoryofcomponents.Forinstance,themass ofthemotorchosenforywheel F 1 is40 g ,whichis66%ofthemassofthemotor inasingle-ywheelsystem.Similarly, F 2 and F 3 havemotorswithmassequalto 26.5 g ,whichis44%ofthesingle-ywheelsystemmotor.Basedontheindicative design,andthecomponentsconsideredhere,itcanbeseenfromTable6-10thatthe three-ywheelsystemisabout20%heaviercomparedtothesingleywheelsystem. Anestimateofthequiescentpowerconsumptionofsingle-,andthree-ywheelsystems 151

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isprovidedinTable6-11.Thepowerconsumptionisbasedontheviscousfrictionin themotorandbearings,andthecoggingresistanceofferedbythemotormagnets.The powerconsumptionduetoviscousfrictionisgivenastheproductofthecoefcientof viscousfrictionin Nms andthesquareoftheangularvelocityin rad = s .Afactorof twoisincludedastherearetwobearingsperywheel.Thepowerconsumptiondue tocoggingresistanceisgivenbytheproductofthecoggingtorqueandtheangular velocity.SimilartoTable6-10,thepowerparametersforthethree-ywheelsystemin Table6-11areexpressedasapercentageofthecorrespondingparametervalueina single-ywheelsystem.Forinstance,theviscousdampingcoefcientofthebearing usedinthethree-ywheelsystemis3 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 Nms ,whichis60%ofthecorresponding valueinasingle-ywheelsystem.Basedontheindicativedesign,andthecomponent characteristicsconsideredhere,itcanbeseenfromTable6-11thatat10000 rpm thethree-ywheelsystemconsumesabout1.5 W morethanthesingle-ywheel system.ForaspacecraftwithfourCMGs,thetotalincreaseinpowerconsumption isabout6 W .Sincethepowerconsumptiondependsonthespeedoftheywheel, aplotofthepowerconsumptionsofsingle-ywheelandthree-ywheelsystemsasa functionofywheelspeedisshowninFigure6-18.Itcanbeseenthatthedifference inpowerconsumptionbetweensingle-ywheelandthree-ywheelsystemsissmaller atlowerangularvelocitiesoftheywheel.Itshouldbenotedthatthepowerandmass comparisonsarebasedonlyontheindicativedesigninFigure6-17,actuatorcapacity, andcomponentcharacteristicsconsideredinTable6-9.Anindependentanalysis mayberequiredforotherdesignsandactuatorcapacities,andthecomparisonresult providedheremaynotbeextrapolated. 6.4.6IdentifyingtheFlywheelPhaseAngle Thekeyaspectofjitterreductionusingthethree-ywheelapproachisaligningthe ywheelphasesasperEq.6sothatstaticanddynamicbalanceisachieved. Knowledgeoftheywheelphaseanglesisthereforecriticaltoimplementingthe 152

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approachonorbit.Sincethephaseoftheywheeldependsonthelocationofthe ywheel'seccentric c.m. ,thereisnodirectmethodofmeasuringthephaseangles. Instead,amethodtoinferthephaseanglesisdevised.Apairofforcesensorsareused inbetweentheywheelsupportsandthespacecraftstructureasshowninFigure6-17. Eachywheelisspunindividuallywhiletheothersarestationarysothattheeffectof jitterduetothatparticularywheelcanbemeasuredattheforcesensors.Sincethe ywheel'sencoderprovidescontinuousinformationabouttheywheelsanglerelativeto aknownreference,theywheelphasecanbeidentiedbycomparingthetimestamped historiesoftheforcesensorandencoderoutputs.Thismethodissimilartotheones usedintraditionalbalancingmachinestoperformrotorbalancing[130].Anadded benetofusingforcesensorstoidentifytheywheelphaseincludestheuseofforce sensordatatomonitorthebearinghealth,andthereforeprovidingtheabilitytopredict bearingfailures. 153

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Table6-1.BOLImbalanceDataofMomentumActuators[17,127,131134] Actuator StaticDynamicSpeedQuality 1 gmm gmm 2 rpmapprox. VectronicVRW-1,RW1806500 < G1 L-3Comm.RWA-15,RW3.69202200 < G0.4 HoneywellM50,CMG615006500 < G0.4 Astrium15-45S,CMG1520003000 < G1 IthacoTW-16B32,RW1540005100 < G2.5 IthacoTW-50E300,RW1860003850 < G2.5 1 QualityiscomputedusingEq.6,andanapproximatevalueofrotormass Table6-2.Single-FlywheelSystemSimulationParameters ParameterValueUnits Solver4thorderRunge-Kutta n = a Timestep0.0001xed s Spacecraftmass m =10 kg Spacecraftinertia J s c = 2 4 0.1586 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0030 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.00300.1586 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.00150.1609 3 5 kgm 2 Flywheelinertia J w o = 2 4 600.0054 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027342.2554 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0027342.2554 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 FlywheelMass m w =0.3 kg Positionofywheel center B r o = [ 0.10.10.05 ] T m Flywheeleccentric mass m e =0.27 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 kg Eccentricity r e = [ 111 ] T 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 m Staticimbalance0.27 gmm DynamicImbalance0.27 gmm 2 Flywheelspeed = [ 1000000 ] T rpm Spacecraftinitial angularvelocity = [ 000 ] T rad = s 154

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Table6-3.IdealThree-FlywheelSystemSimulationParameters ParameterValueUnits Solver4thorderRunge-Kutta n = a Timestep0.0001xed s Spacecraftmass m =10 kg Spacecraftinertia J s c = 2 4 0.1586 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0030 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.00300.1586 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.00150.1609 3 5 kgm 2 InertiaofFlywheel F 1 J F 1 o = 2 4 300.0027 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014155.2839 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014155.2839 3 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 kgm 2 InertiaofFlywheels F 2 and F 3 J F 2 o = J F 3 o = 2 4 150.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.000775.6615 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.000775.6615 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 Massofywheel discs m w 1 =0.15, m w 2 = m w 3 =0.075 kg Positionofpoint O 1 B r o 1 = [ 0.10.10.05 ] T m Positionofpoint O 2 B r o 2 = [ 0.094250.10.05 ] T m Positionofpoint O 3 B r o 3 = [ 0.105750.10.05 ] T m Flywheeleccentric masses m e 1 =0.135 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 m e 2 = m e 3 =0.0675 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheelmass eccentricities F 1 r e 1 = F 2 r e 2 = F 3 r e 3 =10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 [ 111 ] T m Maximumstatic imbalance 0.27 gmm Maximumdynamic Imbalance 0.27 gmm 2 Flywheelspeeds k 1 k = k 2 k = k 3 k =10000 rpm Flywheelinitial phase 1 = 2 = 3 =0 rad Spacecraftinitial angularvelocity = [ 000 ] T rad = s 155

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Table6-4.Three-FlywheelSystemCommonSimulationParametersHi-Fidelity ParameterValueUnits Solver4thorderRunge-Kutta n = a Timestep0.0001xed s Spacecraftmass m =10 kg Spacecraftinertia J s c = 2 4 0.1586 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0030 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.00300.1586 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0015 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.00150.1609 3 5 kgm 2 Massofywheeldiscs m w 1 =0.15, m w 2 = m w 3 =0.075 kg Axialseparationbetween F 1 and F 2 x a =5.75 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 m Axialseparationbetween F 1 and F 3 x a =5.73 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 m Positionofpoint O 1 B r o 1 = [ 0.10.10.05 ] T m Positionofpoint O 2 B r o 2 = [ 0.094250.10.05 ] T m Positionofpoint O 3 B r o 3 = [ 0.105730.10.05 ] T m Nominalywheelspeeds k 1 k = k 2 k = k 3 k =10000 0.5 rpm Peakdifferencebetween ywheelphases i )]TJ/F23 11.9552 Tf 11.955 0 Td [( j = 10 deg Flywheelinitialphase 1 = 2 = 3 =0 rad Spacecraftinitialangular velocity = [ 000 ] T rad = s 156

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Table6-5.G0.4Three-FlywheelSystemSimulationParametersHi-Fidelity ParameterValueUnits Flywheel F 1 eccentric mass m e 1 =0.0525 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 kg Flywheel F 2 eccentric mass m e 2 =0.0269 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 kg Flywheel F 3 eccentric mass m e 3 =0.0256 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 kg Flywheelmass eccentricities F 1 r e 1 = F 2 r e 2 = F 3 r e 3 =10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 [ 111 ] T m InertiaofFlywheel F 1 J F 1 o 1 = 2 4 300.0011 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005155.2823 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0005155.2823 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 2 J F 2 o 2 = 2 4 150.0005 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.000375.6607 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.000375.6607 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 3 J F 3 o 3 = 2 4 150.0005 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.000375.6607 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0003 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.000375.6607 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 Maximumstatic imbalance 0.105 gmm Maximumdynamic Imbalance 0.11 gmm 2 157

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Table6-6.G1Three-FlywheelSystemSimulationParametersHi-Fidelity ParameterValueUnits Flywheel F 1 eccentric mass m e 1 =0.135 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 2 eccentric mass m e 2 =0.0692 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 3 eccentric mass m e 3 =0.0658 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheelmass eccentricities F 1 r e 1 = F 2 r e 2 = F 3 r e 3 =10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 [ 111 ] T m InertiaofFlywheel F 1 J F 1 o 1 = 2 4 300.0027 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014155.2839 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0014155.2839 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 2 J F 2 o 2 = 2 4 150.0014 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.000775.6615 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.000775.6615 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 3 J F 3 o 3 = 2 4 150.0013 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.000775.6615 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0007 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.000775.6615 3 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 kgm 2 Maximumstatic imbalance 0.27 gmm Maximumdynamic Imbalance 0.28 gmm 2 158

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Table6-7.G2.5Three-FlywheelSystemSimulationParametersHi-Fidelity ParameterValueUnits Flywheel F 1 eccentric mass m e 1 =0.3750 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 2 eccentric mass m e 2 =0.1922 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 3 eccentric mass m e 3 =0.1828 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheelmass eccentricities F 1 r e 1 = F 2 r e 2 = F 3 r e 3 =10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 [ 111 ] T m InertiaofFlywheel F 1 J F 1 o 1 = 2 4 300.0075 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037155.2888 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0037155.2888 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 2 J F 2 o 2 = 2 4 150.0038 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0019 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0019 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.001975.6640 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0019 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0019 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.001975.6640 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 3 J F 3 o 3 = 2 4 150.0037 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0018 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0018 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.001875.6638 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0018 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0018 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.001875.6638 3 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 kgm 2 Maximumstatic imbalance 0.75 gmm Maximumdynamic Imbalance 0.74 gmm 2 159

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Table6-8.G6.3Three-FlywheelSystemSimulationParametersHi-Fidelity ParameterValueUnits Flywheel F 1 eccentric mass m e 1 =0.9 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 2 eccentric mass m e 2 =0.4613 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheel F 3 eccentric mass m e 3 =0.4387 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 kg Flywheelmass eccentricities F 1 r e 1 = F 2 r e 2 = F 3 r e 3 =10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 [ 111 ] T m InertiaofFlywheel F 1 J F 1 o 1 = 2 4 300.0180 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090155.2993 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0090155.2993 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 2 J F 2 o 2 = 2 4 150.0092 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0046 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0046 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.004675.6694 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0046 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0046 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.004675.6694 3 5 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 kgm 2 InertiaofFlywheel F 3 J F 3 o 3 = 2 4 150.0088 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0044 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0044 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.004475.6689 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0044 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0044 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.004475.6689 3 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 kgm 2 Maximumstatic imbalance 1.8 gmm Maximumdynamic Imbalance 1.8 gmm 2 Table6-9.MassandPowerParametersofSingle-FlywheelSystem VariableValueUnits FlywheelMass, m w 0.3 kg MotorMass, m m 0.06 kg BearingMass, m b 0.005 kg EncoderMass, m en 0.02 kg Massofhousing, m h 0.3 kg Motorviscousdampingcoefcient, c m 3 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 Nms Bearingviscousdampingcoefcient, c b 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 Nms Motorcoggingtorque, cog 1.2 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 Nm Flywheelspeed, 10000 rpm 160

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Table6-10.MassComparisonBetweenSingle-andThree-FlywheelSystems Component MassofFlywheelSystem kg SingleTriple Numberof ywheels 11+1+1=3 Massof ywheels m w 0.25+0.5+0.25 m w Numberofmotors11+1+1=3 Massofmotors m m 0.4+0.66+0.4 m m Numberof bearings 22+2+2=6 Massofbearings2 m b 2 0.5+0.5+0.5 m b Numberof encoders 11+1+1=3 Massof encoders m en 3 m en Massofhousing andother structures m h 1.2 m h Totalmass m w + m m +2 m b + m en + m h m w +1.46 m m +3 m b +3 m en =0.6900fromTable:6-9+1.2 m h =0.8266fromTable:6-9 Table6-11.PowerComparisonBetweenSingle-andThree-FlywheelSystems ContributingFactor PowerConsumptionofFlywheelSystem Watts SingleTriple Viscousdamping ofmotor 2 c m 2 0.33+0.55+0.33 c m Viscousdamping ofbearings 2 2 c b 2 2 0.6+0.6+0.6 c b Coggingtorqueof motor )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [( cog 0.4+0.6+0.4 cog Totalpower 2 c m +2 c b + )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [( cog 2 1.21 c m +3.6 c b + )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(1.4 cog =2.68WfromTable:6-9=4.13WfromTable:6-9 161

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Figure6-1.SchematicRepresentationofSpacecraftwithUnbalancedFlywheel Figure6-2.SchematicoftheThree-FlywheelSystem 162

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AFullSimulationPlot BZoomedPlot )]TJ/F20 9.9626 Tf 9.962 0 Td [(0.1 s Figure6-3.AngularVelocityofSpacecraftwithSingleUnbalancedFlywheel 163

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Figure6-4.FFTofSpacecraftAngularVelocity 164

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AFlywheelAngularVelocities BSpacecraftAngularVelocity Figure6-5.SimulationofanIdealThree-FlywheelSystem 165

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Figure6-6.FlywheelAngularVelocitiesUsedinsimulationsofaHigh-Fidelity Three-FlywheelSystem Figure6-7.SpacecraftAngularVelocityinaHigh-FidelityG0.4Three-FlywheelSystem 166

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Figure6-8.SpacecraftAngularVelocityinaHigh-FidelityG1Three-FlywheelSystem Figure6-9.SpacecraftAngularVelocityinaHigh-FidelityG2.5Three-FlywheelSystem 167

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Figure6-10.SpacecraftAngularVelocityinaHigh-FidelityG6.3Three-FlywheelSystem Figure6-11.ComparisonofResidualJitteratVariousBalancingQualities 168

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Figure6-12.SimulationofaThree-FlywheelSystemwithaFailedFlywheel F 1 Figure6-13.SimulationofaThree-FlywheelSystemwithaFailedFlywheel F 2 or F 3 169

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Figure6-14.SimulationofaThree-FlywheelSystemwithFailedFlywheels F 1 and F 2 Figure6-15.SimulationofaThree-FlywheelSystemwithFailedFlywheels F 2 and F 3 170

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Figure6-16.ComparisonofFlywheelFailureModesinaThree-FlywheelSystem ASingle-FlywheelDesign BThree-FlywheelDesign Figure6-17.IndicativeCongurationsofSingle-andThree-FlywheelSystems 171

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Figure6-18.PowerConsumptionComparisonBetweenSingle-andThree-Flywheel Systems 172

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CHAPTER7 CONCLUSIONSANDFUTUREWORK AttitudecontrolsystemsACSprovidespacecraftwiththecapabilitytochange itsorientationandpointatanobjectofinterest.Thequalityoftheattitudecontrol mechanisms,sensors,andcontrolalgorithmsdeterminetheprecisionwithwhichthe spacecraftcanperformthesemaneuvers.ControlmomentgyroscopesCMGare favorableastheyprovidehighagilityandpointingprecision.Ambiguityinsomeofthe on-orbitparameterssuchasactuatoralignmentandmechanismimperfectionsleadto deterioratedperformanceoftheCMGbasedACS.TheerrorsintheCMGmechanism, specicallythoseofywheelimbalanceandgimbalmisalignmentthatcontributeto degradationinthepointingperformanceoftheACSwereconsideredinthedissertation. Theknowledgeofangularaccelerationofthespacecraftisbenecialinthedesign ofestimationtechniquessuchasactuatoralignment.Amethodwasdevelopedto estimatethesame.Themethodpresentedinvolvesaspeciccongurationplacement andorientationofsixuniaxialaccelerometerstodeterminetheangularacceleration usingthekinematicrelationsbetweenlinear,andangularaccelerations.Alinear measurementmodelfortheangularaccelerationwasdevelopedtoassesstheeffect ofsensornoiseandbias.ThelinearmodelwasusedtodesignaKalmanlterto determinetheeffectivebiasintheangularaccelerationmeasurements.Inadditionto providingestimatesoftheangularaccelerationbias,thelteralsoprovidessmoothed angularvelocityestimates.Simulationsbasedondatafromcommercialsensors gyroscopesandlinearaccelerometerswerepresentedtoshowtheeffectivenessofthe method.Resultsindicatedapeakerrorintheestimatesofabout 5 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 rad = s 2 Betteraccuraciescanbeobtainedwiththeuseofmoreprecisesensors.Linear accelerometersarewidelyusedonboardspacecraftforinertialnavigation,andby adoptingthespeciccongurationasshowninthisdissertation,theseaccelerometers canbeadditionallyusedfordeterminingtheangularacceleration,andlteredangular 173

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velocityestimates.Theestimationmethodwasdevelopedbasedonalinearized measurementmodel.Whiletheaccuracyoftheestimateobtainedbythisprocedure maybesufcientforcertainapplications,othernon-linearsolutionmethodologiesmay beinvestigatedtofurtherthisresearch.Thesensitivityoftheestimationaccuracyto measurementaxismisalignmentsandsensornoisecanbefurtherinvestigated.Lack ofangularaccelerationknowledgehasturnedawayresearchersfromusingEuler's equationsofmotionforspacecraftdynamicparameterestimationtechniques.The angularaccelerationestimationtechniquepresentedinthisdissertationmaynowpermit theuseofEuler'sequationsofmotiontoestimatespacecraftdynamicparameterssuch asactuatoralignment. TheutilityofEuler'sequationsofmotionwhenangularaccelerationmeasurements areavailableintheestimationactuatoralignmentswasdemonstratedbydeveloping anestimationtechniquetoidentifytheunknowngimbalorientationofaCMG.Itwas shownthat,byextractingouttheunknownelementsinthealignmentmatrix,Euler's equationsofmotioncanbeexpressedinalinearform.Linearleastsquarestechniques werethenusedtoidentifytheunknowngimbalangle.Themeasurementsneeded fortheleastsquaressolutionincludeywheelandspacecraftangularvelocitiesand accelerations.Inertialsensorssuchasgyroscopesandaccelerometerswereused toobtainthesemeasurements.Hi-delitysimulationswereperformedwithemulated sensormeasurementsbasedonsensordatafromcommerciallyavailablesensors. Threevariantsoftheleastsquaressolutionwereusedtoestimatetheunknowngimbal angle.Comparisonbetweenallthreemethodsfordifferentscenarioswaspresented. Specically,theweightedleastsquaresapproachproducedestimateswithanaccuracy of1 arc-minute .AMonte-Carloanalysiswasperformedtoverifytherepeatabilityofthe solutionaccuracy.Unlikepreviousmethods,themethodpresentedhereisinsensitiveto misalignmentofattitudesensors,anddoesnotrequiredifferentiationofangularvelocity measurements.Further,theaccuracyoftheestimationmethodisindependentofthe 174

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magnitudeofmisalignment.Themethodmayprovebenecialinreducingprolonged groundcalibrationprocesses,andthereforecontributetothetechnologiesrequiredfor realizingtheobjectivesof responsivespace .Themethodcanbeextendedtoestimation ofotheractuatore.g.,reactionwheelalignments. ImbalanceinCMGywheelsleadstoattitudestabilityerrorsandaffectthe performanceofcertainpayloads.Atechniquetoreducethemagnitudeofjitteremitted byywheelsinaCMGwasdeveloped.Athree-ywheelapproachthatprovidesthe abilitytoachievestaticanddynamicbalanceon-orbitwasconceptualized.Themethod providestheabilitytomaintainanerbalancecomparedtoasingle-ywheelsystem evenintheeventofon-orbitchangesinywheelimbalance.Detailedsimulations werepresentedtoshowtheeffectivenessoftheapproach.Thethree-ywheelmethod signicantlyreducesthejitterproducedbytheactuatorandcanbeusedtogetherwith otherisolationsystemstofurtherimprovethejitterenvironment.Powerandmass tradesbetweenthesingle-,andthree-ywheelsystemswerealsopresented.Theability toachievesignicanttimesimprovementinpointingstabilityofthespacecraft mayjustifyincreaseinthemassandpowerconsumption.Furtherworkneedstobe performedinconstructingaprototypetoverifyitsimplementationonspacecraft.The useofamechanicalengagementmechanisme.g.,magneticclutchestocouplethe threeywheelsoncebalanceisattainedalsoneedstobeinvestigated.Whilethe three-ywheelsystemwasdescribedwithrelationtoaCMGywheel,thesameconcept canbeappliedtoywheelsystemsofotheractuators. Whilethereareseveralerrorsassociatedwithanattitudeactuator,andneitherone canbecompletelyeliminated,thecontributionsofthisdissertationmayhelpimprovethe stateoftheartintheirdesignanddeployment. 175

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APPENDIXA NOTATIONS Thissectiondescribessomeofthenotationsfrequentlyusedinthisdissertation. 1. Scalars arerepresentedusingitalicizedlowercaseEnglishorGreekalphabets. Whenthescalarcarriesasubscript,thesubscriptrepresentsthebody,point, orcoordinateassociatedwiththescalar.Forinstance, m w representsthemass ofaywheel W x representsthe x -componentoftheangularvelocity,and i representsthemagnitudeangularvelocityofthe i th gimbal. 2. Vectors arerepresentedbylowercase,italicized,andboldfacedEnglishorGreek alphabets. 1 Theyareoftenappendedwithsubscriptsandsuperscripts.The subscriptsrepresentthepointofinterestandthesuperscriptreferstothebody orobjectofinterest.Forinstance, h s c representstheangularmomentumofthe spacecraft S aboutitscenterofmass C 1 representstheangularvelocityof ywheel F 1 ,and a i representstheaccelerationofthepoint i 3. Unitvectors orvectorswithunitmagnitudearerepresentedsimilartoother vectorsbutwitha caret .Forinstance, g i representsthegimbalaxisofthe i th CMG. 4. Coordinatebasis orcoordinatesystemisrepresentedbythescriptedletter C with asubscriptthatindicatesthebodytowhichthecoordinatebasisisattached.For instance, C B isthebodyxedcoordinatebasisofthespacecraft. 5. Coordinatebasisvectors arealsorepresentedsimilarunitvectors,butwith uppercasealphabets X Y ,and Z .Forinstancethespacecraftbodycoordinate basis C B vectorsarerepresentedby h X B Y B Z B i 6. Coordinatizedvectors ormatrixrepresentationsofvectorsaredenotedusing lowercase,upright,andboldfacedEnglishorGreekalphabets. 1 Theyare appendedwithsubscriptsandsuperscriptssimilartovectors.Additionally,they arealsoappendedwithaleftsuperscriptthatidentiestheassociatedcoordinate basis.Forinstance, B a i representstheaccelerationofapoint i expressedinthe bodycoordinatebasis. 7. Inertiadyadics arerepresentedsimilartovectorsbutusinguppercase,italicized, andboldfacedalphabets J and I .Forinstance, J s c representstheinertiadyadicof 1 NotethatGreekalphabetslookidenticalinbothitalicanduprightforms,andinsuch casesthedistinctionbetweenavector,anditsmatrixrepresentationismadebasedon theothertermsinvolvedintheequationorbyspecicmentions. 176

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thespacecraftaboutitscenterofmass C ,and I w o representsthecentroidalabout its c.m.O inertiadyadicofaywheel. 8. Unitdyadics and identitymatrices arerepresentedby 1 .Thedimensionofthe matrixis3 3unlessotherwiseindicated. 9. Inertiamatrices ormatrixrepresentationofinertiadyadicsarerepresentedsimilar toinertiadyadicsbutusinguprightalphabets J and I inlieuofitalicizedalphabets. Additionally,theyarealsoappendedwithaleftsuperscriptthatidentiesthe associatedcoordinatebasis.Forinstance, B J s c representstheinertiamatrixofthe spacecraftaboutitscenterofmasscoordinatizedinthebodycoordinatebasis, and F I w o representsthecentroidalinertiamatrixofaywheelcoordinatizedinthe ywheelcoordinatebasis. 10. Matrices ingeneralarerepresentedbyboldanuprightalphabets.Thealphabets arelowercasewhenthematrixisacolumnorrowmatrix,anduppercasefor rectangularmatrices. 11. Skew-SymmetricMatrices areusedinthisdissertationtorepresentthematrix equivalentofacross-productoperation.Askewsymmetricmatrixisrepresented byenclosingthematrixrepresentationofavectorinsquareparentheseswith a` 'superscript.Forinstance, [ ] r denotesthematrixrepresentationofa cross-productbetweenvectors and r 12. DirectionCosineMatrices arerepresentedusingtheuppercase,uprightand boldfaceletterR.Itisappendedwithasubscriptoftwouppercaseletters.The subscriptindicatesthetransformationfromacoordinatebasisrepresentedbythe rightlettertoacoordinatebasisrepresentedbytheleftletter.Forinstance, R BA isaDCMthatrepresentsthetransformationfrom C A to C B .Insomeinstances, insteadofthesubscripts,therepresentationisappendedwithanargument consistingofaunitvectorandangle.Forexample, R a denotesaDCMthat representstransformationsfromagivencoordinatebasistoanothercoordinate basisobtainedbydisplacingthegivencoordinatebasisbyarotationabout a throughanangle 13. 2-norm ofavector v isdenotedby k v k 14. Measurements ofaquantityarerepresentedbyplacinga tilde ontopofthe quantity.Forinstance e isthemeasurementoftheangularvelocity. 15. Estimates ofaquantityarerepresentedbyenclosingthequantityinangled brackets hi .Forinstance, h i istheestimateoftheangularacceleration. Note: Matrixrepresentationsofvectorsanddyadics,asmentionedabovemaycontain aleftsuperscriptthatindicatesthecoordinatebasisinwhichthevector/dyadichas 177

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beenrepresented.Whenthesuperscriptismissing,unlessotherwiseindicated,matrix representationsofvectorsanddyadicsinthisdissertationareexpressedinacoordinate basisofassociation.Forinstance,measuresofywheelinertia,masseccentricity,and ywheelangularvelocityareexpressedintheywheelcoordinatebasis C F ,andthe spacecraftangularvelocityisexpressedinthespacecraftbodyxedcoordinatebasis C B 178

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APPENDIXB FLYWHEELIMBALANCE B.1StaticandDynamicImbalanceinFlywheels Abriefdiscussionofstaticanddynamicimbalanceinywheelsthatleadtojitteris presentedhere.Concisely, staticimbalance iscausedbytheeccentricityintheywheel c.m. fromtherotationaxis,and dynamicimbalance iscausedbythemisalignment betweentheywheel'sprincipalaxisofinertiaandtherotationaxis. Considerrst,ahomogeneousywheelthatrotatesataconstantspeed ona setofbearingswhicharexedinarigidbodyasshowninFigureB-1.Further,the rigidbodyisxedinertiallyatapoint Q andassumedtobemassless.Thedisturbance causedbyywheelmotionisthereforeresistedbyareactivetorque at Q .Let r b and m denotetheradius,thicknessandmassoftheywheel,respectively.Sincethe ywheelishomogenous,it's c.m. coincideswithitsgeometriccenter O ,andliesonthe axisofrotation.Thusthereisnomasseccentricityintheywheelrelativetotherotation axis.Thepoint O isatadistance r o from Q .Thecoordinatebasis C A givenbythebasis vectors h X A Y A Z A i isxedtotherigidbodysuchthat X A isalongtheywheelrotation axis.Anothercoordinatebasis C F givenbythebasisvectors h X F Y F Z F i isattachedto theywheelsuchthat X F isalongtheywheelrotationaxis.ObservingFigureB-1,the matrixthattransformsrepresentationsin C F tothosein C A isgivenby R AF = 0 B B B B @ 100 0c )]TJ/F20 11.9552 Tf 9.298 0 Td [(s 0s c 1 C C C C A B where,c ands denotethe cosine and sine oftheangle ,respectively.Itshouldbe notedthat,since C A isattachedtoarigidbodythatisinertiallyxed,it C A isaninertial coordinatebasis,andmeasuresexpressedinthiscoordinatebasisareabsolute.The 179

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ywheelinertiamatrix I w o expressedintheywheelcoordinatebasis C F isgivenby F I w o = 0 B B B B B @ mr 2 2 00 0 m )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(3 r 2 + b 2 2 0 00 m )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(3 r 2 + b 2 2 1 C C C C C A B andtheangularvelocityoftheywheelin C F isgivenby F = 00 T B Theywheelrotationaxisisthereforealignedwiththeoneoftheprincipalaxesofthe ywheel.Sincethereisnomasseccentricityintheywheel,andsincetherotationaxis oftheywheelisalignedwithoneofitsprincipalinertiaaxis,theywheeliscompletely balancedanddoesnotcauseanydisturbancetotherigidbody.Weshallnowverifythe same.Theangularmomentumoftherigidbodyincludingtheywheelabouttheinertial point Q expressedinthecoordinatebasis C A isgivenby A h Q = R AF I w o B Therigidbodydoesnotcontributetothedynamicssinceitisinertiallyxed.Substituting theexpressionsinEqs.B,B,andBinEq.B,theexpressionfortheangular momentumcanbegivenas A h Q = mr 2 2 00 T = mr 2 2 B where, = k k .ItcanbeobservedfromEq.Bthattheangularmomentumdirection isthesameasthatoftheangularvelocity,andisinertiallyxed.Theratechangeof angularmomentumcanthereforebegivenby h F Q = [ ] h F Q = 0 B 180

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Thus,fromEq.B,wecanconcludethatthereisnonetchangeintheangular momentumofaspinningywheelwhose c.m. liesontheaxisofrotationandwhose rotationaxisisalongoneoftheprincipalaxesofinertia.Aywheelthatmeetsthese criteriaissaidtobecompletelybalanced. Now,considerthesameywheelbut,withasmalleccentricmass m e located atadistance r A from O suchthat r A isorthogonaltotherotationaxisasshownin FigureB-2.ThedepictionoftherigidbodyinFigureB-2hasbeenomittedforclarity. Thevector r A isinaplaneorthogonaltotheywheelrotationaxis.Thecoordinate basisxedtotheywheelisnowdenedsuchthat X F isalongtherotationaxisand Y F isalong r A asshowninFigureB-2.Wecanthereforewrite r A = r y Y F .Similarto theywheelsconsideredinChapter6,thiscombinationofthehomogeneousywheel andtheeccentricmassisequivalenttoaywheelwithaneccentric c.m. .Theywheel togetherwiththeeccentricmassshallbereferredtoastheywheelsystem.The angularmomentumoftheywheelsysteminFigureB-2canbewrittenas A h w Q = J w o + r o R AF [ ] r A B where, J w o = I w o + m e )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r A T )]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r A 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F58 11.9552 Tf 5.978 -9.684 Td [(r A )]TJ/F58 11.9552 Tf 13.449 -9.684 Td [(r A T SubstitutingtheexpressionfortheDCM R AF r A ,andtheskewsymmetricmatrix [ ] theexpressionfortheangularmomentumcanbewrittenas A h w Q = mr 2 2 + m e r 2 y + r o r y 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(s c T B ItcanbeobservedfromEq.Bthattheangularmomentumnowhasanadditional componentthatdependsontheangle .Since continuallychangesastheywheel rotates,theangularmomentumdirectionalsochangesandisgivenby A h w Q = r o 2 r y 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(c )]TJ/F20 11.9552 Tf 11.955 0 Td [(s T = 181

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Thecontinualchangeintheangularmomentumcreatesanoscillatingtorqueon therigidbodythatisresistedbythereactivetorqueat Q givenby .Theangular momentumchangeresultsfromtheeccentricityof m e .ItcanseenfromEq.Bthatif theeccentricityvectorwas 0 orparalleltotheangularvelocityoftheywheel,thenthere wouldbenonetchangeintheangularmomentum.Thesituationinwhichthechangein theangularmomentumiscausedduetothecomponentoftheeccentricitynormaltothe ywheelangularvelocityvectoriscalled staticimbalance Now,letusconsiderthesamestaticallyunbalancedywheelsystem,butwithan additionaleccentricmassalsoequalto m e atpoint B asshowninFigureB-3.Point B is locatedradiallyoppositetopoint A butalsodisplacedaxiallyalongthe X F directionby r x .Thepositionofpoint B relativeto O canthereforebewrittenas r B = r x X F )]TJ/F59 11.9552 Tf 12.236 0 Td [(r y Y F .It isclearthatbyaddingtheadditionalpointmass m e atpoint B thatisradiallyoppositeto theequalpointmass m e at A ,themasseccentricitynormaltotheywheelrotationaxis vanishes.Thusthesystemhasstaticbalance.However,theinertiamatrixofthesystem expressedintheywheelcoordinatebasisisgivenby F J w o = 0 B B B B @ mr 2 2 +2 m e r 2 y r x r y 0 r x r y r 2 x 0 00 r 2 x +2 r 2 y 1 C C C C A whoseprincipalaxisofinertiaisclearlynotalongthedirectionoftheangularvelocity oftheywheel.Byananalysissimilartothepreviouscases,itcanbeshownthatthe angularmomentumofthesystemisgivenby A h Q = 0 B B B B @ mr 2 2 +2 m e r 2 y )]TJ/F59 11.9552 Tf 5.479 -9.683 Td [(m e r x r y c )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(m e r x r y s 1 C C C C A Theoffdiagonalcomponentsoftheinertiamatrixresultinanangularmomentumvector thatisnotparalleltotheywheelangularvelocityvector,butdependentontheangle 182

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.Thedirectionoftheangularmomentumthereforecontinuallychangesresultinginan oscillatingtorqueontherigidbody.Theratechangeofangularmomentumisgivenby A h Q = m e 2 r x r y 0 B B B B @ 0 )]TJ/F20 11.9552 Tf 9.298 0 Td [(s c 1 C C C C A Althoughthesystemwasstaticallybalanced,imbalancetorqueisproducedontherigid bodyasaresultofthemisalignmentbetweentheaxisofrotationandtheprincipalaxis ofinertia.Thisconditioniscalled dynamicimbalance TheywheelsysteminFigureB-2containedstaticimbalance,resultinginan oscillatorytorque.ThesecondywheelsystemFigureB-3withanadditionalpoint massbalancedoutthestaticimbalancebutresultedinanimbalancetorquedueto misalignmentoftheprincipalaxisofinertiaandtherotationaxis.Withaninsighton thereasonsforstaticanddynamicimbalance,wecancompletelybalancetheywheel systemasfollows.ConsidertheywheelsystemasshowninFigureB-4.Itconsistsofa symmetricandhomogenousywheelsystemwiththreeadditionalpointmasses: m e at point A and0.5 m e atpoints B and C .Thepointmassesareatthesameradialdistance from O andtheiraxialpositionsalongthe X F directionaresuchthatthereexistsaplane ofsymmetrynormaltotherotationaxis.Thisarrangementleadstomasssymmetry abouttherotationaxis.Additionally,therotationaxisisalignedwithoneoftheprincipal axesofinertia.Itcanbeshownthatthelinearandangularmomentaofthisywheel systemisgivenby A h Q = mr 2 2 +2 m e r 2 y 00 T = mr 2 2 +2 m e r 2 y Itcanbeseenthattheangularmomentumofthesystemisparalleltotherotationaxis ortheywheelangularvelocityvector.Thedynamicsdescribingthisywheelsystem areidenticaltothehomogenousywheelshowninFigureB-1,andhencethesystemis completelybalanced. 183

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Threepointmasseswereusedtoillustratetheconceptofbalancingaywheel system.Thus,foranarbitraryywheelwithaneccentric c.m. ,resultinginbothstatic anddynamicimbalance,aminimumoftwobalancingmassesisrequiredistwoseparate parallelplanes.Thesebalancingmasseswillleadtomasssymmetryabouttherotation axistoremovestaticimbalance,andalsocausethecentroidalinertiamatrixtolose itsoff-diagonalterms,thusremovingdynamicimbalance.Thisprocessofbalancing aywheelbyadditionorremovalofmassintwoparallelplanesiscalledtwo-plane balancing[123,124]. B.2BalancingQuality Duetothelimitationsofmachinesandsensorsusedinbalancing,theywheels containsomeamountofresidualbalanceandcannotbeperfectlybalanced.The imbalancequalityoftheywheelisidentiedusingtheISO1940/1balancingstandard thatassignsa`G'numbertheletterGfollowedbyasetofnumbersbasedonthe ywheelimbalanceandthespeedoftheywheel.Alow`G'numberindicatesawell balancedywheel[124].Thepermissibleresidualimbalanceforeachbalancingplane inarotorcanbedeterminedusingISO1940/1balancingqualitychart[125]shown inFigureB-5.Thechartisdrawnona log-log graph.Thespeedoftheywheel/rotor in rpm isgivenonthehorizontalaxis.Theverticalaxisrepresentsthepermissible residualimbalancein gmm per kg ofrotorweight.Alternatively,theverticalaxisalso representsthedisplacementofthe c.m. oftherotorin mm .Forinstance,atG0.4,the permissibleimbalancefora2 kg rotorspinningat2000 rpm is4 gmm .Equivalently,the permissibledisplacementofthe c.m. oftherotoris2 microns .Twoywheelsbalanced tothesamequalitywillhavethesamemagnitudeof c.m. displacement.Thus,a2 kg ywheelanda3 kg ywheelbalancedtothesamequality,sayG0.4,willhavean imbalanceof4 gmm ,and6 gmm ,respectively.Furthermore,thisimbalanceincaused bythedisplacementofeachoftheir c.m. by2 microns .Itshouldalsobenotedthatthe balancingqualitydependsonthespeedoftheywheel.Considertwoywheelswiththe 184

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same c.m. displacement.Theywheelwithahigheroperationalspeedwillhavealower balancingqualityandahigher`G'numbercomparedtotheotherywheelashigher speedresultsinlargerimbalancetorques.Thebalancingqualitythusdeterminesthe abilityoftheywheeltoresultinalowvibrationenvironment. FigureB-1.HomogeneousFlywheel FigureB-2.FlywheelwithOnlyStaticImbalance 185

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FigureB-3.FlywheelwithOnlyDynamicImbalance FigureB-4.BalancedFlywheelSystem 186

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FigureB-5.BalancingQualityasperISO1940/1 187

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BIOGRAPHICALSKETCH VivekNagabhushanobtainedhisbachelorsdegreeinmechanicalengineering fromVivesvarayaTechnologicalUniversity,Bangalore,Indiaintheyear2003.Following which,heworkedasadesignengineeratthetechnologycenterofLarsenandToubro Limited,Mumbai,Indiauntil2007.WhileatLarsenandToubroLimited,heinventedtwo novelmechanismsandledpatentapplicationsforthesame.Hejoinedthegraduate programinthemechanicalandaerospaceengineeringdepartmentattheUniversityof Floridain2007.HejoinedtheSpaceSystemsGroupSSGadvisedbyDr.Norman Fitz-Coyasaresearchassistantin2008andobtainedamaster'sdegreeinaerospace engineeringin2009.HehadtheopportunitytoworkasaninternattheNASAAmes researchcenterinthesummerof2009,andwhileatAmes,Vivekworkedontestingof controlmomentgyroscopesCMG,anddevelopmentofsmallsatelliteattitudecontrol systemsACS.HejoinedthePhDprograminFall2009andcontinuedtoworkasa researchassistantatSSG.WhileatSSG,Vivek'sresearchwasfocusedonspacecraft attitudedynamicsandcontrol,CMGs,smallsatellitedevelopment,hardware-in-the-loop simulationofspacecraftACS,andspacecraftdynamicparameterestimation.Heserved astheprojectmanagerforSwampSatpico-satelliteprojectfrom2008to2011.Healso co-authoredseveralresearchproposalswithhisadvisor.Hehasauthoredoveradozen conferencepapersandhasledthreepatentapplicationsintheareaofspacecraft attitudecontrol.HeworkedasaninternatHoneybeeRoboticsfromFeb2011toMarch 2012.AtHoneybee,VivekworkedonthedevelopmentofsmallsatelliteCMGsandalso authoredseveralsmallbusinessinnovativeresearchSBIRproposals.Hewasawarded theOutstandingInternationalStudentAwardbytheUniversityofFloridaintheyear 2010.Vivekhasalsoservedasareviewerforpeerreviewedjournalsandconferences. 199