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A Novel Integrated Spacecraft Attitude Control System Using Variable Speed Control Moment Gyroscopes

Permanent Link: http://ufdc.ufl.edu/UFE0043645/00001

Material Information

Title: A Novel Integrated Spacecraft Attitude Control System Using Variable Speed Control Moment Gyroscopes a Lyapunov-Based Approach
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Kim, Dohee
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: vscmg
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Abstract A Variable Speed Control Moment Gyroscope (VSCMG) is a momentum exchange device used for attitude control in various objects such as satellite, aircraft, and underwater vehicles. The main body accomplishes the desired attitude maneuver by changing the direction of the angular momentum vector of the momentum exchange device. Momentum exchange devices such as VSCMG, Control Moment Gyroscope (CMG), and Reaction Wheel (RW) achieve precise attitude control since they operate in a continuous manner contrary to the on/off operation of gas jets. The VSCMG is regarded as a hybrid between a CMG and RW since the spinning rotor can be rotated or gimbaled for momentum transfer with an extra degree of freedom (DOF) resulting from the variable speed flywheel. Hence, the VSCMG can take advantage of torque amplification like a conventional CMG and can additionally acquire several benefits like singularity escape, power tracking, internal momentum management, start-up, and so on by its extra DOF resulting from the flywheels with variable speed. The focus of this dissertation is to develop various control laws and further a new singularity detection method for a VSCMG-actuated satellite in the presence of uncertain satellite inertia, uncertain actuator inertia, and uncertain dynamic and static friction. Using novel control laws, this research achieves attitude stabilization as well as energy storage, initial start-up, and power reduction in the presence of friction. A kinematic model quantified by quaternion and a nonlinear VSCMG-actuated satellite dynamic model are developed in Chapter 2. To actively and effectively utilize CMG mode of VSCMG, analysis for CMG singularities is also included in this chapter. In Chapter 3, an adaptive robust integrated power and attitude control system (IPACS) is presented for a VSCMG-actuated satellite. The key concept is that the VSCMG cluster stores kinetic energy by spinning up the flywheels during sunlight periods so that during eclipse periods supporting power for the satellite subsystems by spinning down its flywheels. Such energy storage capability can be used as a mechanical battery. The developed IPACS method is capable of achieving precision attitude control while simultaneously achieving asymptotic power tracking for a rigid-body satellite in the presence of uncertain friction in the VSCMG gimbals and wheels. In addition, the developed controller compensates for the effects of uncertain, time-varying satellite inertia properties. Some challenges encountered in the control design are that the control input is premultiplied by a non-square, time-varying, nonlinear, uncertain matrix and is embedded in a discontinuous nonlinearity. In the presence of uncertain actuator friction, Chapter 4 provides an adaptive attitude controller developed for a satellite that is actuated by a pyramidal arrangement of four single gimbal VSCMGs. From a cascade connection of satellite, gimbal, and wheel dynamics equations, a backstepping method is exploited to develop the controller. Internal friction is included in each torque expressions of the gimbal and the wheel assemblies since friction effects are significant when scaling the size of the VSCMGs. A system utilizing internal friction can reduce the consumption of battery power when decelerating on the developed dynamic structure. A null motion strategy lets the wheels operate in deceleration mode while simultaneously performing the gimbal reconfiguration for singularity avoidance. The applied torques of the wheels containing friction losses contribute to power reduction when in deceleration mode. Chapter 5 develops a new initial start-up method for a satellite actuated by a pyramidal arrangement of VSCMGs despite the effects of uncertain, time-varying satellite inertia properties and uncertain actuator inertia properties. The method provides closed-loop internal momentum tracking control to enable the flywheels to start from rest and reach desired wheel speeds in the transition from safe hold mode (SHM) to initial attitude acquisition mode. The proposed controller functioning as a VSCMG steering law is developed in terms of the gimbal rates and the flywheel accelerations which are weighted by a singularity measure. Specifically, using null motion, a strategy is developed to simultaneously perform gimbal reconfiguration for singularity avoidance and internal momentum management for flywheel start-up. Using obvious benefits of artificial intelligence (AI) techniques which can effectively approximate nonlinearity and complexity, a recurrent neural network (RNN)-based adaptive attitude controller developed in Chapter 6 achieves attitude tracking in the presence of parametric uncertainty, actuator uncertainty, and nonlinear external disturbance torques, which do not satisfy the linear-in-the-parameters assumption (i.e., non-LP). The adaptive attitude controller results from a RNN structure while simultaneously acting as a composite VSCMG steering law. In addition to accurate attitude control, a null motion strategy is developed to simultaneously perform gimbal reconfiguration for singularity avoidance and wheel speed regularization for internal momentum management. Chapter 7 develops a new singularity detection method using fuzzy logic system (FLS). If a specific type of singularity is detected, a VSCMG steering law can efficiently select operation modes such as CMG mode and RW mode corresponding to the singularity type. The FLS-based singularity detection method is based on the passibility condition by null motion near singularity to classify a singularity identity into elliptic and hyperbolic singularity, and then using additional information denoted as a conventional singularity measure index degenerate hyperbolic singularity can also be escaped. The FLS effectively extracts significant information from singularity with nonlinear and complex patterns. By using the FLS-based singularity detection method, all internal singularities can be classified and escaped online.
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 Dohee Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043645:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043645/00001

Material Information

Title: A Novel Integrated Spacecraft Attitude Control System Using Variable Speed Control Moment Gyroscopes a Lyapunov-Based Approach
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Kim, Dohee
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: vscmg
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Abstract A Variable Speed Control Moment Gyroscope (VSCMG) is a momentum exchange device used for attitude control in various objects such as satellite, aircraft, and underwater vehicles. The main body accomplishes the desired attitude maneuver by changing the direction of the angular momentum vector of the momentum exchange device. Momentum exchange devices such as VSCMG, Control Moment Gyroscope (CMG), and Reaction Wheel (RW) achieve precise attitude control since they operate in a continuous manner contrary to the on/off operation of gas jets. The VSCMG is regarded as a hybrid between a CMG and RW since the spinning rotor can be rotated or gimbaled for momentum transfer with an extra degree of freedom (DOF) resulting from the variable speed flywheel. Hence, the VSCMG can take advantage of torque amplification like a conventional CMG and can additionally acquire several benefits like singularity escape, power tracking, internal momentum management, start-up, and so on by its extra DOF resulting from the flywheels with variable speed. The focus of this dissertation is to develop various control laws and further a new singularity detection method for a VSCMG-actuated satellite in the presence of uncertain satellite inertia, uncertain actuator inertia, and uncertain dynamic and static friction. Using novel control laws, this research achieves attitude stabilization as well as energy storage, initial start-up, and power reduction in the presence of friction. A kinematic model quantified by quaternion and a nonlinear VSCMG-actuated satellite dynamic model are developed in Chapter 2. To actively and effectively utilize CMG mode of VSCMG, analysis for CMG singularities is also included in this chapter. In Chapter 3, an adaptive robust integrated power and attitude control system (IPACS) is presented for a VSCMG-actuated satellite. The key concept is that the VSCMG cluster stores kinetic energy by spinning up the flywheels during sunlight periods so that during eclipse periods supporting power for the satellite subsystems by spinning down its flywheels. Such energy storage capability can be used as a mechanical battery. The developed IPACS method is capable of achieving precision attitude control while simultaneously achieving asymptotic power tracking for a rigid-body satellite in the presence of uncertain friction in the VSCMG gimbals and wheels. In addition, the developed controller compensates for the effects of uncertain, time-varying satellite inertia properties. Some challenges encountered in the control design are that the control input is premultiplied by a non-square, time-varying, nonlinear, uncertain matrix and is embedded in a discontinuous nonlinearity. In the presence of uncertain actuator friction, Chapter 4 provides an adaptive attitude controller developed for a satellite that is actuated by a pyramidal arrangement of four single gimbal VSCMGs. From a cascade connection of satellite, gimbal, and wheel dynamics equations, a backstepping method is exploited to develop the controller. Internal friction is included in each torque expressions of the gimbal and the wheel assemblies since friction effects are significant when scaling the size of the VSCMGs. A system utilizing internal friction can reduce the consumption of battery power when decelerating on the developed dynamic structure. A null motion strategy lets the wheels operate in deceleration mode while simultaneously performing the gimbal reconfiguration for singularity avoidance. The applied torques of the wheels containing friction losses contribute to power reduction when in deceleration mode. Chapter 5 develops a new initial start-up method for a satellite actuated by a pyramidal arrangement of VSCMGs despite the effects of uncertain, time-varying satellite inertia properties and uncertain actuator inertia properties. The method provides closed-loop internal momentum tracking control to enable the flywheels to start from rest and reach desired wheel speeds in the transition from safe hold mode (SHM) to initial attitude acquisition mode. The proposed controller functioning as a VSCMG steering law is developed in terms of the gimbal rates and the flywheel accelerations which are weighted by a singularity measure. Specifically, using null motion, a strategy is developed to simultaneously perform gimbal reconfiguration for singularity avoidance and internal momentum management for flywheel start-up. Using obvious benefits of artificial intelligence (AI) techniques which can effectively approximate nonlinearity and complexity, a recurrent neural network (RNN)-based adaptive attitude controller developed in Chapter 6 achieves attitude tracking in the presence of parametric uncertainty, actuator uncertainty, and nonlinear external disturbance torques, which do not satisfy the linear-in-the-parameters assumption (i.e., non-LP). The adaptive attitude controller results from a RNN structure while simultaneously acting as a composite VSCMG steering law. In addition to accurate attitude control, a null motion strategy is developed to simultaneously perform gimbal reconfiguration for singularity avoidance and wheel speed regularization for internal momentum management. Chapter 7 develops a new singularity detection method using fuzzy logic system (FLS). If a specific type of singularity is detected, a VSCMG steering law can efficiently select operation modes such as CMG mode and RW mode corresponding to the singularity type. The FLS-based singularity detection method is based on the passibility condition by null motion near singularity to classify a singularity identity into elliptic and hyperbolic singularity, and then using additional information denoted as a conventional singularity measure index degenerate hyperbolic singularity can also be escaped. The FLS effectively extracts significant information from singularity with nonlinear and complex patterns. By using the FLS-based singularity detection method, all internal singularities can be classified and escaped online.
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 Dohee Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043645:00001


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ANOVELINTEGRATEDSPACECRAFTATTITUDECONTROLSYSTEMUSING VARIABLESPEEDCONTROLMOMENTGYROSCOPES:ALYAPUNOV-BASED APPROACH By DOHEEKIM ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011DoheeKim 2

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Tomyparents,Sang-yeolKimandNamsoonHahn,andmybrother,DukyoonKimfor theirunwaveringsupportandconstantencouragement 3

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ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,WarrenE.Dixon,whose experienceandmotivationhavebeeninstrumentalinthesuccessfulcompletionofmy PhD.Theguidanceandthepatiencehehasshownovertheyearshavehelpedmemature inmyresearchandasaprofessional.Iwouldalsoliketoextendmygratitudetomy committeemembersDr.NormanFitz-Coy,Dr.CarlCrane,andDr.OscarCrisalleforthe timeandhelptheyprovided.Also,Iwouldliketothankmycoworkers,family,andfriends fortheirsupportandencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................8 LISTOFFIGURES....................................9 LISTOFABBREVIATIONS...............................12 ABSTRACT........................................13 CHAPTER 1INTRODUCTION..................................17 1.1MotivationandProblemStatement......................17 1.2Contributions..................................24 2SYSTEMMODEL..................................28 2.1KinematicModel................................28 2.2VSCMG-actuatedSatelliteModel.......................28 2.3Singularities...................................33 2.3.1Whatisasingularity?..........................34 2.3.2SingularityofCMGs..........................34 2.3.2.1Externalsingularity.....................36 2.3.2.2Internalsingularity......................36 3PRECISIONIPACSINTHEPRESENCEOFDYNAMICUNCERTAINTY..38 3.1DynamicModelforIPACSinthePresenceofDynamicUncertainty....38 3.2ControlObjectives...............................39 3.2.1AttitudeControlObjective.......................39 3.2.2PowerTrackingObjective........................40 3.3AdaptiveIPACS................................41 3.3.1AdaptiveAttitudeControlDevelopment...............41 3.3.2AdaptivePowerTrackingControlDevelopment............45 3.3.3StabilityAnalysis............................47 3.4SimulationStudy................................50 3.5Summary....................................59 4INTEGRATEDPOWERREDUCTIONANDADAPTIVEATTITUDECONTROLSYSTEMOFASATELLITE........................64 4.1CoupledDynamics...............................64 4.1.1DynamicsforVSCMGs.........................64 4.1.2DynamicsforasatelliteactuatedbyVSCMGs............67 5

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4.2AttitudeControlObjective...........................69 4.3AttitudeControlDevelopment.........................69 4.3.1StabilityAnalysis............................75 4.4SimulationStudy................................77 4.4.1SimulationSetup............................77 4.4.2SimulationResults...........................78 4.5Summary....................................85 5ANEWINITIALSTART-UPMETHODUSINGINTERNALMOMENTUM MANAGEMENTOFVSCMGS...........................86 5.1ControlObjectives...............................86 5.1.1AttitudeControlObjective.......................86 5.1.2FlywheelAngularMomentumManagementObjective........86 5.2ControllerDevelopment............................87 5.2.1AdaptiveAttitudeControlDevelopment...............87 5.2.2MomentumTrackingControlDevelopment..............92 5.2.3StabilityAnalysis............................93 5.3NumericalExamples..............................95 5.3.1SimulationSetup............................95 5.3.2SimulationResults...........................96 5.4Summary....................................103 6ARNN-BASEDATTITUDECONTROLMETHODFORAVSCMG-ACTUATED SATELLITE.....................................105 6.1DynamicModel.................................105 6.2ControlObjectives...............................107 6.2.1AttitudeControlObjective.......................107 6.2.2FlywheelAngularMomentumManagementObjective........107 6.3AdaptiveRNNController...........................107 6.3.1AdaptiveAttitudeControlDevelopment...............107 6.3.1.1Open-LoopErrorSystem...................107 6.3.1.2Closed-LoopErrorSystem..................112 6.3.2MomentumTrackingControlDevelopment..............115 6.3.3StabilityAnalysis............................115 6.4NumericalExample...............................118 6.5Summary....................................121 7ANEWSINGULARITYDETECTIONMETHODFORVSCMGSUSINGFLS122 7.1SingularityDetectionStrategy.........................123 7.1.1PassabilityConditionbyNullMotionnearSingularity........123 7.1.2Fuzzication...............................125 7.1.3ProductInferenceEngine........................128 7.1.4Defuzzication..............................128 7.2ImplementationofFLS-basedsingularitydetectionindex..........129 6

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7.3NumericalExamples..............................130 7.3.1SimulationSetup............................130 7.3.2SimulationResults...........................130 7.4Summary....................................137 8CONCLUSIONSANDFUTUREWORK......................138 8.1Conclusions...................................138 8.2FutureWork...................................141 REFERENCES.......................................142 BIOGRAPHICALSKETCH................................150 7

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LISTOFTABLES Table page 4-1PhysicalparametersfortheVSCMGsimulation...................77 4-2InitialparametersfortheVSCMGsimulation....................77 5-1PhysicalparametersfortheVSCMGsimulation...................95 5-2InitialparametersfortheVSCMGsimulation....................96 8

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LISTOFFIGURES Figure page 2-1Geometryofsatellitewith i th VSCMG........................29 2-2Pyramidalarrangementof 4 VSCMGssystem....................30 2-3TotalangularmomentumenvelopeofCMGs....................36 3-1Quaterniontrackingerror e t duringclosed-loopoperation............54 3-2Transientresponseofthequaterniontrackingerror e t ..............54 3-3Responseofthequaterniontrackingerror e t duringthesuddenincreasein frictionat 6 ; 000sec ..................................55 3-4Desiredpowerandenergyprolesandactualclosed-looppowertrackingresponse.55 3-5Transientpowerandenergytrackingerror......................56 3-6Powerandenergytrackingerrorresponseduringsuddenfrictionincreaseat 6 ; 000 sec ............................................56 3-7Controlinputgimbalrates t andwheelaccelerations t ...........57 3-8Transientresponseofthecontrolinputgimbalrates t andwheelaccelerations t ........................................58 3-9Responseofthecontrolinputgimbalrates t andwheelaccelerations t duringsuddenincreaseoffrictionparametersat 6 ; 000sec .............58 3-10Wheelspeeds t duringclosed-loopcontrolleroperation.............59 3-11Adaptiveparameterestimates ^ 1 t ^ 2 t ,and ^ 3 t duringclosed-loopoperation...........................................60 3-12Initialtransientresponseofthevectorelementsoftheadaptiveestimate ^ 1 t ..60 3-13Responseofthevectorelementsoftheadaptiveestimate ^ 1 t duringthesuddenfrictionincreaseat 6 ; 000sec ...........................61 3-14Initialtransientresponseofthevectorelementsoftheadaptiveestimate ^ 2 t ..61 3-15Responseofthevectorelementsoftheadaptiveestimate ^ 2 t duringthesuddenfrictionincreaseat 6 ; 000sec ...........................62 3-16Initialtransientresponseoftheadaptiveestimate ^ 3 t ..............62 4-1Quaterniontrackingerror e t duringclosed-loopoperation............78 4-2Actualcontrolinputgimbalrates t andywheelspeeds t ..........79 9

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4-3Close-upofactualcontrolinputgimbalrates t andwheelspeeds t when encounteringextremesingularity...........................79 4-4Singularitymeasurefunction f ,nullmotionswitch S ,andweightfunction W .......................................81 4-5Nullmotion:gimbalrecongurationandwheeldecelerationerror t ......81 4-6Adaptiveparameterestimates ^ 1 t duringclosed-loopoperation.........82 4-7Adaptiveparameterestimates ^ 2 t duringclosed-loopoperation.........82 4-8Appliedgimbaltorque,gimbalfrictiontorque,andtotalgimbaltorqueduring closed-loopcontrolleroperation............................83 4-9Close-upofappliedgimbaltorque,gimbalfrictiontorque,andtotalgimbaltorque duringtheentireclosed-loopoperation........................83 4-10Appliedwheeltorque,wheelfrictiontorque,andtotalwheeltorqueduringclosedloopcontrolleroperation................................84 4-11Close-upofappliedwheeltorque,wheelfrictiontorque,andtotalwheeltorque duringclosed-loopoperationprovidinginputpowerreduction...........84 5-1Quaterniontrackingerror e t forCase1......................96 5-2Controlinputgimbalrates t andwheelaccelerations t forCase1.....97 5-3Flywheelspeed t inducedfrominternalmomentummanagementforCase1.97 5-4Singularitymeasurefunction f andnullmotionweight S forCase1....98 5-5Nullmotion:gimbalrecongurationandinternalmomentumtrackingerror t forCase1........................................99 5-6Adaptiveparameterestimates ^ 1 t and ^ 2 t forCase1..............99 5-7Transientresponseofadaptiveparameterestimate ^ 1 t forCase1........100 5-8Transientresponseofadaptiveparameterestimate ^ 2 t forCase1........101 5-9Controlinputgimbalrates t andwheelaccelerations t forCase2.....102 5-10Flywheelspeed t inducedfrominternalmomentummanagementforCase2.102 5-11Singularitymeasurefunction f andnullmotionweight S forCase2....103 5-12Nullmotion:gimbalrecongurationandinternalmomentumtrackingerror t forCase2........................................104 6-1Quaterniontrackingerror e t duringclosed-loopoperation............118 10

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6-2RNNestimationerror ~ ~ ; ~ duringclosed-loopoperation............119 6-3InducedinnitynormofRNNweightmatrices ^ W t ; ^ V t ............119 6-4Singularitymeasurefunction h ,nullmotionweight S ,andmodeweight W c ..........................................120 7-1Membershipfunctionforsingularitymeasureindex f ..............127 7-2BlockdiagramofFLS-basedsingularitydetectionmethod.............128 7-3Quaterniontrackingerror e t forCase1......................131 7-4Controlinputgimbalrates t andwheelaccelerations t forCase1.....132 7-5Flywheelspeed t forCase1............................132 7-6Singularitydetectionindex t andnullmotioncondition det Q forCase1..133 7-7Singularitymeasureindex f ,nullmotionweight S ,andmodeweight W forCase1........................................133 7-8Nullmotionresultsforgimbalrecongurationandwheelspeedtrackingerror t forCase1.....................................134 7-9Parameterestimates ^ 1 t ^ 2 t forCase1.....................134 7-10Singularitydetectionindex t andnullmotioncondition det Q forCase2..135 7-11Singularitymeasureindex f ,nullmotionweight S ,andmodeweight W forCase2........................................135 7-12Nullmotionresultsforgimbalrecongurationandwheelspeedtrackingerror t forCase2.....................................136 7-13Flywheelspeed t forCase2............................136 11

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LISTOFABBREVIATIONS ACSAttitudeControlSystem ADCSAttitudeDeterminationandControlSystem AIArticialIntelligence C.M.CenterofMass CMGControlMomentGyroscope DCMDirectionCosineMatrix DGCMGsDoubleGimbalControlMomentGyroscopes DOFDegreeOfFreedom FACETSFlywheelAttitudeControlandEnergyTransmissionsSystem FLSFuzzyLogicSystem FNNsFeedforwardNeuralNetworks GUUBGloballyUniformlyUltimatelyBounded IPACSIntegratedPowerandAttitudeControlSystem ISSInternationalSpaceStation MWMomentumWheel NNNeuralNetworks RBFRadialBasisFunction RNNsRecurrentNeuralNetworks RWReactionWheel SGCMGsSingleGimbalControlMomentGyroscopes SHMSafeHoldeMode UFUniversityofFlorida UUBUniformlyUltimatelyBounded VSCMGVariableSpeedControlMomentGyroscope 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ANOVELINTEGRATEDSPACECRAFTATTITUDECONTROLSYSTEMUSING VARIABLESPEEDCONTROLMOMENTGYROSCOPES:ALYAPUNOV-BASED APPROACH By DoheeKim December2011 Chair:WarrenE.Dixon Major:MechanicalEngineering AVariableSpeedControlMomentGyroscopeVSCMGisamomentumexchange deviceusedforattitudecontrolinvariousobjectssuchassatellite,aircraft,andunderwatervehicles.Themainbodyaccomplishesthedesiredattitudemaneuverbychangingthe directionoftheangularmomentumvectorofthemomentumexchangedevice.Momentum exchangedevicessuchasVSCMG,ControlMomentGyroscopeCMG,andReaction WheelRWachievepreciseattitudecontrolsincetheyoperateinacontinuousmanner contrarytotheon/ooperationofgasjets.TheVSCMGisregardedasahybridbetween aCMGandRWsincethespinningrotorcanberotatedorgimbaledformomentum transferwithanextradegreeoffreedomDOFresultingfromthevariablespeedywheel. Hence,theVSCMGcantakeadvantageoftorqueamplicationlikeaconventionalCMG andcanadditionallyacquireseveralbenetslikesingularityescape,powertracking,internalmomentummanagement,start-up,andsoonbyitsextraDOFresultingfromthe ywheelswithvariablespeed. Thefocusofthisdissertationistodevelopvariouscontrollawsandfurtheranew singularitydetectionmethodforaVSCMG-actuatedsatelliteinthepresenceofuncertain satelliteinertia,uncertainactuatorinertia,anduncertaindynamicandstaticfriction. Usingnovelcontrollaws,thisresearchachievesattitudestabilizationaswellasenergy storage,initialstart-up,andpowerreductioninthepresenceoffriction. 13

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AkinematicmodelquantiedbyquaternionandanonlinearVSCMG-actuated satellitedynamicmodelaredevelopedinChapter2.Toactivelyandeectivelyutilize CMGmodeofVSCMG,analysisforCMGsingularitiesisalsoincludedinthischapter. InChapter3,anadaptiverobustintegratedpowerandattitudecontrolsystem IPACSispresentedforaVSCMG-actuatedsatellite.Thekeyconceptisthatthe VSCMGclusterstoreskineticenergybyspinninguptheywheelsduringsunlight periodssothatduringeclipseperiodssupportingpowerforthesatellitesubsystemsby spinningdownitsywheels.Suchenergystoragecapabilitycanbeusedasamechanical battery.ThedevelopedIPACSmethodiscapableofachievingprecisionattitudecontrol whilesimultaneouslyachievingasymptoticpowertrackingforarigid-bodysatellitein thepresenceofuncertainfrictionintheVSCMGgimbalsandwheels.Inaddition,the developedcontrollercompensatesfortheeectsofuncertain,time-varyingsatelliteinertia properties.Somechallengesencounteredinthecontroldesignarethatthecontrolinputis premultipliedbyanon-square,time-varying,nonlinear,uncertainmatrixandisembedded inadiscontinuousnonlinearity. Inthepresenceofuncertainactuatorfriction,Chapter4providesanadaptive attitudecontrollerdevelopedforasatellitethatisactuatedbyapyramidalarrangement offoursinglegimbalVSCMGs.Fromacascadeconnectionofsatellite,gimbal,and wheeldynamicsequations,abacksteppingmethodisexploitedtodevelopthecontroller. Internalfrictionisincludedineachtorqueexpressionsofthegimbalandthewheel assembliessincefrictioneectsaresignicantwhenscalingthesizeoftheVSCMGs. Asystemutilizinginternalfrictioncanreducetheconsumptionofbatterypowerwhen deceleratingonthedevelopeddynamicstructure.Anullmotionstrategyletsthewheels operateindecelerationmodewhilesimultaneouslyperformingthegimbalreconguration forsingularityavoidance.Theappliedtorquesofthewheelscontainingfrictionlosses contributetopowerreductionwhenindecelerationmode. 14

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Chapter5developsanewinitialstart-upmethodforasatelliteactuatedbya pyramidalarrangementofVSCMGsdespitetheeectsofuncertain,time-varyingsatellite inertiapropertiesanduncertainactuatorinertiaproperties.Themethodprovidesclosedloopinternalmomentumtrackingcontroltoenabletheywheelstostartfromrest andreachdesiredwheelspeedsinthetransitionfromsafeholdmodeSHMtoinitial attitudeacquisitionmode.TheproposedcontrollerfunctioningasaVSCMGsteering lawisdevelopedintermsofthegimbalratesandthef1ywheelaccelerationswhichare weightedbyasingularitymeasure.Specically,usingnullmotion,astrategyisdeveloped tosimultaneouslyperformgimbalrecongurationforsingularityavoidanceandinternal momentummanagementforywheelstart-up. UsingobviousbenetsofarticialintelligenceAItechniqueswhichcaneectively approximatenonlinearityandcomplexity,arecurrentneuralnetworkRNN-basedadaptiveattitudecontrollerdevelopedinChapter6achievesattitudetrackinginthepresence ofparametricuncertainty,actuatoruncertainty,andnonlinearexternaldisturbance torques,whichdonotsatisfythelinear-in-the-parametersassumptioni.e.,non-LP.The adaptiveattitudecontrollerresultsfromaRNNstructurewhilesimultaneouslyactingas acompositeVSCMGsteeringlaw.Inadditiontoaccurateattitudecontrol,anullmotion strategyisdevelopedtosimultaneouslyperformgimbalrecongurationforsingularity avoidanceandwheelspeedregularizationforinternalmomentummanagement. Chapter7developsanewsingularitydetectionmethodusingfuzzylogicsystem FLS.Ifaspecictypeofsingularityisdetected,aVSCMGsteeringlawcaneciently selectoperationmodessuchasCMGmodeandRWmodecorrespondingtothesingularity type.TheFLS-basedsingularitydetectionmethodisbasedonthepassibilityconditionby nullmotionnearsingularitytoclassifyasingularityidentityintoellipticandhyperbolic singularity,andthenusingadditionalinformationdenotedasaconventionalsingularity measureindexdegeneratehyperbolicsingularitycanalsobeescaped.TheFLSeectively extractssignicantinformationfromsingularitywithnonlinearandcomplexpatterns. 15

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ByusingtheFLS-basedsingularitydetectionmethod,allinternalsingularitiescanbe classiedandescapedonline. 16

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CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement AcontrolmomentgyroscopeCMGisanattitudeactuatorthatgeneratestorque byexchangingmomentumwithamainbody.Comparedtomomentumexchangedevices suchasareactionwheelRW,CMGsyieldadvantagessuchastorqueamplicationand rapidresponse.CMGscanprovideapreciseattitudecontrolandrapidretargetingina continuousmanner,whereasathrusterforattitudecontroloperateson/odiscontinuous gasjetswithpropellantsaectingmass,power,andvolume. CMGscanbeconguredindierentways.Theextragimbalinadoublegimbal controlmomentgyroscopeDGCMGprovidesanadditionaldegreeoffreedomDOF. AdditionalDOFcanbeusedforsingularityavoidancestrategies,butDGCMGsaremechanicallycomplexandmassive[30,42,82,110].Singlegimbalcontrolmomentgyroscopes SGCMGshaveamechanicallysimplerstructurewhichconsistsofsinglecontrollable gimbalandconstantywheelandcangeneratemoretorqueamplicationthanDGCMGs. However,sinceaCMGsystemonlychangesthedirectionbutnotthemagnitudeofthe angularmomentumvector,SGCMGsinherentlysuerfromsingularityproblems.Ina three-dimensionalworkspace,SGCMGsareunabletoproducetorquealonganarbitrary singulardirectionsincealladmissibletorquedirectionslieonatwo-dimensionalsurface perpendiculartothesingulardirection.SGCMGssingularitiesareclassiedasanexternal/saturationsingularityandaninternalsingularity.Thespecicarrangementofthe gimbalsaectsthetypeandnumberofsingularities. AvariablespeedcontrolmomentgyroscopeVSCMGcombinesthepropertiesof aCMGandaRWinthattheywheelspeedisvariable,providinganextraDOF.This extendedcapabilityenablesaVSCMGclustertoavoidinternalellipticandhyperbolic singularities.Therefore,VSCMGscanbeconsideredageometricallysingularity-free device.IfalltheCMGtorqueaxeslieinaplane,torquegeneratedbythevariable 17

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ywheeloftheVSCMGclusterwillenabletheCMGcongurationtobeoutoftheplane i.e.,RWmode.TheextraDOFpresentinVSCMGsprovidesmorerobustoptions againstsingularityencounterthanCMGs.Forinstance,thegimbalnullsolutionsof CMGsallowthegimbalanglestorecongurewithoutanyrequestoftorquegeneration. Thesingularityavoidancemethodusinggimbalnullmotionsreduces/eliminatesthe amountoftimethattheVSCMGhastooperateinRWmodewhentheCMGJacobian becomessingular.Basedonthequaternion-basedkinematicmodelandtheVSCMGactuatedsatellitedynamicmodeldevelopedinChapter2,thewheelnullmotionsenable dualuseobjectivesincluding:powerstorageasamechanicalbatteryChapter3,wheel decelerationforpowerreductionChapter4,andinternalmomentumtrackingforinitial start-upChapter5whilemaintainingpreciseattitudecontrol. InChapters3-6ofthisdissertation,variousintelligentcontrolmethodsaredevelopedforVSCMG-actuatedsatelliteinthepresenceofuncertainsatelliteinertia,uncertain actuatorinertia,anduncertaindynamicandstaticfriction.InChapter7,anewsingularitydetectionandclassicationmethodisdevelopedusingafuzzylogicsystemFLS underanalysisforCMGsingularitiesinChapter2.Followingtheexistenceofthenull motion,atypeofsingularitycanbeprimarilydetectedintoellipticsingularityandhyperbolicsingularity,andthenusingadditionalinformationwhichisaconventionalsingularity measureindex,themethodcanfurtherdiscriminatedegeneratehyperbolicsingularities whichdonotaecttherankofCMGJacobian. BycombiningtheattitudecontrolcapabilityofCMGwiththeenergystorage capabilityofvariable-speedywheels,VSCMGsoerthepotentialtocombineenergy storageandattitudecontrolfunctionsinasingledevice.Thisintegrationofattitude controlandenergystoragefunctionscanreducethesatellitebusmass,volume,andcost. Inlightoflaunchcostsasafunctionofmass,theadvantageofintegratedfunctionalityis apparent.ThevariablewheelspinratesofVSCMGsendowthemwithadditionalDOF, whichcanbeusedtoachievemultipleobjectivessuchassimultaneousattitudecontrol 18

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andenergystorage.Forthisreason,VSCMGsareoftenutilizedinthedesignofintegrated powerandattitudecontrolsystemIPACS[1,2,21,22,25,41,43,79,8486,89,92,99,100]. However,theenergystorageandattitudecontrolcapabilitiesofVSCMGscandeteriorate overtimeduetochangesinthedynamicssuchasbearingdegradationandincreased frictioninthegimbalsorwheels[68].Forexample,theramicationsoffrictionbuildup includedegradedpowertransfercapabilitiesandpotentialdestabilizingdisturbances. FrictionbuildupintheconstantspeedCMGsontheSkylabspacestationandthe Magellansatellite[32]resultedincatastrophicfailuresofthosesystems.Thepotential forafailureduetofrictionuncertainties/changesinthegimbalandywheeldynamics necessitatesspecialconsiderationindesigningIPACSforVSCMG-actuatedsatellites. MotivatedbythevirtuesofusingVSCMGs,theproblemofdesigningIPACSinthe presenceofuncertaintieshasbeeninvestigatedbyseveralresearchers.In[89],afeedback controllawisdesignedforaVSCMG-actuatedsatellite,whichachievesasymptotic attituderegulationforasatellitewithknowninertiaproperties.Model-basedandadaptive controlstrategiesarepresentedin[15],whichachieveasymptotictrackingforaspacecraft inthepresenceofconstantuncertaintyinthespacecraftinertia,whilesimultaneously trackingadesiredenergy/powerprole.In[100],model-basedandindirectadaptive controllersaredevelopedforaspacecraftwithuncertaininertiaproperties.Anadaptive controlalgorithmisdevelopedin[102],whichachievesattitudecontrolforaVSCMGactuatedsatelliteinthepresenceofunknownmisalignmentsoftheaxisdirectionsofthe VSCMGactuators.Thecontroldevelopmentsin[15,89,100,102]assumethatthesatellite inertiapropertiesareconstant.Whilethisassumptionmaybevalidforlargersatellites, signicantuctuationsintheoverallsatelliteinertiacanoccurinsmallersatellitessmallsatsduetothemotionoftheVSCMGs.Further,thecontrollersin[15,89,100]assume nodynamicuncertaintyintheVSCMGactuators.Whiletheaforementionedcontrollers performwellforapplicationsinvolvinglargesatellites,theymaynotbewellsuitedfor IPACSforVSCMG-actuatedsmall-sats.Thecontroldevelopmentin[47,48]andin 19

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Chapter3ismotivatedbythedesiretoincludetheuncertaindynamicsofthegimbalsand ywheelsinthecontroldesignforimprovedrobustnesstothesedisturbances. ThemajorityofresearchfocusedonVSCMGshasassumedidealconditionssuch asfrictionlessywheelandgimbalbearingsandasystemofVSCMGsasarigidbody. WhenscalingthesizeofCMGs/VSCMGs,theeectsoffrictionpresentinthesystem aresignicanti.e.,duetolessecientbearingsandmotorsandthelackofavailable hardwarecomponents[63,65,68].IPACSand/orywheelattitudecontrolandenergy transmissionsystemFACETSuseVSCMGsforamechanicalbatterybyde-spinningthe ywheelsoftheVSCMGandutilizingtheirkineticenergy[21,40,47,76,84,100].This approachmayseemfeasibleforVSCMGsystemswithmoreecientmotorsandbearings i.e.,largespacecraft;however,forsmallersystemstheeectsoffrictionaremore signicantandcannotbeneglected.Hence,Chapter4explorestheutilizationofbearing and/ormotorfrictiontodeceleratetheVSCMGywheels.Utilizingsystemfrictionin decelerationmodeprovidesanavenuetoreducetheconsumptionofbatterypower.Even ifthefrictioncoecientismodeled,thefrictionmodelmaynotbeabletoreectvariable frictionsinceitisdicultforthefrictioncoecienttobeconstantlypredicted.Hence usinganadaptationmechanismtoestimatetheuncertainfrictioncoecientscanbean alternative.Todevelopacontrollerthatacountsforuncertainfriction,coupleddynamics ofthesatellite,gimbals,andwheelsisdevelopedinChapter4.Basedonthecoupled dynamics,abacksteppingmethodisusedtodesignadaptivecontroltorques.Thewheel decelerationmoderesultingfromthewheelnullsolutioncontributestoinputpower reduction. VariousspacecraftsusemomentumdevicessuchasmomentumwheelsMWs,RWsor CMGstomaintainand/orperformpreciseattitudemaneuvers.Forthesespacecrafts,the operationalspinrateofthewheelmustbeobtained,andseveralwheelinitializationmethodshavebeeninvestigatedforinitialacceleration[11,31,58] ApitchMWmethodcanbe usedduringaninitialattitudeacquisitionmode,wherethewheelrequiresmagnetorquers 20

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tomaintainitsspinratewhileprovidingattitudestabilization[11].ApitchMWmethod canalsobeusedtoacquirethegyroscopicstinessalongtherollandyawaxes[31]. Duringtheinitialwheelaccelerationandholdingthenominalspeed,theattitudedeterminationandcontrolsystemADCSusesasequentialmodechangeoftheMWcontroller andthemagnetorquerwhilemaintaininganadir-pointingattitudewithgravity-gradient stabilization.Forinitialstart-upofRWs,wheelstart-upandattitudestabilizationcanbe achievedwhentransitioningfromsafeholdmodeSHMandinitialattitudeacquisition modebyusingfourRWswithmagnetoquers[58].CMGshavealsobeenemployedin variouslargespacemissionssuchasSkylab,MIR,andinternalspacestationISSto takeadvantageofthetorqueamplicationandpowersavingpropertiesofCMGs.Despitetheirbenets,singularitiesareaninherentproblemforCMGs[26,55,59,70,77,98]. VarioussolutionsaddresstheCMGsingularityissue,andwiththedevelopmentofminiCMGs[6,9,35,49,51,60,63,65,66,68,77],severalrecentspacemissionsusingCMGs havebeenlaunchedorscheduled[8,73,75].Toachievewheelstart-upandstabilization withaCMGsystem,magnetorquerscanbeusedtomaintaincooperationwiththeground stationfordesiredcongurationmodicationtoacceleratetheCMGwheels[73].Ingeneral,previousspacemissionsusingCMGsuseaseparatefeedbackcontrollooptospinup therotortotherequiredspinrateandmaintainitwhilesecuringattitudestabilization usingadditionaldevicessuchasmagnetorquers[88].Theextradegreeoffreedompresent inVSCMGsprovidesanavenuetocondensetheinitialstart-upandinitialattitudeacquisitionmodeintoonestep.Hence,Chapter5ismotivatedbythequestion: Canan additionalDOFofVSCMGbeusedtostartasystemfromrest? TheresultsinChapter 5focusonthedevelopmentofaywheelmomentummanagementstrategytobringthe actualywheelmomentumfromzeromomentumi.e.initialstart-upfromrest. TocompensateforuncertainsystemparametersinChapters3-5,typicaladaptive controlmethodsareused.However,forcomplexandpracticalproblemswhereaccurate mathematicalmodelsmaynotbeavailable,articialintelligentmethodcanbebenecial, 21

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whichcanbeusedtoapproximateanynonlinearsystemwithinanarbitrarilysmall residualerror[3,36].Inparticular,neuralnetworkNNiswidelyutilizedtoapproximate uncertaintiesindynamicsystems[33],andNN-basedcontrolhasproventobeaneective meansofachievingaccurateattitudecontrolofsatellitesinthepresenceofdynamic uncertainty[10,13,29,38,61,62,69,90,93].In[90],aNN-basedcontrollerisusedto compensateforuncertaintiesresultingfromatmosphericeectsandnon-rigiditywhich areoftenneglectedintypicalattitudecontroldesigns.RadialbasisfunctionRBFNNs areutilizedin[93]fordirectadaptivecontrolinthepresenceofunmodeledeectsand parametricuncertainty.In[50],Krishetal.investigatetheuseofback-propagationNNs toproviderobustadaptivenonlinearsatelliteattitudecontrol.Theresultin[50]illustrates thedesignandimplementationofaneuro-controllerforanonlinearspacestationmodel. Attitudetrackingandvibrationstabilizationofaexiblespacecraftareachievedin[38] usingaRBFNN-basedadaptivecontroldesign.Utilizationofanadditionalgaintechnique i.e.,theNussbaumgaintechniquein[38]enabledHuetal.torelaxthesignassumption forhigh-frequencygainforaneuraladaptivecontrol.Althoughfeedforwardneural networksFNNsusingmultilayerperceptronsorradialbasisfunctionnetworksare eectivetocompensateforuncertaintiesandunmodeleddisturbances,emergingresearch showsthatrecurrentneuralnetworksRNNs,whereconnectionsbetweenunitsforma directedcycle,aresuperiortoFNNsinbothmodelingofnonlinearsystemsandprediction oftime-seriesstates[12,19,39,67].Theresearchin[19]illustratestheincreasedcapability ofRNNsoverFNNstocontaintime-varyinganddynamicbehaviorinthepresenceof noise.In[67],Lietal.illustratehowadynamicaltime-variantsystemcanbeeectively approximatedusingtheinternalstateofacontinuous-timeRNN.Theutilizationof dynamicallydrivenRNNscanbemoreecientforidenticationandmodelingofdynamic plantsinacontrol-theoreticframework.ThecapabilityofRNNmodelingfornonlinear dynamicsystemsenablesadynamicalsystemtoevolvethestatescorrespondingto nonlinearstateequations[33,72].Chapter6ismotivatedbythefollowingquestion: 22

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CanRNNmodelingpropertoadynamicaltime-variantsystemeectivelycompensatefor uncertainsystemparametersinVSCMGsystem? Toaddressthismotivatingquestion, anadaptiveRNN-basedattitudecontrollerisdevelopedinChapter6inthepresenceof uncertain,time-varyingsatelliteinertiaproperties,actuatoruncertainties,andnonlinear externaldisturbances. ThesingularityavoidancemethodsusedinChapters4and6maintaintorque amplicationwhileavoidingsingularities.However,tomoreproperlyreacttosingularity correspondingtocharacteristicsofeachsingularity,itisessentialtounderstandsingularity anddetectaspecictypeofsingularity.Toovercomelong-standingsingularityissues whichhaveremainedproblematicsinceMarguliesandAubrun[70]establishedatheory toinvestigateCMGsingularity,variousCMGcongurationsandsteeringlawshave beenproposed[4,14,17,20,34,53,54,78,94,103,104].CMGcongurationscanbe mainlyclassiedintorooftypearrangementandpyramidalarrangement.Therooftype congurationisabletoavoidallinternalsingularityusing6CMGunitsbutinecient sincetheradiusofworkspacei.e.,totalangularmomentumenvelopeissmallinspiteof theredundancyofCMGunits[54,56,57].Ontheotherhand,thepyramidalconguration haslargerworkspacethantherooftypebutellipticsingularitywhichcanbeescaped onlybychangingthemagnitudeofangularmomentummakesitdiculttoactively utilizethisconguration.Hence,ifasteeringmethodforthepyramidalconguration canescapeinternalsingularities,variousmissionswillbeabletoeectivelyutilizeCMG actuators.Toresolvesingularityproblemshasintroducedavarietyofsteeringmethods. Globalavoidancemethodsincludingpathplanning[78],preferredgimbalangle[94], andworkspacerestriction[53,54]cansteerasetofgimbalanglesbasedonapriori knowledgeofthesingularstatesbutsucho-linecalculationlimitstheworkspaceof CMGarrays[53,54,78,94].Gradientmethodsforwhichanullmotionisdeterminedto increaseordecreasedistancetosingularstatesarenoteectiveduetoellipticinternal singularities[14,34,103].Steeringlawsthatallowtorqueerrorsi.e.,asingularityrobust 23

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methodsavoidsingularitiesbutsacricespreciseattitudecontrol[6,27,70,74,97].A hybridsteeringlogicforSGCMGrecentlydevelopedby[64]simultaneouslyutilizesnull motionandtorqueerrorstocopewithaspecictypeofsingularity. ForaVSCMGsystem,acombinationofaSGCMGsystemandaRWsystemcanbe geometricallyconsideredasasingularity-freedevice.Toutilizethetorqueamplication propertyofCMG,thesingularityavoidanceoftheCMGsystemisstillessentialforthe VSCMGsystem[87,101].Further,whenthespecictypeofsingularityisdistinguished online,theCMGsystemcanachievepreciseattitudetrackingperformance[64].Ifthe specictypeofsingularitycanbedetermined,theVSCMGcanacquiremoreeective performancesincetheVSCMGcanmakethebestuseofthetorqueamplicationinCMG modeandalsoutilizethewheelnullmotionse.g.,start-up,powerreduction,etc.while properlyrespondingtoeachtypeofsingularitywhilemaintainingholdingpreciseattitude control.UsingsingularitymetricandtheFLSclassierinChapter7,allofthesingular surfacesinsidetheangularmomentumenvelopecanbeclassiedandescaped/avoided online. 1.2Contributions Thisdissertationfocusesondevelopingavarietyofmulti-functionalintelligentcontrol lawsforVSCMG-actuatedsatellitesinthepresenceofuncertainsatelliteinertia,uncertain actuatorinertia,anduncertaindynamicandstaticfriction,andfurtheranewsingularity detectionmethodforaVSCMG-actuatedsatellite.Usingnovelcontrollaws,thisresearch achievesattitudestabilizationaswellasenergystorage,initialstart-up,andpower reductioninthepresenceoffriction.ThecontributionsofChapters3-7areasfollows. Chapter3, PrecisionIPACSinthePresenceofDynamicUncertainty: ThecontributionofthisworkisthedevelopmentofanIPACSforVSCMG-actuatedsatellitesinthe presenceofuncertaindynamicandstaticfrictionintheVSCMGgimbalsandwheelsso thatthecontrolleriscapableofachievingattitudetrackingwhilesimultaneouslytracking adesiredpowerproleasymptotically.Inaddition,thecontrollercompensatesforthe 24

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eectsofuncertain,time-varyingsatelliteinertiaproperties.Thewheelnullmotions resultingfromtheextendedDOFofVSCMGsallowtheVSCMGsystemtoaccomplisha novelcombinedobjectiveasprecisionattitudetrackingandpowerstoragei.e.,mechanical battery.Thedevelopedcontrollerincludestheuncertaindynamicsofthegimbalsand ywheelsinthecontroldesignforimprovedrobustnesstothesedisturbances.Thedicultiesarisingfromdynamicfrictionanduncertainsatelliteinertiaaremitigatedthrough strategicmanipulationoftheclosed-loopderivedfromaLyapunov-basedanalysis.Inthe presenceofstaticfriction,thecontroldesigniscomplicatedduetothecontrolinputbeing embeddedinadiscontinuousnonlinearity.Thisdicultyisovercomewiththeuseofa robustcontrolelement. Chapter4, IntegratedPowerReductionandAdaptiveAttitudeControlSystemof aSatellite: Thecontributionofthisworkisthedevelopmentofcoupleddynamicsof thesatellite,gimbals,andwheelsincludingstaticanddynamicfrictionongimbaland wheelbearings.WhenscalingthesizeofCMGs/VSCMGs,theeectsoffrictionpresent inthesystemaresignicanti.e.,duetolessecientbearingsandmotorsandthelack ofavailablehardwarecomponents.Inthepresenceofuncertaindynamicandstatic frictionsinboththegimbalsandtheywheels,thecontrollerisdevelopedbyusing aLyapunov-basedbacksteppingtechnique.Thesystemiscapableofachievingglobal asymptoticattitudetrackingwhilesimultaneouslyperformingsingularityavoidanceand wheeldecelerationbythenullmotion.Inthewheeldespinningmode,thewheelfriction torqueisbenecialforthetotalappliedtorquesofthewheels.Sincethewheelfriction providesadditionaltorquesforthewheeldynamics,thedespinningwheelscontributeto powerreductionwithoutanadditionaltorquerequest.Powerreductionresultsfromthe wheeldecelerationmodeandyieldsbothtorqueandpowerreduction.Also,theapplied controltorquecanresponsivelycompensateforuncertainparametersallowingthesystem tomaintainconsistentperformanceinthepresenceofdynamicuncertainty. 25

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Chapter5, ANewInitialStart-upMethodUsingMomentumManagementof VSCMGs: Thecontributionofthisworkisthedevelopmentofanadaptiveattitude controllerforaVSCMG-actuatedsatellitewhichachievesinitialstart-upandinitialattitudestabilizationinthenormaltransitionfromSHMtoinitialattitudeacquisitionmode. PreviousspacemissionsusingCMGshaveusedaseparatefeedbackcontrollooptospin uptherotortotherequiredspinrateandmaintainitwhilesecuringattitudestabilization usingadditionaldevicessuchasmagnetorquers.Inthischapter,theVSCMGsteeringlaw includingtheinternalmomentummanagementallowstheywheeltostartfromrestand toreachthedesiredspeed.Inthepresenceofsatelliteinertiauncertaintyandactuator uncertainty,theproposedattitudecontrolleriscapableofachievingglobalasymptotic attitudetrackingwhilesimultaneouslyperformingsingularityavoidanceandinternal momentummanagement.Thesignicantbenetofthedevelopedsteeringlawistocondenseseveraldiscontinuous,separatefeedbackcontrolstepssuchastheinitialstart-up andinitialattitudeacquisitionmodeintoonecontinuousandsimultaneouscontrolstep. Thecontrolleralsocompensatesfortheeectsofuncertain,time-varyingsatelliteinertia properties.Thedicultiesarisingfromuncertainsatelliteinertiaaremitigatedthrougha Lyapunov-basedstabilityanalysisderivedcontroller. Chapter6, ARNN-basedAttitudeControlMethodforaVSCMG-actuatedSatellite: ThecontributionofthisworkisthedevelopmentofaRNNstructurewhilesimultaneouslyactingasacompositeVSCMGsteeringlawwhichachievesattitudetrackingfor aVSCMG-actuatedsatelliteinthepresenceofuncertaintyinthesatelliteandactuator dynamicsandunmodeledexternaldisturbances.Theinternalstateofacontinuous-time RNNcaneectivelyapproximateadynamicaltime-variantsystem.Sincetheutilizationof dynamicallydrivenRNNscanbemoreecientforidenticationandmodelingofdynamic plantsthanoneofFNNs,thecapabilityofRNNmodelingtoevolvethestatescorrespondingtononlinearstateequationsisexploitedtocompensateforactuatoruncertaintiesof 26

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VSCMGs.SimulationresultsindicatetheRNNsystemproperlyapproximateanonlinearsystemwithinasmallresidualerror.ALyapunov-basedstabilityanalysisisusedto provethecontrollerachievesattitudestabilizationwhilecompensatingfortheeectsof uncertaintime-varyingsatelliteinertiaproperties,parametricuncertainty,andnonlinear externaldisturbancetorques. Chapter7, ASingularityDetectionmethodforVSCMGsUsingFLS: Although variousendeavortoresolvethelong-standingproblemcausedbyCMGsingularityhas beenstudied,therehavebeennoCMGsteeringlawstoescapeallinternalsingularities i.e.,elliptic,hyperbolic,anddegeneratehyperbolicwhileachievingprecisionattitude control.Inthischapter,aFLS-basedsingularitydetectionmethodprovidesanavenueto answerthetroublesomeissues.SinceFLScopeswithcomplexandnonlinearpatternsof singularity,theFLSprovidesaneectivesingularitydetectionstrategy.Thedeveloped singularitydetectionandclassicationmethodcanescape/avoidinternalsingularities, includingthedegeneratehyperbolicsingularity.Thismethodistherstresultthatcan escapeallinternalsingularitiesforthepyramidalarrangement.Bydetectingaspecic typeofsingularity,theVSCMGcanacquiremoreeectiveperformancesincetheVSCMG canmakethebestuseofthetorqueamplicationinCMGmodeandthewheelnull motionse.g.,start-up,powerreduction,etc.whileproperlyrespondingtoeachtypeof singularitywhilemaintainingpreciseattitudecontrol. 27

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CHAPTER2 SYSTEMMODEL 2.1KinematicModel Therotationalkinematicsofasatellitemodelledasarigid-bodycanbeexpressedas q v = 1 2 )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(q v S + q 0 S q 0 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 q T v S ; where S t 2 R 3 denotesthesatelliteangularvelocity,and q t f q 0 t ;q v t g2 R R 3 representstheunitquaterniondescribingtheorientationofthesatellitebody-xedframe F S withrespecttotheinertialreferenceframe I ,subjecttotheconstraint q T v q v + q 2 0 =1 : In2, q v 8 q v =[ q v 1 ;q v 2 ;q v 3 ] T denotesthefollowingskew-symmetricmatrix: q v = 2 6 6 6 6 4 0 )]TJ/F23 11.9552 Tf 9.299 0 Td [(q v 3 q v 2 q v 3 0 )]TJ/F23 11.9552 Tf 9.299 0 Td [(q v 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(q v 2 q v 1 0 3 7 7 7 7 5 : Rotationmatricesthatbring I onto F S and I ontothedesiredbody-xedorientation F S d aredenotedby R q 0 ;q v 2 SO and R d q 0 d ;q vd 2 SO ,respectively,aredenedas R )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(q 2 0 )]TJ/F23 11.9552 Tf 11.955 0 Td [(q T v q v I 3 +2 q v q T v )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 0 q v R d )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(q 2 0 d )]TJ/F23 11.9552 Tf 11.955 0 Td [(q T vd q vd I 3 +2 q vd q T vd )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 q 0 d q vd ; where I 3 denotesthe 3 3 identitymatrix,and q d t f q 0 d t ;q vd t g2 R R 3 represents thedesiredunitquaternionthatdescribestheorientationof F S d withrespectto I 2.2VSCMG-actuatedSatelliteModel Thetotalangularmomentum h S C s B ;! S ; ; 2 R 3 foraVSCMG-actuatedsatellite consistingofabusi.e.,aninfrasturctureofasatellite,usuallyprovidinglocationsforthe 28

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Figure2-1.Geometryofsatellitewith i th VSCMG. payroadandfourVSCMGscanbeexpressedas h S C s = J B C S B + 4 X i =1 I CMG i C S S + 4 X i =1 h CMG i =S C CMG i ; wheresub-andsuperscriptofeachnotationindicateapointofinterestandabody, respectively, B t S t 2 R 3 denotetheangularvelocitiesofthebusandthesatellite, t ; t ; t 2 R 4 arethegimbalangle,gimbalrate,andywheelspeed,respectively. ThegeometryofthedynamicmodelisshowninFigures2-1and2-2.In2, J B C S 2 R 3 3 isthemomentofinertiamatrixofthebusrelativetothecenterofmassC.M.of thesatellite C S and I CMG i C S 2 R 3 3 isthe i th CMGunit'sinertiamatrixrelativeto C S givenby J B C S = J B C B + m B )]TJ/F23 11.9552 Tf 10.461 -9.684 Td [(r T C B r C B I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C B r T C B ; I CMG i C S = D CMG i CMG I CMG i C CMG i D T CMG i + m CMG i h r T C CMG i r C CMG i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C CMG i r T C CMG i i ; where J B C B 2 R 3 3 isthemomentofinertiamatrixofthebusrelativetotheC.M.ofthe bus C B m B 2 R isthemassofthebus, r C B 2 R 3 isthepositionof C B relativeto C S CMG I CMG i C CMG i 2 R 3 3 isexpressedintheCMG-xedframe F CMG i andthemomentof 29

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Figure2-2.Pyramidalarrangementof 4 VSCMGssystem. inertiamatrixofthe i th CMGrelativetotheC.M.ofthe i th CMG C CMG i m CMG i 2 R isthemassofthe i th CMG, r C CMG i 2 R 3 isthepositionof C CMG i relativeto C S ,and D CMG i 2 R 3 3 isthedirectioncosinematrixDCMwhichtransforms F CMG i to F S Specically,themomentofinertiamatrixof i th CMGexpressedin F CMG i isdenedas CMG i I CMG i C CMG i I G i C CMG + m G i h r T C G i r C G i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C G i r T C G i i + I W i C CMG + m W i h r T C W i r C W i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C W i r T C W i i ; where I G i C CMG I W i C CMG 2 R 3 3 arethemomentsofinertiamatrixofeachgimbalandwheel relativeto C CMG i ; m G i m W i 2 R arethemassesof i th gimbalandwheel; r C G i r C W i 2 R 3 arethepositionstotherespectiveC.M.ofthe i th gimbalandofthe i th wheelfrom C CMG i Alsoin2, h CMG i =S C CMG i ; 2 R 3 representstheangularmomentumcontributionsfrom theywheelandthegimbalandisgivenby h CMG i =S C CMG i = CMG i I CMG i C CMG i i ^ a G i + CMG i I CMG i C CMG i i ^ a W i ; wheretheangularmomentumoftheCMGisexpressedintermsofaCMG-xedbasis B = ^ a G i ; ^ a W i ; ^ a T i ; ; ^ a G i isagimbalaxis, ^ a W i isaspinningwheelaxis,and ^ a T i isatransverseaxis.Assumingthat CMG i I CMG i C CMG i isaprincipalinertiamatrix,theCMG 30

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inertiamatrixcanbedenedas CMG i I CMG i C CMG i = diag I CMG i G i I W i I CMG i T i since CMG i I CMG i C CMG i ^ a W i = I W i ^ a W i .Inaddition, t = S t B t in2underthe assumptionofarigidbodysatellite,where t isexpressedin F S .Usingtheprincipal inertiaoftheCMGsystemandtherigidbodyassumption,thetotalangularmomentum ofthesatellitecanberewrittenas h S C s = 4 X i =1 h )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(J B C S + I CMG i C S + h CMG i =S C CMG i i = J S C S + 4 X i =1 I CMG i G i i ^ a G i + I W i i ^ a W i ; where J S C S t = )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(J B C S + P 4 i =1 I CMG i C S t .Theinertialderivativeofthetotal angularmomentum h S C s !; ; isexpressedas h S C s = d dt )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(J S C S + J S C S + J S C S + 4 X i =1 I CMG i G i i ^ a G i + I W i i ^ a W i + I CMG i G i i ^ a G i + I W i i ^ a W i : Expressing t in F CMG i whichisgivenbythe i th CMG-xedbasis B ,thesatellite angularvelocity t canbewrittenas CMG i = D T CMG i = ^ a T G i ^ a T W i ^ a T T i !; = G i ^ a G i + W i ^ a W i + T i ^ a T i ; where G i t =^ a T G i t W i t =^ a T W i t ,and T i t =^ a T T i t .Theinertialderivatives of B in2are ^ a G i = T i ^ a W i )]TJ/F23 11.9552 Tf 11.955 0 Td [(! W i ^ a T i ^ a W i = )]TJ/F23 11.9552 Tf 9.298 0 Td [(! T i ^ a G i + + G i ^ a T i ^ a T i = W i ^ a G i )]TJ/F28 11.9552 Tf 11.955 13.27 Td [( + G i ^ a W i : 31

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Substituting2into2andperformingsomealgebraicmanipulations,theinertial derivativeofthetotalangularmomentumcanberepresentedas h S C s = d dt )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(J S C S + J S C S + J S C S + 4 X i =1 I CMG i G i i )]TJ/F23 11.9552 Tf 11.956 0 Td [(I W i i T i ^ a G i + 4 X i =1 I W i i + I CMG i G i i T i ^ a W i + 4 X i =1 n I W i i i + G i )]TJ/F23 11.9552 Tf 11.955 0 Td [(I CMG i G i i W i o ^ a T i ; where d dt )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(J S C S = P 4 i =1 @ @ i )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(J S C S i t sincethedirectioncosinematrix D CMG i constructingthetotalsatelliteinertia J S C S dependson i t .Hence,thekinetic equationgoverningthemotionofarigidsatellitefollowingEuler'sequationyields h S C s = g E C S ; where g E C S !; !;; ; ; ; 2 R 3 istheexternaltorqueappliedtothesatellite.Using 2and2,theequationofmotionforarigidVSCMG-actuatedsatellitecanbe writtenas g E C S = J! + J + J! + C G I CMG G d )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d [ T ] d + C W [ I W ] d + I CMG G d [ T ] d + C T [ I W ] d [] d +[ I W ] d [ G ] d )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( I CMG G d [ W ] d ; wheretheuncertaintotalsatelliteinertiamatrix J S C S ,henceforthdenotedby J for simplicity,ispositivedeniteandsymmetricsuchthat 1 2 min f J gk k 2 T J 1 2 max f J gk k 2 8 2 R n 32

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where min f J g ; max f J g2 R aretheminimumandmaximumprincipalinertiasof J respectively.In2, C G C W C T 2 R 3 4 fortheCMG-xedaxes ^ a G i ^ a W i ^ a T i aredenedas C G ^ a G 1 ^ a G 2 ^ a G 3 ^ a G 4 = 2 6 6 6 6 4 sin 0 )]TJ/F23 11.9552 Tf 9.299 0 Td [(sin 0 0 sin 0 )]TJ/F23 11.9552 Tf 9.298 0 Td [(sin coscoscoscos 3 7 7 7 7 5 ; C W ^ a W 1 ^ a W 2 ^ a W 3 ^ a W 4 = 2 6 6 6 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(cossin 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(cos 2 cossin 3 cos 4 cos 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(sin 2 cos )]TJ/F23 11.9552 Tf 9.298 0 Td [(cos 3 cossin 4 sinsin 1 sinsin 2 sinsin 3 sinsin 4 3 7 7 7 7 5 ; C T ^ a T 1 ^ a T 2 ^ a T 3 ^ a T 4 = 2 6 6 6 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(coscos 1 sin 2 coscos 3 )]TJ/F23 11.9552 Tf 9.298 0 Td [(sin 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(sin 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(cos 2 cossin 3 coscos 4 sincos 1 sincos 2 sincos 3 sincos 4 3 7 7 7 7 5 ; where isaskewangleforthepyramidalarrangementoffourVSCMGs,andtheinertia matrices I CMG G d [ I W ] d 2 R 4 4 are I CMG G d diag I CMG 1 G I CMG 2 G I CMG 3 G I CMG 4 G ; [ I W ] d diag I W 1 I W 2 I W 3 I W 4 ; where I CMG G d istheunknownconstantpositive-denite,symmetricaboutitsgimbalaxis, gimbalinertiamatrixbut [ I W ] d istheknownconstantpositive-denite,symmetricabout itsspinaxis,ywheelinertiamatrix,andtheangularvelocityprojectedto B isdenotedas [ t ] d diag 1 t 2 t 3 t 4 t : G W T 2 R 4 4 ,and [ t ] d and h t i d denotediagonalmatricescomposedofthevectorelementsofmeasurable t t 2 R 4 ,respectively. 2.3Singularities AVSCMGssystemisageometricallysingularityfreedevicesinceitcangenerate controltorquesalonganarbitrarydirection.TheextraDOFresultingfromavariable 33

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wheelspeedi.e.,RWmodedoesnotallowtheVSCMGtoencounterasingularity. However,tomakethebestuseoftorqueamplicationwhichisasignicantadvantage ofoperatinginaCMGmode,itisimportanttoinvestigatesingularitiesoftheCMGs sincetheexistenceofsingularstatesisanobstacletogenerateatorquealongarbitrary directions. 2.3.1Whatisasingularity? ThetermSingularityhasvariousdenitions.Inthedictionary,asingularityis generallydenedasthestateofbeingsingular,distinct,peculiar,uncommonorunusual. Inphysics,asingularityisdenedasapointorregioninwhichthequantitiese.g., gravitationalforcethatareusedtomeasurethegravitationaleldbecomeinnitei.e., thepointisassociatedwithblackholes.Inmathematics,asingularityisthevalueor rangeofvaluesofafunctionforwhichaderivativedoesnotexist,andthetermSingular matrixisdenedasasquarematrixthatdoesnothaveamatrixinversei.e.,Amatrixis singulariitsdeterminantis 0 .Ofpartiularinterestinthisdissertation,however,isthe denitionofSingularityinaworkspaceorangularmomentumenvelopeofCMGs. 2.3.2SingularityofCMGs SingularityofCMGsisdenedasthecasewherethemappingfromaninputtoan outputspaceanonlinear,vector-valuedmapping H : R n R 3 isnotlocallyonto, orequivalentlyamatrixdoesnothaveafullranki.e.,rank C T < 3 ,where t 2 R 4 is thegimbalanglevectorand C T isaCMGJacobianmatrix.Atsingularstates,inthe three-dimensionalworkspacetheCMGsareunabletoproduceatorquealonganarbitrary singulardirectionsincealladmissibletorquedirectionslieonatwo-dimensionalsurface perpendiculartothesingulardirection.Thespecicarrangementofthegimbalsaects thetypeandnumberofsingularities.TheCMGsingularitiescanbeclassiedaccordingto thelocationofthetotalmomentumvectorrelativetotheworkspace:external/saturation singularityandinternalsingularity. 34

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Foranalyzingthesingularmomentumsurfaces[70,96],anarbitraryvector u is representedintermsofasatellite-xedbasis S f ^ s x ; ^ s y ; ^ s z g as u = u x ^ s x + u y ^ s y + u z ^ s z ; anddenedas U = f u : j u j =1 ;u 6 = ^ a G i ;i =1 ;:::;n g ; where u = ^ a G i onlyhappensinaspecialcongurationsuchasDGCMGssystemora roof-typeconguration.AconditionforwhichtheCMGarrangementcannotgenerateany torquesalongthesingulardirection u isdenedas ^ a T i u =0 ; whereall ^ a T i becomecoplanari.e.,rank C T =2 ,andanarbitraryvector u isperpendiculartothatplane.IntheCMG-xedframe F CMG i asshowninFigure2-1, ^ a G i and u spansaplanenormalto ^ a T i ,and ^ a W i hasamaximalorminimalprojectionontothe singularityvector u i.e., ^ a W i u> 0 or ^ a W i u< 0 .Thesingularityconditionof2 canberewrittenas ^ a T i = i ^ a G i u j ^ a G i u j ; ^ a W i =^ a T i ^ a G i = i ^ a G i u ^ a G i j ^ a G i u j ;i =1 ;:::;n; where i sign ^ a W i u .Hence,thesingularmomentumvectorisexpressedas[54,70,96] H u = n X i =1 ^ a W i = n X i =1 i ^ a G i u ^ a G i j ^ a G i u j : Figure2-3showsthetotalangularmomentumenvelopeconsideringthesingular momentumvectorof2forapyramidalarrangementwith4CMGunits.Theangular momentumenvelopeinFigure2-3includestwotypesofsingularitiesdenotedasexternal andinternalsingularitieswhicharesmoothlyconnected. 35

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Figure2-3.TotalangularmomentumenvelopeofCMGs 2.3.2.1Externalsingularity Thegimbalanglesforwhichthetotalangularmomentumreachestheenvelopeof Figure2-3becomesingularsincetheCMGsareunabletoproduceatorqueoutward theenvelope.ThisisbecauseaCMGsystemchangesonlythedirectionbutnotthe magnitudeoftheangularmomentumvectorandthereforeinexternalsingularitythe CMGsystemexperiencesamaximumworkspaceanddoesnothaveadditionalangular momentumforthesingulardirection.Inotherwords,external/saturationsingularitiesare associatedwiththemaximumprojectionofthetotalangularmomentumalongacertain direction.Thecriteriaforthistypeofsingularitycanbeexpressedas rank C T < 3 ; ^ a W i u> 0 8 i =1 ; 2 ; 3 ; 4 : Externalsingularitiescanbeaddressedinthedesignprocesssincetheycanbeeasily predictedfromsizingoftheCMGactuatorsandmissionprole. 2.3.2.2Internalsingularity Internalsingularityisdenedasacasewherethetotalangularmomentumvector foranysingularstateisinsidetheangularmomentumenvelopeasshownbyFigure2-3. 36

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Theinternalsingularityoccursataspeciccombinationofgimbalangleswhichmakes theCMGJacobiansingularandatthissingularitythetorquevectorslieonthesame planeperpendiculartothesingulardirectionvector.Internalsingularitiescanbeclassied accordingtothepossibilityofthenullmotionintotwotypes:ellipticsingularityand hyperbolicsingularity.Thenullmotionisdenedasamotionthatchangesgimbalangles withoutproducinganytorque.Usingnullmotion,theCMGsystemcanberecongured inacontinuousmannersothattheCMGJacobianbecomesnonsingular.Thesingularity thatcanbeescapedbynullmotionistermedhyperbolicsingularity.However,thecase whereasetofgimbalanglesfortheangularmomentumenvelopehasanisolatedpointis termedellipticsingularity.Ellipticsingularitiescannotbeescapedbynullmotion.Atest forpossibilityofnullmotionisspecicallydiscussedinChapter7.Althoughnullmotion ispossibleathyperbolicsingularity,themereexistenceofnullmotiondoesnotguarantee escapefromthehyperbolicsingularity.Therearedegeneratesolutionswhichdonotaect therankoftheCMGJacobian.Thismeansthatthedegeneratehyperbolicsingularities cannotbeescapedthroughnullmotion[5,6,55,64,96].Asingularitydetectionstrategy whichcanhandleallinternalsingularitiesisintroducedinChapter7. 37

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CHAPTER3 PRECISIONIPACSINTHEPRESENCEOFDYNAMICUNCERTAINTY Inthischapter,anadaptiverobustattitudecontrollerisdeveloped,whichcompensatesforuncertain,time-varyinginertiaandunknownfrictionintheVSCMGgimbals andwheelswhilesimultaneouslyprovidingasymptoticpowertracking.Theinclusionof frictioneectsintheVSCMGgimbalsandwheelscreatessignicantcomplicationsinthe controldevelopment.Thedynamicfrictioneectsmanifestthemselvesasnon-square, time-varying,input-multiplicativeuncertaintyinthetrackingerrordynamics.Thestatic frictioneectsinthedynamicmodelresultinthegimbalangularratecontrolinputbeing embeddedinsideofadiscontinuousnonlinearityi.e.,thestandardsignumfunction.Arobustcontrolmethodisusedtomitigatethedisturbanceresultingfromthestaticfriction, andanadaptivecontrollawisusedtocompensateforthedynamicfrictionandinertiauncertainties.Lyapunov-basedstabilityanalysesareprovided,whichproveattitudetracking andpowertrackinginthepresenceoftheaforementionedVSCMGanomaliesandsatellite inertiauncertainty.Numericalsimulationsareprovidedtoillustratetheperformanceof thecontrollersforsimultaneousattitudecontrolandenergystorage. 3.1DynamicModelforIPACSinthePresenceofDynamicUncertainty Thedynamicmodelin2canbeexpressedforIPACSconsideringfrictionas T = J! + J + J! + C G I CMG G d )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d [ T ] d + C W [ I W ] d + I CMG G d [ T ] d + C T [ I W ] d [] d +[ I W ] d [ G ] d )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( I CMG G d [ W ] d : In3, f t 2 R 4 denotesthetorquegeneratedbytheywheelsandthetorquevector T ; ; 2 R 3 in3isdenedas T = )]TJ/F23 11.9552 Tf 9.299 0 Td [(C T F dg + F sg sgn )]TJ/F23 11.9552 Tf 11.955 0 Td [(C W F dw + F sw sgn : 38

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In3, F dg ;F sg 2 R 4 4 and F dw ;F sw 2 R 4 4 arediagonalmatricescontainingthe uncertaindynamicandstaticfrictioncoecientsforthegimbalsandwheels,respectively, and sgn t 2 R 4 denotesavectorformofthestandard sgn functionwherethe sgn isappliedtoeachelementof t 3.2ControlObjectives 3.2.1AttitudeControlObjective Theattitudecontrolobjectiveistodevelopaywheelaccelerationandgimbalrate controllawtoenabletheattitudeof F totracktheattitudeof F d .Toquantifythe objective,anattitudetrackingerrordenotedby ~ R e v ;e 0 2 R 3 3 isdenedthatbrings F d onto F as ~ R RR T d = )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(e 2 0 )]TJ/F23 11.9552 Tf 11.956 0 Td [(e T v e v I 3 +2 e v e T v )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 e 0 e v ; where R q v ;q 0 and R d q vd ;q 0 d weredenedin2and2,respectively,andthe quaterniontrackingerror e t f e 0 t ;e v t g2 R R 3 isdenedas e 0 q 0 q 0 d + q T v q vd e v q 0 d q v )]TJ/F23 11.9552 Tf 11.955 0 Td [(q 0 q vd + q v q vd : Basedon3,theattitudecontrolobjectivecanbestatedas ~ R e v t ;e 0 t I 3 as t !1 : Basedonthetrackingerrorformulation,theangularvelocityof F withrespectto F d expressedin F ,denotedby ~ t 2 R 3 ,isdenedas ~ )]TJ/F15 11.9552 Tf 14.509 3.022 Td [(~ R! d : Fromthedenitionsofthequaterniontrackingerrorvariables,thefollowingconstraint canbedeveloped[16]: e T v e v + e 2 0 =1 ; 39

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where 0 k e v t k 10 j e 0 t j 1 ; where kk representsthestandardEuclideannorm.From6, k e v t k! 0 j e 0 t j! 1 andhence,3canbeusedtoconcludethatif3issatised,thenthecontrol objectivein3willbeachieved. 3.2.2PowerTrackingObjective Thekineticenergy E t 2 R storedintheywheelsofaVSCMGcanbeexpressed as[24] E t = 1 2 T t I W t : Thepowertrackingcontrolobjectiveistodevelopaywheelaccelerationcontrollaw toenabletheactualVSCMGpowertotrackadesiredpowerprole P d t 2 R while simultaneouslytrackingadesiredtime-varyingattitude.Thedesiredpowerprolecanbe relatedtoadesiredkineticenergyprole E d t 2 R as E d t = t 0 P d d; wherethedesiredkineticenergyandpowerprolesareassumedtobebounded.To quantifytheenergytrackingobjective,akineticenergytrackingerror E t 2 R isdened as E = E d )]TJ/F23 11.9552 Tf 11.955 0 Td [(E: Basedon3,thepowertrackingcontrolobjectivecanbestatedas E 0 as t !1 : 40

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3.3AdaptiveIPACS Inthissection,anadaptiveIPACSisdevelopedthatforcesasatellitetotracka desiredattitudetrajectorywhilesimultaneouslyprovidingasymptoticenergy/power tracking.InSection3.3.1,anadaptiveattitudecontrollerisdevelopedforaVSCMGactuatedsatelliteinthepresenceofgimbalandwheelfriction.InSection3.3.2,apower managementsystemisdevelopedwhichoperatesintandemwiththeattitudecontroller. 3.3.1AdaptiveAttitudeControlDevelopment Tofacilitatethecontrollerdesign,anauxiliarysignal r t 2 R 3 isdenedas[24] r )]TJ/F15 11.9552 Tf 14.509 3.022 Td [(~ R! d + e v ; where 2 R 3 3 isaconstant,positivedenite,diagonalcontrolgainmatrix.After substituting3into3,theangularvelocitytrackingerrorcanbeexpressedas ~ = r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : Motivationforthedesignof r t isobtainedfromthesubsequentLyapunov-basedstability analysisandthefactthat3-3canbeusedtodeterminetheopen-loopquaternion trackingerroras e v = 1 2 )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(e v + e 0 I 3 ~ e 0 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 e T v ~ !: Aftertakingthetimederivativeof3andmultiplyingbothsidesoftheresulting expressionby J ,thefollowingexpressioncanbeobtained: J r = J + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ !; wherethefactthat ~ R = )]TJ/F23 11.9552 Tf 9.298 0 Td [(! ~ R 41

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wasutilized.Underthestandardassumptionthatthegimbalaccelerationterm C G I CMG G d t isnegligible[26,83,89],3,3,3,and3canbeusedtoexpress3as J r = 1 + 2 + Y 1 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr )]TJ/F23 11.9552 Tf 9.298 0 Td [(C T F sg sgn )]TJ/F23 11.9552 Tf 11.955 0 Td [(C W F sw sgn ; wheretheuncertainmatrix 1 e v ;e 0 ;r;; 2 R 3 4 isdenedviatheparameterization 1 )]TJ/F23 11.9552 Tf 10.494 8.087 Td [(@J @ 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T F dg )]TJ/F23 11.9552 Tf 9.299 0 Td [(C W I CMG G d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [] d )]TJ/F28 11.9552 Tf 11.955 9.683 Td [( I CMG G d [ W ] d ; andtheuncertainmatrix 2 2 R 3 4 isdenedas 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(C W [ I W ] d : Alsoin3, Y 1 e v ;e 0 ;r;!;! d ; d ;; 1 isdenedviatheparameterization Y 1 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(C W F dw )]TJ/F23 11.9552 Tf 11.955 0 Td [(C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d )]TJ/F23 11.9552 Tf 9.299 0 Td [(! J! + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(e v + e 0 I ~ !: In3, Y 1 2 R 3 p 1 isameasurableregressionmatrix,and 1 2 R p 1 isavector ofunknownconstants.In3,theauxiliarymatrices 1 and 2 containonly linearlyparameterizableuncertainty,sothetermsaregroupedas 1 + 2 Y 2 2 ; where Y 2 e v ;e 0 ;r;!;! d ;; ; ; 2 R 3 p 2 isameasurableregressionmatrix,and 2 2 R p 2 isavectorofunknownconstants.Someofthecontroldesignchallengesfortheopen-loop systemin3arethatthecontrolinput t ispremultipliedbyanon-square,unknown time-varyingmatrix 1 ,andthegimbalratecontrolinput t isembeddedinsideof adiscontinuousnonlinearityi.e., C T F sg sgn .Toaddressthefactthatthecontrol inputispremultipliedbyanon-square,unknowntime-varyingmatrix,estimatesofthe 42

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uncertaintyin3,denotedby ^ 1 t 2 R 3 4 and ^ 2 t 2 R 3 4 ,aredenedas ^ 1 + ^ 2 Y 2 ^ 2 ; where ^ 2 t 2 R p 2 isasubsequentlydesignedestimatefortheparametricuncertaintyin 1 and 2 .Basedon3and3,3canberewrittenas J r = ^ 1 + ^ 2 + Y 1 1 + Y 2 ~ 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Jr )]TJ/F23 11.9552 Tf 9.298 0 Td [(C T F sg sgn )]TJ/F23 11.9552 Tf 11.955 0 Td [(C W F sw sgn ; wherethenotation ~ 2 t 2 R p 2 isdenedas ~ 2 = 2 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 2 : Basedontheexpressionin3andthesubsequentstabilityanalysis,theywheel accelerationcontrolinputisdesignedas = )]TJ/F15 11.9552 Tf 10.924 3.022 Td [(^ + 2 u c )]TJ/F28 11.9552 Tf 11.956 13.271 Td [( I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 g; where g t isanauxiliarycontrolsignaldesignedtoachievethesubsequentpowertracking objective[24].In3,theauxiliarycontrolinput u c t isdesignedas u c = Y 1 ^ 1 + e v ; andthegimbalratecontrolinputisdesignedas = )]TJ/F15 11.9552 Tf 10.924 3.022 Td [(^ + 1 k + k n r; where k;k n 2 R denotepositivecontrolgains.Sincethematrices ^ 1 t and ^ 2 t are non-square,thepseudo-inverses ^ + i 2 R n 3 8 i =1 ; 2 aredenedsothat ^ i ^ + i = I 3 andthematrix I n )]TJ/F15 11.9552 Tf 13.959 3.022 Td [(^ + i ^ i ,whichprojectsvectorsontothenullspaceof ^ i ,satisesthe 43

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followingproperties: I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i = I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i ^ i I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i =0 I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i T = I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i I n )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + i ^ i ^ + i =0 : Aftersubstituting3-3into3,thefollowingclosed-loopdynamicsfor r t canbeobtained: J r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Jr + Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(kr )]TJ/F23 11.9552 Tf 11.955 0 Td [(k n r )]TJ/F23 11.9552 Tf 9.299 0 Td [(C T F sg sgn )]TJ/F23 11.9552 Tf 11.955 0 Td [(C W F sw sgn )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v ; wherethenotation ~ 1 t 2 R p 1 isdenedas ~ 1 = 1 )]TJ/F15 11.9552 Tf 12.894 3.154 Td [(^ 1 : Basedon3andthesubsequentstabilityanalysis,theparameterestimates ^ 1 t and ^ 2 t aredesignedas ^ 1 = proj )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(1 Y T 1 r ^ 2 = proj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 Y T 2 r ; where )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(1 2 R p 1 p 1 and )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 2 R p 2 p 2 denoteconstant,positive-denite,diagonaladaptation gainmatrices,and proj denotesaprojectionalgorithmutilizedtoguaranteethatthe i th elementof ^ 1 t and ^ 2 t canbeboundedas 1 i ^ 1 i 1 i 2 i ^ 2 i 2 i ; where 1 i 1 i 2 R and 2 i 2 i 2 R denoteknown,constantlowerandupperboundsfor eachelementof ^ 1 t and ^ 2 t ,respectively. 44

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3.3.2AdaptivePowerTrackingControlDevelopment Basedon3and3,thepowertrackingerrorcanbequantiedas E = P d )]TJ/F15 11.9552 Tf 15.987 3.022 Td [(_ E: Todeveloptheclosed-loopdynamicsforthepowertrackingerror,thetimederivativeof 3issubstitutedinto3for E t as E = P d )]TJ/F15 11.9552 Tf 11.955 0 Td [( 3 ; wheretheuncertainvector 3 2 R 1 4 isdenedas 3 T I W : Sincetheuncertaintyin3islinearlyparameterizable,thefollowingparameterization canbedeveloped: 3 Y 3 3 ; where Y 3 ; 2 R 1 p 3 isameasurableregressionmatrix,and 3 2 R p 3 isavectorof unknownconstants.Toaddressthefactthatthecontrolinput t ispremultipliedby anunknowntime-varyingmatrix,anestimateoftheuncertaintyin3,denotedby ^ 3 t 2 R 1 4 isdenedas ^ 3 Y 3 ^ 3 ; where ^ 3 t 2 R p 3 isasubsequentlydesignedestimatefortheparametricuncertaintyin 3 .Basedon3and3,3canberewrittenas E = P d )]TJ/F23 11.9552 Tf 11.955 0 Td [(Y 3 ~ 3 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ 3 ; wherethenotation ~ 3 t isdenedas ~ 3 3 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 3 : 45

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From3andthesubsequentstabilityanalysis,theparameterestimate ^ 3 t is designedas ^ 3 = proj )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(3 Y T 3 E ; where )]TJ/F22 7.9701 Tf 7.315 -1.794 Td [(3 2 R p 3 p 3 denotesaconstant,positive-denite,diagonaladaptationgainmatrix, and proj denotesaprojectionalgorithmutilizedtoguaranteethatthe i th elementof ^ 3 t canbeboundedas 3 i ^ 3 i 3 i ; where 3 i 3 i 2 R denoteknown,constantlowerandupperboundsforeachelementof ^ 3 t ,respectively.Aftersubstituting3into3,thefollowingexpressioncanbe obtained E = P d )]TJ/F23 11.9552 Tf 11.955 0 Td [(Y 3 ~ 3 + ^ 3 ^ + 2 u c + ^ 3 I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 g: Given3, g t isdesignedtosatisfythefollowingrelationship ^ 3 I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 g = )]TJ/F23 11.9552 Tf 9.299 0 Td [(P d )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ 3 ^ + 2 u c )]TJ/F23 11.9552 Tf 11.955 0 Td [(k E E ; where k E 2 R isapositiveconstantcontrolgain.BasedontheMoore-Penrosepseudoinversepropertiesintroducedin3,theminimumnormsolutionof3isgiven as g = I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 ^ T 3 h ^ 3 I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 ^ T 3 i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(P d )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ 3 ^ + 2 u c )]TJ/F23 11.9552 Tf 11.955 0 Td [(k E E : Theresultin3indicatesthatsimultaneousattitudeandpowertrackingispossible anytime I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 ^ T 3 6 =0 .Since I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 6 =0 8 ^ 2 t ,thesimultaneousattitude andpowertrackingobjectivecanbeachievedaslongasthefollowingtwoconditionsare 46

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satisedsimultaneously: ^ T 3 t 6 =0 ^ T 3 t = 2N I 4 )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ + 2 ^ 2 ; where N I 4 )]TJ/F15 11.9552 Tf 13.58 3.022 Td [(^ + 2 ^ 2 denotesthenullspaceofthematrix I 4 )]TJ/F15 11.9552 Tf 13.87 3.022 Td [(^ + 2 ^ 2 .Since ^ 3 t containstheadaptiveelementsof ^ 3 t ,theprojectionfunctionin3canbeselectedto expandthedomainwithinwhichthesimultaneousobjectiveispossible.Aftersubstituting 3into3for g t ,thefollowingclosed-looperrorsystemcanbeobtained: E = )]TJ/F23 11.9552 Tf 9.298 0 Td [(k E E )]TJ/F23 11.9552 Tf 11.955 0 Td [(Y 3 ~ 3 : 3.3.3StabilityAnalysis Theorem2-1: Theywheelcontrolinputof3,3and3alongwith theadaptiveupdatelawsgivenin4andthegimbalratecontrolinputof3 ensuregloballyuniformlyultimatelyboundedGUUBattitudetrackinginthesensethat k e v t k 0 exp )]TJ/F23 11.9552 Tf 9.299 0 Td [(" 1 t + 2 ; where 0 ;" 1 ;" 2 2 R denotepositiveboundingconstantsandasymptoticenergy/power trackinginthesensethat E t 0 as t !1 : Proof: Toprovetheasymptoticpowertrackingresult,let V E E ; ~ 3 ;t 2 R bea nonnegativefunctiondenedas V E 1 2 2 E + 1 2 ~ T 3 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 3 ~ 3 : Afterusing3and3,thetimederivativeof V E t canbeexpressedas V E = )]TJ/F23 11.9552 Tf 9.298 0 Td [(k E 2 E : 47

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Basedon3and3, E t 2L 1 L 2 .Theassumptionthat 3 2L 1 can beusedalongwith33,toshowthat ~ 3 t 2L 1 .Giventhat E t 2L 1 ,6 8,3,3,3,and3canbeusedtoshowthat t 2L 1 .Since t 2L 1 ,3canbeusedalongwith3toconcludethat Y 3 t 2L 1 .Given that E t ; ~ 3 t ;Y 3 t 2L 1 ,3canbeusedtoconcludethat E t 2L 1 i.e., E t isuniformlycontinuous.Barbalat'sLemmacannowbeusedtoshowthat E t 0 as t !1 ToprovetheGUUBattitudetrackingresult,considerthenonnegativefunction V e v ; e 0 ;r; ~ 1 ; ~ 2 ;t 2 R denedas V e T v e v + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 2 + 1 2 r T Jr + 1 2 ~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 1 ~ 1 + 1 2 ~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 2 ~ 2 : Byusingtheboundsgivenin2,6,and3, V t canbeupperandlower boundas 1 k z k 2 + c 1 V t 2 k z k 2 + c 2 ; where 1 2 c 1 c 2 2 R areknownpositiveboundingconstants,and z t 2 R 6 isdened as z e T v r T T : From3,3,3,and3,thetimederivativeof V t canbeexpressedas V = e T v )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 e T v ~ + r T Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(kr )]TJ/F23 11.9552 Tf 11.955 0 Td [(k n r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 9.298 0 Td [(C T F sg sgn )]TJ/F23 11.9552 Tf 11.956 0 Td [(C W F sw sgn )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 1 ^ 1 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 2 ^ 2 : Byusing3,4,andexploitingthefactthat e T v e v ~ =0 ; theexpressionin3canbeupperboundedas V )]TJ/F23 11.9552 Tf 28.56 0 Td [( 3 k z k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(k n k r k 2 + )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [( 0 k F sg k i 1 + 1 k F sw k i 1 k r k ; 48

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where 3 = min f ;k g2 R .Aftercompletingthesquares,3canbewrittenas V t )]TJ/F23 11.9552 Tf 21.918 0 Td [( 3 k z k 2 + 2 4 k n ; where 0 k F sg k i 1 + 1 k F sw k i 1 .Sincetheinequalityin3canbeutilizedtolower bound k z t k 2 as k z k 2 1 2 V t )]TJ/F23 11.9552 Tf 14.046 8.088 Td [(c 2 2 ; theinequalityin3canbeexpressedas V t )]TJ/F23 11.9552 Tf 23.114 8.087 Td [( 3 2 V t + "; where 2 R isapositiveconstantthatisdenedas = 2 4 k n + 3 c 2 2 : Thelineardierentialinequalityin359canbesolvedas V t V exp )]TJ/F23 11.9552 Tf 10.494 8.087 Td [( 3 2 t + 2 3 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F23 11.9552 Tf 10.494 8.087 Td [( 3 2 t : Theexpressionsin3and3canbeusedtoconcludethat r t 2L 1 .Thus, from6,3,and3, ~ t ;z t 2L 1 ,and3canbeusedtoconcludethat t 2L 1 .Equation3thenshowsthat e v t ; e 0 t 2L 1 : Hence,3,3, 3-3,3,3,and3canbeusedalongwiththeassumptionthat E d t ;P d t 2L 1 toprovethatthecontrolinputs t ; t 2L 1 .Standardsignal chasingargumentscanthenbeutilizedtoprovethatallremainingsignalsremainbounded duringclosed-loopoperation.Theinequalitiesin3cannowbeusedalongwith 3and3toconcludethat k z k 2 2 k z k 2 + c 2 1 exp )]TJ/F23 11.9552 Tf 10.494 8.087 Td [( 3 2 t + 2 2 4 k n 3 1 + c 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(c 1 1 : Theresultin3cannowbedirectlyobtainedfrom3. 49

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3.4SimulationStudy TotesttheperformanceoftheproposedIPACS,thedynamicparametersoftheUF CMGtestbedwereusedtocreateanumericalsimulation.WhiletheUFCMGdoes nothavevariablewheelspeedcapabilities,arealisticnumericalsimulationenvironment wascreatedasasteppingstonetoactualexperimentalvalidationoftheproposedcontrollaw.Tothatend,usingthephysicalparametersoftheUFCMGtestbedtocreate theVSCMGmodel,thecontrollerperformancewastestedinasimulationenvironment containingmeasurementnoiseandtimedelays.Moreover,toensurethatthesimulationdemonstratesthecapabilityofthecontrollertoperformsatisfactorilyinpractical implementation,thecontrollerparametersweretunedsuchthatthecontrolobjectiveis achievedusingactuatorcommandsthatarewithinpracticalsaturationandratelimits. UsingthedynamicequationsofmotionintermsoftheVSCMGtestbedin3-3 2andtheVSCMGtestbedinertiamatrix J vscmg 2 R 3 3 isdenedusingtheparallel axistheoremas J vscmg J 0 + 4 X i =1 B J gi + m cmgi )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(r T i r i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r i r T i : In3,theVSCMGparameters J 0 J vscmg = diag f 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ; 6 : 10 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 ; 7 : 64 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 g kg m 2 m cmgi =0 : 265 kg r i 2 R 3 8 i =1 ; 2 ; 3 ; 4 aredenedas r 1 0 : 159100 : 1000 T m r 2 00 : 15910 : 1000 T m r 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 159100 : 1000 T m r 4 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 15910 : 1000 T m; B J gi 2 R 3 3 8 i =1 ; 2 ; 3 ; 4 denotestheinertiamatrixofthe i th gimbalasexpressedin theCMGbody-xedframedenedas B J gi [ C Bgi ] gi J gi [ C Bgi ] T ; 50

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and I n denotesthe n n identitymatrix.Theactualvaluesfortheparameters J 0 I ws I cg F dg F dw F sg F sw m cmgi ,and gi J gi 8 i =1 ; 2 ; 3 ; 4 areusedtogeneratetheplantmodel inthesimulation,buttheyarenotusedinthecontrollaw.Theadaptivecontrollaw compensatesfortheseuncertainparameters.In3,thecoordinatetransformation matrix C Bgi 2 SO 8 i =1 ; 2 ; 3 ; 4 relatesthe i th gimbal-xedframetotheVSCMG body-xedframe,and gi J gi = diag 2 : 80 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 4 : 89 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 2 : 49 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 kg m 2 8 i =1 ; 2 ; 3 ; 4 representstheinertiamatrixofthe i th gimbalasexpressedinthe i th gimbalxedframe.Alsoin3, I CMG G =2 : 80 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 I 4 kg m 2 ,and I W =6 : 95 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 I 4 kg m 2 TheskewangleoftheVSCMGpyramidis =54 : 74deg Theobjectiveistoregulateasatellite'sattitudetothedesiredquaterniondenedby q d = 1000 T withtheinitialquaternionorientationofthesatellitegivenby q = 0 : 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 30 : 80 : 4 T : Theadaptiveestimateswereinitializedtoarbitraryvaluesinthesimulationtotestacase whenlimitedknowledgeoftheparametersisavailable.Theinitialvaluesfortheadaptive 51

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estimateswereselectedasfollows. ^ 1 = 6 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 ; 6 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 ; 9 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ; 7 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 1 : 2 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 6 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 3 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 ; 3 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 ; 3 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 ; 3 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 ; 0 : 3 ; 0 : 3 ; 0 : 3 ; 0 : 3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ^ 2 = 1 : 2 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 6 : 0 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 1 : 5 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ; 1 : 5 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ; 1 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 ; 1 : 5 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 1 : 04 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 1 : 04 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 3 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 3 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 3 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 3 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ^ 3 = 2 : 78 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 2 : 78 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; 2 : 78 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 ; 2 : 78 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 ; andthegimbalangles rad andywheelspeeds rad=s wereinitializedas = 1111 T = 10101010 T ; respectively. Thefrictionmatrices F dg F dw F sg ,and F sw forthesimulatedVSCMGaree.g., see[66] F dg =2 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 I 4 F sg =4 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 I 4 F dw =2 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 I 4 F sw =4 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 I 4 : 52

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Totestthescenariowhenasuddenincreaseinthefrictionoccurs,aninstantaneousjump i.e.,stepfunctionof 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 inthe F dg F dw F sg ,and F sw parametersisprogrammedto occur 1 secondintothesimulation.Additionally,asuddenincreaseof 10% inthefriction parametersisprogrammedtooccuraftertherstperiodofpowertrackingat 6 ; 000sec 1 Toimprovetherealismofthesimulationenvironment,Gaussiandistributedrandom numbernoiseof 10% wasaddedtoallsensormeasurementsinthesimulation,andaxed timestepof 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 sec wasused. Figures3-1-3-16showthesimulationresultsoftheclosed-loopsystem.Figure3-1 showsthequaterniontrackingerrorduringclosed-loopoperationfortheentireduration 15 ; 000sec ofthenumericalsimulation.Toshowtheinitialtransientresponse,Figure 3-2showsaplotoftherst 200sec ofthequaterniontrackingerrorplot.Figure3-3 highlightsthequaterniontrackingerrorresponseduringthesuddenfrictionincreaseat 6 ; 000sec .Figures3-4and3-6showthepowerandenergytrackingachievedduring closed-loopcontrolleroperation.Figure3-4showsthepowerandenergyprolesand thecorrespondingclosed-loopsystemresponsefortheentiredurationofthesimulation, andFigure3-5showstherst 10sec tohighlighttheinitialtransientresponse.Figure 3-6highlightsthepowerandenergytrackingerrorresponsesduringthestepincreasein frictionoccurringat 6000sec .Figures3-7-3-9showthecontrolinputgimbalrates t andwheelaccelerations t duringclosedloopoperation.Thespikesshownin thecontrolinputresponses t and t occuratinstantswherethedesiredpower proleinstantaneouslychangessignfromnegativetopositiveseeFigure3-4.Although thespikesappeartoexceedpracticalratelimitationsoftheactuators,thisisdueto theresolutionofFigure3-7.Theeectofthesuddenfrictionincreaseonthecontrol 1 Inarealisticsituation,thegimbalfrictionwouldmostlikelyincreasegraduallyover timee.g.,duetobearingdegradation,corrosion,etc.,sotheinstantaneousincreasein frictioninthesimulationtestsaworstcasescenario. 53

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Figure3-1.Quaterniontrackingerror e t duringclosed-loopoperation. Figure3-2.Transientresponseofthequaterniontrackingerror e t 54

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Figure3-3.Responseofthequaterniontrackingerror e t duringthesuddenincreasein frictionat 6 ; 000sec Figure3-4.Desiredpowerandenergyprolesandactualclosed-looppowertracking response. 55

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Figure3-5.Transientpowerandenergytrackingerror. Figure3-6.Powerandenergytrackingerrorresponseduringsuddenfrictionincreaseat 6 ; 000sec 56

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Figure3-7.Controlinputgimbalrates t andwheelaccelerations t inputresponsesisshowninFigure3-9.Thecommandedcontrolinputsremainwithin reasonableratesthroughoutcontrolleroperation.Thewheelspeedsusedduring closed-loopoperationareshowninFigure3-10.Themaximumwheelspeedcommanded wasapproximately 9 ; 500 rpm ,whichisareasonablevalueforpracticalimplementation. ThetimevariationoftheadaptiveparameterestimatesisshowninFigures3-11-3-16. Someoftheparameterestimatesdidnotvarymuchduringclosed-loopoperation,but theestimatesremainedboundedforthedurationofthenumericalsimulation.Figure 3-11showstheelementsoftheadaptiveparametervectors ^ 1 t ^ 2 t ,and ^ 3 t for theentiredurationofclosed-loopcontrolleroperation.Figures3-12,3-14,and3-16show thetransientresponseoftheadaptiveparameterestimatesduringtherst 10 secondsof closed-loopoperation.Figure3-12depictsthevectorelementsof ^ 1 t ,Figure3-14depicts theelementsof ^ 2 t ,andFigure3-16depicts ^ 3 t .Figures3-13and3-15highlightthe responsesof ^ 1 t and ^ 2 t duringthesuddenincreaseinfrictionat 6 ; 000sec .Since dierentvectorelementsof ^ 1 t and ^ 2 t haddierentinitialconditionsandamountsof variation,theplotsinFigures3-12-3-15weredividedintomultiplewindowsforclarity. 57

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Figure3-8.Transientresponseofthecontrolinputgimbalrates t andwheel accelerations t Figure3-9.Responseofthecontrolinputgimbalrates t andwheelaccelerations t duringsuddenincreaseoffrictionparametersat 6 ; 000sec 58

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Figure3-10.Wheelspeeds t duringclosed-loopcontrolleroperation. Someoftheparameterestimatesvarybysmallmagnitudesontheorderof 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 .However, afterutilizingequations3,3,3,3,3-3,and3,itcanbe shownthattheequivalentcontroltorqueresultingfromthefeedforwardterms ^ 1 t ^ 2 t and ^ 3 t allhavesimilarmagnitudes,whichisofthesameorderofmagnitudeasthatof thefeedbackcontrolterms. 3.5Summary Inthischapter,anIPACSdesignmethodforaVSCMG-actuatedsatelliteispresented.InthepresenceofuncertaindynamicandstaticfrictionintheVSCMGgimbals andwheels,thecontrolleriscapableofachievingGUUBattitudetrackingwhilesimultaneouslytrackingadesiredpowerproleasymptotically.Inaddition,thecontroller compensatesfortheeectsofuncertain,time-varyingsatelliteinertiaproperties.The dicultiesarisingfromdynamicfrictionanduncertainsatelliteinertiaaremitigated throughinnovativedevelopmentoftheerrorsystemalongwithaLyapunov-basedadaptive law.Inthepresenceofstaticfriction,thecontroldesigniscomplicatedduetothecontrol inputbeingembeddedinadiscontinuousnonlinearity.Thisdicultyisovercomewiththe 59

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Figure3-11.Adaptiveparameterestimates ^ 1 t ^ 2 t ,and ^ 3 t duringclosed-loop operation. Figure3-12.Initialtransientresponseofthevectorelementsoftheadaptiveestimate ^ 1 t 60

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Figure3-13.Responseofthevectorelementsoftheadaptiveestimate ^ 1 t duringthe suddenfrictionincreaseat 6 ; 000sec Figure3-14.Initialtransientresponseofthevectorelementsoftheadaptiveestimate ^ 2 t 61

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Figure3-15.Responseofthevectorelementsoftheadaptiveestimate ^ 2 t duringthe suddenfrictionincreaseat 6 ; 000sec Figure3-16.Initialtransientresponseoftheadaptiveestimate ^ 3 t 62

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useofarobustcontrolelement.Theattitudeandpowertrackingresultsareprovenvia Lyapunovstabilityanalysisanddemonstratedthroughnumericalsimulations. 63

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CHAPTER4 INTEGRATEDPOWERREDUCTIONANDADAPTIVEATTITUDECONTROL SYSTEMOFASATELLITE Inthischapter,anattitudecontrollerisdevelopedtoadaptivelyestimateactuator frictionwhereattitudecontroltorquesaregeneratedbymeansofapyramidalarrangementoffoursinglegimbalVSCMGs.Todevelopthecontrollerfromcoupleddynamics composedofthesatellite,gimbals,andwheels,abacksteppingmethodisexploited.Inputpowerreductionresultsfromawheeldecelerationmodeinducedbynullsolutionin additiontointernalfrictionoftheywheels.Internalfrictionreducesthepotentialinput torquetobeproducedbytheywheelssincethefrictionindecelerationcanplayarole asadditionaltorquegenerationtotheresultanttorquewhichconsistsoftheappliedcontroltorqueandfriction.ThedevelopedVSCMGsteeringlawisafunctionofthegimbal ratesandtheywheelvelocitieswhichareweightedbythesingularitymeasure[97]to activelyexploittheadditionaldegreeoffreedomaordedbytheVSCMGs[55,89,100].In astationarystateaftercompletinganarbitraryattitudemaneuver,thewheelspeedscan potentiallybereduced.Abenetofthecontrolleristhatpowerreductioncanbeachieved whenthewheelsdeceleratewhilethegimbalsnullmotionsimultaneouslyperformsthe gimbalrecongurationforsingularityavoidance.Thesingularityavoidancemethodreduces/eliminatestheamountoftimethattheVSCMGhastooperateinRWmodewhen theCMGJacobianbecomessingularinimpassable/passablesingularsurfacethroughthe useofnullmotion[55,64,96,100].Numericalsimulationsdemonstratetheperformance oftheadaptiveVSCMGsteeringlawtoindicatetheinputpowerreductionforcontrolof VSCMGsinthepresenceoffrictionandtheecacyofsingularityavoidancewithreduced reactionwheelmodes. 4.1CoupledDynamics 4.1.1DynamicsforVSCMGs ThegimbalandywheelofaVSCMGassemblyFigure2-1haveacenterofmass CM, C G i and C W i ,respectively.ByplacingtheCMGassemblyinsideasatellite,the 64

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angularmomentum h CMG i C CMG i !; i ; i 2 R 3 ofthe i th CMGwithrespecttoitsCM, C CMG i iswrittenas h CMG i C CMG i = I CMG i C CMG i + h CMG i =S C CMG i ; wherethesub-andsuperscriptnotationindicateapointofinterestandabody,respectively, t 2 R 3 denotestheangularvelocityofthesatellite, i t ; i t ; i t 2 R arethegimbalangle,thegimbalrate,andtheywheelvelocityofthe i th CMG,and I CMG i C CMG i 2 R 3 3 isthemomentofinertiatensorofthe i th CMGrelativeto C CMG i .Specically,themomentofinertiatensorofthe i th CMGis I CMG i C CMG i = I G i C CMG + m G i h r T C G i r C G i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C G i r T C G i i + I W i C CMG + m W i h r T C W i r C W i I 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r C W i r T C W i i ; where I G i C CMG I W i C CMG 2 R 3 3 arethemomentsofinertiatensorofeachgimbalandwheel relativeto C CMG i ,and m G i m W i 2 R aremassesofthe i th gimbalandwheel,where r C G i r C W i 2 R 3 arethepositionsoftheCMfor C G i and C W i withrespectto C CMG i .Alsoin 4, h CMG i =S C CMG i i ; i representstheangularmomentumcontributionsfromtheywheel andthegimbalandisgivenby h CMG i =S C CMG i = I CMG i C CMG i i ^ a G i + I CMG i C CMG i i ^ a W i ; wheretheangularmomentumoftheCMGisexpressedintermsofaCMG-xedbasis B = ^ a G i ; ^ a W i ; ^ a T i ; ; ^ a G i isagimbalaxis, ^ a W i isaspinningwheelaxis,and ^ a T i isa transverseaxis.Toobtainthekineticequationgoverningthemotionofthe i th CMG,the timederivativeof4yields h CMG i C CMG i = I CMG i C CMG i + d dt I CMG i C CMG i + h CMG i =S C CMG i = g CMG i ; where g CMG i t = CMG i t )]TJ/F23 11.9552 Tf 12.098 0 Td [(g CMG i Friction i ; i 2 R 3 representstheappliedgimbaland/or ywheelmotortorques CMG i t andtheassociatedfrictiontorques g CMG i Friction i ; i 65

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GivenaCMG-xedframe F CMG i ,thetransporttheoremindicatesthat h CMG i C CMG i = I CMG i C CMG i + ^ a G i + i ^ a W i + + ^ a G i h G i + + ^ a G i + i ^ a W i h W i ; where h G i I CMG i C CMG i + ^ a G i ,and h W i I CMG i C CMG i i ^ a W i 2 R 3 .Consideringacoodinate systemwithorigin C CMG i andbasis B xedineachCMGunit,4isexpressedas CMG i h CMG i C CMG i = CMG i I CMG i C CMG i D T CMG i + ^ a G i + i ^ a W i + D T CMG i + ^ a G i h G i + D T CMG i + ^ a G i + i ^ a W i h W i ; where CMG i I CMG i C CMG i 2 R 3 3 isthemomentofinertiamatrixofthe i th CMGexpressedin theCMG-xedframe F CMG i ,andassumingthat CMG i I CMG i C CMG i isaprincipalinertiamatrix, theCMGinertiamatrixcanbedenedas CMG i I CMG i C CMG i = diag I CMG i G i I W i I CMG i T i since CMG i I CMG i C CMG i ^ a W i = I W i ^ a W i sothat h W i I W i i ^ a W i ,and D T CMG i i 2 R 3 3 isthe directioncosinematrixwhichtransforms F S to F CMG i .Thegimbalmotortorqueand bearingfrictioncontributionsareobtainedfrom CMG i h CMG i C CMG i ^ a G i =^ a G i h CMG i I CMG i C CMG i D T CMG i + ^ a G i i +^ a G i D T CMG i + ^ a G i h W i = CMG i Gimbal motor )]TJ/F23 11.9552 Tf 11.955 0 Td [(g CMG i Gimbal Friction ; where CMG i Gimbal motor t g CMG i Gimbal Friction i 2 R aretheappliedtorqueandfrictiontorqueof the i th gimbal.Similarly,theywheelcontributionscanbeexpressedas CMG i h CMG i C CMG i ^ a W i =^ a W i h CMG i I CMG i C CMG i D T CMG i + ^ a G i i +^ a W i + ^ a G i h G i +^ a W i CMG i I CMG i C CMG i i ^ a W i = CMG i Wheel motor )]TJ/F23 11.9552 Tf 11.955 0 Td [(g CMG i Wheel Friction ; where CMG i Wheel motor t g CMG i Wheel Friction i 2 R aretheappliedtorqueandfrictiontorqueof the i th wheel.Theequationsofmotionofthegimbalandywheelinmatrixformcanbe 66

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expressedas = I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 h A +[ I W ] d [] d [ ]+ CMG Gimbal motor )]TJ/F23 11.9552 Tf 11.956 0 Td [(g CMG Gimbal Friction i = [ I W ] d )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 B + CMG Wheel motor )]TJ/F23 11.9552 Tf 11.955 0 Td [(g CMG Wheel Friction ; where A !; !; B !; !;; 2 R 4 areresidualtermsobtainedbyalgebraicmanipulationsof4and4,and [ i ] 2 R 4 isavectorresultingfrom ^ a G i D T CMG i i t ^ a W i ;theinertiamatrices I CMG G d [ I W ] d 2 R 4 4 are I CMG G d diag \002 I CMG 1 G I CMG 2 G I CMG 3 G I CMG 4 G [ I W ] d diag I W 1 I W 2 I W 3 I W 4 where I CMG G d istheunknowngimbalinertiamatrix,and [ I W ] d istheknownywheelinertiamatrix;thesatelliteangularvelocity t projectedto B isdenoted as [ t ] d diag 1 t 2 t 3 t 4 t : G W T 2 R 4 4 and [ ] d denotesadiagonalmatrixexpressedbycomponentsofeachVSCMG;and t ; t 2 R 4 aregimbalrateandywheelvelocityvectors,respectively.In4, CMG Gimbal motor t CMG Wheel motor t 2 R 4 aretheappliedcontroltorques,andthefrictionvectors g CMG Gimbal Friction g CMG Wheel Friction 2 R 4 are g CMG Gimbal Friction F dg + F sg sgn g CMG Wheel Friction F dw + F sw sgn ; where F dg ;F sg 2 R 4 4 and F dw ;F sw 2 R 4 4 arediagonalmatricescontainingtheuncertain dynamicandstaticfrictioncoecientsforthegimbalsandwheels,respectively,and sgn 2 R 4 denotesavectorformofthestandardsignumfunctionwherethe sgn is appliedtoeachelementof t and t 4.1.2DynamicsforasatelliteactuatedbyVSCMGs Thetotalangularmomentum H t !; ; 2 R 3 ofarigidVSCMG-actuatedsatellite canbewrittenas H t = J! + C G I CMG G d + C W [ I W ] d 67

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wheretheangularmomentumoftheCMGisexpressedintermsofaCMG-xed basis B = ^ a G i ; ^ a W i ; ^ a T i ; ;and C G C W C T 2 R 3 4 aredenedas C G ^ a G 1 ^ a G 2 ^ a G 3 ^ a G 4 C W ^ a W 1 ^ a W 2 ^ a W 3 ^ a W 4 C T ^ a T 1 ^ a T 2 ^ a T 3 ^ a T 4 .In4,theuncertaintotalsatelliteinertiamatrix J 2 R 3 3 ispositive deniteandsymmetricsuchthat 1 2 min f J gk k 2 T J 1 2 max f J gk k 2 8 2 R n where min f J g ; max f J g2 R aretheminimumandmaximumprincipalinertiasof J and t 2 R 4 isthegimbalangle.TheequationofmotionforarigidVSCMG-actuated satellitecanbewrittenas g E C S = J! + J + J! + C G I CMG G d )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d [ T ] d + C W [ I W ] d + I CMG G d [ T ] d + C T [ I W ] d [] d +[ I W ] d [ G ] d )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( I CMG G d [ W ] d ; where g E C S !; !;; ; ; ; 2 R 3 istheexternaltorqueappliedtothesatellite, t ; t 2 R 4 arethegimbalaccelerationandywheelacceleration,and [ t ] d and h t i d denotediagonalmatricescomposedofthevectorelementsof t t 2 R 4 respectively.Basedon4and4,thecoupleddynamicequationsforaVSCMGactuatedsatellitecanberepresentedas = I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 h A +[ I W ] d [] d [ ]+ CMG Gimbal motor )]TJ/F23 11.9552 Tf 11.956 0 Td [(g CMG Gimbal Friction i = [ I W ] d )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 B + CMG Wheel motor )]TJ/F23 11.9552 Tf 11.955 0 Td [(g CMG Wheel Friction = )]TJ/F23 11.9552 Tf 9.299 0 Td [(J )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 d dt J + J! + C G I CMG G d )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d [ T ] d + C W [ I W ] d + I CMG G d [ T ] d + C T [ I W ] d [] d +[ I W ] d [ G ] d )]TJ/F28 11.9552 Tf 11.956 9.684 Td [( I CMG G d [ W ] d i : 68

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4.2AttitudeControlObjective Theattitudecontrolobjectiveistodevelopagimbalrateandywheelvelocity controllawtoenabletheattitudeof F S totracktheattitudeof F S d .Theobjectiveis quantiedin3.2.1inChapter3. 4.3AttitudeControlDevelopment TofacilitatethedesignofacompositeVSCMGsteeringlaw,anauxiliarysignal r t 2 R 3 isdenedas[23] r )]TJ/F15 11.9552 Tf 14.509 3.022 Td [(~ R! d + e v ; where 2 R isaconstantcontrolgain.Using3and4,theangularvelocity trackingerrorcanbeexpressedas ~ = r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : Intheabsenceofexternaldisturbances,thesatellitedynamicsof4canbeusedto rewritetheopen-looperrordynamicsfor r t as J r = G + D + H )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr; wheretheauxiliaryterms G e 0 ;e v ;!;! d ; d ; 2 R 3 D e 0 ;e v ;!;! d ;r;; 2 R 3 4 ,and H !; 2 R 3 4 aredenedas G )]TJ/F23 11.9552 Tf 9.299 0 Td [(! J! + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ !; D )]TJ/F23 11.9552 Tf 9.299 0 Td [(C W I CMG G d [ T ] d + C T I CMG G d [ W ] d )]TJ/F23 11.9552 Tf 13.15 8.088 Td [(@J @ 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 9.299 0 Td [(C T [ I W ] d [] d ; H C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d ; 69

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andtheterms C G I CMG G d t and C W [ I W ] d t areassumedtobenegligible [26,83,89].Aftersomealgebraicmanipulation,theerrordynamicsin4are J r = G + D d + H d + D ~ + H ~ )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr = G + Q d + D ~ + H ~ )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr; where Q e 0 ;e v ;r;!;; D e 0 ;e v ;r;!;; H !; 2 R 3 8 d t T d T d T 2 R 8 ,andthebacksteppingerrors ~ t ~ t 2 R 4 aredenedas ~ )]TJ/F15 11.9552 Tf 13.796 3.155 Td [(_ d ; ~ )]TJ/F15 11.9552 Tf 11.955 0 Td [( d : Basedontheerrordynamicsof4,theauxiliarybacksteppinginput Q e 0 ;e v ;r;!;; d t isdesignedas Q d = )]TJ/F23 11.9552 Tf 9.298 0 Td [(G )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 1 r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v ; where k 1 2 R isapositiveconstantgain.Substituting4into4,theclosed-loop errordynamicsfor r t canbedeterminedas J r = D ~ + H ~ )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 1 r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : Inadditiontoattitudecontrol,thecontroldevelopmentalsotargetsreducedpower consumptionduringywheeldespinning.Theywheelcanachievedecelerationby nullmotioncontrol.Tofacilitatethedevelopmentofanullmotioncontroller,thenew compositeVSCMGsteeringlawin4isexpressedas d = )]TJ/F23 11.9552 Tf 9.299 0 Td [(Q + w G + k 1 r + e v )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q S = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(R 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(N 1 s g k @ @ )]TJ/F23 11.9552 Tf 9.299 0 Td [(R 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(N 2 s w g 3 7 5 ; where Q + w e 0 ;e v ;r;!;; W Q T Q W Q T )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,and W 2 R 8 8 denotes amodeweightmatrixthatdetermineswhethertheVSCMGssystemoperatesinaCMG 70

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modeoraRWmode,and S 2 R 8 8 isanullmotionweight[45,46]toarbitratethenull motion t 2 R 8 1 ofthegimbalandywheeldenedas 2 6 4 k @ @ g 3 7 5 ; wherethenullmotion t generatesgimbalrecongurationandwheeldeceleration, k 2 R denotesapositiveconstant,and g t isanauxiliarycontrolsignaltotrackthedesired ywheelspeedtrajectory.Inthepartitionedsteeringlawof4, R i e 0 ;e v ;r;!;; = [ Q + w f G e 0 ;e v ;r;!; + k 1 r t + e v t g ] i indicatescomponentsofeach R 4 term, N i e 0 ;e v ;r;!;; 2 R 4 8 istheprojectoronto N Q ,and s g s w 2 R 8 4 are composedofcomponents s g s w 2 R 4 4 ofnullmotionweight S ,denedas s g s g 0 4 4 T s w 0 4 4 s w T : Themodeweightmatrix W 2 R 8 8 isdesignedas[87,89,101] W 2 6 4 W I 4 4 0 4 4 0 4 4 W I 4 4 3 7 5 ; where W isdenedas W W exp 1 f ; where 1 ;W ;W 2 R arepositiveconstants,andthesingularitymeasuringobjective function f is f = )]TJ/F15 11.9552 Tf 11.291 0 Td [(det )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(C T C T T : Thecontrolinput g t of4isdesignedas g = )]TJ/F15 11.9552 Tf 11.291 0 Td [([ N 2 s w ] )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 R 2 + f ; where f t 2 R 4 isadesired,givenywheeldecelerationtrajectory.Inastationary stateaftercompletinganarbitraryattitudemaneuver,thewheelsdonotneedtomaintain 71

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highspeeds.Deceleratingthewheelscanreducethepotentialforwheelspeedsaturation, buttypicalstrategiesrequirebatterypowertodeceleratethewheelswhilemaintaining adesiredattitude.Theauxiliarycontrolinput g t resultsfromaninternalmomentum managementstrategythatenablesinternalfrictiontobeusedalongwithnullspace motionasapowerreductionmethodtodeceleratetheywheelswhilemaintaininga desiredattitude. Remark4.1. Thebracketedtermin4isinvertible,andsimultaneousattitudeand wheeldecelerationispossiblewhen N 2 s w = I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q 4 8 s w 6 =0 4 4 .Since I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q 4 8 isnonzeroforall Q N 2 6 =0 4 8 andsimultaneousattitudeand wheeldecelerationcanbeachievedprovided s w 6 =0 and s w = 2N N 2 ; where N N 2 denotesthenullspaceofthematrix N 2 .Hence, s w = 2N N 2 except when s w =0 8 4 ,whichmeanstheCMG'sJacobianitselfissingular,whichdoesnot occurunlessitisinitiallysingular.Accordingly,aslongasthesystemdoesnotstartina singularconguration,thecontrolinput g t existsandsimultaneousattitudeandwheel decelerationcanbeachieved. Thecompositesteeringlawin4hasanextradegreeoffreedomresultingfrom thevariablespeedywheel,enablingtheVSCMGsystemtoescapeanellipticsingularity andagimballocksingularity[89,100].Sincethesteeringlawof4isproducedbyfour VSCMGunits,4isalsoasolutiontoanunderdeterminedsystemcontainingfour gimbalratesandfourywheelvelocitiesasunknowns[55,89,100].In4,theterm I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q S t generatestheVSCMGnullmotionforsingularityavoidance andwheeldeceleration.Thesingularityavoidancemethodreduces/eliminatestheusageof RWmodeincontinuousmannerwhenencounteringelliptic/hyperbolicsingularity.Since Q isnonsquare,thepseudo-inverse Q + w 2 R 8 3 isdenedsothat Q Q + w = I 3 andthematrix I 8 )]TJ/F23 11.9552 Tf 10.032 0 Td [(Q + w Q ,whichprojectsvectorsontothenullspaceof Q ,satises 72

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theproperties )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q )]TJ/F23 11.9552 Tf 12.952 -9.684 Td [(I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q = I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q a Q )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(I 8 )]TJ/F23 11.9552 Tf 11.955 0 Td [(Q + w Q =0 3 8 : b ThecompositeVSCMGsteeringlawin4providesadesiredgimbalandywheel velocity.Sincetheactualcontrolinputsaregimbalandywheeltorques,abackstepping approachisusedtodesign CMG Gimbal motor t and CMG Wheel motor t byexaminingtheopen-loop errorsystemforthemismatchbetweenthedesiredandactualgimbalandywheel velocities,denotedbythebacksteppingerrors ~ t ~ t .Specically,afterusing4 and4andperformingsomealgebraicmanipulation,theopen-looperrorsystemfor ~ t ~ t is ~ = I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 h A +[ I W ] d [] d [ ]+ CMG Gimbal motor )]TJ/F23 11.9552 Tf 11.956 0 Td [(g CMG Gimbal Friction i + D + D d + G + k 1 r ~ = [ I W ] d )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 B + CMG Wheel motor )]TJ/F23 11.9552 Tf 11.955 0 Td [(g CMG Wheel Friction + H + H d +_ e v ; where G )]TJ/F15 11.9552 Tf 11.531 0 Td [(_ J! )]TJ/F23 11.9552 Tf 11.955 0 Td [(! J! + J + J! + J ~ R! d + J! ~ R! d + ~ R d )]TJ/F15 11.9552 Tf 13.413 3.022 Td [(_ J ~ R d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 Je v + J e v ~ + Je v ~ + Je 0 I 3 + J e 0 I 3 ~ + Je 0 I 3 ~ ; D )]TJ/F23 11.9552 Tf 9.298 0 Td [(C T h i d I CMG G d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C W I CMG G d [_ T ] d )]TJ/F23 11.9552 Tf 9.298 0 Td [(C W h i d I CMG G d [ W ] d + C T I CMG G d [_ W ] d )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ 2 J @ 2 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 13.15 8.088 Td [(@J @ 1 2 r + ~ R! d + ~ R d )]TJ/F23 11.9552 Tf 11.955 0 Td [( e v + C W h i d [ I W ] d [] d )]TJ/F23 11.9552 Tf 11.956 0 Td [(C T [ I W ] d h i d ; H C G [ I W ] d [_ T ] d )]TJ/F23 11.9552 Tf 15.967 8.088 Td [(@ @ C T [ I W ] d [ G ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [_ G ] d : 73

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Theinternalfrictions g CMG Gimbal Friction and g CMG Wheel Friction in4areparameterizedin termsofknownregressionmatrices Y 1 ;Y 2 2 R 4 8 andvectorsofeightunknown constants 1 ; 2 2 R 8 as Y 1 1 F dg + F sg sgn Y 2 2 F dw + F sw sgn : Basedontheopen-looperrordynamicsof4,theappliedcontroltorques CMG Gimbal motor t and CMG Wheel motor t aredesignedas CMG Gimbal motor = )]TJ/F23 11.9552 Tf 9.298 0 Td [(A )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d [] d [ ]+ Y 1 ^ 1 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( I CMG G d n D + D d + G + k 1 r o )]TJ/F28 11.9552 Tf 11.291 9.684 Td [( I CMG G d D T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 2 I CMG G d ~ CMG Wheel motor = )]TJ/F23 11.9552 Tf 9.298 0 Td [(B + Y 2 ^ 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d n H + H d +_ e v o )]TJ/F15 11.9552 Tf 11.955 0 Td [([ I W ] d H T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 [ I W ] d ~ ; where k 2 ;k 3 2 R arepositiveconstantgains,and ^ 1 t ^ 2 t 2 R 8 areestimatesforthe parametricuncertainty F dg F sg F dw ,and F sw .Theparameterupdatelawsfor ^ 1 t ^ 2 t aredesignedas ^ 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(1 Y T 1 I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ~ ^ 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F22 7.9701 Tf 7.314 -1.794 Td [(2 Y T 2 [ I W ] d )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ~ ; where )]TJ/F22 7.9701 Tf 7.315 -1.794 Td [(1 )]TJ/F22 7.9701 Tf 7.314 -1.794 Td [(2 2 R 8 8 denoteconstant,positive-denite,diagonaladaptationgainmatrices. Aftersubstitutingthecontroltorquesof4into4,theclosed-looperrordynamics for ~ t ~ t are ~ = )]TJ/F23 11.9552 Tf 9.299 0 Td [(D T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 2 ~ )]TJ/F28 11.9552 Tf 11.955 13.27 Td [( I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Y 1 ~ 1 ~ = )]TJ/F23 11.9552 Tf 9.299 0 Td [(H T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 ~ )]TJ/F28 11.9552 Tf 11.955 13.27 Td [( [ I W ] d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Y 2 ~ 2 ; where ~ 1 t ~ 2 t 2 R 8 aredenedas ~ 1 1 )]TJ/F15 11.9552 Tf 12.895 3.155 Td [(^ 1 ; ~ 2 2 )]TJ/F15 11.9552 Tf 12.895 3.155 Td [(^ 2 : 74

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4.3.1StabilityAnalysis Theorem4.1. Theweightedsteeringlaw4includingthegimbalrateandtheywheel velocity,andtheappliedcontroltorques4alongwiththeadaptiveupdatelawsgiven in4ensureglobalasymptoticattitudetrackinginthesensethat k e v t k! 0 as t !1 : Proof: Toprovetheattitudetrackingresult,let D R 31 beadomaincontaining e v ;e 0 ;r; ~ ; ~ ; ~ 1 ; ~ 2 =0 31 ,where e v ;e 0 ;r; ~ ; ~ ; ~ 1 ; ~ 2 2 R 31 isdenedas e T v e 0 r T ~ T ~ T ~ 1 ~ 2 T : Let V : D [0 ; 1 R denoteacontinuouslydierentiable,positivedenitefunction denedas V = e T v e v + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 2 + 1 2 r T Jr + 1 2 ~ T ~ + 1 2 ~ T ~ + 1 2 ~ T 1 )]TJ/F26 7.9701 Tf 7.315 4.936 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 1 ~ 1 + 1 2 ~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 ~ 2 ; Afterdierentiating4andusing4,3,4,and4,theresulting expressionisgivenas V = e T v )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 e T v ~ + r T D ~ + H ~ )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 1 r )]TJ/F23 11.9552 Tf 11.956 0 Td [(e v + ~ T )]TJ/F23 11.9552 Tf 9.299 0 Td [(D T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 2 ~ )]TJ/F28 11.9552 Tf 11.955 13.27 Td [( I CMG G d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Y 1 ~ 1 + ~ T )]TJ/F23 11.9552 Tf 9.299 0 Td [(H T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 ~ )]TJ/F28 11.9552 Tf 11.955 13.27 Td [( [ I W ] d )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Y 2 ~ 2 )]TJ/F15 11.9552 Tf 10.237 3.155 Td [(~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.937 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 1 ^ 1 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 2 ^ 2 : Afterperformingsomealgebraicmanipulations,andusing4,4andexploiting thefactthat e T v e v ~ =0 ,theexpressionin4canbeupperboundedas V )]TJ/F23 11.9552 Tf 21.917 0 Td [( k z k 2 ; 75

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where = min f ;k 1 ;k 2 ;k 3 g2 R and z e v ;r; ~ ; ~ 2 R 14 isdenedas z e T v r T ~ T ~ T T : From4and4, r e v ;e 0 ;! ;e v t ; ~ t ; ~ t ; ~ 1 t ; ~ 2 t 2L 1 .Thus,from 6,4,and4, e v t ;e 0 t ; ~ t ;z e v ;r; ~ ; ~ 2L 1 ,and4canbeusedto concludethat t 2L 1 .Theopen-loopquaterniontrackingerrorin3canbeused toconcludethat e v t ; e 0 t 2L 1 .Thefactthat t -dependentfunctionsresultfrom directioncosinematricesindicatesthatthefunctionscontain t withinboundedtrigonometricfunctions.Since e v t ;e 0 t ;r e v ;e 0 ;! ;! t ; t 2L 1 ,4canbeusedto showthat d t ; d 2L 1 .Through4and ~ t ; d t ; ~ t ; d t 2L 1 ,then t ; t 2L 1 .Thefactthat r e v ;e 0 ;! ;e v t ;e 0 t ;! t ; t ; ~ t ; ~ t ; t 2L 1 canbeusedwith48toconcludethat r e v ;e 0 ;r; ; ~ ; ~ 2L 1 .Thetimederivative of4showsthat t 2L 1 with t ;e v t ; e v t ;e 0 t ;! d t ; d t 2L 1 From4, ^ 1 t ; ^ 2 t 2L 1 .In4, CMG Wheel motor t 2L 1 basedonthefact that t ; t ; t ; d t ; ~ t ;r e v ;e 0 ;! ; e v t ; ^ 2 t 2L 1 .Fromthefactthat t ; t ; t ; t ; CMG Wheel motor t 2L 1 ,4indicatesthat t 2L 1 .In4 26, CMG Gimbal motor t 2L 1 basedonthefactthat t ; t ; t ; d t ; ~ t ; t ; t ;r e v ;e 0 ;! ; r e v ;e 0 ;r; ; ~ ; ~ ;e v t ; e v t ; ^ 1 t 2L 1 .Under4andthefact that t ; t ; t ; t ; CMG Gimbal motor t 2L 1 t 2L 1 .From4and4, itissaidthat ~ t ; ~ t 2L 1 .Standardsignalchasingargumentscanthenbeutilized toprovethatallremainingsignalsareboundedduringclosed-loopoperation.Thefact that e v t ;r e v ;e 0 ;! ; e v t ; r e v ;e 0 ;r; ; ~ ; ~ ; ~ t ; ~ t ; ~ t ; ~ t 2L 1 issufcienttoshowthat e v t ;r e v ;e 0 ;! ; ~ t ,and ~ t areuniformlycontinuous.Since e v t ;r t ; ~ t ,and ~ t areuniformlycontinuous, e v t ;r e v ;e 0 ;! ; ~ t ; ~ t 2L 1 L 2 from4,Barbalat'sLemmacanbeusedtoprove e v t ;r e v ;e 0 ;! ; ~ t ; ~ t 0 as t !1 76

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4.4SimulationStudy 4.4.1SimulationSetup Numericsimulationsarepresentedtoexaminetheperformanceofthedeveloped controller.Parametersusedforthesatellitearebasedonahighdelitymodeland aregiveninTable4-1.Thedesiredangularvelocitytrajectories d t are d t = 0 : 05sin t= 300000 rad= sec andthedesireddecreasingywheelspeedtrajectoryforeachwheelisdesigned f = in )]TJ/F15 11.9552 Tf 11.956 0 Td [( fi exp )]TJ/F23 11.9552 Tf 9.298 0 Td [(ct + fi ; where c isacoecientthatcandetermineadegreeofdeceleration, in isaninitialdesired ywheelspeedandissetas 50 rad=s 500 rpm ,and fi isanaldesiredywheel speedandissetas 10 rad=s 100 rpm .TheinitialconditionsaregiveninTable4-2. PhysicalParameter Value J kg m 2 diag 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 7 : 64 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 m cmg kg 0 : 165 I G kg m 2 2 : 80 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 I 4 I W kg m 2 6 : 95 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 I 4 skewangle 54 : 74 Table4-1.PhysicalparametersfortheVSCMGsimulation. InitialParameter Value q 0 : 10 : 30 : 80 : 4 rad=s 0 ^ 1 ^ 2 0 : 02 0 rad )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 33520 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 33520 rad=s 50 Table4-2.InitialparametersfortheVSCMGsimulation. 77

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Figure4-1.Quaterniontrackingerror e t duringclosed-loopoperation. 4.4.2SimulationResults Figures4-1-4-11showthesimulationresultsoftheclosed-loopsystemtoachieve thecontrolobjectivedenotedin3duringa 200 s simulationperiod.Figure4-1shows thequaterniontrackingerrorresultsduringtheentire 200 s .Figure4-2showstheactual gimbalrates t andtheactualwheelvelocities t .Figure4-4depicts f S W whichindicatethesingularitymeasure,thenullmotionweight,andthemodeweight, respectively.Basedonthesingularitymeasure, f S and W generatespecicvaluesto properlyperformtheirdualtasksfortheVSCMG.Foranormalenvironment,themode weight W allowstheVSCMGtobeaconventionalCMGsothatitcantakeadvantage ofatorqueamplicationas W hasabiggervaluethan W asseenbyFigure4-4.Only whenencounteringasingularityaboveanacceptablelevel, W becomesbiggerthan W .AdditionalsolutionsprovidedbytheVSCMGasanunderdeterminedsystembrings aboutadditionalnullsolutionswhichenablewheeldeceleration.Hence,tousebothnull motionbenetssuchasgimbalrecongurationandwheeldeceleration,thenullmotion weight S isintroduced.Correspondingtothevariationof f s g generatespropergimbal recongurationand s w allowssteadywheeldecelerationdepictedinFigure4-5.Forthe 78

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Figure4-2.Actualcontrolinputgimbalrates t andywheelspeeds t Figure4-3.Close-upofactualcontrolinputgimbalrates t andwheelspeeds t when encounteringextremesingularity. 79

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nullmotions,theroleofnullmotionweight S showninFigure4-4becomesremarkable becausethe S caninhibitthewheeldecelerationwhentheCMGsJacobianapproaches singularitysothattheywheelcanworkinRWmode.Although S cannotdistinguish betweendierentsingularities,theweightmatrixcanadjusttheinterventionofthe ywheelwhenapproachingasingularity.Hence,thecompositeweightedsteeringlawin 4cancopewithanellipticsingularitywhilemaintainingprecisionattitudecontrol sincethesteeringlawgeneratestherequiredtorqueinRWmodetopassthroughor escapeaninternalsingularitydierentlyfromthesingularityescapemethodsforCMGs thatrequireaddedtorque[6,27,64,77,95,97].Thisobservationindicatesabenetthat resultsfromtheextracontrollabledegreeoffreedomoftheVSCMG.Specically, s w depictedinFigure4-4allowswheeldecelerationwheninnormaloperationandrestricts thewheeldecelerationwhenapproachingasingularity.Thus,inmostregionsgimbal recongurationisresponsibleforsingularityavoidancebyexploitinggimbalratenull solution.Thenullmotionweight S servesasaswitchthatactsasanalternativetocope withbothellipticandhyperbolicsingularitieswhileachievingwheeldeceleration.The extremesingularityencounterduring 80 )]TJ/F15 11.9552 Tf 11.803 0 Td [(85 s showninFigure4-4causesthewheeltorque generationtoescapethesingularityasseeninFigure4-3.Suchsingularityescapeis consideredasanadvantageresultingfromanextraDOFoftheVSCMGsbutnotpossible intheconventionalCMG.Thetimevariationoftheadaptiveparameterestimatesis showninFigures4-6and4-7whichshowstheelementsoftheadaptiveparametervectors ^ 1 t and ^ 2 t .InFigures4-8and4-10,thefrictiontorqueconsistingofthedynamic frictionandthestaticfrictiondenotedin4haveasteadyimpactontheentiresystem. TheabsolutevalueofthegimbaltorqueinFigure4-9indicatesconsistentperformance despitetheuncertainfrictioneects.ThewheeltorqueinFigure4-11inducedbythe wheeldecelerationshowstheinputpowerreductioneectinthatthewheelshavemore torquesaddedbyboththedynamicandstaticfriction.Sucheectcanyieldtotalpower reductionbyattitudecontrolsystemACS. 80

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Figure4-4.Singularitymeasurefunction f ,nullmotionswitch S ,andweight function W Figure4-5.Nullmotion:gimbalrecongurationandwheeldecelerationerror t 81

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Figure4-6.Adaptiveparameterestimates ^ 1 t duringclosed-loopoperation. Thenotation ^ 1 t x denotesthe x -elementof ^ 1 t Figure4-7.Adaptiveparameterestimates ^ 2 t duringclosed-loopoperation. Thenotation ^ 2 t x denotesthe x -elementof ^ 2 t 82

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Figure4-8.Appliedgimbaltorque,gimbalfrictiontorque,andtotalgimbaltorqueduring closed-loopcontrolleroperation. Figure4-9.Close-upofappliedgimbaltorque,gimbalfrictiontorque,andtotalgimbal torqueduringtheentireclosed-loopoperation. 83

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Figure4-10.Appliedwheeltorque,wheelfrictiontorque,andtotalwheeltorqueduring closed-loopcontrolleroperation. Figure4-11.Close-upofappliedwheeltorque,wheelfrictiontorque,andtotalwheel torqueduringclosed-loopoperationprovidinginputpowerreduction. 84

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4.5Summary Inthischapter,abacksteppingmethodisusedtodevelopthecontrollerfroma cascadedynamicsconnectionforaVSCMG-actuatedsatellite.Inthepresenceofuncertain dynamicandstaticfrictionsinboththegimbalsandtheywheels,thecontrolleris capableofachievingglobalasymptoticattitudetrackingwhilesimultaneouslyperforming singularityavoidanceandwheeldeceleration.Simulationsshowthattheappliedtorques ofthewheelscontainingfrictioncontributetopowerreductioninthatthefrictionenables thewheeltoobtainmoretorqueswithoutanadditionaltorquerequest.Suchbenet isinducedbythedecelerationmoderesultingfromthenullmotionandcangivethe actuatorbothtorqueandpowerreductioneect.Also,theappliedcontroltorquecan responsivelyrealizemorerealistictorqueconsideringfrictionlossinpracticaluseandsuch considerationcanallowthesystemtomaintainconsistentperformanceinthepresenceof dynamicuncertainty. 85

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CHAPTER5 ANEWINITIALSTART-UPMETHODUSINGINTERNALMOMENTUM MANAGEMENTOFVSCMGS Anadaptivecontrollerisdevelopedinthischapterthatuseinternalmomentum managementandsingularityavoidancewhilesimultaneouslyyieldingattitudecontrol. Thenullsolutionoftheclosed-loopywheelcontrolinputachievesinternalmomentum managementsothattheywheelstartsfromrestandreachesthedesiredwheelspeed withoutaseparatefeedbackcontrolloop.Theclosed-loopVSCMGssteeringlawyields simultaneousasymptoticattitudetracking,exponentialinternalmomentumtracking,and singularityavoidance.Thesingularityavoidancemethodreduces/eliminatestheamount oftimethattheVSCMGhastooperateinRWmodewhentheCMGJacobianbecomes singularthroughtheuseofnullmotion.Thecontrollerisdevelopedforasatellitewith anuncertain,state-dependentinertiaalongwithanuncertaininertiaintheVSCMG actuators.Numericalsimulationsillustratetheperformanceofthedevelopedadaptive controllerasaVSCMGssteeringlaw. 5.1ControlObjectives 5.1.1AttitudeControlObjective Theattitudecontrolobjectiveistodevelopaywheelaccelerationandgimbalrate controllawtoenabletheattitudeof F S totracktheattitudeof F S d .Theobjectiveis quantiedin3.2.1inChapter3. 5.1.2FlywheelAngularMomentumManagementObjective Theangularmomentum h 2 R 4 generatedbytheywheelsofthefourVSCMGs canbeexpressedas h =[ I W ] d : Theywheelangularmomentummanagementobjectiveistodevelopaninternalmomentumtrackingcontrollawresultingfromthenullsolutionoftheywheelcontrolinputso thattheactualangularmomentumtracksadesiredconstantangularmomentum h d 2 R 4 whilesimultaneouslytrackingadesiredtime-varyingattitude.Toquantifythemomentum 86

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managementobjective,anangularmomentumtrackingerror 2 R 4 isdenedas h d )]TJ/F23 11.9552 Tf 11.955 0 Td [(h; wherethedesiredangularmomentumisdenedas h d [ I W ] d d 5.2ControllerDevelopment 5.2.1AdaptiveAttitudeControlDevelopment Tofacilitatethecontroldesign,anauxiliarysignal r !;e 0 ;e v 2 R 3 isdenedas[23] r )]TJ/F15 11.9552 Tf 14.509 3.022 Td [(~ R! d + e v ; where 2 R 3 3 isaconstant,positivedenite,diagonalcontrolgainmatrix.After substituting5into3,theangularvelocitytrackingerrorcanbeexpressedas ~ = r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : Takingthetimederivativeof5andmultiplyingbothsidesoftheresultingexpression by J yields J r = J + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(e v + e 0 I 3 ~ !; wherethefactthat ~ R = )]TJ/F23 11.9552 Tf 9.299 0 Td [(! ~ R wasutilized.Aftersubstituting2into5,theopen-looperrordynamicsfor r !;e 0 ;e v canbewrittenas J r = )]TJ/F23 11.9552 Tf 9.298 0 Td [(Q + Y 1 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d ; 87

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underthestandardassumptionthatthegimbalaccelerationterm C G I CMG G d t is negligible[26,83,89].In5, Q e 0 ;e v ;r;!;; 2 R 3 8 isdenedas Q C W [ I W ] d C W I CMG G d [ T ] d + C T [ I W ] d [] d )]TJ/F28 11.9552 Tf 11.291 9.683 Td [( I CMG G d [ W ] d + @J @ 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v ; = T T T 2 R 8 1 isacompositecontrolinputconsistingoftheywheelaccelerationsandthegimbalrates,and Y 1 e 0 ;e v ;r;!; 1 representslinearlyparametrizable uncertaintyintermsofameasurableregressionmatrix Y 1 e 0 ;e v ;r;!; 2 R 3 p 1 anda vectorof p 1 unknownconstants 1 2 R p 1 denedas Y 1 1 )]TJ/F23 11.9552 Tf 9.298 0 Td [(! J! + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(e v + e 0 I 3 ~ !: Tocompensateforthelinearlyparametrizableuncertaintypresentin Q e 0 ;e v ;r;!;; anotherregressionmatrixdenotedby Y 2 e 0 ;e v ;r;!;; ; ; 2 R 3 p 2 andavectorof p 2 unknownconstants 2 2 R p 2 aredenedas Y 2 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(Q : Toaddressthefactthatthecontrolinput t ispremultipliedbythenonsquare,statedependentuncertainmatrix Q e 0 ;e v ;r;!;; ,anestimateoftheuncertaintyin5, ^ Q t 2 R 3 8 isdenedas Y 2 ^ 2 )]TJ/F15 11.9552 Tf 11.983 3.022 Td [(^ Q ; where ^ 2 t 2 R p 2 isasubsequentlydesignedestimatefortheparametricuncertaintyin Q e 0 ;e v ;r;!;; .Basedon5and5,theexpressionin5canbewrittenas J r = Y 2 ~ 2 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + Y 1 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d ; wherethenotation ~ 2 t 2 R p 2 isdenedas ~ 2 2 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 2 : 88

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Basedontheopen-looperrordynamicsof5,thecompositeweightedVSCMG steeringlawisdesignedas = ^ Q + w Y 1 ^ 1 + kr + e v + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d + I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q S; where k 2 R denotesapositivecontrolgain, ^ Q + w ;t = W ^ Q T t ^ Q t W ^ Q T t )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 and W 2 R 8 8 denotesaweightmatrixthatdeterminesiftheVSCMGssystemusesa CMGmodeoraRWmode.Specically, W isdesignedas[87,89,101] W 2 6 4 W I 4 4 0 4 4 0 4 4 W I 4 4 3 7 5 ; where W 2 R isdenedas W W exp 1 f ; 1 ;W ;W 2 R arepositiveconstants,andtheobjectivefunction f measuringthe singularityisdenedas f )]TJ/F15 11.9552 Tf 11.291 0 Td [(det )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(C T C T T : Thecontrolinputin5hasanextradegreeoffreedomresultingfromthevariable speedywheel,andthusenablestheVSCMGsystemtoescapeagimballocksingularity [89].Sincethecontrolinputin5isproducedbyVSCMGunits,5isalsoa solutiontoanunderdeterminedsystemcontainingywheelaccelerationsandgimbalrates asunknowns[55,89,100].Theterm I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ;t ^ Q t S t in5generates theVSCMGnullmotionforinternalmomentummanagementandsingularityavoidance. Sincethematrices ^ Q + w ;t and ^ Q t arenonsquare,thepseudo-inverse ^ Q + w ;t 2 R 8 3 isdenedsothat ^ Q t ^ Q + w ;t = I 3 ,andthematrix I 8 )]TJ/F15 11.9552 Tf 15.086 3.022 Td [(^ Q + w ;t ^ Q t ,whichprojects 89

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vectorsontothenullspaceof ^ Q t ,satisestheproperties I 8 )]TJ/F15 11.9552 Tf 14.639 3.022 Td [(^ Q + w ^ Q I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q = I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q ^ Q I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q =0 : Togeneratenullmotionformomentumtrackingandgimbalreconguration,thenull motion t 2 R 8 isdenedas k w g T k @ @ T T ; where k w k 2 R denotepositiveconstants, g t 2 R 4 isasubsequentlydesignedauxiliary controlsignaltotrackthedesiredywheelangularmomentum,andthesecondrowallows thenullmotiontoperformthegimbalrecongurationcorrespondingtoavariationofthe singularitymeasureindex ,whichisdenedas[97] = 0 exp 2 f ; wheretheobjectivefunction f isdenedin5,and 2 ; 0 2 R denotepositive constants.Thematrix S 2 R 8 8 in5isusedasanullmotionweightforthe VSCMGnullmotion,whichcanweighapropernullmotionbasedonthesingularity measure.Specically, S isdesignedas S diag [ s w ;s g ]= 2 6 4 sech 1 k d det C T C T T + 0 4 4 0 4 4 sech )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(k g det )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(C T C T T 3 7 5 ; where k d ;k g ;" 2 R .In59, s w and s g selectivelyarbitratebetweeninternal momentumtrackingandgimbalreconguration,correspondingtohowapproximateor fartheCMGcongurationistoasingularity.Gimbalrecongurationresultsfromnull motionwhentheCMGJacobianbecomessingulari.e.,ahyperbolicsingularity,allowing simultaneousattitudeandinternalmomentumtrackingwiththebenetsoftorque amplicationinCMGmode.WhentheCMGexperiencesadegeneratesingularityi.e., 90

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eventhoughthenullmotionsexist,thesingularitycannotbeescapedbythenullmotions orwhenagimballocksingularityi.e.,anellipticsingularityoccurs,then s w 0 whichinhibitsmomentumtrackingsothattheVSCMGwilloperateinRWmode. Thecontrolinput t in5canbepartitionedas = 2 6 4 3 7 5 = 2 6 4 R 1 + N 1 s w k w g R 2 + N 2 s g k @ @ 3 7 5 ; where R i = h ^ Q + w Y 1 ^ 1 + kr + e v + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d i i indicates componentsofeach R 4 controlinput, N 1 t N 2 t 2 R 4 8 aredenedas N 2 6 4 N 1 N 2 3 7 5 = 2 6 4 I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q 3 7 5 and s w s g 2 R 8 4 aredenedas s w s w 0 4 4 T s g 0 4 4 s g T : Aftersubstituting5into5andusing5,closed-loopdynamicsfor r !;e 0 ;e v aregivenby J r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Jr + Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(kr )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v ; wherethenotation ~ 1 t 2 R p 1 isdenedas ~ 1 = 1 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 1 : Basedon5andthesubsequentstabilityanalysis,theparameterestimates ^ 1 t and ^ 2 t aredesignedas ^ 1 = proj )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(1 Y T 1 r ^ 2 = proj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 Y T 2 r ; 91

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where )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(1 2 R p 1 p 1 and )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 2 R p 2 p 2 areconstant,positive-denite,diagonaladaptation gainmatrices,and proj denotesaprojectionalgorithmutilizedtoguaranteethatthe i th elementof ^ 1 t and ^ 2 t canbeboundedas 1 i ^ 1 i 1 i 2 i ^ 2 i 2 i ; where 1 i 1 i 2 R and 2 i 2 i 2 R areknown,constantlowerandupperboundsforeach elementof ^ 1 t and ^ 2 t ,respectively. 5.2.2MomentumTrackingControlDevelopment Theopen-loopdynamicsforthemomentumtrackingerrorcanbeobtainedbytaking thetimederivativeof5as = )]TJ/F23 11.9552 Tf 9.298 0 Td [(I W : Multiplying5bytheknownpositive-denitesymmetricmatrix I )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 W ,andsubstituting 5intotheresultingexpansionfor t yields I )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 W = )]TJ/F23 11.9552 Tf 9.299 0 Td [(R 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(N 1 s w k w g: Basedonthestructureof5,thenull-spacecontrol g t isdesignedtosatisfy k w N 1 s w g = )]TJ/F23 11.9552 Tf 9.298 0 Td [(R 1 + k m ; where k m 2 R isapositiveconstantcontrolgain.Theminimumnormsolutionof5is g =[ k w N 1 s w ] )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 11.9552 Tf 9.298 0 Td [(R 1 + k m : Theresultin5indicatesthatsimultaneousattitudeandmomentumtrackingis possiblewhen k w N 1 ;t s w isinvertible.Aftersubstituting5into5,the closed-looperrorsystemis I )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 W = )]TJ/F23 11.9552 Tf 9.299 0 Td [(k m : 92

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Remark5.1. Thebracketedtermin5isinvertible,andsimultaneousattitudeand momentumtrackingispossiblewhen N 1 ;t s w = I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ;t ^ Q t 4 8 s w 6 =0 Since I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ;t ^ Q t isnonzeroforall ^ Q t N 1 ;t 6 =0 andsimultaneousattitude andmomentumtrackingcanbeachievedprovided s w 6 =0 and s w = 2N N 1 ; where N N 1 ;t denotesthenullspaceofthematrix N 1 ;t .Hence, s w = 2N N 1 exceptfor s w =0 ,whichmeanstheCMG'sJacobianitselfissingular,but s w =0 doesnotoccurfortheCMGsJacobianunlessitstartsoutthatwayinitially.Accordingly, aslongasthesystemdoesnotstartinasingularconguration,theminimumnorm solutionof5existsandsimultaneousattitudeandmomentumtrackingcanbe achieved. 5.2.3StabilityAnalysis Theorem5-1: Theweightedcontrolinput5alongwiththeadaptiveupdate lawsgivenin5ensuresglobalasymptoticattitudetrackingsuchthat k e v t k! 0 as t !1 ; alongwithexponentialmomentumtrackinginthesensethat k t k exp )]TJ/F23 11.9552 Tf 9.298 0 Td [(I W k m t : Proof: Theexponentialinternalmomentumtrackingresultisevidentfrom5. Toprovetheattitudetrackingresult,let D R 8 beadomaincontaining e v ;e 0 ;r;P = 0 ,where e v ;e 0 ;r;P 2 R 8 isdenedas t e T v t e 0 t r T e v ;e 0 ;! P ~ 1 t ; ~ 2 t T 93

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andtheauxiliaryfunction P ~ 1 t ; ~ 2 t 2 R isdenedas P 1 2 ~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.937 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 1 ~ 1 + 1 2 ~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.937 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 ~ 2 ; andlet V ; : D [0 ; 1 R beacontinuouslydierentiable,positivedenitefunction denedas V e T v e v + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 2 + 1 2 r T Jr + P: Afterusing3,5,5,5,and5,thetimederivativeof V ; can beexpressedas V = e T v )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(e v + e 0 I ~ + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 e T v ~ + r T Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(kr )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F15 11.9552 Tf 10.237 3.154 Td [(~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 1 ^ 1 )]TJ/F15 11.9552 Tf 12.895 3.154 Td [(~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 2 ^ 2 : Byusing5,5,andexploitingthefactthat e T v t e v t ~ t =0 ,theexpressionin 5canbeupperboundedas V )]TJ/F23 11.9552 Tf 21.917 0 Td [( k z k 2 ; where = min f ;k g2 R ,and z e v ;r 2 R 6 isdenedas z e T v t r T e v ;e 0 ;! T : From5and5, r e v ;e 0 ;! ;e v t ; ~ 1 t ; ~ 2 t 2L 1 .Thus,from6,5, and5, e v t ;e 0 t ; ~ t ;z e v ;r 2L 1 ,and5canbeusedtoconcludethat t 2L 1 .Theopen-loopquaterniontrackingerrorin3canbeusedtoconclude that e v t ; e 0 t 2L 1 .From5, t 2L 1 andthen5and5canbeused toindicatethatthefactthat t 2L 1 .In2,conservationofangularmomentum, t ,and t 2L 1 showsthat t 2L 1 .Thefactthat t -dependentfunctionresult fromdirectioncosinematricesindicatesthatthefunctionscontain t withinbounded trigonometricfunctions.Since e v t ;e 0 t ;r e v ;e 0 ;! ;! t ; t 2L 1 ,5and5 canbeusedtoshowthat g t 2L 1 .Thefactthat r e v ;e 0 ;! ;e v t ; t ; ~ 1 t ; ~ 2 t 2 94

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L 1 canbeusedwith522toconcludethat r e v ;e 0 ;r; ; ~ 1 ; ~ 2 2L 1 .From5and 5, ^ 1 t ; ^ 2 t 2L 1 .Hence,5,5,5,5,and z e v ;r ; t 2 L 1 canbeusedtoprovethatthecontrolinput t 2L 1 .Standardsignalchasing argumentscanthenbeutilizedtoprovethatallremainingsignalsareboundedduring closed-loopoperation.Since e v t ;r e v ;e 0 ;! ; e v t ; r e v ;e 0 ;r; ; ~ 1 ; ~ 2 2L 1 e v t and r e v ;e 0 ;! areuniformlycontinuous.Since e v t and r e v ;e 0 ;! areuniformly continuous,and e v t ;r e v ;e 0 ;! 2L 1 L 2 ,Barbalat'sLemmacanbeusedtoprove r e v ;e 0 ;! ;e v t 0 as t !1 5.3NumericalExamples 5.3.1SimulationSetup Numericsimulationsillustratetheperformanceofthedevelopedcontroller.The satelliteparametersarebasedonamodelofaprototypepico-satelliteandaregivenin Table5-1.Agimbalratelimitwasincludedinthemodelas sat i = 8 > < > : i ; for i 25 rad= sec 8 i =1 ; 2 ; 3 ; 4 25 sgn i ; for i > 25 rad= sec ; where sgn denotesthestandardsignumfunction.Thedesiredangularvelocitytrajectories d t are d t = 0 : 004sin t= 200000 rad= sec ,andthedesiredywheel speedforeachwheelis d =200 rad=s 2 ; 000 rpm .Theinitialconditionsaregivenin Table5-2. PhysicalParameter Value J kg m 2 diag 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 7 : 64 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 m cmg kg 0 : 165 I G kg m 2 2 : 80 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 I 4 I W kg m 2 6 : 95 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 I 4 skewangle 54 : 74 Table5-1.PhysicalparametersfortheVSCMGsimulation. 95

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InitialParameter Value q 0 : 10 : 30 : 80 : 4 rad=s 0 ^ 1 ^ 2 0 rad 0 : 54980 : 23330 : 54980 : 2333 rad=s 0 Table5-2.InitialparametersfortheVSCMGsimulation. Figure5-1.Quaterniontrackingerror e t forCase1. 5.3.2SimulationResults Thesimulationresultsaredevelopedfortwocases.Case1isincludedtoillustrate howthecontrollerrespondswhenthemomentumtrackinghasalongtransient.Forthis casetheVSCMGhastooperateinRWmodeduringthetransient,resultingingimbal ratesaturation.TheresultsaregiveninFigures5-1-5-8.Case2isincludedtoillustrate amorefavorableconditionwhichcanbeachievedthroughcontrolgainswherethe momentumtrackingerrorhasashorttransientresponse.ResultsinFigures5-9 -5-12illustratethatforthiscase,gimbalratesaturationisavoidedandtheVSCMG operatesinRWmodelessthanforCase1.Figure5-1showsthequaterniontracking errorresultstoachievethecontrolobjectivedenotedin3duringa 500 s simulation 96

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Figure5-2.Controlinputgimbalrates t andwheelaccelerations t forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse,includinggimbalratesaturation. Figure5-3.Flywheelspeed t inducedfrominternalmomentummanagementforCase 1. 97

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Figure5-4.Singularitymeasurefunction f andnullmotionweight S forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse,includinggimbalratesaturation. period.Figure5-2showsthecontrolinputgimbalrates t andwheelaccelerations t Whilethecontrolinputsachieveattitudestabilization,thewheelaccelerationcontrol input t contributestotheinternalmomentummanagementbyregulatingthewheel speedafterstartingfromrest.Figure5-3indicatestheinitialstart-upofywheelsfrom rest.Tocompensateforthelackoftorquegeneratedbytheywheelduetotheslow momentumtracking,thegimbalratesinFigure5-2areshowntogeneratemoretorques includingtorquesaturationduringthetransientresponse.Theincreasedtransient responseofthegimbalrateyieldsincreasedsingularitiesintheCMGJacobian,but thesesingularitiesareeectivelyavoidedasshowninFigures5-4and5-5.Asaresultof approachingsingularities,theywheelsarerequiredtooperateinRWmodeasshownin Figure5-3.Thenullmotionweight S in5,depictedinFigure5-4,hasanincreased transientinthiscasebecause S inhibitsthemomentumtrackingwhentheCMGJacobian approachesasingularitysothattheywheelcanworkinRWmode.Although S cannot distinguishbetweendierentsingularities,theweightmatrixcanadjusttheintervention 98

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Figure5-5.Nullmotion:gimbalrecongurationandinternalmomentumtrackingerror t forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse,includinggimbalratesaturation. Figure5-6.Adaptiveparameterestimates ^ 1 t and ^ 2 t forCase1. 99

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Figure5-7.Transientresponseofadaptiveparameterestimate ^ 1 t forCase1. Thenotation ^ 1 t x denotesthe x -elementof ^ 1 t oftheywheelwhenapproachingasingularity.Hence,thecompositeweightedsteering lawin5cancopewithanellipticsingularitywhilemaintainingprecisionattitude controlsincethesteeringlawgeneratestherequiredtorqueinRWmodetopassthrough orescapeaninternalsingularitydierentlyfromthesingularityescapemethodsforCMGs thatrequireaddedtorque[6,27,64,77,95,97].Thisobservationindicatesabenetthat resultsfromtheextracontrollabledegreeoffreedomoftheVSCMG.Specically, s w in 5,depictedinFigure5-4,allowsmomentumtrackingwheninnormaloperationand restrictsthemomentumtrackingwhenapproachingasingularity.Thus,inmostregions gimbalrecongurationisresponsibleforsingularityavoidancebyexploitinggimbalrate nullsolution.Thenullmotionweight S servesasaswitchthatactsasanalternativeto copewithbothellipticandhyperbolicsingularitieswhileachievinginternalmomentum management.Thisbenetprovidesanavenuetoeectivelyacquiretheinitialstart-up withoutaseparatefeedbackloop.Thetimevariationoftheadaptiveparameterestimates isshowninFigures5-6-5-8.Figure5-6showstheelementsoftheadaptiveparameter vectors ^ 1 t and ^ 2 t .Figures5-7and5-8,dividedintomultiplewindowsforclarity, 100

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Figure5-8.Transientresponseofadaptiveparameterestimate ^ 2 t forCase1. Thenotation ^ 2 t x denotesthe x -elementof ^ 2 t highlighttheadaptiveparameterestimatesof ^ 1 t ^ 2 t ,respectively.Theadaptation mechanismadjuststheuncertainparameterestimatesothatattitudetrackingcanbe asymptoticallyachieved.Moreover,theinitialvaluesoftheuncertainparametersare chosentobezero,indicatingnoaprioriparameterknowledge.Eventhoughsomeofthe parameterestimatesindicatethevariationbysmallmagnitudes,theequivalentcontrol torquesarisingfromthefeedforwardterms ^ 1 t ^ 2 t ,resultingfrom2,5, 5,5,and5,havesimilarmagnitudestothatofthefeedbackcontrolterms. Case2isdevelopedfortheclosed-loopoperationsinfastmomentumtrackingand thesimulatedresultsareprovidedinFigures5-9-5-12.Figure5-9showsthegimbalrate t andthewheelacceleration t controlinputs.Figure5-10showsthattheywheel speedtracksthedesiredwheelspeedfromrest.Themomentumtrackinggain k m denoted in5controlsthemomentumtrackingspeed.Sincetheywheelsrapidlyarriveat thedesiredconstantspeed,fastmomentumtrackingallowstheVSCMGsteeringlawto operatelongerintheCMGmodewhichprovidestorqueamplicationandpowersavings. 101

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Figure5-9.Controlinputgimbalrates t andwheelaccelerations t forCase2. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse,includinggimbalratesaturation. Figure5-10.Flywheelspeed t inducedfrominternalmomentummanagementforCase 2. 102

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Figure5-11.Singularitymeasurefunction f andnullmotionweight S forCase2. Figure5-11illustratesthesingularitymeasurefunction f andthenullmotionweight S Correspondingtothemoderatevariationof f s w allowssteadymomentumtracking,and s g generatesthepropergimbalrecongurationasdepictedinFigure5-12. 5.4Summary Inthepresenceofsatelliteinertiauncertaintyandactuatoruncertainty,thedevelopedattitudecontrollerinthischapteriscapableofachievingglobalasymptoticattitude trackingwhilesimultaneouslyperformingsingularityavoidanceandinternalmomentum management.Thebenetssuchassingularityavoidanceandinternalmomentummanagementemergefromthenullsolutionofthecontrolinputs.Inparticular,theinternal momentummanagementallowstheywheeltostartfromrestandtoreachthedesired speed.TomaximizeoperationinCMGmode,thesteeringlawexploitsthesingularity avoidancestrategyresultingfromthegradientmethod,andthenullmotionweightadjusts theinternalmomentumtrackingoftheywheelswhenapproachinganinternalsingularity. TheVSCMG-actuatedsatellitecanaccomplishasymptoticattitudetrackingandexponentialinternalmomentumtrackingwhilesimultaneouslyachievingsingularityavoidance. 103

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Figure5-12.Nullmotion:gimbalrecongurationandinternalmomentumtrackingerror t forCase2. Thecontrolleralsocompensatesfortheeectsofuncertain,time-varyingsatelliteinertiaproperties.Thedicultiesarisingfromtheuncertainsatelliteinertiaaremitigated throughaninnovativedevelopmentoftheerrorsystemalongwithaLyapunov-based adaptivelaw.Theattitudetrackingandmomentumtrackingresultsareprovenviaa Lyapunovstabilityanalysisanddemonstratedthroughnumericalsimulations. 104

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CHAPTER6 ARNN-BASEDATTITUDECONTROLMETHODFORAVSCMG-ACTUATED SATELLITE AnadaptiveRNN-basedsatelliteattitudecontrollerisdevelopedtoachieveprecision attitudecontrolofaVSCMG-actuatedsatelliteinthepresenceofuncertain,time-varying satelliteinertiapropertiesandactuatoruncertaintiesinadditiontounmodeledexternal disturbances.Thechallengeencounteredinthecontroldesignisthatthecontrolinput ispremultipliedbyanon-square,time-varying,nonlinear,uncertainmatrix.TheRNN estimatorresultingfromtheRNNstructureservesasacompositeVSCMGsteeringlaw forthesatellite,whichcompensatesforsatelliteandactuatoruncertaintiespresentinthe nonlineardynamics.Usingnullmotion,astrategyisdevelopedtosimultaneouslyperform gimbalrecongurationandwheelspeedregularization.Numericalsimulationsdemonstrate theperformanceoftheadaptiveRNN-basedVSCMGsteeringlawandtheRNNtraining. 6.1DynamicModel TheequationofmotionforarigidbodyVSCMG-actuatedsatellitecanbewrittenas L r = J! + J + J! + d ; where t 2 R 3 isanangularvelocityofthesatellite,and J 2 R 3 3 isatotalsatellite inertiamatrixcontainingabusandasetofCMGunits.Theinertiamatrix J is positivedeniteandsymmetricandsatises 1 2 min f J gk k 2 T J 1 2 max f J gk k 2 8 2 R n ; where min f J g ; max f J g2 R aretheminimumandmaximumeigenvaluesof J respectivelyand t 2 R 4 isagimbalangularpositionvector.In4, d t 2 R 3 representsunknown,smoothdisturbancetorquesactingonthesystem,whichareassumed tobeboundedas k d k 1 ; 105

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where 1 2 R isaknownpositiveboundingconstant,andthecontroltorque L r t 2 R 3 producedbyasetoffourVSCMGscanbeexpressedas L r = )]TJ/F23 11.9552 Tf 9.298 0 Td [(Q 0 + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.956 0 Td [(C T [ I W ] d [ G ] d )]TJ/F23 11.9552 Tf 9.298 0 Td [(C G I CMG G d : In6, Q 0 !;; 2 R 3 8 isdenedas Q 0 h C W [ I W ] d C W I CMG G d [ T ] d + C T [ I W ] d [] d )]TJ/F28 11.9552 Tf 11.955 9.683 Td [( I CMG G d [ W ] d i ; t 2 R 4 denotesthewheelangularvelocity,and t 2 R 8 denotesthetimederivativeof t = T t T t T .Since t dependsonreactioncausedbythesatellitetracking correspondingtoconservationofangularmomentum,providedthatthesatellitetrajectory isbounded, t canbeboundedas k k 2 ; where 2 2 R isaknownpositiveboundingconstant.In6and6, C G ^ a G 1 ^ a G 2 ^ a G 3 ^ a G 4 C W ^ a W 1 ^ a W 2 ^ a W 3 ^ a W 4 ,and C T ^ a T 1 ^ a T 2 ^ a T 3 ^ a T 4 2 R 3 4 where ^ a G ^ a W ,and ^ a T aregimbal,wheel,andtransverseaxes. Alsoin6and6,theinertiamatrices I CMG G d [ I W ] d 2 R 4 4 ofCMGunit aredenotedas I CMG G d diag I CMG 1 G I CMG 2 G I CMG 3 G I CMG 4 G and [ I W ] d diag I W 1 I W 2 I W 3 I W 4 ,andtheangularvelocityprojectedtothegimbal-xed axesisdenotedas [ ] d diag 1 2 3 4 : G W T 2 R 4 4 ,and [ t ] d and h t i d denotediagonalmatricescomposedofthevectorelementsof t t 2 R 4 ,respectively.Theexpressionin6representstheactuatordynamics.The subsequentdevelopmentfocusesondesigningthecompositeVSCMGcontrolinputs t and t toimpartadesiredtorqueonthesatellitebody. 106

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6.2ControlObjectives 6.2.1AttitudeControlObjective Theattitudecontrolobjectiveistodevelopaywheelaccelerationandgimbalrate controllawtoenabletheattitudeof F totracktheattitudeof F d .Thetrackingerror formulationisquantiedin3.2.1inChapter3.Fromthedenitionsofthequaternion trackingerrorvariables,thefollowingconstraintcanbedeveloped[16]: e T v e v + e 2 0 =1 ; where 0 k e v t k 10 j e 0 t j 1 ; where kk representsthestandardEuclideannorm. 6.2.2FlywheelAngularMomentumManagementObjective Theywheelangularmomentummanagementobjectiveinthischapteristo developaywheelaccelerationcontrollawsothattheactualangularmomentum h = I W t 2 R 4 tracksadesiredconstantangularmomentum h d = I W d while simultaneouslytrackingadesiredtime-varyingattitude.Theangularmomentumtracking error 2 R 4 isquantiedinthesensethat5ofChapter5isdetermined. 6.3AdaptiveRNNController 6.3.1AdaptiveAttitudeControlDevelopment 6.3.1.1Open-LoopErrorSystem Tofacilitatethecontroldesign,anauxiliarysignal r t 2 R 3 isdenedas[23] r )]TJ/F15 11.9552 Tf 14.509 3.022 Td [(~ R! d + e v ; where 2 R 3 3 isaconstant,positivedenite,diagonalcontrolgainmatrix.Motivation forthedesignof r t isbasedonthesubsequentLyapunov-basedstabilityanalysis.After multiplyingthetimederivativeof6by J andusing6and6,theopen-loop 107

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errordynamicsfor r t canbewrittenas J r = )]TJ/F23 11.9552 Tf 9.299 0 Td [(Q + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [( d + Y 1 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Jr; wheretheterm C G I CMG G d isassumedtobenegligible[26,83,89],and Q e v ;r;!;; 2 R 3 8 isdenedas Q h C W [ I W ] d C W I CMG G d [ T ] d + C T [ I W ] d [] d )]TJ/F28 11.9552 Tf 11.291 9.684 Td [( I CMG G d [ W ] d + @J @ 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : In6, Y 1 e v ;e 0 ;!; 2 R 3 p 1 isameasurableregressionmatrix,and 1 2 R p 1 isa vectorof p 1 unknownconstantsdenedviatheparameterization Y 1 1 )]TJ/F23 11.9552 Tf 9.298 0 Td [(! J! + J! ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(J ~ R d + 1 2 J )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ !: Tocompensateforthelinearlyparametrizableuncertaintyin Q e v ;r;!;; ,thefollowing parameterizationwillbedened: Y 2 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(Q ; where Y 2 e v ;r;!;; ; ; 2 R 3 p 2 isameasurableregressionmatrix,and 2 2 R p 2 isavectorof p 2 unknownconstants.Toaddressthefactthatthecontrolinput t ispremultipliedbythenonsquare,time-varyinguncertainmatrix Q e v ;r;!;; ,an estimate ^ Q t 2 R 3 8 oftheuncertaintyin6isdenedvia Y 2 ^ 2 )]TJ/F15 11.9552 Tf 11.983 3.022 Td [(^ Q ; where ^ 2 t 2 R p 2 isasubsequentlydesignedestimatefortheparametricuncertaintyin Q e v ;r;!;; .Basedon6and6,theexpressionin6canbewrittenas J r = Y 2 ~ 2 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + Y 1 1 + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d )]TJ/F23 11.9552 Tf 11.956 0 Td [( d )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr; 108

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wherethenotation ~ 2 t 2 R p 2 isdenedas ~ 2 = 2 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 2 : Basedontheopen-looperrordynamicsof6andthecompositeVSCMGsteeringlaw of6,therequiredcontroltorque L r e v ;r; 2 R 3 of6isdesignedas L r = )]TJ/F23 11.9552 Tf 9.298 0 Td [(Y 1 ^ 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( k 1 + k a r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v + @J @ 1 2 r + ~ R! d )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v : Aftersubstituting6into6andperformingsomealgebraicmanipulations,the resultingexpressionforthecompositeVSCMGsteeringlawcanbewrittenas Y 2 ~ 2 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d = )]TJ/F23 11.9552 Tf 9.299 0 Td [(Y 1 ^ 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 1 + k a r )]TJ/F23 11.9552 Tf 11.956 0 Td [(e v ; where k 1 k a 2 R arepositiveconstantgains.Byexploitingtheuniversalapproximation propertyofNNs,theterms C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 10.548 0 Td [(C T [ I W ] d [ G ] d in6canberepresented as C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d = f ; where e T v r T T T 2 R 14 ,and f isdenedas f W T )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(V T + : In6, 2 R N +1 isabasisfunctionvectorwithsmooth,bounded,monotonically increasingelements[3,18,28,37], W 2 R N +1 3 and V 2 R 14 N areconstantunknown matricesofidealRNNweights, N isthenumberofhiddenlayers,and 2 R 3 denotes thefunctionalreconstructionerror. Remark6.1. Foranypositiveconstantrealnumber N 2 R f iswithin N oftheNN rangeifthereexistnitehiddenneurons N andconstantweightssothatforallinputsin thecompactset S ,theapproximationholdswith k k N : 109

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TheStone-Weierstrasstheoremindicatesthatanysucientlysmoothfunctioncanbe approximatedbyasuitablylargenetwork.Therefore,thefactthattheapproximationerror isboundedfollowsfromtheuniversalapproximationpropertyofNNs. Substituting6into6yields J r = Y 2 ~ 2 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + Y 1 1 + W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T + )]TJ/F23 11.9552 Tf 11.955 0 Td [( d )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 Jr: Basedon6andthesubsequentstabilityanalysis,thedesiredsteeringlawcanbe designedas = ^ Q + w n W T )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(V T + + Y 1 ^ 1 + k 1 + k a r + e v o )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 + k b + I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q S'; where ^ Q + w ;t = W c ^ Q T t ^ Q t W c ^ Q T t )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,and k 2 k b 2 R arepositive constantgains,and W c 2 R 8 8 denotesaweightmatrixdeterminingwhetherthe VSCMGsystemusesaCMGmodeoraRWmodedesignedas[87,89,101] W c 2 6 4 W I 4 4 0 4 4 0 4 4 W I 4 4 3 7 5 ; where W 2 R isdenedas W W exp 1 h ; where 1 ;W ;W 2 R arepositiveconstants,andtheobjectivefunction h 2 R measuringthesingularitycanbedenotedas h )]TJ/F15 11.9552 Tf 11.291 0 Td [(det )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(C T C T T : In6,thesecondterm I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w t ^ Q t S t generatestheVSCMGnull motionformomentummanagementandsingularityavoidance.Sincethematrices ^ Q + w t and ^ Q t arenonsquare,thepseudo-inverse ^ Q + w t 2 R 8 3 isdenedsothat ^ Q t ^ Q + w t = 110

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I 3 ,andthematrix I 8 )]TJ/F15 11.9552 Tf 15.163 3.022 Td [(^ Q + w t ^ Q t ,whichprojectsvectorsontothenullspaceof ^ Q t andsatisestheproperties I 8 )]TJ/F15 11.9552 Tf 14.639 3.022 Td [(^ Q + w ^ Q I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q = I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q ^ Q I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q =0 : Togeneratenullmotionformomentumtrackingandgimbalreconguration,thenull motion t 2 R 8 1 isdenedas 2 6 4 k d g k @ @ 3 7 5 ; where k d 2 R denotesapositiveconstant,and g t 2 R 4 isanauxiliarycontrolsignal totrackthedesiredf1ywheelangularmomentum.In6, k 2 R denotesapositive constantandthesecondrowallowsthenullmotiontoperformthegimbalreconguration correspondingtoavariationofsingularitymeasureindex ,whichisdenedas[97] 0 exp 2 h ; wheretheobjectivefunction h isdenedin6and 0 ; 2 2 R denotepositive constants.Alsoin6, S 2 R 8 8 selectsapropernullmotionbasedonthe singularitymeasure.Specically, S diag [ s w ;s g ] isdesignedas 2 6 4 sech 1 k w det C T C T T + s 0 4 4 0 4 4 sech )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(k g det )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(C T C T T 3 7 5 ; where k w ;k g ;" s 2 R arepositiveconstants.In6, s w s g weighteitherthemomentum trackingorthegimbalrecongurationcorrespondingtohowapproximateorfartheCMG congurationistoasingularity. Thesteeringlawin6cannotbeimplementedbecauseitdependsontheunknownidealweightmatrices W and V andthereconstructionerror .Theimplemented steeringlaw ^ t 2 R 8 isdevelopedbasedontherecursive,internodefeedback-basedRNN 111

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structureas ^ = ^ Q + w n ^ W T ^ V T ^ + Y 1 ^ 1 + k 1 + k a r + e v o )]TJ/F15 11.9552 Tf 11.956 0 Td [( k 2 + k b ^ + I 8 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q + w ^ Q S'; where ^ f ^ ^ W T ^ V T ^ 2 R 3 ,and ^ t isdenedas e T v r T ^ T T 2 R 14 ^ W t 2 R N +1 3 and ^ V t 2 R 14 N areestimatedweights.Theerrorequationforthe RNNestimatorcanbeobtainedfrom6and6as ~ = ^ Q + w n W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ V T ^ + o )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 + k b ~ ; wheretheestimationmismatch ~ t 2 R 8 isdenedas ~ )]TJ/F15 11.9552 Tf 12.781 0 Td [(^ : Tofacilitatethesubsquentclosed-loopanalysis,theopenloopequationfor r t in6 canberewrittenas J r = Y 2 ~ 2 )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q ^ )]TJ/F15 11.9552 Tf 14.64 3.022 Td [(^ Q ~ + Y 1 1 + W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T + )]TJ/F23 11.9552 Tf 11.955 0 Td [( d )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Jr: 6.3.1.2Closed-LoopErrorSystem Thesteeringlaw ^ t functioningasacontrolinputisdesignedasaself-tuning adaptivecontrollerconstructedintermsoftheestimatefunction ^ W T ^ V T ^ resulting fromtheRNNstructurewheretheweights ^ W t ^ V t areupdatedonlineviaadaptation laws.Substituting6and6into6yields J r = Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( k 1 + k a r + k 2 + k b ^ Q )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 11.955 0 Td [( d )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Jr; wherethenotation ~ 1 t 2 R p 1 isdenedas ~ 1 = 1 )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(^ 1 : 112

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Basedon6andthesubsequentstabilityanalysis,theparameterestimates ^ 1 t and ^ 2 t aresolutionstotheupdatelaws ^ 1 = proj )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(1 Y T 1 r ^ 2 = proj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 Y T 2 r ; where )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(1 2 R p 1 p 1 and )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(2 2 R p 2 p 2 denoteconstant,positive-denite,diagonaladaptation gainmatrices.In6,thefunction proj denotesaprojectionalgorithmutilizedto guaranteethatthe i th elementof ^ 1 t and ^ 2 t canbeboundedas 1 i ^ 1 i 1 i ; 2 i ^ 2 i 2 i ; where 1 i 1 i 2 i 2 i 2 R denoteknownconstantlowerandupperboundsforeach elementof ^ 1 t and ^ 2 t ,respectively.Minimizingtheestimationerror ~ t ensures thatthestateestimate ^ t in6dynamicallyapproximatesthesystemstate t in 6.Afteraddingandsubtractingtheterms W T ^ V T ^ and W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T ^ insidethe bracketedexpressionin6,thefollowingisobtained: ~ = ^ Q + w n W T h )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T ^ )]TJ/F23 11.9552 Tf 11.955 0 Td [( ^ V T ^ i + W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T )]TJ/F23 11.9552 Tf 11.955 0 Td [(W T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T ^ + ~ W T ^ V T ^ + o )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 + k b ~ : TheTaylorseriesofthevectorfunction )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(V T ^ intheneighborhoodof ^ V T ^ is )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T ^ = ^ V T ^ + 0 ^ V T ^ ~ V T ^ + O 2 ~ V T ^ ; where 0 ^ V T ^ d )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(V T ^ =d )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(V T ^ j V T ^ = ^ V T ^ ~ V t V )]TJ/F15 11.9552 Tf 14.585 3.022 Td [(^ V t ,and O 2 ~ V T ^ denoteshigherorderterms.UsingtheTaylorseriesof )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(V T ^ givenin6,6can berewrittenas ~ = ^ Q + w n ^ W T 0 ^ V T ^ ~ V T ^ + ~ W T ^ V T ^ + w o )]TJ/F15 11.9552 Tf 11.956 0 Td [( k 2 + k b ~ ; 113

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where ~ W t 2 R N +1 3 and ~ V t 2 R 14 N denotetheRNNweightestimatemismatches denedas ~ W W )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W ~ V V )]TJ/F15 11.9552 Tf 13.741 3.022 Td [(^ V; andthedisturbanceterm w e v ;r; ^ ;t 2 R 3 isdenedas w W T )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(V T )]TJ/F23 11.9552 Tf 11.955 0 Td [(W T )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(V T ^ + ~ W T 0 ^ V T ^ ~ V T ^ + W T O 2 ~ V T ^ + ": Thedisturbance w e v ;r; ^ ;t in6canbeupperboundedas k w k 3 + 4 k z k ; where 3 4 2 R arepositiveboundingconstants,and z t 2 R 14 isdenedas z e T v r T ~ T T : Also,basedon6,6,and6,thefollowinginequalityholds: ^ Q i 1 1 ^ Q + w i 1 2 ; where 1 ; 2 2 R areknownpositiveboundingconstantsand kk i 1 istheinducedinnity normofamatrix.Basedonthesubsequentstabilityanalysis,theweightupdatelawsfor theRNNaredesignedvia ^ W =)]TJ/F22 7.9701 Tf 26.381 -1.793 Td [(3 proj h ^ V T ^ ~ T ^ Q + w i ^ V =)]TJ/F22 7.9701 Tf 26.381 -1.793 Td [(4 proj h ^ ~ T ^ Q + w ^ W T 0 ^ V T ^ i ; where )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(3 2 R N +1 N +1 )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(4 2 R 14 14 denoteconstant,positivedenite,diagonal adaptationgainmatrices,and proj denotesaprojectionalgorithmutilizedtoguarantee thatthe i th elementof ^ W t and ^ V t canbeboundedas W i ^ W i W i ;V i ^ V i V i ; 114

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where W i W i V i ,and V i 2 R denoteknown,constantlowerandupperboundsforeach elementof ^ W t and ^ V t 6.3.2MomentumTrackingControlDevelopment Toachievetheywheelangularmomentummanagementobjectivewhilemaintaining theattitudestabilization,theclosed-looperrorsystemisdevelopedin5inChapter5. 6.3.3StabilityAnalysis Theorem6-1: Theadaptivecontrollawgivenin6resultingfromtheRNN structureensuresuniformlyultimatelyboundedUUBattitudetrackinginthesensethat k e v k 0 exp f)]TJ/F23 11.9552 Tf 15.276 0 Td [(" 1 t g + 2 where 0 1 2 2 R denotepositiveboundingconstantswhilealongwithexponential momentumtrackinginthesensethat k t k exp )]TJ/F23 11.9552 Tf 9.298 0 Td [(I W k m t : Proof: Theexponentialmomentumtrackingresultisevidentfrom5ofChapter 5. Proof: Let V e v ;e 0 ;r; ~ ; ~ 1 ; ~ 2 ; ~ W; ~ V;t 2 R bedenedasthefollowingnonnegative function: V e T v e v + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 2 + 1 2 r T Jr + 1 2 ~ T ~ + 1 2 ~ T 1 )]TJ/F26 7.9701 Tf 7.315 4.937 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 1 ~ 1 + 1 2 ~ T 2 )]TJ/F26 7.9701 Tf 7.314 4.937 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 ~ 2 + 1 2 tr ~ W T )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 3 ~ W + 1 2 tr ~ V T )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 4 ~ V : Basedon6,6,6,6-6,6,6,and6,6canbe upperandlowerboundedas c 0 k z k 2 + c 1 V t c 2 k z k 2 + c 3 ; 115

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where c 0 c 1 c 2 c 3 2 R areknownpositiveboundingconstants.Afterusing3,6, and6,thetimederivativeof V t canbeexpressedas V = e T v )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(e v + e 0 I 3 ~ + )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 0 e T v ~ + r T n Y 1 ~ 1 + Y 2 ~ 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e v )]TJ/F23 11.9552 Tf 11.955 0 Td [( d )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 1 + k a r + k 2 + k b ^ Q o +~ T h ^ Q + w n ^ W T 0 ^ V T ^ ~ V T ^ o + ^ Q + w n ~ W T ^ V T ^ o + ^ Q + w w )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 + k b ~ ] )]TJ/F15 11.9552 Tf 12.894 3.154 Td [(~ T 1 )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 1 ^ 1 )]TJ/F15 11.9552 Tf 12.895 3.154 Td [(~ T 2 )]TJ/F26 7.9701 Tf 7.315 4.936 Td [()]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 ^ 2 )]TJ/F23 11.9552 Tf 9.299 0 Td [(tr ~ W T )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 3 ^ W )]TJ/F23 11.9552 Tf 11.956 0 Td [(tr ~ V T )]TJ/F26 7.9701 Tf 7.314 4.936 Td [()]TJ/F22 7.9701 Tf 6.587 0 Td [(1 4 ^ V : Byusing6and6,andexploitingthefactthat e T v e v ~ =0 ,6canbe rewrittenas V = )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 1 + k a r T r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e T v e v )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 + k b ~ T ~ +~ T ^ Q + w w + k 2 + k b r T ^ Q )]TJ/F23 11.9552 Tf 11.955 0 Td [(r T d : Using6,6,6,and6,theresultingexpressionof6canbeupper boundedas V )]TJ/F15 11.9552 Tf 30.552 0 Td [( k 1 + k a k r k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( k e v k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 + k b k ~ k 2 + 5 k z k + 6 k z k 2 ; where 5 k 2 + k b 1 2 + 2 3 + 1 and 6 2 4 .Basedon6, V t canbewritten as V )]TJ/F23 11.9552 Tf 21.918 0 Td [( 1 k z k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 k z k 2 + 5 k z k ; where 1 min k 1 ;= 2 ;k 2 and 2 min k a ;= 2 ;k b )]TJ/F23 11.9552 Tf 12.089 0 Td [( 6 .Aftercompletingthesquares in6,theupperboundof V t canbeexpressedas V )]TJ/F23 11.9552 Tf 21.918 0 Td [( 1 k z k 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [( 2 k z k)]TJ/F23 11.9552 Tf 25.107 8.087 Td [( 5 2 2 2 + 3 ; 116

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where 3 2 5 4 2 .Since650canbeusedtolowerbound k z k 2 as k z k 2 1 c 2 V t )]TJ/F23 11.9552 Tf 13.151 8.088 Td [(c 3 c 2 : Thelowerboundof k z k 2 in6yields V )]TJ/F23 11.9552 Tf 23.113 8.088 Td [( 1 c 2 V + 4 ; where 4 isdenedas 4 1 c 3 c 2 + 3 : Thelineardierentialinequalityin656canbesolvedas V t V exp )]TJ/F23 11.9552 Tf 10.494 8.087 Td [( 1 c 2 t + 4 c 2 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F23 11.9552 Tf 10.494 8.087 Td [( 1 c 2 t : Theexpressionsin6,6,and6canbeusedtoprovethat r t ~ t 2L 1 Thus,6canbeusedtoconcludethat t 2L 1 ,andfrom3and6, ~ t 2 L 1 .Theattitudekinematicsin3canthenbeusedtoshowthat e v t ; e 0 t 2L 1 Since ^ W t ; ^ V t 2L 1 from6,theassumptionthat W;V 2L 1 canbeusedalong with6toprovethat ~ W t ; ~ V t 2L 1 .Thefactsthat e v t ;! t 2L 1 canbe usedalongwith6toshowthat Y 1 t 2L 1 .Basedontheassumptionthat 1 2L 1 6and6canbeusedtoprovethat ~ 1 t 2L 1 .From6, t 2L 1 and then5and5ofChapter5canbeusedtoindicatethat t 2L 1 .Thefact that t -dependentfunctionsareresultingfromdirectioncosinematricesindicatesthat thefunctionscontain t withinboundedtrigonometricfunctions.Hence,theinput set e T v r T T T 2L 1 inRNNnetworkof6followingtheUniversal approximationtheorem[3,18].Giventhat ~ t ;W;V; ^ 1 t ;r t ;e v t 2L 1 ,6 20,6,and6canbeusedtoprovethat t 2L 1 .Since t 2L 1 t t 2L 1 .Giventhat Y 1 t ; ^ 1 t ;r t ;e v t ; t 2L 1 L r t 2L 1 from6 16.Giventhat e v t ;r t ; t 2L 1 ,theRNNinputvector t 2L 1 .Since t ~ t 2L 1 ,6canbeusedtoconcludethat ^ t 2L 1 .Byutilizingthefactthat 117

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Figure6-1.Quaterniontrackingerror e t duringclosed-loopoperation. t -dependentfunctionsonlycontain t withinboundedtrigonometricfunctionsand e v t ;r t ;! t ; t ; t ; t 2L 1 ,6canbeusedtoshowthat Y 2 t 2L 1 Giventhat ^ W t ; ^ V t ;r t ;e v t ; ^ 1 t ; ^ 2 t ;Y 1 t v t 2L 1 ,6and6 canbeusedtoprovethatthecontrolinputvector ^ t 2L 1 .Since r t e v t t Y 1 t Y 2 t ; ~ 1 t ; ~ 2 t 2L 1 andby6,6canbeusedtoshowthat r t 2L 1 Standardsignalchasingargumentscanthenbeusedtoprovethatallothersignalsremain boundedduringclosed-loopoperation.Theinequalitiesin6canbeusedalongwith 6and6toconcludethat k z k 2 c 2 k z k 2 + c 3 c 0 exp )]TJ/F23 11.9552 Tf 10.494 8.088 Td [( 1 c 2 t + 2 1 c 3 + 3 1 c 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(c 1 c 0 : Theresultin6cannowbedirectlyobtainedfrom6. 6.4NumericalExample Numericsimulationsillustratetheperformanceofthedevelopedcontroller. Thesatelliteparametersarebasedonamodelofaprototypepico-satellitewhich hasapyramidalarrangementoffourVSCMGs.Themodelhasthetotalinertiaof 118

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Figure6-2.RNNestimationerror ~ ~ ; ~ duringclosed-loopoperation. Figure6-3.InducedinnitynormofRNNweightmatrices ^ W t ; ^ V t 119

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Figure6-4.Singularitymeasurefunction h ,nullmotionweight S ,andmodeweight W c J total = diag 6 : 10 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 6 : 10 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 7 : 64 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 andtheVSCMGunitmassof m vscmg =0 : 165 kg .Thecontrolobjectiveistostabilizeasatellite'satttiudewhiletracking thedesiredangularvelocitytrajectory.ThesimulationresultsaregiveninFigures6-1-64.Figure6-1showsthequaterniontrackingerrorresultsduringa 500 s simulationperiod. TheRNNestimationerror ~ ~ ; ~ shownbyFigure6-2illustratesasteady-stateresponse whiletheRNNestimatorresultingfrom6compensatesfortheactuatoruncertainty. Sincethestate ; ofthenonlinearfunction f isdynamicallyupdatedintheRNN structureconsistingof 10 hiddenlayers,thestatefeedbackactivelycontributestotraining ofweightestimates, ^ W ^ V showninFigure6-3.Also,theRNNestimatorfunctionsasthe compositeVSCMGsteeringlawwhicharbitratesbetweentheCMGandRWmodecorrespondingtomodeweight W c inducedbythesingularitymeasure h asshowninFigure 6-4.Thenullmotionstrategyperformsgimbalrecongurationforsingularityavoidance andwheelspeedregularizationforinternalmomentummanagementrespondingtonull motionweight S inFigure6-4.AlthoughtheVSCMGisageometricallysingularity-free deviceandthesingularityavoidanceisnotalwaysnecessary,thesingularityavoidance 120

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methodreduces/eliminatestheamountoftimethattheVSCMGhastooperateinRW modewhentheCMGJacobianbecomessingularthroughtheuseofnullmotion. 6.5Summary Inthischapter,aRNN-basedcontroltechniqueispresented,whichachievesattitude trackingforaVSCMG-actuatedsatelliteinthepresenceofuncertaintyinthesatellite andactuatordynamicsandunmodeledexternaldisturbances.ALyapunov-basedstability analysisisusedtoprovethecontrollerachievesUUBattitudetrackingwhilecompensating fortheeectsofuncertaintime-varyingsatelliteinertiaproperties,parametricuncertainty, andnonlinearexternaldisturbancetorques.Innovativedevelopmentoftheerrorsystem alongwithaLyapunov-basedadaptivelawmitigatesthedicultiesresultingfromsatellite inertiauncertainty.NumericsimulationresultsillustrateperformanceofRNNestimator whichfunctionsasacompositeVSCMGsteeringlawaswellasbenetsprovidedby gimbalandwheelnullmotions. 121

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CHAPTER7 ANEWSINGULARITYDETECTIONMETHODFORVSCMGSUSINGFLS AsseeninChapter4-6,thenullmotionsofVSCMGcanprovidebenecialeects suchasgimbalrecongurationandinternalmomentummanagementforspacecraftoperationwhilethemotionsgeneratenonettorquefromtheVSCMGs.EveniftheVSCMGis geometricallyasingularity-freedeviceexceptsaturation,thesingularityavoidancemethod usingthegimbalnullmotioncanreduce/eliminatetheamountoftimethattheVSCMG hastooperateinRWmodewhentheCMGJacobianbecomessingular.Accordingto whetherthetotalangularmomentumvectorisinsideoroutsidethemomentumenvelope, thesingularstateisdenedasinternalorexternal.TheCMGsystemencountersexternal saturationsingularitieswhentheindividualangularmomentumhasmaximummagnitudeforitsdirection.Internalsingularitiescanbeclassiedintoellipticorhyperbolic singularitybywhetherthenullmotionispossiblearoundthesingularstate.Thenull motionscanbegeneratedatthehyperbolicsingularitybutnotattheellipticsingularity[5,52,55,64,70,91,96].However,thefactthatthenullmotionexistsdoesnotguarantee escapefromthehyperbolicsingularity.Therearedegeneratesolutionswhichdonotaect therankoftheCMGJacobian.Thismeansthatthedegeneratehyperbolicsingularities cannotbeescapedthroughnullmotion[5,55,64,96].Ifthespecictypeofsingularitycan bedetermined,theVSCMGcanacquiremoreeectiveperformancesincetheVSCMGcan makethebestuseofthetorqueamplicationinCMGmodeandalsoutilizethewheel nullmotionse.g.,start-up,powerreduction,etcwhileproperlyrespondingeachtypeof singularityaswellasholdingpreciseattitudecontrol. Inthischapter,anewsingularitydetectionmethodisdevelopedusingafuzzy logicsystemFLS.TheFLS-basedsingularitydetectionandclassicationmethodcan determineaspecictypeofsingularityusingthenecessaryconditionforthenullmotion andasingularitymeasure.Numericalsimulationsdemonstratetheperformanceofthe 122

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adaptiveVSCMGsteeringlawwithFLS-basedsingularitydetectionmethodintheecacy ofsingularityavoidancewithreducedreactionwheelmodes. 7.1SingularityDetectionStrategy ForthepyramidalarrangementoftheVSCMGsystem,allinternalsingularitescan beescapedthroughtorquegenerationinRWmodewhilemaintainingpreciseattitude trackingperformance.Thepossibilityofescapebynullmotionnearsingularitythathas beenstudiedin[5,55,64,96]canbeadesirablecriteriatodistinguishsingularstates. However,thesingularregionsinsidethemomentumenvelopeinFigure2-3arenonlinear andcomplex.Fordegeneratehyperbolicsingularities,thesurfacelookslikeahyperbolic singularitywhichcanbeescapedbythenullmotions,buthaveafeatureofanelliptic singularityi.e.,thenullmotionsdonotaecttherankofCMGJacobian..Inthis singularity,thenulldisplacementsdonotallowthesingularcongurationtobedisturbed sincethesingularstateislocatedonapointonnulltrajectoriesforasetofgimbalangle. Hence,singularityescapebytheadmissiblenullvariationsisnotpossible.Basedonthe giveninformationsuchasthepassabilityconditionbythenullmotionnearsingularity andthesingularitymeasureindex,aFLScanbeanappropriatetool.AFLScandeal withnonlinearandcomplexproblemsintherealmsofsearch,question-answeringdecision andcontrol,andprovidesafoundationforthedevelopmentofnewtoolsfordealingwith linguisticinformationandknowledgerepresentation[7,71,105109].Thebasicstructureof aFLSiscomposedofafuzzier,fuzzyproductinferenceengine,anddefuzzier. 7.1.1PassabilityConditionbyNullMotionnearSingularity FortheVSCMGcase,anullmotionalwaysexistssinceaVSCMGJacobianinduced byacombinationof 4 gimbalsand 4 wheelsalwaysspansthethree-dimensionalspacei.e., therankdeciencyoftheVSCMGJacobianisescapable.[101].Thepassabilitycondition bynullmotionsnearsingularityisusedinCMGmodeoftheVSCMGsystemwhich maintainstorqueamplication.Thetestforexistenceofthenullmotionisestablishedby aTaylorseriesexpansionaboutasingularstate s 2 R 4 [5,55,64,96].Supposethat s is 123

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asingularpointand H s 2 R 3 isonthesingularsurface,asecondorderTaylorseries expansionof H intheneighborhoodyields H = H s + d )]TJ/F23 11.9552 Tf 11.955 0 Td [(H s = 4 X i =1 @h i @ i i + 1 2! @ 2 h i @ 2 i 2 i + ; where i isthegimbalangledisplacementbetweenagimbalangle i andasingularpoint s i .Byusingthefollowingrelations @h i @ i = f i and @ 2 h i @ 2 i = )]TJ/F23 11.9552 Tf 9.299 0 Td [(h i ; thenullmotionconstraintcanbeexpressedas H = 4 X i =1 f i i )]TJ/F15 11.9552 Tf 14.777 8.087 Td [(1 2! h i 2 i + =0 3 1 ; where H 0 nearasingularitysincethetotalangularmomentum H isnotaectedby nullmotion. Toobtaintheconstraintequationfornullmotion,theinnerproductof H withan arbitrarysingularvector u 2 R 3 is u H ')]TJ/F15 11.9552 Tf 31.381 8.088 Td [(1 2! u 4 X i =1 h i 2 i = )]TJ/F15 11.9552 Tf 12.12 8.088 Td [(1 2! 4 X i =1 u h i 2 i ; where u = null )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(C T T and u f i =0 i =1 ; 2 ; 3 ; 4 bythedenitionofasingularvector. Hence,thesecondordernecessaryconditionfornullmotioniswrittenas 4 X i =1 P 2 i =0 ; where P = u h i denotestheprojectionmatrix.Inmatrixform,theconditioncanbe rewrittenas T P =0 : 124

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Byusingthenull-spacebasisvector n i 2 R 4 oftheCMGJacobianmatrix C T 2 R 3 4 thenullmotionofgimbalanglescanberepresentedas = 4 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 X i =1 c i n i = Nc; where c isasetofweightingcoecientsdenotedas c = c 1 ;c 2 ,andthenullspace N 2 R 4 [4 )]TJ/F24 7.9701 Tf 6.586 0 Td [(rank C T ] iswrittenas N C T = n i 2 R 4 j C T n i =0 3 1 : Substituting7into7,anecessaryconditionfornullmotionisobtainedas c T Qc =0 ; where Q = N T PN .Thequadraticformof7representsanullmotionconstraint equationinthevicinityofasingularstate,andcanbeasingularityclassicationcriteria accordingtopropertiesof7:1denite Q ,or2indeniteorsingular Q [5,96].If Q isadenitematrix, c =0 istheonlysolutionandthenullmotionisnotpossible.The denitematrix Q indicatesanellipticsingularitywhichhasnonullmotions.When Q is anindeniteorsingularmatrix, Q indicatesahyperbolicsingularityandthenullmotion ispossible.Althoughtheclassicationcriteriabasedon Q impliestheexistenceofnull motion,themerepossibilityofnullmotiondoesnotguaranteeescapefromasingularity. DegeneratehyperbolicsingularitieswhichdonotaecttherankoftheCMGJacobian mustbeexcluded[5,96].Forthedetectionofdegeneratehyperbolicone,theconventional singularitymeasureindexdenotedas f )]TJ/F15 11.9552 Tf 11.291 0 Td [(det )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(C T C T T canbeusedtoprovide additionalinformation.Suchconvolutedcriteriatodetermineatypeofsingularitycanbe eectivelyclassiedbyIF-THENrulesofFLS. 7.1.2Fuzzication Therststepinfuzzylogicsystemistoconvertthemeasuredsignalsintoasetoffuzzy variables.Specically,thefuzzicationprocessturnseach 125

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sign multiplicationofeigenvalues of Q in7forthenullmotionpassabilityconditions nearsingularityandameasuredconventionalsingularityindex f intoasetoffuzzy variables.Basedonwhethernullmotionsarepossible,thetypeofsingularityis determinedas sign multiplicationofeigenvalues > 0 ; Ellipticsingularity sign multiplicationofeigenvalues 0 ; Hyperbolicsingularity : However,inthiscriteria,it'simpossibletodetectdegeneratehyperbolicsingularity. Asanadditionalinformationfordegeneratehyperbolicsingularity,theconventional singularitymeasureindex f isused.Thisinformationsupplementsthenullmotion passabilityconditionof7.Forexample,letasingularstatebeincludedtoacriterion ofhyperbolicsingularity.However,ifthesingularitymeasureindexstaysaroundzero i.e.,eventhoughthenullmotionsexist,therankofCMGJacobianisstilldecientand itoccursthevariationofangularmomentumbythetorquegenerationofRW,itcanbe consideredtobeadegeneratehyperbolicsingularity.TheintegratedIF-THENrulesfor singularitydetectionareasfollows: IF sign multiplicationofeigenvalues > 0 ; THENEllipticSingularity IF sign multiplicationofeigenvalues 0 AND f< 0 : 3 ; THENHyperbolicSingularity IF sign multiplicationofeigenvalues 0 AND f< 0 : 1 ; THENDegenerateHyperbolicSingularity ; wherefordetectingellipticsingularity,itisnotessentialforthesingularitymeasureindex f tobeusedsincetheIFclauseof7clariesthespecictypeofsingularity.The degeneratehyperbolicsingularitywhichdoesnotaecttherankofCMGJacobianmakes theindex f approachtozeroalthoughthenullmotionexists.Hence,IF-THENrules ofFLSin7detectsatypeofsingularityandsingularitymeasure,andbyusingthe determinedsingularityidentitythesteeringlawcandetermineanecientoperationmode. 126

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Figure7-1.Membershipfunctionforsingularitymeasureindex f Specically, f < 0 : 3 in7isdeterminedasareferencetostartgimbalnullmotions; otherwise,thesteeringlawdoesnotneedevengimbalnullmotionsbutneedsCMGmode asaminimumnormpseudo-inversesolution.Also,in sign multiplicationofeigenvalues 0 and f < 0 : 1 ,thesingularityisreferredtoasadegeneratehyperbolicsingularity andthesteeringlawrequiresthetorquegenerationofRW.Thesingularitymeasureand thresholdreferenceforreninghyperbolicsingularityaredeterminedandfuzziedby membershipfunctionsasshownbyFigure7-1.Thedottedlinefuzziesthesingularity measureindex f toadditionallydetectdegeneratehyperbolicsingularityandthe regionintersectedbytwotrianglesreectsfuzzinessbetweenhyperbolicsingularityand degeneratehyperbolicsingularityinthesingularitymeasureindex f .Specically,when asingularityisdeterminedashyperbolicsingularity sign multiplicationofeigenvalues 0 andthereexistsnullmotions,thefactthatsingularitymeasureindex f stilldecrease to 0 : 1 under 0 : 3 allowsRWtoinducetorquegenerationfollowingtheconsecutiveFLS process. 127

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Figure7-2.BlockdiagramofFLS-basedsingularitydetectionmethod. 7.1.3ProductInferenceEngine TheFLS-basedsingularitydetectionmethodusesfuzzyequivalentsoflogical ANDoperationstobuildupfuzzylogicrules.If sign isthemembershipofclass sign multiplicationofeigenvalues forindicatingatypeofsingularityand f i isthe membershipofclass f i foraconventionalsingularitymeasure,thenthefuzzyANDis obtainedasthemultiplicationofthemembershipvalues: signANDf i = sign ^ f i = sign f i wherethesymbol ^ isusedtodenotethefuzzyANDoperationand sign hasavalueof 0 or 1 i.e.,Ifa sign in7isdetermined, sign has 1 7.1.4Defuzzication Thelaststepinbuildingafuzzylogicsystemisturningthefuzzyvariablesgenerated bythefuzzylogicrulesintoasingularitydetectionindex.Thedefuzziercombinesthe informationinthefuzzyinputssothatitobtainsasinglecrispoutputvariableusingthe centerofgravitymethod.Tobemorespecic,ifthefuzzylevelsgivenin7have membershipvalues sign and f i ,thenthecrispoutputsignal D j isdenedas D j = P k i =1 W i i P k i =1 i 128

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where D j j =1 ; 2 isadefuzziedvalue: D 1 and D 2 areresultingfromtherulesfor ellipticsingularityandhyperbolicsingularity,respectively, i isafuzziedvaluelike sign f i ,and signANDf i resultingfromtheIF-THENruleortheproductinferenceengine, W i isaweightvaluedesignedbyheuristicinformation,and k isthenumberoffuzzied values.AnewsingularitydetectionindexasseeninFigure7-2isdenedas = sech D 1 + D 2 + ; where denotesashiftconstanttoadjustafunctionoutputsignal. 7.2ImplementationofFLS-basedsingularitydetectionindex Basedontheproposedsingularitydetectionindex t ,thecompositeVSCMG steeringlawofChapter5canbewrittenas = ^ Q + w Y 1 ^ 1 + kr + e v + C G [ I W ] d [ T ] d )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T [ I W ] d [ G ] d + I 8 )]TJ/F15 11.9552 Tf 14.639 3.022 Td [(^ Q + w ^ Q S; whereamodeweightmatrix W isdesignedas W 2 6 4 W I 4 4 0 4 4 0 4 4 W I 4 4 3 7 5 ; where W 2 R isdenedas W W 0 ; W 0 ;W 2 R arepositiveconstants,andthenewsingularitydetectionindex t is obtainedas7.Sincetheindex t detectsthesingularitytypeandpropertyof thesingularity,theywheelscangeneratetherequiredtorqueinRWmodeatelliptic singularityanddegeneratehyperbolicsingularitybasedontheFLS-basedsingularity detection.In7,togeneratenullmotionforinternalmomentumtrackingandgimbal reconguration,thenullmotion t 2 R 8 isdenedas k w g T k @ @ T T ; 129

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where k w k 2 R denotepositiveconstants, g t 2 R 4 isanauxiliarycontrolsignalto trackthedesiredywheelangularmomentum,andthesecondrowisthegradientmethod forthegimbalreconguration.Thematrix S 2 R 8 8 in7isanullmotionweight fortheVSCMGnullmotiondenotedas S I 8 8 ; where I 8 8 2 R 8 8 isanidentitymatrix.Whenthesingularitydetectionindexdetects ellipticsingularityordegeneratehyperbolicsingularity,theindex t inhibitsboth gimbalandwheelnullmotionssothattheVSCMGcanbeoperatedasRW.When encounteringhyperbolicsingularityandinnormalworkspace,thesteeringlawworksfora conventionalCMGwiththetorqueamplicationandinternalmomentumtracking. 7.3NumericalExamples 7.3.1SimulationSetup Numericsimulationsillustratetheperformanceofthedevelopedcontroller.The satelliteparametersandinitialconditionsarebasedonamodelofaprototypepicosatelliteinTable5-1and5-2ofChapter5.Theinitialanddesiredywheelspeedfor eachwheelare 0 = d =200 rad=s 2 ; 000 rpm .Thesimulationparametersfor defuzzicationare W 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ;W 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 ; =3 where W 1 and W 2 areweightvaluesforeach membershipfunctionofFigure7-1,and isashiftconstant.Dierentlyfromtheinitial startupofChapter5,thewheelnullmotionachievestheinternalmomentumtrackingto maintainthedesiredywheelspeedsothattheVSCMGsystemcansteadilygaintorque amplicationinCMGmode. 7.3.2SimulationResults ThesimulationresultsaredevelopedtoshowtheperformanceoftheFLS-based singularitydetectionindexfortwocases.Toillustratetheperformanceofthesingularity detectionandescapeatinternalellipticsingularity,Case1isstartedataninternalelliptic singularpointdenotedas = )]TJ/F15 11.9552 Tf 9.299 0 Td [(900900 deg andtheresultsaregivenin 130

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Figure7-3.Quaterniontrackingerror e t forCase1. Figures7-3-7-9.Case2isstartedataninternalhyperbolicsingularpointdenotedas = 090180 )]TJ/F15 11.9552 Tf 9.298 0 Td [(90 deg andtheresultsaregiveninFigures7-10-7-13. Figure7-3showsthequaterniontrackingerrorresultsduringa 100 s simulation period.Figure7-4showsthecontrolinputgimbalrates t andwheelaccelerations t .Thewheelaccelerationcontrolinput t contributestotheinternalmomentum managementbyequalizingthewheelspeedafterescapingfromtheinternalelliptic singularityataninitialpointwhilethecontrolinputsachieveattitudestabilization. SincethesimulationforCase1startsataninternalellipticsingularpoint,the det Q regardedasmultiplicationofeigenvaluesshowsapositivesigninFigure7-6andthe singularitymeasureindex f showsthestartat 0 inFigure7-7.Also,theFLS-based singularitydetectionindex t inhibitsbothgimbalandwheelnullmotionswhileit allowsatorquegenerationofRWasshownbyFigure7-6,7-7,and7-5.OncetheCMG Jacobianescapestheinternalellipticsingularity,thegimbalnullmotionshelpthesystem escapethesingularitywhilethewheelnullmotionsachievewheelspeedequalization internalmomentumtrackingerror t 0 asshowninFigure7-8.Figure7-9shows theelementsoftheadaptiveparametervectors ^ 1 t and ^ 2 t 131

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Figure7-4.Controlinputgimbalrates t andwheelaccelerations t forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse. Figure7-5.Flywheelspeed t forCase1. 132

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Figure7-6.Singularitydetectionindex t andnullmotioncondition det Q forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse. Figure7-7.Singularitymeasureindex f ,nullmotionweight S ,andmodeweight W forCase1. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse. 133

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Figure7-8.Nullmotionresultsforgimbalrecongurationandwheelspeedtrackingerror t forCase1. Figure7-9.Parameterestimates ^ 1 t ^ 2 t forCase1. 134

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Figure7-10.Singularitydetectionindex t andnullmotioncondition det Q forCase 2. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse. Figure7-11.Singularitymeasureindex f ,nullmotionweight S ,andmodeweight W forCase2. Theleftcolumnillustratestheresponsefortheentireduration,andtherightcolumn illustratesthetransientresponse. 135

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Figure7-12.Nullmotionresultsforgimbalrecongurationandwheelspeedtrackingerror t forCase2. Figure7-13.Flywheelspeed t forCase2. 136

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ThesimulationresultsforCase2startedataninternalhyperbolicsingularpoint areproviedinFigure7-10-7-13.InFigure7-10,the det Q showingthenullmotion conditionillustratesanegativesignatthestartingpointwhichmeansthestartfromthe internalhyperbolicsingularity.Thesingularitydetectionindexshowstwovariationswhich areshowninFigure7-10.Thesecondvariationisresultingfromaccesstoanothersingular pointrightafterescapingfromtheinternalhyperbolicsingularity.Correspondingtothe secondvariationof t and f ,thesingularityencounterisescapedbythegimbalnull motionswithoutthetorquegenerationofRWshowninFigure7-12and7-13. 7.4Summary Inthischapter,aFLS-basedsingularitydetectionmethodisdevelopedforthepyramidalarrangementoftheVSCMGsystem.TheFLSfuzziesthepossibilitybynull motionnearsingularitytoprimarilyclassifysingularityintoellipticandhyperbolicone, andthenuseadditionalinformationdenotedastheexistingsingularitymeasureindexto evendetectdegeneratehyperbolicsingularity.Basedonthedeterminedsingularityidentity,thesteeringlawcandetermineanecientoperationmodedependingonsingularity detectioninputandsteadilymaintaintorqueamplication.Numericsimulationshave twocasesdividedintoellipticandhyperbolicsingularitycase,andeachcaseillustrates boththesingularitydetectionperformanceandthesingularityavoidanceoftheFLS-based singularitydetectionmethod. 137

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CHAPTER8 CONCLUSIONSANDFUTUREWORK 8.1Conclusions TomakethebestuseofanextraDOFprovidedbyVSCMGs,variousmultifunctionalsteeringlawsaredevelopedinChapters3-6.SincetheVSCMGsystem includessatelliteandactuatoruncertaintiessuchasdynamicandstaticfriction,inertia, etc.,thedicultiesresultingfromuncertainpropertiesaremitigatedthroughinnovative developmentoftheerrorsystemalongwithaLyapunov-basedadaptivelaw.ThepreviouschaptersprovidetheLyapunov-basedstabilityanalysistoproveprecisionattitude trackingwhilesimultaneouslyachievingadditionaltrackingobjectivesdenotedaspower trackingandinternalmomentumtracking.Chapter7promotestheeectiveutilizationof hybridmodei.e.,CMGmodeandRWmoderesultingfromanextraDOFofVSCMGs. AlthoughtheVSCMGisgeometricallyasingularity-freedevice,thesingularityavoidance methodusingthegimbalnullmotioncanreduce/eliminatetheamountoftimethatthe VSCMGhastooperateinRWmodewhentheCMGJacobianbecomessingular.Moreover,detectingaspecictypeofsingulairtycanprovidethemaximumuseofCMGmode toutilizetorqueamplicationandastable,ecientsingularityavoidanceceect.Hence,a FLS-basedsingularitydetectionmethoddevelopedinChapter7distinguishesthespecic typeofsingulairty,whicharbitratesgimbalandwheelnullmotionssothatallinternal singularitiescanbeescaped/avoided. InChapter3,indynamicandstaticfrictionintheVSCMGgimbalsandwheels,the controlleriscapableofachievingGUUBattitudetrackingwhilesimultaneouslytrackinga desiredpowerproleasymptotically.Inaddition,thecontrollercompensatesfortheeects ofuncertain,time-varyingsatelliteinertiaproperties.Thewheelnullmotionsresulting fromtheextendedDOFofVSCMGsallowtheVSCMGsystemtoaccomplishanovel combinedobjectiveasprecisionattitudetrackingandpowerstoragei.e.,mechanical 138

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battery.TheattitudeandpowertrackingresultsareprovenviaLyapunovstability analysisanddemonstratedthroughnumericalsimulations. InChapter4,themajorityofcontrolresearchfocusedonVSCMGshasassumedideal conditionssuchasfrictionlessywheelandgimbalbearingsandasystemofVSCMGsas arigidbody.WhenscalingthesizeofCMGs/VSCMGs,theeectsoffrictionpresentin thesystemaresignicant.Toactivelyconsiderfrictioneectinsidedynamics,acoupled dynamicsconnectionofasatellite,gimbals,andwheelsisdeveloped.Abackstepping methodisusedtodevelopthecontrollerfromacascadedynamicsconnectionfora VSCMG-actuatedsatellite.Inthepresenceofuncertaindynamicandstaticfriction inboththegimbalsandtheywheels,thecontrolleriscapableofachievingglobally asymptoticalattitudetrackingwhilesimultaneouslyperformingsingularityavoidanceand wheeldeceleration.Simulationsshowthattheappliedtorquesofthewheelscontaining frictioncontributetopowerreductioninthatthefrictionenablesthewheeltoobtainmore torqueswithoutanadditionaltorquerequest.Suchbenetisinducedbythedeceleration moderesultingfromthenullmotionandcangivetheactuatorbothtorqueandpower reductioneect. PreviousspacemissionsusingCMGshaveusedaseparatefeedbackcontrolloop tospinuptherotortotherequiredspinrateandmaintainitwhilesecuringattitude stabilizationusingadditionaldevicessuchasmagnetorquers.InChapter5,theVSCMG steeringlawincludingtheinternalmomentummanagementallowstheywheeltostart fromrestandtoreachthedesiredspeed.Inthepresenceofsatelliteinertiauncertainty andactuatoruncertainty,thedevelopedattitudecontrollerinthischapteriscapableof achievingglobalasymptoticattitudetrackingwhilesimultaneouslyperformingsingularity avoidanceandinternalmomentummanagement.Thesignicantbenetofthedeveloped steeringlawistocondenseseveraldiscontinuous,separatefeedbackcontrolstepssuch astheinitialstart-upandinitialattitudeacquisitionmodeintoonecontinuousand simultaneouscontrolstep.Theattitudetrackingandinternalmomentumtracking 139

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resultsareprovenviaaLyapunovstabilityanalysisanddemonstratedthroughnumerical simulations. Chapter6showsaRNN-basedcontroltechniquewhichachievesattitudetrackingfor aVSCMG-actuatedsatelliteinthepresenceofuncertaintyinthesatelliteandactuator dynamicsandunmodeledexternaldisturbances.ThecapabilityofRNNmodelingto evolvethestatescorrespondingtononlinearstateequationsisexploitedtocompensate foractuatoruncertaintiesofVSCMGs.ALyapunov-basedstabilityanalysisisusedto provethecontrollerachievesUUBattitudetrackingwhilecompensatingfortheeectsof uncertaintime-varyingsatelliteinertiaproperties,parametricuncertainty,andnonlinear externaldisturbancetorques.Numericsimulationresultsillustrateperformanceofthe RNNestimatorwhichfunctionsasacompositeVSCMGsteeringlawaswellasbenets providedbygimbalandwheelnullmotions. InChapter7,aFLS-basedsingularitydetectionmethodisdevelopedforthepyramidalarrangementoftheVSCMGsystem.SincetheFLScopeswithcomplexandnonlinear patternsofsingularity,theFLSprovidesaneectivesingularitydetectionstrategyconsideringthepassabilityconditionforthenullmotionandthesingularitymeasureindex.The FLSfuzziesthepassabilityconditionofthenullmotiontoprimarilyclassifysingularity intoellipticandhyperbolicone,andthenuseadditionalinformationdenotedastheexistingsingularitymeasureindextodetectadegeneratehyperbolicsingulairty.Basedonthe determinedsingularityidentity,thesteeringlawcandetermineanecientoperationmode dependingonsingularitydetectioninputandsteadilymaintaintorqueamplication.The developedsingularitydetectionmethodscanescapeallofinternalsingularitiesincluding degeneratehyperbolicsingularityandthismethodistherstresultthatcanescapeall internalsingularitiesforthepyramidalarrangement.Numericsimulationshavetwocases dividedintoellipticandhyperbolicsingularitycase,andeachcaseillustratesboththesingularitydetectionperformanceandthesingularityavoidanceoftheFLS-basedsingularity detectionmethod. 140

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8.2FutureWork TheIPACSmethodinChapter3needsaelectronicdevicetostoremechanicalenergy inpractice.Itisimportanttocomparefrictionenergylossofwheelwithkineticenergy growthtograspreliablityofIPACS.InthesamemannerofChapter4,IPACSmethed canalsoconsideractuatorfrictioninaphysicaltorquelevel.ForChapter4,thesteering lawisdevelopedbyacombinationbetweengimbalrateandwheelspeed.Apracticaltest forthissteeringlawcanillustratearealisticeectaboutpowerreductionindespinning modeandgiveacomparisonwithasteeringlawdevelopedbygimbalrateandwheel acceleration.ConsideringexternaldisturbancefortheinitialstartupmethodinChapter 5,thestart-upmethodcanprovidemoreattractiveresultstousethestartupsteering lawinpractice.Thequantiedinvestigationofstart-upmethodusingjustonefeedback loopcanillustrateviableeectsofthestart-upmethoddevelopedinChapter5comparing withotherstart-upmethodsusingseparatefeedbackloopandextradevices.Hence,the hardwareimplementingandcomparisonstudyarerequiredtoensurereliabilityforvarious multi-functionalsteeringlaws. OneofshortcomingsoftheRNN-basedsteeringlawisthatUUBstabilityisachieved. Furtherinvestigationwhichincludesaddtionofrobustfeedbackcontroltermorother intelligentmethodisneededtoimprovethestabilityresult.Avenuestoobtainasemiglobalasymptoticresultareprovidedin[44,80,81]underasetofassumptions. TheFLS-basedsingularitydetectionmethoddevelopedinChapter7canbealso appliedtoSGCMGwithoutlossofgenerality.Hence,thedevelopmentofCMGsteering lawutilizingtheFLS-basedsingularitydetectionmethodisneededandfutureeortscan focusoncomparingthemethodofChapter7withavarietyofestablishedCMGsteering laws. 141

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BIOGRAPHICALSKETCH DoheeKimwasborninSeoul,Korea.HereceivedhisBachelorofSciencedegree in2006andhisMasterofSciencedegreein2008intheDepartmentofAstronomyfrom theYonseiUniversity,Korea.Inhismaster'sdegreethesis,hedesignedanddeveloped Hardware-In-the-LoopHILsimulatorforspacecraftattitudecontrol.Hethenjoinedthe NonlinearControlsandRoboticsNCRresearchgroupoftheDepartmentofMechanical andAerospaceEngineering,UniversityofFloridaintheFallof2008topursuehisdoctoral researchundertheadvisementofWarrenE.Dixon. 150