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Lyapunov-Based Neuromuscular Electrical Stimulation Control

Permanent Link: http://ufdc.ufl.edu/UFE0044949/00001

Material Information

Title: Lyapunov-Based Neuromuscular Electrical Stimulation Control
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Wang, Qiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: closed-loop -- control -- electrical -- lyapunov-based -- neuromuscular -- stimulation
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Neuromuscular electrical stimulation (NMES) or functional electrical stimulation (FES) is a widely used technique for rehabilitation and restoration of motor function. Millions of people suffering from disability and paralysis caused by neural disorders such as a stroke, spinal cord injury, multiple sclerosis, cerebral palsy, or traumatic brain injury can benefit from NMES. Open-loop methods, using grouped electrical pulses with fixed parameters, are widely accepted in clinical settings, primarily for strength training related rehabilitation treatments. The development of closed-loop NMES can provide new rehabilitation treatments where accurate limb movement is essential. Contributions of this dissertation result from the development of robust closed-loop NMES controllers that account for uncertainties and nonlinearities in the muscle and activation dynamics. Specifically, this dissertation examines an optimal trade-off between performance and muscle fatigue, the effects of modulating the control input, the effects of time delay in the muscle contraction, and switching controllers during the gait cycle. In Chapter 2 and 3, inverse and direct optimal NMES controller are designed which consider potential overstimulation by NMES controllers. Muscle fatigue is a multiple-factor problem which affects all aspects of NMES. Overstimulation is an avoidable fact that leads to early onset of muscle fatigue. An optimal control framework provides practitioners a useful tool to balance between stimulation dosage and tracking performance. Experiments are provided to illustrate the performance. In Chapter 4, a muscle activation model with a pulse modulated control input is developed to capture the discontinuous nature of muscle activation, and a closed-loop NMES controller is designed for the uncertain pulse muscle activation model. The pulse modulated control input in the model results in an explicit condition that relates performance, pulse magnitude, and pulse frequency. Higher frequency results in more rapid muscle fatigue. Given the important role of modulation frequency in managing muscle fatigue, this contribution illustrates how stimulation frequency can be included in the analysis of the closed-loop controller design, which provides a starting point for designing more frequency efficient NMES controllers. Experiments are provided to illustrate the performance. In Chapter 5, an identifier based control structure is developed. Muscle force output from electrical stimulation exhibits large time delays from the muscle contraction dynamics. Previous results in literature had to use known (or guessed) model parameters to compensate for the muscle contraction dynamics. By using acceleration feedback uncertain muscle contraction model can be used for closed-loop controller design. The use of limb acceleration is problematic for control implementation due to noise. In this chapter, a closed-loop controller is developed which can be implemented only using position and velocity signals. In Chapter 6, by combining the approaches in Chapter 4 and Chapter 5, an identification-based controller is developed which includes an uncertain muscle contraction dynamics with pulse modulated control input. The controller is implemented only using position and velocity signals and the pulse modulation effect is included in the analysis. Experiments are provided to illustrate the performance. Ankle motion is important for maintaining normal gait. For individuals who lost their ability to control the ankle, NMES can be used to help restore normal gait. In Chapter 7, a sliding mode based controller is developed to control ankle motion during gait. Ankle motion is modeled as a hybrid system and a switched sliding mode controller is designed to enable the ankle to track desired trajectories during gait.
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 Qiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044949:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044949/00001

Material Information

Title: Lyapunov-Based Neuromuscular Electrical Stimulation Control
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Wang, Qiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: closed-loop -- control -- electrical -- lyapunov-based -- neuromuscular -- stimulation
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Neuromuscular electrical stimulation (NMES) or functional electrical stimulation (FES) is a widely used technique for rehabilitation and restoration of motor function. Millions of people suffering from disability and paralysis caused by neural disorders such as a stroke, spinal cord injury, multiple sclerosis, cerebral palsy, or traumatic brain injury can benefit from NMES. Open-loop methods, using grouped electrical pulses with fixed parameters, are widely accepted in clinical settings, primarily for strength training related rehabilitation treatments. The development of closed-loop NMES can provide new rehabilitation treatments where accurate limb movement is essential. Contributions of this dissertation result from the development of robust closed-loop NMES controllers that account for uncertainties and nonlinearities in the muscle and activation dynamics. Specifically, this dissertation examines an optimal trade-off between performance and muscle fatigue, the effects of modulating the control input, the effects of time delay in the muscle contraction, and switching controllers during the gait cycle. In Chapter 2 and 3, inverse and direct optimal NMES controller are designed which consider potential overstimulation by NMES controllers. Muscle fatigue is a multiple-factor problem which affects all aspects of NMES. Overstimulation is an avoidable fact that leads to early onset of muscle fatigue. An optimal control framework provides practitioners a useful tool to balance between stimulation dosage and tracking performance. Experiments are provided to illustrate the performance. In Chapter 4, a muscle activation model with a pulse modulated control input is developed to capture the discontinuous nature of muscle activation, and a closed-loop NMES controller is designed for the uncertain pulse muscle activation model. The pulse modulated control input in the model results in an explicit condition that relates performance, pulse magnitude, and pulse frequency. Higher frequency results in more rapid muscle fatigue. Given the important role of modulation frequency in managing muscle fatigue, this contribution illustrates how stimulation frequency can be included in the analysis of the closed-loop controller design, which provides a starting point for designing more frequency efficient NMES controllers. Experiments are provided to illustrate the performance. In Chapter 5, an identifier based control structure is developed. Muscle force output from electrical stimulation exhibits large time delays from the muscle contraction dynamics. Previous results in literature had to use known (or guessed) model parameters to compensate for the muscle contraction dynamics. By using acceleration feedback uncertain muscle contraction model can be used for closed-loop controller design. The use of limb acceleration is problematic for control implementation due to noise. In this chapter, a closed-loop controller is developed which can be implemented only using position and velocity signals. In Chapter 6, by combining the approaches in Chapter 4 and Chapter 5, an identification-based controller is developed which includes an uncertain muscle contraction dynamics with pulse modulated control input. The controller is implemented only using position and velocity signals and the pulse modulation effect is included in the analysis. Experiments are provided to illustrate the performance. Ankle motion is important for maintaining normal gait. For individuals who lost their ability to control the ankle, NMES can be used to help restore normal gait. In Chapter 7, a sliding mode based controller is developed to control ankle motion during gait. Ankle motion is modeled as a hybrid system and a switched sliding mode controller is designed to enable the ankle to track desired trajectories during gait.
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 Qiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044949:00001


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LYAPUNOV-BASEDNEUROMUSCULARELECTRICALSTIMULATIONCONTROL By QIANGWANG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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2012QiangWang 2

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ACKNOWLEDGMENTS Iwouldliketosincerelythankmyadviser,WarrenE.Dixon,forhelpingmesuccessfullygothroughthejourneyofmyPhD.Thanksfortheguidancehehasprovided overtheyears,whichledmeenteringtheresearcheldandencouragedmewhenIgot frustrated.Thanksforthepatience,timeandeffortshehasgiventohelpmemature inmyresearch.Thanksforhisrolemodelsasaadviser,ateacher,acolleagueanda friend,whichwillbenetmeintheyearscoming.Iwouldalsoliketoextendmygratitude toDr.Gregory,Dr.Sharma,Dr.Johnson,Kamalapurkar,Dinh,Downey,Bellman,andall mycolleaguesinNCRforthevarioushelpsfromtechnicaldiscussions,commentsand reviews,andwarmwordsandsmilestheyprovided.Iwouldliketothankmywifeforher endlesslove,support,andpatience.Also,Iwouldliketothankmyparentsandsons, whoaremyconstantsourceforloveandstrength. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................3 LISTOFFIGURES.....................................6 ABSTRACT.........................................8 CHAPTER 1INTRODUCTION...................................11 1.1Motivation....................................11 1.2Contribution...................................17 1.3Outline......................................19 2ADAPTIVEINVERSEOPTIMALNEUROMUSCULARELECTRICALSTIMULATION........................................20 2.1MuscleActivationandLimbModel......................20 2.2ControlDevelopment..............................22 2.3StabilityAnalysis................................24 2.4CostFunctionalMinimization.........................26 2.5ExperimentResults...............................28 2.5.1TrackingExperiments..........................30 2.5.2PerformanceTrade-offs.........................31 2.5.3ChangingGravityLoad.........................32 2.6Discussion...................................34 3ASYMPTOTICOPTIMALNEUROMUSCULARELECTRICALSTIMULATION.37 3.1ControlObjective................................37 3.2FeedbackLinearizingOptimalControlDesign................38 3.3AdaptiveControlDesign............................41 3.4StabilityAnalysis................................45 3.5ExperimentResults...............................48 3.5.1TrackingExperiments..........................48 3.5.2PerformanceTrade-offs.........................48 3.6Discussion...................................49 4NEUROMUSCULARELECTRICALSTIMULATIONLIMBTRACKINGWITH APULSEDMODULATEDCONTROLINPUT...................53 4.1MuscleActivationandLimbModel......................53 4.2ControlDevelopment..............................55 4.3StabilityAnalysis................................56 4.4Experiments...................................60 4

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5NEUROMUSCULARELECTRICALSTIMULATIONWITHAUNCERTAINMUSCLECONTRACTIONMODEL...........................66 5.1MuscleActivationandLimbModel......................66 5.2ControlDevelopment..............................66 5.3ObserverDesign................................70 5.4StabilityAnalysis................................74 5.5Simulation....................................77 6IDENTIFICATION-BASEDCLOSED-LOOPNEUROMUSCULARELECTRICALSTIMULATIONLIMBTRACKINGWITHAPULSEDMODULATEDCONTROLINPUT.....................................80 6.1MuscleActivationandLimbModel......................80 6.2ControllerDevelopment............................80 6.3ObserverDesign................................83 6.4StabilityAnalysis................................83 6.5Experiments...................................87 7HYBRIDNEUROMUSCULARELECTRICALSTIMULATIONTRACKINGCONTROLOFANKLE..................................91 7.1MuscleActivationandLimbModel......................91 7.2ControlDesign.................................93 7.3StabilityAnalysis................................94 7.4Simulation....................................98 8CONCLUSIONANDFUTUREWORK.......................101 8.1Conclusion...................................101 8.2FutureWork...................................103 REFERENCES.......................................105 BIOGRAPHICALSKETCH................................111 5

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LISTOFFIGURES Figure page 2-1Trackingerrorforarepresentativetrial.......................31 2-2Steptestforarepresentativetrial.Thesolidlinedepictsthedesiredangle andthedottedlinedepictstheactualtrajectory..................32 2-3Experimentswith Q =1 where R variedfrom 8 to 10000 .............33 2-4Experimentswhere Q variedfrom 10 to 600 and R =2000 ............33 2-5Trajectoriesofthestandingexperiment.Thesolidlinedepictsthedesiredtrajectoryandthedashedlinedepictstheactualtrajectory.............34 3-1Trackingtrajectoriesdashedline-desired,solidline-actualforarepresentativetrialonanable-bodiedindividual........................49 3-2Trackingerrorforarepresentativetrialonanable-bodiedindividual.......50 3-3Typicalexperimentsonanable-bodiedsubjectwithvaried R from 5 to 120 for axed 1 =1 .....................................51 3-4Typicalexperimentsonanable-bodiedsubjectwithvaried 1 between 0 : 5 to 4 foraxed R =20 ...................................52 3-5Single-sidedamplitudespectrumsofthetotalcontrolinputfrom R =5 solid lineand R =90 dottedlineforthesameexperimentsinFig.3onanablebodiedperson....................................52 4-1Desiredsolidlineandmeasureddashedlinetrajectories.Thestimulation pulsefrequencyis30Hz...............................61 4-2Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. Thestimulationpulsefrequencyis30Hz......................62 4-3ControlinputVoltageforarepresentativesinusoidaltrajectorytrackingexperiment.Thestimulationpulsefrequencyis30Hz................63 4-4Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. Thestimulationpulsefrequencyis100Hz......................64 4-5ControlinputVoltageforarepresentativesinusoidaltrajectorytrackingexperiment.Thestimulationpulsefrequencyis100Hz...............64 4-6Constantvoltageresponseat30Hzsolidlineand100Hzdashedline...65 5-1Actualsolidanddesireddashedtrajectories..................77 5-2Limbpositiontrackingerror.............................78 6

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5-3Unmodulatedinputcontrolvoltage.........................78 5-4Estimatedsolidandactualdashedaccelerations...............79 6-1Performanceofarepresentativesinusoidaltrajectorytrackingexperiment. Thedesiredtrajectoryisplottedasasolidlineandthemeasuredtrajectory isplottedasadashedline..............................88 6-2Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. ThetrackingerrorisplottedasasolidlineandtheRMSerroroverevery4 secondsisplottedasadashedline.........................89 6-3ControlinputVoltageforarepresentativesinusoidaltrajectorytrackingexperiment.Thetotalcontrolinputisplottedasasolidlineandthecontribution ofNNisplottedasadashedline...........................89 6-4Trajectorytrackingperformanceofanirregulartrajectory.Thedesiredtrajectoryisplottedasasolidlineandthemeasuredtrajectoryisplottedasadashed line...........................................90 7-1Actualsolidlineanddesireddashedlinetrajectories.............99 7-2Trackingerror....................................99 7-3Computedvoltageascontrolinputforthedosiexationmusclegroup......100 7-4Computedvoltageascontrolinputfortheplantaexationmusclegroup.....100 7

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LYAPUNOV-BASEDNEUROMUSCULARELECTRICALSTIMULATIONCONTROL By QiangWang December2012 Chair:WarrenE.Dixon Major:ElectricalandComputerEngineering NeuromuscularelectricalstimulationNMESorfunctionalelectricalstimulation FESisawidelyusedtechniqueforrehabilitationandrestorationofmotorfunction. Millionsofpeoplesufferingfromdisabilityandparalysiscausedbyneuraldisorders suchasastroke,spinalcordinjury,multiplesclerosis,cerebralpalsy,ortraumaticbrain injurycanbenetfromNMES.Open-loopmethods,usinggroupedelectricalpulseswith xedparameters,arewidelyacceptedinclinicalsettings,primarilyforstrengthtraining relatedrehabilitationtreatments.Thedevelopmentofclosed-loopNMEScanprovide newrehabilitationtreatmentswhereaccuratelimbmovementisessential. ContributionsofthisdissertationresultfromthedevelopmentofrobustclosedloopNMEScontrollersthataccountforuncertaintiesandnonlinearitiesinthemuscle andactivationdynamics.Specically,thisdissertationexaminesanoptimaltrade-off betweenperformanceandmusclefatigue,theeffectsofmodulatingthecontrolinput,the effectsoftimedelayinthemusclecontraction,andswitchingcontrollersduringthegait cycle. InChapter2and3,inverseanddirectoptimalNMEScontrolleraredesigned whichconsiderpotentialoverstimulationbyNMEScontrollers.Musclefatigueisa multiple-factorproblemwhichaffectsallaspectsofNMES.Overstimulationisan avoidablefactthatleadstoearlyonsetofmusclefatigue.Anoptimalcontrolframework 8

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providespractitionersausefultooltobalancebetweenstimulationdosageandtracking performance.Experimentsareprovidedtoillustratetheperformance. InChapter4,amuscleactivationmodelwithapulsemodulatedcontrolinputis developedtocapturethediscontinuousnatureofmuscleactivation,andaclosed-loop NMEScontrollerisdesignedfortheuncertainpulsemuscleactivationmodel.The pulsemodulatedcontrolinputinthemodelresultsinanexplicitconditionthatrelates performance,pulsemagnitude,andpulsefrequency.Higherfrequencyresultsinmore rapidmusclefatigue.Giventheimportantroleofmodulationfrequencyinmanaging musclefatigue,thiscontributionillustrateshowstimulationfrequencycanbeincluded intheanalysisoftheclosed-loopcontrollerdesign,whichprovidesastartingpointfor designingmorefrequencyefcientNMEScontrollers.Experimentsareprovidedto illustratetheperformance. InChapter5,anidentierbasedcontrolstructureisdeveloped.Muscleforceoutput fromelectricalstimulationexhibitslargetimedelaysfromthemusclecontractiondynamics.Previousresultsinliteraturehadtouseknownorestimatedmodelparameters tocompensateforthemusclecontractiondynamics.Byusingaccelerationfeedback uncertainmusclecontractionmodelcanbeusedforclosed-loopcontrollerdesign.The useoflimbaccelerationisproblematicforcontrolimplementationduetonoise.Inthis chapter,aclosed-loopcontrollerisdevelopedwhichcanbeimplementedonlyusing positionandvelocitysignals. InChapter6,bycombiningtheapproachesinChapter4andChapter5,an identication-basedcontrollerisdevelopedwhichincludesanuncertainmusclecontractiondynamicswithpulsemodulatedcontrolinput.Thecontrollerisimplemented onlyusingpositionandvelocitysignalsandthepulsemodulationeffectisincludedinthe analysis.Experimentsareprovidedtoillustratetheperformance. Anklemotionisimportantformaintainingnormalgait.Forindividualswholosttheir abilitytocontroltheankle,NMEScanbeusedtohelprestorenormalgait.InChapter 9

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7,aslidingmodebasedcontrollerisdevelopedtocontrolanklemotionduringgait. Anklemotionismodeledasahybridsystemandaswitchedslidingmodecontrolleris designedtoenabletheankletotrackdesiredtrajectoriesduringgait. 10

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CHAPTER1 INTRODUCTION 1.1Motivation AnuppermotorneuronlesionUMNLleadstodisabilityandparalysis,affecting millionsofpeople.UMNLisaconditionusuallycausedbyneuraldisorderssuchasa stroke,spinalcordinjury,multiplesclerosis,cerebralpalsy,ortraumaticbraininjury.The overallreportedprevalenceis37,000people/million/year[1].SincethelowermotorneuronsystemandmusclesareintactinthosewithanUMNL,musclecontractionscanbe evokedbydirectlyapplyingelectricalstimulustothemuscles.Thistechniqueiswidely usedforrehabilitationandrestorationofmotorfunctionandisreferredtoasfunctional electricalstimulationFES,ormoregenerallyasneuromuscularelectricalstimulation NMES.ChallengesforNMEScontroldesignincludethenonlinearresponsefrom muscletoelectricalinput,loadchangingduringfunctionalmovement,unmodeled disturbances,delayedmuscleresponse,andmusclefatigue. Open-loopmethods,whichapplygroupedelectricalpulseswithxedparameters, arewidelyusedinclinicalsettings.Closed-loopNMEScontrolispromisingbasedonits abilitytoachieveaccuratelimbmovementwhichisessentialforfunctionalrehabilitation tasks.SeveralPID-basedlinearNMEScontrollershavebeendeveloped[26],but thesemethodsaretypicallybasedonanassumedlinearmusclemodelorlacka stabilityanalysis.NeuralnetworkNNbasedNMEScontrollers[721]havealsobeen developedbasedontheideathattheuniversalapproximationpropertyofNNscanbe usedtoapproximatethenonlinearunstructureddynamics.RobustNMESmethods havealsorecentlybeendevelopedin[13]and[22]thatachieveguaranteedasymptotic limbtracking. Musclefatiguehasbeendescribedasafailuretomaintaintherequiredorexpectedforcefromamuscle[23].Theonsetofmusclefatigueduringelectricalstimulationisfasterthanthatduringvolitionalcontractions,whichhinderstheapplicationof 11

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functionalandtherapeuticNMES.Biologicalfactorssuchasthereversedrecruitment orderofmotorunits,synchronousrecruitmentofmotorunitsandstimulationsettings relatedtostimulationintensity,frequency,andthegroupingofelectricalpulseshelp toexplainthepossiblemechanismsfortherapidonsetofmusclefatigueduringarticialelectricalstimulation[24].Closed-loopcontrolofmusclehasbeenproventoyield accuratelimbpositioning,butcontinuousexternalstimulationofmusclecanleadto rapidfatigueespeciallyifthecontrollerrequireshighgainstoincluderobustnessto disturbances/uncertaintyinthedynamics.Rehabilitativeproceduresseektomaximize thenumberofrepetitivesteps,somusclefatigueisacriticalconcern.Whilevarious stimulationstrategieshavebeeninvestigatedcf.[2528]suchaschoosingdifferent stimulationpatternsandparameters,improvingfatigueresistancethroughmuscleretraining,sequentialstimulation,andsizeorderrecruitment,reducingtheonsetoffatigue remainsalargelyopenresearchtopic. Closed-loopcontrolcanachievepreciselimbtrackingdespiteunpredictable perturbationsduetomusclespasmsandcentralneuralsystemCNSfeedbackloops [24].However,closed-loopmethodsoftenneedhighgaintoguaranteeperformancein thepresenceofuncertainty,andhigh-gainfeedbackcanamplifyhighfrequencyeffects, whichleadtomusclefatigue.Fordifferentrehabilitationtasks,thepractitionermay valuelimbtrajectoryaccuracyoverdosagei.e.numberofrepetitionsorviceversa. Motivatedbytheneedtoarbitratebetweenthesepotentiallyconictingobjectives,an optimalcontrolmethodisdevelopedthatprovidesacostfunctionwhichcanbeadjusted toplacegreateremphasisonaccuracyversusdosageforuncertainnonlinearmuscle dynamics. Theunderlyingideaofoptimalcontrolistodevelopavaluefunctionthatisthe steadystatesolutiontotheHamilton-Jacobi-BellmanequationHJB,stabilizesanonlinearsystem,andguaranteesoptimalitybyminimizingacostfunctional.Nonlinearitiesin thesystemdynamicsposechallengesindevelopingcontrollersthatcanguaranteeboth 12

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stabilityandoptimality,thusinverseoptimalcontrol[29,30]isusedtoavoidthecomplexityofsolvingthesteadystateHJBequations.Ratherthanminimizingagivencost functional,inverseoptimalcontrolaimstoparameterizeafamilyofstabilizingcontrollers thatminimizeameaningful derived costfunctional.Thederivedcostfunctionalismeaningfulinthesensethatitcontainsapositivefunctionofthestateandapositivedenite functionofthefeedbackcontrol.Thegeneralformofthemeaningfulcostfunctionalis givenas J =lim t !1 t 0 l x + u x T R x u x dx; where l x isasemi-positivedeniteandradiallyunboundedfunctionofthestate x t u x denotesthecontrolinputand R x isapositiverealvaluedfunction. Feedbacklinearizationisananothercommonlyusedtechniquetoachieveoptimal control.Aquadraticcostfunctionalcanbeexpressedas J = 1 0 1 2 x T Qx + 1 2 u T Rudt; where x t denotesthestate,and u t denotesthecontroleffort,respectivelyand Q and R arepositivesemi-deniteandpositivedenite,respectively.Thetradeoffbetween limbpositionaccuracyanddosagecanbeachievedbytuning Q and R withguaranteed robustness.TheseparametersareincludedintheHamiltonianofoptimization,andthe optimalcontrollerisderivedbysolvingtherespectiveHamilton-Jacobi-BellmanHJB equation.Fornonlineardynamicsystems,solvingtheHJBequationcanbeintractable. Feedbacklinearizationisacommonlyusedtechniquetocancelnonlinearelements, leavingaresiduallinearsystemwheretheHJBequationreducestoanAlgebraicRiccati EquationARE.Combinedwithadaptiveandlearning-basedapproaches,methods havebeendevelopedtosuccessfullyminimizeacostfunctionaldespiteuncertaintyina dynamicsystem[31,32]. Chapter2and3examineinverseanddirectasymptoticoptimalNMEScontrollers. 13

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NMESisdeliveredintheformofelectricalcurrentpulseswhichcreatealocalized electriceldtoelicitactionpotentialsinthenearbyneurons.Theoutputofmuscleforce isdeterminedbythepulseamplitude,duration,frequency,andthemusclefatiguestate. Pulsedurationandamplitudedeterminetheactivationregion,i.e.,howmanymotor unitsarerecruited,andareequivalentregardingthetotalappliedelectriccharge.This effectisoftenreferredtoasspatialsummation.Eachelectricpulsecausesatwitch inthemusclebers.Ifasecondpulseisappliedbeforethersttwitchnishes,the twotwitchessumandahigherforceoutputfromthemuscleisachieved.Thiseffect isoftenreferredtoastemporalsummation.Whenthepulsefrequencyishigherthan athresholdcalledthefusionfrequency,continuousmuscleforceoutputisobserved. Largerforcecanbeachievedwithhigherfrequencies.However,higherstimulation frequenciescausethemuscletofatiguefaster.Inpractice,muscleforceiscontrolled bymodulatingthepulseamplitudeorpulseduration,andthefrequencyissettoa constantvaluethatisaslowaspossibletomaintainfusedforceoutputwhileavoiding fatiguingthemuscleprematurely[33].Recentresultsdemonstratedthatfrequencymodulationcanyieldbetterperformanceforbothpeakforcesandforce-timeintegrals thanpulse-duration-modulation,whileproducingsimilarlevelsofmusclefatigue[34]. SincetheNMEScontrolinputisimplementedasaseriesofpulses,hybridsystems theoryprovidesamathematicalframeworktoinvestigatetheeffectofthediscontinuous modulationstrategy.Sincethemodulationstrategyhassignicantimpactonthemuscle performanceandfatigue,theabilitytoexaminetheimpactofthecontrolsignaland modulationstrategyinanalysismayopennewinsightintothedevelopmentofNMES controllers.Chapter4investigatestheuseofahybridcontrolmethodthatexplicitly accountsforthemodulationstrategy. Thereexistsatimedelaybetweentheapplicationofelectricalstimulationand muscleforceoutput.Musclecontractiondynamicsaccountforthemajorityofthe totaltimedelayinmuscleforceoutput.Thecontractiondynamictimedelayis20ms 14

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forfastglycolyticFGand120msforslowoxidativeSObers[35];however,since thistimedelayisrelatedtothecontractiondynamics,itvarieswiththeinputsignal frequency.Toaccountforthisdelay,someauthorshavedevelopedcontrollersforthe delaywithoutconsideringtheunderlyingdelaymechanism[36].Otherresultsseekto compensateforthedelaybymodelingtheunderlyingdelaycausingmusclecontraction dynamics.In[5]and[37],musclecontractiondynamicsweremodeledasarstorder systemwithknownparameters.Theexactparametersofthemuscledynamicsare noteasytodetermine,andtheseparametersarelikelytochangeovertimedueto musclefatigueormusclestrengthtraining.Inaddition,musclecontractionstatesarenot directlymeasurable.Accelerationmeasurementscanbeusedasasubstituteasin[5] and[37].Inpractice,limbpositionandvelocityaremeasurablewithacceptablenoise levels,whileaccelerationisnotdirectlyavailableandoftencontaminatedwithnoise. Derivativeestimationischallenging.Forexample,numericaldifferentiationapproaches suchasbackwarddifferencingareverysensitivetosensornoise.Considerablehigh frequencynoisecanbeintroducedtotheaccelerationsignal.AlowpasslterLPF canbeusedtosuppressthenoise,buttheuseofaLPFintroducesextraphaselag. Anidentier/observerapproachisshownlesssensitivetonoisein[38].Chapter5 exploresthepotentialtoimprovethetrackingperformanceandrobustnessofcontrollers thatincludethemuscledynamicsintheclosed-loopcontroldesignwithouttheuseof accelerationmeasurements.. NeurologicaldisordersresultingfromanUMNLcanhavelastingimpairments.For example,ofthe730,000individualswhosurviveastrokeeachyear,73%haveresidual disability[39].Strokehassignicantimpactonwalkingabilityresultingincharacteristic poststrokegait.Inadequatedorsiexionduringswingphaseanddecreasedplantarexionforcegenerationduringthestancephasearebothcommonimpairmentsofpost strokegaitcausingfootdraggingandslapping,largermetaboliccostonwalking,slow walkingspeed,andgaitasymmetry.Bydeliveringpulsedelectricalcurrentintoaffected 15

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musclesornerves,desiredmusclecontractionscanbeobtained.FESiscommonly usedasaneffectiverehabilitationtooltoenablemuscletrainingandgaitcorrection toimprovepoststokerecoveryandachievedailylifeindependence.Traditionallywith thehelpoffootswitchesoremployingtiltsensors,FEScanbedeliveredtoactivate ankledorsiexormusclesduringtheswingphaseofthegaittocorrectfootdrop,a commonsymptomcausedbystroke,spinalcordinjuryandotherneurologicaldiseases. However,stimulatingankledorsiexormusclesonlyduringtheswingphasedoesnot preventfootslap,decreasedswingphasekneeexion,orslowwalkingspeed.Resent research[40,41]hasshownthatdeliveringFEStoboththeplantarexoranddorsiexor musclesduringgaitcanhelptocorrectpoststrokegaitdecitsatmultiplejointsankleandkneeduringboththeswingandstancephasesofgaitresultinginimproved functionalambulation.TheseFEStreatmentsaretypicallyappliedopen-loopandhave ledtosomepromisingresultsincludingsomecommercialproducts.However,such approachesofferlimitedprecisionandpredictabilitywithoutfeedback,andtypicallyover stimulatethemusclepotentiallyleadingtoamorerapidonsetoffatigue. Previouscontrollershavebeendevelopedwithassociatestabilityanalysiswithout consideringthediscretenatureofawalkinggait.Asdescribedin[42],theanklemotion iscontinuousduringnormalgait,buttheplantarexoranddorsiexormusclesalternate andthemovingsegmentisthefootduringinitialstancephaseandswingphasewhile themovingsegmentistheshankfromtoestriketotoeoff.Theanklemovementis acontinuousevolutionoftheanglebetweenfootandshank,yetanisolateddiscrete signalisneededtodenotethetransitionbetweenplantarexionanddorsiexion.The transitionisimportanttomaintainacontinuousanklemotion.Theswitchingpropertyof gaitcontrolsuggeststheneedtomodelandanalyzetheanklemotioncontrolsystem usinghybridsystemscontroltheory.Generally,coexistenceandinteractionbetween continuousdynamicsanddiscreteeventssuchasswitchinginasystemresultin uniquepropertiesthatarenotinheritedfromindividualsubsystems.Awellknown 16

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examplecanbefoundin[43]thatshowsswitchingbetweengloballyexponentiallystable subsystemsdoesnotguaranteethestabilityofthehybriddynamicsystem.Thestability ofswitchedsystemsdependsontheinterplayofthedynamicsofsubsystemsandthe propertiesoftheswitchingsignals.Chapter7addresstheswitchingdynamicsduring gaitthroughthedevelopmentofahybridNMEScontroller. 1.2Contribution InCharter 2 ,aNN-basedinverseoptimalNMEScontrollerisdevelopedtoenablethelowerlimbtotrackadesiredtrajectorythroughelectricalstimulationofthe quadricepsdespiteuncertaintiesintheconsideredmuscleactivationandlimbmodel. Experimentalresultsfortrackingadesiredtrajectoryandafunctionalexperimentstandingillustratedtheperformanceofthecontroller.Motivationforthisresultisaframework thatcanbeusedtoexaminetheinterplaybetweentheperformanceandthecontrol authorityforrehabilitationclinicians.Aninverseoptimalmethodwasusedtoensure optimalityforaderivedmeaningfulcostfunctional.TheframeworkillustratesthatNN controllersaugmentedbyaPDfeedbackmechanismcanminimizeacostfunctional whichcanbeadjustede.g.,throughQandRtoplacemoreemphasisontrackingerror performanceversusthefeedbackcontrolinput.Whilethisworkmakesacontribution astherstanalysistoexploreanoptimalcontrollerforNMESgivenanonlinearuncertainmusclemodel,thedevelopmentislimitedbytherestrictiontouseaderivedcost functional. InCharter 3 ,anNMEScontrollerisdesignedwhichminimizesaquadraticcost functionalwhilealsoyieldingasymptoticlimbpositiontracking.Thecontrollerhas thepotentialtoreducetheeffectofoverstimulationbypenalizingthetrackingperformanceandthecontrolinput,whichprovidesaframeworkforclinicianstoexaminethe balance/interplaybetweenperformanceandcontrolauthority.Experimentsillustrate trackingperformanceofthecontrollerandtheabilitytoachieveadjustmentbetween trackingerrorsandthefeedbackcontrolthrougherrorpenaltyandcontrolpenalty.This 17

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workexplorestheapplicationofanoptimalcontrollerwhichisobtainedbyfeedback linearizingtheuncertainnonlineardynamicsthroughaNNandimplicitlearningmethod. InChapter 4 ,forthersttime,amusclecontractionmodelwithapulsemodulated controlinputisdevelopedtocapturethediscontinuousnatureofmuscleactivation,and aclosed-loopNMEScontrollerisdesignedfortheuncertainpulsemuscleactivation model.Semi-globallyuniformlyultimatelyboundedSUUBtrackingisguaranteed. Theclosed-loopsystemisanalyzedthroughLyapunov-basedmethodsandapulse frequencyrelatedgainconditionisobtained.Simulationresultsareprovidedtoillustrate theperformanceofthedevelopedcontroller.Forthersttime,thischapterbrings togetherananalysisofthecontrollerandmodulationscheme. InChapter 5 ,forthersttime,theuncertaintiesinthemusclecontractiondynamics weretakenintoconsiderationwhencompensatingforthemusclecontractiondynamics. Acontrollerisdesignedtogetherwithanidentier/observer,whichcouldpotentially improvethetrackingperformanceandachievemorerobustcontrol.Sincethemuscle contractionstateisnotmeasurable,thedynamicsaremanipulatedtoremovethedependenceonmusclecontractionstate.Bydesigningaidentier/observertogenerate thesecondorderderivativeoftheestimatedposition,thecontrollercanbeimplemented withoutaccelerationmeasurements.Theoverallstabilityoftheidentier-controllersystemisanalyzedthroughLyapunovmethods.Semi-globalUUBtrackingandestimation areachieved.Simulationresultsillustratethecontrollerperformance. InChapter6,anaccelerationfreeNMEScontrollerisdevelopedbasedonan identier/observerframeworkincorporatingmodulatedcontrolinputswithmuscle contractiondynamics.Theoverallstabilityoftheidentier-controllersystemisanalyzed throughLyapunovmethods.Semi-globalUUBtrackingandestimationareachieved. Experimentresultsillustratethecontrollerperformance. InChapter 7 ,aswitchedslidingmodebasedcontrollerisdevelopedtoaddress thechallengethatatdifferentphaseofthegait,themovinglimbsegmentsandthe 18

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musclegroupsareswitchingbackandforth,ensuringthattheankleasymptoticlytracks adesignedorrecordedtrajectoryduringgaitwhichcanbeusedforgaitretraining. Semi-globalasymptotictrackingisobtainedfortheswitchedcontrollerduringgait,which isanalyzedbasedonmultipleLyapunovfunctionsandtheperformanceisillustrated thoughsimulations. 1.3Outline Chapter1servesasanintroduction.Themotivation,problemstatementandthe contributionsofthedissertationarediscussedinthischapter. InChapters2and3,inverseanddirectoptimalNMEScontrollersaredeveloped withguaranteedstability,whichaddresstheproblemofpossibleoverstimulationby balancingtheperformanceandthecontroleffort,potentiallyreducingmusclefatigue. Chapter4considersamodulatedcontrolinput.Usinghybridanalysis,anexplicit frequencyconditionisdeveloped. InChapter5,anuncertainmusclecontractionmodelisincludedinthecontrol designtoaddresstheproblemofmusclecontractiontimedelay.Tomitigatetheproblem thatthelimbaccelerationisnotalwaysavailable,anidentication-basedframeis designedtoimplementanaccelerationfreeNMEScontroller. Chapter6developedanidentication-basedaccelerationfreeNMEScontroller consideringamusclecontractiondynamicswithapulsemodulatedinput. Chapter7examinestheproblemofapplyingNMEScontrollertothesystemof ankleduringwalking.Ahybridcontrollerisdesignedtoaddressthechallengeof controllingtheanklemotionduringthegait. Chapter8providesaconclusionandfutureworks. 19

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CHAPTER2 ADAPTIVEINVERSEOPTIMALNEUROMUSCULARELECTRICALSTIMULATION Effortsinthischapterfocusonthedevelopmentofanadaptiveinverseoptimal NMEScontroller.Thecontrolleryieldsdesiredlimbtrajectorytrackingwhilesimultaneouslyminimizingacostfunctionalthatispositiveintheerrorstatesandstimulation input.Thedevelopmentofthisframeworkallowstrade-offstobemadebetweentrackingperformanceandcontroleffortbyputtingdifferentpenaltiesonerrorstatesand controlinputdependingontheclinicalgoalorfunctionaltask.ThecontrollerisexaminedthroughaLyapunov-basedanalysis.Experimentsonable-bodiedindividualsare providedtodemonstratethefunctionalityandperformanceofthedevelopedcontroller. 2.1MuscleActivationandLimbModel Thedynamicsofafreeswingingshankwhenthesubjectisseatedcanbesegregatedintobodysegmentaldynamicsandmuscleactivationandcontractiondynamics. Thecompletedynamicmodelisgivenby[5] M I + M e + M g + M v + d = : In2, M I q 2 R denotestheinertialeffectsoftheshank-footcomplexaboutthe knee-joint, M e q 2 R denotesthenonlinearelasticeffectsduetojointstiffness, M g q 2 R denotesthegravitationalcomponent, M v q 2 R denotesthenonlinearviscous effectsduetodampinginthemusculotendoncomplex[44], d t 2 R isconsideredas anunknownboundeddisturbancewhichrepresentsanunmodeledreexactivationof themusclee.g.,musclespasticityandotherunknownunmodeledphenomenae.g., dynamicfatigue,and t 2 R denotesthetorqueproducedatthekneejoint,where q t q t q t 2 R denotethegeneralizedangularposition,velocity,andaccelerationof thelowerlimbabouttheknee-joint,respectively.Theinertialcomponent M I q 2 R is denedas M I q t = J q t : 20

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TheelasticeffectsaremodeledontheempiricalndingsbyFerrarinandPedottiin[44] as M e = )]TJ/F25 11.9552 Tf 9.299 0 Td [(k 1 e )]TJ/F26 7.9701 Tf 6.587 0 Td [(k 2 q q )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 3 ; where k 1 ;k 2 ;k 3 2 R areunknownpositivecoefcients.Asshownin[5],theviscous moment M v q canbemodeledas M v = B 1 tanh )]TJ/F25 11.9552 Tf 9.298 0 Td [(B 2 q )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 3 q; where B 1 ;B 2 ; and B 3 2 R areunknownpositiveconstants.Thetorqueproducedatthe kneejointcanbemodeledas t = V t ; where V t 2 R istheelectricalstimulusappliedtothequadricepsmusclegroup, q; q 2 R isamappingfunctionbetweenthegeneratedkneetorqueandtheappliedelectrical stimulusonquadriceps.Forcompletedetailsofthedynamicsin2,see[22]. Assumption1 :Basedontheresultsin[45],thenonlinearfunction q; q is assumedtobecontinuouslydifferentiable,positive,andaboundedfunction. Assumption2 :Thedisturbanceterm d t anditsrsttimederivativeareassumed tobebounded.Thisassumptionisreasonablefortypicaldisturbancessuchasmuscle spasticity,fatigue,andloadchangesduringfunctionaltasks. Tofacilitatethesubsequentanalysis,theexpressionin2isrewrittenas J q t + M + d = V t ; where J q; q ;M q; q d q; q 2 R aredenedas J = )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 J;M = )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 M e + M g + M v ; d = )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 d : BasedonAssumptions1and2,thefollowinginequalitiescanbedeveloped 0 j J j 1 j d j 2 ; 21

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where 0 1 2 2 R areknownpositiveconstants. 2.2ControlDevelopment ArehabilitativegoalofNMESistoelicitadesiredmuscleresponsethatcanleadto restoredindependentfunction.Forrehabilitativeoutcomes,repetitivetrainingisessential;yet,electricallystimulatedmusclecanoftenfatiguequicklyduetooverstimulation andvariousotherfactorssuchassynchronousexcitationandnon-physiologicalmotor unitrecruitmentorder.Asaninroadtoaddresstheseconcerns,thecontrolobjectiveis tostimulatethequadricepsmusclegrouptoenabletheshanktotrackadesiredtimevaryingtrajectory,denotedby q d t 2 R ,despiteuncertaintiesinthedynamicmodel, whilealsominimizingagivenperformanceindexthatincludesapenaltyonthetracking errorandthecontroleffort. Toquantifythetrackingobjective,lowerlimbangularpositiontrackingerrorandan auxiliarytrackingerrordenotedby e t ;r t 2 R ,respectively,aredenedas e = q d )]TJ/F25 11.9552 Tf 11.955 0 Td [(q;r =_ e + e; where 2 R isapositiveconstantgain. Aftertakingthetimederivativeof r t ,multiplyingitby J q; q ; andutilizing Equations2and2thefollowingopen-looperrorsystemcanbeobtained: J r = d + f 1 + f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(V; where f 1 t ;f 2 t 2 R aredenedas f 1 = J e;f 2 = J q d + M : Basedon2andthesubsequentstabilityanalysisgiveninTheorem1,thevoltage controlinput V t isdesignedas V = u 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(u 1 = ^ f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(u 1 ; 22

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where u 1 t 2 R issubsequentlydesignedcontrolinput,and u 2 t = ^ f 2 t 2 R isaNN estimateof f 2 t .Athree-layerNNcanbeusedtorepresent f 2 as f 2 y = W T )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(U T y + y where U 2 R N 1 +1 N 2 and W 2 R N 2 +1 1 areboundedconstantidealweightmatrices, : R N 1 +1 R N 2 +1 isanNNactivationfunction, y t 2 R N 1 +1 isaninputvector denedas y t =[1 q t q t q d t ] T ; and y : R N 1 +1 R isafunctionalreconstructionerrorthatcanbeupperboundedas j y j ; where 2 R isaknownpositiveconstant.Theestimate ^ f 2 t isdesignedas ^ f 2 = ^ W T ^ U T y ; where, ^ U t 2 R N 1 +1 N 2 ; ^ W t 2 R N 2 +1 1 areweightestimatematrices.Theideal weightmatrixestimates ^ U t and ^ W t areupdatedon-lineusingtheprojectionalgorithm ^ W = proj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [()]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w ^ r T ; ^ U = proj )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u y ^ 0 T ^ Wr T ; where )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w 2 R N 2 +1 N 2 +1 and )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u 2 R N 1 +1 N 1 +1 areconstant,positivedenite,symmetricgainmatrices, ^ = ^ U T y ; and ^ 0 = 0 ^ U T y = d )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(U T y =d )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(U T y j U T y = ^ U T y : Theweightmismatcherrors ~ U t 2 R N 1 +1 N 2 and ~ W t 2 R N 2 +1 1 aredenotedas ~ W = W )]TJ/F15 11.9552 Tf 15.368 3.022 Td [(^ W; ~ U = U )]TJ/F15 11.9552 Tf 13.953 3.022 Td [(^ U; andthehidden-layeroutputmismatch ~ y 2 R N 2 +1 foragiven y t isdenedas ~ = )]TJ/F15 11.9552 Tf 12.57 0 Td [(^ = )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(U T y )]TJ/F25 11.9552 Tf 11.955 0 Td [( ^ U T y : 23

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ByusingaTaylorSeriesapproximation,thehidden-layeroutputmismatch ~ y canbe expressedas ~ =^ 0 ~ U T y + O ~ U T y 2 ; where O ~ U T y 2 denotesthehigherorderterms. Substituting2into2andperformingsomealgebraicmanipulationyields J r = N + ~ W T ^ + ^ W T ^ 0 ~ U T y + u 1 ; wheretheauxiliaryterm N ~ W; ~ U;y 2 R isdenedas N = f 1 + ~ W T ^ 0 ~ U T y + W T O ~ U T y 2 + y + d : BasedonEquations2,2,and2 N ~ W; ~ U;y canbeupperboundedas[46] k N k c 1 + c 2 k z k ; where c 1 ;c 2 2 R areknownpositiveconstants,and z t 2 R 2 isdenedas z t =[ er ] T : Basedon2andthesubsequentstabilityanalysis,thestabilizingPDcontroller u 1 in 2isdesignedas u 1 = )]TJ/F25 11.9552 Tf 9.299 0 Td [(R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 r = )]TJ/F15 11.9552 Tf 11.291 0 Td [( k s 1 + k s 2 + k s 3 r; where R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;k s 1 ;k s 2 ;k s 3 2 R denotepositiveadjustablegains. FromAssumption1,Equations2 )]TJ0 0 1 rg 0 0 1 RG/F22 11.9552 Tf 9.299 0 Td [(2and2,itcanbeshownthat 1 2 J k z k ; where k z k 2 R isapositive,globalinvertiblefunction. 2.3StabilityAnalysis Theorem2.1. Thecontrollergivenin2,2,and2ensuresthatall closed-loopsignalsarebounded,andthepositiontrackingerrorissemi-globaluniformly 24

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ultimatelyboundedSUUBinthesensethat j e t j 0 exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 t + 2 ; where 0 ; 1 ; 2 2 R denotepositiveconstantsin D z R 2 jk z k )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ 5.48 0.25 Td [(p k s 3 providedthecontrolgains ,and k s 2 introducedin2and2areselectedbased onthesufcientconditions min k s 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 >c 2 ;k s 2 ;> 1 2 : Proof. Considerapositivedenite,continuouslydifferentiable,andradiallyunbounded function V L e;r; ~ W; ~ U 2 R denedas V L = 1 2 e 2 + 1 2 J r 2 + 1 2 tr ~ W T )]TJ/F29 7.9701 Tf 7.314 4.937 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 w ~ W + 1 2 tr ~ U T )]TJ/F29 7.9701 Tf 7.315 4.937 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 u ~ U : Byusing2andtypicalNNproperties[47], V L t canbeupperandlowerboundedas 1 k z k 2 V L 2 k z k 2 + 3 ; where 1 ; 2 ; 3 2 R areknownpositiveconstants.Takingthetimederivativeof2, utilizing2and2,andcancelingcommontermsyields V L = e e + 1 2 J r 2 + rN + ru 1 : Using2andYoung'sinequality,theexpressionin2canbeboundedas V L )]TJ/F31 11.9552 Tf 23.91 16.857 Td [( )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 e 2 + r 2 1 2 J + 1 2 + rN + ru 1 : Byutilizing2,2,and2,theexpressionin2canbeupperbounded as V L )]TJ/F31 11.9552 Tf 23.91 16.857 Td [( )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 e 2 )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( k s 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 r 2 )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k s 1 r 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 1 j r j + c 2 k z k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( k s 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [( k z k r 2 : 25

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Applyingnonlineardampingandneglectingnegativeterms,theexpressionin2 canbeupperboundedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( 4 k z k 2 + c 2 1 4 k s 1 ; 8k z k2D ; and 4 =min k s 2 )]TJ/F24 7.9701 Tf 13.043 4.707 Td [(1 2 ; )]TJ/F24 7.9701 Tf 13.043 4.707 Td [(1 2 )]TJ/F25 11.9552 Tf 11.847 0 Td [(c 2 > 0 providedthesufcientgainconditionsin2are satised.Theinequalityin2canbeusedtorewrite2as V L )]TJ/F25 11.9552 Tf 23.113 8.088 Td [( 4 2 V L + "; 8k z k2D ; where 2 R isapositiveconstant.Thelineardifferentialinequalityin2canbe solvedas V L t V L e )]TJ/F27 5.9776 Tf 7.782 4.324 Td [( 4 2 t + 2 4 h 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F27 5.9776 Tf 7.782 4.324 Td [( 4 2 t i ; 8k z k2D : Providedthesufcientconditionsgivenin2aresatised,theexpressionsin2 29and2canbeusedtoprovethecontrolinputandalltheclosed-loopsignals areboundedin D .Largervalueof k s 3 and k s 1 willexpandthesizeofthedomain D to includeanyinitialconditionsi.e.,asemi-globaltypeofstabilityresultandreducethe residualerror.From2and2,theresultin2canbeobtained. 2.4CostFunctionalMinimization Aninverseoptimalcontroller[29,48,49]isoptimalwithrespecttoan aposteriori costfunctionalthatisderivedfromaLyapunov-basedanalysisincomparisonto minimizingan apriori givencostfunctionalindirectoptimalcontrol.Duetotheuseof aNNtocompensatefortheunstructureduncertaintyinthemusclemodel,aresidual disturbanceispresentinthesystemi.e.,theUUBstabilityresult.Giventhisresidual disturbance,thefollowinganalysisisformulatedinthespiritofatwoplayerzero-sum differentialgamewheretheobjectiveistominimizethecostfunctionalwithrespectthe controlinputinthepresenceofthemaximum"worst-case"disturbance.Thefeedforward NNelementestimatesthenon-LPdynamics,whilethefeedbackelementispenalizedby thecostfunctional. 26

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Theorem2.2. Thefeedbacklawgivenby u o = )]TJ/F25 11.9552 Tf 9.298 0 Td [(R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 r; withthescalargainconstantselectedas > 2 andtheupdatelawgivenin2, minimizesthecostfunctional J =lim t !1 2 V L t + t 0 l + u 2 1 R )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 c 2 1 4 k s 1 d ; where l z;t 2 R isapositivefunctionofthetrackingerror l = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 e e + 1 2 J r 2 + rf 1 + rN )]TJ/F25 11.9552 Tf 18.519 8.088 Td [(c 2 1 4 k s 1 + 2 r 2 R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; providedthesufcientconditionsin2aresatised. Thecostfunctionalin2issaidtobemeaningfulifthebracketedtermsin2 39arepositivei.e.,positivestateandcontrolfunctions.Toexaminethesignof l z;t ; theexpressionsin2,2,2andtheconditionin2canbeusedto determinethat e e + 1 2 J r 2 + rf 1 + rN )]TJ/F25 11.9552 Tf 18.519 8.088 Td [(c 2 1 4 k s 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 2 R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 0 : Aftermultiplyingbothsidesby )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 andadding )]TJ/F15 11.9552 Tf 12.169 0 Td [(2 r 2 R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ,theexpressionin2 canberewrittenas l )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 r 2 R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 = Qr 2 ; where Q 2 R isapositiveconstant.Theinequalityin2indicatesthat l z;t is positivesince R ispositiveand > 2 .Therefore J t isameaningfulcostfunctionalthat penalizestheerrorfunctionin z t andthefeedbackcontrol u 1 t .Thecostfunctional in2andtheresultin2indicatesthatlargervaluesof Q placeagreater penaltyonthetrackingerror,whereaslargervaluesof R placeagreaterpenaltyonthe feedbackcontrol.Theeffectsofselectingdifferentvaluesfor Q and R areillustratedin Section2.5. 27

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Toshowthat u o minimizes J t ,theauxiliarysignal v t 2 R isdenedas v = u 1 + R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 r: Substituting2and2into2andperformingsomealgebraicmanipulation yields J =lim t !1 2 V L t + t 0 v 2 Rd )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 t 0 V L d : Afterintegrating2,thecostfunctional J t canbeexpressedas J =2 V L +lim t !1 t 0 v 2 Rd : Bysubstituting2into2,itcanbeshownthat u o stabilizesthesystem.Since J t isminimizedif v t =0 ,thencontrollaw u 1 = u o isoptimalwithrespecttothe meaningfulcostfunctionalin2. 2.5ExperimentResults Theproposedinverseoptimalcontrollerwasimplementedonhealthynormal volunteerstoevaluatetheperformanceofthecontroller.Thefocusofthispaperisto developandanalyzeaninverseoptimalcontrollerasameanstoprovideamethodfor understandingthetradeoffofthecontrolparametersQandRthatareincludedinacost functionalcomposedoftermssuchasthelimbtrackingerrorandstimulationinput.This sectiondescribestheperformanceofthedevelopedstrategywhenimplementedona groupofhealthynormalvolunteers.Theperformanceofthedevelopedmethodmay varywhenimplementedinpopulationsofindividualsaffectedwithdifferentneurological disorders:clinicaltrialsonspecicaffectedpopulationsofinterestaremotivatedas futureworktofurthertheclinicalimplicationsofthefollowingoutcomes.Theresults obtainedfromhealthynormalsubjectsinthissectionmayprovidesomeinsightinto furtherclinicaltrials.Forexample,in[50]andinresultssuchas[10,17,51]which directlyorindirectlycitetheworkin[50]arelaxedlimbisshowntobehavelikea recentlyparalyzedlimb.However,therearedifferencesinthemuscleresponsethat 28

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areassociatedwithdifferentconditions.Forexample,alimbthathasbeenparalyzed forsometimewillexhibitmuscleatrophywithdisuse,willfatiguemorerapidly,and mayexhibitclonusandmusclespasticity[51].Theinverseoptimalcontrollerhasbeen developedandanalyzedwhileincludingaddedunmodeleddisturbancesi.e., d t theaforementionedeffectsforparalyzedmusclearenotpresentinthehealthynormal volunteersubjects.Able-bodiedvolunteerscanpotentiallyexecuteunintentionalmuscle contractionswhichmayalsobecapturedby d t thatcanaidorhinderthedesired limbmotion.Tomitigatethispotential,volunteerswereinstructedtorelaxandtoallow thestimulationtocontrolthelimbmotioni.e.,thesubjectswerenotsupposedto inuencethelegmotionvoluntarilyandwerenotallowedtoobservethedesiredlimb trajectory. Thestudyvolunteerswereseatedinanon-motorizedlegextensionmachineLEM. ThefreeswinginglegsofthevolunteerwereattachedtothemovablearmoftheLEM wherethepositionoftheLEMarmismeasuredbyanopticalencoderandusedasa feedbacksignal.Adjustmentsweremadebeforeeachtrialtoensurethecentersofthe kneeandtheencoderwerealigned.A4.5kglb.weightwasattachedontheweight baroftheLEMarm,andamechanicalstopwasusedtopreventhyperextension.Selfadhesivereusableneuromuscularstimulationelectrodeswereusedintheexperiments. Oneelectrodewasplacedoverthedistal-medialportionofthequadricepsfemoris musclegroupsandtheotherwasplacedovertheproximal-lateralportion.Electrical pulsesweredeliveredthroughacustombuiltstimulator.Dataacquisitionwasperformed at1000Hzandtwodigital-to-analogsignalswereusedasinputstothestimulation circuitrythatproducesapositivesquarepulsebetween3-100Hzwithavoltageoutput between1-50voltspeak. Themodulatedpulsewidthwassettoaconstant 400 sec andthefrequencyofthe pulsesequencewas28Hz.Themotivationforchoosinga 400 sec pulseisduetothe factthatitgeneratesreliableoutputbasedonitsforce-frequencyandforce-amplitude 29

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relationshiprelativetootherpulsewidths.Thestimulationfrequencywasselected basedonforce-frequencycurves[52],whichshowthatasstimulationfrequencyis increasedmuscleforceincreasestoasaturationlimit.Higherfrequenciescanbe chosentogeneratemoreforceuptoasaturationlimitbutmusclestendtofatiguefaster athigherfrequencies.The 28 Hzpulsewaveyieldsreducedfatigueincomparisonto higherfrequenciesbutlowerfrequenciestendtoproducerippledkneemotion[52,53]. 2.5.1TrackingExperiments Trackingexperimentswereconductedonvevolunteerstwofemalesandthree males,ages 22 )]TJ/F15 11.9552 Tf 9.298 0 Td [(40 yrs..Thedesiredangulartrajectoryforthekneejointwas q d = 8 > < > : 35 2 +sin 2 T t + 3 2 ;t< T 2 ; 15+sin 2 T t + 3 2 +5 ;t T 2 ; withafrequencyof1.5HzandrangeofmotionROMbetween5and35.Thereason toselectthistrajectoryisthesinusoidalfunctionsaresufcientsmoothandeasyto implemented.ThevalueswereselectedtoapproximatethefrequencyandROMofthe lowerlimbduringwalking.Anysufcientlysmoothdesiredtrajectorycouldhavebeen selected.Forthetrackingexperiments,QandRareadjustedbytrialanderrorforeach individualtoyieldthebestperformance.Eachindividualwasstimulatedfor5to10trials withaminimumrestof5minutesbetweentrials.Eachtrialwas30s.Thesteadystate RootMeanSquareRMSandPeaki.e., max j e t j trackingerroriscalculatedfrom3s to30s.Table1summarizestheRMSandPeakerrorsforgivenQandRgains.Figure 2-1illustratesatypicalknee/limbtrackingerror.TheQandRgainswereadjusted toobtaintheresultsinTable1.ThemeansteadystateRMSis 1 : 92 withastandard deviationSTDof 0 : 18 ,andthemeanPeakerroris 6 : 57 withaSTDof 1 : 29 Unitsteptestswereconductedonthreevolunteersonefemalesandtwomales, ages25-40yrs..TheresultsaresummarizedinTable2.Arepresentativetrialisshown inFig.2-2. 30

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Table2-1.Steady-staterootmeansquareRMSandpeakerrorforvehealthynormal individuals SubjectPeakRMSPeak-30sRMS-30sQR A9.672.348.912.1044.92000 B6.141.895.061.8179.82000 B9.692.206.071.5919.92000 C15.473.377.182.139.732000 D13.723.076.881.9819.92000 E8.472.315.341.91499.52000 Mean10.532.546.571.92 STD3.450.591.410.20 Figure2-1.Trackingerrorforarepresentativetrial 2.5.2PerformanceTrade-offs Todemonstratetheabilityforacliniciantochoosedifferentcombinationsof Q and R toplaceagreateremphasisontrackingperformanceorfeedbackcontrolinput, trackingexperimentswereconductedonthreehealthynormalvolunteersages 25 )]TJ/F15 11.9552 Tf 9.299 0 Td [(40 yrs.TwogroupsofexperimentswereconductedwithxedNNupdategains.Therst experimentsxed Q =1 andvaried R from 8 to 10000 toillustratetheeffectofpenalizing thecontrolinput.Additionalexperimentsvaried Q between 8 to 600 foraxed R =2000 toshowtheeffectofpenalizingtheperformance.Eachsessionwas20s,andRMS valueswerecalculatedfortheerror,totalcontrolinput,andoptimalcontrolinputi.e. u 1 t ,respectively.Figure2-3illustratesthatthefeedbackcontrolinputdecreases 31

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Figure2-2.Steptestforarepresentativetrial.Thesolidlinedepictsthedesiredangle andthedottedlinedepictstheactualtrajectory andthetrackingerrorincreasesbyincreasing R .Figure2-4illustratesthattheerror decreasesandthefeedbackcontrolinputincreaseswithincreasing Q .Theresults inFigure2-3andFigure2-4representtheoutcomeforonevolunteer,buttheresults showedthesametrendsfortheothertwosubjectswithdifferentadjustableranges. 2.5.3ChangingGravityLoad Toillustratethattheproposedinverseoptimalcontrollercanbeusedforatask whichinvolvesachangingloadi.e.themomentarmofgravityforcechanges,asitto-standtransition-likeexperimentwasconductedona38yearoldhealthynormal male.Notethataphysiologicalsit-to-standtransactioninvolvesamixedeccentricconcentriccontractionofthemusclesduetothebiarticularnatureofthequadriceps group.Forthisexperiment,theelectrodeswereplacedononeleg,andthevolunteer wasseatedontheedgeofachair.Thekneejointanglewasmeasuredbyagoniometer BiometricsLtd.,VA,wherethegoniometermeasured 90 intheseatedpositionand approximately180 inthestandingposition.Giventhelargeinitialconditionoferrori.e., 90 anexperimentallydetermineddesiredrisetostandingtrajectorywasdesignedas q d =135+45sin 2 5 t + 3 2 ; if t< 5 2 and 180 if t 5 2 .Fig.2-5,depictstheactualversus desiredtrajectoryforthestandingexperiment.Themaximumpositiveandnegative 32

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Table2-2.Thesteptestforarepresentativetrial SubjectLegFinalValueRisingTimeSettingTimeRMS Aleft203.253.460.40 Bleft200.310.460.54 Cleft203.663.910.49 Mean2.412.610.48 Aleft402.552.801.31 Bleft402.312.541.10 Cleft401.431.621.72 Mean2.321.38 Figure2-3.Experimentswith Q =1 where R variedfrom 8 to 10000 Figure2-4.Experimentswhere Q variedfrom 10 to 600 and R =2000 33

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transienterrorsare +6 : 0 and )]TJ/F15 11.9552 Tf 9.299 0 Td [(5 : 3 ,respectively.Thesteadystateerroris )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 63 0 : 17 withamaximumstimulationvoltageof 30 volts. Figure2-5.Trajectoriesofthestandingexperiment.Thesolidlinedepictsthedesired trajectoryandthedashedlinedepictstheactualtrajectory 2.6Discussion ANN-basedinverseoptimalcontrollerisproposedandevaluated.Thecontroller isproventoachieveuniformlyultimatelyboundedtrackinginthepresenceofbounded unmodeleddisturbances.Thestructureofthecontrollerisorganizedasacombination ofaNNfeedforwardandaPDfeedbackelement.TheNNelementcompensatesfor thenonlinearuncertaintiespresentinthedynamicssuchaspassiveconstraintsonjoint movementandmusclestimulationwhichincludenonlinearrecruitment,torque-angle, torque-velocityscaling,etc.Acostfunctionalisconstructedtoallowgainstobeadjusted toscaletherelativepenaltyofthetrackingerrororthefeedbackcontrolportionofthe control.AsindicatedinTable1,ameanRMSerrorof1.92.2for3-30seconds wasachievedforthegivendesiredtrajectory.Thesit-to-standtransition-likeexperiment showsthatthecontrolleralsoyieldspromisingresultswherethemaximumpositiveand negativetransienterrorsare+6.0and-5.3,respectively,withasteadystateerror within-0.63.17.Thecontrolaccuracyfromtheseexperimentsissufcientfor typicalfunctionaltasks.Inadditiontodevelopingacontrollerandassociatedstability 34

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proofthatcanyielddesiredtrackingerrorperformance,acontributionofthiseffortis todevelopaframeworktoadjusttheperformanceversuscontroleffort.Thetrade-off betweentrackingperformanceandfeedbackcontroleffortcanbeachievedbychoosing differentvaluesofQandR.LargervaluesofQyieldbettertrackingperformanceat theexpenseofalargerfeedbackcontroleffortwhilelargervaluesofRyieldreduced feedbackcontroleffortwithlargertrackingerrors.AsillustratedinFigure2-3,withQ=1, increasingRfrom8to10000resultsinareductionoftheRMSfeedbackcontrolinput from8.2voltsto0.5volts.Figure2-4illustratesthatwithR=2000,increasingQfrom20 to600reducestheRMStrackingerrorfrom7to3.5.Additionaldevelopmentremains toexaminetheeffectsonfatigueofincreasedcontrolinput.Moreover,thecurrent developmentisnotabletoincludetheentirecontrolinputinthecostfunctionali.e., onlythefeedbackportionofthecontrollerisincluded.Theresultsvalidatetheabilityto directlyalterthefeedbackcontrolthroughR,buttheresultsdonotshowacorrelation betweenchangesinthefeedbackportionversuschangesintheoverallcontroli.e.,the overallcontroloutputwasrelativelyinvarianttochangesinR.Thiscanbeexplainedby theNNfeedforwardcomponentcompensatingforthedifferences.However,heuristically, itiswellacceptedthatlargerfeedbackgainsresultinnoiseamplicationandhigher frequencycontrol.Itisalsowellacceptedthathigherfrequencystimulationcanlead tomorerapidfatigue.Theseresultspointtotheneedforfurtherstudiesinfuturework toinvestigatetherelationshipbetweenfatigueasafunctionoffeedbackcontrolversus feedforwardcontrol.Fromatheoreticalperspective,theapproachin[54]provides aninroadtodevelopinganinverseoptimalcontrollerthatincludesaportionofthe feedforwardcomponentinthecostfunctionalfortheparametricstrict-feedbacksystems. Able-bodiesindividualsareaheterogeneousgroupduetomusclesize,strength andfatigabilityvaryinggreatlywhichisseenintheexperimentgroup.Theresults reecttherobustnessofthecontrollerthatthecontrollerisbaletoaccountforindividual differencesinresponsetoelectricalstimulation.Theresponseofmusclestoelectrical 35

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stimulationcouldbedifferentbetweenable-bodiedindividualsandindividualswith variousdisorders.Forexample,apersonwithSCIthatoccurredpriortoafewweeks hasmusclesthatareatrophied,experiencemorerapidfatigue,andarepotentially subjecttodisturbancessuchasspasticityandclonus.Themuscleatrophyandrapid fatiguecanbeimprovedthroughmusclere-conditioningusingelectricalstimulation. Theexperimentalpopulationusedinthisstudyproducesaproof-of-conceptthatthe controllerworkstoregulateelectricallystimulatedlimbtracking,buttheresultsshould notbeextrapolatedtothepotentialperformanceofthesysteminindividualswhohave disorderswithoutclinicaltrialsinsuchapplication. 36

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CHAPTER3 ASYMPTOTICOPTIMALNEUROMUSCULARELECTRICALSTIMULATION Musclefatigueduringelectricalstimulationonsetsearlyandiscomparatively moresubstantialthanduringvolitionalcontractions,hinderingsuccessfulapplication ofFES/NMES.Oneoftheavoidablecausesofmusclefatiguecanbeattributedtothe overstimulationduringNMES.Inthischapter,aNMEScontrollerisdevelopedtominimizeaquadraticcostfunctionaltobalanceasymptotictrajectorytrackingperformance andcontroleffort,potentiallyreducingoverstimulationofthemuscle.ALyapunov-based analysisisusedtoprovetheasymptoticconvergenceofclosed-looptrackingerrorand asymptoticminimizationofthegivencostfunctional.Experimentsonhealthnormal individualsareprovidedtofurthervalidatetheperformanceofthedevelopedcontroller. 3.1ControlObjective Trajectorytrackingisanessentialtaskinmanyrehabilitativeexercisesandfunction restorationtasks.Therefore,thecontrolobjectiveistoensurethekneeangle q t tracks adesiredtrajectory,denotedby q d t 2 R .Thefollowingdevelopmentisbasedon theassumptionthat q t and q t aremeasurable.Thedesiredtrajectorycanbeany continuoussignalorasimpleconstantsetpoint. Toquantifythetrackingobjective,alowerlimbangularpositiontrackingerror, denotedby e 1 t 2 R ,isdenedas e 1 q d )]TJ/F25 11.9552 Tf 11.955 0 Td [(q; where q d t isanaprioritrajectory,designedsuchthat q d t q i d t 2L 1 ,where q i d t denotesthe i th derivativefor i =1 ; 2 ; 3 ; 4 : Tofacilitatethesubsequentcontroldesignand stabilityanalysis,lteredtrackingerrorsdenotedby e 2 t ;r t 2 R arealsodenedas e 2 =_ e 1 + 1 e 1 ; r =_ e 2 + 2 e 2 ; 37

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where 1 2 2 R arepositiveconstantgains.Thelteredtrackingerror r t isnot measurablesincetheexpressionin3dependson q t : 3.2FeedbackLinearizingOptimalControlDesign Tomotivatethecontroldevelopmentfortheuncertainmuscledynamics,the developmentinthissectionassumesthatallthesystemparametersanddisturbance areknownthisassumptionisrelaxedinthenextsection.Acontrollerisdevelopedthat minimizesaquadraticcostfunctionalwhichpenalizesthestatesandcontrolinput. Multiplyingthetimederivativeof3by J q andusing2and3,yields J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 J e 2 + h + d )]TJ/F25 11.9552 Tf 11.955 0 Td [(V; wherethefunction h q; q; q d ; q d 2 R isdenedas h = 1 2 J e 2 + J 1 e 1 + J q d + M : Theunmodulatedvoltageappliedtothemuscleisdesignedas V = h + d )]TJ/F25 11.9552 Tf 11.955 0 Td [(u; toyieldthefeedbacklinearizeddynamics J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 J e 2 + u; where u t 2 R isanauxiliarycontrolinputthatwillbedesignedtominimizethegiven costfunctional J z;u 2 R denedas J = 1 0 1 2 z T Qz + 1 2 Ru 2 dt; where z t 2 R 2 is z = e 1 e 2 T ; 38

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Q 2 R 2 2 isapositivesemi-denitesymmetricconstantmatrixand R 2 R isapositive gaintoweighttheinuenceofthestatesandpartialcontroleffort,respectively.The resultingoptimalcontrolvoltageinput,denotedby V t ,forthefeedbacklinearized systemis V = h + d )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ; wheretheoptimalvalueof u t = u t isdesignedas u t = )]TJ/F25 11.9552 Tf 9.298 0 Td [(R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e 2 tominimize3withrespecttothedifferentialconstraintsin3and3. TheexpressioninEquations3and3canberewritteninstatespaceformas z = A q; q z + B q; q u; where A q; q 2 R 2 2 and B q; q 2 R 2 1 are A = 2 6 4 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 1 0 )]TJ/F24 7.9701 Tf 10.494 4.708 Td [(1 2 J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 J 3 7 5 ;B = 2 6 4 0 J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 3 7 5 : Theoptimalcontrollaw u t minimizes3subjectto3ifandonlyifthereexists avaluefunction V o z;t where )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(@V o @t = 1 2 z T Qz + 1 2 u T Ru + @V o @z z satisestheHJBequation @V o @t +min u H z;u; @V o @t ;t =0 ; wheretheHamiltonianofoptimization H )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(z;u; @V o @t ;t 2 R isdenedas H = 1 2 z T Qz + 1 2 u T Ru + @V o @z z: 39

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Theminimumof3isobtainedfortheoptimalcontroller u t = u t ; wherethe respectiveHamiltonianis H =min u H z;u; @V o @t ;t = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(@V o @t : Tofacilitatethesubsequentdevelopment,let P q; q 2 R 2 2 bedenedas P = 2 6 4 K 0 0 J 3 7 5 ; where K 2 R isapositiveconstantgain,andlet Q in3bepartitionedas Q = 2 6 4 Q 11 Q 12 Q 12 Q 22 3 7 5 : If 1 R ,and K introducedin3,3,and3,satisfythefollowingalgebraic relationships K = )]TJ/F25 11.9552 Tf 9.299 0 Td [(Q 12 > 0 ; Q 11 =2 1 K; R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 = Q 22 ; andthen P q satisesthedifferentialRiccatiequation PA + A T P T )]TJ/F25 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 B T P + P + Q =0 ; andthevaluefunction V o z;t 2 R V o = 1 2 z T Pz; satisestheHJBequationin3.Lemma1of[55]canbeusedtoconcludethatthe optimalcontrol u t thatminimizes3subjectto3is u = )]TJ/F25 11.9552 Tf 9.298 0 Td [(R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 B T @V o z;t @z T = )]TJ/F25 11.9552 Tf 9.298 0 Td [(R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e 2 : 40

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Sothefeedbacklinearizingoptimalcontrolvoltageinputisgivenby3. Toassociatetheerrorpenaltywithcontrollergains,using3,3,319and 3, z T t Qz t 2 R canbedevelopedas z T Qz =2 1 Ke 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 Ke 1 e 2 + R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e 2 2 : 3.3AdaptiveControlDesign Intheprevioussection, h q; q; q d ; q d and d q;t areassumedtobeknownto developtheoptimalcontroller u fortheresidualdynamicsin3.Theassumption ofknowndynamicsisrelaxedbyusinganadaptivecontrollerthatcombinestheuniversalapproximationpropertyofNNswiththeimplicitlearningcharacteristicsofRISE feedbacktoasymptoticallyconvergeto V Multiplying3by J q andusing3yields J r = h + f d + d )]TJ/F25 11.9552 Tf 11.955 0 Td [(V; where f d q d ; q d ; q d ; h q; q;q d ; q d ; q d 2 R aredenedas f d = J q d q d + M q d ; q d ; h = 2 J e 2 + J 1 e 1 + J q d + M )]TJ/F25 11.9552 Tf 11.955 0 Td [(f d : TheNNestimate ^ f d t 2 R isdenotedas ^ f d = ^ W T ^ U T x d ; where ^ U t 2 R N 1 +1 N 2 ; ^ W t 2 R N 2 +1 1 areweightestimatematricesfortheideal weightsbetweentherst-to-secondandthesecond-to-thirdlayersofaNN,respectively. Theinputvector x d t 2 R 4 isdenedas x d t =[1 q d ; q d ; q d ] T : 41

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Basedon3andthesubsequentstabilityanalysis,thecontrollerin3is redesignedas V = ^ f d + )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ; whichconsistsofoptimalcontrol u t givenin3,where ^ f d t isdesignedin 3,andtheRISEfeedback t isdenedas = k s e 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(k s e 2 + ; where e 2 t 2 R isthesolutiontothegeneralizedequation = k s 2 e 2 + sgn e 2 ; where k s 2 R arepositivecontrolgains.UsingFilippov'stheoryofdifferential inclusions[5659],theexistenceofsolutionscanbeestablishedfor 2 K [ h 1 ] e 2 ;t where h 1 e 2 ;t 2 R isdenedastheright-handsideof in3and K [ h 1 ] T > 0 T S m =0 coh 1 B ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ,where T S m =0 denotestheintersectionoverallsets S m of Lebesguemeasurezero, co denotesconvexclosure,and B ; = f & 2 R jk )]TJ/F25 11.9552 Tf 11.955 0 Td [(& k < g [60,61].Thedifferentialequationgivenin3iscontinuousexceptfortheLebesgue measurezerosetwhen e 2 e 1 ; e 1 ;t =0 Amulti-layerNNisusedtoexpress f d q d ; q d ; q d as f d = W T )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(U T x d + x d ; where U 2 R N 1 +1 N 2 and W 2 R N 2 +1 1 areboundedconstantidealweightmatrices fortherst-to-secondlayerandsecondtothirdlayer,respectively; N 1 N 2 ; 1 arethe numbersofneuronsintherst,second,andthirdlayeroftheNN,respectively; : R N 1 +1 R N 2 +1 isanactivationfunctionfortheNN,and : R 4 R isafunctional reconstructionerror.Theinputvector x d t 2 R isintroducedin3. 42

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Basedontheassumptionthatthedesiredtrajectoryisbounded,thefollowing inequalitieshold j x d j 0 ; j x d ; x d j 1 ; j x d ; x d ; x d j 2 ; where 0 ; 1 ; and 2 2 R areknownconstants.TheNNestimate ^ f d t isdenedin 3.Theupdatelawisdesignedas ^ W = proj )]TJ/F26 7.9701 Tf 7.315 -1.793 Td [(w ^ 0 ^ U T x d e T 2 ; ^ U = proj )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u x d ^ 0 T ^ We 2 T ; where )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w 2 R N 2 +1 N 2 +1 and )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u 2 R N 1 +1 N 1 +1 areconstant,positivedeniteand symmetricgainmatrices, ^ = ^ U T x d ; and ^ 0 = 0 ^ U T x d d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(U T x d j U T x d = ^ U T x d : Theprojectionalgorithmensuresthatthe ^ U t and ^ W t remainboundedinsideknown boundedconvexregions[62]. Theweightmismatcherrors ~ U t 2 R N 1 +1 N 2 and ~ W t 2 R N 2 +1 1 aredenotedas ~ W = W )]TJ/F15 11.9552 Tf 15.368 3.022 Td [(^ W; ~ U = U )]TJ/F15 11.9552 Tf 13.953 3.022 Td [(^ U: Byusing3 )]TJ/F22 11.9552 Tf 9.298 0 Td [(3,theclosedlooperrorsystemcanbeexpressedas J r = 2 J e 2 + J 1 e 1 + J q d + M + d )]TJ/F15 11.9552 Tf 15.368 3.022 Td [(^ W T ^ U T x + + R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e 2 : Takingthetimederivativeof3yields J r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 J r )]TJ/F25 11.9552 Tf 11.956 0 Td [(k s r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 r + N + ~ N; 43

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wheretheunmeasurableauxiliaryfunctions ~ N e 1 ;e 2 ;r;t and N ^ W; ^ V;x d ; x d 2 R are denedas ~ N = M )]TJ/F25 11.9552 Tf 11.955 0 Td [(M q d ; q d + M )]TJ/F15 11.9552 Tf 17.592 3.022 Td [(_ M q d ; q d ; q d +_ d )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ d q d ; q d ;t + J )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r + 1 e 1 + 2 e 2 + q d )]TJ/F15 11.9552 Tf 16.069 3.022 Td [(_ J q d ; q d q d + 2 J e 2 + J 1 e 1 )]TJ/F29 7.9701 Tf 17.117 13.715 Td [( ^ W T ^ U T x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T 0 ^ U T x d ^ U T x d ; N = N D + N B : In30, N D x d ; x d ;t 2 R isdenedas N D = d +_ d q d ; q d ;t + J ... q d + J q d ; q d q d + M q d ; q d + M q d ; q d ; q d ; while N B ^ W; ^ U;x d ; x d ;t 2 R isdenedas N B = N B 1 + N B 2 ; where N B 1 ^ W; ^ U;x d ; x d ;t 2 R and N B 2 ^ W; ^ U;x d ; x d ;t 2 R aredenedas N B 1 = )]TJ/F25 11.9552 Tf 9.298 0 Td [(W T ^ 0 ^ U T x d ^ U T x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ 0 ^ U T x d ~ U T x d ; and N B 2 = ^ W T ^ 0 ^ U T x d ~ U T x d + ~ W T ^ 0 ^ U T x d ^ U T x d : Inasimilarmannerasin[63],theMeanValueTheoremcanbeusedtodevelopthe followingupperbound ~ N k y k k y k ; where y t 2 R 3 isdenedas y = e T 1 e T 2 r T T ; andtheboundingfunction k y k isapositivegloballyinvertiblenon-decreasing function.ThefollowinginequalitiescanbedevelopedbasedonAssumption2,3 44

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and3, k N D k 2 ; k N B k 3 ; N D 4 ; N B 5 + 6 e 2 ; where i 2 R ;i =2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 areknownpositiveconstants. 3.4StabilityAnalysis Theorem3.1. Thecontrollawgivenin3and3 )]TJ/F50 11.9552 Tf 9.299 0 Td [(3ensuresallclosedloopsignalsareboundedandthekneejointtrackingerrorisregulatedinthesensethat j e 1 t j! 0 as t !1 ; providedthesufcientconditions 1 > 1 2 ; 2 > 6 +1 ; > 2 + 3 + 1 2 4 + 1 2 5 ; aresatised,where k; 1 ; 2 ; and arecontrollergains,respectively,and i ;i =2 )]TJ/F15 11.9552 Tf 12.312 0 Td [(6 areknownboundsofthetermsinthedynamicsystem.Furthermore,thecontroller asymptoticallyminimizesthecostfunctionin3providedconditionsin3and 3aresatised. Proof. Let D R 5 beadomaincontaining t =0 ; where t 2 R 5 isdenedas y t T p P v t p G t T ; where P v t 2 R isdenedasthegeneralizedFilippovsolutiontothefollowingdifferentialequation P v = r N B 1 + N D )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 +_ e 2 N B 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 6 e 2 2 ; P v e 2 t 0 ;t 0 j e 2 j)]TJ/F25 11.9552 Tf 17.933 0 Td [(e 2 N ; where 2 R isknownpositivecontrolgain.Similartothedevelopmentin3,existenceofsolutionsfor P v e 2 ;t canbeestablishedusingFilippov'stheoryofdifferential inclusionsfor P v 2 K [ h 2 ] r; e 2 ;e 2 ;t ,where h 2 r; e 2 ;e 2 ;t 2 R isdenedastherighthandsideof Pv .When ischosenaccordingtothesufcientconditionin3,then 45

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P v e 2 ;t 0 See[64]forproof.Theauxiliaryfunction G t 2 R in3isdenedas G 2 2 tr ~ W T )]TJ/F26 7.9701 Tf 7.315 -1.793 Td [(w ~ W + 2 2 tr ~ U T )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u ~ U ; where )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(u 2 R arepositivedenitematrices,and 2 2 R isapositivecontrolgain. Let V ;t : D [0 ; 1 R beapositive-denite,Lipschitzcontinuous,regular functiondenedas V L e 2 1 + 1 2 e 2 2 + 1 2 r 2 J + P v + G: V L ;t canbeupperandlowerbounded 1 k k 2 V L ;t 2 k k 2 ; where 1 2 2 R aresomeknownconstantsdenedas 1 = 1 2 min f 1 ; 0 g ; 2 = 1 2 max f 2 ; 1 g : In3, 0 and 1 2 R aredenedin2. UnderFilippov'sframework,ageneralizedLyapunovstabilitytheorycanbeused toestablishstrongstabilityoftheclosed-loopsystem y = h 3 ;t ,where h 3 ;t 2 R denotestheright-handsideoftheclosed-looperrorsignals.Thetimederivativeof 3existsalmosteverywherea.e.and V ;t a:e: 2 ~ V ;t where ~ V = 2 @V ;t T K e 1 e 2 r 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 v P v 1 2 G )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 G 1 T where @V isthegeneralizedgradientof V ;t [65].Since V ;t isaLipschitz continuousregularfunction, ~ V 2 e 1 e 2 rJ 2 P 1 2 v 2 G 1 2 1 2 J r 2 K [ ] T : Usingthecalculusfor K [ ] from[61], V L ;t canbedeterminedas ~ V 2 e 1 e 1 + e 2 e 2 + 1 2 r 2 J + rJ r + P v + G: 46

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Byutilizing3,3,and3,substitutingforthetimederivativeof P v and G; usingYoung'sInequality,and3,3,3canbedevelopedas ~ V a:e: )]TJ/F25 11.9552 Tf 31.804 0 Td [( k y k 2 + 2 k y k k y k 2 4 k s ; where =min f 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 6 ;R )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 g .Theexpressionin3canbefurther upperboundedbyacontinuous,positivesemi-denitefunction ~ V a:e: )]TJ/F25 11.9552 Tf 29.888 0 Td [( 3 k y k 2 8 y 2 D forsomepositiveconstant 3 2 R anddomain D = t 2 R 5 jk k < )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 p k s Largervaluesof k s willexpandthesizeofthedomain D .Theinequalitiesin3and 3canbeusedtoshowthat V L ;t 2L 1 in D .Thus, e 1 t ;e 2 t ;r t 2L 1 in D .Theclosed-looperrorsystemcanbeusedtoconcludethattheremainingsignalsare boundedin D ,andthedenitionsfor t canbeusedtoshowthat t isuniformly continuousin D .Let S D D denoteasetdenedas S D t D j 2 k k 2 < 1 )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 p k s 2 : Theregionofattractionin3canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k s .Theinequationin3canbeusedto indicatethat 3 k y t k 2 0 as t !18 y 2S D Basedonthedenitionof y t ,3canbeusedtoshowthat j e 1 t j! 0 as t !18 y 2S D Theresultsin3indicatesthatas t !1 ,3reducesto ^ f d + = h + d : 47

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Therefore,thedynamicsin3convergestothestatespacesystemin3. Hence, u t convergestoanoptimalcontrollawthatminimizes3subjectto3, providedtheconditionsin3,3,and3aresatised. 3.5ExperimentResults Thedevelopedcontrollerwasimplementedonable-bodiedvolunteerstoevaluate theperformance.ThesametestbedandprocedurewereusedasinChapter2. 3.5.1TrackingExperiments TrackingexperimentsfortheadaptivecontrollerinSectionVwereconducted onfourvolunteersonefemaleandthreemales,ages 22 )]TJ/F15 11.9552 Tf 9.299 0 Td [(40 yrs.usingthedesired trajectorywithafrequencyof 1 : 5 Hz andrangeofmotionROMbetween 5 and 35 ThesevalueswereselectedtoapproximatethefrequencyandROMofthelowerlimb duringwalking,butanysufcientlysmoothdesiredtrajectorycouldhavebeenselected. TheRootMeanSquareRMStrackingerroriscalculatedfrom 5 sto 20 s.The Q and R gainswereadjustedtoobtainthebestperformancewithoutregardtothecontrol input.ThemeanRMSerroris 4 : 2 withastandarddeviationSTDof 1 : 3 .Themean peak-to-peakabsoluteerroris 6 : 6 withaSTDof 1 : 7 .Theseresultsdemonstratethe performanceofthetrackingabilityoftheproposedcontroller.Figure3-2illustratesa typicalknee/limbtrackingerror. 3.5.2PerformanceTrade-offs Todemonstratetheabilityforacliniciantochoosedifferentcombinationsof Q and R toplaceagreateremphasisontrackingperformanceorfeedbackcontrolinput, trackingexperimentswereconducted.Twogroupsofexperimentswereconducted. Therstexperimentsxed 1 =1 andvaried R from 5 to 120 toillustratetheeffectof penalizingthecontrolinput.Additionalexperimentsvaried 1 between 0 : 5 to 4 foraxed R =20 toshowtheeffectofpenalizingtheperformance.Eachsessionwas20s,and RMSvalueswerecalculatedfortheerror,totalcontrolinput,andfeedbackcontrolinput, respectively.Figure3-3illustratesthatthefeedbackcontrolinputdecreasesandthe 48

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Figure3-1.Trackingtrajectoriesdashedline-desired,solidline-actualfora representativetrialonanable-bodiedindividual trackingerrorincreasesbyincreasing R .Figure3-4illustratesthattheerrordecreases andthefeedbackcontrolinputincreaseswithincreasing 1 .TheresultsinFigures 3-3and3-4representtheoutcomeforonevolunteer,butthesametrendswereobtained fortheothertwosubjectswithdifferentadjustableranges. 3.6Discussion AnadaptivecontrollerwhichincludesaNNtermandaRISEtermisusedto asymptoticallyminimizeagivencostfunction.Theoverallcontrollerisprovento achieveasymptotictrackinginthepresenceofboundedunmodeleddisturbances.The asymptoticadaptivecontrollerimplicitlycompensatesforthenonlinearuncertainties presentinthedynamicssuchaspassiveconstraintsonjointmovementandmuscle stimulationwhichincludenonlinearrecruitment,torque-angle,andtorque-velocity scaling,etc.Aquadraticcostfunctionalisadjustedtoscaletherelativepenaltyofthe trackingerrororthefeedbackcontrolportionoftheoverallcontrolinput. Asindicatedinthetrackingexperiments,ameanRMSerrorof 4 : 2 1 : 3 for5-20 secondswasachievedforthegivendesiredtrajectory.Thelimbpositionaccuracyfrom theseexperimentsissufcientfortypicalfunctionaltasks.Inadditiontodevelopinga 49

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Figure3-2.Trackingerrorforarepresentativetrialonanable-bodiedindividual controllerandassociatedstabilityproofthatcanyielddesiredtrackingerrorperformance,acontributionofthiseffortistodevelopaframeworktoadjusttheperformance versusdosage. Thetrade-offbetweentrackingperformanceandfeedbackcontroleffortcanbe achievedbychoosingdifferentvaluesof Q and R .Largergainsin Q yieldbettertracking performanceattheexpenseofalargerfeedbackcontroleffortwhilelargervaluesof R yieldreducedfeedbackcontroleffortwithlargertrackingerrors.Sincetheerrorterms z T t Qz t 2 R in3canberelatedtocontrollergainsas3,where K 2 R isnot includedinthecontroller,theerrorpenalty Q canbeincreasedbyincreasingeither 1 or R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 .Sinceincreasing R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 isequivalenttodecreasingthecontrolpenalty,whenthe controlpenaltyiskeptconstant,increasingtheerrorpenaltyonlycanbeimplemented byincreasing 1 .AsillustratedinFigure3-3,with 1 =1 ,increasing R from 5 to 120 resultsinareductionoftheRMSofthefeedbackcontrolinputfrom 5 : 18 voltsto 2 : 35 volts.Figure3-4illustratesthatwith R =20 ,increasing 1 from 0 : 5 to 4 reducesthe RMStrackingerrorfrom 13 : 94 to 5 : 48 .Furtherresearchisneededtoexaminethe effectsoffatigueduetoincreasedcontrolinput.Furthermore,thecurrentdevelopment doesnotincludetheentirecontrolinputinthecostfunctionali.e.,onlythefeedback 50

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Figure3-3.Typicalexperimentsonanable-bodiedsubjectwithvaried R from 5 to 120 for axed 1 =1 portionofthecontrollerisincluded.Theseresultsvalidatethetheoreticalabilityto directlyalterthefeedbackcontrolthrough R ,buttheresultsdonotshowacorrelation betweenchangesinthefeedbackportionversuschangesintheoverallcontroli.e.,the overallcontroloutputwasrelativelyinvarianttochangesin R However,heuristically,itiswellacceptedthatlargerfeedbackgainsresultin noiseamplicationandhigherfrequencycontrol.TheRMStotalcontrolinput R =5 haslessvaluethanthatwhen R =90 inFigure3-3.However,thetotalcontrolinput R =5 hasmuchhigherhighfrequency>6Hzcomponentsthanthatwhen R =90 in Figure3-5.Itisalsowellacceptedthathigherfrequencystimulationleadtomorerapid musclefatigue.Theseresultspointtotheneedforfurtherstudieswhichinvestigate therelationshipbetweenfatigueasafunctionoffeedbackcontrolversusfeedforward control. 51

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Figure3-4.Typicalexperimentsonanable-bodiedsubjectwithvaried 1 between 0 : 5 to 4 foraxed R =20 Figure3-5.Single-sidedamplitudespectrumsofthetotalcontrolinputfrom R =5 solid lineand R =90 dottedlineforthesameexperimentsinFig.3onan able-bodiedperson 52

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CHAPTER4 NEUROMUSCULARELECTRICALSTIMULATIONLIMBTRACKINGWITHAPULSED MODULATEDCONTROLINPUT Typically,stabilityanalysisforclosed-loopNMESignorethemodulatedimplementationofNMES.However,electricalstimulationisappliedtomuscleasmodulatedseries ofpulsesandthemodulationstrategyhassignicantimpactonthemuscleperformance andfatigue,theabilitytoexaminetheimpactofthecontrolsignalandmodulationstrategyinanalysismayopennewinsightintothedevelopmentofNMEScontrollers..Inthis chapter,forthersttime,amuscleactivationmodelwithapulsemodulatedcontrolinput isdevelopedtocapturethediscontinuousnatureofmuscleactivation,andaclosed-loop NMEScontrollerisdesignedandanalyzedfortheuncertainpulsemodulatedmuscle activationmodel.Semi-globaluniformlyultimatelyboundedSUUBtrackingisguaranteed.Thestabilityoftheclosed-loopsystemisanalyzedwithLyapunov-basedmethods, andapulsefrequencyrelatedgainconditionisobtained.Simulationresultsareprovided tovalidatethecontroller.Forthersttime,thispaperbringstogetherananalysisofthe controllerandmodulationscheme. 4.1MuscleActivationandLimbModel Thebodysegmentaldynamicsconsideredinthischapterarethesameasthose consideredinChapter2and3.Inthischapter,thetotalmuscletorque t generated attheknee-jointisconsideredasproductofanunknownnonlinearfunction q 2 R momentarmandthemusclecontractionforce x f q generatedbyelectricstimulation as x f : Aftersubstituting2and4,anddividingbothsidesby ; theexpressionin2 canbeexpressedas x f = J q + f 1 + 1 ; 53

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where J q ;f 1 q; q ; 1 q;t 2 R aredenedas J J I )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ;f 1 M e + M g + M v )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; 1 ds )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 : Musclecontractiondynamicscanbemodeledasarstorderdynamicsystem cf.[5,37,66,67],whichcanbeexpressedas x f + A f x f + f f + f = bu; where A f q ;f f q ;b q ; f t 2 R areuncertainfunctions.Theintroductionofthe unknownnonlinearfunctions A f q and f f q enablethemusclecontractiontobe consideredundergeneralconditionsinthesubsequentcontroldevelopment,and u t 2 R istheappliedelectricstimulationvoltage.Bysubstituting x f t and x f t ,the dynamicsin4canbeexpressedas J ... q = )]TJ/F25 11.9552 Tf 9.299 0 Td [(f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 + u; where J q ;f 2 q; q; q ; 2 t 2 R aredenedas J = b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 J ; f 2 b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 J + A f J q + b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 A f f 1 + f 1 + f f + b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 ; 2 b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f + A f 1 : Thefollowingassumptionsareusedtofacilitatethesubsequentcontroldevelopmentandstabilityanalysis. Assumption3 :Thefunction q isacontinuouslydifferentiable,nonzero,positive, monotonic,andboundedfunction[45]. Assumption4 :Thefunction b q isthemusclegainmusclerecruitmentwhich canbeassumedtobeacontinuouslydifferentiable,nonzero,positive,monotonic,and boundedfunction. 54

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Assumption5 :Thefunctions A f q ;f 2 m q ; 2 m t arecontinuouslydifferentiable andboundedfunctions. BasedonAssumptions2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(5,thefollowinginequalitycanbedeveloped 0 J 1 ; j 2 j 2 where 0 1 2 areknownpositiveconstants. Theelectricalpulseinput u t 2 R canbemodeledas u = 8 > < > : v;nT t < > : )]TJ/F25 11.9552 Tf 9.299 0 Td [(f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 + v;nT t
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and3.Acompositiveerrorsignal z e 1 t ;e 2 t ;r t 2 R 3 isdenedas z e 1 e 2 r T : Using4and3 )]TJ/F22 11.9552 Tf 9.299 0 Td [(3,theopen-looperrorsystemfor e 3 t canbedevelopedas J r = f 3 + 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(u; where f 3 q; q; q;q d ; q d ; q d ; ... q d 2 R isdenedas f 3 J ... q d + 1 + 2 Je 3 )]TJ/F31 11.9552 Tf 11.955 9.683 Td [()]TJ/F25 11.9552 Tf 5.48 -9.683 Td [( 2 1 + 1 2 + 2 2 Je 2 + 3 1 e 1 + f 2 : ByAssumptions2-5, f 3 canbeboundedas k f 3 k c + k z k k z k ; where c 2 R isaknownpositiveconstantand k z k 2 R isapositive,globalinvertible function. Basedon4andthesubsequentstabilityanalysis,theNMEScontrolleris designedas v = kr; where k 2 R isapositivegain.Theclosed-looperrorsystemfor r t is J r = 8 > < > : f 3 + 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(kr;nT t
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conditions: 1 > 1 2 ; 2 > 1 ;k> 1 ; 1 > 2 T )]TJ/F25 11.9552 Tf 11.955 0 Td [(d d ; where 1 isgivenin4, T and d areintroducedin4, 2 isaknownbounding constantthatdependson 0 ; 2 ; and c denedin4and4,and 1 isagain constantthatcanbemadearbitrarilylargebyselecting 1 ; 2 ; and k in3,3,and 4arbitrarilylarge. Proof. Let V z t 2 R beacontinuouslydifferentiablepositivedenitefunctiondened as V 1 2 z T z: From3 )]TJ/F22 11.9552 Tf 9.298 0 Td [(3,4,and4,thetimederivativeof4is V = )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 e 2 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 e 2 2 + e 1 e 2 + e 2 r + rJ )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f 3 + rJ )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(rJ )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 u: UsingYoung'sinequality, V z t canbeboundedas V )]TJ/F31 11.9552 Tf 23.91 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 e 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 e 2 2 + 1 2 r 2 + rJ )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 3 + rJ )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(rJ )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 u: Thefunction V z t canbeexpressedinsegments V n z; ,where V n z; 2 R is denedas V n z; V z;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(nT ; where t )]TJ/F25 11.9552 Tf 11.955 0 Td [(nT;n b t=T c : Ontheinterval 0
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where 0 1 2 isgivenin4,and c and k z k isgivenin4.Whentheconditionsin4and4hold,aftercompletingthesquares V n )]TJ/F15 11.9552 Tf 23.113 8.088 Td [(1 2 1 k z k 2 + 1 ; 8k z k2 D D = z t 2 R 3 jk z k < )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 0 q 0 k )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 1 where 0 2 R isdenedas 0 min 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 2 k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ; 1 2 R isapositiveconstant,whichcanbemadearbitrarylargebyincreasingthe controlgains 1 2 ,and k ,and 1 2 R isapositiveconstantthatcanbemade arbitrarilysmallbyselecting k arbitrarilylarge.Largervaluesof 1 2 ,and k willexpand thesizeofthedomain D toincludeanyinitialconditionsi.e.,asemi-globaltypeof stabilityresult.Ontheinterval d < > : )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 V n + 1 ; 0
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Byusing4and4,andthefactthat V n +1 z; 0= V n z;T ; thedifference between V n +1 z; 0 and V n z; 0 denedas ~ V n z is ~ V n = V n +1 z; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(V n z; 0 V n z;d e 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 11.9552 Tf 11.955 0 Td [(V n z; 0+ 2 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(e 2 T )]TJ/F26 7.9701 Tf 6.587 0 Td [(d )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; V n z; 0 )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d e 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + 1 1 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d e 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d + 2 2 )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(e 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 : Basedon4, ~ V n z < 0 i.e., V z >V z T >V z T > when V n z; 0 > d 2 where d 2 R isdenedas d> s 2 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 T )]TJ/F26 7.9701 Tf 6.587 0 Td [(d + 1 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 1 d e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d : wherethegainconditionin4determinesthesizeof d basedontheperiod,pulse width,andcontrolgains.Basedon4, z t 2 D uniformlyconvergestotheultimate bound k z t k < p 2 d providedthesufcientconditionsin4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(4aresatised. k e t k issemi-global uniformlyultimatelybounded[68,Theorem4.18]inthesensethat k e t kk z t k < p 2 d; 8 t T )]TJ/F15 11.9552 Tf 7.545 -6.529 Td [( d; k z k ; 8k z k2D ; where T )]TJ/F15 11.9552 Tf 7.546 -6.529 Td [( d; k z k 2 R isapositiveconstantthatdenotestheultimatetimetoreach theball. Remark 4.1 Basedon4and6,theinterplaybetweenthemodulation strategyandthecontrollercanbedetermined.Tominimizemusclefatigue,oneis motivatedtodecreasethestimulationfrequencyi.e.,increaseT.From46and 6,decreasingthestimulationfrequencyindicatesthatthecontrolgainsshould beselectedlargermaking 1 largerandthattheultimateerrorwillbelarger.ifthe 59

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frequencyisincreasedleadingtofastermusclefatiguethenthecontrolgainscanbe selectedlowerandalowerultimateboundcanbeobtained. Theultimateerrorboundin4hasterms e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 T )]TJ/F26 7.9701 Tf 6.586 0 Td [(d and e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d relatedtothe stimulationfrequencyandcontrolgains,respectively.Thecontrolgainsonlyreducethe contributionoftheterm e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d intheultimateerrorbound,whiletheterm e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 T )]TJ/F26 7.9701 Tf 6.587 0 Td [(d inthe ultimateerrorboundcanonlybereducedviaincreasingthestimulationfrequency. 4.4Experiments Theproposedcontrollerwasimplementedonable-bodiedvolunteersmales, ages26-42yrs.toevaluatetheperformance.Thesametestbedandprocedurewere usedasinChapter2.Theelectricalstimulationisdeliveredwithaconstantpulsewidth of 400 s andapulsefrequencyof 30 Hz or 100 Hz .Theamplitudeoftheelectricalpulses ismodulatedbytheoutputofthecontroller.AButterworthlowpasslterwithcutoff frequencyof1000Hzwasusedtoreducethenoisein e 2 t .Noweightwasattachedto theweightingbar.Anysufcientlysmoothdesiredtrajectorycouldhavebeenselected. Thedesiredangulartrajectorywasselectedas q d = 8 > < > : 65 2 +sin 1 ; 25 t + 3 2 ;t< 1 : 25 30+sin 1 : 25 t + 3 2 +5 ;t 1 : 25 ; Thetrajectorywasasinusoidaltrajectorywithaperiodof2.5sandrangeofmotion ROMbetween5and60seeFigure4-1.Thelargestanglemotioncanbeachieved is80ontheLEM.TheselectionofthisROMwastogetalargeROMandleavesome roomforovershooting. TheexperimentresultsaresummarizedinTable4-1,wherethepeaktrackingerror wascalculatedas max j e t j .InTable4-1,thetotalRMSrootmeansquareerroris recorded,aswellastheRMSerrorin10secondintervals.Arepresentativetriali.e. C-leftinTable4-1isshowninFigures4-1 )]TJ0 0 1 rg 0 0 1 RG/F22 11.9552 Tf 12.622 0 Td [(4-3. 60

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Table4-1.Trackingerrorsforasinusoidaltrajectorywithaperiodof2.5sandROM between5and60.Thestimulationfrequencyis30Hz. Subject RMSPeak Total0-10s10-20sTotal0-10s10-20s A-left4.014.643.2610.8710.876.42 A-right4.074.633.4313.6413.647.10 B-left4.415.203.4411.1811.187.23 B-right3.914.523.1914.9614.969.78 C-left3.834.173.839.599.599.27 C-right4.034.173.898.698.698.69 Mean4.044.553.5111.495.608.08 STD0.180.350.272.181.491.24 Figure4-1.Desiredsolidlineandmeasureddashedlinetrajectories.Thestimulation pulsefrequencyis30Hz Forcomparison,thecontrollerwasimplementedat100Hzstimulationpulse frequencyonthesamegroupofsubjects.ThetrackingerrorsarelistedinTable4-2. Arepresentativetriali.e.A-rightinTable4-2isshowninFigures4-4and4-5.The RMSandpeakerrorsfrom0to10sand0to20smarkedby*sinTable4-2areboth statisticallylowerstudentT-test,onetail,paired,p<0.05thantheresultsobtainedusing 30Hzstimulationpulsefrequency.NostatisticaldifferencewasdeterminedintheRMS andpeakerrorintherange10to20s.Thelowererrorduringtheinitial10seconds andfrom0-20secondsispredictedbythetheoreticalanalysis.Thatis,highfrequency stimulationyieldsastrongercontractionthatcanbeusedtodecreasethetrackingerror. 61

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Figure4-2.Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. Thestimulationpulsefrequencyis30Hz. Thelackofstatisticdifferencefrom10-20sisduetothefactthatthehigherfrequency stimulationresultsinamorerapidmusclefatigue.TheerrorsinFigure4-4showa continuousincreasingtrend.Toillustratethefactthathighfrequencyfatiguesthemuscle morerapidlythanlowfrequency[23],A30vpulsetrainisappliedtoSubjectB-leftat 100Hzfor20s,andafter5minutesrest,a35vpulsetrainat30Hzwasappliedfor20s. Theappliedvoltageat30Hzwereselectedtoinitiallyhavesimilaroutputsastheresult at100Hzi.e.,theresponseof30vat100Hzwas21.7-25.4andtheresponseof35vat 30Hzwas22.3-24vinthetimeintervalof1-2s.Thepositionofthelowerlegisgivenin Figure4-6,whichillustratetherapidonsetoffatiguewithhighfrequencypulsetrains. 62

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Figure4-3.ControlinputVoltageforarepresentativesinusoidaltrajectorytracking experiment.Thestimulationpulsefrequencyis30Hz. Table4-2.Thetrackingerrorsforasinusoidaltrajectorywithaperiodof2.5sandROM between5and60.Thestimulationfrequencyis100Hz.The*indicatesa statisticallysignicantdifferencep<0.05whencomparedtothe30Hz experiments. Subject RMSPeak Total0-10s10-20sTotal0-10s10-20s A-left3.082.763.367.565.727.56 A-right2.602.232.916.274.976.27 B-left3.663.213.998.026.718.02 B-right3.982.794.8711.696.1811.69 C-left3.283.393.167.066.537.06 C-right2.812.902.726.755.816.75 Mean3.23*2.88*3.507.89*5.99*7.89 STD0.470.370.731.790.581.79 63

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Figure4-4.Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. Thestimulationpulsefrequencyis100Hz. Figure4-5.ControlinputVoltageforarepresentativesinusoidaltrajectorytracking experiment.Thestimulationpulsefrequencyis100Hz. 64

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Figure4-6.Constantvoltageresponseat30Hzsolidlineand100Hzdashedline 65

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CHAPTER5 NEUROMUSCULARELECTRICALSTIMULATIONWITHAUNCERTAINMUSCLE CONTRACTIONMODEL Inthischapter,uncertaintiesinthemusclecontractiondynamicsaretakeninto considerationwhencompensatingforthemusclecontractiondynamics.Accountingfor themusclecontractiondynamicsisachallengebecauseofuncertainty,nonlinearityand thefactthatthecontractionstatesarenotmeasurable.Aneural-networkNN-based controllertogetherwithadynamicNN-basedidentierisdesignedtoenablesemiglobaluniformlyultimatelyboundedtrackingofadesiredlimbtrajectoryandon-line estimationofthelimbacceleration.Theoverallstabilityoftheidentier-controllersystem isanalyzedthroughLyapunovmethods.Simulationresultsareprovidedtoillustratethe controllerperformance. 5.1MuscleActivationandLimbModel Thesamebodysegmentalandmusclecontractiondynamicsareusedasthosein Chapter4exceptthatinput u t 2 R isnotmodulated.Thedynamicscanbeexpressed as J ... q = )]TJ/F25 11.9552 Tf 9.299 0 Td [(f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 + u: 5.2ControlDevelopment Thecontrolobjectiveistoensurethekneeangle q t tracksadesiredtrajectory, denotedby q d t 2 R ,whichisanessentialtaskinmanyrehabilitativeexercisesand functionrestorationtasks.Toquantifythetrackingobjective,alowerlimbangular positiontrackingerror,denotedby e t 2 R ,isdenedasin2.Tofacilitatethe subsequentcontroldesignandstabilityanalysis,lteredtrackingerrorsdenotedby e 1 t ;e 2 t 2 R ,arealsodenedas e 1 e + 1 e; e 2 e 1 + 2 e 1 ; 66

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where 1 and 2 2 R arepositiveconstantcontrolgains.Using2,5 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5, e 2 t canbeexpressedas e 2 = q d )]TJ/F15 11.9552 Tf 12.814 0 Td [( q + 1 + 2 q d )]TJ/F15 11.9552 Tf 14.115 0 Td [(_ q + 1 2 e: Thesubsequentdevelopmentisbasedontheassumptionthat q t and q t aremeasurable.Theerrordynamicsin5dependontheunmeasurablelimbacceleration.To compensatefortheaccelerationdependency,anerrorestimationisdesignedbasedon 5as ^ e 2 q d )]TJ/F15 11.9552 Tf 12.814 3.155 Td [( ^ q + 1 + 2 q d )]TJ/F15 11.9552 Tf 14.115 3.155 Td [(_ ^ q + 1 2 e:; where ^ q t ; ^ q t 2 R denotesthesubsequentlydesignedobserveroutput. Tofacilitatethesubsequentanalysis,let f 2 d q d ; q d ; q d 2 R bedenedas f 2 d b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d ; q d f 1 q d ; q d )]TJ/F25 11.9552 Tf 11.955 0 Td [(J I b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 q d q d ; q d q d + J I b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d A f q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d q d + b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d A f q d ; q d M e q d + M g q d + M v q d + b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d M e q d ; q d + M g q d ; q d + M v q d ; q d )]TJ/F25 11.9552 Tf 11.955 0 Td [(b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 q d q d ; q d M e q d + M g q d + M v q d : Basedontheuniversalfunctionapproximationproperty[69],theunknownfunctionin 5canbeapproximatedbyamulti-layerNNwhichisdenedas f 2 d = W T )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V T X d + "; where X d q d ; q d ; q d 2 R 4 isdenedas X d 1 q d q d q d T ; and W 2 R n 1 V 2 R 4 n denotetheidealweightsforthehiddenlayerneuronsandthe inputlayerneurons,respectively,wherethenumberofhiddenlayerneuronsisselected 67

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as n X d 2 R denotesthereconstructionerror.Since X d q d ; q d ; q d dependsonthe knownboundeddesiredtrajectory,theNNinputisguaranteedtolieonacompactset. Assumption6 :Theactivationfunction anditsrstorderderivativewith respecttoitsarguments 0 areboundedbyknownconstants[47]. Assumption7 :Thereconstructionerror X d anditsrstorderderivativewith respecttoitsarguments 0 X d areboundedbyknownconstants[47]. Byusing4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5and5,theopen-looperrorsystemfor e 2 t canbe obtainedas J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Je 2 + f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(f 2 d + W T )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V T X d + 1 2 Je 2 + J ... q d + 2 + )]TJ/F25 11.9552 Tf 11.955 0 Td [(u: Let ^ W t 2 R n 1 ^ V t 2 R 4 n betheestimatedweightsfor W;V ,and X d ; ^ X d ;t ; ^ 0 X d ;t ; ~ X d ;t ; ~ W t 2 R n 1 ~ V t 2 R 4 n bedenedas V T X d ; ^ ^ V T X d ; ^ 0 @ ^ V T X @ ^ V T X j ^ V T X = ^ V T X d ; ~ )]TJ/F15 11.9552 Tf 12.57 0 Td [(^ ; ~ W W )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W; ~ V V )]TJ/F15 11.9552 Tf 13.742 3.022 Td [(^ V: ByusingaTaylorseriesapproximation, ~ X d ;t canbeexpressedas ~ =^ 0 ~ V T X d + o ~ V T X d 2 ; where o 2 2 R denoteshigherorderterms.Byusing5 )]TJ/F22 11.9552 Tf 9.299 0 Td [(7, W T V T X d can beexpressedas W T V T X d = ^ W T ^ 0 ~ V T X d + W T ^ + ~ W T ^ 0 ~ V T X d + W T o ~ V T X d 2 : 68

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Basedon5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(5,theerrorsystemin5canbeexpressedas J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Je 2 + f 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(u + W T ^ + ^ W T ^ 0 ~ V T X d ; where f 3 q; q; q;q d ; q d ; q d ;t 2 R aredenedas f 3 f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(f 2 d + ~ W T ^ 0 ~ V T X d + W T o ~ V T X d 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(J 2 1 e + J 1 + 2 e 1 + 1 2 Je 2 + J ... q d + 2 + ": Iftheupdatelaws ^ W t 2 R n 1 ; ^ V t 2 R 4 n areselectedas ^ W proj )]TJ/F26 7.9701 Tf 11.866 -1.793 Td [(w ^ ^ e 2 ; ^ V proj )]TJ/F26 7.9701 Tf 11.867 -1.793 Td [(v X d ^ e 2 ^ W T ^ 0 ; where proj isasmoothprojectionoperator[62]. ^ e 2 ~ W T ^ +^ e 2 ^ W T ^ 0 ~ V T X d + G 1 =0 ; where G 1 t 2 R isdenedas G 1 1 2 tr ~ W T )]TJ/F29 7.9701 Tf 7.315 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 w ~ W + 1 2 tr ~ V T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 v ~ V ; where )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w 2 R n n ; )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(v 2 R 4 4 arepositivedenitematricesand tr denotesthetrace ofamatrix.Itisstraightforwardtoshowthat G 1 t 0 .Since proj guarantees ^ W t ^ V t tobebounded, ~ W t ; ~ V t areboundedbyusing5andthefactthat W and V arebounded.Since X d q d ; q d ; q d isbounded,thefollowingboundcanbeobtainedas ~ W T ^ + ^ W T ^ 0 ~ V T X d a 1 ; where a 1 2 R isaknownpositiveconstant.Since W;V; ; ^ W t ; ~ W t ; ~ V t ; ^ X d ; 0 ^ V T X d ;o ~ V T X d ; and X d arebounded,usingtheMeanValueTheorem,5 andtheassumptionthat ... q d t isbounded, f 3 q; q; q;q d ; q d ; q d ;t 2 R canbeboundedas j f 3 j a 2 + 1 k z f k k z f k ; 69

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where a 2 2 R isapositiveconstant, z f e t ;e 1 t ;e 2 t 2 R 3 isdenedas z f ee 1 e 2 T ; and 1 k z f k 2 R isapositive,globallyinvertiblefunction.Basedon5,5,and thesubsequentstabilityanalysis,thecontrolinputisdesignedas u = k f ^ e 2 + ^ W T ^ ; where k f 2 R isapositivecontrolgain.Aftersubstituting5into5,the closed-looperrorsystemcanbeobtainedas J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Je 2 + f 3 + ~ W T ^ + ^ W T ^ 0 ~ V T X d )]TJ/F25 11.9552 Tf 11.955 0 Td [(k f ^ e 2 : 5.3ObserverDesign Theobjectiveofthissectionistodesignanobserver/identiertogeneratethe estimationof ^ q t ,whichisusedin ^ e 2 t in5,sothatthecontrollerin5canbe implementedwithonlymeasurementsof q t and q t Tofacilitatethefollowingobserverdesign,dene x t ^ x t ~ x t r t 2 R 2 z t 2 R 4 as x q q T ; ^ x ^ q ^ q T ; ~ x x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x; r ~ x + ~ x = r 1 r 2 T ; z ~ x T r T T ; 70

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where 1 + 2 2 R isaconstantcontrolgain.Byusing5and5,the differencebetween e 2 t and ^ e 2 t yieldstheaccelerationestimationerroras ^ e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 = r 2 ; whichcanbeboundedas j r 2 jk r k : Aftersubstituting2,and4,thedynamicsin2canbeexpressedas J I q + M e + M g + M v + d = x f ; whichcanberewrittenas x = )]TJ/F25 11.9552 Tf 9.298 0 Td [(x + g 1 + d; where x t isdenedin5and g 1 q; q ;d t 2 R 2 aredenedas g 1 x + 0 B @ q )]TJ/F25 11.9552 Tf 9.298 0 Td [(J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 I M e + M g + M v + J )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 I x f 1 C A ; d 0 B @ 0 )]TJ/F25 11.9552 Tf 9.299 0 Td [(J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 I d 1 C A : Let g 1 d q d ; q d 2 R 2 bedenedas g 1 d x d + 0 B @ q d f d 1 C A ; where f d q d ; q d 2 R isdenedas f d J )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 I q d x f q d ; q d )]TJ/F25 11.9552 Tf 11.956 0 Td [(J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 I M e q d + M g q d + M v q d ; and x d q d q d 2 R 2 isdenedas x d q d q d T : 71

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Theunknownfunction g 1 d q d ; q d 2 R 2 canbeapproximatedbyamulti-layerNNwitha reconstructionerroras g 1 d = W T 1 1 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V T 1 x d + 1 ; where W 1 2 R n 1 2 V 1 2 R 2 n 1 denotetheidealweightsforthehiddenlayerneurons andtheinputlayerneurons,respectively,wherethenumberofhiddenlayerneuronsis selectedas n 1 ; 1 x d 2 R denotesthereconstructionerror. Assumption8 :Theactivationfunction 1 anditsrstorderderivativewith respecttoitsarguments 0 1 areboundedbyknownconstants[47]. Assumption9 :Thereconstructionerror 1 x d anditsrstorderderivativewith respecttoitsarguments 0 1 x d areboundedbyknownconstants[47]. Thedynamicsin5canberewrittenas x = )]TJ/F25 11.9552 Tf 9.299 0 Td [(x + g 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(g 1 d + W T 1 1 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V T 1 x d + 1 + d: Basedon5,amulti-layerdynamicNNobserverisdesignedas ^ x = )]TJ/F25 11.9552 Tf 9.298 0 Td [( ^ x + ^ W T 1 1 ^ V T 1 ^ x + ; where ^ W 1 t 2 R n 1 2 ^ V 1 t 2 R 2 n 1 denotetheestimatedweightsfor W 1 ;V 1 ,and ~ x 2 R 2 isdenedas k ~ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(k ~ x + t 0 k ~ xd; where k 2 R ispositivecontrolgain. Basedon5and5,theobservererrordynamicscanbewrittenas ~ x = )]TJ/F25 11.9552 Tf 9.298 0 Td [( ~ x + 1 + 2 + g 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(g 1 d + d )]TJ/F25 11.9552 Tf 11.956 0 Td [(; where 2 x 2 R 2 isdenedas 2 W T 1 1 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(V T 1 x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T 1 1 ^ V T 1 ^ x : 72

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Aftersomealgebraicmanipulation,thetimederivativeof5canbewrittenas 2 = W T 1 0 1 V T 1 x d )]TJ/F29 7.9701 Tf 17.117 13.715 Td [( ^ W T 1 ^ 1 ^ V T 1 ^ x )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T 1 ^ 0 1 ^ V T 1 ^ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(W T 1 ^ 0 1 V T 1 ^ x + ~ W T 1 ^ 0 1 ~ V T 1 ^ x + ^ W T 1 ^ 0 1 ~ V T 1 ^ x + ~ W T 1 ^ 0 1 ^ V T 1 ^ x; where ~ W 1 t W 1 )]TJ/F15 11.9552 Tf 17.285 3.022 Td [(^ W 1 t 2 R n 1 2 ~ V 1 t V 1 )]TJ/F15 11.9552 Tf 14.106 3.022 Td [(^ V 1 t 2 R 2 n 1 denotetheestimated mismatchesfortheidealweightestimates.Basedonthesubsequentstabilityanalysis, theupdatelaws ^ W 1 t 2 R n 1 2 ; ^ V 1 t 2 R 2 n 1 aredesignedas ^ W 1 proj )]TJ/F26 7.9701 Tf 11.866 -1.794 Td [(w 1 ^ 0 1 ^ V T 1 ^ x ~ x T ; ^ V 1 proj )]TJ/F26 7.9701 Tf 11.867 -1.794 Td [(v 1 ^ x ~ x T ^ W T 1 ^ 0 1 ; where )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w 1 2 R n 1 n 1 )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(v 1 2 R 2 2 arepositivedenitegainmatrices.Basedon5, ~ x T ^ W T 1 ^ 0 1 ~ V T 1 ^ x + ~ W T 1 ^ 0 1 ^ V T 1 ^ x + G 2 =0 ; where G 2 t 2 R isdenedas G 2 1 2 tr ~ W 1 T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 w 1 ~ W 1 + 1 2 tr ~ V T 1 )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 v 1 ~ V 1 : ByusingtheMeanValueTheorem,Assumptions 8 )]TJ/F15 11.9552 Tf 12.551 0 Td [(9 and5,thefollowing inequalitiescanbeobtained N 1 2 k y k k y k + a 3 ; N 2 a 4 k z k + a 5 k z f k + a 6 where N 1 y ;N 2 y 2 R aredenedas N 1 W T 1 0 1 V T 1 x d )]TJ/F29 7.9701 Tf 17.118 13.715 Td [( ^ W T 1 ^ 1 ^ V T 1 ^ x )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T 1 ^ 0 1 ^ V T 1 ^ x )]TJ/F25 11.9552 Tf 11.956 0 Td [(W T 1 ^ 0 1 V T 1 ^ x + ~ W T 1 ^ 0 1 ~ V T 1 ^ x +_ 1 + d +_ g 1 q; q; q )]TJ/F15 11.9552 Tf 13.672 0 Td [(_ g 1 q d ; q d ; q d ; N 2 ^ W T 1 ^ 0 1 ~ V T 1 ^ x + ~ W T 1 ^ 0 1 ^ V T 1 ^ x; 73

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and a i 2 R i =4 )]TJ/F15 11.9552 Tf 12.277 0 Td [(7 arepositiveconstants, 2 k y k 2 R ispositive,globallyinvertible functionand y t 2 R 7 isdenedas y t z T t z T f t T .Byusing5,5, 5,5,and5,theobservererrorsystemin5canberewrittenas r = )]TJ/F25 11.9552 Tf 9.299 0 Td [(kr + N 1 + N 2 : 5.4StabilityAnalysis Theorem5.1. Theclosed-loopsystemin5andtheobserversystemin5 togetherwiththecontrollawin5and5andtheupdatelawsin5and 5ensurethatallclosed-loopsignalsarebounded,andthetrackingerrorissemiglobaluniformlyultimatelyboundedSUUBprovidedthecontrolgains k k f 1 2 areselectedaccordingtothefollowingconditions: 1 > 1 2 ; 2 ;k> 1 ;k f > 2 ; min 1 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 4 k f )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 >a 5 ; min )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ; 1 2 k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k f > 2 a 4 + a 5 ; where a 4 ;a 5 2 R arepositiveconstantsintroducedin5. Proof. Let D R 9 beadomaincontaining t =0 ; where t 2 R 9 isdenedas y T p G 1 p G 2 T ; andconsidertheLyapunovcandidatefunction V L : D! R ,whichiscontinuously differentiablepositivedenitefunctiondenedas V L 1 2 e 2 + 1 2 e 2 1 + 1 2 Je 2 2 + 1 2 ~ x T ~ x + 1 2 r T r + G 1 + G 2 ; whichsatisesthefollowinginequalities U 1 V L ';t U 2 : 74

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In5, U 1 ;U 2 2 R arecontinuouspositivedenitefunctionsdenedas U 1 1 2 min ; 0 k k 2 ;U 2 max ; 1 2 1 k k 2 : Takingtimederivativeof5,substitutingthedynamicsin5and5,and using5yields V L = ee 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 e 2 + e 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e 2 1 +^ e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 2 ~ W T ^ + ^ W T ^ 0 ~ V T X d + e 2 f 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k f e 2 e 2 + r 2 + r T N 1 + r T N 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(kr T r +~ x T r )]TJ/F15 11.9552 Tf 12.679 0 Td [(~ x T ~ x + G 1 + G 2 : UsingtheYoung'sInequalitytogetherwith5 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5,5and5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(5 yields V L )]TJ/F31 11.9552 Tf 30.552 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 e 2 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k f )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 e 2 2 + a 1 k r k + a 2 k z f k + 1 k z f k k z f kj e 2 j + 1 2 k f k r k 2 + 2 k y k k y kk r k + a 3 k r k +2 k z k a 4 k z k + a 5 k z f k + a 6 )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 r T r )]TJ/F31 11.9552 Tf 11.956 16.857 Td [( )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ~ x T ~ x; whichcanberewrittenas V L )]TJ/F15 11.9552 Tf 28.56 0 Td [(2 1 k z f k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 k z k 2 + a 2 k z f k + a 1 +2 a 6 k z k)]TJ/F15 11.9552 Tf 21.785 8.088 Td [(1 4 kr T r + a 3 k r k )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 4 kr T r + 2 k y k k y kk r k)]TJ/F15 11.9552 Tf 21.785 8.088 Td [(1 4 k f e 2 2 + 1 k z f k k z f kj e 2 j ; where 1 ; 2 2 R arepositiveconstantsthatcanbemadearbitrarilylargebyincreasing thecontrolgains k k f 1 2 ,whicharedenedas 1 1 2 min 1 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 4 k f )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(a 5 ; 2 1 2 min )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ; 1 2 k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k f )]TJ/F15 11.9552 Tf 11.956 0 Td [( a 4 + a 5 : 75

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Bycompletingthesquares,thefollowinginequalitycanbeupperboundedas V L )]TJ/F31 11.9552 Tf 30.552 16.857 Td [( 1 )]TJ/F25 11.9552 Tf 13.532 8.088 Td [( 2 1 k f k z f k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 k z k 2 + 2 2 k k y k 2 + 3 )]TJ/F25 11.9552 Tf 28.56 0 Td [( 4 k y k 2 + 3 ; where 3 ; 4 2 R aredenedas 3 a 2 2 4 1 + a 1 +2 a 6 2 4 2 + a 2 3 k ; 4 min 1 )]TJ/F25 11.9552 Tf 13.532 8.088 Td [( 2 1 k f )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k ; 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k : If k y k 2 > 3 4 andthesufcientconditionsin5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(5aresatised, V t isasymptoticallydecreasinguntil k y t k 2 enterstheultimatebound 3 4 .Theregionof attraction D isdenedas D n t R 9 jk k min )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 kk f ; )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p 2 k o ; where )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 R istheinverseofthefunction k 2 1 + k f 2 2 2 R ,.Theregion ofattraction D in5canbemadearbitrarilylargetoincludeanyinitialcondition byincreasingthecontrolgain k and k f i.e.,asemi-globaltypeofstabilityresult.The ultimatebound 3 4 canbemadearbitrarilysmallbyincreasingthecontrolgain k and k f Hence,thetrackingerrorissemi-globaluniformlyultimatelyboundedSUUB. Theinequalityin5andthefactthat V t 0 when k y t k 2 > 3 4 canbeused toshowthat e t ;e 1 t ;e 2 t ; ~ x t ;r t ;G 1 t ;G 2 t 2L 1 in D .Given e t ;e 1 t ; e 2 t ; ~ x t ;r t 2L 1 in D andusing5,5,and5,standardlinearanalysis methodscanbeusedtoprovethat e 1 t e 2 t ; ~ x t 2L 1 in D .Using4,5, 5,andtheassumptionthat q d t anditsderivativesareboundeduptothirdorder, q t ; q t ; q t 2L 1 in D canbeproven.Given q t ; q t ; q t ~ x t 2L 1 andusing 5,itcanbeshownthat ^ q t ; ^ q t ; ^ q t 2L 1 in D .Using5,itcanbeshown ^ e 2 t 2L 1 in D .Given W;V;W 1 ;V 1 areboundedbyassumptionsand ^ W t ; ^ V t ; ^ W 1 t ; ^ V 1 t areboundedbyusing proj u t 2L 1 in5in D canbeshown : 76

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Given e 1 t e 2 t ; ~ x t 2L 1 andusing5,itcanbeshownthat t 2L 1 in D Thedenitionfor t canbeusedtoprovethat t iscontinuousin D 5.5Simulation Simulationsareperformedusingamodiedmusclemodelgivenin[70].The controllercomputesavoltageastheinputtothesimulatedmuscledynamics.The simulationresultsareshowninFigures5-1 )]TJ0 0 1 rg 0 0 1 RG/F22 11.9552 Tf 9.299 0 Td [(5-4.forthecontrolgains k f =0 : 05 ; 1 =20 ; 2 =20 ; )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w = diag [1 ; 1 ; 1 ; 1 ; 1 ; 0 : 1] 0 : 01 ; )]TJ/F26 7.9701 Tf 7.314 -1.794 Td [(v = diag [1 ; 1 ; 0 : 1 ; 0 : 01] 0 : 01 ; k = diag [96 ; 112] ; =20 ; )]TJ/F26 7.9701 Tf 7.314 -1.793 Td [(w 1 =[0 : 02 ; 0 : 02 ; 0 : 02 ; 0 : 02 ; 0 : 02] T ; )]TJ/F26 7.9701 Tf 7.314 -1.794 Td [(v 1 =[0 : 02 ; 0 : 02 ; 0 : 02 ; 0 : 02 ; 0 : 02] : Figure5-1andFigure5-2depictthetrackingperformance.Thecontrolinputdepicted inFigure5-3iswithinatypicalrangeforquadricepsstimulation.Figure5-4depictsthe accelerationestimation. Figure5-1.Actualsolidanddesireddashedtrajectories 77

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Figure5-2.Limbpositiontrackingerror Figure5-3.Unmodulatedinputcontrolvoltage 78

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Figure5-4.Estimatedsolidandactualdashedaccelerations 79

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CHAPTER6 IDENTIFICATION-BASEDCLOSED-LOOPNEUROMUSCULARELECTRICAL STIMULATIONLIMBTRACKINGWITHAPULSEDMODULATEDCONTROLINPUT InChapter4,aNMEScontrollerisdesignedandanalyzedconsideringthepulse modulatedcontrolinputwhichbringstogetherananalysisofthecontrollerandmodulationscheme.Alimitationofthedesignisthataccelerationhastobeusedtoimplementationofthecontrollerbecausetheaccelerationobtainedfrompositionderivationis noisy.InChapter5,anidentication-basedcontrollerisdevelopedforthemuscle-limb modelwhichincludesanuncertainrstorderdynamicsystemthatmodelsmuscle contractiondynamics.Thecontrollerdevelopedcanbeimplementedbyonlyusing positionandvelocitysignals.Inthischapter,basedonthetwoapproachesinChapters 4and5,amuscleactivationmodelwithapulsemodulatedcontrolinputisdeveloped tocapturethediscontinuousnatureofmuscleactivation,andanidentication-based closed-loopNMEScontrollerisdesignedandanalyzedfortheuncertainpulsemuscle activationmodel.AfeedforwardNNtermisincludedinthecontrollertoachievebetter trackingperformance.Semi-globaluniformlyultimatelyboundedSUUBtrackingis guaranteed.Theclosed-loopsystemisanalyzedthroughLyapunov-basedmethodsand apulsefrequencyrelatedgainconditionisobtained.Experimentresultsareprovidedto illustratetheperformanceofthedevelopedcontroller. 6.1MuscleActivationandLimbModel ThedynamicsusedinthischapterarethesameasthoseintheChapter4. 6.2ControllerDevelopment Thecontrolobjectiveistoensurethekneeangle q t tracksadesiredtrajectory, denotedby q d t 2 R ,whichisanessentialtaskinmanyrehabilitativeexercisesand functionrestorationtasks.Toquantifythetrackingobjective,alowerlimbangular trackingerror,denotedby e 1 t 2 R ,isdenedin3.Tofacilitatethesubsequent controldesignandstabilityanalysis,lteredtrackingerrorsdenotedby e 2 t ;e 3 t 2 R 80

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arealsodenedin3and3.Anerrorestimationisdesignedbasedon5as in5. Tofacilitatethesubsequentanalysis,let f 2 d q d ; q d ; q d 2 R bedenedas f 2 d b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d f 1 q d ; q d )]TJ/F25 11.9552 Tf 11.956 0 Td [(J I b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 q d q d ; q d q d + J I b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d A f q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d q d + b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d A f q d ; q d M e q d + M g q d + M v q d + b )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d M e q d ; q d + M g q d ; q d + M v q d ; q d )]TJ/F25 11.9552 Tf 11.956 0 Td [(b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q d ; q d )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 q d q d ; q d M e q d + M g q d + M v q d : Basedontheuniversalfunctionapproximationproperty[69],theunknownfunctionin 6canbeapproximatedbyamulti-layerNNwhichisdenedas f 2 d = W T )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V T X d + "; where X d q d ; q d ; q d 2 R 4 isdenedasin5,and W 2 R n 0 1 V 2 R 4 n 0 denote theboundedidealweightsforthehiddenlayerneuronsandtheinputlayerneurons, respectively,wherethenumberofhiddenlayerneuronsisselectedas n 0 ,and X d 2 R denotesthereconstructionerror.Since X d q d ; q d ; q d dependsontheknownbounded desiredtrajectory,theNNinputisguaranteedtolieonacompactset. Assumption10: Theactivationfunction anditsrstorderderivativewith respecttoitsarguments 0 areboundedbyknownconstants[47]. Assumption11: Thereconstructionerror X d anditsrstorderderivative 0 X d areboundedbyknownconstants[47]. Byusing4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(5and6,theopen-looperrorsystemfor e 2 t is J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Je 2 + f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(f 2 d + W T )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V T X d + 1 2 Je 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(J 2 1 e + J 1 + 2 e 1 + J ... q d + 2 + )]TJ/F25 11.9552 Tf 11.955 0 Td [(u: 81

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Let ^ W t 2 R n 0 1 ^ V t 2 R 4 n 0 betheestimatedweightsfor W;V ,and X d ;t ; ^ X d ;t ; ^ 0 X d ;t ; ~ X d ;t ; ~ W t 2 R n 0 1 ~ V t 2 R 4 n 0 bedenedas5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(7. ByusingaTaylorseriesapproximation, ~ X d ;t canbeexpressedas ~ =^ 0 ~ V T X d + o ~ V T X d 2 ; where o 2 2 R n 0 1 denoteshigherorderterms.Byusing5 )]TJ/F22 11.9552 Tf 9.298 0 Td [(7, W T V T X d canbeexpressedas W T V T X d = ^ W T ^ 0 ~ V T X d + W T ^ + ~ W T ^ 0 ~ V T X d + W T o ~ V T X d 2 : Theupdatelaws ^ W t 2 R n 1 ^ V t 2 R 4 n canbearbitrarilyselectedas ^ W proj ; ^ V proj ; where proj isasmoothprojectionoperator[62].Gradient-basedupdatelawswere usedinthefollowingexperiments.Since proj guarantees ^ W t ; ^ V t arebounded, ^ W T ^ a 1 ; where a 1 2 R isaknownpositiveconstant. Theerrorsystemin6canbeexpressedas J e 2 = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 Je 2 + f 3 + ^ W T ^ )]TJ/F25 11.9552 Tf 11.955 0 Td [(u; where f 3 q; q; q;q d ; q d ; q d ;t 2 R isdenedas f 3 f 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(f 2 d + ~ W T ^ + ^ W T ^ 0 ~ V T X d + ~ W T ^ 0 ~ V T X d + W T o ~ V T X d 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [(J 2 1 e + J 1 + 2 e 1 + 1 2 Je 2 + J ... q d + + 2 : Since W;V; ; and X d arebounded,usingtheMeanValueTheorem,6,and theassumptionthat ... q d t isbounded, f 3 q; q; q;q d ; q d ; q d ; ... q d ;t 2 R canbeboundedas j f 3 j a 2 + 1 k z f k k z f k ; 82

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where a 2 2 R isapositiveconstant, z f e t ;e 1 t ;e 2 t 2 R 3 isdenedas z f ee 1 e 2 T ; and 1 k z f k 2 R isapositive,globallyinvertiblefunction.Basedon 6,6,andthesubsequentstabilityanalysis,thecontrolinputisdesignedas v = k f ^ e 2 + ^ W T ^ ; where k f 2 R isapositivecontrolgain.Aftersubstituting4,6into6,the closed-looperrorsystemcanbeobtainedas J e 2 = 8 > < > : )]TJ/F24 7.9701 Tf 10.494 4.707 Td [(1 2 Je 2 + f 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k f ^ e 2 ;nT t 1 2 ; 2 ;k> 1 ;k f > 2 ; min 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 4 k f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 > 1 2 a 5 ; min )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ; 1 2 k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 a 5 >a 4 ; 1 d T )]TJ/F25 11.9552 Tf 11.955 0 Td [(d > 3 ; 83

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where a 4 ;a 5 2 R arepositiveconstantsintroducedin5, T d 2 R areintroduced in4, 3 2 R isaknownpositiveboundingconstant,and 1 2 R isagainconstant thatcanbemadearbitrarilylargebyselecting 1 ; 2 ;k f k 2 R in5,5,6,and 5arbitrarilylarge. Proof. ConsidertheLyapunovcandidatefunction V : R 7 R ,whichiscontinuously differentiablepositivedenitefunctiondenedas V 1 2 e 2 + 1 2 e 2 1 + 1 2 Je 2 2 + 1 2 ~ x T ~ x + 1 2 r T r; whichsatisesthefollowinginequalities 1 k k 2 V 2 k k 2 ; where 1 ; 2 2 R arepositiveconstantsdenedas 1 1 2 min ; 0 ; 2 1 2 max ; 1 : Takingtimederivativeof6,substitutingthedynamicsin6and5andusing 5and5yields V = ee 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 e 2 + e 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e 2 1 + e 2 f 3 + e 2 ^ W T ^ )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 u + r T N 1 + r T N 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(kr T r +~ x T r )]TJ/F15 11.9552 Tf 12.68 0 Td [(~ x T ~ x: Thefunction V t canbeexpressedinsegments V n '; ,where V n '; 2 R is denedas V n '; V nT + ; where t )]TJ/F25 11.9552 Tf 12.163 0 Td [(nT;n b t=T c : UsingtheYoung'sInequalitytogetherwith6,5, 5,and5ontheinterval 0
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where 1 ; 2 2 R aredenedas 1 1 2 min 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 4 k f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 a 5 ; 2 min )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; 1 2 k )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 k f )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 a 5 )]TJ/F25 11.9552 Tf 11.955 0 Td [(a 4 ; providedthesufcientconditionsin6 )]TJ/F22 11.9552 Tf 9.298 0 Td [(6aresatised.Completingthe squaresyields V n '; )]TJ/F31 11.9552 Tf 30.552 16.857 Td [( 1 )]TJ/F25 11.9552 Tf 13.532 8.088 Td [( 2 1 k f k z f k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 k z k 2 + 2 2 k k k 2 + a 2 2 4 1 + a 3 + a 6 2 k : Letaset D bedenedas D n t R 7 jk k min )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 kk f ; )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p 2 k o ; where )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 R istheinverseofthefunction k 2 1 + k f 2 2 2 R .Using6, 6canberewrittenas V n '; )]TJ/F25 11.9552 Tf 21.918 0 Td [( 1 V n + 2 ; 8k k2D ; where 1 ; 2 2 R aredenedas 1 1 2 min 1 )]TJ/F25 11.9552 Tf 13.533 8.088 Td [( 2 1 k f )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k ; 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k ; 2 a 2 2 4 1 + a 3 + a 6 2 k : Theregionofattraction D in6canbemadearbitrarilylargetoincludeanyinitial conditionbyincreasingthecontrolgain k and k f i.e.,asemi-globalresult. Likewise,ontheinterval d
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Using6and6, V n '; is V n '; 8 > < > : )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 V n '; + 2 ; 0 1 d 2 ,then V nT V T >V T > where d 2 R isdenedas d s 4 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 3 T )]TJ/F26 7.9701 Tf 6.587 0 Td [(d + 2 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 1 d 1 e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 3 T )]TJ/F26 7.9701 Tf 6.587 0 Td [(d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 1 d ; andthesizeof d isdeterminedbasedontheperiod,pulsewidth,andcontrolgains. Given6,6,6,and6, k e t k issemi-globaluniformlyultimately bounded[68,Theorem4.18]inthesensethat k e t kk t k < d; 8 t T )]TJ/F15 11.9552 Tf 7.545 -6.529 Td [( d; k k ; 8k k2D ; 86

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Table6-1.Thetrackingerrorsforasinusoidaltrajectorywithaperiodof2.5sandROM between5and60DataarecomingfromTable4-1.Thestimulation frequencyis30Hz. SubjectRMSSteadystateRMS-20sSteadystatepeak-20s A-left4.013.265.18 A-right4.073.433.75 B-left4.413.446.35 B-right3.913.195.27 C-left3.833.838.44 C-right4.033.894.61 Mean4.043.515.60 STD0.180.271.49 where T )]TJ/F15 11.9552 Tf 7.546 -6.529 Td [( d; k k 2 R isapositiveconstantthatdenotestheultimatetimetoreach theball. 6.5Experiments Theproposedcontrollerwasimplementedonable-bodiedvolunteersmales, ages26-42yrs.toevaluatetheperformance.Thesametestbedandprocedurewere usedasinChapter2.Thestimulationfrequencywas30Hzandnoweightwasattached totheweightbar.Nolow-passlterwasusedtosmooththefeedbacksignals.Any sufcientlysmoothdesiredtrajectorycouldhavebeenselected.Onetrajectorywasa sinusoidaltrajectorywithaperiodof2.5sandrangeofmotionROMbetween5and 60whichwasusedinChapter4seeFigure6-1. Toillustratethattheproposedcontrollercanachievebetterresults,thetracking errorsusingthecontrollerproposedinChapter4isre-listedinTable6-1.NotethatTable 6-1isthesamedateasinTable4-1,printedhereforcomparison.Thesteadystate RMSandpeakerrorwerecalculatedfromtheinterval10-20s. TheexperimentsresultsfromtheproposedcontrollerinthischapteraresummarizedinTable6-2.SamesubjectswereusedasinChapter4.Thepeaktracking errorwascalculatedas max j e t j .ThesteadystateRMSandpeakerrorsusingthe proposedcontrollerarebothstatisticallylowerstudentT-test,onetail,paired,p<0.05 thantheresultsobtainedusingthecontrollerinChapter4.Nostatisticaldifferencewas 87

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determinedinthetotalRMSerrorbecausethecontrollerwithafeedforwardNNterm hadbiggertransienterrors.Arepresentativetriali.e.,A-leftinTable6-2isshownin Figures6-2and6-3. Table6-2.Thetrackingerrorsforasinusoidaltrajectorywithaperiodof2.5sandROM between5and60.Thestimulationfrequencyis30Hz.The*indicatesa statisticallysignicantdifferencep<0.05whencomparedtotheresultsfrom Chapter4Table6-1. SubjectRMSSteadystateRMSSteadystatepeak A-left4.691.825.18 A-right4.471.423.75 B-left2.912.226.35 B-right3.242.435.27 C-left6.053.188.44 C-right4.952.134.61 Mean4.372.20*5.60* STD1.050.541.49 Figure6-1.Performanceofarepresentativesinusoidaltrajectorytrackingexperiment. Thedesiredtrajectoryisplottedasasolidlineandthemeasuredtrajectory isplottedasadashedline. Toillustratetheabilityoftrackingmorecomplextrajectories,anirregularcontinuous trajectorywithvariedperiodandROMwasselectedseeFigure6-4.Thetracking errorsaresummarizedinTable6-3.Acomparabletrackingperformancewasachieved. Arepresentativetriali.e.D-leftinTable6-3isgiveninFigure6-4. 88

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Figure6-2.Trackingerrorofarepresentativesinusoidaltrajectorytrackingexperiment. ThetrackingerrorisplottedasasolidlineandtheRMSerroroverevery4 secondsisplottedasadashedline. Table6-3.Thetrackingerrorsforanirregulartrajectory SubjectRMSPeak A-left3.8910.1 A-right3.4513.6 B-left3.8913 B-right3.4513.15 D-left2.738.49 Mean3.4811.67 STD0.422.01 Figure6-3.ControlinputVoltageforarepresentativesinusoidaltrajectorytracking experiment.Thetotalcontrolinputisplottedasasolidlineandthe contributionofNNisplottedasadashedline. 89

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Figure6-4.Trajectorytrackingperformanceofanirregulartrajectory.Thedesired trajectoryisplottedasasolidlineandthemeasuredtrajectoryisplottedas adashedline. 90

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CHAPTER7 HYBRIDNEUROMUSCULARELECTRICALSTIMULATIONTRACKINGCONTROLOF ANKLE NMESisaneffectiverehabilitationtoolforgaitretrainingforindividualssuffering fromvariousneurologicaldisorders.Traditionally,NMESisonlydeliveredtoactivate ankledorsiexormusclesduringtheswingphaseofthegaittocorrectfootdrop. Recentresearchindicatesthatimprovedfunctionalambulationcanbeachievedby deliveringNMEStoboththeplantarexoranddorsiexormusclesduringgait.Closedloopelectricalstimulationhasthepotentialtoyieldpositiverehabilitativeoutcomesby enablingaccurateandpreciselimbmotionsduringgaitretraining.Naturally,themotion ofankleduringgaitisanevent-drivensystemcombiningcontinuousevolutionofthe anglebetweenthefootandshank,alternatemovingsegmentsofthefootandshank, andalternateactivationoftheplantarexoranddorsiexormuscles.Inthischapter,a switchedslidingmodebasedcontrollerisdevelopedtoensurethattheankletracksa designedorrecordednormaltrajectoryduringgaitwhichcanbeusedforgaitretraining. Semi-globalasymptotictrackingofthehybridcontrollerisanalyzedusingmultiple Lyapunovfunctionsandtheperformanceisillustratedthoughsimulations. 7.1MuscleActivationandLimbModel Whilethearcsofanklemotionduringwalkingarenotlarge,theyarecriticalfor progressionandshockabsorptionduringstance[42].Duringnormalgait,thearcsof anklemotioncontinuouslyplantarexandthendorsiex.Duringthestancephase, theankleplantarexes,dorsiexesandthenplantarexesagain.Duringtheswing phasetheankleonlydorsiexes.Theactivatedmusclesswitchbetweendorsiexorand plantarexormusclesduringeachgaitcycle. Eachgaitcyclestartsfromheelstrike,thebeginningofthestancephase.The anklepositionstartsatneutral.Thedorsiexormuscletibialisanteriorbecomes activeimmediatelyafterheelstriketotoestriketodeceleratetherateofplantarexion, whichcontributestoshockabsorption,bodyweightacceptance,andlimbprogression. 91

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Throughoutheelstriketotoestrikephase,themovingsegmentisthefootandtheleg remainsrelativelystationary.Aftertoestrike,theforefootcontactstheoorandthe footbecomesstationary.Themovingsegmentisthelegshank.Theplantarexor calfmusclesgraduallyincreaseseccentriccontractiontocontrolankledorsiexion andprovidecriticalstabilizationthatallowsboththefootandtibiatomoveforwardand provideforwardpropulsionpush-off.Bytheendofterminalstance,withtheweight shiftingtotheotherleg,thestabilizingforceinthefootgoesaway,andthefootisfree toplantarexcorrespondingtotheactivationofplantarexormusclegastrosoleus whiletheonsetofdorsiexormuscletibialisanterioractivitydeceleratesthefootfall thiscoactivationofdorsiexorandplantarexormusclesismodeledastheactivation ofplantarexormusclesonlyforsimplicity.Attoe-off,theswingphasebeginsand thesecondarcofdorsiexionstarts.Dorsiexormusclesactivateagaintoliftthefoot clearoftheground.Attheendoftheswingphasetheanklepositionreturnstoneutral preparingforheelcontact[42].AsummaryofthegaitcycleisprovidedinTable7-1. Table7-1.Summaryofanklemotions,activatedmusclegroups,andlimbmovements duringgaitcycle GaitCycle StancePhaseSwingPhase AnkleMotionPlanta-Dorsi-Planta-DorsiActivatedMuscleGroupDorsi-Planta-Planta-DorsiMovingPartFootShankShankFoot SwitchingSignal1234 Thedynamicsofamuscle-limbsystemismodeledasthesameasthatinChapter 2.Tocapturetheswitchingpropertyofgaitcycle,theankledynamicscanbemodeled as J q = )]TJ/F25 11.9552 Tf 9.298 0 Td [(M )]TJ/F25 11.9552 Tf 11.955 0 Td [( d + u ; 92

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where :[0 ; 1 !P ,denotesapiecewiseconstantswitchingsignalwhichcanbe expressedas = p;p 2P 1 ; 2 ; 3 ; 4 Forexample, t couldbethesignalfromfootswitchesindicatingthetransitionofgait phases. 7.2ControlDesign Thecontrolobjectiveistoensurethattheanklefollowsdesignedorrecordedankle trajectoriesduringnormalgait,whichisessentialinrehabilitativeexercisesandfunction restoration. Toquantifythetrackingobjectiveandfacilitatethesubsequentcontroldesignand stabilityanalysis,anangularpositiontrackingerror,denotedby e t 2 R andaltered trackingerror,denotedby r t 2 R ,aredenedasin2.Thecontrolobjectiveisto ensurethattheanklefollowsdesignedorrecordedankletrajectoriesduringnormalgait, whichisessentialinrehabilitativeexercisesandfunctionrestoration. Controllersaredesignedforeachsubsystemindividually,whichisindicatedbya subscript p 2P 1 ; 2 ; 3 ; 4 .Takingthederivativeof r t in2,multiplying bothsidesby J p; andusing7and2,yields J p r = J p q d + e + M p + dp )]TJ/F25 11.9552 Tf 11.955 0 Td [(u p : Basedon70andthesubsequentstabilityanalysis,thecontrollawisdenedas u p = k s p r + e + p sgn r ; where k s p ; p 2 R areadjustablegains,and y t 2 R 2 isdenedas y = er : 93

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Aftersubstituting7into7andperformingsomealgebraicmanipulation,the closed-looperrordynamicscanbeexpressedas J p r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 J p r + ~ N p + N D p )]TJ/F25 11.9552 Tf 11.955 0 Td [(k s p r )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 11.9552 Tf 11.956 0 Td [( p sgn r ; wheretheauxiliaryfunctions N p e;r;q; q; q d ;N D p q d ; q d ; q d ;t ; ~ N p e;r;q; q;q d ; q d ; q d ;t 2 R aredenedas N p 1 2 J p r + J p q d + e + M p ; ~ N p N p )]TJ/F25 11.9552 Tf 11.955 0 Td [(J p q d q d )]TJ/F25 11.9552 Tf 11.956 0 Td [(M p q d ; q d ; N D p J p q d q d + M p q d ; q d + dp : ThefollowinginequalitycanbedevelopedbasedonAssumption1and2, N D p 4 p ; where 4 p 2 R isaknownpositiveconstant.Thecontrolgain k c canbeadjustedto reduce 4 p .TheMeanValueTheoremcanbeusedtodevelopthefollowingupperbound ~ N p p k y k k y k ; where p k y k 2 R isapositive,globallyinvertiblefunction. 7.3StabilityAnalysis Foreachgaitcycle,let t 0 =0 and t 4 = T; where T 2 R denotestheperiodofa gaitcycle.Thetimeintervalbetweentwoswitches,denotedby T i 2 R i =1 ; 2 ; 3 ; 4 ; is denedas T i = t i )]TJ/F25 11.9552 Tf 11.955 0 Td [(t i )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; 94

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where t i denotetheswitchingtimes.Dene 1 p ; 2 p 2 R ;p 2P as 1 p 1 2 min ; 0 p ; 2 p 1 2 max ; 1 p ; where 0 p ; 1 p areintroducedin2. Theorem7.1. Thecontrollaw u t = u t t ensuresallclosed-loopsignalsare boundedandsemi-globalasymptotictrackinginthesensethat j e t j! 0 as t !1 ; providedthecontrolgainsareselectedsufcientlylargebasedontheinitialconditionsofthestatesseethesubsequentstabilityanalysisandthefollowingsufcient conditionsaresatised: p > 4 p ;p 2P ; 4 X p =1 3 p 2 p T p > 4 X p =1 log 2 p 1 p ; aresatised,where p 2 R isintroducedin7, 4 p isintroducedin7 1 p ; 2 p 2 R ; areintroducedin7,and7,and 3 p 2 R arepositiveconstantsdeterminedby theinitialconditionofthesystemandthecontrolgains and k s p Proof. Foreachphaseindicatedby t ,consideracontinuouslydifferentiable,radially unbounded,positivedenitefunction V p e;r;t 2 R p 2P denedas V p = 1 2 e 2 + 1 2 r 2 J p : Using2, V p e;r;t canbeupperandlowerboundedas 1 p k y k 2 V p 2 p k y k 2 ; where 1 p ; 2 p 2 R aredenedin7and7.Aftertakingthetimederivativeof 7,andusing2and7, V p e;r;t canbeexpressedas V p = )]TJ/F25 11.9552 Tf 9.298 0 Td [(e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k s p r 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( p j r j + ~ N p r + N D p r: 95

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Using7and7, V p t canbeupperboundedas V p )]TJ/F25 11.9552 Tf 28.56 0 Td [(e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k s p r 2 + p k y k j r jk y k)]TJ/F25 11.9552 Tf 20.59 0 Td [( p j r j + 4 p j r j ; whichcanbefurtherupperboundedas V p )]TJ/F31 11.9552 Tf 23.91 16.857 Td [( min ; 3 4 k s p )]TJ/F25 11.9552 Tf 13.151 9.168 Td [( 2 p k y k k s p k y k 2 )]TJ/F25 11.9552 Tf 21.918 0 Td [( 3 p k y k 2 ; providedthesufcientconditionsin7issatised,andwhere 3 p 2 R isapositive constantprovided and k s p areselectedsufcientlylargebasedontheinitialcondition oftheactivatedsubsystem.Theregionofattraction D p isdenedas D p y t R 2 jk y k )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 p r min ; 3 4 k s p k s p ; Thatis,theregionofattractioncanbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain and k s p i.e.,asemi-globaltypeofstability result.Using7and7,andsolvingtheresultingdifferentialequationyields V p t V p e )]TJ/F26 7.9701 Tf 6.587 0 Td [( p t ; where p 2 R isaconstantdenedas p 3 p 2 p : Providedtheconditionin7issatised,thecontrolinputandalltheclosed-loop signalsareboundedduring t = p in D p Eventhougheachcontrollerisexponentiallystable,additionaldevelopmentis requiredtoexaminethestabilityofthecompositesystem.Tothisend,7canbe usedtoconclude V t i t i i V t i )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 t i ; 96

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where t i = i +1 ,andtheconstant i 2 R i =1 ; 2 ; 3 ; 4 isdenedas i 2 t i 1 t i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 : From7and7, V t i t i i V t i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 t i i V t i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 t i )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F26 7.9701 Tf 6.586 0 Td [( t i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 T i : Iterating7for i =1 to 4 yields V t 4 t 4 4 V t 3 t 4 4 V t 3 t 3 e )]TJ/F26 7.9701 Tf 6.587 0 Td [( 4 T 4 V t 0 t 0 e )]TJ/F26 7.9701 Tf 6.587 0 Td [( ; where ; 2 R aredenedas 1 2 3 4 ; 1 T 1 + 2 T 2 + 3 T 3 + 4 T 4 : Basedon7and7,thefollowinginequalitycanbedeveloped: V 1 t 4 )]TJ/F25 11.9552 Tf 11.955 0 Td [(V 1 t 0 )]TJ/F31 11.9552 Tf 30.552 9.683 Td [()]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( V 1 t 0 )]TJ/F31 11.9552 Tf 30.552 9.684 Td [()]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 1 1 k y t 0 k 2 : Thisresultcanbegeneralizedas V t i t i + T )]TJ/F25 11.9552 Tf 11.955 0 Td [(V t i t i )]TJ/F31 11.9552 Tf 23.91 9.684 Td [()]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [( 1 t i k y t i k 2 : Provided k s and p areselectedsufcientlarge,and 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.587 0 Td [( > 0 ; 7,7,7,and7canbeusedtoconcludeasymptotictracking[43,71]. Theswitchedcontrollaw u t = u t t ensuresallclosed-loopsignalsarebounded, 97

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and j e t j! 0 as t !1 in D .Theregionofattraction D isdenedas D D 1 D 2 D 3 D 4 : Theregionofattractioncanbemadearbitrarilylargetoincludeanyinitialconditionsby increasingthecontrolgain and k s p i.e.,asemi-globaltypeofstabilityresult. 7.4Simulation Simulationsareperformedusingamodiedmodelbasedonthemodelgiven in[70].Thecontrollercomputesavoltageasaninputtothesimulatedmusclemodel. ThesimulationresultsareshowninFigs.7-1 )]TJ0 0 1 rg 0 0 1 RG/F22 11.9552 Tf 9.299 0 Td [(7-3.Adesiredtrajectoryisdesignedto simulateanaveragetrajectorycf.theexperimentaldatain[42]whichisgivenas q d =90+ pf + ps )]TJ/F25 11.9552 Tf 11.955 0 Td [(pf tt d 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(15 tt d 4 4+6 tt d 5 ; where pf = A i ;ps = A i +1 ;tt = t mod T )]TJ/F25 11.9552 Tf 12.688 0 Td [(T t i ;d = T t i +1 )]TJ/F25 11.9552 Tf 12.688 0 Td [(T t i ; where T 2 R isthegaitperiod; T t 2 R 5 isthetimeperiodforeachphase; A 2 R 4 isthe amplitudeofeachphase;and i 2 R isthephaseindicatordenedas T =2 : 5 s;T t = [ 00 : 12 T 0 : 48 T 0 : 62 TT ] ;A =[ 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(710 )]TJ/F15 11.9552 Tf 9.298 0 Td [(20 ] ; and i = 8 > > > > > > > < > > > > > > > : 10 t mod T < 0 : 12 T 20 : 12 T t mod T < 0 : 48 T 30 : 48 T t mod T < 0 : 62 T 40 : 62 T t mod T
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of 15 )]TJ/F15 11.9552 Tf 12.157 0 Td [(30 V .Thecontrolinputsarewithinatypicalrangeandthetrackingerrorindicate thecontrollercanyieldfunctionalgaits. Figure7-1.Actualsolidlineanddesireddashedlinetrajectories Figure7-2.Trackingerror 99

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Figure7-3.Computedvoltageascontrolinputforthedosiexationmusclegroup Figure7-4.Computedvoltageascontrolinputfortheplantaexationmusclegroup 100

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CHAPTER8 CONCLUSIONANDFUTUREWORK 8.1Conclusion Millionsofpeoplesufferingfromdisabilityandparalysiscouldbenetfromadvancementsinclosed-loopNMES.Potentially,moreefcientrehabilitationandmore precisemotorfunctioncanbeachievedthroughclosed-loopcontrolofNMES.Inthis dissertation,Lyapunov-basedmethodsareusedtodesignandanalyzeNMEScontrollers.Variedproblemsareaddressedtobuildtheframeworkforfutureresearchandto improvetheefcacyandtheperformanceoftheclosed-loopNMEScontroller. InChapter 2 ,anadaptiveinverseoptimalNMEScontrollerisdevelopedforlower limbtrajectorytrackinginthepresenceofparametricuncertaintyandexternaldisturbancesinthemuscleactivationandlimbdynamicsmodel.ALyapunov-basedstability analysisisusedtoprovethatthedevelopedNN-basedcontrolleryieldsuniformlyultimatelyboundedUUBtrackingwhilesimultaneouslyminimizingacostfunctional. Experimentsonablebodiedvolunteersvalidatetheperformanceoftheproposed controllerforalimbtrackingtaskforwalkingspeedtrajectoriesandafunctionalstand fromasittingpositiontask.Experimentsalsoillustratetheabilitytoalterthecontrol performancethroughweightingthecostfunction. InChapter 3 ,aNMEScontrollerthatachievesasymptotictrackingandminimizes aquadraticcostfunctionalisachieved.Theoverallcontrolstructureasymptotically convergestoanoptimalcontrollerforaspecicdynamicsystem.Theresultingcontroller achievesasymptoticerrortrackingwhileconvergingtoanoptimalcontroller.ANN feedforwardcomponentisincorporatedwithaRISEfeedbackcontrollertoimprovethe transientandsteadystateresponseandreducethecontroleffort.ALyapunov-based analysisisusedtoprovethelowerlimbasymptoticallytracksadesiredtimevarying angulartrajectory.Thecontrollerisalsoproventoasymptoticallyminimizeagivencost functional.Experimentsonhealthynormalvolunteersvalidatetheperformanceofthe 101

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proposedcontrollerforalimbtrackingtaskforwalkingspeedtrajectories.Experiments alsoillustratetheabilitytoalterthecontrolperformancethroughweightingthecost function. InChapter 4 ,amuscleactivationmodelwithapulsemodulatedcontrolinputis analyzed.Aclosed-loopNMEScontrollerisdesignedbasedonanuncertainmuscle activationmodel.Thecontrollerensuresthatthekneeangletracksadesiredtrajectory whichcanbeusedforrehabilitationandfunctionalrestorationpurposes.ALyapunovbasedmethodisusedtoconcludeUUBtrackingbasedonsufcientconditionsonthe gainsandmodulationparameters.Forthersttime,ananalysisofthecontrollerwith modulationschemeisillustrated. InChapter 5 ,asystemidentierisdevelopedtoidentifythelimbdynamicsand estimateacceleration.Thecontrollercanbeimplementedonlyusingpositionand velocitysignals.Accelerationisnotrequiredtoimplementthecontrollerasitwas requiredin[5]and[37].Forthersttime,anidentication-basedcontrollerisdeveloped forthemuscle-limbmodelwhichincludesanuncertainrstorderdynamicsystemthat modelsmusclecontractiondynamics.Theparametersofthelimbdynamicsandthe musclecontractionmodelareunknown.Thedesignedidentier-controllersystemis analyzedthroughLyapunovmethods.Semi-globaluniformlyultimatelyboundedSUUB trackingandasymptoticidenticationareguaranteed.Simulationresultsareprovidedto illustratethecontrollerperformance. InChapter 6 ,anklemotioncontrolismodeledasahybridsystemandaswitched controllercomprisedofmultipleslidingmodebasedcontrollersisdesignedfortherst timetoenabletheankletotrackdesiredtrajectoriesduringgait.Semi-globalasymptotic trackingresultoftheswitchedcontrollerduringgaitisanalyzedbasedonmultiple Lyapunovfunctionsandtheperformanceisillustratedthoughsimulations. 102

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8.2FutureWork Theworkinthisdissertationaddressessomeimportantchallengesinclosed-loop controlofNMES.Frameworksaredevelopedtofacilitatefutureresearch.However, severalremainingquestionsneedforfurtherinvestigation: 1.Optimalframeworksaredevelopedtofacilitateapractitioner'sabilitytobalance controleffortsandperformance.Howcandifferentrehabilitationgoalsbeachieved byselectingproceduresthatemphasizeontrackingperformanceoverthat emphasizeonrepetitionnumbers?Moreworkhastobedonetodevelopoptimal controllerthatincludesoverallcontrolinputinsteadofjustpartialfeedbackofthe totalcontroleffortlikethecurrentapproaches. 2.Reducingmusclefatigueisstillanopenquestion.Approachesbyselecting differentmodulationstrategiesrelatedtostimulationintensity,frequency,and intervalhavebeenpublished.Howtoincorporatetheseopen-loopresultsintothe closed-loopcontrollerdesigntoimprovethefatigueresistanceperformancecould beapathworthtopursue.InChapter4,ahybridanalysisisdemonstratedand futureresultscanbuildonfrequencyoptimization. 3.Onecauseoftheearlyonsetofmusclefatigueisduetosynchronousrecruitment ofmotorunits.Asynchronousexcitementofmotorunitscanbeachievedbyusing multiplesurfaceorimplantedelectrodes.Controllersthatcanhandletherapid switchingbetweenpairsofelectrodesshouldbedevelopedtoinvestigatethe possibilitytoreducefatigue. 4.Mostmusclesworkcollaboratively.Acontrollerthatcanorganizetheagonistand theantagonistandutilizeco-contractionswouldbehighlyvaluedintheapplication offunctionalrestoration. 5.Currentapproachesarebasedononejointmodels,theseresultscanbeextended tomulti-jointsituationstosolvemorerealisticclinicproblems. 103

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6.ModernbiomedicaltechniquessuchasMRImagneticresonanceimaging,NIRS nearinfraredspectroscopycanprovideinsidemusclemetabolicinformation. Incorporatingtheseusefulinformationintoclosed-loopcontroldesignisinteresting andpromisingtoreducingmusclefatigue. 104

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[67]F.Zajac,Muscleandtendon:properties,models,scaling,andapplicationto biomechanicsandmotorcontrol, Crit.Rev.Biomed.Eng. ,vol.17,no.4,pp. 359,1989. [68]H.K.Khalil, NonlinearSystems ,3rded.PrenticeHall,2002. [69]G.Cybenko,Approximationbysuperpositionsofasigmoidalfunction, Math. ControlSignalsSyst. ,vol.2,pp.303,1989. [70]M.Ferrarin,F.Palazzo,R.Riener,andJ.Quintern,Model-basedcontrolofFESinducedsinglejointmovements, IEEETrans.NeuralSyst.Rehabil.Eng. ,vol.9, no.3,pp.245,Sep.2001. [71]R.Goebel,R.Sanfelice,andA.Teel,Hybriddynamicalsystems, IEEEContr. Syst.Mag., ,vol.29,no.2,pp.28,2009. 110

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BIOGRAPHICALSKETCH QiangWangwasborninHubei,China.Hereceivedhisbachelor'sdegreein biomedicalengineeringfromHuazhongUniversityofScienceandTechnology.He receivedhismaster'sandPhDdegreesinbiomedicalengineeringfromPekingUnion MedicalCollege.HejoinedtheNonlinearControlsandRoboticsNCRresearchgroup topursuehisseconddoctoraldegreeinelectricalengineering,undertheadvisementof Dr.WarrenE.Dixon.HereceivedhisPhDinelectricalandcomputerengineeringfrom theUniversityofFloridaintheFallof2012. 111