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Control of Cycling Induced by Functional Electrical Stimulation

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Title:
Control of Cycling Induced by Functional Electrical Stimulation a Switched Systems Theory Approach
Creator:
Bellman, Matthew J
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (129 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
DIXON,WARREN E
Committee Co-Chair:
CRANE,CARL D,III
Committee Members:
BANKS,SCOTT ARTHUR
WHEELER,BRUCE
Graduation Date:
12/18/2015

Subjects

Subjects / Keywords:
Cadences ( jstor )
Control systems ( jstor )
Electric motors ( jstor )
Electrical stimulation ( jstor )
Knee joint ( jstor )
Sine function ( jstor )
Sufficient conditions ( jstor )
Torque ( jstor )
Trajectories ( jstor )
Velocity ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
controls -- cycling -- fes -- lyapunov -- rehabilitation -- robotics -- switching
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
Functional electrical stimulation (FES) can be used to activate the dysfunctional lower limb muscles of individuals with some neurological disease or injury to produce cycling as a means of exercise, rehabilitation, and transportation. However, FES-cycling is metabolically inefficient and yields lower power output at the cycle crank than able-bodied cycling. Previous literature suggests that these problems are symptomatic of poor muscle control and non-physiological muscle fiber recruitment. The latter is a known problem with FES in general, and the former motivates investigation of better control methods for FES-cycling. In Chapter 1, motivation for FES-cycling is introduced along with a survey of the challenges, approaches, and shortcomings of FES-cycling in the literature. In Chapter 2, a nonlinear model of a stationary FES-cycling system is developed that includes parametric uncertainty and an unknown, bounded, time-varying disturbance. A stimulation pattern for the gluteal, quadriceps femoris, and hamstrings muscle groups is designed based on the kinematic effectiveness of the rider's hip and knee joints to produce a forward torque about the cycle crank. In Chapter 3, a switched sliding mode controller is designed for the uncertain, nonlinear cycle-rider system with autonomous, state-dependent switching. The switched controller yields ultimately bounded tracking of a desired cadence, provided sufficient conditions on the control gains, desired trajectory, initial conditions, and stimulation pattern are satisfied. Stability is derived via a common Lyapunov-like function, and experimental results demonstrate the performance of the switched controllers under typical FES-cycling conditions. In Chapter 4, the results of Chapter 3 are extended to include the control objective of crank position tracking, and experimental results demonstrate improved performance over cadence tracking alone. Chapter 5 builds on the results of Chapter 4 by including an electric motor in the FES-cycling system. In Chapter 6, a control system is developed that utilizes the electric motor to track a desired crank trajectory while FES of the rider's muscle groups is used to achieve a desired power output at the cycle crank. Chapter 7 concludes with a summary of the developed control systems, discussion of the challenges encountered, and guidance for future research. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: DIXON,WARREN E.
Local:
Co-adviser: CRANE,CARL D,III.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-12-31
Statement of Responsibility:
by Matthew J Bellman.

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UFRGP
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Applicable rights reserved.
Embargo Date:
12/31/2016
Classification:
LD1780 2015 ( lcc )

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CONTROLOFCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION:A SWITCHEDSYSTEMSTHEORYAPPROACH By MATTHEWJOHNBELLMAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2015

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c 2015MatthewJohnBellman 2

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TomyparentsformakingmewhoIam,andtomywifeforbuildingontheirsuccess 3

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ACKNOWLEDGMENTS Iwouldliketoacknowledgemyadvisor,Dr.WarrenE.Dixon,formorethansix yearsofguidance,withoutwhichthisdissertationand,moreimportantly,mycurrent positioninlifewouldnotbepossible.Dr.Dixon,alongwithmycolleaguesandthegiants uponwhoseshouldersweallstand,haveallplayedtheirpartsinforgingmeintothe engineerandscientistthatIamtoday.Noamountofgratitudecouldbesufcientto repaytheircontribution.AllIcandoistotakewhattheyhavegivenmeandusethose giftstoleavetheworldabetterplacethanIfoundit.Thankyouall. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................10 CHAPTER 1INTRODUCTION...................................12 2PRELIMINARIES...................................17 2.1Notation.....................................17 2.2StationaryCycleandRiderDynamicModel.................17 2.2.1Kinematics................................17 2.2.2Dynamics................................22 2.2.3MotorizedCycle-RiderSystem.....................27 2.3SwitchedSystemModel............................31 2.3.1StimulationPatternDevelopment...................31 2.3.2SwitchedControlInput.........................33 2.3.3SwitchingStatesandTimes......................35 3SWITCHEDCONTROLOFCADENCEDURINGSTATIONARYCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION............37 3.1ControlDevelopment..............................37 3.1.1Open-LoopErrorSystem........................37 3.1.2Closed-LoopErrorSystem.......................38 3.2StabilityAnalysis................................38 3.3Experiments...................................50 3.3.1Methods.................................51 3.3.2Results..................................55 3.3.2.1ProtocolAresults......................55 3.3.2.2ProtocolBresults......................56 3.3.3Discussion................................59 3.4ConcludingRemarks..............................62 4SWITCHEDCONTROLOFCRANKPOSITIONANDCADENCETRACKING DURINGSTATIONARYCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION.................................63 4.1ControlDevelopment..............................63 4.1.1Open-LoopErrorSystem........................63 5

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4.1.2Closed-LoopErrorSystem.......................64 4.2StabilityAnalysis................................65 4.3Experiments...................................76 4.3.1Methods.................................76 4.3.2Results..................................78 4.3.3Discussion................................81 4.4ConcludingRemarks..............................84 5SWITCHEDCONTROLOFSTATIONARYCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATIONWITHELECTRICMOTORASSISTANCE........................................85 5.1SwitchedControlInput.............................85 5.2ControlDevelopment..............................86 5.2.1Open-LoopErrorSystem........................87 5.2.2Closed-LoopErrorSystem.......................88 5.3StabilityAnalysis................................88 5.4Experiments...................................91 5.4.1Methods.................................92 5.4.2Results..................................96 5.4.3Discussion................................97 5.5ConcludingRemarks..............................99 6SWITCHEDCONTROLOFCADENCEANDPOWEROUTPUTDURINGSTATIONARYCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION.........................................101 6.1CadenceControl................................101 6.1.1ControllerDevelopment.........................101 6.1.2StabilityAnalysis............................102 6.2PowerControl..................................104 6.2.1ControllerDevelopment.........................104 6.2.2StabilityAnalysis............................107 6.3Experiments...................................110 6.3.1Methods.................................110 6.3.2Results..................................114 6.3.3Discussion................................117 6.4ConcludingRemarks..............................118 7CONCLUSION....................................119 APPENDIX:DETAILEDEXPRESSIONSFORCYCLE-RIDERDYNAMICS.....123 REFERENCES.......................................124 BIOGRAPHICALSKETCH................................129 6

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LISTOFTABLES Table page 3-1Comparisonofcadencetrackingerrorforallable-bodiedsubjects'volitional andFES-cycling....................................58 3-2ComparisonofcadencetrackingerrorofthesubjectwithPDduringvolitional andFES-assistedcycling..............................60 4-1Experimentalprotocolsused.............................76 4-2 T max Knee foreachsubject,protocol,andcorresponding " Quad usedforeachtrial..77 4-3TrackingperformanceinrevolutionsperminuteRPMandcomfortscaleratingfromallsubjectsoveralltrials..........................80 5-1SummaryofmotorizedFES-cyclingperformanceforallvesubjectsduring Protocol1.......................................98 5-2SummaryofmotorizedFES-cyclingperformanceforallvesubjectsduring Protocol2.......................................98 6-1PassivetorqueestimateparametersforSubject1.................113 6-2Meanandstandarddeviationoftrackingperformanceforallsubjects......117 7

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LISTOFFIGURES Figure page 2-1Diagramofthecycle-ridersystem..........................18 2-2Examplestimulationpatterndepictingintervalsofthecrankcycleoverwhich themusclegroupsofonelegarestimulated....................36 3-1FES-cyclingtestbed.................................52 3-2Onesubject'scadencetrackingerrorandcontrolinputtoeachmusclegroup duringtheFES-cyclingtrialofProtocolA......................56 3-3ControlinputoverasinglecrankcycleduringtheFES-cyclingtrialofProtocol AforSubjectAB3...................................57 3-4CadencetrackingerrorforthevoluntaryandFES-cyclingphasesofProtocol AforSubjectAB3...................................57 3-5CadencetrackingerrorandcontrolinputtoeachmusclegroupduringtheFESassistedphaseofProtocolB.............................59 3-6ControlinputoverasinglecrankcycleduringtheFES-assistedcyclingtrialof ProtocolB.......................................60 3-7CadencetrackingerrorofthesubjectwithPDduringthevoluntaryandFESassistedphasesofProtocolB............................62 4-1Illustrationofconvergencebehavioroftheboundingsequence..........74 4-2StimulationregionsusedforSubject4.......................77 4-3Cadencetrackingerrorandswitchedcontrolinputacrossthethreetrialsfor Subject4........................................79 4-4SwitchedcontrolinputfromthersttrialofSubject4................81 4-5ComparisonofthepositionandcadencetrackingerrorsduringFES-cycling andvolitionalcyclingforSubject4..........................82 5-1Illustrationofthebehaviorof V L ...........................91 5-2MotorizedFES-cyclingtestbed...........................93 5-3TrackingperformanceforSubject1duringProtocol1andProtocol2.......96 5-4FEScontrolinputsandmotorcurrentinputfromonemotorizedFES-cycling trialoverasinglecrankcycle............................97 6-1Measuredversusestimatedpassiveridertorque.................113 8

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6-2Electricmotorcontroller'strackingperformance...................114 6-3Desiredversusactualactivetorqueaveragedoverthecrankcycle........115 6-4FEScontroller'storquetrackingperformance....................116 6-5FEScontrolinputoverasinglecrankcycle.....................116 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CONTROLOFCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION:A SWITCHEDSYSTEMSTHEORYAPPROACH By MatthewJohnBellman December2015 Chair:WarrenE.Dixon Major:MechanicalEngineering FunctionalelectricalstimulationFEScanbeusedtoactivatethedysfunctional lowerlimbmusclesofindividualswithsomeneurologicaldiseaseorinjurytoproduce cyclingasameansofexercise,rehabilitation,andtransportation.However,FES-cycling ismetabolicallyinefcientandyieldslowerpoweroutputatthecyclecrankthanablebodiedcycling.Previousliteraturesuggeststhattheseproblemsaresymptomaticof poormusclecontrolandnon-physiologicalmuscleberrecruitment.Thelatterisa knownproblemwithFESingeneral,andtheformermotivatesinvestigationofbetter controlmethodsforFES-cycling. InChapter1,motivationforFES-cyclingisintroducedalongwithasurveyof thechallenges,approaches,andshortcomingsofFES-cyclingintheliterature.In Chapter2,anonlinearmodelofastationaryFES-cyclingsystemisdevelopedthat includesparametricuncertaintyandanunknown,bounded,time-varyingdisturbance.A stimulationpatternforthegluteal,quadricepsfemoris,andhamstringsmusclegroups isdesignedbasedonthekinematiceffectivenessoftherider'shipandkneejointsto produceaforwardtorqueaboutthecyclecrank.InChapter3,aswitchedslidingmode controllerisdesignedfortheuncertain,nonlinearcycle-ridersystemwithautonomous, state-dependentswitching.Theswitchedcontrolleryieldsultimatelyboundedtrackingof adesiredcadence,providedsufcientconditionsonthecontrolgains,desiredtrajectory, initialconditions,andstimulationpatternaresatised.Stabilityisderivedviaacommon 10

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Lyapunov-likefunction,andexperimentalresultsdemonstratetheperformanceofthe switchedcontrollersundertypicalFES-cyclingconditions.InChapter4,theresults ofChapter3areextendedtoincludethecontrolobjectiveofcrankpositiontracking, andexperimentalresultsdemonstrateimprovedperformanceovercadencetracking alone.Chapter5buildsontheresultsofChapter4byincludinganelectricmotorin theFES-cyclingsystem.InChapter6,acontrolsystemisdevelopedthatutilizesthe electricmotortotrackadesiredcranktrajectorywhileFESoftherider'smusclegroups isusedtoachieveadesiredpoweroutputatthecyclecrank.Chapter7concludeswith asummaryofthedevelopedcontrolsystems,discussionofthechallengesencountered, andguidanceforfutureresearch. 11

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CHAPTER1 INTRODUCTION FunctionalelectricalstimulationFESistheapplicationofelectricalcurrent acrossmuscleberstoarticiallyinduceamusclecontractionandachieveafunctional outcomee.g.,limbmotion.Sincethe1980s,FEShasbeenappliedtothelowerlimb musclesofpeoplewithuppermotorneuronlesionse.g.,followingspinalcordinjuryor stroketoenablethemtopedalastationarycycle[1],andnumerousphysiologicaland psychologicalbenetshavesincebeendemonstrated[2].Tosupplementthesebenets withenhancedlocomotion,mobileFES-cyclingdeviceshavebeendeveloped[3–9]and evencommercialized 1 .Despiteitssuccessasarehabilitativeintervention,FES-cycling isstilllimitedbylowerefciencyandpoweroutputthanvolitionalcyclingbyable-bodied individuals[10].Lowefciencyandpoweroutputlimitthedurationandworkload achievableinFES-cycling,therebylimitingthetrainingeffectofstationaryFES-cycling andthepracticalityofmobileFES-cycling[11],[12].Thedirectcauseoftheselimitations isstillunknown,butithasbeensuggestedthataprimaryfactoraffectingefciencyand poweroutputispoorcontrolofthestimulationparameterse.g.,frequency,intensity, timing[13]. EarlyFES-cyclingstudiesusedopen-looporsimpleproportional-derivativePD feedbackcontrolofthestimulationintensitytoachieveadesiredcyclingcadence[3–5,7, 14–16].Inthelate1990s,researchersbeganusingmoresophisticatedcontrolmethods ineffortstoimproveFES-cyclingperformance,includinglinearmodelidentication andpoleplacementmethods[8,17,18],fuzzylogiccontrol[19,20],neuralnetwork feedforwardinadditiontoPDfeedbackcontrol[21,22],andhigher-orderslidingmode combinedwithfuzzylogiccontrol[23].AllofthesepreviousFES-cyclingcontrolstudies 1 http://www.hasomed.de/en/products/rehabike-cycling-with-fes.html; http://www.berkelbike.co.uk/ 12

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usedaswitchedcontrolinputthatalternatedstimulationacrossdifferentmusclegroups accordingtoapredenedstimulationpattern.Thestimulationpatterndenesthe segmentsofthecrankcycleoverwhicheachmusclegroupisstimulatedtoachievethe desiredcyclingmotionandis:manuallydetermined[1,3,7,8,15,24],determinedfrom ofinenumericaloptimization[14,21,25–27],analyticallydetermined[19],orbasedon able-bodiedelectromyographyEMGmeasurements[4,5,16,28]. FES-cyclingsystemsthatincludeelectricmotorassistancehavebeendesigned tofacilitatecontrollability[4,8,15,18,20,23,29–31],asanelectricmotorhascontrol authorityacrosstheentirejointspacei.e.,notlimitedbydeadpoints.In[4],an electricmotorwasaddedtoensurethattheFES-cyclingcadencedidnotfallbelow25 revolutionsperminuteRPMandsuppliedaconstant5Wattsofpowertocompensate forlossesingearing.Similarly,in[20],afuzzylogiccontrolschemewasusedtocontrol amotorandFEStoachieveadesiredcadence.In[15],amotorcontrolledthecycling cadencewhileopen-loopstimulationwasappliedtotherider'smusclestomaximize poweroutput.Similarly,in[8,18,23,29–31],electricmotorswereusedtomaintaina desiredcadencewhileFESwasusedtotrackadesiredpoweroutput. Switchingthestimulationcontrolinputbetweenmultiplemusclegroupsandan electricmotoraccordingtothecyclecrankanglemakestheoverallFES-cyclingsystem aswitchedcontrolsystemwithautonomous,state-dependentswitching[32],Section 1.1.3.Ingeneral,duringFES-cycling,thereexistperiodsduringwhichoneormore musclegroupsareactivefollowedbyperiodsduringwhichnomusclegroupsareactive. Whenmusclegroupsareactivelycontrolledbystimulation,thesystemmaystablytrack thedesiredtrajectory,butwhennomusclegroupsareactive,thesystemmaybecome unstable.ThisbehavioriscomplicatedbythefactthatthedynamicsofFES-cycling arenonlinear,time-varying,anduncertain,sothatthesystem'sstatetrajectoriese.g., cadenceareunknownapriori.Noneoftheaforementionedstudieshaveexplored FES-cyclingcontrolwhileconsideringthesepropertiesoftheFES-cyclingsystem. 13

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InvestigatingFES-cyclinginthelightofswitchedsystemstheorymayyieldcontrol strategiesthatimproveFES-cyclingperformance,therebyincreasingthesafetyand effectivenessofFES-cycling. Chapter2providesadetailedmodelofthestationarycyclingsystemandtwoleggedriderundergoingFES-inducedcycling.Anonlinearmodelofthecycle-rider systemisdevelopedthatincludesparametricuncertaintyandunknown,bounded, time-varyingdisturbances.Astimulationpatternforthegluteal,quadricepsfemoris, andhamstringsmusclegroupsisdesignedbasedonthekinematiceffectivenessofthe rider'shipandkneejointstoproduceaforwardtorqueaboutthecyclecrank.Basedon thedevelopedstimulationpattern,theFEScontrolinputisconsideredasaswitched controlinput,makingtheoverallcycle-ridersystemaswitchedsystemwithautonomous, state-dependentswitching. Chapter3considersthecontrolobjectiveofcrankvelocitycadencetracking. Aswitchedslidingmodecontrollerisdesignedbasedonthemodelandstimulation patterndevelopedinChapter2.AcommonLyapunov-likefunctionisusedtoprove thatthecadencetrackingerrorisboundedbyanexponentiallydecayingenvelope inregionswheremusclegroupsareactivatedandbyanexponentiallyincreasing envelopeinregionswherenomusclegroupsareactivated.Theoverallerrorsystem isdemonstratedtobeultimatelyboundedprovidedsufcientconditionsonthecontrol gains,desiredtrajectory,andstimulationpatternaresatised.Experimentalresults onable-bodiedsubjectsandonesubjectwithParkinson'sdiseasedemonstratethe switchedcontroller'sperformanceundertypicalFES-cyclingconditions. Chapter4extendstheresultsofChapter3toincludethecontrolobjectiveofcrank positiontracking,motivatedbythedesiretominimizethesteadystatetrackingerror observedintheresultsofChapter3.Aswitchedslidingmodecontrollerisdesigned toyieldsimultaneoustrackingofadesiredcrankpositionandcadence.Acommon 14

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Lyapunov-likefunctionisusedtoprovethatthetrackingerrorisboundedbyanexponentiallydecayingenvelopeinregionswheremusclegroupsareactivatedandby atangentiallyincreasingenvelopewithniteescapetimeinregionswherenomusclegroupsareactivated.Theoverallerrorsystemisdemonstratedtobeultimately bounded,providedsufcientconditionsonthecontrolgains,thedesiredcadence,the initialconditions,andthestimulationpatternaresatised.Experimentalresultsonve able-bodiedsubjectsdemonstratetheswitchedcontroller'sperformanceusingFESof onlythequadricepsfemorismusclegroupsundertypicalFES-cyclingconditions.While thetheoreticaloutcomeofthestabilityanalysis,ultimatelyboundedtrackingerror,isthe sameinChapters3and4,thecontrollerdevelopedinChapter4demonstratesimproved trackingperformance,likelyduetotheinclusionofanintegratorandlteredtracking errorinthecontrolinput. Chapter5buildsontheresultsofChapter4byincludinganelectricmotorinthe FES-cyclingsystem.Theelectricmotor'scurrentiscontrolledusingthecontroller developedinChapter4withtheobjectiveoftrackingthedesiredcrankpositionand cadenceintheregionsofthecrankcyclewherenomusclegroupsareactivated.This approachremovesthesufcientconditionsfromChapter4thatconstrainedthedesired cadence,theinitialconditions,andthestimulationpattern,astheelectricmotorgrants controllabilityofthesystemthroughouttheentirecrankcycle.Theresultisglobally, exponentiallystabletrackingofthedesiredcrankpositionandcadence,provided sufcientgainconditionsaresatised.Experimentalresultsfromveable-bodied, passiveridersdemonstratethatthemotorizedFES-cyclingcontrolsystemachievesan averagecadencetrackingerrorof0.00 2.91revolutionsperminuterpmforadesired trajectoryof50rpm. Chapter6considersthecontrolobjectiveofmotor-controlledcrankpositionand velocityandFES-controlledpoweroutputatthecrank.AsinChapter5,aslidingmode controllerfortheelectricmotor'scurrentisdevelopedtoachievestabletrackingofthe 15

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desiredcranktrajectory,butinChapter6,itisassumedthatameasureofthecrank torqueisavailable,whichisthenusedasafeedforwardtermintheelectricmotor controller.BasedonthemodeldevelopedinChapter2,aslidingmodecontrolleris designedfortheFEScontrolinputwiththeobjectiveoftrackingadesiredactivetorque, averagedoverthecrankcycle.ALyapunovfunctionisusedtoprovethatthemotor controlsystemyieldsglobally,exponentiallystabletrackingofthedesiredcrankposition andcadence,providesufcientgainconditionsaresatised.AdiscreteLyapunov functionisusedtoprovethattheaveragetorquetrackingerrorisultimatelybounded, providedsufcientconditionsonthecontrolgainsaresatised.Experimentalresults fromthreeable-bodied,passiveridersdemonstratethatthedevelopedcontrolsystemis abletomaintainanactivepowertrackingerrorof 0 : 46 2 : 6 Wattsforadesiredtrajectory of20Wattsat50rpm. Chapter7concludesthedissertation.Asummaryofthedissertationisprovided alongwithadiscussionofchallengesthatwereencounteredthroughoutthedevelopmentofChapters2-6butthatliebeyondthescopeofthisdissertation.Concluding remarksaremadethatmaybeusefulindirectingfutureresearchonFES-cycling. 16

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CHAPTER2 PRELIMINARIES Inthischapter,anonlinearmodelofastationaryFES-cyclingsystemisdeveloped thatincludesparametricuncertaintyandanunknown,bounded,time-varyingdisturbance.Astimulationpatternforthegluteal,quadricepsfemoris,andhamstringsmuscle groupsisdesignedbasedonthekinematiceffectivenessoftherider'shipandknee jointstoproduceaforwardtorqueaboutthecyclecrank. 2.1Notation Throughoutthedissertation,functionaldependenciesareexplicitlystatedwhen termsarerstintroduced.Thereafter,fornotationalbrevity,functionaldependencies aresuppressed,unlessrequiredforclarityofexposition.Forexample,theequation _ x t = f x t + g x t ;t ,wheredotnotationisusedtodenotederivativeswith respecttotime,isstatedassuchwhenrstintroducedbutstatedsimplyas _ x = f + g or _ x = f x + g x;t thereafter.Thefunctionsin _ x = f + g aredenotedas x : R 0 ! R n ; f : R n ! R n ,and f : R n R 0 ! R n ,where R n denotes n -dimensionalEuclidean space, R >a denotesthesetofrealnumbersstrictlygreaterthan a 2 R ,and R a denotes thesetofrealnumbersgreaterthanorequalto a 2 R . 2.2StationaryCycleandRiderDynamicModel 2.2.1Kinematics Atwo-leggedriderpedalingarecumbentstationarycyclecanbemodeledas asingledegree-of-freedomDOFsystem[33]comprisedoftwocoupled,parallel, closedkinematicchains,asdepictedinFig.2-1.Eachkinematicchainincludesfour revolutejoints,representingthehip,knee,pedal,andcrankjoints,connectedbyfour rigidbodies,representingthethighandshanklimbsegmentsaswellasthecycle crankarmandframe.Theanklejointisassumedtobexedintheanatomicallyneutral positioninaccordancewithcommonclinicalcyclingpracticesforsafetyandmediolateral stability[34],andtherider'sfeetareassumedtobexedtothepedals.Therigid 17

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Figure2-1.Diagramofthecycle-ridersystem. body,orlink,representingthecycleframetowhichtherider'shipjointsandthecycle crankjointsareconnected,isassumedtobexedtothegroundwheretheground isassumedtobeaninertialreferenceframeforstationarycycling.Cyclingtypically involvesmotioninonlythesagittalplane,sothesubsequentdevelopmentassumesthat bothkinematicchainstranslateandrotateinplanesparalleltotheanatomically-dened sagittalplane.Aclosedkinematicchainoffourplanarlinks,oneofwhichisxedtothe ground,eachconnectedbyfourrevolutejoints,hasonlyoneDOF.Sincebothkinematic chainshaveoneDOF,thetotalcycle-ridersystemwouldhavetwoDOFs,exceptthat cyclestypicallyhavetheircrankarmsxed180degreesoutofphasewithrespect toeachother.Couplingthecrankarmsinthismannerleavesthetotalcycle-rider systemwithoneDOF.Sincethecycle-ridersystemhasonlyoneDOF,thepositionand orientationofallthelinkscanbecalculatedusingonlyknowledgeofthelinklengthsand onejointanglewithrespecttoground.Inpractice,itissimplesttomeasurethecrank angle,sothecrankangle q : R 0 !Q willbeusedinthesubsequentdevelopmentto parametrizethesystem,where Q R denotesthesetofcrankangles. Sincethecycle-ridersystemcanberepresentedbyaclosed-loopmechanism, theclosed-loopequationdescribingthejointpositionsrelativetooneanotherandthe 18

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groundforonekinematicchain,oronesideofthecycle-ridersystem,canbeexpressed as ~r s H=G + ~r s K=H q t + ~r s P=K q t + ~r s C=P q t + ~r s G=C = ~ 0 ;s 2S ; where ~r s H=G 2 R 2 representsthevectorfromapoint O xedintheinertialreference frametothehipjoint,wherethesuperscript s 2S , f R;L g indicatestheright R orleft L sideofthecycle-ridersystem; ~r s K=H ;~r s P=K ;~r s C=P : Q! R 2 representvectorsfrom thehipjointtothekneejoint,thekneejointtothepedaljoint,andthepedaljointtothe crankjoint,respectively; ~r s G=C 2 R 2 representsthevectorfromthecrankjointbackto O ; and ~ 0 2 R 2 representsthezerovector.Thesubsequentdevelopmentconsiderstheright sideofthecycle-ridersystemrst,thenextendstheresultstoincludetheleftside,and thesuperscript s isomittedunlessitaddsclarity.Usingthecoordinatesystemdepicted inFig.2-1,thevectorsin2canbeexpressedas ~r s H=G , 0^ x + l s y ^ y; ~r s K=H q t , l s t cos q s h q t ^ x + l s t sin q s h q t ^ y; ~r s P=K q t , l s l cos q s k q t ^ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(l s l sin q s k q t ^ y; ~r s C=P q t , l s c cos q t ^ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(l s c sin q t ^ y; ~r s G=C , )]TJ/F25 11.9552 Tf 9.299 0 Td [(l s x ^ x +0^ y: for s 2S ,where q s h ;q s k : Q! R denotethehipandkneejointangles,measuredas inFig.2-1;theconstants l s t ;l s l ;l s c 2 R > 0 denotethelengthsofthethigh,shank,and cranklinks,respectively;theconstants l s x ;l s y 2 R > 0 denotethe x -and y -components, respectively,ofthedistancefromthecrankjointtothehipjoint;and ^ x; ^ y 2 R 2 represent unitvectorsparalleltothe x -and y -axes.Substitutingtheexpressionsfrom2-2 into2andgroupingthe x -and y -componentsyields l t cos q h + l l cos q k + l c cos q )]TJ/F25 11.9552 Tf 11.956 0 Td [(l x ^ x + l t sin q h )]TJ/F25 11.9552 Tf 11.956 0 Td [(l l sin q k )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c sin q + l y ^ y =0^ x +0^ y; 19

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whichcanbeseparatedintothefollowingindependent,closed-loopkinematicequations: l t cos q h + l l cos q k + l c cos q )]TJ/F25 11.9552 Tf 11.955 0 Td [(l x =0 ; l t sin q h + l l sin q k )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c sin q + l y =0 : Toparametrizethecycle-ridersystemintermsofthecrankangle,itisnecessary tosolvefor q h and q k intermsof q ,andthisdevelopmentbeginsbysolvingfor q h . Movingthetermsincluding q k totheright-handsidesof2and2,squaringboth equations,andaddingthemtogetheryieldsasingleequationoftheform k 1 q t sin q h q t + k 2 q t cos q h q t + k 3 q t =0 ; wherethecoefcients k i : Q! R , i 2f 1 ; 2 ; 3 g ,aredenedas k 1 q t , )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 l t l c sin q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(l y ; k 2 q t , 2 l t l c cos q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(l x ; k 3 q t , l 2 x + l 2 y + l 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(l 2 l + l 2 c )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 l c l x cos q t + l y sin q t : Equation2canbesolvedfor q h usingthetrigonometricsolutiontechniquedescribed in[35,Section6.7].Thistechniqueispreferredoverotherclosed-formsolutiontechniqueslikethetan-half-angletechniquebecauseitwillyieldpropersolutionsinspecial caseswheretheothertechniqueswillnot.Dividing2by p k 2 1 + k 2 2 yields k 1 p k 2 1 + k 2 2 sin q h + k 2 p k 2 1 + k 2 2 cos q h + k 3 p k 2 1 + k 2 2 =0 : 20

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Recognizingthat p k 2 1 + k 2 2 isthehypotenuseofarighttrianglewithsidesoflength k 1 and k 2 ,auniqueangle : Q! R canbedenedwiththefollowingrelations: q t , arctan k 1 q t k 2 q t ; sin = k 1 p k 2 1 + k 2 2 ; cos = k 2 p k 2 1 + k 2 2 : Substituting2-2into2yields cos cos q h + sin sin q h + k 3 p k 2 1 + k 2 2 =0 : Usingthetrigonometricidentitycos u )]TJ/F25 11.9552 Tf 12.199 0 Td [(v = cos u cos v + sin u sin v in2and groupingtermsyields cos q h )]TJ/F25 11.9552 Tf 11.955 0 Td [( = )]TJ/F25 11.9552 Tf 9.299 0 Td [(k 3 p k 2 1 + k 2 2 : Twosolutionsfor q h )]TJ/F25 11.9552 Tf 11.955 0 Td [( canbefoundfor2,and,because isunique,twovalues for q h cansubsequentlybefoundwhichareequaltothetwosolutionsof q h = arccos )]TJ/F25 11.9552 Tf 9.299 0 Td [(k 3 p k 2 1 + k 2 2 ! + : Thetwosolutionsof2canviewedfromananatomicalperspective,withthe negativesolutionrepresentinghipexionandthepositivesolutionrepresentinghyperextensionofthehipwithrespecttoFig.2-1.Therefore,thesubsequentdevelopment proceedsusingthenegativesolutionin2. Thekneeanglecanbefoundasafunctionofboththehipangleandthecrank angle.Isolatingthetermscontaining q k in2and2,dividing2by2,and applyingtheinversetangentyields q k = )]TJ/F22 11.9552 Tf 9.299 0 Td [(arctan l t sin q h )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c sin q + l y l t cos q h + l c cos q )]TJ/F25 11.9552 Tf 11.955 0 Td [(l x : 21

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Toparametrizethevelocitiesofthecycle-ridersystemintermsofthecrankangle, thersttimederivativeofboth2and2aredetermined,thetermscontaining q areisolated,andtheresultingequationsareexpressedinmatrixformas 2 6 4 )]TJ/F25 11.9552 Tf 9.298 0 Td [(l t sin q h )]TJ/F25 11.9552 Tf 9.298 0 Td [(l l sin q k l t cos q h )]TJ/F25 11.9552 Tf 9.299 0 Td [(l l cos q k 3 7 5 2 6 4 _ q h _ q k 3 7 5 = 2 6 4 l c sin q l c cos q 3 7 5 _ q: Solving2for _ q h and _ q k intermsof _ q yields 2 6 4 _ q h q t ; _ q t _ q k q t ; _ q t 3 7 5 = 2 6 4 S 1 q t S 2 q t 3 7 5 _ q t ; where S 1 ;S 2 : Q! R denotevelocitytransformationtermsdenedas S 1 q t , l c l t sin q k q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(q t sin q k q t + q h q t ;S 2 q t , )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(l c l l sin q h q t + q t sin q k q t + q h q t : Thesolutionin2isachievedbyperformingamatrixinversion,whichisonly possibleif q k + q h 6 = n;n 2 Z ,whichisalwayssatisedprovidedthattherideris positionedsuchthatthekneejointneverreachesfullextension.Itisusefultonotethat thejointvelocitiesmayalsobeexpressedas _ q h = @q h @q _ q; _ q k = @q k @q _ q: Combining2with2demonstratesthat S 1 = @q h @q ;S 2 = @q k @q : Theserelationshipswillbeusedinthesubsequentdevelopmenttodeterminethe contributionoftorqueappliedabouttherider'sjointstotheresultanttorqueaboutthe crankaxis. 2.2.2Dynamics Nowthatkinematicsofthecycle-ridersystemhavebeenparametrizedinterms ofthecrankangle,thedynamicsofthecycle-ridersystemcanbeexpressedinterms 22

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ofthecrankangle,crankvelocity,andcrankacceleration.Thekineticandpotential energyoftherightsideofthecycle-ridersystemaresubsequentlyderivedandusedina Lagrangianformulationtodeterminethetotalsystem'sequationofmotion. Thekineticenergyoftherightsideofthecycle-ridersystemcanbeexpressedas T q t ; _ q t , 1 2 m t ~v t q t ; _ q t ~v t q t ; _ q t + 1 2 I t _ q 2 h q t ; _ q t + 1 2 m l ~v l q t ; _ q t ~v l q t ; _ q t + 1 2 I l _ q 2 k q t ; _ q t + 1 2 m c ~v c q t ; _ q t ~v c q t ; _ q t + 1 2 I c _ q 2 t ; where T : Q R ! R denotesthekineticenergyofthekinematicchain; m t ;m l ;m c 2 R > 0 denotethemassofthethigh,shank,andcrankarm,respectively; ~v t ;~v l ;~v c : Q R ! R 2 representthevelocityvectorsofthecenterofmassofthethigh,shank, andcrankarm,respectively,andareobtainedbydifferentiationofthepositionsofeach centerofmasswithrespecttotheinertialreferenceframe;and I t ;I l ;I c 2 R > 0 denote theprinciplemomentsofinertiaaboutthecenterofmassofthethigh,shank,and crankarm,respectively.Thethighandshankcanbemodeledasconicalcylinderswith principlemomentsofinertiagivenas[36] I i , 2 A i m 2 i i l i ;i 2f t;l g ; where A i ; i ; i 2 R > 0 , i 2f t;l g ,aredenedas A i , 9 20 " 1+ i + 2 i + 3 i + 4 i + i + 2 i 2 # ; i , r i =R i ; i , 3 m i l i R 2 i + R i r i + r 2 i ; and r i ;R i 2 R > 0 denotethedistalandproximalcircumferencesofeachlimbsegment,respectively.Thecrankarmismodeledasaslenderrodwithauniformmass 23

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distribution,anditsprinciplemomentofinertiawithrespecttoitsmasscenteris I c , 1 12 m c l 2 c : Dening l c;t ;l c;l ;l c;c 2 R > 0 todenotethedistancesfromthehipjoint,kneejoint, andcrankjointtothethigh,shank,andcrankarmcentersofmass,respectively,the positionsofthecenterofmassofeachlinkcanbeexpressedas ~r c;t q t , l c;t cos q h q t ^ x + l c;t sin q h q t ^ y; ~r c;l q t , l t cos q h q t + l c;l cos q k q t ^ x + l t sin q h q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c;l sin q k q t ^ y; ~r c;c q t , l x )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c;c cos q t ^ x + l c;c sin q t + l y ^ y; where ~r c;t ;~r c;l ;~r c;c : Q! R 2 representthepositionsofthecenterofmassofthethigh, shank,andcrankoftherstleg,respectively.Then,thevelocitiesofthecenterofmass ofeachlinkcanbeexpressedas ~v t q t ; _ q t = )]TJ/F25 11.9552 Tf 9.299 0 Td [(l c;t sin q h q t _ q h q t ; _ q t ^ x + l c;t cos q h q t _ q h q t ; _ q t ^ y; ~v l q t ; _ q t = )]TJ/F25 11.9552 Tf 9.298 0 Td [(l t sin q h q t _ q h q t ; _ q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(l c;l sin q k q t _ q k q t ; _ q t ^ x + l t cos q h q t _ q h q t ; _ q t )]TJ/F25 11.9552 Tf 9.299 0 Td [(l c;l cos q k q t _ q k q t ; _ q t ^ y; ~v c q t ; _ q t = l c;c sin q t _ q t ^ x + l c;c cos q t _ q t ^ y: Substitutingtheexpressionsin2-2intothekineticenergyin2gives T = 1 2 )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(m t l 2 c;t + I t _ q 2 h + 1 2 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(m l l 2 t _ q 2 h + )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(m l l 2 c;l + I l _ q 2 k )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 m l l t l c;l cos q k )]TJ/F25 11.9552 Tf 11.956 0 Td [(q h _ q h _ q k + 1 2 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(m c l 2 c;c + I c _ q 2 ; 24

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whichcanberewrittenusing2as T q t ; _ q t = J 1 q t + J 2 q t + J 3 _ q 2 t ; where J 1 ;J 2 : Q! R > 0 ;J 3 2 R > 0 denotetheeffectivemomentofinertiaofthethigh, shank,andcrankarmaboutthecrankjointaxisandaredenedas J 1 q t , 1 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(m t l 2 c;t + I t S 2 1 q t ; J 2 q t , 1 2 m l l 2 t S 2 1 q t + 1 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(m l l 2 c;l + I l S 2 2 q t )]TJ/F25 11.9552 Tf 9.298 0 Td [(m l l t l c;l cos q k q t )]TJ/F25 11.9552 Tf 11.956 0 Td [(q h q t S 1 q t S 2 q t ; J 3 , 1 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(m c l 2 c;c + I c : Thepotentialenergyoftherightsideofthecycle-ridersystem, U : Q! R 0 ; can beexpressedas U q t , )]TJ/F25 11.9552 Tf 8.903 0 Td [(~r c;t q t ~g t )]TJ/F25 11.9552 Tf 11.56 0 Td [(~r c;l q t ~g l )]TJ/F25 11.9552 Tf 11.56 0 Td [(~r c;c q t ~g c ; where ~g i , m i g )]TJ/F15 11.9552 Tf 10.091 0 Td [(^ y ;i 2f t;l;c g ; and g 2 R > 0 denotestheaccelerationduetogravity. Equation2canbeexpressedusing2-2as U = m t l c;t sin q h + m l l t sin q h + m l l c;l sin q k + m c l c;c sin q + m c l y g: TheLagrangianfortherightsideofthecycle-ridersystemcanbedenedas L q t ; _ q t , T q t ; _ q t )]TJ/F25 11.9552 Tf 12.071 0 Td [(U q t : Deningthecrankangle q asthegeneralized coordinate,Lagrange'sequationcanbeappliedas d dt @L q t ; _ q t @ _ q t )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(@L q t ; _ q t @q t = crank t ; where crank : R 0 ! R denotesexternaltorquesappliedaboutthecrankjointaxis e.g.,torquesappliedbyfriction,musclecontractions,etc..Thestep-by-stepsolutionof 25

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2is @L @q = @T @q )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(@U @q = @J 1 @q + @J 2 @q _ q 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(@U @q : @L @ _ q = @T @ _ q =2 J 1 + J 2 + J 3 _ q: d dt @L @ _ q =2 dJ 1 dt + dJ 2 dt _ q +2 J 1 + J 2 + J 3 q; =2 @J 1 @q + @J 2 @q _ q 2 +2 J 1 + J 2 + J 3 q: d dt @L @ _ q )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(@L @q =2 J 1 + J 2 + J 3 q + @J 1 @q + @J 2 @q _ q 2 + @U @q = crank : Equation2canbewrittencompactlyas M R q t q t + V R q t ; _ q t _ q t + G R q t = R crank t ; where M s : Q! R > 0 ;V s : Q R ! R ;G s : Q! R ;s 2S ,denoteinertial, centripetalandCoriolis,andgravitationaleffects,respectively,ofonesideofthecycleridersystemaboutthecrankjointaxisandaredenedas M s q t , 2 J s 1 q t + J s 2 q t + J s 3 ; V s q t , @J s 1 q t @q t + @J s 2 q t @q t _ q t ; G s q t , )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(@U s q t @q t : Detailedexpressionsfor M s , V s ,and G s areprovidedinAppendix7. Nowthattheequationofmotionfortherightsideofthecycle-ridersystemhas beendenedin2,theleftsideofthesystemcanbeeasilyincludedintheequation ofmotion,sinceitisalsodenedbythecrankangle,thoughlaggedby radiansas depictedinFig.2-1.Repeatingthesamepreviousdevelopmentusingtheparametersof theleftsideofthecycle-ridersysteme.g., m L t ;l L l andsubstituting q + for q yields theequationofmotionfortheleftsideofthesystemas M L q + V L _ q + G L = L crank ; 26

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Sincethetwosidesofthecycle-ridersystemarecoupledtogetheratthecrankjoint, andsincethecrankangleisusedasthegeneralizedcoordinateparametrizationthe equationofmotionforeachside,2and2canbeaddedtogethertoyieldthe overallsystemequationofmotionas M q t q t + V q t ; _ q t _ q t + G q t = crank t ; where M : Q! R > 0 ;V : Q R ! R ;G : Q! R ; aredenedas M , M R + M L ; V , V R + V L ; G , G R + G L ; crank , R crank + L crank : 2.2.3MotorizedCycle-RiderSystem Theequationofmotionin2describesthedynamicsofasetofrigidbodies conguredasacycle-ridersystem.AnFES-cyclingsystemincludestheeffectsofthe cycleitselfe.g.,frictioninthecrankbearings,viscoelastictissueintherider'sjoints, activationoftherider'smuscles,anauxiliaryelectricmotor,andunknowndisturbances. Specically,amotorizedFES-cyclingsystemcanbemodeledas cycle _ q t ; q t ;t + rider q t ; _ q t ; q t ;t = motor t ; where cycle : R R R 0 ! R denotestheeffectsofinertia,friction,anddisturbances inthemotorizedcycle; rider : Q R R R 0 ! R denotestheeffectsoftherider's passivelimbdynamics,activemusclecontractions,anddisturbancesfromtherider;and motor : R 0 ! R denotesthetorqueappliedaboutthecyclecrankaxisbytheelectric motor.Thenetcycletorque cycle _ q t ; q t ;t canbemodeledas cycle _ q t ; q t ;t , J cycle q t + b cycle _ q t + d cycle t ; 27

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where J cycle 2 R > 0 istheunknowncycleinertia, b cycle 2 R > 0 istheunknowncoefcientof viscousdampinginthecycle,and d cycle : R 0 ! R denotesunknowndisturbancessuch aschangesinload.Thenetridertorque rider q t ; _ q t ; q t ;t canbemodeledas rider q t ; _ q t ; q t ;t = p q t ; _ q t ; q t )]TJ/F25 11.9552 Tf 11.955 0 Td [( a q t ; _ q t ;t + d rider t ; where p : Q R R ! R denotesthenettorqueproducedbytherider'spassivelimb dynamics, a : Q R R 0 ! R denotesthenettorqueproducedbyactivecontractions oftherider'smuscles,and d rider : R 0 ! R denotestheeffectsofunknowndisturbances fromtheridersuchasmusclespasms. Assumption1. Theunknowndisturbances d cycle t and d rider t andtheirrsttime derivativeshaveknownboundssuchthat j d cycle t j c d;cycle and j d rider t j c d;rider ; where c d;cycle ;c d;rider 2 R > 0 areknownconstants,and _ d cycle t c _ d;cycle and _ d rider t c _ d;rider ; where c _ d;cycle ;c _ d;rider 2 R > 0 areknownconstants. Therider'spassivelimbdynamicscanbemodeledasasingledegree-of-freedom system[33]ofrigidlinkswithviscoelasticpinjointsrotatinginthesagittalplane,which canbeexpressedas p q t ; _ q t ; q t , M q t q t + V q t ; _ q t _ q t + G q t + P q t ; _ q t ; where P : Q R ! R denotestheeffectsofpassiveviscoelastictissueforcesinthe rider'sjoints.Thepassiveviscoelasticeffectsofthetissuessurroundingthehipand kneejointscanbeexpressedas P q t ; _ q t , X j 2J T j q t p j q t ; _ q t ; where T j : Q! R aretheknownjointtorquetransferratios[26]withsubscript j 2J , f RHip;RKnee;LHip;LKnee g indicatingrightandlefthipandkneejoints, and p j : Q R ! R denotestheresultanttorqueabouttherider'sjointfromviscoelastic 28

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tissueforces.ThejointtorquetransferratiosaretheJacobianelementsthattranslate torquefromonejointtoanotherandcanbeexpressedas T Hip q t = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(@q Hip q t @q t = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(l c l t sin q Knee q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(q t sin q Knee q t + q Hip q t ; T Knee q t = @q Hip q t @q t + @q Knee q t @q t ; = l c l l l t l l sin q Knee q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(q t )]TJ/F25 11.9552 Tf 11.956 0 Td [(l t sin q Hip q t + q t sin q Knee q t + q Hip q t ; wherethemeasurableconstants l t ;l l ;l c 2 R > 0 arethethighandshanklengthsof theriderandthecyclecrankarmlength,respectively,assumingthattheselengthsare symmetricwithrespecttotheleftandrightsidesofthesystem; q j : Q! R denotesthe rider'sjointangleswithrespecttothehorizontalplane;andthenotation indicatesthat theexpressionholdsforbothrightandleftsidesofthemodeli.e., canbereplaced by R or L tocreatedistinctexpressions.Basedon[37]and[38], p j q t ; _ q t canbe modeledas p j q t ; _ q t , k j; 1 exp k j; 2 j q t j q t + k j; 3 + b j; 1 tanh b j; 2 _ j q t ; _ q t + b j; 3 _ j q t ; _ q t ; for j 2J ,where k j;i ;b j;i 2 R , i 2f 1 ; 2 ; 3 g ; areunknown,constantcoefcients,and j : Q! R denotestherider'srelativehipandkneejointangles,denedas Hip q t , q Hip q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(q t + ; Knee q t , q Hip q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(q Knee q t ; where q t 2 R isthemeasurable,constanttrunkangle. Theactivetorqueresultingfromtherider'smusclescanbeexpressedas a , X m 2M B m q t ; _ q t u m t + r t ; where B m : Q R ! R denotestheuncertaincontroleffectivenessofamusclegroup withsubscript m 2M , f RGlute;RQuad;RHam;LGlute;LQuad;LHam g indicating 29

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theright R andleft L gluteal Glute ,quadricepsfemoris Quad ,andhamstrings Ham musclegroups,respectively; u m : R 0 ! R denotestheelectricalstimulation intensityappliedtoeachmusclegroup;and r : R 0 ! R denotestorqueappliedabout thecrankaxisthroughtherider'svolitionalcyclingeffort.Thecontroleffectivenessfor eachmusclegroupcanbedenedas B m q t ; _ q t , m q t ; _ q t T m q t ; or m 2M ,where m : Q R ! R denotestheuncertainrelationshipbetween stimulationintensityandamusclegroup'sresultanttorqueaboutthejointitspans, and T m : Q! R denotesthetorquetransferratioforamusclegroup,whichcanbe determinedaccordingtotheprimaryjointthateachmusclegroupspansas T Glute q t = T Hip q t ;T Quad q t = T Ham q t = T Knee q t ; giventhatthefollowingassumptionholds. Assumption2. Thebiarticulareffectsoftherectusfemorisandhamstringmusclesare negligiblei.e.,activationofthequadricepsandhamstringsmusclegroupsproduces torqueaboutthekneejointsonly. Theuncertainfunction m q t ; _ q t canbemodeledasin[39],neglectingthe effectsofmusclefatigue,as m q t ; _ q t , m q t m q t ; _ q t cos a m q t ; for m 2M ,where m : Q! R denotestheuncertainmomentarmofamuscle's outputforceaboutthejointitspans; m : Q R ! R denotestheuncertain,nonlinear functionrelatingstimulationintensitytomuscleberforce;and a m : Q! R denotesthe uncertainpennationangleofthemusclebers. 30

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Property1. Themomentarmofthemusclegroupaboutthejointitspans m q t dependsonthejointangleandisnonzeroandcontinuouslydifferentiablewitha boundedrsttimederivativeforall m 2M [40]. Property2. Thefunctionrelatingstimulationvoltagetomuscleberforce m q t ; _ q t dependsontheforce-lengthandforce-velocityrelationshipsofthemusclebeingstimulatedandislowerandupperboundedbyknownpositiveconstants ; 2 R > 0 ; respectively,forall m 2M ,providedthemuscleisnotfullystretched[41]orcontracting concentricallyatitsmaximumshorteningvelocity. Property3. Themuscleberpennationangle a m q t 6 = n + = 2 ;n 2 Z ,i.e., cos a m q t 6 =0 forall m 2M [42]. Property4. Basedon2andProperties1-3,thefunctionrelatingtheelectrical stimulationintensityappliedtoamusclegroupandtheresultingtorqueaboutthejointis nonzeroandbounded.Inotherwords, 0 < < j m q t ; _ q t j ,where ; 2 R > 0 areknownpositiveconstants,forall m 2M . Torqueaboutthecrankaxisprovidedbyanelectricmotorismodeledas motor t , B e u e t ; where B e 2 R isaconstantrelatingthecurrentintheelectricmotor'swindingstothe resultingtorqueaboutthecrankaxis,and u e : R 0 ! R isthecontrolcurrentappliedto theelectricmotorwindings.Thecontroleffectivenessfortheelectricmotorisdenedas B e , K r g ,where K 2 R istheconstantcoefcientrelatingarmaturecurrenttotorque aboutthecrankaxis,and r g 2 R > 0 isthegearratiobetweenthemotoroutputandthe crankaxis.Itisassumedthat 0 0 isaknownconstant. 2.3SwitchedSystemModel 2.3.1StimulationPatternDevelopment Themuscletorquetransferratios T m indicatehoweachmusclegroupshould beactivatedtoinduceforwardpedaling.Multiplyingthejointtorqueyieldedbya 31

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musclecontractionwith T m transformsthattorquetoaresultanttorqueaboutthe crank.Therefore,ifonlyforwardpedalingisdesired,theneachmusclegroupshould onlybeactivatedwhenityieldsaclockwisewithrespecttoFig.2-1torqueaboutthe crank.Inotherwords,stimulationshouldonlyactivatethequadricepswhen T Quad is negative,thehamstringswhen T Ham ispositive,andtheglutealmuscleswhen T Glute ispositive.However,thisstimulationpatternwouldrequirestimulationofthemuscle groupsforvanishinglysmallvaluesof T m i.e.,neartheso-calleddeadpointsofthe crankcycleandmaythereforeactivatethemusclesinefciently,inthesensethatlarge valuesofstimulationandmetabolicpoweroutputwouldresultinlittlepoweroutputat thecyclecrank.Therefore,toincreaseFES-cyclingefciency,motivationexiststoonly stimulateamusclegroupwhenitstorquetransferratioissufcientlylarge.Indeed, evidencein[26]suggeststhatthestimulationintervalforeachmusclegroupshouldbe minimizedtooptimizemetabolicefciency. Thecontrolinputcanbegeneratedbystimulationofthemusclegroupsorby anelectricmotor.Acommonquestionthatarisesinhuman-machineinteractionis: Howshouldtheperson'seffortbebalancedwiththemachinetoaccomplishatask cooperatively?Inthiscase,theperson'seffortistheelectricallystimulatedmuscleinput andthemachine'sistheelectricmotorinput.Inthepresentdevelopment,theelectric motorcontributionisconsideredtobeagenerictorqueinput,butinChapter5,the human-machineeffortisbalancedbyonlyactivatingthemusclegroupswherethey caneffectivelycontributetopedalingandactivatingtheelectricmotoreverywhereelse, andinChapter6,thehuman-machineeffortisbalancedbyassigningeachsystema differentcontrolobjectivei.e.,theelectricmotoristaskedwithtrackingadesiredcrank positionandcadencewhileFESisusedtotrackadesiredcranktorquetoachievea desiredpoweroutput.Switchingthecontrolinputinthismanneryieldsanautonomous, state-dependent,switchedcontrolsystem[32]. 32

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Theportionofthecrankcycleoverwhichaparticularmusclegroupisstimulatedis denoted Q m Q for m 2M ,andtheunionofthestimulationregionsisthecontrolled region,denoted Q c , [ m 2M Q m .Theuncontrolledregioni.e.,theregionofthecrank cycleinwhichnomusclegroupsarestimulatedisdenoted Q u , QnQ c .Similarly,the portionofthecrankcycleoverwhichtheelectricmotoractivelycontributestorqueis denoted Q e Q .Inthisdevelopment, Q m isdenedforeachmusclegroupas Q Glute , f q t j T Glute q t >" Glute t g ; Q Quad , f q t j)]TJ/F25 11.9552 Tf 20.589 0 Td [(T Quad q t >" Quad t g ; Q Ham , f q t j T Ham q t >" Ham t g ; where " m : R 0 ! 0 ; max T m q t i isatime-varyingsignaldenedfor m 2M . Deningthestimulationregionsasin2-2limitsstimulationtoportionsof thecrankcyclewheretheratioofthetorqueproducedbystimulationofthemuscle groupandtheresultanttorqueaboutthecrankaxisisboundedbelowby " m t ,which isdesignedapriori,andpreventsbackpedaling,asthemusclegroupsmayonlybe stimulatedwhentheresultanttorqueaboutthecrankaxisispositivei.e.,forward pedaling.Notethatanegativesignisincludedin2becausekneeextensortorque isdenedtobenegative. 2.3.2SwitchedControlInput Tostimulatetherider'smusclegroupsaccordingtothestimulationpatterndened by2-2,thestimulationinputtoeachmusclemustbeswitchedonandoffat appropriatepointsalongthecrankcycle.Basedontheseswitchinglaws,piecewise constantswitchingsignalscanbedevelopedfor m 2M , m : Q!f 0 ; 1 g ; andforthe 33

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electricmotor, e : Q!f 0 ; 1 g ,as m q t , 8 > > < > > : 1 if q t 2Q m 0 if q t = 2Q m ; e q t , 8 > > < > > : 1 if q t 2Q e 0 if q t = 2Q e : Usingthesestate-dependentswitchingsignals,thestimulationinputtothemuscles groupsandthecurrentinputtothemotorwindingscanbedenedas u m t , k m m q t u FES t ;u e t , k e e q t u motor t ; where k m ;k e 2 R > 0 ;m 2M ; arepositive,constantcontrolgains,and u FES ;u motor : R 0 ! R arethesubsequentlydesignedstimulationandelectricmotorcontrolinputs, respectively.Substituting2-2,2,and2into2andrearranging termsyields J cycle q t + b cycle _ q t + M q t q t + V q t ; _ q t _ q t + G q t + P q t ; _ q t + d cycle t + d rider t = r t + B FES q t ; _ q t u FES t + B motor q t u motor t ; where B FES : Q R ! R > 0 ;B motor : Q! R arelumped,switchedcontrol effectivenesstermsdenedas B FES , X m 2M B m k m m ;B motor , B e k e e : Theswitchedsystemin2hasthefollowingproperties. Property5. J J cycle J; where J ; J 2 R > 0 areknownconstants Property6. 0 0 isaknownconstant. Property7. M M q t M; where M ; M 2 R > 0 areknownconstants. Property8. j V q t ; _ q t j c V j _ q t j ,where c V 2 R > 0 isaknownconstant. Property9. j G q t j c G ; where c G 2 R > 0 isaknownconstant. 34

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Property10. j P q t ; _ q t j c P 1 + c P 2 j _ q t j ,where c P 1 ;c P 2 2 R > 0 areknown constants. Property11. Forall m 2M , B B m q t ; _ q t B for q t 2Q m ,where B ; B 2 R > 0 areknownconstants, j B m q t ; _ q t j c B forall q t 2Q ; where c B 2 R > 0 isaknown constant,and _ B m q t ; _ q t ; q t c _ B 1 j _ q t j + c _ B 2 j q t j forall q t 2Q ; where c _ B 1 ;c _ B 2 2 R > 0 areknownconstants. Property12. j r t j c r and j _ r t j c _ r ,where c r ;c _ r 2 R > 0 areknownconstants. Property13. _ M q t ; _ q t )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 V q t ; _ q t =0 : 2.3.3SwitchingStatesandTimes Assumingthattheinitialcrankangle q on 0 2Q isanelementof Q c ,theknown sequenceofswitchingstatesfortheelectricalstimulation,whicharepreciselythe limitpointsof Q u ,isdenedas q on n ;q off n 2Q 1 n =0 ; wherethesuperscripts on and off indicatethatthesumofsignals m isswitchingfromzerotononzeroornonzero tozero,respectively.Thecorrespondingsequenceofunknownswitchingtimes t on n ;t off n 2 R 0 1 n =0 isdenedsuchthateachon-time t on n andoff-time t off n denotes theinstantwhen q reachesthecorrespondingon-angle q on n andoff-angle q off n ; respectively.Fig.2-2exempliesthestimulationpatternandtheassociatedswitchingstates. 35

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Figure2-2.Examplestimulationpatterndepictingintervalsofthecrankcycleoverwhich themusclegroupsofonelegarestimulated. 36

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CHAPTER3 SWITCHEDCONTROLOFCADENCEDURINGSTATIONARYCYCLINGINDUCEDBY FUNCTIONALELECTRICALSTIMULATION Inthischapter,aswitchedslidingmodecontrolinputisdeveloped,basedonthe stimulationpatterndevelopedinChapter2,withtheobjectivethattheriderpedal atadesiredcadencecrankvelocity.AcommonLyapunov-likefunctionisused,in conjunctionwiththeswitchedFES-cyclingsystemmodeldevelopedinChapter2,to provethatthecadencetrackingerrorisboundedbyanexponentiallydecayingenvelope inregionswheremusclegroupsareactivatedandbyanexponentiallyincreasing envelopeinregionswherenomusclegroupsareactivated.Theoverallerrorsystem isshowntobeultimatelyboundedprovidedsufcientconditionsonthecontrolgains, desiredtrajectory,andstimulationpatternaresatised.ExperimentalresultsonablebodiedsubjectsandonesubjectwithParkinson'sdiseasedemonstratetheswitched controller'sperformanceundertypicalFES-cyclingconditions. 3.1ControlDevelopment Inthisdevelopment,noelectricmotorinputisprovidedi.e., u motor =0 andthe cycleinertia J cycle islumpedtogetherwiththeriderinertia M i.e.,here M , M + J cycle . 3.1.1Open-LoopErrorSystem Thecontrolobjectiveistotrackadesiredcrankcadencewithperformancequantiedbythetrackingerrorsignal r : R 0 ! R ,denedas r t , _ q d t )]TJ/F15 11.9552 Tf 14.115 0 Td [(_ q t ; where _ q d : R 0 ! R denotesthedesiredcadence,designedsothatitsderivatives existand _ q d ; q d 2L 1 .Withoutlossofgenerality, _ q d isdesignedtobenon-negative,i.e., backpedalingisnotdesired. Takingthetimederivativeof3,multiplyingby M; andusing2and3 yieldstheopen-looperrorsystem 37

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M q t _ r t = q t ; _ q t ;t )]TJ/F25 11.9552 Tf 10.233 0 Td [(V q t ; _ q t r t )]TJ/F25 11.9552 Tf 10.233 0 Td [(B FES q t ; _ q t u FES t ; wheretheauxiliaryterm : Q R R 0 ! R isdenedas , M q d + V _ q d + b cycle _ q + G + P + d cycle + d rider )]TJ/F25 11.9552 Tf 11.955 0 Td [( r : BasedonAssumption1andProperties5-12, canbeboundedas j j c 1 + c 2 j r j ; where c 1 ;c 2 2 R > 0 areknownconstantsdenedas c 1 , Mc d 2 + c G + c P 1 + c d;cycle + c d;rider + c r + c V c 2 d 1 + )]TJETq1 0 0 1 423.723 460.506 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F25 11.9552 Tf 423.723 450.531 Td [(b + c P 2 c d 1 ; c 2 , c V c d 1 + b + c P 2 : 3.1.2Closed-LoopErrorSystem Basedon3andthesubsequentstabilityanalysis,thecontrolinputisdesigned as u FES , k 1 r + k 2 sgn r ; wheresgn : R ! [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] denotesthesignumfunctionand k 1 ;k 2 2 R > 0 areconstant controlgains.Aftersubstituting3intotheopen-looperrorsystemin3,the followingswitchedclosed-looperrorsystemisobtained: M _ r = )]TJ/F25 11.9552 Tf 11.955 0 Td [(Vr )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES k 1 r + k 2 sgn r : 3.2StabilityAnalysis Let V L : R ! R 0 beapositivedenite,continuouslydifferentiable,common Lyapunov-likefunctiondenedas V L r t , 1 2 M q t r 2 t : 38

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ThecommonLyapunov-likefunction V L isradiallyunboundedandsatisesthefollowing inequalities: 1 2 M r 2 V L 1 2 M r 2 : Theorem3.1. For q 2Q c ,thetrackingerrorisboundedbyanexponentiallydecaying envelopegivenby j r t j q MM )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 j r t on n j exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 c t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n ; forall t 2 t on n ;t off n andforall n ,where c 2 R > 0 isdenedas c , 2 M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 k 1 B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 ; providedthefollowingsufcientgainconditionsaresatised: k 1 >c 2 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;k 2 c 1 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 : Proof. Let r t for t 2 t on n ;t off n beaFilippovsolutiontothedifferentialinclusion _ r t 2 K [ h ] r t ; where K [ ] isdenedasin[43]andwhere h : R ! R isdenedby 3as h , M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Vr )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES k 1 r + k 2 sgn r : Thetimederivativeof3existsalmosteverywherea.e.,i.e.,foralmostall t 2 t on n ;t off n ; and _ V L a:e: 2 _ ~ V L ; where _ ~ V L isthegeneralizedtimederivativeof3alongthe Filippovtrajectoriesof _ r 2 h r andisdenedas[44] _ ~ V L , 2 @V L T K 2 6 4 h 1 3 7 5 ; where @V L isthegeneralizedgradientof V L .Since V L iscontinuouslydifferentiablein r , @V L = fr V L g ; thus, _ ~ V L 2 6 4 Mr 1 2 _ Mr 2 3 7 5 T K 2 6 4 h 1 3 7 5 : 39

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Usingthecalculusof K [ ] from[44]andsubstituting3intotheresultyields _ ~ V L )]TJ/F25 11.9552 Tf 11.955 0 Td [(Vr )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 K [ B FES ] r )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 2 K [ B FES sgn ] r r + 1 2 _ Mr 2 : First,3isevaluatedbetweeninstantswhere B FES switchestoprovethat V L isa commonLyapunovfunctioninthecontrolledregions.Inotherwords,consider q 2Q m foranarbitrary m 2M ; where B FES iscontinuousandnonzero,sothat3canbe expressedas _ ~ V L )]TJ/F25 11.9552 Tf 11.955 0 Td [(Vr )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 B FES r )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 B FES K [ sgn ] r r + 1 2 _ Mr 2 ; for q 2Q m ; where K [ B FES sgn ] r = B FES SGN r and SGN r , 8 > > > > > > > < > > > > > > > : 1 if r> 0 [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if r =0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if r< 0 : UsingProperty13allows3toberewrittenas _ ~ V L r )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 B FES r 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 2 B FES SGN r r: Thesingletonsupremumof3canbefoundusingProperty11,3,and3, andthefactthat _ V L a:e: 2 _ ~ V L cansubsequentlybeusedtoprovethat _ V L a:e: )]TJ/F15 11.9552 Tf 32.468 0 Td [( k 2 min k m B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 1 j r j)]TJ/F15 11.9552 Tf 17.933 0 Td [( k 1 min k m B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 r 2 ; whereSGN r r wasreplacedwith j r j sinceSGN r isonlyset-valuedfor r =0 .From Property11,itcanbedemonstratedthattheinequalityin3holdsforallsubsets Q m ofthecontrolledregion Q c ; soitcanbeconcludedthat V L isacommonLyapunov functioninthecontrolledregion.Providedtheconditionsonthecontrolgainsin3 40

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aresatised,3canbeusedtoupperbound3as _ V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( c V L ; where c wasdenedin3.Theinequalityin3canberewrittenas exp c t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n _ V L + c V L a:e: 0 ; whichisequivalenttothefollowingexpression: d dt V L exp c t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n a:e: 0 : TakingtheLebesgueintegralof3andrecognizingthattheintegrandonthelefthandsideisabsolutelycontinuousallowstheFundamentalTheoremofCalculustobe usedtoyield V L r t V L r t on n exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( c t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n ; forall t 2 t on n ;t off n andforall n .Rewriting3using3andperformingsome algebraicmanipulationyields3. Remark 3.1 . Theorem3.1guaranteesthatthedesiredcadencecanbetrackedwith exponentialconvergence,providedthatthecrankangledoesnotexitthecontrolled region.Thus,ifthestimulationpatternanddesiredcadencearedesignedsuchthat thecrankisnotrequiredtoexitthecontrolledregion,thecontrollerin3mayyield exponentialtrackingforalltime.Ifthedesiredcadenceisdesignedsuchthatthecrank mustexitthecontrolledregion,thesystemmaybecomeuncontrolledandthefollowing theoremdetailstheresultingerrorsystembehavior. Theorem3.2. For q 2Q u ,thetrackingerrorcanbeboundedas j r t j M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJETq1 0 0 1 236.102 126.157 cm[]0 d 0 J 0.478 w 0 0 m 12.574 0 l SQBT/F25 11.9552 Tf 236.102 116.314 Td [(Mr 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n +1 exp )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( u )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n )]TJ/F25 11.9552 Tf 11.955 0 Td [(M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 1 2 ; 41

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forall t 2 t off n ;t on n +1 andforall n ,where u 2 R > 0 isdenedas u , 2 max c 1 q 2 M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; 2 c 2 M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 : Proof. Intheuncontrolledregion,thetimederivativeof3canbeexpressedusing 3,Property11,andProperty13as _ V L = r; whichcanbeupperboundedusing3and3as _ V L c 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 V L + c 1 q 2 M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 V 1 2 L : Theright-handsideof3canbeupperboundedinapiecewisemanneras _ V L 8 > > < > > : 1 2 u V L +1 if V L 1 u V L if V L > 1 ; where u wasdenedin3.Sinceboth V L and u arepositive,3canalways beupperboundedas _ V L u V L + 1 2 : Thesolutionto3overtheinterval t 2 t off n ;t on n +1 yieldsthefollowingupperbound on V L intheuncontrolledregion: V L r t V L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n exp )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( u )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n + 1 2 )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(exp )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [( u )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; forall t 2 t off n ;t on n +1 andforall n .Rewriting3using3andperformingsome algebraicmanipulationyields3. Remark 3.2 . Theexponentialboundin3indicatesthatintheuncontrolledregions, theerrornormisboundedbyanexponentiallyincreasingenvelope.Sincetheerror normdecaysatanexponentialrateinthecontrolledregions,asdescribedby3, 42

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sufcientconditionsforstabilityoftheoverallsystemcanbedevelopedbasedonthe exponentialtimeconstants c and u andthetimethatthecrankdwellsineachregion dwell-times t on n , t off n )]TJ/F25 11.9552 Tf 12.206 0 Td [(t on n and t off n , t on n +1 )]TJ/F25 11.9552 Tf 12.206 0 Td [(t off n .However,achallengeisthatthe dwell-time t on n andreversedwell-time t off n nomenclaturederivedfrom[45]depend ontheswitchingtimes,whichareunknownapriori.Thefollowingpropositionsintroduce boundsontheuncertaindwell-times. Proposition3.1. The n th dwell-time t on n hasaknown,constantlowerbound t on min 2 R > 0 suchthat min n t on n t on min ; where t on min isdenedas t on min , min n )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(q off n )]TJ/F25 11.9552 Tf 11.955 0 Td [(q on n _ q max ; provided _ q t _ q max ; forall t 2 t on n ;t off n ,where _ q max 2 R > 0 isaselectedsafetylimit. Proof. Thedwell-time t on n canbepreciselyexpressedas t on n = q off n )]TJ/F25 11.9552 Tf 11.956 0 Td [(q on n _ q on avg;n ; where _ q on avg;n 2 R istheaveragecadencethroughthe n th controlledregion.The numeratorof3dependsonthechoiceof " m ,introducedin2-2,and theinitialconditions;therefore, q off n and q on n areknownapriori.However,theaverage cadencethrougharegionofthecrankcycledependsonfuturevelocitystatesandis thereforeunknownapriori.Then,theexpressionin3canbelowerboundedusing thecadencelimit _ q max as t on n q off n )]TJ/F25 11.9552 Tf 11.956 0 Td [(q on n _ q max ; provided3issatisedforall t 2 t on n ;t off n .Finally,3canbeboundedforall n asin3. 43

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Assumption3. Ifthevelocityateachoff-timeisgreaterthanorequaltosomeknown criticalvelocity _ q crit 2 R > 0 ,thenthecrankwillexittheuncontrolledregioninanite amountoftimeforall n .Inotherwords, _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n _ q crit t off n t off crit ; 8 n ,where t off crit 2 R > 0 isaknownconstant.Thecriticalvelocityandthecorrespondingcritical reversedwell-time t off crit canbenumericallydeterminedforawiderangeofsystem congurationsorexperimentallydeterminedforaspecicconguration. Proposition3.2. The n th reversedwell-time t off n hasaknown,constantupperbound t off max 2 R > 0 suchthat max n t off n t off crit = t off max ; provided _ q )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n _ q crit ; forall n . Proof. AccordingtoAssumption3,each t off n isboundedbyaknownconstant, providedtheinitialcrankvelocityissufcientlylargeuponenteringtheuncontrolled regions.Therefore,if3issatisedforall n ,thenmax n t off n t off crit .Thecritical off-durationisthereforealsothemaximumoff-duration,i.e., t off crit = t off max ,providedthe criticalvelocityisalsotheminimumvelocityforalloff-times,i.e.,3. Remark 3.3 . Propositions3.1and3.2introducesufcientconditions3and3 that,ifsatised,guaranteeknownboundsonthedwell-times t on n andreversedwelltimes t off n ,despiteuncertaintyintheswitchingtimes.Thefollowingtheoremsdescribe howthesesufcientconditionscanbesatisedapriori. Theorem3.3. Thecrankvelocitysatises3forall t 2 t on n ;t off n ,providedthe followingconditionsaresatisedforall n : max t 2 [ t on n ;t off n ] _ q d t < _ q max ; j r t on n j q 2 D M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; 44

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where D 2 R 0 isdenedas D , 1 2 M _ q max )]TJ/F48 11.9552 Tf 24.898 0 Td [(max t 2 [ t on n ;t off n ] _ q d t ! 2 : Proof. Theconditionsin3and3canbeusedwith3todemonstratethat V L r t on n D ,whichcanthenbeusedwith3toupperbound V L r t as V L r t D exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( c t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t on n ; forall t 2 t on n ;t off n : Using3,3,and3,andassumingtheworstcase scenariowhere r t < 0 ,thefollowingupperboundonthecrankvelocityinthe controlledregionscanbeobtained: _ q t _ q d t + q 2 D M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( c t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n ; forall t 2 t on n ;t off n .Tosatisfy3,itissufcienttodemonstratethattheright-hand sideof3isupperboundedby _ q max : Theright-handsideof3canbeupper boundedfurtheras _ q t max t 2 [ t on n ;t off n ] _ q d t + q 2 D M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; forall t 2 t on n ;t off n .Fromthedenitionin3,theright-handsideof3isequal to _ q max .Therefore,itcanbeconcludedvia3andthedenitionin3that3 issatisedforall t 2 t on n ;t off n ,whichallowsthedwell-timetobelowerboundedasin Proposition3.1. Theorem3.4. Thecrankvelocityatthe n th off-timesatises3providedthe sufcientconditionsinTheorem3.3andthefollowingconditionsaresatisedforall n : _ q d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n > _ q crit ; c )]TJ/F15 11.9552 Tf 41.824 8.087 Td [(1 t on min ln D D )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; 45

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where D 2 R 0 isdenedas D , 1 2 M min n _ q d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n )]TJ/F15 11.9552 Tf 14.114 0 Td [(_ q crit 2 : Proof. Toensurethat3issatisedforall n ,3mustbesatised;otherwise, thetrackingerrormayconvergetozeroattheoff-time,accordingto3,which couldresultinacrankvelocityattheoff-timethatisslowerthanthecriticalvelocity.In addition,3alsoguaranteesthat D isnonzero,whichwouldotherwiserequire c ! 1 via3.If3issatised,thenitissufcienttoshowthattheactualvelocityis sufcientlyclosetothedesiredvelocityattheoff-time,whichcanbeguaranteedbya sufcientlylargeconvergencerate c : Specically,iftheconditionsinTheorem3.3are satised,theworstcasescenariowhere t on n = t on min forall n canbeconsidered,so that3canbeusedtodemonstratethat V L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n D exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( c t on min : Then,3canbeusedtodemonstratethattheright-handsideof3isupper boundedby D ,sothat3canbefurtherboundedas V L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n D : Assumingtheworstcasescenariowhere r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n > 0 andusing3with3,the crankvelocityatthe n th off-timecanbelowerboundedas _ q )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(t off n min n _ q d )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(t off n )]TJ/F30 11.9552 Tf 11.955 15.283 Td [(q 2 D M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 : Fromthedenitionin3,theright-handsideof3isequalto _ q crit : Therefore,it canbeconcludedthattheconditionsinTheorem3.3,Theorem3.4,3–40,and3 aresufcienttosatisfy3foranarbitrary n ,which,fromAssumption3,allowsthe reversedwell-time t off n tobeupperboundedasinProposition3.2. 46

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Remark 3.4 . Theorems3.3and3.4givesufcientconditionswhichensureboundednessofthecrankvelocity,andtherebythedwell-time t on n andreversedwell-time t off n , overasinglecrankcycle.Toensureboundednessacrossallcrankcycles,itmustbe shownthatthetrackingerrordoesnotgrowsomuchintheuncontrolledregionsthatthe crankvelocitycouldexceed _ q max uponre-enteringthecontrolledregions.Inotherwords, giventhatthesufcientconditionsintheprevioustheoremsaresatisedfor n =0 ,andif r )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t on n +1 p 2 D M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; thentheresultsofTheorems3.3and3.4holdforall n: Theorem3.5. Thetrackingerroruponexitingtheuncontrolledregions r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t on n +1 is boundedas r )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t on n +1 q 2 D M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 forall n; providedthesufcientconditionsinTheorems3.3and3.4andthefollowing conditionsaresatised: c u t off max t on min )]TJ/F15 11.9552 Tf 25.22 8.088 Td [(1 t on min ln M M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(b D )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; D>b MM )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; where b 2 R > 0 isdenedas b , 1 2 )]TJ/F48 11.9552 Tf 5.479 -9.684 Td [(exp )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( u t off max )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Consequently,thedwell-times t on n andreversedwell-times t off n areboundedasin Propositions3.1and3.2forall n: Proof. Assumingtheworstcasescenarioforthe n th cyclewhere t on n = t on min and t off n = t off max i.e.,minimumdecayandmaximumgrowthofthetrackingerror,3, 3,3,and3canbeusedtodemonstratedthat 1 2 M r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t on n +1 2 a D MM )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 + b; 47

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where a 2 R > 0 isdenedas a , exp )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( u t off max )]TJ/F25 11.9552 Tf 11.955 0 Td [( c t on min : Provided3issatised,itcanbedemonstratedthattheright-handsideof3is boundedaboveby D ,i.e.,3.Provided3issatised,itcanbedemonstrated thattheargumentofln in3ispositive,sothat c canalwaysbeselectedto satisfy3.Finally,if3issatised,Theorems3.3-3.5canbeappliedforall n . Remark 3.5 . Theconditionin3arisesduetothefactthatthetrackingerrorwill alwayshaveanonzeroupperbound,derivedfrom3,whenexitingtheuncontrolled region.Therefore, D mustbesufcientlylarge.Basedonthedenitionin3, D can bemadearbitrarilylargebyselectionof _ q max andthedesignof _ q d : Remark 3.6 . Propositions3.1and3.2proposeknownboundsonthedwell-timesand reversedwell-times,andTheorems3.3-3.5givesufcientconditionstovalidatethe Propositionsforall n .Withknownboundsonthetimebetweenswitchesandknown ratesofconvergenceanddivergenceofthetrackingerror,aknownultimatebound onthetrackingerrorcanbecalculated.ThefollowingTheoremgivesthevalueofthis ultimateboundalongwithasufcientconditionforconvergenceofthetrackingerrorto thatbound. Theorem3.6. Thetrackingerrorisultimatelyboundedinthesensethat j r t j converges toaballwithconstantradius d 2 R > 0 asthenumberofcrankcyclesapproachesinnity i.e.,as n !1 ,where d isdenedas d , s 2 b M )]TJ/F25 11.9552 Tf 11.955 0 Td [(a ; providedthesufcientconditionsinTheorems3.3-3.5andthefollowingconditionare satised: c > u t off max t on min : 48

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Proof. Using3and3sequentiallyandassumingtheworstcasescenariofor eachcyclewhere t on n = t on min and t off n = t off max ,anupperboundfor V L r t on n after N 2 N > 0 cyclescanbedevelopedas V L r t on N V L r t on 0 a N + b N )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 X n =0 a n : Theright-handsideof3canbeviewedasaserieswhoseelementsformthe sequence f x n 2 R 0 g N 0 ; denedfor n 2f 1 ; 2 ;:::;N g as x n , x 0 a n + b n )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 X k =0 a k ; where x 0 , V L r t on 0 .Then,thesequence f V L r t on n g isupper-boundedby f x n g foreach n .Thesequence f x n g ispositive,monotonic,andbounded,provided3is satisedi.e., a< 1 ;therefore,thelimitof f x n g existsandcanbeexpressedas lim n !1 x n = d; where d 2 R > 0 isaknownconstantdenedas d , b 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(a : Therefore,since V L r t on n x n foreach n , V L r t on n isultimatelyboundedby d inthe sensethatas n !1 ;V L r t on n x n ! d: Similarly, V L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(r )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n isultimatelybounded by d 2 R 0 ,whichisaknownconstantdenedas d , d exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( c t on min : Monotonicityoftheboundsin3and3canbeusedtodemonstratethat V L r t max f x n ;x n +1 g for t 2 t on n ;t on n +1 ,soitcanbeconcludedthat V L is ultimatelyboundedby d .Itcanalsobeconcludedthat V L r t d forall t t off N when V L r t off N d ,or,similarly,forall t t on N when V L r t on N d: Inotherwords, ifthecontrollerisswitchedonwhen V L r t on N d , V L willremainwithinthatballfor 49

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allsubsequenttime.Using3,itcanthenbedemonstratedthatas n !1 ; j r t j convergestoaballwithconstantradius d ,where d wasdenedin3. Remark 3.7 . Thesizeoftheultimatebound d dependsontheboundsoftheinertial termgiveninProperty7,theboundsof in3,theminimumdwell-time t on min ,the maximumreversedwell-time t off max ; andthecontrolgain k 1 via c .However,theeffectof thecontrolgainislimited;considerthefollowing: lim c !1 d = b; lim c !1 d =0 : Basedon3and3,thelowerlimitof V L canbedriventozerobychoosing thegainstobearbitrarilylarge,but V L willalwayshaveanonzeroupperbound.The sizeoftheultimateboundcanbefurtherminimizedbydecreasingthelengthofthe uncontrolledregionsvia " m orbyensuringlarger _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n foreach n toreduce t off max . Notealsothat3isasufcientconditionfor d D: Remark 3.8 . Theconditionsin3,3,and3,togetherwith3,all imposeconstraintsontheselectionof k 1 via c andcanthereforebecombinedas k 1 max c 2 min k m B )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJETq1 0 0 1 323.291 322.526 cm[]0 d 0 J 0.478 w 0 0 m 12.574 0 l SQBT/F25 11.9552 Tf 323.291 312.683 Td [(M 2 min k m B t on min ln D D )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; c 2 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 + M 2 min k m B t on min u t off max )]TJ/F22 11.9552 Tf 11.956 0 Td [(ln 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(b D )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; wherethefactthat3isasufcientconditionfor3wasused. 3.3Experiments FES-cyclingexperimentswereconductedwiththeprimaryobjectiveofevaluating theperformanceoftheswitchedcontrollergivenin3anddistributedtothegluteal, quadricepsfemoris,andhamstringsmusclegroupsaccordingto2.TheexperimentsweredividedintoProtocolAandProtocolB.Theobjectiveofbothprotocols wastodemonstratethecontroller'scadencetrackingperformanceinthepresence 50

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ofparametricuncertaintyandunmodeleddisturbances.TheFES-cyclingtrialswere stoppedifthecontrolinputsaturated,thesubjectreportedsignicantdiscomfort,the cadencefellbelow0RPM,thetrialruntimeexpired,orthecadenceexceeded60RPM. Theexperimentscouldalsobeendedatanytimebythesubjectsviaanemergencystop switch. 3.3.1Methods Fourable-bodiedmalesubjects25-27yearsoldwererecruitedfromthestudent populationattheUniversityofFlorida,andonemalesubjectwithPD,60yearsold,with amodiedHoehnandYahrdisabilityscoreof2.5[46],wasrecruitedfromtheUniversity ofFloridaCenterforMovementDisordersandNeurorestoration.Eachsubjectgave writteninformedconsentapprovedbytheUniversityofFloridaInstitutionalReview Board.Able-bodiedsubjectswererecruitedtovalidatethecontrollerdesign,andthe subjectwithPDwasrecruitedtodemonstratefeasibilityoftheproposedapproachina potentialpatientpopulation. ThesubjectwithPDinthisexperimentexhibitedmildbilateralmotorimpairment withevidenttremor.Itwasobservedduringpreliminarytestingthatthesubject'sright sidewasmoreaffectedi.e.,greatertremorandexhibitedbradykinesiaduringcycling i.e.,whentherightlegwassupposedtopedal,cadencedecreasedsignicantly. Inaddition,preliminarytestingrevealedthatsubjectcouldnottoleratethelevelof stimulationintensitynecessaryforFES-inducedcyclingi.e.,FES-cyclingwithout volitionaleffortfromtherider.ItwashypothesizedthatFES-assistedcyclingi.e., FES-cyclingwithvolitionaleffortfromtheriderwouldbeamoreappropriateprotocol forsubjectswithPD.ItwasfurtherhypothesizedthatFES-assistancewouldprovide sensorycuesandmuscleactivationassistanceduringcyclingandtherebydecrease variabilityinthesubject'scadence. Acommerciallyavailable,stationary,recumbentexercisecycleAudioRiderR400, NordicTrackwasmodiedforthepurposesoftheFES-cyclingexperimentsandis 51

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Figure3-1.FES-cyclingtestbed.Photocourtesyofauthor. depictedinFigure3-1.Thecycleoriginallyhadaywheelwhichwasdrivenbya freewheel.Thefreewheelwasthenreplacedwithaxedgearsothatthecrankshaft wasdirectlycoupledtotheywheel,allowingtheywheeltocontributeitsmomentum tothecycle-ridermomentumandimprovingthesystemenergetics[47].Thecyclehas anadjustableseatandamagnetichysteresisbrakeontheywheelwith16incremental levelsofresistanceresistancewassettoLevel1unlessotherwisenoted.Custom pedalswereconstructedthatallowedhigh-toporthoticbootsReboundAirWalker, ssurtobeafxedtothem;theseorthoticpedalsservedtoxtherider'sfeettothe pedals,preventdorsiexionandplantarexionoftheankles,andmaintainsagittal alignmentofthelowerlegs.Anoptical,incrementalencoderHS35F,BEISensors, resolution 0 : 018 wasaddedtothecycleandcoupledtothecrankshafttomeasurethe cyclingcadence.ThecyclewasequippedwithaHalleffectsensorandmagnetonthe crankthatprovidedanabsolutepositionreferenceoncepercycle. 52

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Acurrent-controlledstimulatorRehaStim,Hasomeddeliveredbiphasic,symmetric,rectangularpulsestothesubject'smusclegroupsviabipolar,self-adhesive electrodesPALS3x5. 1 Apersonalcomputerequippedwithdataacquisitionhardware andsoftwarewasusedtoreadtheencodersignal,calculatethecontrolinput,andcommandthestimulator.Stimulationfrequencywasxedat60Hztoleveragetheresults foundin[48].Stimulationintensitywascontrolledbyxingthepulseamplitudeforeach musclegroupandcontrollingthepulsewidthaccordingto3.Pulseamplitudewas determinedforeachsubject'smusclegroupsinpreliminarytestingandrangedfrom 50-110mA. Electrodeswereplacedoverthesubjects'gluteal,quadricepsfemoris,andhamstringsmusclegroups,accordingtoAxelgaard'selectrodeplacementmanual 2 ,while subjectswerestandingupright.Subjectswerethenseatedonthestationarycycle,and theirfeetwereinsertedsecurelyintotheorthoticpedals.Thecycleseatpositionwas adjustedforeachsubject'scomfortwhileensuringthathyperextensionoftheknees couldnotbeachievedwhilecycling.Thesubject'shippositionrelativetothecyclecrank axiswasmeasuredalongwiththedistancesbetweenthesubjects'greatertrochanters andlateralfemoralcondyles l t andbetweenthesubjects'lateralfemoralcondyles andthepedalaxesofrotation l l .Thesedistanceswereusedtocalculatethetorque transferratiosforthesubjects'musclegroupsandtotherebydeterminethestimulation pattern. Thedesiredcrankvelocitywasdenedinradianspersecondas _ q d , 5 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F25 11.9552 Tf 9.298 0 Td [(t ; 1 SurfaceelectrodesforthisstudywereprovidedcomplimentsofAxelgaardManufacturingCo.,Ltd. 2 http://www.palsclinicalsupport.com/videoElements/videoPage.php 53

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where 2 R > 0 wasaselectableconstantusedtocontroltheaccelerationofthedesired trajectoryand t on 0 =0 seconds.Thetrajectoryin3ensuredthatthedesiredvelocity startedat0RPMandsmoothlyapproached50RPM.Thecontrolgains,introduced in2and3,weretunedtoyieldacceptabletrackingperformanceforeach subjectinpreliminarytestingandrangedasfollows: k 1 2 [70 ; 150] ;k 2 2 [7 ; 15] , k Glute 2 [0 : 5625 ; 1 : 125] , k Quad 2 [0 : 9 ; 1 : 125] , k Ham 2 [0 : 816 ; 1 : 2375] : ProtocolAwascompletedbyallable-bodiedsubjectsandconsistedofavoluntary cyclingphasefollowedbyveminutesofrestandasubsequentFES-cyclingphase. Duringthevoluntarycyclingphase,subjectswereshownacomputerscreenwitha real-timeplotoftheiractualcadence,asmeasuredbytheencoder,versusthedesired cadencegivenin3,andeachsubjectwasaskedtovoluntarilypedalsothatthe twoplotscoincidedwithoneanotheri.e.,minimizethetrackingerror r .After175 secondshadelapsed,theywheelresistancewasincreasedfromLevel1directlyto Level9foraperiodof30seconds,afterwhichtheresistancewasdecreasedbackto Level1fortheremainderofthecyclingphase.Thevoluntarycyclingphaselastedve minutes. Followingveminutesofrest,theFES-cyclingphasewasinitiated,whereincycling wasonlycontrolledbystimulationofthegluteal,quadricepsfemoris,andhamstrings musclegroupsi.e.,acompletelypassiverider.Thestimulationpatterni.e.,therange ofcrankanglesoverwhicheachmusclewasstimulatedforProtocolAwasdened byselecting " Glute =0 : 2 ;" Quad =0 : 3 ;" Ham =0 : 38 ; whichwasfoundtoyield satisfactoryperformanceinpreliminarytesting.Whilethesamevaluesof " m were usedforallsubjects,thestimulationpatternresultingfromthechoiceofeach " m was slightlydifferentforeachsubjectbecauseeachsubjecthaddifferentleglengthsand preferredseatingpositions.Thesubjects'limbswerethenpositionedmanuallyso thattheinitialcrankpositionwasinthecontrolledregion,andthenthecontrollerwas activated.Thesubjectswereinstructedtorelaxasmuchaspossiblethroughoutthis 54

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phaseandtomakenoefforttovoluntarilycontrolthecyclingmotion;additionally,the subjectswerenotgivenanyindicationofthecontrolperformancei.e.,subjectscould nolongerseetheactualordesiredtrajectory.Asinthevoluntarycyclingphase,the ywheelresistancewasincreasedfromLevel1toLevel9for t 2 [175 ; 205] seconds todemonstratethecontroller'srobustnesstoanunknown,bounded,time-varying disturbance.TheFES-cyclingphaselastedveminutes. ProtocolBwascompletedbythesubjectwithPDandwasthesameasProtocol A,withtheexceptionthatthesubjectwasallowedtovoluntarilypedalduringthe FES-cyclingphasei.e.,FES-assistedcyclingandcouldseetheactualanddesired cadence.WhileProtocolAwasintendedtodemonstratethecontroller'sperformance withacompletelypassiverider,aswouldbethecasewithasubjectwithmotorcomplete spinalcordinjury,ProtocolBdemonstratesfeasibilityofthedevelopedcontrollerfor abroaderpatientpopulationwithintact,albeitdiminished,motorcontrol,suchas thosewithincompletespinalcordinjury,hemipareticstroke,traumaticbraininjury, andPD.Thisvoluntaryassistancefromtheriderismodeledas r in2.Although disturbancesaregenerallyneitherassistivenorresistive,voluntaryeffortfromtherider duringFES-cyclingisgenerallyassistiveandisthereforeexpectedtodecreasethe controlinputneededtotrackthedesiredcadence. 3.3.2Results 3.3.2.1ProtocolAresults Fig.3-2depictsoneable-bodiedsubject'strackingperformance,quantiedbythe cadencetrackingerror r ,andthestimulationintensitypulsewidthinputtoeachmuscle group u m duringtheFES-cyclingphaseofProtocolA.InFig.3-2andsubsequent gures,theshadedregionmarkstheperiodoftimeduringwhichtheergometer's resistancewasincreasedfromLevel1toLevel9.Fig.3-3providesanenhancedview ofthecontrolinputoverasinglecrankcycletoillustratethecontrollerswitchingand distributionofthecontrolinputacrossthemusclegroups.Table3-1comparesthe 55

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Figure3-2.Onesubject'scadencetrackingerrorandcontrolinputtoeachmusclegroup duringtheFES-cyclingtrialofProtocolA.ACadencetrackingerror.B Stimulationintensityinputtoeachmusclegroup. subjects'volitionalandFES-inducedtrackingperformance,quantiedbythemeanand standarddeviationofthecadencetrackingerrorinRPM,overthetotaltrial t 2 [0 ; 300] secondsandduringseveralphasesofeachtrial:thetransientphase t 2 [0 ; 40] seconds,thesteadystatephase t 2 ; 175 seconds,theaddeddisturbancephase t 2 [175 ; 205] seconds,andthenalphase t 2 ; 300] seconds.Fig.3-4compares anothersubject'scadencetrackingerrorinthevoluntaryandFES-inducedcycling phases.Alltrialswenttocompletion. 3.3.2.2ProtocolBresults Fig.3-5depictsthetrackingperformanceofthesubjectwithPD,quantiedbythe cadencetrackingerror r ,andthestimulationintensitypulsewidthinputtoeachmuscle group u m duringtheFES-assistedphaseofProtocolB.Anenhancedviewofthecontrol 56

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Figure3-3.ControlinputoverasinglecrankcycleduringtheFES-cyclingtrialof ProtocolAforSubjectAB3. Figure3-4.CadencetrackingerrorforthevoluntaryandFES-cyclingphasesofProtocol AforSubjectAB3.AVoluntarytrackingphase.BFES-cyclingphase. 57

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Table3-1.Comparisonofcadencetrackingerrorforallable-bodiedsubjects'volitional andFES-cycling. SubjectPhase VolitionalerrorRPMFESerrorRPM AB1 Transient1.04 1.582.41 1.08 Steadystate-0.06 1.593.12 1.04 Disturbance0.04 2.013.39 1.61 Final-0.43 1.703.47 1.21 Totaltrial-0.02 1.733.16 1.22 AB2 Transient0.28 3.266.03 2.07 Steadystate-0.07 1.159.78 1.58 Disturbance0.01 1.4314.56 1.78 Final-0.01 1.1712.68 1.21 Totaltrial0.01 1.6310.68 2.92 AB3 Transient0.62 1.682.31 2.54 Steadystate0.01 1.063.12 1.70 Disturbance-0.21 1.514.14 1.52 Final-0.31 1.383.19 1.63 Totaltrial-0.03 1.343.14 1.85 AB4 Transient1.22 2.723.51 2.75 Steadystate-0.01 1.363.93 1.78 Disturbance0.40 1.564.74 3.52 Final0.18 1.344.41 2.97 Totaltrial0.25 1.674.11 2.57 All Transient0.79 2.313.56 2.11 Steadystate-0.03 1.294.99 1.53 Disturbance0.06 1.636.71 2.11 Final-0.14 1.405.94 1.76 Totaltrial0.05 1.595.27 2.14 58

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Figure3-5.Cadencetrackingerrorandcontrolinputtoeachmusclegroupduringthe FES-assistedphaseofProtocolB.ACadencetrackingerror.BStimulation intensityinputtoeachmusclegroup. inputoverasinglecrankcycleisprovidedinFig.3-6toillustratethecontrollerswitching anddistributionofthecontrolinputacrossthemusclegroups.Table3-2summarizes thevolitionalandFES-assistedcadencetrackingperformanceofthesubjectwithPD usingthesamemetricsasdescribedinSection3.3.2.1.Fig.3-7comparesthesubject's cadencetrackingerrorinthevoluntaryandFES-assistedcyclingphases.Alltrialswent tocompletion. 3.3.3Discussion TheresultsofProtocolAsuccessfullydemonstratetheabilityofthecontrollerin 3,distributedacrossthemusclegroupsaccordingto2,toachieveultimately boundedtrackingofthedesiredcadencedespiteparametricuncertaintye.g.,uncertain riderlimbmassandunknowndisturbances.Ultimatelyboundedtrackingwasachieved 59

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Figure3-6.ControlinputoverasinglecrankcycleduringtheFES-assistedcyclingtrial ofProtocolB. Table3-2.ComparisonofcadencetrackingerrorofthesubjectwithPDduringvolitional andFES-assistedcycling. SubjectPhase VolitionalerrorRPMFESerrorRPM PD Transient-1.28 7.411.28 4.87 Steadystate0.80 3.210.07 2.82 Disturbance2.10 3.881.15 2.91 Final0.11 2.65-0.46 2.32 Totaltrial0.43 4.060.17 3.11 60

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evenacrossarangeofstimulationpatterns.Althoughtheultimateboundonthetracking errorwashigherforFES-cyclingthanvolitionalcyclingbyallsubjectsinProtocolA, thiswaslikelyduetothesteadystateoffsetinthetrackingerrorandnotduetolarge variationsincyclingcadence,asquantiedinTable3-1.Thecadencetrackingerrorof allable-bodiedsubjectsduringvoluntaryandFES-inducedcyclingwas 0 : 05 1 : 59 RPM and 5 : 27 2 : 14 RPM,respectively.ThesteadystateerrorobservedintheFES-cycling phasemaybecausedbyalackofadaptationintheFES-cyclingcontroller.During volitionalcycling,riderscanlearnhowtomodulatetheforceoutputofthemuscles involvedincyclingtoimprovetrackingperformanceovertime.Therefore,toachieve cadencetrackingperformanceduringFES-cyclingthatissimilartothatobservedduring volitionalcycling,motivationarisestouseadaptivecontrolmethodsduringthecontrolled regions.However,thisischallengingbecauseadaptivecontrolmethodsusuallyonly achieveasymptoticconvergenceofthetrackingerror,butstabilityofaswitchedsystem withstableandunstablesubsystemscanonlybeguaranteediftheconvergenceand divergenceratesareknownasisthecasewithexponentialconvergence,forexample. TheresultsofProtocolBdemonstratethecontroller'strackingperformancedespite thepresenceofanadditionalunknowndisturbancemanifestedasvolitionaleffort fromthesubjectwithPD.ThedatagiveninTable3-2indicatethattheadditionof FES-assistancetothesubject'svolitionaleffortimprovedcadencetrackingperformance measurably.5%and23.4%improvementinmeanandstandarddeviationofthe cadencetrackingerroracrossthetotaltrial.Theimprovementintrackingperformance maybeduetothebiasofthestimulationinputtowardsthesubject'saffectedright legasdepictedinFig.3-6,providingbothassistanceinactivatingtheappropriate musclegroupsandasensorycuetovolitionallypedalfaster.Moredataisneededto determineiftheseresultsarestatisticallysignicant,buttheresultsnonethelessindicate thepotentialofFES-assistancetoimprovetheabilityofapersonwithPDtopedalata desiredcadence. 61

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Figure3-7.CadencetrackingerrorofthesubjectwithPDduringthevoluntaryand FES-assistedphasesofProtocolB.AVoluntarytrackingphase.B FES-cyclingphase. 3.4ConcludingRemarks AcommonLyapunov-likefunctionwasusedtoprovethatthedevelopedcontroller, givenin3,yieldsultimatelyboundedtrackingofadesiredcadencei.e.,crank velocity,providedthedesiredcadence,controlgains,andstimulationpatternsatisfy sufcientconditions.Experimentswereconductedonfourable-bodiedsubjects,and theresultsbothdemonstratetherobustnessandstabilityofthedevelopedswitched controllerandcorroboratethetheoreticalrelationships,describedinTheorems3.3-3.5, betweenthedesiredcadence,controlgains,andstimulationpattern.Experimentswere alsoconductedononesubjectwithPD,andtheresultssuggestthatFES-assisted cyclingusingthedevelopedswitchedcontrollermayimprovetheabilityofpeoplewith PDtotrackadesiredcadence. Motivatedbytheneedtominimizethesteadystatecadencetrackingerrorobserved inthischapter,thenextchapterwilladdcrankpositiontrackingtothecontrolobjectiveof cadencetracking. 62

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CHAPTER4 SWITCHEDCONTROLOFCRANKPOSITIONANDCADENCETRACKINGDURING STATIONARYCYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION Inthischapter,aswitchedslidingmodecontrollerisdesignedfortheswitched systemin2withtheobjectivethattheFEScontrolinputachievestrackingofa desiredcrankpositionandcadence,simultaneously.AcommonLyapunov-likefunction isusedtodemonstratethatthetrackingerrorisultimatelybounded,despiteuncertainty andboundeddisturbancesinthesystemanddespiteautonomous,state-dependent switching,providedsufcientconditionsonthecontrolgains,thedesiredcadence, theinitialconditions,andthestimulationpatternaresatised.Experimentalresults areprovidedwhichdemonstratethecontroller'strackingperformanceforarangeof stimulationpatterns. 4.1ControlDevelopment Inthisdevelopment,noelectricmotorinputisprovidedi.e., u motor =0 andthe cycleinertia J cycle islumpedtogetherwiththeriderinertia M i.e.,here M , M + J cycle . 4.1.1Open-LoopErrorSystem Thecontrolobjectiveistosimultaneouslytrackadesiredcrankpositionand cadence,withperformancequantiedbythetrackingerrorsignals e 1 ;e 2 : R 0 ! R , denedas e 1 t , q d t )]TJ/F25 11.9552 Tf 11.956 0 Td [(q t ; e 2 t , _ e 1 t + e 1 t ; where q d : R 0 ! R isthedesiredcrankposition,designedsothatitsderivativesexist and j _ q d j c d 1 ; j q d j c d 2 ,where c d 1 ;c d 2 2 R > 0 areknownconstants,and 2 R > 0 isa selectableconstant.Withoutlossofgenerality, q d isdesignedtomonotonicallyincrease, i.e.,backpedalingisnotdesired.Takingthetimederivativeof4,multiplyingby M; andusing2with4and4yieldsthefollowingopen-looperrorsystem: 63

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M q t _ e 2 t = q t ; _ q t ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(V q t ; _ q t e 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES q t ; _ q t u FES t ; wheretheauxiliaryterm : Q R R 0 ! R isdenedas , M q d + _ e 1 + V _ q d + e 1 + b cycle _ q + G + P + d cycle + d rider )]TJ/F25 11.9552 Tf 11.955 0 Td [( r : BasedonAssumption1andProperties5-12, canbeboundedas j q t ; _ q t ;t j c 1 + c 2 k z t k + c 3 k z t k 2 ; where c 1 ;c 2 ;c 3 2 R > 0 areknownconstantsdenedas c 1 , Mc d 2 + c G + c P 1 + c d;cycle + c d;rider + c r + c V c 2 d 1 + )]TJETq1 0 0 1 423.723 431.119 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F25 11.9552 Tf 423.723 421.143 Td [(b + c P 2 c d 1 ; c 2 , 2 M + )]TJETq1 0 0 1 208.69 404.09 cm[]0 d 0 J 0.478 w 0 0 m 12.574 0 l SQBT/F25 11.9552 Tf 208.69 394.247 Td [(M +2 c V c d 1 + b + c P 2 + c d 1 c V + b + c P 2 ; c 3 , c V )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 2 + ; andtheerrorvector z : R 0 ! R 2 isdenedas z , e 1 e 2 T : 4.1.2Closed-LoopErrorSystem Basedon4andthesubsequentstabilityanalysis,thecontrolinputisdesigned as u FES , k 1 e 2 + )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 ; wheresgn : R ! [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] denotesthesignumfunctionand k 1 ;k 2 ;k 3 ;k 4 2 R > 0 are constantcontrolgains.Aftersubstituting4intotheopen-looperrorsystemin4, 64

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thefollowingswitchedclosed-looperrorsystemisobtained: M _ e 2 = )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ve 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 1 e 2 + )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 : 4.2StabilityAnalysis Let V L : R 2 ! R denoteacontinuouslydifferentiable,positivedenite,radially unbounded,commonLyapunov-likefunctiondenedas V L z t , 1 2 z T t W q t z t ; wherethepositivedenitematrix W : Q! R 2 2 isdenedas W , 2 6 4 10 0 M 3 7 5 : V L satisesthefollowinginequalities: 1 k z k 2 V L 2 k z k 2 ; where 1 ; 2 2 R > 0 areknownconstantsdenedas 1 , min 1 2 ; 1 2 M ; 2 , max 1 2 ; 1 2 M : Theorem4.1. For q 2Q c ,thetrackingerrorisboundedbyanexponentiallydecaying envelopegivenby k z t k r 2 1 k z t on n k exp )]TJ/F25 11.9552 Tf 13.799 8.088 Td [( 1 2 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n ; forall t 2 t on n ;t off n andforall n ,where 1 2 R > 0 isdenedas 1 , min )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ;k 1 min k m B )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ; 65

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providedthefollowinggainconditionsaresatised: > 1 2 ;k 1 > 1 2 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;k 2 c 1 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; k 3 c 2 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;k 4 c 3 min k m B )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 : Proof. Let z t for t 2 t on n ;t off n beaFilippovsolutiontothedifferentialinclusion _ z t 2 K [ h ] z t ; where K [ ] isdenedasin[43]andwhere h : R 2 ! R 2 isdenedby 4and3as h , 2 6 4 _ e 1 _ e 2 3 7 5 = 2 6 4 e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e 1 M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ve 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 1 e 2 + )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 3 7 5 : Thetimederivativeof4existsalmosteverywherea.e.,i.e.,foralmostall t 2 t on n ;t off n ; and _ V L a:e: 2 _ ~ V L ; where _ ~ V L isthegeneralizedtimederivativeof4alongthe Filippovtrajectoriesof _ z 2 h z andisdenedas[44] _ ~ V L , 2 @V L T K 2 6 4 h 1 3 7 5 ; where @V L isthegeneralizedgradientof V L .Since V L iscontinuouslydifferentiablein z , @V L = fr V L g ; thus, _ ~ V L 2 6 4 z T W 1 2 z T _ Wz 3 7 5 T K 2 6 4 h 1 3 7 5 : Usingthecalculusof K [ ] from[44],substituting4,andusing4tosimplifythe resultingexpressionyields _ ~ V L )]TJ/F25 11.9552 Tf 28.56 0 Td [(e 2 1 + e 1 e 2 + e 2 + 1 2 _ M )]TJ/F25 11.9552 Tf 11.956 0 Td [(V e 2 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(K [ B FES ] )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 1 e 2 2 )]TJ/F30 11.9552 Tf 11.291 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 K [ B FES sgn ] e 2 e 2 : Theexpressionin4isevaluatedbetweeninstantswhere B FES switchestoprove that V L isacommonLyapunovfunctioninthecontrolledregions.Inotherwords, 66

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consider q 2Q m foranarbitrary m 2M ; where B FES iscontinuousandnonzero,sothat 4canbeexpressedas _ ~ V L )]TJ/F25 11.9552 Tf 28.559 0 Td [(e 2 1 + e 1 e 2 + e 2 + 1 2 _ M )]TJ/F25 11.9552 Tf 11.955 0 Td [(V e 2 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 1 e 2 2 )]TJ/F30 11.9552 Tf 11.291 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 B FES SGN e 2 e 2 ; where K [ B FES sgn ] e 2 = B FES SGN e 2 .UsingProperty13allows4tobe rewrittenas _ ~ V L )]TJ/F25 11.9552 Tf 28.559 0 Td [(e 2 1 + e 1 e 2 + e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(k 1 e 2 2 )]TJ/F30 11.9552 Tf 11.291 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 B FES SGN e 2 e 2 : Since _ V L a:e: 2 _ ~ V L ; 4canbeusedtodemonstratethat _ V L a:e: = )]TJ/F25 11.9552 Tf 9.298 0 Td [(e 2 1 + e 1 e 2 + e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 1 e 2 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(B FES )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 j e 2 j ; whereSGN e 2 e 2 wasreplacedwith j e 2 j sinceSGN e 2 isonlyset-valuedfor e 2 =0 .By usingYoung'sinequality,Property11,and3,4canbeupperboundedas _ V L a:e: )]TJ/F30 11.9552 Tf 32.468 16.857 Td [( )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 e 2 1 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( k 1 min k m B )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 2 e 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 min k m B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 1 j e 2 j )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 3 min k m B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 k z kj e 2 j)]TJ/F15 11.9552 Tf 17.933 0 Td [( k 4 min k m B )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 3 k z k 2 j e 2 j : From11,itcanbedemonstratedthattheinequalityin4holdsforallsubsets Q m ofthecontrolledregion Q c ; soitcanbeconcludedthat V L isacommonLyapunov functioninthecontrolledregion.Providedthegainconditionsin4–14aresatised, 4canbeusedtorewrite4as _ V L a:e: )]TJ/F25 11.9552 Tf 25.407 8.087 Td [( 1 2 V L ; 67

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where 1 wasdenedin4.Theinequalityin4canberewrittenas exp 1 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n _ V L + 1 2 V L a:e: 0 ; whichisequivalenttothefollowingexpression: d dt V L exp 1 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n a:e: 0 : TakingtheLebesgueintegralof4andrecognizingthattheintegrandonthelefthandsideisabsolutelycontinuousallowstheFundamentalTheoremofCalculustobe usedtoyield V L z t V L z t on n exp )]TJ/F25 11.9552 Tf 10.872 8.088 Td [( 1 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n ; forall t 2 t on n ;t off n andforall n .Using4torewrite4andperformingsome algebraicmanipulationyields4. Remark 4.1 . Theorem4.1guaranteesthatdesiredcranktrajectoriescanbetracked withexponentialconvergence,providedthatthecrankangledoesnotexitthecontrolled region.Thus,ifthecontrolledregionsanddesiredtrajectoriesaredesignedappropriately,thecontrollerin4yieldsexponentialtrackingofthedesiredtrajectoriesforall time.Ifthecrankpositionexitsthecontrolledregion,thesystembecomesuncontrolled andthefollowingtheoremdetailstheresultingerrorsystembehavior. Theorem4.2. For q 2Q u ,thetrackingerrorcanbeboundedasfollows: k z t k 1 p 1 tan 3 2 )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n + tan )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 z )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(t off n 2 1 2 ; forall t 2 t off n ;t on n +1 andforall n ,where 2 2 R > 0 isaknownconstantdenedas 2 , max c 3 1 1 3 2 ; c 2 + 1 2 1 1 ;c 1 1 1 1 2 ; providedthereversedwell-time t off n , t on n +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n issufcientlysmallsuchthat t off n < 6 2 ; 68

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forall n ,andprovided 1 ,denedin4,issufcientlylargesuchthat 1 > 2 t on n ln )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V L t on n tan )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(3 2 t off n ; forall n ,where t on n , t off n )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on n . Proof. Thetimederivativeof3for t 2 t off n ;t on n +1 canbeexpressedusing4, 3,Property11,andProperty13as _ V L = )]TJ/F25 11.9552 Tf 9.298 0 Td [(e 2 1 + e 1 e 2 + e 2 : Young'sinequality,3,and4allow4tobeupperboundedas _ V L c 3 k z k 3 + c 2 + 1 2 k z k 2 + c 1 k z k : Using3,4canbeupperboundedas _ V L c 3 1 1 V L 3 2 + c 2 + 1 2 1 1 V L + c 1 1 1 V L 1 2 : Tosimplifytheresultingexpression,4canbeupperboundedas _ V L 3 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V 2 L +1 ; where 2 wasdenedin4.Utilizing4andtheComparisonLemmain[49],the followingupperboundon V L intheuncontrolledregioncanbefound: V L z t tan )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(3 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t off n + tan )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(z )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n ; forall t 2 t off n ;t on n +1 andforall n .Theboundin4exhibitsniteescapetimeand thereforeisonlyvalidiftheargumentoftan in4liesontheinterval 0 ; 2 ,i.e., 3 2 )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t off n + tan )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(V L )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(z )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(t off n < 2 : 69

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Using3,itcanbedemonstratedthat4isasufcientconditionfor4. Also,4issufcienttoensurethat4canbesatised.Finally,using3to rewrite4andperformingsomealgebraicmanipulationyields4. Remark 4.2 . Theboundin4hasaniteescapetime,so V L maybecomeunboundedunlessthereversedwell-timeconditionin4issatised.Thefollowing assumptionandsubsequentremarkdetailhowthereversedwell-timeconditionmaybe satised. Assumption4. Thetimespentinthe n th controlledregion t on n , t off n )]TJ/F25 11.9552 Tf 12.769 0 Td [(t on n hasa known,constantlowerbound t on min 2 R > 0 suchthat min n t on n t on min > 0 : Likewise,thetimespentinthe n th uncontrolledregion t off n hasaknown,constant upperbound t off max 2 R > 0 thatsatisesmax n t off n t off max and4forall n; i.e., t off max < 6 2 : Remark 4.3 . Assumption4canbevalidatedthroughappropriatedesignofthedesired crankvelocity _ q d .Firstconsider t on min .Thetimespentinthe n th controlledregion t on n canbedescribedusingtheMeanValueTheoremas t on n = q on n _ q on avg;n ; where q on n , q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n )]TJ/F25 11.9552 Tf 11.268 0 Td [(q t on n isthelengthofthe n th controlledregion,whichisconstant forall n 1 inthisdevelopmentandissmallestfor n =0 ,and _ q on avg;n 2 R istheaverage crankvelocitythroughthe n th controlledregion.Using4and4andassuming _ e 1 < 0 , _ q canbeupperboundedas _ q _ q d ++ k z k : 70

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Usingthefactthat k z k monotonicallydecreasesinthecontrolledregionstogetherwith 4and4allowstheaveragecrankvelocity _ q on avg;n tobeupperboundedas _ q on avg;n max t 2 [ t on n ;t off n ] _ q d t ++ r 2 1 k z t on n k : Therefore,4canbelowerboundedusing4as t on n q on n max t 2 [ t on n ;t off n ] _ q d t ++ k z t on n k : Supposenowthatadesired t on min wasspecied.Thentheright-handsideof4 wouldneedtobegreaterthanorequaltotheselected t on min ; i.e., q on n max t 2 [ t on n ;t off n ] _ q d t ++ k z t on n k t on min ; whichcanberewrittenasfollows: max t 2 [ t on n ;t off n ] _ q d t q on n t on min )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ k z t on n k : From4anupperboundonthedesiredvelocityateachon-timeisgivenwhich guaranteesthaton-duration t on n isgreaterthanaspeciedminimumon-duration t on min . Therefore,designingthedesiredvelocitytosatisfy4forall n issufcienttovalidate therstpartofAssumption4. Nowconsider t off max : Thecrank'sentranceintotheuncontrolledregioncanbe likenedtoaballisticevent,wherethecrankiscarriedbythecontrollerto q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n andreleasedwithinitialvelocity _ q )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n : Inthatsense,specifyingadesired t off max is equivalenttorequiringthecranktoballisticallyi.e.,onlyundertheinuenceofpassive dynamicstraversethelengthoftheuncontrolledregion, q off n ,inasufcientlyshort amountoftime.Sincetheonlycontrollablefactorsaffectingthebehaviorofthecrank intheuncontrolledregionaretheinitialconditions,ofwhich q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n ispredetermined byselectionof " m ,thenonlytheinitialvelocity _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n canbeusedtoguaranteethat 71

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thetotaltimespentinthe n th uncontrolledregionislessthan t off max : Ifitisreasonably assumedthat t off n / _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ,thenitcanbeassumedthatthereexistsasufciently largeinitialvelocity–acriticalvelocity _ q crit 2 R > 0 –whichguarantees t off n t off max < 6 2 : Morespecically,supposethereexists _ q crit suchthat _ q )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n _ q crit t off n t off max < 6 2 : Using4and4andassuming _ e 1 > 0 ; _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n canbelowerboundedas _ q )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n _ q d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ z )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n : Then,combining4and4,thefollowingsufcientconditionforthedesired crankvelocityatthe n th off-timewhichguarantees4canbedeveloped: _ q d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n _ q crit ++ z )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n : Furthermore,4canbeusedtoobtainasufcientconditionfor4intermsof theinitialconditionsofeachcycleas _ q d )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t off n _ q crit ++ r 2 1 k z t on n k exp )]TJ/F25 11.9552 Tf 13.798 8.088 Td [( 1 2 2 t on min : Notethatincreasing 1 viaselectionof and k 1 relaxes4toalimitwhereonly _ q d )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n _ q crit isrequired. Therefore,satisfactionofboth4and4forall n issufcienttovalidate Assumption4.However,4dependsonknowledgeof _ q crit ; leadingtothefollowing additionalassumption. Assumption5. Thecriticalvelocity _ q crit isknownaprioriforall n .Thisassumption ismildinthesensethatthecriticalvelocitycanbeexperimentallydeterminedforan individualsystemcongurationornumericallycalculatedforawiderangeofindividualor cyclecongurations. 72

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Theorem4.3. Thetrackingerrorisultimatelyboundedinthesensethat,asthenumber ofcrankcyclesapproachesinnityi.e.,as n !1 , k z t k convergestoaballwith constantradius d 2 R > 0 ,denedas d , v u u t 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B )]TJ/F30 11.9552 Tf 11.955 14.799 Td [(q )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 A 2 B 2 AB 1 where A;B 2 R 0 areconstantsdenedas A , tan )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(3 2 t off max ;B , exp )]TJ/F25 11.9552 Tf 13.799 8.088 Td [( 1 2 2 t on min ; providedAssumption4holdsandthefollowingconditionsaresatised: k z t on 0 k < v u u t 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(B + q )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 A 2 B 2 AB 2 ; A< 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 2 p B : Proof. Assumingtheworstcasescenariowhere t on n = t on min and t off n = t off max forall n; 3and3canbeusedtowrite V L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(z )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(t off n V L z t on n exp )]TJ/F25 11.9552 Tf 13.798 8.088 Td [( 1 2 2 t on min ; V L )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(z )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(t on n +1 tan 3 2 t off max + tan )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 V L z t on n exp )]TJ/F25 11.9552 Tf 13.799 8.087 Td [( 1 2 2 t on min ; provided 0 < 3 2 t off max + tan )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 V L z t on n exp )]TJ/F25 11.9552 Tf 13.799 8.088 Td [( 1 2 2 t on min < 2 ; whichissufcientlysatisedby4and ABV L z t on n < 1 : Then,thesequenceofswitchingpoints f V z t on n 2 R 0 g N n =0 ,for N 2f 0 ; 1 ; 2 ;::: g ,can beupperboundedas f V L z t on n gf x n g ; 73

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Figure4-1.Illustrationofconvergencebehavioroftheboundingsequence. wherethesequence f x n 2 R 0 g N n =0 ,for N 2f 0 ; 1 ; 2 ;::: g ,isdenedsuchthat x 0 , V L z t on 0 ;x n +1 = A + Bx n 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(ABx n : Thesequence f x n g convergesifitispositive,bounded,andmonotonic.Foradecaying convergence,itcanbedemonstratedthatthefollowingconditionissufcienttosatisfy x n +1 x n : ABx 2 n )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.956 0 Td [(B x n + A 0 : Theinequalityin4issatisedfortherangeof x n forwhichaconcaveupward paraboladenedby f x n = ABx 2 n )]TJ/F15 11.9552 Tf 11.634 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(B x n + A isnon-positive,i.e.,for x n suchthat d x n d; wherethebounds d ; d 2 R > 0 aredenedas d , 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B )]TJ/F30 11.9552 Tf 11.956 14.799 Td [(q )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 A 2 B 2 AB ; d , 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B + q )]TJ/F25 11.9552 Tf 11.955 0 Td [(B 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 A 2 B 2 AB : 74

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Thepolynomialrootsin4and4arerealanddistinctif4issatised. Therefore,provided4,4,and4aresatised,andgiventhat d x 0 d; thenthesequence f x n g hasalimitthatcanbecalculatedas lim n !1 x n +1 = 8 > > < > > : d if x 0 = d d if d x 0 < d : Inaddition,itcanbedemonstratedthat,givenany x n 0 canbeupperboundedas L V 8 > > < > > : d if V L z t on 0 = d d if V L z t on 0 < d : Using4,itcanthenbedemonstratedthat k z k convergestoaballwithconstant radius d ,denedin4,as n !1 ,provided4,4,and4aresatised, notingthat4isasufcientconditionfor4. Remark 4.4 . Theconditionin4canberewrittenas tan )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(3 2 t off max < sinh 1 2 2 t on min ; whichcouldleadtoanotherreversedwell-timecondition,gainconditionon 1 ,ordwelltimeconditionon t on min ; dependingontheapproach.Considerareversedwell-time approach,sothat4canberewrittenas t off max < 1 3 2 tan )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 sinh 1 2 2 t on min : 75

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Table4-1.Experimentalprotocolsused. Protocol " s quad A 0 : 5 T max Knee B 0 : 7 T max Knee C 0 : 9 T max Knee Theright-handsideof4isupper-boundedby 6 2 ,sothat4isasufcient conditionfor4. 4.3Experiments Fiveable-bodiedsubjectsage25-42yearsparticipatedinthischapter'sexperiments.Subjectrecruitment,theFES-cyclingtestbed,andtheexperimentalsetupwere thesameinthischapterasinChapter3,withtheexceptionthatonlythequadriceps femorismusclegroupswerestimulatedintheseexperiments. 4.3.1Methods Thegoaloftheseexperimentswastodemonstratethetrackingperformanceand robustnessofthecontrollerin3andtocomparetheeffectof " Quad onthecontrol inputandperformance.Foreachtrial, " Quad waschosenrandomlyfromTable4-1, where T max Knee , max q 2Q T Knee q wasthemaximumvalueofthesubject'storque transferratio.Theprotocolsusedforeachsubjectandtrialaretabulatedin4-2,andthe stimulationregionsusedforSubject4areillustratedin4-2. Priortothetrialswasatwominutewarm-upperiod,duringwhichthesubject wasshownagraphofthedesiredcrankvelocityonacomputerscreenalongwiththe measuredcrankvelocity.Eachsubjectwasaskedtotrackthedesiredcrankvelocity tothebestoftheirability.Thispre-trialprovidedameasureofvolitionalcyclingability, whichwasusedasaperformancebenchmark. DuringeachFES-cyclingtrial,thesubjectswereinstructedtorelaxandwerenot shownthecomputerscreen,andatleastveminutesofrestwasallottedbetween eachtrialtomitigatetheeffectsoffatigue.Stimulationwasconductedatafrequencyof 76

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Table4-2. T max Knee foreachsubject,protocol,andcorresponding " Quad usedforeachtrial. Subject T max Knee TrialProtocol " Quad 10.49 1B0.34 2A0.25 3C0.44 20.55 1A0.27 2B0.38 3C0.49 30.53 1B0.37 2C0.47 3A0.26 40.50 1C0.45 2B0.35 3A0.25 50.52 1C0.47 2B0.36 3A0.26 Figure4-2.StimulationregionsusedforSubject4. 77

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40Hzwithaconstantamplitudeof100mAandavariablepulsewidthdictatedbythe controllerin3.Trialswereendedif 90 revolutionshadbeencompleted,thecontrol inputsaturatedat 400 microseconds,orthesubjectreportedsignicantdiscomfort.After eachFES-cyclingtrial,subjectswereaskedthefollowingcomfortratingquestion:“Ona scalefromonetoten,onebeingascomfortableaspedalingbyyourselfandtenbeing painful,howcomfortablewastheFES-inducedpedaling?” Thedesiredcrankpositionandvelocityweregiveninradiansandradiansper second,respectively,as q d , t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t on 0 )]TJ/F15 11.9552 Tf 14.114 0 Td [(_ q d + q on 0 ; _ q d , )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp t on 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ; where 2 R > 0 wasaselectableconstantdenedas , 7 6 radianspersecond.The trajectoriesin4and4ensuredthatthedesiredvelocitystartedat0RPMand exponentiallyapproached35RPM.Thefollowingcontrolgainswerefoundinpreliminary testingandusedforallsubjects: =7 ;k 1 =10 ;k 2 =0 : 1 ;k 3 =0 : 1 ;k 4 =0 : 1 ;k Quad =1 : 4.3.2Results Fig.4-3depictsrepresentativecadencetrackingerror, _ e 1 ; andswitchedcontrol input, u Quad ,acrossthethreetrialsforonesubjectSubject4.Fig.4-4presentsthe controlinputfromthersttrialforthesamesubjectovertheintervalbetween 20 and 25 seconds,forthepurposeofillustratingthebehavioroftheswitchedcontrolinput.Fig. 4-5comparesthepositionandcadencetrackingerrors, e 1 and _ e 1 ,respectively,during FES-cyclingandvolitionalcycling,againforSubject4.Themean,standarddeviation, androotmeansquareRMScadencetrackingerrorfromallsubjectsoveralltrials, includingthevolitionalwarm-up,arepresentedinTable4-3,alongwiththereported comfortscaleratingforeachtrial. 78

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Figure4-3.Cadencetrackingerrorandswitchedcontrolinputacrossthethreetrialsfor Subject4.ACadenceerror,Trial1,ProtocolC.BControlinput,Trial1, ProtocolC.CCadenceerror,Trial2,ProtocolB.DControlinput,Trial2, ProtocolB.ECadenceerror,Trial3,ProtocolA.FControlinput,Trial3, ProtocolA. 79

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Table4-3.TrackingperformanceinrevolutionsperminuteRPMandcomfortscale ratingfromallsubjectsoveralltrials _ e 1 standardComfortscale SubjectProtocol _ e 1 meanRPMdeviationRPM _ e 1 RMSRPMrating 1 Volitional-0.532.352.411.0 A0.1923.123.134.0 B0.1222.662.663.0 C0.413.653.685.0 2 Volitional-0.0031.5111.5111.0 A0.0611.921.922.5 B0.0792.122.123.5 C0.483.023.066.0 3 Volitional0.473.543.581.0 A0.544.214.254.0 B0.242.582.595.0 C0.242.392.403.0 4 Volitional-0.581.8301.921.0 A0.1533.313.325.0 B0.1323.433.444.0 C0.1312.822.823.0 5 Volitional-0.191.451.461.0 A0.253.163.173.0 B0.494.234.266.0 C0.694.174.237.0 80

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Figure4-4.SwitchedcontrolinputfromthersttrialofSubject4. 4.3.3Discussion TheFES-cyclingcontrollerin4yieldssimultaneouscrankpositionandcadence trackingwithultimatelyboundedtrackingerrorsimilartothatofvolitionalcyclingby able-bodiedsubjectsacrossallthreeprotocolsandsubjects,despiteusingthesame controlgainsforeachsubjectandonlystimulatingthequadricepsfemorismuscle groups.Whilethereisnotenoughdatatodeterminestatisticalsignicancebetweenthe trackingperformanceoftheFES-cyclingandvolitionaltrials,thesamecharacteristics canbeobservedinboththeFES-cyclingandvolitionaltrackingexperiments.However, althoughbothFES-cyclingandvolitionaltrackingexperimentsdemonstratedmean trackingerrorsnearzeroRPM,thevolitionaltrackingerrorhadlessstandarddeviation forallsubjectsexceptSubject3,whosevolitionaltrackingperformancewaspoorwhen comparedtothatoftheothersubjects.Lowerstandarddeviationinthetrackingerror suggestsbetterperformanceduetosmoothnessinpedaling,whichistobeexpected consideringthatvolitionalcyclingcanmakeuseofallthemusclegroupsinthelower 81

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Figure4-5.ComparisonofthepositionandcadencetrackingerrorsduringFES-cycling andvolitionalcyclingforSubject4.AFES-cyclingcrankpositionerror.B Volitionalcyclingcrankpositionerror.CFES-cyclingcadenceerror.D Volitionalcyclingcadenceerror. 82

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extremities,whereastheseFES-cyclingtrialsonlyusedthequadricepsfemorismuscle groups.Motivationarises,therefore,torepeattheseexperimentsusingstimulationofthe hamstringsandglutealmusclegroupsinadditiontothequadricepsfemoris.Itshould benotedthat,whiletheFES-cyclingtrialsyieldedsignicantlyreducedcrankposition trackingerrorwhencomparedtovolitionalcycling,subjectswerenotaskedtominimize thecrankpositiontrackingerrorduringvolitionalcyclingtrials. Itwasobservedthattheeffectivenessofeachprotocolinminimizingthetracking error,aswellasthesubject'scomfortduringeachprotocol,wassubject-dependent. Forexample,ProtocolCforSubject4yieldedthebesttrackingperformanceofthe threeprotocols,and,perhapsasaconsequence,wasalsoreportedbythesubjectas beingthemostcomfortable,whereasProtocolAyieldedthebesttrackingperformance andgreatestcomfortforSubject5.FromProtocolAtoProtocolC,thelengthofthe stimulationregiondecreased,requiringanincreaseinstimulationintensitytomakeup forthedecreaseinstimulationduration.Somesubjectsappearedtotoleratehigher intensitiesbutnotlongerdurations,makingitlikelythattheywouldreportProtocolC asbeingthemostcomfortable.Ontheotherhand,somesubjectspreferredlonger durationsoflowerintensitystimulation,makingitlikelythattheywouldreportProtocol Basbeingmostcomfortable.Regardlessofthesubject'spreference,itwasalways truethatthesubjectreportedtheprotocolthatyieldedthebesttrackingperformanceas beingthemostcomfortable,suggestingacorrelationbetweentrackingperformanceand subjectcomfortduringFES-cycling.Moredataisneededtoexplorethiscorrelationas wellastheeffectoftheselectionof " m ontrackingperformance. WhencomparedtothecontrolinputandtrackingperformancefromChapter3,the controlinputinthischapterwassmootherandyieldedbettertrackingperformanceby reducingsteadystatetrackingerror.ThecontrollerinChapter3usedaslidingmode strategywith r =_ q d )]TJ/F15 11.9552 Tf 14.201 0 Td [(_ q asitsinput,causingthecontrolinputtochangesignsfrequently, whereasthecontrollerinthischapterusedalteredtrackingerror, e 2 ; astheinputto 83

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theslidingmodetermandsorarelydidthecontrolinputchangesignsonlyduringthe uncontrolledregions.Itwasobservedthattuning couldincreaseordecreasethe frequencyofsignchanges.Theresultwasbettertrackingperformanceherethanin Chapter3,sincetheactualvelocitycouldoscillatearoundthedesiredvelocitywithout chattering. 4.4ConcludingRemarks AcommonLyapunov-likefunctionwasusedtoprovethatthedevelopedcontroller, givenin4,yieldsultimatelyboundedtrackingofbothadesiredcrankpositionand cadencei.e.,crankvelocity,providedthedesiredcadence,controlgains,initialconditions,andstimulationpatternsatisfysufcientconditions.Experimentswereconducted onveable-bodiedsubjects,andtheresultsbothdemonstratetherobustnessand stabilityofthedevelopedswitchedcontrollerandcorroboratethetheoreticalrelationshipsbetweenthedesiredcadence,controlgains,andstimulationpatternthatwere developedinthestabilityanalysis.Theexperimentalresultssuggestthatthereexist correlationsbetweenthesizeofthestimulationregions,determinedbyselectionof " m ; thetrackingperformance,andsubjectcomfortduringFES-cycling,thoughmore dataisneededtodeterminestatisticalsignicance.Whilethetheoreticaloutcomeof thestabilityanalysisinthischapter,ultimatelyboundedtrackingerror,isthesameas inChapter3,thecontrollerdevelopedinthischapterexperimentallydemonstrated improvedtrackingperformanceoverthecontrollerdevelopedinChapter3,likelydueto theadditionofanintegratorintothecontrolinputaswellastheuseofalteredtracking errorfortheslidingmodecontrolterm. Thenextchapterincludesanelectricmotortoprovidecontroleffortduringthe uncontrolledregions,removingtheconstraintsonthedesiredcadence,theinitial conditions,andthestimulationpatternfromChapter3andthischapter,sincethe electricmotorgrantscontrollabilityofthesystemthroughouttheentirecrankcycle. 84

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CHAPTER5 SWITCHEDCONTROLOFSTATIONARYCYCLINGINDUCEDBYFUNCTIONAL ELECTRICALSTIMULATIONWITHELECTRICMOTORASSISTANCE BuildingontheresultsinChapters3and4,whichdonotconsiderelectricmotor input,inthischapter,anovelstrategyforelectricmotorassistanceisdevelopedthatonly providescontrolinputintheregionsaroundthedeadpointswherenoFEScontrolinput isprovided.Basedonthismodel,aswitched,sliding-modecontrollerisdevelopedfor boththeFESandthemotorthatyieldsglobal,exponentiallystabletrackingofadesired cranktrajectory,despitetheswitchingeffects,uncertaintyinthesystemparameters,and thepresenceofunknown,boundeddisturbances.Experimentalresultswithveablebodiedsubjectsarepresentedtovalidatethecontrollerandtodemonstratepractical applicationofthetheoreticalinsights. 5.1SwitchedControlInput FormotorizedFES-cyclingsystemsinarehabilitativesetting,itispreferredthat themusclesexertasmuchworktocompletethecyclingtaskaspossibletomaximize therapeuticeffect;therefore,motivationarisestoactivatetheelectricmotoronlyas needed.Inthepresentdevelopment,thehuman-machineeffortisbalancedbyonly activatingthemusclegroupswheretheycaneffectivelycontributetopedalingand activatingtheelectricmotoreverywhereelse.Inotherwords,theelectricmotorcontrol regionisdenedas Q e , QnQ c ,whichwasdenedtobetheuncontrolledregion Q u in thepreviouschapters.Dening Q e inthiswayallows2toberewrittenas J cycle q t + b cycle _ q t + M q t q t + V q t ; _ q t _ q t + G q t + P q t ; _ q t + d cycle t + d rider t = r t + B q t ; _ q t u t ; where u , u FES = u motor and B : Q R ! R > 0 isthelumped,switchedcontrol effectivenesstermdenedas B , B FES + B motor : 85

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Ingeneral,stimulationofsixmusclegroupsallowsfor 2 6 =64 possiblecombinations ofactivemusclegroups,includingtheemptyseti.e.,theregionwherenostimulation isapplied.Thedenitionsin2-2introduceconstraintsthatpermitatmost 28 differentsubsystemsi.e., B mayswitchupto 28 timesoveracrankcycle,sothat anauxiliaryswitchingsignalcanbedenedas 2P , f 1 ; 2 ; 3 ;:::; 28 g ,wherethe rst 27 subsystemsrepresentsomecombinationofactivemusclegroupsandthe 28 th representsonlyelectricmotoractivation.Theswitchingsignal speciestheindexof B andswitchesaccordingtothecrankposition.Forexample,ifonlytherightandleft quadricepsfemorismusclegroupswerestimulatedaccordingto2andtheelectric motorwasactivatedelsewhere,therewouldbeonlythreesubsystems,and wouldbe denedas , 8 > > > > > > > < > > > > > > > : 1 if q 2Q RQuad 2 if q 2Q LQuad 3 if q 2Q e : Theswitchedcontroleffectiveness B hasthefollowingproperty. Property14. B B B , 8 2P ; where B ; B 2 R > 0 areknownconstants. Inthisdevelopment,theknownsequenceofswitchingstates,whicharethelimit pointsof Q m ,forall m 2M ,isdenedas q n 2Q 1 n =0 ; andthecorresponding sequenceofunknownswitchingtimes t n 2 R 0 1 n =0 isdenedsuchthateach t n denotestheinstantwhen q reachesthecorrespondingswitchingstate q n .Theswitching signal isassumedtobecontinuousfromtherighti.e., q = lim q ! q + n q . 5.2ControlDevelopment Inthisdevelopment,thecycleinertia J cycle islumpedtogetherwiththeriderinertia M i.e.,here M , M + J cycle . 86

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5.2.1Open-LoopErrorSystem AsinChapter4,thecontrolobjectiveistotrackadesiredcranktrajectorywith performancequantiedbythetrackingerrorsignals e 1 ;e 2 : R 0 ! R ,denedas e 1 t , q d t )]TJ/F25 11.9552 Tf 11.956 0 Td [(q t ; e 2 t , _ e 1 t + e 1 t ; where q d : R 0 ! R isthedesiredcrankposition,designedsothatitsderivativesexist and j _ q d j c d 1 ; j q d j c d 2 ,where c d 1 ;c d 2 2 R > 0 areknownconstants,and 2 R > 0 isa selectableconstant.Withoutlossofgenerality, q d isdesignedtomonotonicallyincrease, i.e.,backpedalingisnotdesired.Takingthetimederivativeof5,multiplyingby M; andusing5-5yields M q t _ e 2 t = q t ; _ q t ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 1 t )]TJ/F25 11.9552 Tf 9.298 0 Td [(V q t ; _ q t e 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(B q t ; _ q t u t ; wheretheauxiliaryterm : Q R R 0 ! R isdenedas , M q t q d t + _ e 1 t + V q t ; _ q t _ q d t + e 1 t + G q t + P q t ; _ q t + b cycle _ q t + d cycle t + d rider t + e 1 t )]TJ/F25 11.9552 Tf 11.956 0 Td [( r t : BasedonAssumption1andProperties5-12, canbeboundedas j q t ; _ q t ;t j c 1 + c 2 k z t k + c 3 k z t k 2 ; 87

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where c 1 ;c 2 ;c 3 2 R > 0 areknownconstantsdenedas c 1 , Mc d 2 + c G + c P 1 + c d;cycle + c d;rider + c r + c V c 2 d 1 + c P 2 c d 1 + bc d 1 ; c 2 , M + + c V c d 1 + c V c d 1 + c P 2 + b +1 ; c 3 , c V ; andtheerrorvector z : R 0 ! R 2 isdenedas z , e 1 e 2 T : 5.2.2Closed-LoopErrorSystem Basedon5andthesubsequentstabilityanalysis,thecontrolinputisdesigned as u , k 1 e 2 + )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 ; wheresgn : R ! [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] denotesthesignumfunctionand k 1 ;k 2 ;k 3 ;k 4 2 R > 0 are constantcontrolgains.Substituting5into5yields M _ e 2 = )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ve 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 1 e 2 + )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 : 5.3StabilityAnalysis Let V L : R 2 ! R denoteacontinuouslydifferentiable,positivedenite,radially unbounded,commonLyapunov-likefunctiondenedas V L z t , 1 2 z T t W q t z t ; wherethepositivedenitematrix W : Q! R 2 2 isdenedas W , 2 6 4 10 0 M 3 7 5 : 88

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V L satisesthefollowinginequalities: 1 k z k 2 V L 2 k z k 2 ; where 1 ; 2 2 R > 0 areknownconstantsdenedas 1 , min 1 2 ; 1 2 M ; 2 , max 1 2 ; 1 2 M : Theorem5.1. Thetrackingerrorisboundedbyanexponentiallydecayingenvelope givenby k z t k r 2 1 k z t 0 k exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 s t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 ; forall t 2 [ t 0 ; 1 ,where t 0 2 R 0 istheinitialtime,and s 2 R > 0 isdenedas s , 1 2 min ;c B p k 1 ; where c B p 2 R > 0 isaknownconstantdenedas c B p , min min m 2M B FES ;B motor ; providedthefollowinggainconditionsaresatised: k 2 c 1 c B p ;k 3 c 2 c B p ;k 4 c 3 c B p : Proof. Consider = p forsomearbitrary p 2P suchthat B p iscontinuous.Because ofthesignumfunctionin u ,thetimederivativeof5existsalmosteverywherea.e., i.e.,foralmostall t 2 [ t n ;t n +1 , n 2f 0 ; 1 ; 2 ;::: g ,asdetailedintheanalysisinChapter 4.Therefore,aftersubstituting5,utilizingProperty13,andrearrangingterms,the timederivativeof5canbeexpressedas _ V L a:e: =_ e 1 e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 1 e 2 + e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(B p )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(k 1 e 2 + )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(k 2 + k 3 k z k + k 4 k z k 2 sgn e 2 e 2 : 89

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Using5,5,andProperty11,itcanbedemonstratedthat _ V L a:e: )]TJ/F25 11.9552 Tf 30.476 0 Td [(e 2 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c B p k 1 e 2 2 )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(c B p k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 1 j e 2 j )]TJ/F30 11.9552 Tf 11.291 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(c B p k 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 k z kj e 2 j)]TJ/F30 11.9552 Tf 17.933 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(c B p k 4 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 3 k z k 2 j e 2 j : Providedthegainconditionsin5aresatised,5canbeusedtorewrite 5as _ V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( s V L ; where s wasdenedin5.Theinequalityin5canberewrittenas exp s t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t n _ V L + s V L a:e: 0 ; for t 2 [ t n ;t n +1 ,whichisequivalenttothefollowingexpression: d dt V L exp s t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t n a:e: 0 : TakingtheLebesgueintegralof5andrecognizingthattheintegrandonthelefthandsideisabsolutelycontinuousallowstheFundamentalTheoremofCalculustobe usedtoyield V L V L t n exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( s t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t n ; for t 2 [ t n ;t n +1 .Since5wasproventoholdforanarbitrary ,5holdsforall 2P .Therefore, V L isindeedacommonLyapunovfunction,and5holdsforall t 2 [ t 0 ; 1 .Inotherwords, V L V L t 0 exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( s t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 : Using5tobound5andperformingsomealgebraicmanipulationyields 5. Remark 5.1 . Theexponentialdecayrate s representsthemostconservativei.e., smallestdecayratefortheclosed-loop,switchederrorsystem.Inpractice,each 90

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Figure5-1.Illustrationofthebehaviorof V L . subsystemhasitsowndecayratedependentonthelowerboundofthecorresponding B ,butintheprecedingstabilityanalysis, c B p wasusedasthelowerboundon B forall 2P .Figure5-1illustrateshow V L maybehaveinpracticedottedredlineversusthe conservativeboundgivenin5solidblueline. 5.4Experiments Experimentswereconductedwiththeprimaryobjectiveofevaluatingtheperformanceofthecontrollergivenin5anddistributedasFESandelectricmotorcurrent accordingto2-2.Fiveable-bodiedsubjectsfourmale,onefemale21-43 yearsoldparticipatedintheexperiments.Eachsubjectgavewritteninformedconsent approvedbytheUniversityofFloridaInstitutionalReviewBoard.Duringthesubsequent experiments,thesubjectswereinstructedtorelaxandmakenovolitionalefforttoeither assistorinhibittheFESortheelectricmotorinputi.e.,passiveriders. 91

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5.4.1Methods AcommerciallyavailablerecumbenttricycleTerraTrikeRoverwasmodiedfor thepurposesoftheFES-cyclingexperiments.A250Watt,brushed,24VDCelectric motorUniteMotorCo.Ltd.MY1016Z2wasmountedtotheframeandcoupledtothe drivechain.Orthoticboots OssurReboundAirTallwereafxedtocustompedals; theseorthoticpedalsservedtoxtherider'sfeettothepedals,preventdorsiexion andplantarexionoftheankles,andmaintainsagittalalignmentofthelowerlegs.An opticalencoderwithanangularresolutionof0.018degreesUSDigitalH1wascoupled tothecyclecrankviaspurgearstomeasurethecrankposition.Tomakethesystem stationary,astationarycyclingtrainerandriserringsKineticbyKurtwereusedto liftthetricycle'sdrivewheelofftheground.Currentcontrolofthecycle'smotorwas enabledbyageneralpurposelinearamplierAETechronLVC5050interfacingwith thedataacquisitionhardwareQuanserQ8-USB,whichalsomeasuredtheencoder signal.Thecontrollerwasimplementedonapersonalcomputerrunningreal-time controlsoftwareQUARC,MATLAB/Simulink,Windows7atasamplingrateof500Hz. Figure5-2depictsthemotorizedFES-cyclingtestbed. StimulationwasappliedusingthestimulatorandelectrodesdescribedinChapter 3.Thestimulationamplitudeswerexedat90mAforthequadricepsand80mAfor thehamstringsmusclegroups,andthestimulationpulsewidthforeachmusclegroup wasdeterminedby u m andcommandedtothestimulatorbythecontrolsoftware. Stimulationfrequencywasxedat60Hztoleveragetheresultsfoundin[48].Forsafety, anemergencystopswitchwasattachedtothetricyclethatenabledthesubjecttostop theexperimentimmediatelyifnecessary,thoughnosubjectsfounditnecessary. Electrodeswereplacedoverthesubjects'quadricepsfemorisandhamstrings musclegroupsaccordingtoAxelgaard'selectrodeplacementmanual 1 .Inthese 1 http://www.palsclinicalsupport.com/videoElements/videoPage.php 92

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Figure5-2.MotorizedFES-cyclingtestbed.Photocourtesyofauthor. experiments,onlythequadricepsandhamstringsmusclegroupswerestimulatedto betterdemonstratethebalancebetweentheFESandmotorinputs.Eachsubject'slegs weremeasuredtoobtainthedistancefromthegreatertrochantertothelateralfemoral condyleandfromthelateralfemoralcondyletothesoleofthefootwhiletheanklewas heldintheanatomicallyneutralposition.Subjectswerethenseatedonthetricycle,and theirfeetwereinsertedsecurelyintotheorthoticpedals.Thetricycle'sseatpositionwas adjustedforeachsubject'scomfortwhileensuringthatfullextensionofthekneescould notbeachievedwhilecycling,andthedistancefromthecyclecranktothesubject's rightgreatertrochanterwasmeasured.Thesemeasurementswereusedtocalculate thetorquetransferratiosforthesubjects'musclegroupsandtotherebydeterminethe stimulationpattern. Twoexperimentalprotocolswereconductedwitheachsubject,andeachtriallasted 180seconds.InProtocol1,thedesiredcadenceroseto50rpmandremainedtherefor thedurationoftheexperiment,whileinProtocol2thedesiredcadencerstroseto50 93

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rpm,thenvariedsinusoidallyfrom40rpmto60rpmtodemonstratetherobustnessof thedevelopedcontrolsystem.ForProtocol1thedesiredcrankvelocity _ q d andposition q d weredesignedas _ q d , 5 3 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2 5 t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t 0 ; q d , 5 3 t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t 0 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(5 2 _ q d + q t 0 ; where t 0 =0 seconds.Thetrajectoriesin5and5ensuredthatthedesired cadencestartedat0rpmandsmoothlyapproached50rpm.ForProtocol2,thedesired crankvelocity _ q d andposition q d weredesignedas _ q d , 8 > > > > > > > > > > > > < > > > > > > > > > > > > : 5 3 1 )]TJ/F30 11.9552 Tf 11.955 13.271 Td [( t )]TJ/F26 7.9701 Tf 6.587 0 Td [(t 1 t 1 4 t > > > > > > > > > > < > > > > > > > > > > > : 5 3 t )]TJ/F24 7.9701 Tf 13.151 6.18 Td [( t )]TJ/F26 7.9701 Tf 6.587 0 Td [(t 1 5 + t 5 1 5 t 4 1 + q t 0 t
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where 2 R wasascalingfactordesignedas , 8 > > > > > > > < > > > > > > > : 1 t
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Figure5-3.TrackingperformanceforSubject1duringProtocol1andProtocol2.A Protocol1positiontrackingerror.BProtocol1cadencetrackingerror.C Protocol1stimulationinput.DProtocol1motorinput.EProtocol2position trackingerror.FProtocol2cadencetrackingerror.GProtocol2stimulation input.HProtocol2motorinput. in2and5,andtheconstant ,introducedin5,weretunedtoyield acceptabletrackingperformancepriortoeachtrialandrangedasfollows: 2 [7 ; 10] , k m =0 : 25 8 m 2M , k e 2 [5 : 75 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 ; 13 : 2] , k 1 2 [80 ; 100] , k 2 2 [4 ; 100] , k 3 =0 : 01 , k 4 =0 : 001 . 5.4.2Results Figure5-3depictsonesubject'strackingperformanceduringbothprotocols, quantiedby e 1 , _ e 1 ,thestimulationintensityinputtoeachmusclegroup u m ,andthe electricmotorcurrentinput u e .Figure5-4providesanenhancedviewofthedistribution ofthecontrolinputbetweenFESandthemotoracrossonecrankcycle.Table5-1 summarizesthemeanandstandarddeviationst.dev.ofthepositionandcadence 96

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Figure5-4.FEScontrolinputsandmotorcurrentinputfromonemotorizedFES-cycling trialoverasinglecrankcycle. trackingerrorforeachsubjectduringthemotor-only t 2 [ t 0 ;t 1 seconds,transitory t 2 [ t 1 ;t 2 seconds,andFES/motor t 2 [ t 2 ; 180] secondsperiodsofProtocol 1.Similarly,Table5-2summarizesthetrackingperformanceforeachsubjectduring Protocol2. 5.4.3Discussion Theexperimentalresultssuccessfullydemonstratetheabilityofthecontrollerin 5,distributedbetweenFESoftherider'smusclegroupsandelectricmotorcurrent accordingto2,toachieveexponentiallystabletrackingperformancedespite parametricuncertaintye.g.,uncertainriderlimbmassandunknowndisturbances. However,theresultsindicateexponentialconvergencetoanultimateboundonthe trackingerror,insteadofconvergencetozero,whichcouldbeduetounmodeledeffects suchaselectromechanicaldelaybetweenmuscleactivationandforceproduction[51]. 97

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Table5-1.SummaryofmotorizedFES-cyclingperformanceforallvesubjectsduring Protocol1. Motor-onlyTransitoryFES/Motor SubjectErrormeanst.dev.meanst.dev.meanst.dev. 1 e 1 deg.13.702.3618.722.8624.382.90 _ e 1 rpm0.341.320.112.120.002.07 2 e 1 deg.12.042.1918.643.9126.133.69 _ e 1 rpm0.331.340.201.970.002.68 3 e 1 deg.10.612.7315.133.2917.482.80 _ e 1 rpm0.311.730.132.770.003.44 4 e 1 deg.13.832.5721.853.7733.254.59 _ e 1 rpm0.361.750.242.560.012.82 5 e 1 deg.5.982.5510.743.3615.182.69 _ e 1 rpm0.252.110.133.090.013.52 Table5-2.SummaryofmotorizedFES-cyclingperformanceforallvesubjectsduring Protocol2. Motor-onlyTransitoryFES/Motor SubjectErrormeanst.dev.meanst.dev.meanst.dev. 1 e 1 deg.11.002.8414.992.9220.554.62 _ e 1 rpm0.331.750.062.710.012.98 2 e 1 deg.6.701.8110.282.4413.965.23 _ e 1 rpm0.281.350.122.380.013.11 3 e 1 deg.9.322.2313.242.7418.955.59 _ e 1 rpm0.301.530.152.040.002.79 4 e 1 deg.8.993.1913.962.5322.045.15 _ e 1 rpm0.312.080.132.520.013.28 5 e 1 deg.5.612.5110.263.0614.724.29 _ e 1 rpm0.251.850.142.970.003.62 98

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TheresultsforSubject1,presentedinFigure5-3,demonstratetypicalperformance duringthemotorizedFES-cyclingtask,ascorroboratedbythedatainTable5-1and Table5-2.Ofparticularnoteisthemeanandstandarddeviationofthecadencetracking errorduringtheFES/motorperiodforallsubjects,wheretheaveragecadencetracking erroracrossallvesubjectswas 0 : 00 2 : 91 rpmi.e.,theactualcadencewascentered aboutthedesiredcadencewithlessthan3rpminstandarddeviationforProtocol1 and 0 : 01 3 : 15 rpmforProtocol2.Incomparison,theaveragepositiontrackingerror acrossallvesubjectswas 23 : 28 3 : 33 degreesforProtocol1and 18 : 05 4 : 98 degrees forProtocol2,indicatingthattheactualcranktrajectorylaggedthedesiredtrajectory consistentlyacrossallexperiments.Thesteadystateoffsetinthepositiontracking errorwaslikelycausedbyabiasinthetuningofthecontrolgainstowardsimproving cadencetrackingperformance,ascadenceerrorisgenerallyconsideredtobethemain performancecriterionduringcyclingtasks. AsindicatedintheexperimentaldataplottedinFigure5-4,theelectricmotor providedassistanceasneededintheregionsoftheFES-cyclingjointspacewherethe rider'storquetransferratiosweresmall,andstabilitywasmaintainedthroughoutthe trialdespitethediscontinuousswitchinginthetorqueinputtothesystem.Thesubjects reportedthatthecyclingmotionfeltcomfortableandnaturalandthattheyperceived theirmusclesascontributingsignicantlytothecyclingtask,thoughneithermetabolic norrelativetorquecontributioni.e.,comparingFEStorqueinputtomotortorqueinput measurementswereavailabletoquantifytheseeffects. 5.5ConcludingRemarks AnovelswitchingstrategywasdevelopedthatappliesFEStotherider'smuscle groupsinregionsofthecrankcyclewheretherider'smusclescontributesignicantly tothecyclingtaskandutilizesanelectricmotorforassistanceonlyasneededi.e.,in regionsofpoorkinematicefciency.Aswitchedsliding-modecontrollerwasdesigned toyieldglobal,exponentiallystabletrackingofadesiredcranktrajectory,provided 99

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sufcientgainconditionsaresatised.Thecontroldesignwasvalidatedinexperiments withveable-bodiedsubjects,whereanaveragecadencetrackingerrorof 0 : 00 2 : 91 rpm 0 : 00 5 : 82% errorwasdemonstratedwhentrackingaconstantdesiredcadenceof 50rpm. ThedevelopedcontrolsystemformotorizedFES-cyclingsystemshasthepotential toenhancetherapeuticoutcomesinarehabilitativesettingandtoimprovetheperformanceofassistivecyclingdevices;however,clinicalimplementationofthedeveloped controlsystemmaypresentadditionalchallengesnotconsideredinthischapter.While thetheoreticaldevelopmentinthischapterconsidersageneralizedcycle-ridersystem, applyingthedevelopedcontrolsystemtoaparticularpatientpopulationmayrequire disorder-specictuningofthesystemparametersortheadditionofdisorder-specic functionality.Asdiscussedintheconcludingchapterofthisdissertation,futureworkwill focusonapplyingthedevelopedcontrolsystemtopeoplewithneurologicaldisorders andwillnecessarilyconsidersuchdisorder-specicchallengestoimplementation.Furtherworkalsoneedstoinvolveclinicaltrialsinclinicallyrelevantpatientpopulationsto investigatetrainingbenets. 100

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CHAPTER6 SWITCHEDCONTROLOFCADENCEANDPOWEROUTPUTDURINGSTATIONARY CYCLINGINDUCEDBYFUNCTIONALELECTRICALSTIMULATION ThischapterpresentsthedevelopmentofanFES-cyclingcontrolsystemthat utilizesanelectricmotortomaintainadesiredcadencewhileFESisusedtocontrol thepoweroutputatthecyclecrankbytherider'selectricallystimulatedmuscles.As describedin[8],forFESexercisetesting,itiscrucialtocontrolboththecyclingcadence andtheworkdonebythemusclegroups.Inthepresentdevelopment,theelectric motoristaskedwithtrackingadesiredcranktrajectorywhileFESofthemusclegroups isusedtotrackadesiredtorqueaboutthecrankaxis.Itisassumedthatthetorque appliedbytherider rider ismeasurablealongwiththecrankposition q andvelocity _ q . Aslidingmodecontrollerisdevelopedfortheelectricmotorthatguaranteesglobally, exponentiallystabletrackingofthedesiredcranktrajectory.Similarly,aslidingmode controllerisdevelopedfortheFEScontrolinputthatguaranteesultimatelybounded trackingofthedesiredactivetorque,averagedoverthecrankcycle,providedsufcient gainconditionsaresatised,despitetheswitchingeffects,uncertaintyinthesystem parameters,andthepresenceofunknown,boundeddisturbances.Experimentalresults fromthreeable-bodiedindividualsdemonstratethecontrolsystem'sperformanceunder typicalFES-cyclingconditions. 6.1CadenceControl 6.1.1ControllerDevelopment Theelectricmotortorque motor canbemodeledasin2where,here, B e is assumedtobeaknownconstant.Thecontrolobjectivefortheelectricmotortorqueisto trackadesiredcranktrajectorywithperformancequantiedbythetrackingerrorsignals e i : R 0 ! R ;i 2f 1 ; 2 g ; denedas e 1 t , q d t )]TJ/F25 11.9552 Tf 11.956 0 Td [(q t ; e 2 t , _ e 1 t + e 1 t ; 101

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where 2 R > 0 isaselectableconstant,and q d : R 0 ! R denotesthedesired crankposition,designedsuchthatitsrsttwotimederivativesexistandarebounded as j _ q d j c q 1 and j q d j c q 2 ; where c qi 2 R 0 ;i 2f 1 ; 2 g ; areknownconstants.Taking thetimederivativeof6,multiplyingby J cycle ,substituting2,2,2and 6,andrearrangingtermsyields J cycle _ e 2 = J cycle q d + _ e 1 + b cycle _ q + d cycle + rider )]TJ/F25 11.9552 Tf 11.955 0 Td [(B e u e : Basedon6andthesubsequentstabilityanalysis,theelectricmotorcontrolcurrent isdesignedas u e , B )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 e rider + k 1 e 2 + k 2 + k 3 j e 1 j sgn e 2 ; where k 1 ;k 2 2 R > 0 areconstantcontrolgains,andsgn : R ! [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] isthesignum function.Substituting6into6yields J cycle _ e 2 = J cycle q d + _ e 1 + b cycle _ q + d cycle )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 + k 3 j e 1 j sgn e 2 : 6.1.2StabilityAnalysis Let V q : R 2 ! R 0 denoteacontinuouslydifferentiable,positivedenite,Lyapunov functioncandidatedenedas V q z t , 1 2 e 2 1 t + 1 2 J cycle e 2 2 t ; wheretheerrorvector z : R 0 ! R 2 isdenedas z t , e 1 t e 2 t T : TheLyapunovfunctioncandidate V q satisesthefollowinginequalities: 1 k z k 2 V q 2 k z k 2 ; 102

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where 1 ; 2 2 R > 0 areknownconstantsdenedas 1 , min 1 2 ; 1 2 J ; 2 , max 1 2 ; 1 2 J : Theorem6.1. Thetrackingerrorisboundedbyanexponentiallydecayingenvelope givenby k z t k r 2 1 k z t 0 k exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 ; forall t 2 [ t 0 ; 1 ,where t 0 2 R 0 istheinitialtime,and q 2 R > 0 isdenedas q , 1 2 min ; J )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 ; providedthefollowinggainconditionsaresatised: k 1 > J;k 2 Jc q 2 + bc q 1 + c d;cycle ;k 3 1+ )]TJETq1 0 0 1 406.532 442.939 cm[]0 d 0 J 0.478 w 0 0 m 7.577 0 l SQBT/F25 11.9552 Tf 406.532 433.096 Td [(J + b : Proof. Becauseofthesignumfunctionin6,thetimederivativeof V q existsalmost everywherea.e.,i.e.,foralmostall t 2 [ t 0 ; 1 .Using6and6,thetime derivativeof6canbeexpressedas _ V q a:e: = )]TJ/F25 11.9552 Tf 9.298 0 Td [(e 2 1 + e 1 e 2 + J cycle q d + _ e 1 + b cycle _ q + d cycle )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 + k 3 j e 1 j sgn e 2 e 2 : UsingAssumption1andProperties5-6,6canbeboundedaboveas _ V q a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [(e 2 1 + )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( J )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 e 2 2 + )]TJETq1 0 0 1 231.024 239.719 cm[]0 d 0 J 0.478 w 0 0 m 7.577 0 l SQBT/F25 11.9552 Tf 231.024 229.876 Td [(Jc q 2 + bc q 1 + c d;cycle )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 j e 2 j + )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1+ J + b )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 3 j e 1 jj e 2 j : Providedtheconditionsin6aresatised,6canbeusedtobound6as _ V q a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( q V q ; where q wasdenedin6.From6,itcanbedemonstratedthat V q z t V q z t 0 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( q t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 103

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for t 2 [ t 0 ; 1 .Using6tobound6andperformingsomealgebraicmanipulationyields6. Remark 6.1 . Using6,6,6,and6,itcanbedemonstratedthatthecrank velocity _ q andacceleration q areboundedas j _ q j c q 1 + +1 r 2 1 k z t 0 k ; j q j J )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 1 + k 3 + b r 2 1 k z t 0 k + k 2 + c d;cycle ! : 6.2PowerControl 6.2.1ControllerDevelopment Inthisdevelopment,thecontrolobjectivefortheFEScontrolinputis:totrack adesiredactivetorquetrajectorythatyieldsadesiredactivepoweroutputatthecycle crankinthecontrolledregionsi.e.,for q 2Q c andtoensureboundednessofthe torquetrackingerrorsystemintheuncontrolledregionsi.e.,for q 2Q u .Here,as in[8,29,52,53],thedesiredtorqueisdenedbasedontheaveragetorqueoverone crankcycleinsteadoftheinstantaneoustorque,because,asnotedin[52],thelimited bandwidthofelectricallystimulatedmusclegroupsmakestrackinganinstantaneous cranktorqueunfeasible.Ideally,toachievetheactive-torque-trackingobjective,anerror systemandcontrollerwouldbedesignedbasedonthedifferencebetweenthedesired torqueand a .However,noninvasivetorque-sensingdevicescanonlymeasurethenet torqueproducedbytherider, rider ; whichincludestherider'spassivelimbdynamics anddisturbances,asin2.Therefore,inthisdevelopment,ameasureof rider is takenwhiletherider'sactivetorque a =0 i.e.,theriderispassiveandisnotelectrically stimulated,therebyachievingameasureof p alongadesiredtrajectory,asdone in[10,29,52].Thisestimateof p ,denotedhereas ^ p : Q R ! R ,isaddedtothe torquesensormeasurementtoobtainanestimateoftheactiveridertorquethereafter. 104

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Then,theactivetorqueestimate ^ a : R 0 ! R isdenedas ^ a t , ^ p q t ; _ q c )]TJ/F25 11.9552 Tf 11.955 0 Td [( rider q t ; _ q t ; q t ;t ; where _ q c 2 R isaknown,constantcrankvelocity.Thefollowingassumptionismade aboutthepassivetorqueestimate ^ p . Assumption6. Anestimateoftherider'spassivelimbdynamics ^ p canbegenerated duringpreliminarytestingsuchthattheestimateerror ~ p : R 0 ! R ,denedas ~ p t , p q t ; _ q t ; q t )]TJ/F15 11.9552 Tf 12.563 0 Td [(^ p q t ; _ q c ; anditsrsttimederivativecanbeboundedas j ~ p j c ~ p ; and _ ~ p c _ ~ p ; where c ~ p ;c _ ~ p 2 R 0 areknownconstants.Thisassumptionisreasonableprovidedthat a and d rider aresufcientlysmallwhen ^ p isgeneratedandthatthevalueof _ q c usedwhen generating ^ p isthesameusedwhensubsequentlyapplyingtheestimate. Theaverageoftheactivetorqueestimateoverthecrankcycle ^ avg a : R 0 ! R isa discretesignaldenedas ^ avg a t = kT , 1 T kT k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 T ^ a t dt; where k 2 Z denotesthenumberofcrankcyclesandisincrementedwhenthecrank reacheszerodegrees,and T 2 R > 0 isthenominalcyclingperiod,denedas T , 2 _ q c . Hereafter,forsimplicityofnotation,discretesignalsaredenotedasfunctionsof k; ratherthan kT e.g., ^ avg a k , ^ avg a t = kT .Inthisdevelopment,thedesiredtorque d : R 0 ! R isbasedonthedesired,averagepoweroutputatthecrank d : R 0 ! R andisdenedas d t , 8 > > < > > : 0_ q d t < _ q d d t _ q d t _ q d t _ q d ; 105

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where _ q d 2 R > 0 isaselectable,positiveconstant.Thedesiredtorqueisdesignedtobe continuousandboundedas j d j c d withitsrsttimederivativeboundedas j _ d j c _ d . Theaveragetorquetrackingerrorsignal e : R 0 ! R isdenedas e k , d k )]TJ/F15 11.9552 Tf 12.564 0 Td [(^ avg a k : Substituting2,2,6,6,and6into6andrearranging termsyields e k = d k + D k )]TJ/F15 11.9552 Tf 14.467 8.087 Td [(1 T kT k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 T X m 2M B m q t ; _ q t u m t ! dt; where D : Z ! R isdenedas D k , 1 T kT k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 T ~ p t + d rider t )]TJ/F25 11.9552 Tf 11.955 0 Td [( r t dt: UsingAssumptions1and6,Property12 j D k j c _ ~ p + c _ d;rider + c _ r T; wheretheoperator indicatestheforwarddifferencei.e., D k , D k +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(D k . Becausethefeedbacksignal,torqueaveragedoverthecrankcycle,isonlyupdated oncepercrankcycle,thetorquetrackingerrorsystemisdiscretized.Therefore,the controlinputisalsodiscretizedinthesensethataconstantcontrolinputisapplied throughouteachcrankcycleandisupdatedonceanewmeasurementofthetracking errorisavailable.If u m weredenedasin2,thestimulationappliedtoeachmuscle groupwouldessentiallybeasquarewave.InpreviousFES-cyclingstudies,aramping envelopeforthestimulationwasusedtoavoidthelargejumpdiscontinuitiesthat arisefromasquareinputtothemusclegroupse.g.,[4,19].Motivatedtodolikewise whilesimultaneouslymaximizingefciencyoftheFES-evokedjointtorques,inthis 106

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development, u m isdenedas u m t , k m T m q t m q t u FES k ; for t 2 [ kT; k +1 T ,where k m , T m , m ,and u FES weredenedinChapter2.Substituting6into6andrearrangingtermsyields e k = d k + D k )]TJ/F15 11.9552 Tf 14.468 8.088 Td [(1 T B k u FES k ; where B : Z ! R > 0 isdenedas B k , kT k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 T X m 2M B m q t ; _ q t k m T m q t m q t ! dt: Notethatthedesignoftheswitchingsignals m ensuresthat B ispositive. Assumption7. B = B = B = ::: = B k = B k ,where B k 2 R > 0 isan uncertainconstantand B B k B ,where B ; B 2 R > 0 areknownconstants.This assumptionisreasonableprovidedthatthecadenceisnearlyconstantfromcycleto cycle. UsingAssumption7,theforwarddifferenceof6canbeexpressedas e k = d k + D k )]TJ/F15 11.9552 Tf 14.467 8.088 Td [(1 T B k u FES k : Basedon6andthesubsequentstabilityanalysis,theelectricalstimulationcontrol inputisdenedas u FES k , k 4 e k + k 5 + k 6 j d k j sgn e k ; where k 4 ;k 5 ;k 6 2 R > 0 areconstantcontrolgains. 6.2.2StabilityAnalysis Theorem6.2. Thetorquetrackingerrorisultimatelyboundedinthesensethat j e k j convergestoaballwithconstantradius d 2 R > 0 asthenumberofcrankcycles approachesinnityi.e.,as k !1 ,where d isdenedas 107

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d , r 2 c ; where c ; 2 R > 0 areconstantsdenedas c , 1 2 1+ 1 T Bk 6 c _ d + c _ ~ p + c _ d;rider + c _ r + 1 T 2 Bk 5 2 T 2 ; , 2 T B k k 4 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 1 T B k k 4 2 ; c , 1 2 1+ 1 T Bk 6 c _ d + c _ ~ p + c _ d;rider + c _ r + 1 T 2 Bk 5 2 T 2 ; providedthefollowinggainconditionsaresatised: k 4
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Substituting6and6into6andperformingsomealgebraicmanipulation yields V e k = 1 2 1 T B k 2 k 2 4 )]TJ/F15 11.9552 Tf 14.468 8.088 Td [(1 T B k k 4 ! e 2 k + 1 T B k 2 k 4 k 5 )]TJ/F15 11.9552 Tf 14.467 8.088 Td [(1 T B k k 5 ! j e k j + 1 )]TJ/F15 11.9552 Tf 14.467 8.087 Td [(1 T B k k 4 d k e k + 1 )]TJ/F15 11.9552 Tf 14.467 8.087 Td [(1 T B k k 4 D k e k + 1 T B k 2 k 4 k 6 )]TJ/F15 11.9552 Tf 14.468 8.088 Td [(1 T B k k 6 ! j d k jj e k j + 1 2 d k + D k )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 5 + k 6 j d k j 1 T B k sgn e k 2 : Providedthegainconditionsin6-6aresatised,6canbebounded using6as V k )]TJ/F25 11.9552 Tf 21.918 0 Td [( V k + c ; where and c weredenedin6and6,respectively.Applying6 sequentiallyfromthersttothe k th crankcycleyieldsthefollowingboundon V k : V k )]TJ/F25 11.9552 Tf 11.955 0 Td [( k V + k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 X n =0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( n c : Thesummationin6isageometricseriesthatconvergesfor 2 ; 2 ,whichis satisedprovided6issatised,sothat6canbeexpressedas V k )]TJ/F25 11.9552 Tf 11.955 0 Td [( k V )]TJ/F25 11.9552 Tf 14.045 8.088 Td [(c + c : Then,asthenumberofcrankcyclesapproachesinnityi.e.,as k !1 , V k is ultimatelyboundedinthesensethat lim k !1 V k c ; 109

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and,afterusing6andperformingsomealgebraicmanipulation,itcanbedeterminedthat j e k j isultimatelyboundedinthesensethat lim k !1 j e k j d; where d wasdenedin6. 6.3Experiments Experimentswereconductedwiththeprimaryobjectiveofevaluatingtheperformanceofthecontrollersdenedby6and6,withtheFESdistributedto themusclegroupsaccordingto2,2,and6.Threeable-bodied,male subject-26yearsoldparticipatedintheexperimentsaftergivingwritteninformed consentapprovedbytheUniversityofFloridaInstitutionalReviewBoard.Duringthe subsequentexperiments,thesubjectswereinstructedtorelaxandmakenovolitional efforttoeitherassistorinhibittheFESortheelectricmotorinputi.e.,passiveriders. 6.3.1Methods ThetestbedusedfortheseexperimentswasthesameoneusedfortheexperimentsdescribedinChapter5,exceptthatatorque-measuringcranksetSRM PowerMeterwasaddedtoprovideinstantaneousfeedbackoftheridertorque rider : The torquesignalwaslteredusingadiscrete,second-orderlowpasslterwithacut-offfrequencyof25radianspersecondtheprincipalfrequencyofthecycle-riderdynamicsis equaltothecrankvelocity,whichremainedwellbelow8radianspersecondthroughout theexperiments.Thelteredtorquesignalwasusedin6,whileitsaveragevalue, obtainedbyaveragingovertheperiod T andonlysamplingwhenthecrankpassed throughzerodegrees,wasusedforfeedbackin6. 110

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Thedesiredcrankvelocity _ q d andposition q d weredesignedas _ q d , 8 > > > < > > > : _ q c 1 )]TJ/F30 11.9552 Tf 11.955 13.27 Td [( t )]TJ/F26 7.9701 Tf 6.587 0 Td [(t 1 t 1 4 t 0 t > < > > : _ q c t )]TJ/F24 7.9701 Tf 13.151 6.18 Td [( t )]TJ/F26 7.9701 Tf 6.587 0 Td [(t 1 5 + t 5 1 5 t 4 1 + q t 0 t 0 t > > > > > > < > > > > > > > : 0 t 0 t
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whenstimulated,primarilybecauseoflowtoleranceofthestimulationbytherider; therefore,onlythequadricepsandglutealmusclegroupswerestimulatedinthese experiments.Thestimulationpatternwasdesignedaccordingto2and2with " m , 0 : 1 max T m ,where T m wasdeterminedbasedonmeasurementsoftherider's thigh l t andshank l l lengthsaswellashisseatposition l x and l y ,asdescribed inChapter5.Thestimulationpatternwasshiftedbackwardwithrespecttothecrank rotationtoaccountfordelayinthetorqueresponseofthemusclegroupstotheelectrical stimulation,aswasdonein[7,8,15,19,23,54].Here,themagnitudeofthebackward shiftwas 0 : 1_ q radians,assumingadelayof100milliseconds.Thestimulationamplitude wassetto90mAforthequadricepsand80mAfortheglutealmusclegroups,andthe stimulationfrequencywassetto60Hztoleveragetheresultsin[48].Thegainsfor theFEScontrolinputweretunedforeachsubjectandrangedasfollows: k 4 =[4 ; 10] ; k 5 =[2 : 5 ; 5] , k 6 =[25 ; 70] , k Quad =1 , k Glute =[1 ; 1 : 50] . Priortobeginningthetrackingtrial,acalibrationprocedurewasconductedtoobtain ^ p q t ; _ q c foreachsubject.Theelectricmotorwasusedtotrackthetrajectoriesin6– 39and6while rider wasmeasuredusingthecycle'storquesensorandtherider remainedpassivei.e., a =0 .Thecollectedtorquedatawasanalyzedfor t 1 t t f i.e.,for _ q = _ q c ,andaneight-termFourierserieswasttothedatatoobtainthepassive torqueestimateas ^ p = 8 X n =0 a n cos n!q + b n sin n!q : where a i 2 R ;i 2f 0 ; 1 ; 2 ;:::; 8 g ;b j 2 R , j 2f 1 ; 2 ; 3 ;:::; 8 g ,and ! 2 R aret parametersgiveninTable6-1forSubject1.Theresultsofthetoverasinglecrank cycleforSubject1areillustratedinFigure6-1,wherethethickdashedlinedepictsthe ridertorqueaveragedoverthecalibrationtrial,thethindashedlinesdepictonestandard deviationfromthemean,andthethicksolidlinedepicts ^ p .Theestimateobtainedfor eachsubjectwasusedinthesubsequenttrialtocalculate ^ a . 112

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Table6-1.PassivetorqueestimateparametersforSubject1. ParameterValueParameterValue a 0 -1.1108 b 1 0.1286 a 1 -0.1226 b 2 0.4559 a 2 -0.4834 b 3 0.0020 a 3 0.0112 b 4 -0.1664 a 4 -0.4055 b 5 0.0121 a 5 0.0131 b 6 -0.0370 a 6 -0.0763 b 7 0.0068 a 7 0.0142 b 8 -0.0011 a 8 -0.0102 ! 1 Figure6-1.Measuredversusestimatedpassiveridertorque. 113

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Figure6-2.Electricmotorcontroller'strackingperformance.APositiontrackingerror. BCadencetrackingerror.CMotorinput. 6.3.2Results Figure6-2depictstheelectricmotorcontroller'strackingperformanceforSubject 1,quantiedby e 1 , _ e 1 ,andtheelectricmotorcurrentinput u e .Notethatintherst30 secondsofthetriali.e.,for t
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Figure6-3.Desiredversusactualactivetorqueaveragedoverthecrankcycle. Figure6-3depictsthedesiredtorqueandtheactual,activetorqueestimate, averagedoverthecrankcycle ^ avg a k .Figure6-4depictstheFEScontroller'storque trackingperformance,quantiedby e k andthestimulationappliedtoeachmuscle group u m .Notethattorquetrackingdidnotbeginuntil t 1 =30 seconds,sonodatais displayedintherst30secondsofFigure6-4.Figure6-5providesanenhancedviewof thestimulationinput,depictedoverasinglecrankcycle.AcrossthetrialforSubject1, theFEScontrollermaintainedanaverageactivetorquetrackingerrorof 0 : 031 0 : 38 Nm. Toquantifytheoverallperformanceofthepowertrackingsystem,theaverage, activepowertrackingerror e : R 0 ! R isdenedas e k , d k )]TJ/F15 11.9552 Tf 12.563 0 Td [(^ avg a k _ q avg k ; 115

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Figure6-4.FEScontroller'storquetrackingperformance.ATorquetrackingerror.B Stimulationinput. Figure6-5.FEScontrolinputoverasinglecrankcycle. 116

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Table6-2.Meanandstandarddeviationoftrackingperformanceforallsubjects. Subject e 1 deg. _ e 1 rpm e Nm e Watts 12.7 5.80.0165 1.50.031 0.380.12 2.1 23.9 5.60.0199 2.30.066 0.560.29 3.0 32.9 6.30.0198 2.00.193 0.520.96 2.8 where _ q avg : R 0 ! R isdenedas _ q avg k , kT k )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 T _ q t dt: AcrossthetrialforSubject1,theelectricmotorandFEScontrollersworkedtogether tomaintainanaverage,activepowertrackingerrorof 0 : 12 2 : 1 Wattsforadesired trajectoryof20Wattsat50rpm. Table6-2reportsthemeanandstandarddeviationoftheposition,cadence,torque, andpowertrackingerrorsacrossthetrialforeachsubject. 6.3.3Discussion Theexperimentalresultsdemonstrateultimatelyboundedtrackingofadesired activepoweroutput,averagedoverthecrankcycle,atadesiredcadence.Tocompare theresultswitharecentstudywiththesametrackingobjectives,thenormalizedrootmean-squareNRMSofthepowertrackingerrorwascalculatedforeachsubject as10.3%,15.3%,and14.7%,respectively,foradesiredtrajectoryof20Wattsat50 rpm,whilein[23]theNRMSofthepowertrackingerrorrangedfrom4.6%to11.1% foradesiredtrajectoryof5-10Wattsat40-45rpm.In[23],thesubjectshadmotor andsensorycompletespinalcordinjuries,andafuzzylogicschemewasusedtovary thepulseamplitudeoverthecourseofeachtrial.Thesedifferences,especiallythe useofspinalcordinjuredsubjects,alongwiththeslowerandlesspowerfuldesired trajectories,likelyleadtoslightlybettertrackingperformance.Intheexperiments describedthroughoutthisdissertation,itwasobservedthatable-bodiedsubjectsoften havedifcultyremainingcompletelypassiveduringtheexperimentsandsometimes 117

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involuntarilyresistedthestimulationbecauseofdiscomfort,whichwasmoreeasily observedinthischaptergivenfeedbackoftheridertorque.Moreexperimentsmust beconductedundersimilarconditionstotrulycomparetheresultswithotherworks. Nevertheless,theresultsdescribedhereindemonstratesuccessfulachievementofthe controlobjective,andthedevelopedcontrolsystemprovidesasteppingstoneforfuture work. 6.4ConcludingRemarks ControllersforanelectricmotorandFEScontrolinputsweredevelopedthat guaranteeultimatelyboundedtrackingofadesired,average,activepoweroutputata desiredcyclingcadence.Asinthepreviouschapters,slidingmodecontrollersareused toachieveexponentialconvergenceofthetrackingerrorwhenthesystemiscontrolled. Unlikethepreviouschapters,theswitchingdynamicsareduetoquantizationofthe feedbacksignal,whichisonlymeasurableoncepercrankcycle,andtheresulting discretesystemhasnouncontrolledregions.Stabilityisprovenbymeansofdiscrete Lyapunovmethods,andexperimentalresultsonthreesubjectsdemonstratetheelectric motorandFEScontrollers'abilitytotrackthedesiredtrajectories. Futureworkmustinvestigatethepracticalimplicationsofthedevelopedcontrol system,especiallywhenappliedtoimpairedindividualse.g.,spinalcordinjured persons,asneurologicalimpairmentsarelikelytoinuencethemagnitudeofthepower outputthatcanbeachievedbytherider.Anotherfuturedirectionalongthislineofwork wouldbetoinvestigatevaryingthestimulationpattern,inadditiontothestimulation intensity,toachievethedesiredtorqueoutput,assuggestedin[26]. 118

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CHAPTER7 CONCLUSION InvestigatingthecontrolofFES-cyclingfromtheperspectiveofswitchedsystems theorymayyieldinsightsintomethodsforimprovingtheefciencyandpoweroutputof FES-cycling.InChapter2,anonlinear,uncertainmodelofthecycle-ridersystemwith pedalinginducedbyaswitchedFEScontrolinputwasdeveloped.InChapters3and4, switchedslidingmodecontrollersweredevelopedthatyieldedexponentialconvergence ofthetrackingerrorduringperiodswheremusclegroupswereactivatedanddivergence ofthetrackingerrorduringperiodswheremusclegroupswerenotactivated.Chapter5 builtontheresultsofChapters3and4byincludingelectricmotorinputintheregions ofthecrankcyclewherethemusclegroupsoftheriderwerenotactivated,thereby guaranteeingglobal,exponentiallystabletrackingofthedesiredcrankpositionand cadenceandremovingtheconstraintsonthedesiredcadence,initialconditions,and stimulationpatternthatarosewhenunstablemodeswerepresentintheswitched system.Chapter6detailedthedevelopmentofanFES-cyclingcontrolsystemthatuses anelectricmotortocontrolthecyclingcadencewhileFESisusedtoachieveadesired activepoweroutputatthecrank,andultimatelyboundedtrackingofadesiredcycling workrateisachieved.Experimentswereconductedonable-bodiedindividualsandan individualwithParkinson'sdiseasetovalidatethedevelopedcontrolsystemsandto demonstratehowthecontrolsystemsmightimpactclinicaloutcomesinarehabilitative setting. Throughoutthedevelopmentofthesecontrolsystems,severalchallengeswere encounteredthatcouldnotbeaddressedinthescopeofthisdissertation. Delayintheactivationofstimulatedmusclegroupsandtheirrelaxationpresentsa signicantchallengetoeffectiveFES-cycling.Forexample,ifthedelayis100millisecondsandthecrankisrotatingat60rpm,thenthecrankwillhavetraveled36degrees beforethestimulatedmuscleoutputsthedesiredforceorbeforetheforcedissipates. 119

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Thelattercaseisespeciallyproblematic,astheforcemaycontinueuptoandbeyonda deadpointandcausebackpedaling.SeveralpastFES-cyclingstudieshaveattempted toaddressthisissuebyshiftingthestimulationregionsbackwardsproportionaltothe cadencescaledbyanestimateddelay[8,15,19],aswasdoneinChapter6,orby usingmoresophisticatedmethods[21].Noneofthesestudiesconsideredthatdelay changesasafunctionoffatigueoranalyzedtheeffectsofthedelayonthestabilityof theFES-cyclingcontrolsystem,sothereismuchroomforfurtherexplorationsintohow thestimulationpatternmightbemodulatedasafunctionofthecyclingcadence,fatigue, orotherfactors. Whilethecontrolsystemsdevelopedinthisdissertationutilizedamodelofboth sidesofthecycle-ridersystemaswellascontrolgainsforeachmusclegroup,asymmetriesinthesystemoftencauseddegradationsintheexperimentalcontrolperformance. Asymmetriesinthesystemarisefromdifferencesintherider'sleglengthsandmass, musclebertypecomposition,musclestrength,locationofmotorunits,electrodeplacement,jointstiffnessandimpedance,andotherfactors.Practically,themostsignicant factoraffectingperformancewasthedifferencein B FES betweentherightandleftmusclegroups,astheresponsebythemuscletoelectricalstimulationwasalwaysdifferent whenstimulating,forexample,thequadricepsoftherightlegversustheleft.Thisissue wouldbeexacerbatedifasubjectwithextremeasymmetry,aswouldbethecasefora subjectwithhemipareticstroke,weretoparticipateinFES-cycling.Experienceindicates thattuningofthecontrolgainsforeachmusclegroupcouldaidincombatingtheissue ofasymmetry,aswasdoneintheexperimentsforChapter3,butmoresophisticated strategies,suchastheoneemployedin[55],arelikelytoyieldgreaterimprovementsin thecontrolperformanceandshouldbeexploredfurtherinthecontextofthemethods developedinthisdissertation. Asignicantlimitationoftheswitchedsystemsapproachappliedinthisdissertation isthenecessityforstabilityresultswheretherateofconvergencetoordivergence 120

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fromthedesiredtrajectoryhasaknownbounde.g.,exponentialconvergence.This requirementledtotheuseofslidingmodecontrolthroughoutthisdissertation,which, whiletheoreticallyeffectiveandrobust,iscertainlynottheoptimaloreventhesafest controlstrategyforFES-cycling.Adaptivecontrolmethodse.g.,neuralnetworks, iterativecontrolhavethepotentialtoimprovetheexperimentalresults,despitethe inevitablyweakertheoreticalresulti.e.,asymptoticallystabletracking,becauseofthe modelknowledgetheycontributetothecontrolsystembymeansoflearning.Iterative learningcontrolisapromisingtechniqueforFES-cyclingbecauseofthecyclicalnature ofthesystemandisespeciallypromisingforatrackingobjectivelikethatdescribed inChapter6.However,applyingadaptivecontrolmethodstoswitchedsystemswith uncertaintyanddisturbancesrepresentsasignicantchallengeintermsofcontroller developmentandanalysis.Nevertheless,explorationsintothetopicshouldbemade, asadaptivecontrolmethodscouldnotonlyimprovethecontrolperformancebutmay alsofacilitatetheadoptionofFES-cyclingintotheclinicalsetting,asadaptivecontrollers couldaccommodateawiderrangeofsubjectswithoutrequiringthetuningofcontrol gainsandothersubject-specicparametersofthesystem. Inadditiontoinvestigatingthechallengesencounteredinthisdissertation,there remainchallengesbeyondthescopeofthisworkthatshouldbeundertaken.Inthis work,onlystationarycyclingwasconsidered,butmobileFES-cyclinghasthepotential toprovidepeoplewithparalysisanewformofmobilityandrecreationinadditionto thealreadystatedhealthbenetsofFES-cycling.Mobilecyclingpresentsitsown setofengineeringchallenges,particularlywhenconsideringtheeffectsofsloped terrain,turning,andsafetydespiteenvironmentalfactorssuchasrain.Theworkin thisdissertationcouldalsoserveasasteppingstonetowardstheconceptofFESwalking.AnaturalprogressionfromcyclingtowalkingcouldoccurthroughFES-cycling withankleactuation,bodyweight-supportedFES-cyclingonanellipticalmachine, bodyweight-supportedFES-walkingonatreadmill,andnallytobodyweight-supported 121

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FES-walkingoverground,likelywithsupportfromanexoskeletonorsimilarbracing system.Whateverthepathsmaybethatleadfromthisdissertation,theclinicalimpact ofFES-enhancedtechnologiesshouldremainattheforefrontoftheresearcher's mind,astheredoesexistagreatneedintheworldbeyondthelaboratoryforassistive technologiesthatcanimprovethequalityoflifeformillionsofpeoplewithparalysis. 122

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APPENDIX:DETAILEDEXPRESSIONSFORCYCLE-RIDERDYNAMICS TheEuler-Lagrangedynamicsforoneside s ofthecycle-ridersystemcanbe expressedasin2as M s q t q t + V s q t ; _ q t _ q t + G s q t = s crank t : Detailedexpressionsfor M s , V s ,and G s areprovidedasfollows: M s = m s t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(l s c;t 2 + I s t + m s l l s t 2 S s 1 2 + m s l )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(l s c;l 2 + I s l S s 2 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 m s l l s t l s c;l cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 1 S s 2 + m s c )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(l s c;c 2 + I s c ; V s = m s t )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(l s c;t 2 + I s t + m s l l s t 2 S s 1 l s c l s t S s 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q sin q s k + q s h sin 2 q s k + q s h m s t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(l s c;t 2 + I s t + m s l l s t 2 S s 1 l s c l s t )]TJ/F22 11.9552 Tf 9.298 0 Td [(sin q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q cos q s k + q s h S s 2 + S s 1 sin 2 q s k + q s h )]TJ/F30 11.9552 Tf 11.291 13.27 Td [( m s l )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(l s c;l 2 + I s l S s 2 l s c l s l S s 1 +1 cos q s h + q sin q s k + q s h sin 2 q s k + q s h )]TJ/F30 11.9552 Tf 11.291 13.271 Td [( m s l )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(l s c;l 2 + I s l S s 2 l s c l s l )]TJ/F22 11.9552 Tf 9.299 0 Td [(sin q s h + q cos q s k + q s h S s 2 + S s 1 sin 2 q s k + q s h + m s l l s t l s c;l sin q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(S s 1 S s 1 S s 2 + m s l l s t l s c;l cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 1 l s c l s l S s 1 +1 cos q s h + q sin q s k + q s h sin 2 q s k + q s h + m s l l s t l s c;l cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 1 l s c l s l )]TJ/F22 11.9552 Tf 9.298 0 Td [(sin q s h + q cos q s k + q s h S s 2 + S s 1 sin 2 q s k + q s h )]TJ/F25 11.9552 Tf 9.299 0 Td [(m s l l s t l s c;l cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 2 l s c l s t S s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 cos q s k )]TJ/F25 11.9552 Tf 11.956 0 Td [(q sin q s k + q s h sin 2 q s k + q s h )]TJ/F25 11.9552 Tf 9.298 0 Td [(m s l l s t l s c;l cos q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q s h S s 2 l s c l s t )]TJ/F22 11.9552 Tf 9.298 0 Td [(sin q s k )]TJ/F25 11.9552 Tf 11.955 0 Td [(q cos q s k + q s h S s 2 + S s 1 sin 2 q s k + q s h _ q; G s = )]TJ/F30 11.9552 Tf 11.291 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(m s t l s c;t cos q s h S s 1 + m s l l s t cos q s h S s 1 + m s l l s c;l cos q s k S s 2 + m s c l s c;c cos q g: 123

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[50]C.FornusekandG.M.Davis,“Maximizingmuscleforcevialow-cadencefunctional electricalstimulationcycling,” J.Rehabil.Med. ,vol.36,pp.232,2004. [51]N.Sharma,C.Gregory,andW.E.Dixon,“Predictor-basedcompensationfor electromechanicaldelayduringneuromuscularelectricalstimulation,” IEEETrans. NeuralSyst.Rehabil.Eng. ,vol.19,no.6,pp.601,2011. [52]T.Schauer,“Feedbackcontrolofcyclinginspinalcordinjuryusingfunctional electricalstimulation,”Ph.D.dissertation,UniversityofGlasgow,2006. [53]J.Szecsi,A.Straube,andC.Fornusek,“Comparisonofthepedallingperformance inducedbymagneticandelectricalstimulationcycleergometryinable-bodied subjects,” Med.Eng.Phys. ,vol.36,no.4,pp.484,2014. [54]L.Comolli,S.Ferrante,A.Pedrocchi,M.Bocciolone,G.Ferrigno,andF.Molteni, “Metrologicalcharacterizationofacycle-ergometertooptimizethecyclinginduced byfunctionalelectricalstimulationonpatientswithstroke,” Med.Eng.Phys. ,vol.32, no.4,pp.339–348,2010. [55]E.Ambrosini,S.Ferrante,G.Ferrigno,F.Molteni,andA.Pedrocchi,“Cycling inducedbyelectricalstimulationimprovesmuscleactivationandsymmetryduring pedalinginhemipareticpatients,” IEEETrans.NeuralSyst.Rehabil.Eng. ,vol.20, no.3,pp.320,May2012. 128

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BIOGRAPHICALSKETCH MatthewBellmanhasbeenamemberoftheNonlinearControlsandRobotics groupsinceMay2009,andhassincereceivedhisbachelor's,master's,anddoctoral degreesinmechanicalengineeringfromtheUniversityofFloridaundertheguidance ofDr.WarrenE.Dixon.MatthewcompletedhisdoctoraldegreeasaNationalDefense ScienceandEngineeringFellow,withafocusonthetheoreticaldevelopmentofrobust controlsystemsforapplicationsinvolvingfunctionalelectricalstimulation,specically thoseinvolvingrehabilitationandmobilityofthelowerextremities.Matthewhasstarted acompany,MYOLYN,tobringtheresultsofhisresearchtothemillionsofpeoplewho couldbenetfromit,anditistherethatheservesasMYOLYN'sCo-founderandChief TechnologyOfcer. 129