Citation
Isometric Torque Control for Neuromuscular Electrical Stimulation with Fatigue-Induced Delay

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Title:
Isometric Torque Control for Neuromuscular Electrical Stimulation with Fatigue-Induced Delay
Creator:
Merad, Manelle
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (57 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
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
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Delay lines ( jstor )
Electric stimulation therapy ( jstor )
Electrical stimulation ( jstor )
Fatigue ( jstor )
Isometric contraction ( jstor )
Muscle fatigue ( jstor )
Propagation delay ( jstor )
Quadriceps muscle ( jstor )
Signals ( jstor )
Torque ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
delays -- fatigue -- nmes
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, M.S.

Notes

Abstract:
Neuromuscular electrical stimulation (NMES) is the application of an electric potential field across a muscle to produce a contraction. When the stimulus produces a functional limb motion, it is often called functional electrical stimulation (FES). NMES is used for postoperative rehabilitation, muscle strengthening for patients with motor function impairment, and it can also be implemented in a closed-loop feedback manner to assist a person in activities of daily living. External muscle stimulation is known to yield rapid muscle fatigue. When the muscle fatigues, the torque production decreases and the muscle response to electrical stimulation is delayed. Such changes in the torque and electromechanical delay (EMD) can lead to limit cycle oscillations and other undesirable or unpredictable behaviors. This thesis focuses on controlling the isometric torque evoked by external electrical stimulation of the quadriceps femoris muscle group. After highlighting the necessity of delay compensation in NMES closed-loop control, EMD and uncertainties are considered in the Lyapunov-based stability analysis that is used to design closed-loop controllers. The developed controllers are tested in NMES experiments on healthy individuals. The control designs are based on the assumption that the EMD is known and first experiments highlighted the time-varying aspect of the EMD; therefore, a delay estimation algorithm is developed to measure the delay between the stimulation and torque signals in real time. The results show that the developed controller enabled the reaction torque evoked by external electrical stimulation to track a desired torque despite time-varying input delays and uncertain dynamics. ( 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.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: DIXON,WARREN E.
Local:
Co-adviser: CRANE,CARL D,III.
Statement of Responsibility:
by Manelle Merad.

Record Information

Source Institution:
UFRGP
Rights Management:
Copyright Merad, Manelle. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
969976986 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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ISOMETRICTORQUECONTROLFORNEUROMUSCULARELECTRICAL STIMULATIONWITHFATIGUE-INDUCEDDELAY By MANELLEMERAD ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2014

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2014ManelleMerad 2

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Tomylovingparents. 3

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ACKNOWLEDGMENTS Firstofall,IwouldliketosincerelythankPr.WarrenE.Dixon,myadvisoratthe UniversityofFlorida,whoacceptedmeintheNonlinearControlandRoboticsgroup andguidedmetowardsthesuccessofthisthesis.Thesupportandhelpprovidedby Pr.BernardBayle,myadvisoratT el ecomPhysiqueStrasbourg,forthepastthree yearshavebeenverymuchappreciated.Iwouldliketooffermyspecialthanksto RyanDowneyforhiscontributiontothisthesis.Iamverygratefultoallthepeoplewho allowedmetotakepartinthisexperience:Pr.ChristopheCollet,Pr.Tran-Son-Tay,Lyn Straka,ShellyBurleson,AbigailNelsonandAlexandreDabrowski.Iwouldliketothank theNCRgroup2013/2014andDr.AysegulGunduzformakingthisexperiencebetter.I wouldliketothankSarahTonelloforherdailysupport.Finally,Iwouldliketothankmy parents,mybrothersandmyboyfriendfortheirsupportandloveacrosstheocean. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................6 LISTOFFIGURES.....................................7 ABSTRACT.........................................8 1INTRODUCTION...................................10 1.1Context.....................................10 1.2LiteratureReview................................12 1.3Outline......................................14 2ISOMETRICTORQUETRACKINGWITHOUTINPUTDELAYS.........16 2.1ModelPresentation...............................16 2.2ControlDevelopment..............................17 2.3StabilityAnalysis................................17 2.4Simulations...................................19 2.5Conclusion...................................19 3TORQUECONTROLWITHCONSTANTEMDCOMPENSATION........23 3.1SystemPresentation..............................23 3.2ControlDevelopment..............................24 3.3Experiments...................................29 3.3.1Methods.................................29 3.3.2Results..................................31 3.4Conclusion...................................33 4TORQUECONTROLWITHTIME-VARYINGEMDCOMPENSATION......36 4.1ControlDesign.................................36 4.2Experiments...................................42 4.3ComparisonwithConstantDelayCompensation..............47 4.4Conclusion...................................47 5CONCLUSION....................................49 5.1Achievements..................................49 5.2FutureWork...................................49 REFERENCES.......................................50 BIOGRAPHICALSKETCH................................57 5

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LISTOFTABLES Table page 3-1Controleffectivenessestimates...........................33 3-2TrackingerrorandEMDfortrialswithconstantdelaycompensation.......35 4-1Trackingerror,EMDandcontroleffectivenessmeasurements..........46 4-2Trackingerrorsforconstantandtime-varyingdelaycompensation........48 6

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LISTOFFIGURES Figure page 1-1Electrodesintranscutaneouselectricalstimulation................12 1-2IllustrationofisometricFES-inducedcontractions.................13 2-1TrackingperformancewithoutEMD.........................20 2-2TrackingperformancewithEMD..........................21 3-1Exampleofstimulationsignal............................31 3-2Recruitmentcurvemeasurement..........................32 3-3ConstantEMDestimation..............................32 3-4Torquetrackingperformancewithconstantdelaycompensation.........34 4-1Torquetrackingperformancewithtime-varyingdelaycompensation.......44 4-2EMDestimationinrealtime.............................45 4-3EvolutionoftheEMDwithinalltrials........................45 4-4Comparisonbetweenconstantandtime-varyingdelaycompensation......47 7

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience ISOMETRICTORQUECONTROLFORNEUROMUSCULARELECTRICAL STIMULATIONWITHFATIGUE-INDUCEDDELAY By ManelleMerad August2014 Chair:WarrenE.Dixon Major:MechanicalEngineering NeuromuscularelectricalstimulationNMESistheapplicationofanelectric potentialeldacrossamuscletoproduceacontraction.Whenthestimulusproduces afunctionallimbmotion,itisoftencalledfunctionalelectricalstimulationFES.NMES isusedforpostoperativerehabilitation,musclestrengtheningforpatientswithmotor functionimpairment,anditcanalsobeimplementedinaclosed-loopfeedbackmanner toassistapersoninactivitiesofdailyliving. Externalmusclestimulationisknowntoyieldrapidmusclefatigue.Whenthe musclefatigues,thetorqueproductiondecreasesandthemuscleresponsetoelectrical stimulationisdelayed.SuchchangesinthetorqueandelectromechanicaldelayEMD canleadtolimitcycleoscillationsandotherundesirableorunpredictablebehaviors. Thisthesisfocusesoncontrollingtheisometrictorqueevokedbyexternalelectrical stimulationofthequadricepsfemorismusclegroup.Afterhighlightingthenecessity ofdelaycompensationinNMESclosed-loopcontrol,EMDanduncertaintiesare consideredintheLyapunov-basedstabilityanalysisthatisusedtodesignclosed-loop controllers.ThedevelopedcontrollersaretestedinNMESexperimentsonhealthy individuals.ThecontroldesignsarebasedontheassumptionthattheEMDisknown andrstexperimentshighlightedthetime-varyingaspectoftheEMD;therefore,adelay estimationalgorithmisdevelopedtomeasurethedelaybetweenthestimulationand torquesignalsinrealtime.Theresultsshowthatthedevelopedcontrollerenabledthe 8

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reactiontorqueevokedbyexternalelectricalstimulationtotrackadesiredtorquedespite time-varyinginputdelaysanduncertaindynamics. 9

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CHAPTER1 INTRODUCTION 1.1Context Motorneuronsinnervatemusclebersandcontroltheircontractionsbytransmitting electricalpotentialsalongtheiraxonsfromthebraintothemuscle.Skeletalmuscles areessentialforlocomotion:theypullontendonsand,bycontractingtheyallowthe jointtorotateandthus,thebodytomove.Adisruptioninthemotorsignalpathor damageinthemotorcortexcanimpairmotorfunctionse.g.,followingstrokeorspinal cordinjury.BasedonGalvani'sandVolta'sdiscoveries[1],anelectricalcurrent propagatingalongmusclebersbetweentwoelectrodescancausethemuscleto contract.NeuromuscularelectricalstimulationNMEShadbeendevelopedbasedon thisphenomenonandfunctionalelectricalstimulationistheapplicationofNMESto performfunctionaltasks[2].Theelectrodesusedtostimulatethemusclecanbeplaced overtheskin transcutaneous stimulation,noninvasiveorbeneaththeskin,closerto themotorneuron percutaneous stimulation,invasive.Anexampleoftranscutaneous stimulationisillustratedinFigure1-1. NMESisatechniqueprimarilyusedinpostoperativerehabilitation[3–5]and formusclestrengthening[6].However,itcanalsobeimplementedinaclosed-loop feedbackmechanismwheretheelectricalstimuliaredesignedtoachievevarious rehabilitationoutcomesinvolvingdynamicsorisometriccontractions[7–16];autonomy forparaplegicpatients[17],obstacleavoidance[11]anddailyassistance[18–20]are examplesofFESapplications.TwotypesofexercisesareperformedduringNMES trainingdependingonthetargetedtypeofcontractions:in isometric contractions, themusclelengthisconstantandthejointangleisxed,whereasthemusclelength shortensandthejointanglevariesin dynamic contractions.AnisometricNMES trainingexampleisgiveninFigure1-2.Manyrehabilitationoutcomesmandatedynamic trainingleadingtolimbmotion.Forotheroutcomes,isometriccontractionsaremore 10

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advantageous:itisconsideredsaferthandynamictrainingsincethejointisnotmoving, developsresistance,candecreasethepainduringtherehabilitationtime[21,22]and maylowerbloodpressureofpatientswithhighrisksofcardiovasculardiseaseby providinghigherintensityexercises[23,24]. Experimentalevidenceexiststodemonstratethatthereisatimelag,termed electromechanicaldelayEMD,betweenthemuscleelectricalactivationandtheonset ofmuscleforce[25–29].Thisdelayprimarilyresultsfromthetimerequiredtostretch theelasticcomponentsinserieswiththecontractileelementsofamuscle[30].There aremanydiscrepanciesintheliteraturerelativetothedelayvalueduetodifferent measurementmethods[31].Controlinstabilitymaybecausedbytheinputdelay,and therefore,EMDshouldbeconsideredinthesystemmodelandinthecontrolstrategy. InadditiontotheEMD,thereareotherphysiologicalmechanismsoccuringduring musclecontractions.Muscleresponsetoelectricalimpulsesisnonlinearanddepends ontheuser'sphysiologicalconditionsandthestimulationparameters[32–35].Experimentalresultsshowevidenceoffatigueduringmusclecontractionsthatlimitsthecontrol performance.Musclefatigueisaprocesswherebythemuscleforcedecreaseseven thoughthestimulationsignalismaintained[32,36,37]andmusclefatigueisknownto occurfasterwithNMEStrainingthanvoluntarycontractions.Failureofmotorneuron excitationorimpairmentofactionpotentialpropagationaresuggestedexplanationsof musclefatigue[38].TherearevarioussuggestedcausesforNMES-inducedfatigue suchasareversaloftheHenneman'ssizeprinciple[39]aswellasspatiallyxedand temporallysynchronousberrecruitment[40]andstimulationfrequency[41].Further, musclefatiguehasbeenshowntolengthentheEMD[42–44].Manyotherfactors aresuspectedtoinuencetheEMDincluding:typeofexerciseperformed[26,27], temperature[45],gender[46]andage[43,47]. 11

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Figure1-1:Exampleoftranscutaneouselectricalstimulationofthequadricepsfemoris musclegroup:thestimulationsignalisowingbetweenthetwoelectrodes,evoking musclebercontractions.PhotoCourtesyoftheAuthor. Thefocusofthisthesisistodesignclosed-loopfeedbackcontrolmethodsthat compensatesforNMES-induceddelays;isometriccontractionsofthequadriceps femorismusclegroupwereconsidered. 1.2LiteratureReview NMES-induceddelaysarearesultofthemuscleactivationprocess;therefore,they areintroducedinthedynamicsviaadelayedinput[48,49].Comparedtoopen-loop shapingofthemodulationstrategy,closed-loopcontrolmethodstocompensateforEMD havereceivedlessattention. NMESclosed-loopcontrollersweredevelopedin[50–52]tocompensateforEMD assuminglineardynamicsandaconstantknowndelay.Thepreviousstabilityresults ofnonlinearmethodsin[53–55]thatactivelycompensatefortheknowninputdelay havefocusedonmoregeneraldynamicalsystemsassumingexactmodelknowledge. Theconstantinputdelayissueinuncertainnonlineardynamicalsystemsisaddressed in[56–59]wherethedelayisassumedtobeknown,andin[56,60]wherethedelay 12

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Figure1-2:Exampleofelectrically-inducedcontractionofthequadricepsfemorismuscle groupevokingakneeextension.Inisometriccontractions,thelowershankisxedand thekneejointangleisconstantwhileanelectricalsignalstimulatesthemusclebers betweentheelectrodes.Thecontractionresultsinatorqueattheknee-joint.The correspondingforceismeasuredwithaforcetransducer.PhotoCourtesyofthe Author. 13

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isunknown.ResultsofknownconstantEMDcompensationarepresentedin[61]for knownmuscledynamicsandin[29]foruncertainEuler-Lagrangedynamicsappliedto themuscle. WhilepreviousresearchfocusedonNMESclosed-loopstabilizationinthepresenceofconstantinputdelaysinthedynamics,thetime-varyingaspectofthedelayis highlightedinexperiments.Eventhoughresearchershavepursuedmethodstoslowthe rateofNMES-inducedfatigue,forinstancebydecreasingthestimulationfrequencyor pattern[62,63],orbymodulatingtheinputsignal[64],theonsetoffatigueisinevitable. Tocompensatefortheknowntime-varyinginputdelay,methodsweredevelopedto stabilizeinputdelayedsystemsin[65]and[66]assumingexactmodelknowledgeand in[67]wheresemi-globaluniformlyultimatelyboundedpositiontrackingisachievedfor uncertainEuler-Lagrangedynamics. PreviousNMESresultshaveachievedclosed-loopcontrolyieldingdynamiccontractions.Fewerstudiesinvestigatedclosed-loopcontrolyieldingisometriccontractions. In[68],alineartorquetrackingcontrollerwasdevelopedforclosed-loopNMES.Agroup ofrecentresultsdeveloptorquetrackingcontrollersforisometriccontractionsusing fatigueprediction[69–71]:musclefatigueisestimatedbasedonelectromyographic EMGsignalsinordertocontroltheankletorque.However,surfaceEMGaredifcult todissociatefromtheinputstimulationsignalduringtranscutaneouselectricalstimulation.Further,EMDwasnotconsideredintheaforementionedtorquetrackingresults, althoughithasinuenceonthereactiontorqueandsystemstability. 1.3Outline ThisthesisisfocusedonthedevelopmentofatorquetrackingcontrollerforisometricNMESonthequadricepsfemorismusclegroupwithtime-varyingknowninput delaysinthedynamics.Lyapunov-basedstabilitymethodsyieldglobalexponentially boundedtorquetrackingerrorwhenEMDisnotconsideredinthedynamicsandglobal uniformlyultimatelyboundedtorquetrackingerrordespitethepresenceofuncertainties, 14

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nonlinearitiesandEMD.Experimentswereconductedonhealthysubjectstoassessthe performanceofthedevelopedcontrollers. Chapter1presentsthecontextandthemotivationsofthiswork.Previousstudies thatinvestigatedNMESclosed-loopcontrolarereviewed. Chapter2isanintroductiontoNMEStorquecontrol.TheEMDisnotconsidered inthemodeldynamicsinordertofocusonthechallengesduetotorquetracking.The theoreticalstabilityproofyieldsexponentialtracking,whereasthesimulationsshow instabilityofthetorquetrackingerrorwhendelaysareintroducedinthesystemplant. InChapter3,aconstantEMDisincludedinthesystemmodelalongwithadelay compensationterminthecontrolinput.Thedevelopedcontrollerachievesglobal uniformlyultimatelyboundedtorquetracking.Experimentswereconductedona modiedlegextensionmachinetoillustratetheperformanceofthecontrollerwherethe EMDwasestimatedbeforeeachtrial. InChapter4,acontrollerisdevelopedthatcompensatesforatime-varyinginput delayduringisometriccontractions.ALyapunov-basedanalysisisusedtoprovea globaluniformlyultimatelyboundedtorquetrackingerror.Thecontrollerwastested inexperimentsonhealthyindividualswheretheEMDwasestimatedinrealtime.At theendofthischapter,constantandtime-varyingdelaycompensationmethodsare compared. 15

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CHAPTER2 ISOMETRICTORQUETRACKINGWITHOUTINPUTDELAYS 2.1ModelPresentation Anelectricpotentialeldacrossamusclecandepolarizeamotorneuronifits amplitudeisgreaterthantheexcitationthreshold,evokingacontractionofthemuscle bersandproducingatorque[2].Thisthesisfocusesonelectrically-evokedisometric contractionsofthequadricepsfemorismusclegroup.Areactiontorqueisproducedat theknee-jointandisconsideredasthetime-varyingstateofthesystem.Theindividual isseatingwiththelegxedandthetorqueoutputismeasuredduringthetrials.The uncertainnonlinearmodelin[49]isadaptedtoisometriccontractionsbyxingthejoint angleandaddingtothemodelequationthereactiontorqueduetotheforceexertedby theforcetransducerontheleg,yielding R t = f q + D t + t V t ; where R 2 R denotesthereactiontorque, f q 2 R denotesthegravityandtheelastic components,whichdependonlyontheconstantknee-jointangle q 2 R , D 2 R denotes time-varyingexogenousunknowndisturbances,and 2 R isanunknownnonzero time-varyingfunctionrelatingtheinputvoltage V 2 R tothetorque. Assumption1. Thedisturbance D isboundedanditsrsttime-derivativeexistsandis bounded[29]. Assumption2. Thepositivenonzerounknownfunction isboundedsuchthat t forall t ,where and arepositiveconstants.Thersttime-derivativeof existsandisbounded. Remark 1 . Asmusclefatigues,thereactiontorquedecays,butonlytoaminimumvalue. Assumption2ismildinthesensethatitassuresaknownconservativelowerbound whichrelatestotheminimumtorquethatcanbeproducedforagiveninput.Thisbound couldbedeterminedexperimentally. 16

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Thecontrolobjectiveistodesignacontinuouscontrollerthatensuresthereaction torque R t oftheinputdelayedsystemin2tracksadesiredtorque R d t 2 R despiteuncertaintiesandadditiveboundeddisturbances. 2.2ControlDevelopment Toquantifytheobjective,thetorquetrackingerrorbetweenadesiredtorque R d 2 R andthemeasuredtorqueisdenedas e , R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(R: Theopen-looperrorsystemisobtainedbymultiplying2by )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 andtakingits time-derivativesuchthat d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e dt = S )]TJ/F15 11.9552 Tf 15.042 3.022 Td [(_ V; wheretheauxiliaryterm S , d dt )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(f q )]TJ/F24 11.9552 Tf 11.955 0 Td [(D canbeupperboundedbya positiveconstant " 2 2 R usingAssumptions1and2as j S j " 2 : Basedonthesubsequentanalysisand2,thecontrolinputisdesignedasthe solutionto _ V = k a 1 e + k a 2 sgn e ;V t 0 = V 0 ; where V 0 2 R isaselectableconstant,and k a 1 , k a 2 2 R aretwopositivecontrolgains. Substituting2into2,theclosed-looperrorsystemiswrittenas d )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e dt = S )]TJ/F24 11.9552 Tf 11.955 0 Td [(k a 1 e )]TJ/F24 11.9552 Tf 11.955 0 Td [(k a 2 sgn e : 2.3StabilityAnalysis Theorem1. Giventhemodelin2,thecontrollawin2ensuresaglobalexponentiallyboundedtorquetrackingerrorinthesensethat j e t j 1 e )]TJ/F25 7.9701 Tf 6.587 0 Td [( 0 t ; 17

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where 0 , 1 2 R + denoteconstants,providedthefollowingsufcientconditionis satised k a 2 >" 2 : Proof: Let V L : R [0; 1 ! R beacontinuouslydifferentiable,positivedenite functionaldenedas V L = 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e 2 ; whichcanbeboundedas 1 e 2 V L 2 e 2 ; where 1 , 1 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 and 2 , 1 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 aretwopositiveconstants.Using2,thetimederivativeof2canbeexpressedas _ V L = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e S )]TJ/F24 11.9552 Tf 11.956 0 Td [(k a 1 e )]TJ/F24 11.9552 Tf 11.955 0 Td [(k a 2 sgn e : Using2,2canbeupperboundedas _ V L )]TJ/F24 11.9552 Tf 28.56 0 Td [(k a 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k a 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(" 2 j e j _ V L )]TJ/F24 11.9552 Tf 28.56 0 Td [(k a 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e 2 : Using2andAssumption2,theexpressionin2canbewrittenas _ V L )]TJ/F15 11.9552 Tf 21.918 0 Td [(2 k a 1 V L : Therefore, V L ispositivedenite,and _ V L isnegativedenite.Thedifferentialequationin 2canbesolved,andusing2,thetorquetrackingerrorisboundedas j e j p 2 V L t 0 exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(k a 1 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 ; yieldingtheresultin2.From2,thecontrolinput V isbounded. 18

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2.4Simulations Thecontrollerdevelopedinthischapteristestedinsimulations.Aplantmodelwas implementedaccordingto2where f q = 2N m ,thedisturbanceisdesignedas aGaussiannoise = 0 , 2 = 0 : 01 andthecontroleffectiveness isapproximated byaconstantofvalue 0 : 5N m s )]TJ/F47 7.9701 Tf 6.587 0 Td [(1 .Theplantinputisthestimulationsignalwhich isimplementedbasedon2.Hence,thereactiontorque,outputoftheplant,is controlledtotrackadesiredtorquethatisconstitutedofhightiringandlowresting plateaus.Figure2-1illustratestheperformanceofthesimulatedtorquetracking controllerwithoutdelayinthesystem.Thecontrolledtorquetracksthedesiredtorque despitenoiseandthetrackingerrordecreasesexponentiallytozeroRMSerror:0.035 N m . Forcomparison,thesamecontrollerisusedwithaninputdelayedplantmodel. TheEMDchosenisincreasingexponentiallyfrom50msto65ms,anddelaystheplant stimulationinput.TheEMDvaluesarewithintherangeofvaluesfornaturally-and articially-inducedcontractionsfoundintheliterature:32.6msto41.9msafterexercise in[72],86msin[73],17.2msin[27]areexamples.Thesamegainsareusedforthe twotrials: k a 1 = 50 and k a 2 = 50 .PlotsinFigure2-2representtheresultswiththesame controllerbutwithtime-varyinginputdelaysimplementedinthesystemmodel.Since thecontrollerisnotdesignedtocompensatefortheEMD,thetrackingerrorisrapidly unstableandgrowswithtimeRMSerror:35 N m . 2.5Conclusion AtorquetrackingcontrollerwasdesignedforisometricNMES-inducedcontractions withoutconsideringtheEMDinthemusclemodel.Thetorquetrackingerrorwas proventodecayexponentiallytozerowithoutassumingsystemmodelknowledge. But,experimentalresultsshowthatthemusclecontractionprocessdelaysthemuscle response,thereforethemodelinput.GiventhesimulationresultswhentheEMD isimplementedinthemusclemodel,thetorquetrackingerrorexhibitsanunstable 19

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a b Figure2-1:Simulationresultsforatorquetrackingtrialwithnondelayedmodeled NMESsystem.Theplotinaillustratesthetorquetrackingperformancewithoutinputdelaysinthesystem.Thetorquetrackingerrorisrepresentedinbandconverges inlessthanonesecond. 20

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a b Figure2-2:SimulationresultsforatorquetrackingtrialofamodeledFNMESexperimentwithnonzeroEMD t = 0 : 05e 0 : 002t .Thetorquetrackingperformanceisillustrated inawhereasthetrackingerrordepictedinbdiverges. 21

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behavior.However,thisstudyhelpedtoidentifytheissuesofFEStorquetracking.The controlstrategyshouldcompensateforthefatigue-inducedEMD,whichisintroduced inthedynamicsviathecontrolinput,andshouldalsotakeintoaccounttheunknown controleffectivenesswhichmultipliestheelectricalinput.Theerrorsystemandthe Lyapunov-basedstabilityanalysisdevelopedinthischaptercannotbeusedwitha delayedinputsystem.Motivatedbythisoutcome,theresultsinthesubsequentchapters presentclosed-loopcontrollersthatactivelycompensateforthedelays,withassociated stabilityanalysisandexperimentalresults. 22

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CHAPTER3 TORQUECONTROLWITHCONSTANTEMDCOMPENSATION Experimentalevidenceexiststoprovethatthereisadelaybetweenthemuscle electricalactivationandtheonsetoftorque.Inthischapter,themusclemodelismodiedtocapturethisphenomenonbyinsertingaconstantinputdelayinthedynamics. TheaimofthischapteristodevelopacontrolmethodthatensuresstableNMEStorque trackingdespitetheEMD.Themotivationtoincludetheinputdelayinthesystemmodel andinthecontrolstrategyistoimproveNMEScontrolperformanceandreliabilityin torquetracking. 3.1SystemPresentation Aconstantinputdelayisintroducedinthesystem,describedpreviouslyin2, andiswrittenas R t = f q + D t + t V t )]TJ/F24 11.9552 Tf 11.956 0 Td [( ; where 2 R denotestheknownconstantEMDandwhere R , f , D , and V were introducedin2. Thecontrolobjectiveistodesignacontinuouscontrollerthatensuresthestate R t oftheinput-delayedsystemin3tracksadesiredtorque R d t 2 R despite uncertainties,knownconstantinputdelayandadditiveboundeddisturbances.To quantifythisobjective,thetorquetrackingerrorisdenedas e , R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(R: Tofacilitatethesubsequentanalysis,anauxiliarytrackingerrorisdenedas r , e )]TJ/F24 11.9552 Tf 11.955 0 Td [(Be z ; wheretheauxiliarysignal e z 2 R isdenedas e z , t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V d: 23

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In3, B 2 R + isaconstant,bestguessestimateof .Themismatchbetween B and isdenedas , B )]TJ/F15 11.9552 Tf 11.955 0 Td [( ; whichsatisesthefollowinginequality j j ; basedonAssumption2,where 2 R isapositiveknownconstant. Notation Throughoutthestudyfornotationalbrevity,atimedependentdelayedfunction :[0 ; 1 ! R correspondingto isdenedas t , 8 > > < > > : t )]TJ/F24 11.9552 Tf 11.955 0 Td [( t t t 0 t< t : . 3.2ControlDevelopment Multiplying3by )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 andusing3and3yields )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(f q )]TJ/F24 11.9552 Tf 11.955 0 Td [(D )]TJ/F24 11.9552 Tf 11.955 0 Td [(V )]TJ/F24 11.9552 Tf 11.955 0 Td [(B )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e z : Theopen-looperrorsystemcanbeobtainedbytakingthersttime-derivativeof3 andusing3-3as )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 _ r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r + N + S )]TJ/F15 11.9552 Tf 15.042 3.022 Td [(_ V )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 _ V )]TJ/F15 11.9552 Tf 15.042 3.022 Td [(_ V t )]TJ/F25 7.9701 Tf 6.587 0 Td [( ; wheretheauxiliarysignals S 2 R and N 2 R aredenedas S , d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(f q )]TJ/F24 11.9552 Tf 11.955 0 Td [(D ; N , )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r )]TJ/F24 11.9552 Tf 11.955 0 Td [(B d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e z : Basedon3-3,theopen-looperrorsystemin3nowcontainsadelay-free controlinput.Fromthesubsequentanalysisand3,thecontrolinputisdesignedas 24

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thesolutionto _ V = k b r;V t 0 = V 0 ; where V 0 2 R isaselectableconstantand k b 2 R isacontrolgainsuchthat k b = k b 1 + k b 2 + k b 3 where k b 1 , k b 2 and k b 3 2 R arepositiveconstants.Substituting3into3yields thefollowingclosed-looperrorsystem )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 _ r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r + N + S )]TJ/F24 11.9552 Tf 11.956 0 Td [(k b r )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F24 11.9552 Tf 11.956 0 Td [(r : Theauxiliarysignal S canbeupperboundedbyaknownconstant " 2 2 R + accordingto Assumptions1and2suchas j S j " 2 ; andusingAssumption2,theexpressionin3canbeupperboundedas j N j 1 jj z jj ; where 1 2 R isapositiveknownconstant,and z 2 R 2 isdenedas z = re z T : Tofacilitatethesubsequentstabilityanalysis,let y 2 R 4 bedenedas y = re z p P p Q T ; where P , Q 2 R areLyapunov-Krasovskiifunctionalsdenedas P , ! t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( t s _ V 2 d ds; Q , k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 4 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( r 2 d; 25

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where ! and 2 2 R + areselectableconstants.Basedonthesubsequentanalysis,the constant 1 2 R + isdenedas 1 , min f m 1 ;m 2 g ; where m 1 , inf k b 3 )]TJ/F24 11.9552 Tf 13.151 8.847 Td [(k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(8 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + 2 1 + k b 2 2 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [(k 2 b ! ; m 2 , inf 1 ! )]TJ/F24 11.9552 Tf 11.955 0 Td [( k b 2 1 + 2 2 2 ; where m 1 , m 2 2 R arepositiveconstantsand 1 2 R + isaselectableconstant. Theorem2. Thecontrollaw,denedin3,ensuresglobalultimatelyuniformly boundedtorquetrackinginthesensethat j e t j p a 11 exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(a 0 t + a 12 + p a 21 exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(a 0 t + a 22 ; where a 0 , a 11 , a 12 , a 21 , a 22 2 R + denoteconstants,providedthefollowingsufcient conditionsaresatised k b 3 > sup k 2 b ! + k b )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(8 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + 2 1 + k b 2 2 4 ; !> sup k b 2 1 + 2 2 2 ; 1 > 2 1 4 k b 1 : Proof: Let V L : R [0; 1 ! R beacontinuouslydifferentiable,positivedenite functionaldenedas V L , 1 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r 2 + 1 2 e z + P + Q; whichcanbeboundedas 1 jj y jj 2 V L 2 jj y jj 2 ; 26

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wheretheconstants 1 , 2 2 R aredenedas 1 , 1 2 min )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 2 , max 1 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ; 1 ; where and aredenedinAssumption2.ApplyingtheLeibnizRuletodetermine thetimederivativeof3and3,andutilizing3,3and3,thetime derivativeof3canbewrittenas _ V L = )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b r 2 + Sr + Nr )]TJ/F24 11.9552 Tf 11.956 0 Td [(k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r )]TJ/F24 11.9552 Tf 11.955 0 Td [(e z r )]TJ/F24 11.9552 Tf 11.955 0 Td [(r + k b )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 4 r 2 + ! j _ V j 2 )]TJ/F24 11.9552 Tf 13.151 8.846 Td [(k b )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 4 r 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(! t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( V 2 d: UsingYoung'sInequality,thefollowinginequalitiescanbedeveloped k b j e z jj r j k b 2 1 4 r 2 + 1 2 1 e 2 z ; k b j e z jj r j k 2 b 2 2 4 r 2 + 1 2 2 e 2 z ; k b )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 j r jj r j k b )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [(r 2 + r 2 : ByapplyingtheCauchy-SchwarzInequality, j e z j 2 = t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 1 d 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( j 1 j 2 d t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: ByusingAssumption2,3,3,3,3and3-3,theexpression in3canbeupperboundedas _ V L )]TJ/F24 11.9552 Tf 23.911 0 Td [(k b r 2 + " 2 j r j + 1 jj z jjj r j + k b 2 1 4 r 2 + k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(8 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 4 r 2 + k 2 b !r 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(! t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d + k b 2 1 + 1 2 2 e 2 z : Using3and3,3isupperboundedandgroupedas _ V L )]TJ/F52 10.9091 Tf 21.818 0 Td [(k b 1 r 2 + 1 jj z jjj r j)]TJ/F52 10.9091 Tf 16.364 0 Td [(k b 2 r 2 + " 2 j r j)]TJ/F30 10.9091 Tf 16.364 18.655 Td [( k b 3 )]TJ/F52 10.9091 Tf 12.105 8.073 Td [(k b )]TJ/F17 10.9091 Tf 5 -8.836 Td [(8 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + 2 1 + k b 2 2 4 )]TJ/F52 10.9091 Tf 10.909 0 Td [(k 2 b ! ! r 2 )]TJ/F30 10.9091 Tf 10.909 15.382 Td [( ! )]TJ/F52 10.9091 Tf 10.909 0 Td [( k b 2 1 + 2 2 2 t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d )]TJ/F52 10.9091 Tf 14.596 7.38 Td [( 2 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: 27

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Completingthesquaresandusing3,3canbeupperboundedas _ V L )]TJ/F30 10.9091 Tf 21.818 18.655 Td [( k b 3 )]TJ/F52 10.9091 Tf 12.105 8.073 Td [(k b )]TJ/F17 10.9091 Tf 5 -8.836 Td [(8 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + 2 1 + k b 2 2 4 )]TJ/F52 10.9091 Tf 10.91 0 Td [(k 2 b ! ! r 2 + " 2 2 4 k b 2 + 2 1 4 k b 1 jj z jj 2 )]TJ/F17 10.9091 Tf 12.379 7.381 Td [(1 ! )]TJ/F52 10.9091 Tf 10.909 0 Td [( k b 2 1 + 2 2 2 e z )]TJ/F52 10.9091 Tf 14.596 7.38 Td [( 2 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: Usingthedenitionof z in3and 1 in3,theexpressionin3canbe upperboundedas _ V L )]TJ/F30 11.9552 Tf 23.911 16.857 Td [( 1 )]TJ/F24 11.9552 Tf 16.374 8.088 Td [( 2 1 4 1 jj z jj 2 + " 2 2 4 k b 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 2 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: Afterusing3,3,3andthefollowinginequality t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( t s _ V 2 d ds t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( sup s 2 [ t )]TJ/F25 7.9701 Tf 6.586 0 Td [(;t ] t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d ds = sup s 2 [ t )]TJ/F25 7.9701 Tf 6.587 0 Td [(;t ] t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d = t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d; Theexpressionin3canbeboundedas _ V L )]TJ/F30 11.9552 Tf 23.91 16.857 Td [( 1 )]TJ/F24 11.9552 Tf 17.883 8.088 Td [( 2 1 4 k b 1 jj z jj 2 + " 2 2 4 k b 2 )]TJ/F15 11.9552 Tf 22.737 8.087 Td [(1 2 2 2 ! P )]TJ/F15 11.9552 Tf 48.069 8.087 Td [(2 k b 2 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 Q: Fromthedenitionof y in3,3canbeupperboundedas _ V L )]TJ/F24 11.9552 Tf 21.918 0 Td [( 2 jj y jj 2 + " 2 2 4 k b 2 ; where 2 2 R + isdenedas 2 , min 1 )]TJ/F24 11.9552 Tf 17.883 8.088 Td [( 2 1 4 k b 1 ; 1 2 2 2 ! ; inf 2 k b 2 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ! : Using3,theexpressionin3canbewrittenas _ V L )]TJ/F24 11.9552 Tf 23.223 8.088 Td [( 2 2 V L + " 2 2 4 k b 2 : Finally,thedifferentialequationin3canbesolvedas V L V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 4 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.956 0 Td [(t 0 : 28

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From3, V L isgloballyuniformlyultimatelybounded.Using3, r and e z arealso boundedas j r j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 ; j e z j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 : From3,3and3, j e j canbeupperboundedas j e j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.956 0 Td [(t 0 + B s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 ; yieldingtheresultin3.Finallyfrom3and3,thecontrolinput V is bounded. 3.3Experiments Thecontrolperformancewasevaluatedinclosed-loopNMESexperiments.Surface electricalstimulationwasappliedtothequadricepsmusclegrouptoinduceisometric kneeextensiontorque,whichwasmeasuredbyaforcetransducer.Basedon3,the implementedcontrollawusedthetorqueandpre-trialEMDmeasurementstocalculate thestimulationsignal. 3.3.1Methods FourhealthysubjectsAge25 2.5years participatedinthetrialsaftergivingwritteninformedconsent,asapprovedbytheInstitutionalReviewBoardattheUniversity ofFlorida.ParticipantswereaskedtositinamodiedLegExtensionMachineLEM. Figure1-2illustratestheexperimentalsetupusedtomeasuretheforcewhilethestimulationisdelivered.Usingthemeasuredmomentarmlength,theforceexertedonthe LEMfromthelowershankwasconvertedinatorqueandusedinthecontroller.Two torquetrackingexerciseswereperformedoneachparticipant'srightlegwitharesting periodof15minutesbetweenthetrials. 29

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Acurrent-controlledstimulatorRehaStim,Hasomed,GmbH,Germanywas usedtodeliverthestimulationpatterntoeachparticipant'squadricepsfemorismuscle groupviabipolarsurfaceelectrodes” 5”PALSPlatinumovalelectrodeswhile theparticipantwasaskedtoremainpassive.Theelectricalstimulationpatternwas composedofpulseswithaconstantpulsefrequencyof30Hzandaconstantpulse amplitudeof90mA.Figure3-1illustratesanexampleofstimulationsignal;inthe experiments,allparametersremainedconstant,exceptforthepulsewidththatwas modulatedbasedon3.Asmoothperiodictrajectory,thesameasinChapter2,was selectedasthedesiredtorque. Implementationofthecontrollerrequiredtheestimationoftwoparameters.The rstparameteristhecontroleffectiveness introducedin3thatrelatestheinput voltagetothetorqueabouttheknee-joint.Asdescribedin3, isapproximatedby theconstant B ,thatisestimatedbeforeeachtorquetrackingtrialasthelinearslopeof therecruitmentcurve,asillustratedonFigure3-2.Themusclestartedtoproducedforce onlyafterthepulsewidthwentoveraminimumthreshold;thisvalue,measuredfromthe recruitmentcurve,wasaddedtothecontrolinput. ThesecondparameteristheconstantEMDusedinthecontroller.Basedon themeasurementmethoddevelopedin[73],thetimelagbetweenmuscleelectrical activationandtorqueonsetisestimatedbyperformingthecross-correlationofthese twosignals.Thecorrelationmeasurestheresemblanceoftwosignalsbyshiftingintime oneofthesignalswithrespecttotheother.Thecross-correlationismaximumwhenthe twosignalsarealigned.Thetimeshiftthatalignsthesignalsandreachesthemaximum ofcorrelationcorrespondstothedelaybetweenthem.Therefore,theEMDcanbe foundbycorrelatingtheelectricalinputsignalandthetorqueoutput.Fortheconstant EMDmeasurements,muscleswerestimulatedwithaseriesofveshortpulsesthat evokedstrongtorquesofshortduration,asshownonFigure3-3.Thestimulationsignal parameterswerethesameasforthetorquetrackingtrialswithapulsefrequencyof 30

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Figure3-1:Exampleofastimulationsignal,withthethreemainuser-denedparameters:thepulsefrequency,thepulseamplitudeandthepulsewidth. 30Hzandapulseamplitudeof90mA.Thepulsewidthwasselectedtomatchatorque outputof10 N m basedontherecruitmentcurves.Thisprotocolwaspreformedbefore eachtrialandcorrelationofthepulsewidthinputwiththemeasuredtorqueyieldeda constantdelayestimation.ThetwominutetorquetrackingtrialbeganaftertheEMDand controleffectiveness B wereestimated.Therecruitmentcurvewasmeasuredagainat theendofthesession. 3.3.2Results TheresultsshowthatincludingaconstantEMDcompensationterminthecontroller allowstheclosed-loopNMEStrackingtobestable.Thetrackingperformance,the trackingerrorandthecontrolinputforSubject4aredepictedonFigure3-4.Theoverall increaseinthepulsewidthillustratestheeffectsoffatigue:attheendofthetrial,the 31

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Figure3-2:Exampleofthreerecruitmentcurvesobtainedduringatorquetracking sessiononSubject4.Thepulseinputwasincreasedwhilethereactiontorquewas measured.Theconstantestimatedcontroleffectivenesswasdeterminedbylinearregression. Figure3-3:Thetorqueismeasuredwhilethemuscleisstimulatedwithanelectrical signalcomposedofveconsecutivepulseswithaconstantamplitude90 mA ,aconstant pulsefrequency30 Hz andapulsewidthaimedtomatch10 N m .Theplotrepresents thenormalizeddata. 32

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Table3-1:Estimationofthecontroleffectivenessasthelinearslopeoftherecruitment curve.Themeasurementtestisperformedbeforeeachtorquetrackingtrialandatthe endofthesession.Thedecreasingvaluesdemonstratetheeffectsoffatigueonthe recruitmentproperties. Subject Estimationof B N m s )]TJ/F47 7.9701 Tf 6.587 0 Td [(1 1 st trial 2 nd trialEndofsession 10.41470.32920.3317 20.66170.63970.6426 30.49960.39120.4664 40.36610.29620.2797 musclesrequiredmoreelectricalinputtoproducethesametorquethanatthebeginning ofthetrial.ResultsforthecontroleffectivenessmeasurementsarepresentedinTable 3-1wheretheeffectsoffatigueonthemuscledynamicsarevisible:thelinearslope oftherecruitmentcurvesisvaryingwithinasession.Therefore,itwasnecessaryto estimate B beforeeachtrialgivenitsgreatvariability.Thelastvalueofthecontrol effectivenessapproximationwasgreaterthanthesecondvalueinsomesessions.A possibleexplanationisthatthemusclebersrecruitedduringtherstandsecondtrials reachedaleveloffatiguethatpreventedthemtocontract.Hence,newnonfatigued berscouldhavebeenrecruitedduringthelasttrial,restoringtheoverallstateofthe muscle.AllconstantEMDestimationsandRMSerrorsareprovidedinTable3-2:the meanvaluesare111.8 ms 8.5 ms and1.659 N m 0.395 N m fortheEMDand theRMSerror,respectively.Inmostcases,theEMDvalueisgreaterinthesecondtrial. 3.4Conclusion Inthischapterthedevelopedtorquetrackingcontrollerensuresthatthereaction torquetracksadesiredtorquedespiteconstantinputdelay.Thetorquetrackingerror wasproventoexponentiallydecaytoaballinthepresenceofconstantinputdelay. Experimentsonfourhealthysubjectswereperformedtotesttheperformanceonthe controller.Thequadricepsfemorismusclegroupwasstimulatedbasedontorquefeedback,constantEMDestimate,andcontroleffectivenessapproximationinordertotrack 33

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a b c Figure3-4:Evolutionofthemeasuredtorquesolidlineduringclosed-looptorquetrackingwithconstantdelaycompensationina.Thecorrespondingtorquetrackingerroris representedinb.Theplotinccorrespondstothecontrolinputduringthetrial. 34

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Table3-2:Resultsfortorquetrackingtrialswithconstantdelaycompensation. Subject-TrialRMSerrorsconstantEMD 1-a2.377 N m 116ms 1-b2.156 N m 123ms 2-a1.315 N m 105ms 2-b1.657 N m 123ms 3-a1.466 N m 102ms 3-b1.445 N m 114ms 4-a1.530 N m 103ms 4-b1.328 N m 108ms adesiredtorque.Thetwolastparameterswereestimatedbeforeeachtrialanditwas inferredfromthemeasurementsthatfatiguehadmajoreffectsonthesystemdynamicsandespeciallythedelay,yieldingconsiderablevariationsintheparametervalues betweenthetrials.Thus,theEMDinthemusclemodelshouldalsobeconsideredas time-varyinginthemusclemodel. 35

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CHAPTER4 TORQUECONTROLWITHTIME-VARYINGEMDCOMPENSATION BasedonthefactthattheEMDdependsonthemusclestate,thecontrolmethod developedinChapter3ismodiedinthischapterforatime-varyinginputdelayed system.Inthischapter,theimplicationsofatime-varyingdelayareinvestigatedand resultinanewcontroldesignandstabilityanalysis.Newexperimentsareconducted toassessthecontrolperformance.Basedonexperimentalresults,bothconstantand time-varyingdelaycompensationstrategiesarecompared. 4.1ControlDesign Thenonlineardelayedmodelisdenedas R t = f q + D t + t V t )]TJ/F24 11.9552 Tf 11.955 0 Td [( t ; where 2 R denotesthetimevaryinginputdelayandwhere R , f , D , and V were denedin2. Assumption3. TheEMD t isboundedsuchthat 0 < t <' 1 forall t ,where ' 1 2 R + isaknownconstant.Therateofchangeofthedelayisboundedsuchthat j _ j < 1 )]TJ/F24 11.9552 Tf 12.62 0 Td [(" ,where " 2 R + satises 0 <"< 1 anditssecondtimederivativeisalso boundedsuchthat j j <' 2 ,where ' 2 2 R + isaknownconstant. Remark 2 . Assumption3ismildinthesensethatitimpliesthatthedelayisbounded andthatthechangeinthedelayisaslowprocess.Thisassumptionisdemonstrated fromthesubsequentexperimentalresults. Thecontrolobjectiveisthesameasinthepreviouschapter,exceptthatthe reactiontorque R t shouldtrackthedesiredtorque R d t despitetime-varyinginput delays.Thesystemerrordenedbythesignals r , e , e z and wasdenedinChapter3. Multiplying3by )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 andusing3and4yields )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(f )]TJ/F24 11.9552 Tf 11.955 0 Td [(D )]TJ/F24 11.9552 Tf 11.956 0 Td [(V )]TJ/F24 11.9552 Tf 11.955 0 Td [(B )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e z : 36

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Theopen-looperrorsystemcanbeobtainedbytakingthetime-derivativeof4and using3and3suchthat )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 _ r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r + N + S )]TJ/F15 11.9552 Tf 15.042 3.022 Td [(_ V )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 _ V )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ _ V ; wheretheauxiliarysignals S 2 R and N 2 R aredenedas S , d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 R d )]TJ/F24 11.9552 Tf 11.955 0 Td [(f )]TJ/F24 11.9552 Tf 11.956 0 Td [(D ; N , )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r )]TJ/F24 11.9552 Tf 11.955 0 Td [(B d dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e z : Basedon3-3,theopen-looperrorsystemin4nowcontainsadelay-free controlinput.Fromthesubsequentanalysisand4,thecontrolinputisdesignedas asolutionto _ V = k b r;V = V 0 ; where V 0 2 R isaselectableconstant,and k b 2 R isaselectableconstantcontrolgain suchthat k b , k b 1 + k b 2 + k b 3 ; where k b 1 , k b 2 and k b 3 2 R + .Substituting4into4yieldstothefollowingclosedlooperrorsystem )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 _ r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 _ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r + N + S )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b r )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ r : Theauxiliarysignal S in4canbeupperboundedbyaknownconstant " 2 2 R + accordingtoAssumptions1and2suchthat j S j " 2 ; andusingAssumption2,theexpressionin4canbeupperboundedas j N j 1 jj z jj ; 37

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where 1 2 R isapositiveknownconstant,and z 2 R 2 isdenedas z = re z T : Tofacilitatethesubsequentstabilityanalysis,let y 2 R 4 bedenedas y = re z p P p Q T ; where P , Q 2 R areLKfunctionalsdenedas P , ! t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( t s _ V 2 d ds; Q , k b )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( r 2 d; where ! and 2 2 R + areselectableconstants.Basedonthesubsequentanalysis,the constant 1 2 R + isdenedsuchthat 1 , min f m 1 ;m 2 g ; with m 1 , inf ; _ k b 3 )]TJ/F24 11.9552 Tf 13.151 8.088 Td [(k b 2 1 4 )]TJ/F24 11.9552 Tf 13.15 8.847 Td [(k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2_ + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F24 11.9552 Tf 11.955 0 Td [(k 2 b ! ; m 2 , inf ; _ 1 ! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F24 11.9552 Tf 11.955 0 Td [( k b 2 1 + 4 2 2 )]TJ/F30 11.9552 Tf 13.151 18.531 Td [()]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 ' 2 2 k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 ! ; where m 1 , m 2 2 R arepositiveconstantsand 1 2 R + isaselectableconstant. Theorem3. Giventhestaticmodelin4,thecontrollawin4ensuresglobal uniformlyultimatelyboundedtorquetrackinginthesensethat j e t p c 11 exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(c 0 t + c 12 + p c 21 exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(c 0 t + c 22 ; 38

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where c 0 , c 11 , c 12 , c 21 and c 22 2 R + denoteconstants,providedthefollowingsufcient conditionsaresatised k b 3 > sup ; _ k 2 b ! + k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2_ + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ + k b 2 1 4 ; !> sup ; _ 1 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ k b 2 1 + 4 2 2 + )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ' 2 2 k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 3 ; 1 > 2 1 4 k b 1 : Proof: Let V L : R [0; 1 ! R beacontinuouslydifferentiable,positivedenite functionalonanopenset D R ,denedas V L , 1 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r 2 + 1 2 e z + P + Q; whichcanbeboundedas 1 jj y jj 2 V L 2 jj y jj 2 ; wheretheconstants 1 , 2 2 R aredenedas 1 , 1 2 min )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 2 , max 1 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ; 1 ; where and aredenedinAssumption2.ApplyingtheLeibnizRuletodetermine thetimederivativeof4and4,andutilizing3,4and4,thetime derivativeof4canbewrittenas _ V L = )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b r 2 + Sr + Nr )]TJ/F24 11.9552 Tf 11.955 0 Td [(k b )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F24 11.9552 Tf 11.955 0 Td [(e z r )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ r + k b )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ r 2 + ! j _ V j 2 )]TJ/F24 11.9552 Tf 13.15 8.088 Td [(k b 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 r 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( V 2 d + k b )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( r 2 d: 39

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UsingYoung'sInequalityandAssumption3,thefollowinginequalitiescanbedeveloped k b j e z jj r j k b 2 1 4 r 2 + 1 2 1 e 2 z ; k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ j e z jj r j k 2 b 2 2 2 r 2 + 2 2 2 e 2 z ; k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 j r jj r j k b )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [(r 2 + r 2 : ByapplyingtheCauchy-SchwarzInequality, j e z j 2 = t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 1 d 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( j 1 j 2 d t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: ByusingAssumptions2and3,3,4,4,4and4-4–23,the expressionin4canbeupperboundedas _ V L )]TJ/F24 11.9552 Tf 23.911 0 Td [(k b r 2 + " 2 j r j + 1 jj z jjj r j + k b 2 1 4 r 2 + k b )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2_ + k b 2 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ r 2 + k 2 b !r 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d + k b 2 1 + 2 2 2 e 2 z + )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ' 2 2 k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: Using4and4,4isupperboundedandgroupedas _ V L )]TJ/F52 10.9091 Tf 21.818 0 Td [(k b 1 r 2 + 1 jj z jjj r j)]TJ/F52 10.9091 Tf 16.364 0 Td [(k b 2 r 2 + " 2 j r j)]TJ/F30 10.9091 Tf 16.364 18.655 Td [( k b 3 )]TJ/F52 10.9091 Tf 12.105 7.38 Td [(k b 2 1 4 )]TJ/F52 10.9091 Tf 12.104 8.073 Td [(k b )]TJ/F17 10.9091 Tf 5 -8.836 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F17 10.9091 Tf 10.909 0 Td [(2_ + k b 2 2 2 )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ )]TJ/F52 10.9091 Tf 10.909 0 Td [(k 2 b ! ! r 2 )]TJ/F30 10.9091 Tf 10.909 18.655 Td [( ! )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ )]TJ/F52 10.9091 Tf 10.909 0 Td [( k b 2 1 + 4 2 2 )]TJ/F30 10.9091 Tf 12.105 16.91 Td [()]TJ/F17 10.9091 Tf 5 -8.837 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ' 2 2 k b )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ 2 ! t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d )]TJ/F17 10.9091 Tf 12.105 7.38 Td [(2 2 2 t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d: Completingthesquaresandusing4,4canbeupperboundedas _ V L )]TJ/F30 10.9091 Tf 21.818 18.655 Td [( k b 3 )]TJ/F52 10.9091 Tf 12.105 7.38 Td [(k b 2 1 4 )]TJ/F52 10.9091 Tf 12.104 8.073 Td [(k b )]TJ/F17 10.9091 Tf 5 -8.836 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F17 10.9091 Tf 10.909 0 Td [(2_ + k b 2 2 2 )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ )]TJ/F52 10.9091 Tf 10.909 0 Td [(k 2 b ! ! r 2 + " 2 2 4 k b 2 + 2 1 4 k b 1 jj z jj 2 )]TJ/F17 10.9091 Tf 12.38 7.38 Td [(1 ! )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ )]TJ/F52 10.9091 Tf 10.909 0 Td [( k b 2 1 + 4 2 2 )]TJ/F30 10.9091 Tf 12.105 16.909 Td [()]TJ/F17 10.9091 Tf 5 -8.836 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ' 2 2 k b )]TJ/F17 10.9091 Tf 12.699 0 Td [(_ 2 ! e z )]TJ/F17 10.9091 Tf 12.105 7.38 Td [(2 2 2 t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d: 40

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Usingthedenitionof z in4and 1 in4,theexpressionin4canbe upperboundedas _ V L )]TJ/F30 11.9552 Tf 23.911 16.857 Td [( 1 )]TJ/F24 11.9552 Tf 16.374 8.088 Td [( 2 1 4 1 jj z jj 2 + " 2 2 4 k b 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 2 2 t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d: Afterusing4,4,4andthefollowinginequality t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( t s _ V 2 d ds t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( sup s 2 [ t )]TJ/F25 7.9701 Tf 6.586 0 Td [(;t ] t t )]TJ/F25 7.9701 Tf 6.587 0 Td [( _ V 2 d ds = sup s 2 [ t )]TJ/F25 7.9701 Tf 6.587 0 Td [(;t ] t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d = t t )]TJ/F25 7.9701 Tf 6.586 0 Td [( _ V 2 d; Theexpressionin4canbeboundedas _ V L )]TJ/F30 11.9552 Tf 23.91 16.857 Td [( 1 )]TJ/F24 11.9552 Tf 17.883 8.088 Td [( 2 1 4 k b 1 jj z jj 2 + " 2 2 4 k b 2 )]TJ/F15 11.9552 Tf 19.81 8.088 Td [(1 2 2 ! P )]TJ/F15 11.9552 Tf 29.079 8.088 Td [(2 k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 + k b 2 2 Q: Fromthedenitionof y in4,4canbeupperboundedas _ V L )]TJ/F24 11.9552 Tf 21.918 0 Td [( 2 jj y jj 2 + " 2 2 4 k b 2 ; where 2 2 R + isdenedas 2 , min 1 )]TJ/F24 11.9552 Tf 17.883 8.088 Td [( 2 1 4 k b 1 ; 1 2 2 ! ; inf ; _ 2 k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + k b 2 2 ! : Using4,theexpressionin4canbewrittenas _ V L )]TJ/F24 11.9552 Tf 23.223 8.088 Td [( 2 2 V L + " 2 2 4 k b 2 : Finally,thedifferentialequationin4canbesolvedas V L V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 4 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.956 0 Td [(t 0 : 41

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From4, V L isgloballyuniformlyultimatelybounded.Using4, r and e z arealso boundedas j r j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 ; j e z j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 : From3,4and4, j e j canbeupperboundedas j e j s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.956 0 Td [(t 0 + B s 2 V L t 0 exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 + " 2 2 2 2 k b 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 10.604 8.088 Td [( 2 2 t )]TJ/F24 11.9552 Tf 11.955 0 Td [(t 0 ; yieldingtheresultin4.Finallyfrom3and4,thecontrolinput V is bounded. 4.2Experiments Thecontrollerin4wasimplementedtoyieldclosed-loopisometriccontractions usingtorquefeedback.Theprotocolusedinthisexperimentalsessionisthesameas inChapter3,withatime-varyingdelayestimationinsteadofaconstantapproximation. Fourhealthysubjectsparticipatedinthetrialsaftergivingtheirwrittenconsent.The stimulationparameterswere30Hzfortheconstantpulsefrequency,90mAforthe constantamplitudeandthepulsewidthvariedaccordingtothecontrollawin4. Arecruitmentcurvewasmeasuredbeforeeachtrialandattheendofeachsession inordertoupdatethecontroleffectivenessestimate.Theminimalpulsewidthvalue requiredtoproducetorquewasmeasuredusingtherecruitmentcurvesandwasadded tothecontrolinput.Fifteenminutesseparatedeachtrialsthatlastedtwominutes. TheEMDwasmeasuredinreal-timebasedonthefollowingalgorithm:1theinput pulsewidthandmeasuredtorquedatawerebufferedinavectorforoneperiodof thedesiredtrajectory,3thetwovectorsarenormalized,3thecross-correlation betweenthetwonormalizedvectorsiscalculated,4theindexthatmaximizedthe 42

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cross-correlationwasconvertedintosecondsmultipliedbythesamplingtimetoobtain thetimedelaybetweenthetwosignals. Remark 3 . Anothercommonmethodtomeasurethedelaybetweentwosignalsisto useathresholdandtodenethedelayasthedifferencebetweenthetimesatwhich thesignalsgooverthethreshold.Thismethodwasrstimplementedandprovided satisfyingresults.However,thethresholdvaluedependsontheuserandonthe measurementnoise.Itrequirespre-trialteststondasatisfyingthreshold,whichleads toprematuremusclefatigue.Thecross-correlationmethodispreferredbecauseitdoes notrequireuser-dependentparameters.Giventhegreatvariabilitybetweenindividuals, theEMDwasestimatedbycross-correlatingthecontrolinputandthereactiontorque, insteadofusingthethresholdmethod. Table4-1.aprovidesthecontroleffectivenessmeasurementsatthebeginningand attheendofeachsessionandtheRMSerrorsforeachtrials,whereasnotablevalues fortheEMDestimatesareprovidedinTable4-1.b.Asexpected, B wasvaryingbetween thetrials,mostlydecreasing.Thefactthat B couldincreaseonthelastmeasurement wasdiscussedinthepreviouschapter.ThetrackingperformanceforatrialwithSubject 4wheretheEMDwasestimatedinrealtimeisdepictedonFigure4-1.Amongalltrials, themeanRMSerroris1.379 N m 0.275 N m andthemeanEMDis94.5 ms 10.7ms.Thislastvalueiswithintherangeofotherresultsintheliterature,forexample in[73],whereVos etal. foundanEMDof86 ms inthevastuslateralis.Figure4-3 representsthevariationsoftheEMDforalltrials.Inmostcases,theEMDdecreased betweentherstandthesecondtrial.Itissuggestedthatfatiguepreventsmusclebers fromproducingsupplementarytorqueafteracertainstateofmusclefatigue;thus,new bersarerecruited,decreasingtheoverallmusclefatigue.Onceallbersarefatigued, allEMDcurvesareincreasing. 43

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a b c Figure4-1:Themeasuredtorque,representedbyasolidlineina,tracksadesired torquewhenthecontrolinputinccompensatesforthetime-varyingdelay.Inb,the torquetrackingerrorisstable. 44

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Figure4-2:Thedelayestimation,averagedwithamovingwindowof30secondsoff-line solidline,isrepresentedwiththeonlineEMDmeasurementsdashedlineusedinthe controlinput. Figure4-3:TemporalEMDevoluationbasedontherealtimeestimatesamongalltrials. 45

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Table4-1:Resultsfortorquetrackingtrialswithtime-varyingdelaycompensation.The variationoftheEMDforonetrialisobtainedbysubtractingtheminimumestimated valuetothemaximumvalueanddividingbytheminimumvalue. Subject-Trial B estimationRMSerror 1-a0.3732 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.810 N m 1-b0.2529 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.694 N m 2-a0.7353 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.448 N m 2-b0.6251 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.324 N m 3-a0.4948 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.439 N m 3-b0.4313 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.039 N m 4-a0.3191 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.167 N m 4-b0.2376 N m s )]TJ/F47 7.9701 Tf 6.586 0 Td [(1 1.110 N m a Subject-Trial EMDestimation MeanvalueStartvalueEndvalueMinvalueMaxvalueVariation 1-a101.0ms88.3ms105.8ms80.3ms115.3ms45% 1-b102.0ms98.7ms131.6ms87.1ms130.9ms50% 2-a99.5ms83.0ms111.3ms80.5ms116.6ms45% 2-b93.7ms93.9ms97.4ms79.9ms103.8ms30% 3-a82.4ms80.9ms75.5ms74.5ms091.4ms23% 3-b77.7ms82.7ms85.1ms68.9ms085.1ms24% 4-a90.2ms63.7ms124.6ms58.4ms123.9ms112% 4-b116.2ms94.5ms141.6ms88.8ms145.1ms63% b 46

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Figure4-4:RMSerrors,computedusinga16-smovingwindowtoaveragetheraw trackingerrors:comparisonbetweenconstantandtime-varyingdelayestimation.The dashedlinescorrespondtothestandarddeviations. 4.3ComparisonwithConstantDelayCompensation Torquetrackingtrialswereperformedwithtwodifferentcontrolmethods:eighttrials withconstantdelaycompensation,eightwithtime-varyingdelaycompensation.The resultsarenowcomparedtohighlighttheimprovedperformancewhentheinputdelay isconsideredtobetime-varying.Table4-2providestheRMSerrorsforbothprotocols. Foreachtrial,atime-varyingRMSerrorwascomputedbyusinga16-smovingwindow toaveragerawtrackingerrors.Time-varyingRMSerrorswerethenaveragedacrossall subjects:Figure4-4representstheaveragedRMSforconstantandtime-varyingdelay compensation.Thecurveforthetime-varyingdelaycompensationtendtobebelowthe RMSforconstantEMDcompensation.Basedontheseresults,thetime-varyingEMD compensationyieldedabettertrackingthantheconstantdelaycompensationmethod. 4.4Conclusion TorquetrackingwasachievedinisometricNMESexperiementsonhealthyindividualsusingatime-varyingdelayestimationtocompensatefortheEMDinthemuscle dynamics.Theperformanceswerestabledespitethetime-varyinginputdelayand 47

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Table4-2:RMSerrorcomparisonbetweenconstantandtime-varyingEMDestimation kkkkkkkkk Subject-Trial RMSerror constantdelay estimation RMSerror time-varyingdelay estimation 1-a2.377 N m 1.810 N m 1-b2.156 N m 1.694 N m 2-a1.315 N m 1.448 N m 2-b1.657 N m 1.324 N m 3-a1.466 N m 1.439 N m 3-b1.445 N m 1.039 N m 4-a1.530 N m 1.167 N m 4-b1.328 N m 1.110 N m unknowndynamics.Thecomparisonbetweenconstantandtime-varyingdelaycompensationmethodssuggeststhattime-varyingEMDcompensationyieldsbetterresults. However,moretrialsarerequiredtoprovethisstatement.Inthestabilityanalysis,the EMDwasassumedtobeknownandcouldbeusedinthecontrollaw.Atthetimeof thedevelopmentofthework,noEMDmodelwasavailableyet;thedelayhadtobe estimated.Therefore,thetrackingerrorbetweenthedesiredandthereactiontorque dependsalsoonthequalityoftheEMDestimation. 48

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CHAPTER5 CONCLUSION 5.1Achievements Thisthesisaimedtodesignandtestisometrictorquetrackingcontrollersfor electrically-evokedcontractionsofthequadricepsfemorismusclegroup.Adapting thepositiontrackingcontrolstrategytotheisometricsystem,theworkinChapter2 helpedtounderstandtheissuesoftorquetrackingcontrolandemphasizedtheneed ofconsiderationfortheEMD.Atorquetrackingcontrollerwasdesignedforisometric FESsystemswithoutinputdelaycompensationandfailedtostabilizethesysteminthe presenceofinputdelays. InChapters3and4,acontrollerwasdevelopedtocompensateforinputdelays: constantandtime-varyingEMDcompensationwerestudiedseparately.Controllers withconstantortime-varyingEMDcompensationensuredthetheoreticalstabilityofthe torquetrackingerror,whichwasillustratedinexperiments.Bothstrategiesyieldedgood trackingresults.However,consideringtheEMDastime-varyinginthemusclemodel providesbettertorquecontrolasillustratedinChapter4. 5.2FutureWork OutcomesofthisthesisprovedthatNMES-inducedisometrictorquetrackingcanbe achieveddespiteuncertaintiesandtime-varyingEMD,buttheresultscouldbeimproved. Moretrialswouldcontributetoprovethattime-varyingEMDcompensationintorque trackingleadstobetterresultsthanconstantdelaycompensation.BecausetheEMD wasassumedtobeknowninthecontroldevelopment,controlimplementationrequired EMDestimation.Therearetwomethodstocomputethedelayinthecontroller:byusing amodel,whichstillneedstobedeveloped,orbymeasuringtheEMDinrealtime.The resultsobtaineddependedonthequalityoftheEMDestimationmethod.Futureefforts shouldconsiderunknowndelaysinthesystem. 49

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BIOGRAPHICALSKETCH ManelleMeradwasborninFranceandreceivedherscienticFrenchBaccalaur eat in2009.WithastronginterestforMathematicsandPhysics,shepursuedtwoyearsof PreparatoryClassesbeforeEntrancetoFrenchEngineeringSchools,whichledherto jointherstclassofT el ecomPhysiqueStrasbourginInformationandCommunication TechnologyappliedtoHealthin2011.StudyingintheMasterImagerie,Robotiqueet Ing enieriepourleVivantImaging,RoboticsandEngineeringappliedtoHealthsince 2012,shetooktheopportunitytopursueintheAtlantisCRISPDualDegreeprogramat theUniversityofFlorida.Foroneyear,shehasbeenstudyinganddoingresearchon neuromuscularelectricalstimulationandnonlinearcontrolinthelaboratoryofPr.Dixon. 57