Citation
Non-Isometric Neuromuscular Electrical Stimulation via Non-Model Based Nonlinear Control Methods

Material Information

Title:
Non-Isometric Neuromuscular Electrical Stimulation via Non-Model Based Nonlinear Control Methods
Creator:
Stegath, Keith
Publisher:
University of Florida
Publication Date:
Language:
English

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 Members:
Banks, Scott A.
Fregly, Benjamin J.
Graduation Date:
12/14/2007

Subjects

Subjects / Keywords:
Electric potential ( jstor )
Electric stimulation therapy ( jstor )
Electrical stimulation ( jstor )
Experimental results ( jstor )
Extrema ( jstor )
Knee joint ( jstor )
Knees ( jstor )
Legs ( jstor )
Pulse duration ( jstor )
Signals ( jstor )
controller, cord, electrical, fes, injruy, neuromuscular, nmes, nonlinear, numeric, spinal, stimulation, stroke

Notes

General Note:
For people afflicted with neuromuscular disorders such as stroke or spinal cord injuries, there has been limited success by engineers and biological researchers to artificially control the afflicted person's muscles with neuromuscular electrical stimulation (NMES). NMES is the application of an electrical current via internal or external electrodes which results in a muscle contraction. NMES is currently prescribed to treat muscle atrophy and impaired motor control associated with orthopedic and neurological damage, circulatory impairments, joint motion dysfunction, postural disorders, swelling and inflammatory reactions, slow-to-heal wounds and ulcers, and incontinence. The development of NMES as a neuroprosthesis has grown rapidly because of the potential improvement in the activities of daily living for individuals with movement disorders such as stroke and spinal cord injuries. Researchers have mainly focused on two methods to deliver the electrical signal to the muscle, direct nerve stimulation with electrodes inserted through the skin or implanted under the skin, and surface stimulation where adhesive-backed electrodes are placed on the skin. While each method has its advantages and disadvantages, a barrier to both methods that limits further application of NMES is that an unknown mapping exists between the stimulation parameters (e.g., voltage, frequency, and pulse width) to muscle force production. In one situation, different stimulation parameters will yield significantly different contraction forces, while in another situation an infinite combination of the parameters can yield the same contraction force. In addition to the variability in stimulation parameters, the muscle contraction force is difficult to predict due to a variety of uncertainties related to muscle physiology (e.g., architecture, temperature, and pH) and the ability to consistently deliver the stimulation (e.g., electrode placement, resistance variations due to subcutaneous fat). The practical limitations imposed for some NMES applications due to the uncertain relationship between stimulation parameters and the force produced by the muscle provided the motivation to explore methods that can compensate for these uncertainties.

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Source Institution:
UFRGP
Rights Management:
Copyright Stegath, Keith. 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.
Embargo Date:
12/31/2009

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NON-ISOMETRICNEUROMUSCULARELECTRICALSTIMULATIONVIA NON-MODELBASEDNONLINEARCONTROLMETHODS By KEITHSTEGATH ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2007

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c 2007KeithStegath

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Tomywifewhosesupportenabledmetoreturntocollege,myfatherwhose integrityneverfaltered,andmymotherwhoseshortlifewas lledwithloveand compassion.

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ACKNOWLEDGMENTS Mygratitudegoestothenumerouspe oplewhoallowedmethefreedomto improveexistingskillsandguidanceindevelopingnewones. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFTABLES........... ......................vii LISTOFFIGURES.... ............................viii ABSTRACT.............. ......................x CHAPTER 1INTRODUCTION.. ............................1 2EXTREMUMSEEKINGCONTROLSCHEME.............8 2.1ControlObjective............................10 2.2ExtremumGeneration.........................10 2.3ExperimentalResults..........................13 2.3.1ExperimentalTestbed......................13 2.3.2ExperimentalSetup.......................15 2.3.3OptimalVoltageSeekingResults................15 2.3.4OptimalFrequencySeekingResults..............18 2.4Discussion.... ............................22 2.5ConcludingRemarks..........................22 3NONLINEARCONTROLSCHEME....................24 3.1RobustIntegralSignoftheError...................24 3.2MuscleActivationandLimbModel..................26 3.3ControlDevelopment..........................28 3.4ExperimentalResults..........................31 3.4.1ExperimentalSetup.......................32 3.4.2RegulationResults.......................32 3.4.3TrackingResults.........................34 3.5Discussion.... ............................37 3.6ConcludingRemarks..........................40 4CONCLUSIONSANDRECOMMENDATIONS..............41 APPENDIX ELECTRICALDESIGNANDINTERFACIN G ............ 4 3 v

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A.1CircuitDesignfor100 secWidePulse................43 A.2CircuitDescription...........................43 A.2.1HighVoltagePowerOp-AmpPA866EU............45 A.2.2VoltagetoFrequencyConversion-VFC32..........47 A.2.3Transistors2N3565.......................48 A.2.4VoltageRegulator7805.....................49 A.2.5CircuitBehavior.........................49 A.3Interfacingthe 100 secWidePulsePCBtotheComputer.....49 A.4CircuitDesignforMultiplesof 25 secWidePulse.........50 A.4.1CircuitDescription.......................50 A.5IsometricAttachment..........................53 REFERENCES....... ............................55 BIOGRAPHICALSKETCH............................59 vi

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LISTOFTABLES Table page 21Determiningwhethertochangetheupperorlowerbound.........13 22Knee-jointanglecontrolledbyvoltage...................17 23RMSerrorandsteady-stateerrorforBrentsMethodusingVM.....19 24Knee-jointanglecontrolledbyfrequency..................21 25RMSerrorandsteady-stateerrorforBrentsMethodusingFM.....22 31RMSandsteady-stateerrorforRISEregulationexperiments.......36 32RMSandsteady-stateerrorforRISEtrackingexperiments........39 A1Inputvoltagestocircuit...........................45 A2Partslistforstimulatorcircuit........................48 A3Digitalinputstothecircuit.........................52 A4Thee ectoftherelayonthecorrespondingpulsewidth.........52 A5Partslistforpulsewidthcontroller.....................53 vii

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LISTOFFIGURES Figure page 11Leganglegeneratedwithaconstantvoltageforonesecondwitha100 swidepulsedelivedat20Hz.Stimulationvoltageswere25,30,35,and 40volts............... ......................3 12Leganglegeneratedwithaconstantfrequencyforonesecondwitha100 swidepulsedeliveredat40volts.Thestimulationfrequencieswere1, 5,10,and20Hz.... ............................4 21Legcurlandextensionmachineaftermodi cations.............14 22Onlinecomputedvoltage(longdashed),desiredlegangle(shortdashed), andactuallegangle(solid)..........................17 23BrentsMethodovershootingthedesiredanglebeforeconvergingtowards thesolutionusingVM.............................18 24Onlinecomputedfrequency(boldsolid),desiredlegangle(shortdashed), andactuallegangle(solid)..........................20 25BrentsMethodconvergingonthesolutionusingFM............21 3Knee-jointanglede nedby .........................27 32Typicalmuscleexcursionofthetestsubjectsusedfortheregulationand trackingexperimen ts.................. ............33 33RegulationofkneejointangleusingtheRISEcontroller..........34 34RegulationvoltageusingtheRISEcontroller................35 3Regulationerrorofkneejointangle(desiredangleminusactualangle)..35 3Desiredtrackingpro leextendedto20seconds...............37 37KneejointtrackingusingtheRISEcontroller................38 38TrackingvoltageusingtheRISEcontroller.................38 3Trackingerrorofkneejointangle(d esiredangleminusactualangle)...39 A1Schematicofcircuitryusedtodeliverthecomputedstimulationpulse train................. ......................44 viii

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APCBlayoutofcircuitryusedtodeliverthecomputedpulsetrain.....45 ACircuitboardusedtogenerateandamplifya100 secpulse........46 A4Shapeofstimulationpulse...........................47 A5Circuitforadjustingthepulsewidthwithstepsof 25 sec .........51 A6CircuitdiagramforS-beamloadcell.....................54 ix

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul llmentofthe RequirementsfortheDegreeofMasterofScience NON-ISOMETRICNEUROMUSCULARELECTRICALSTIMULATIONVIA NON-MODELBASEDNONLINEARCONTROLMETHODS By KeithStegath December2007 Chair:WarrenE.Dixon Major:MechanicalEngineering Forpeoplea ictedwithneuromusculardisorderssuchasstrokeorspinalcord injuries,therehasbeenlimitedsuccessbyengineersandbiologicalresearchers toarti ciallycontrolthea ictedpersonsmuscleswithneuromuscularelectrical stimulation(NMES).NMESistheapplicationofanelectricalcurrentviainternal orexternalelectrodeswhichresultsina musclecontraction.NMESiscurrently prescribedtotreatmuscleatrophyandimpairedmotorcontrolassociatedwith orthopedicandneurologicaldamage,circulatoryimpairments,jointmotiondysfunction,posturaldisorders,swellingandin ammatoryreactions,slow-to-heal woundsandulcers,andincontinence.ThedevelopmentofNMESasaneuroprosthesishasgrownrapidlybecauseofthepotentialimprovementintheactivitiesof dailylivingforindividualswithmovementdisorderssuchasstrokeandspinalcord injuries. Researchershavemainlyfocusedontwomethodstodelivertheelectrical signaltothemuscle,directnervestimulationwithelectrodesinsertedthroughthe skinorimplantedundertheskin,andsurfacestimulationwhereadhesive-backed electrodesareplacedontheskin.Whileeachmethodhasitsadvantagesand x

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disadvantages,abarriertobothmethodsthatlimitsfurtherapplicationofNMES isthatanunknownmappingexistsbetw eenthestimulationparameters(e.g., voltage,frequency,andpulsewidth)tomuscleforceproduction.Inonesituation, di erentstimulationparameterswillyieldsigni cantlydi erentcontraction forces,whileinanothersituationanin nitecombinationoftheparameterscan yieldthesamecontractionforce.Inadditiontothevariabilityinstimulation parameters,themusclecontractionforceisdi culttopredictduetoavarietyof uncertaintiesrelatedtomusclephysiology(e.g.,architecture,temperature,andpH) andtheabilitytoconsistentlydeliverthestimulation(e.g.,electrodeplacement, resistancevariationsduetosubcutaneous fat).Thepracticallimitationsimposed forsomeNMESapplicationsduetotheuncertainrelationshipbetweenstimulation parametersandtheforceproducedbyt hemuscleprovidedthemotivationto exploremethodsthatcancompens atefortheseuncertainties. xi

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CHAPTER1 INTRODUCTION Neuromuscularelectricalstimulation(NMES)istheapplicationofapotential eldacrossamuscleviainternallyorex ternallyplacedelectrodesinorderto produceadesiredmusclecontraction.NMESisaprescribedtreatmentfora numberofneurologicaldysfunctions.Becauseofthepotentialforimprovements indailyactivitiesbypeoplewithmovementdisorderssuchasstrokeandspinal cordinjuries,thedevelopmentofNMESasa neuroprosthesishasgrownrapidly[1]. However,theapplicationandgrowthofNMEStechnologieshavebeenstymied byseveraltechnicalchallengesrelatedtothedesignofanautomaticstimulation strategy.Speci cally,duetoavarietyofuncertaintiesinmusclephysiology(e.g., temperature,pH,andarchitecture),predictingtheexactcontractionforceexerted bythemuscleisdi cult.Onecauseofthisdi cultyisthatthereisanunknown mappingbetweenthegeneratedmuscleforceandstimulationparameters.There areadditionalproblemswithdeliveringconsistentstimulationenergytothemuscle duetoelectrodeplacement,percentageofsubcutaneousbodyfat,musclefatigue,as wellasoverallbodyhydration.Therearealsotimedelaysbetweenthedeliveryof thestimulationsignalandthecontractionofthemuscle. Giventheuncertaintiesinthestructureofthemusclemodelandtheparametricuncertaintyforspeci cmuscles,someinvestigatorshaveexploredvariouslinear PID-basedpurefeedbackmethods[2].Typically,theseapproacheshaveonly beenempiricallyinvestigatedandnoanalyticalstabilityanalysishasbeendevelopedthatprovidesanindicationoftheperformance,robustnessorstabilityofthese controlmethods.Somerecentstudies[7]pointtoevidencethatsuggestslinearcontrolmethodsdonotyieldacceptableperformance.Thedevelopmentofastability 1

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2 analysisforpreviousPID-basedNMEScontrollershasbeenevasivebecauseofthe factthatthegoverningequationsformusclecontraction/limbmotionarenonlinear withunstructureduncertainties.Somee ortshavefocusedonanalyticalcontrol developmentforlinearcontrollers[5,8,9];however,thegoverningequationsare typicallylinearizedtoaccommodateagainschedulingorlinearoptimalcontroller approach. MotivatedbythelackofcontroldevelopmentforPID-basedfeedbackmethods,signi cantresearche ortshavefocusedontheuseofneuralnetwork-based controllers[10].Nonlinearneuralnetworkmethodsprovidedaframeworkthat allowedtheperformance,robustness,andstabilityofthedevelopedNMEScontrollerstobeinvestigatedwithoutlinearizationassumptions.However,allprevious neuralnetwork-basedNMEScontrollersarelimitedtoauniformlyultimately boundedresultbecauseoftheinevitableresidualnonlinearfunctionapproximation error.Additionally,neuralnetworksmay exhibitperformancedegradationduring thetransientphasewhiletheestimatesupdate. ThemajorbarrierforfurtherapplicationofNMESisthatanunknown mappingexistsbetweenthestimulationparameters(e.g.,amplitude,frequency, pulseshape,pulsewidth,pulsetrain)andthemuscleforceproduction.Thisissue occursforbothisometricandnon-isometricmusclecontraction.Inadditionto anin nitecombinationofstimulationparametersthatyieldequivalentmuscle contractionforces,modulatingdi erentstimulationparametersyieldsdi erent contractionforcepro les(Fig.1and1).These guresarerepresentativeofa non-isometricNMESexperimentalcomparisonperformedwithapulsetrainofonesecondappliedtothequadriceps.Fig.11showstheknee-jointanglegenerated withdi erentvoltagesusingastimulationfrequencyof 20 Hzwitha100 swide pulse,whileFig.1measurestheknee-jointanglegeneratedwithconstantvoltage andpulsewidth(40voltsand100 srespectively)butvaryingthefrequency.

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3 Figure1:Leganglegeneratedwithaconstantvoltageforonesecondwitha100 swidepulsedelivedat20Hz.Stimulationvoltageswere25,30,35,and40volts.

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4 Figure1:Leganglegeneratedwithaconstantfrequencyforonesecondwitha 100 swidepulsedeliveredat40volts.Thestimulationfrequencieswere1,5,10, and20Hz.

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5 Toaddresstheuncertaintiesofnon-model-basedNMES,twocategoriesof nonlinearcontrollerswereimplemented;anextremumseekingnumericalmethod, andarobustmethod. Extremumseekingisanalternativenon-model-basedmethodthathasbeen appliedtoavarietyofengineeringsystemsbeginningatleast vedecadesago [19]thathasexperiencedaresurgenceoverthepasttwodecades[24].The returntopopularityofextremumseekin gmethodsisbasedonthesimplicityof approach(e.g.,comparedtoimplementinganeuralnetwork)andthenon-modelbasedcharacteristic.Extremumseekingisanattractiveapproachforapplications wheretheresponseofthesystemisdictatedbyanonlinearbehaviorthatis di culttomodelandhasalocalminimumormaximum[24]. Thisthesisexplorestheuseofanextremumseekingmethodtodetermine NMESparametersforsetpointregulationofahumanknee/lowerlimb(themethod couldbeappliedtoothermusclegroupswithoutlossofgenerality).Speci cally, anerrorsignalisde nedbetweentheactualangleofthekneeandsomeknown desiredangle.Theangularpositionoftheknee/lowerlimbisrelatedtosome setofstimulationparametersthroughsomeunknownmapping(i.e.,amuscle model).Ifmultipleparameterswerevariedsimultaneously,therewouldexist anin nitecombinationofstimulationparametersyieldingthesameangular positionofthelimb.Determiningwhich parametersshouldbe varied(e.g.,some parametervariationshaveadditionalbene tssuchasreducedfatigue[25,26]) isatopicofon-goingresearch.Theappr oachhereistosetallthestimulation parameterstoaconstantexceptonesothatauniquerelationshipexistsbetween thesingleparameterandtheknee/limbposition.Speci cally,e ortsfocuson usinganextremumseekingalgorithmtodeterminetheoptimalvoltage(amplitude modulation)oroptimalfrequency(freq uencymodulation)toyieldadesired knee/limbposition.Othermodulationschemessuchaspulsewidthmodulation

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6 couldalsobeexploredusingtheexactsameapproach.Anadvantageofthe developedapproachisthatonlytheupperandlowerboundsonthevoltage andfrequencyarerequired(i.e.,amusclemodelisnotrequired).Experimental resultsareprovidedthatindicatedthedesiredknee/limbpositionisobtained within 0 7oferrorforbothfrequencyandvoltagemodulation.Theexperiments weredevelopedasaself-test.Thatis,givenasetofstimulationparametersthe extremumseekingalgorithmwoulddetermineanoptimalvoltageamplitude.The computedvoltageamplitudewasthenusedwiththesamestimulationparameters inasecondexperimentwheretheextremumseekingalgorithmdeterminedthe correspondingfrequency.Theexperimentshowedthatthecomputedfrequency matchedthepresetfrequencyofthe rstexperimentwithinonehertz Recently,anewcontinuousfeedbackmethod(coinedRISEforRobustIntegral oftheSignoftheErrorin[27,28])hasbeendevelopedthatwasproventoyield asymptotictrackingofnonlinearsystemswithunstructureduncertaintyand boundedadditivedisturbances.ThecontributionhereistoillustratehowtheRISE controllercanbeappliedforNMESsystems.ImplementingtheRISEmethod requireddeveloping,andthenrewriting,amusclemodelinaformthatadheres topreviousRISE-basedLyapunovstabilityanalyses.Theperformanceofthe nonlinearcontrollerisexperimentallyveri edforboththetrackingandregulation ofahumanshank/footcomplexbyapplyingNMESacrossexternalelectrodes attachedtothedistal-medialandproximal-lateralportionofthequadricepsfemoris musclegroup.TheRISEcontrollerusedavoltagemodulationschemewitha xed frequencyanda xedpulsewidth.Othermodulationstrategies(e.g.,frequencyor pulse-widthmodulation)couldhavealsobeenimplemented(andappliedtoother skeletalmusclegroups)withoutlossofgenerality.Fortheseinitialresults,the regulationexperimentindicatesthatthedesiredknee-jointanglecanberegulated

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7 within 0 5oferror,andthetrackingexperimentcanbecontrolledwithin 3 5of steady-stateerror. Inadditiontodevelopingtwononlinearcontrolschemes,theelectronic circuitryusedforgeneratingtheelectricalimpulsedeliveredtothemusclewasalso developed.Arequirementforthecircui trywasthatitneededtodeliverapulse trainwithapositivesquarepulsewhosewidthwasbetween 100 675 sec s.It musthaveafrequencyrangebetween 10 1000 Hzandanamplitudebetween 1 150 volts.Anadditionalrequirementwasthatthefrequencyandvoltagemust beabletorespondtochangesinthecomputedvaluesatthecontrol-samplingrate (1000Hz).Thecircuitrydescribedintheappendixalsoincorporatestheinterfacing ofaloadcellforisometricexperiments.Allexperimentswerenon-isometricanduse apulsetrainwitha100 sec positivesquarepulseandthenmodulateeitherthe frequencyorthevoltage.

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CHAPTER2 EXTREMUMSEEKINGCONTROLSCHEME Anoptimalextremumseekingapproachis developedinthischaptertoidentify frequencyandvoltagemodulationparametersforaneuromuscularelectrical stimulationcontrolobjective.Thecontrolobjectiveistoexternallyapplyoptimally variedvoltageorfrequencymodulationparameterstoahumanquadricepsmuscle togenerateadesiredknee-jointangle.Experimentalresultsareprovidedto illustratethelimbpositioningperformanceofareal-timeextremumseekingroutine (i.e.,BrentsMethod). ThefocusofthisresearchwastodeterminetheNMESparametersfora regulationexperimentofahumanknee-joint(themethodcouldbeappliedtoother musclegroupswithoutlossofgenerality).Speci cally,anerrorsignalwasde ned betweentheactualangleofthekneeandsomeknowndesiredangle.Theangular positionoftheknee-jointisrelatedtosomesetofstimulationparametersthrough someunknownmapping(i.e.,amusclemodel).Acombinationofstimulation parametersyieldsthesameangularpositionofthelimbifmultipleparametersare variedsimultaneously.Whichparametersshouldbevaried(e.g.,someparameter variationshaveadditionalbene tssuchasreducedfatigue[25,26])isatopic ofon-goingresearch.Theapproachpresentedfocusesonsettingallbutone stimulationparameterconstantsothatauniquerelationshipexistsbetweenthe singleparameterandtheknee-jointangle. Theextremumseekingmethodisbasedonaniterativenumericalmethod. Thecontrollerregulatesthetestsubjectsquadricepsmuscleviafrequencyor voltagemodulationinordertomovetheknee-jointto 45.Extremumseekingis 8

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9 analternativenon-model-basedmethodthathasbeenappliedtoavarietyofengineeringsystemsbeginningatleast vedecadesago[19]thathasexperienced resurgenceoverthepasttwodecades[24].Thereturntopopularityofextremum seekingmethodsisbasedonthesimplicityoftheapproach(e.g.,comparedto implementinganeuralnetwork)anditsnon-model-basedcharacteristic.Itisalso anattractiveapproachforapplicationswheretheresponseofthesystemisdictated byanonlinearbehaviorthatisdi culttomodelandwherethenonlinearityhasa localminimumoramaximum[24].Thecontrolmethodisusedtodeterminethe optimalvoltageorfrequencyneededtogen eratethedesiredknee-jointangle.Other modulationschemessuchaspulsewidthmodulationcouldalsobeexploredwith theexactsameapproach. Threeextremumsearchalgorithmswereinvestigatedascandidatesforcontrol ofNMES;BrentsMethod[29],aDownhillSimplexMethod[29],andKrstics PerturbationMethod[30].Thedecisionforthecontrolmethodwasbasedon ve criteria: Itmustnotrequireaplant(amusclemodel). Itdoesnotrequiretuningofcontrolgains. Itmustquicklyconvergeonasolution. Itmustberobustinthesensethatitsperformanceisindependentofthe patientsmuscledynamics. Itmustallowforatleastoneindependentvariable. Usingtheabovecriteria,theSimplexmethodwasnotchosenasitismore appropriateforsystemswithmultipleindependentvariables.ThePerturbation Methodwasnotchosen(eventhoughithasprovenstabilityresults)becauseitis veryslowtoconvergeonasolution.Brentsiterativenumericalmethodwaschosen becauseitadheredtoalloftheabovecriteria.

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10 2.1ControlObjective Theobjectiveofthecontrolleristoregulatetheangleofapersonsknee-joint throughNMESofthequadricepsmuscleundergoingnon-isometriccontractions. Toquantifythisobjective,anangularkneepositionerror,denotedby ( ) R ,is de nedas ( )= ( ) ( ) (21) where ( ) R denotesaconstantknowndesiredangularkneeposition.Topositiontheknee(andhencelimb)atthedesiredanglerequiresauniquecontraction forcebeelicitedbyacombinationofstimulationparameters.Foranamplitude modulationstrategy,thestimulationfrequencyisheldconstantandthevoltageamplitudeisvaried.Therefore,achallengeinachievingtheobjectivein(21)isthat adesiredvoltagemustbedeterminedthatensures ( ) where R isan unknownpositiveconstantrepresentingtheunknowndesiredvoltagecorresponding tothedesiredknee-jointangle ( ) .Thesubsequentdevelopmentisnotbasedon anassumedmusclemodel,butarequirementforextremumseekingmethodsisthat auniquevoltage(orfrequency)existsthatwillminimizetheregulationerror ( ) (i.e.,thecontractionforceofthemuscleisnotsaturated).Thefollowingdevelopmentisprovidedforamplitudemodulationwithoutlossofgenerality.Frequency andamplitudemodulationmethodsarepresentedintheexperimentalresultsin Section2.3. 2.2ExtremumGeneration Severalextremumsearchalgorithms(e.g.BrentsMethod[29],aDownhill SimplexMethod[29],andKrsticsPerturbationMethod[30],etc.).canbeutilized toshowthatif ( ) ,thentheangularkneepositionerrorisminimized.For example,BrentsMethodonlyrequiresmeasurementoftheoutputfunction(i.e., ( ) in(2))andtwoinitialguessesthatenclosetheunknownvaluefor (the twoinitialguessesarenotrequiredtobeclosetothevalueof ).BrentsMethod

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11 thenusesaninverseparabolicinterpolationalgorithmandmeasurementsof ( ) togenerateestimatesfor untiltheestimatesconverge.Speci cally,anobjective function,denotedby ( ( )) R ,isde nedas ( ) 1 2 ( )( ) (22) Theobjectivefunctionin(2)hasauniqueminimumat ( )= ( ) .The unknownmapping 1( ): R R betweentheappliedvoltageandresultinglimb positioncanbeusedtorewrite(2)as ( )= 1 2 ( 1( ) 1( ))( 1( ) 1( )) (23) Undertheassumptionthat 1( ) ismonotonic,auniqueminimumat ( )= correspondstoauniqueminimumat ( )= ( ) .Avarietyofstandard optimizationroutines(e.g.,FMINUNCfromtheMATLABoptimization)could potentiallybeutilizedtolocatetheminimumof ( ( )).However,because ( ) cannotbedirectlymanipulatedandbecausethelimbhasassociateddynamics,a delayfunctionmustbeincludedintheoptimizationroutine.Speci cally,oncethe optimizationroutinegeneratesanewvoltage ( ) ,theroutinemustpauseuntil thedynamicsreachsteady-stateatwhichpointtheresultingknee-jointangleis evaluated.Inthefollowingexperimentalresults,theoptimizationroutineincluded adelaythatwasexperimentallydeterminedtobesu cientforthelimbdynamics toreachsteady-state.Moresophisticatedmethodssuchasaslidingwindowcould alsobeexplored. Thenumerically-basedextremumgenerationformulaforcomputingthe optimalvoltageamplitude(foragivenfrequency,pulsewidth,andwaveform)that minimizestheangularkneepositionerrorcanbedescribedasfollows[31]. Step1.Threeinitialbest-guessestimates,denotedby 1, 2, 3 R ,are selectedwhere 1isthebest-guessestimateforalowerboundontheoptimal

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12 voltage, 3isthebest-guessestimateforanupperboundontheoptimal voltage,and 2isthebest-guessestimatefortheoptimalvoltage,where 2 ( 13) .Themuscleisstimulatedwith ( )= 2.Intheexperimental resultspresentedinthispaper,apositivesquarewavewitha100 sec pulse widthwasappliedfor vesecondsatapresetfrequency(i.e., 20 Hz). Step2.Thealgorithmwaitsforthelimbdynamicstoreachsteady-state. Step3.Thenextvoltageamplitudeisdeterminedfromthefollowingexpression 4= 2 1 2 1 2(24) where 1, 2 R areconstantsde nedas 1=( 2 1)2[ ( 2) ( 3)] (25) ( 2 3)2[ ( 2) ( 1)] 2=( 2 1)[ ( 2) ( 3)] (26) ( 2 3)[ ( 2) ( 1)] where =1 2 3 aredeterminedfromthe rsttwosteps.Speci cally, and ( ) aresubstitutedinto(2)-(2)andtheresultingexpressionyields thenextbest-guessfor denotedby 4 R .Themuscleisstimulatedwith ( )= 4. Step4.Thealgorithmwaitsforthelimbdynamicstoreachsteady-state. Step5.Theresultingsteady-statelimbpositioncorrespondingto ( )= 4(denotedby ( 4) )iscomparedtotheresultinglimbpositioncorresponding to ( )= 2(denotedby ( 2) ).BasedontheconditionsshowninTable 21thestimulationboundsaremodi ed.If ( 4) ( 2) and 24orif ( 2) ( 4) and 42,thenthethreenewestimatesusedtoconstructa newparabolaare 2, 3, 4.If ( 4) ( 2) and 42orif ( 2) ( 4)

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13 Table21:Determiningwhethertochangetheupperorlowerbound. Condition1:LowerBoundtooLow CurrentError PreviousError AND PreviousVoltage CurrentVoltage OR PreviousError CurrentError AND CurrentVoltage PreviousVoltage Condition2:UpperBoundtooHigh CurrentError PreviousError AND CurrentVoltage PreviousVoltage OR PreviousError CurrentError AND PreviousVoltage CurrentVoltage and 24,thenthethreenewestimatesusedtoconstructanewparabola are 1, 2, 4. Step6.RepeatSteps3-5forsuccessive =5 6 ,wherethethree estimatesdeterminedfromStep5areusedtoconstructanewparabola. Steps3-5arerepeateduntilthedi erencebetweenthenewupperandlower estimatesisbelowsomeprede ned,arbitrarilysmallthreshold. 2.3ExperimentalResults TwoNMESexperimentswereperformedusingBrentsMethodasthecontroller.The rstexperimentinvolvedpositioningtheknee-jointtoadesired anglevia.voltagemodulation(VM).Thesecondexperimentinvolvedfrequency modulation(FM)whosepurposewasaself-testofthecontrolmethod. 2.3.1ExperimentalTestbed Alltheexperimentswereconductedonamodi edcommerciallegcurland extensionmachine(LEM)andacustomcomputercontrolledstimulationcircuit. ThepictureofthetestbedisshowninFig.21.TheLEMwasmodi edtoinclude two 5000 pulse-per-revolutionopticalencoder swithincrementalquadratureoutput of Aand Bchannels(oneencoderperleg).Theprecisionoftheencoders

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14 Figure21:Legcurlandextensionmachineaftermodi cations. allowsforaresolutionof 0 018withafrequencyresponseof 150 kHz.TheLEM allowsseatingadjustmentstoensuretherotationofthekneeisabouttheencoder axis.Fortheexperimenta 4 5 kg( 10 lb.)loadwasattachedtotheweightbarof theLEM,andamechanicalstopwasusedtopreventhyperextension. AcustomstimulationcircuitwasinterfacedwithaServoToGodataacquisition card.Thedataacquisitionwasperformedat1000Hzandconsistedofasingle encoderwhoseoutputwasusedtodeterminethekneeangle,andtwodigital-toanalogsignalswereusedasinputtothecustomstimulationcircuitrythatproduces a 100 sec positivesquarepulsebetween 3 1000 Hzwithavoltageoutput between 1 100 voltspeak.TheI/OcardiscontainedinaPentiumIVPChosting thereal-timeoperatingsystemQNX.TheRISEalgorithmwasimplementedin C++,andtheresultingreal-timeexecutablewasaccessedthroughtheQMotor3.0 GraphicalUserInterface[32].

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15 Intheexperiment,bipolarself-adhesiveneuromuscularstimulationelectrodes wereplacedoverthedistal-medialandproximal-lateralportionofthequadriceps femorismusclegroupandconnectedtothec ustomstimulationcircuitry.Priorto participatinginthestudy,writteninformedconsentwasobtainedfromallsubjects, asapprovedbytheInstitutionalReviewBoardattheUniversityofFlorida.All testsubjectswerehealthymalesbetweentheagesof 24 and50.Eachtestsubject wasinstructedtorelaxasmuchaspossibleandtoallowthestimulationtocontrol thelimbmotion(i.e.,thesubjectswerenotsupposedtoin uencethelegmotion voluntarily). 2.3.2ExperimentalSetup AnexperimentwasperformedusingBrentsMethod(Section2.2)todetermine theoptimalvoltageamplitude,givenapositivesquarewavewithapulsewidth of 100 sec andfrequencyof 20 Hz.Oncetheseekingroutinedeterminedthe optimalvoltageamplitude,afrequencymodulationexperimentwasperformed usingthecomputedvoltagefromthe rstexperimentalongwiththesame 100 sec pulsewidth.Theextremumseekingmethodwasusedinthesecondexperiment todeterminetheoptimalcorresponding frequency.Sincethevoltagemagnitude fromthe rstexperimentisusedinthesecondexperiment,theoptimalfrequency inthesecondexperimentshouldbeapproximately 20 Hz.Thefollowingresults indicatethatinbothteststheextremumseekingalgorithmwasabletominimize theangularkneepositionregulationerror,andthatthefrequencyseekingstrategy convergednearthefrequencyusedinthe rstexperiment. 2.3.3OptimalVoltageSeekingResults FollowingStep1intheprocedureoutlinedinSection2.2,thethreeinitial best-guessestimatevoltages, 1, 2, 3wereselectedas 1=20 0 2=30 0 3=55 0

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16 Themusclewasstimulatedwitha 30 voltpositivesquarewavepulsetrainwith a xed 100 sec pulsewidthat 20 Hz.Thepulsetrainwasappliedfor 5 seconds toensurethelimbdynamicsreachsteady-state.The 5 -seconddelayisasimple methodtoensurethedynamicsreachsteady-statebasedonpreviousexperience withtheexperimentaltestbedandtestsubject;however,severalalternativemethodscouldhavealsobeenusedsuchasaslidingwindowmethodthatmonitorspeak topeakoscillations.Duringthe vesecondsofstimulation,theknee-jointangle measurementswererecordedat 1000 Hz.Theanglerecordedattheendofthe veseconds(i.e.,thesteady-statevalue)wasrecordedandusedasaninput(i.e., Step3inSection2.2)toBrentsMethodtocomputethenextstimulationvalue, 4.Thekneeanglemeasurementwasrecordedafter vesecondsofstimulation with ( )= 4andusedtocomputethenextstimulationvaluebasedonStep5 inSection2.2.AccordingtoStep6,Steps3-5wererepeateduntilthealgorithm convergedwithinatoleranceofthedesiredangle.Representativeresults(Fig.2) show veiterationsofSteps3-5wereimplementeduntilthealgorithmconvergedto 44 7 volts.Fig.22indicatesthedesiredkneeangle(shortdashed)of 45,theactuallegangle(solid),andtheoutputvoltage(longdashed)computedfromBrents Method. AsshowninFig.22the rstbest-guessfor 2was 30 voltswhichyielded asteady-statekneeangleofapproximately 6 3.Thenextstimulationvoltage, determinedfromthejointangleerroras 4=37 6 volts,generatedakneejointangleof 29 7.Afterthreeadditionaliterationstheknee-jointanglewas approximately 44 1whichwaswithinthedesiredtolerance.Table22summarizes thecomputedvoltagelevelsandtheresultingkneeangle.Usingfourtestsubjects, atotalofsevenVMexperimentswereperformed.Fig.23showsasecondexample ofBrentsMethodovershootingthedesiredanglebeforeconvergingtothesolution. TheRMSerrors,standarddeviation,andsteady-stateerrorsforthesevenVM

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17 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 50 Time (s)Angle (deg) Voltage (V)50 30 25 20 15 10 5 45 40 35 Actual Leg Angle Desired Leg Angle Computed Voltage Figure2:Onlinecomputedvoltage(longdashed),desiredlegangle(short dashed),andactuallegangle(solid). Table22:Knee-jointanglecontrolledbyvoltage Time[s]Angle [deg] Voltage [V] 0.000.030.0 5.006.337.6 10.0029.742.4 15.0042.142.5 20.0042.844.4 25.0044.346.5 30.0046.443.7 35.0044.144.7

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18 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 50 Time [s]Angle [deg]5 10 15 20 25 30 35 40 45 50Voltage [v]Figure23:BrentsMethodovershootingthedesiredanglebeforeconvergingtowardsthesolutionusingVM. experimentsareshowninTable2.Thepointintimewhenthesystemachieved steady-statewasestimatedtooccurat 3 4 ofthetotalexperimenttime,hencethe steady-stateerrorsusedatafromthe nal 1 4 ofthestimulationperiod. 2.3.4OptimalFrequencySeekingResults Asecondexperimentwasperformedwheretheextremumseekingalgorithm wasusedtodeterminethedesiredfrequencyusingFMforagivenvoltageamplitude,waveform,andpulsewidth.MotivationfortheFMexperimentwasaself-test todemonstratetheabilityofusingthee xtremumseekingmethodforfrequency modulationandtocomparetheresultsbetweenthetwoexperiments.Speci cally, usingafrequencyof 20 HzinthepreviousVMexperiment,theextremumseeking algorithmconvergedto 44 7 volts.Therefore,inordertodeterminethevalidity ofBrentsMethodasacontroller,thevoltagefortheFMexperimentwassetto 44 7 volts.Usingthepreviousvoltage,theextremumseekingalgorithmintheFM

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19 Table2:RMSerrorandsteady-stateerrorforBrentsMethodusingVM TestLegRMSErrorSteady-stateMax.Steady-state Subject Error(RMS)Error(deg.) 1Left20.7660.771 1.062 1Right19.3963.188 3.528 2Left19.1075.664 6.876 2Right19.8565.278 6.372 3Left16.1301.490 1.404 3Right14.2552.894 3.600 4Left12.2933.762 1.062 17.4003.293 3.414Mean Standard 2.9721.672 2.264Deviation experimentshouldconvergeto20Hz.TheFMexperimentwasperformedafterthe testsubjectwasgivena 2 minuterestperiod. FollowingStep1oftheprocedureoutlinedinSection2.2,thethreeinitial best-guessestimatefrequencies, 1, 2, 3wereselectedas 1=15 2=18 3=25 Themusclewasstimulatedwitha 44 7 volt(i.e.,theoptimalvoltagecorresponding toa20HzpulsetrainfromtheVMexperiment)positivesquarewavepulsetrain witha xed 100 sec pulsewidthatafrequencyof ( )=18 Hzfor 5sec .The anglerecordedattheendofthe veseconds(i.e.,thesteady-statevalue)was recordedandusedasaninput(i.e.,Step3inSection2.2)toBrentsMethodto computethenextstimulationvalue, 4.Thekneeanglemeasurementwasrecorded after vesecondsofstimulationwiththestimulationfrequency ( )= 4andused tocomputethenextstimulationvaluebasedonStep5inSection2.2.According toStep6,Steps3-5wererepeateduntilthealgorithmconvergedwithinatolerance ofthedesiredangle.AsindicatedinFigure2, veiterationsofSteps3-5were implementeduntilthealgorithmconvergedtowithinatolerance.Figure24

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20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 50 Time [s]Angle [deg] Frequency [Hz]40 35 30 25 20 15 10 5 50 45 Desired Leg Angle Computed Frequency Actual Leg Angle Figure24:Onlinecomputedfrequency(boldsolid),desiredlegangle(short dashed),andactuallegangle(solid). indicatesthedesiredkne eangle(shortdashed)of 45,theactualkneeangle(solid), andtheoutputfrequency(boldsolid)computedfromBrentsMethod. Fig.24indicatesthattheinitialfrequencyof 18 Hzgeneratedasteady statejointangleof 45 4.After veiterations,thealgorithmconvergedto 19 0 Hz and 45 2.Table24summarizestheresultsfromthefrequencyexperimentand illustratesthatsmallchangesinthefrequencyproducemeasurablechangesinthe jointangle.ThesamefourtestsubjectsusedintheVMexperimentswereused forsevenFMexperiments.Fig.2showsasecondexampleofBrentsMethod overshootingthedesiredanglebeforeconvergingtothesolution.TheRMSerrors, standarddeviation,andsteady-stateerrorsforthesevenFMexperimentsare showninTable25.Thepointintimewhenthesystemachievedsteady-statewas estimatedtooccurat 3 4 ofthetotalexperimenttime,hencethesteady-state errorsusedatafromthe nal 1 4 ofthestimulationperiod.

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21 Table2:Knee-jointanglecontrolledbyfrequency Time[s]Angle [deg] Frequency [Hz] 0.000.018.0 5.0045.419.9 10.0045.719.9 15.0046.317.1 20.0045.218.6 25.0045.219.4 30.0045.219.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 50 Time [s]Angle [deg]5 10 15 20 25 30 35 40 45 50Frequency [Hz]Figure2:BrentsMethodconvergingonthesolutionusingFM.

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22 Table25:RMSerrorandsteady-stateerrorforBrentsMethodusingFM TestLegRMSErrorSteady-stateMax.Steady-state Subject Error(RMS)Error(deg.) 1Left4.9060.855 1.494 1Right5.0911.885 0.594 2Left4.9841.819 1.368 2Right5.1350.940 3.078 3Left5.7401.914 2.088 3Right4.8941.786 2.268 4Left4.6610.377 0.522 5.0581.638 1.630Mean Standard 0.3130.582 0.854Deviation 2.4Discussion Theresultsfrombothexperimentswerepromising.Speci cally,theexperimentalresultsindicatedthatwithnomusclemodel(onlyupperandlowerfrequency andvoltageamplitudeswererequired),theextremumseekingalgorithmcould determinetheappropriatestimulationwithinapproximately veiterations.Theextremumseekingalgorithmwasappliedforbothvoltagemodulationandfrequency modulationtoobtainalessthan 0 9degreesteady-statelimb/kneepositioning error.Theexperimentswereconstructedtoperformaself-test.Byusingthe voltageamplitudedeterminedfromthe rstexperiment,thefrequencyalgorithm inthesecondexperimentconvergedtowit hinonehertzofthefrequencyusedin the rstexperiment.Di erentextremumseekingalgorithmscouldbeappliedfor di erentresults.Hence,theideaofusinganoptimalextremumseekingalgorithm todeterminestimulationparametersforNMESapplicationsseemspromising. 2.5ConcludingRemarks AnextremumseekingNMESapproachwasimplementedtostimulatethe humanquadricepsmusclegroup.Amodi edversionofBrentsMethodwasimplementedastheextremumseekingroutinetodeterminevoltageorfrequency

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23 modulationparametersthatwouldyieldadesiredangularkneeposition.This methodonlyrequiredmeasurementsoftheresultingkneeangle,andsomeknowledgeofupperandlowerboundsonthevoltageorfrequencysettings.Experimental resultswereobtainedthatindicatedthedesiredknee/limbpositioncouldbeobtainedwithina 0 9tolerance.Inoneexperimentthealgorithmwasappliedto determinethevoltageamplitudewheretheremainingstimulationparameterswere xed,andasecondexperimentwasperformedwherethealgorithmdeterminedthe desiredfrequency.

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CHAPTER3 NONLINEARCONTROLSCHEME Anonlinearcontrolmethodisdevelopedinthischapterthatusesneuromuscularelectricalstimulationtocontrolthehumanquadricepsfemorismuscle undergoingnon-isometriccontractions.Theobjectiveofthecontrolleristoposition thelowerlimbofahumanalongatime-varyingtrajectoryortoadesiredsetpoint. Thedevelopedcontrollerdoesnotrequireamusclemodelandcanbeprovento yieldasymptoticstabilityforanonlinearmusclemodelinthepresenceofbounded nonlineardisturbances.Performanceofthecontrollerisillustratedintheprovided experimentalresults. 3.1RobustIntegralSignoftheError TheresearchpresentedhereillustratestheperformanceofacomputercontrolledNMESmethodforbothtrackingandregulationofahumanknee-joint angle.Themethodcouldbeappliedtoothermusclegroupswithoutlossofgenerality.TheNMEScontrollerisbasedontherecentlydevelopedRobustIntegral SignoftheError(RISE)techniquethatusesanerrorsignalde nedbetweenthe actualangleofthekneeandsomeknowndesiredangle.OneofthemotivatingfactorsforimplementingtheRISEcontrolleristhatthemethoddoesnotdependon muscle-modelknowledge,andLyapunov-basedstabilityanalysismethodshavebeen developedthatproveasymptoticstabilityfordynamicsystemssubjecttogeneral boundeddisturbances[27,28].Signi cantresearche ortshavefocusedontheuse ofneuralnetwork-basedcontrollers[10].Nonlinearneuralnetworkmethods providedaframeworkthatallowedtheperformance,robustness,andstabilityofthe developedNMEScontrollerstobeinvestigatedwithoutlinearizationassumptions. However,allofthepreviousneuralnetwork-basedNMEScontrollersarelimited 24

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25 toauniformlyultimatelyboundedresultbecauseoftheinevitableresidualnonlinearfunctionapproximationerror.Addi tionally,neuralnetworksmayexhibit performancedegradationduringthetransientphasewhiletheestimatesupdate.In comparisontoothernon-model-basedapproachessuchasPD/PIDcontrollers[2], theRISEmethodisarobustcontrollerthatwasproventoyieldasymptotic trackingofnonlinearsystemswithunstructureduncertaintyandboundedadditivedisturbances.Incomparisontooptimalmethodssuchasextremumseeking, theRISEcontrollerdoesnotrequiretheglobalmaximumassumptionforthe torque/voltagecurve,anddoesnotrequir eiterativestepsthataredelayedbythe transientresponseofthemuscleandlimbdynamics.Inordertoimplementthe RISEcontrolleramusclemodelisdevelopedandthenrewritteninaformthat adherestopreviousRISE-basedLyapunovstabilityanalyses.Theperformance ofthenonlinearcontrollerisexperimentallyveri edforboththetrackingand regulationofahumanleg/shankbyapplyingthecontrollerasavoltagepotential acrossexternalelectrodesattachedtothedi stal-medialandproximal-lateralportion ofthequadricepsfemorismusclegroup.TheRISEcontrollerisimplementedbya voltagemodulationschemewitha xedfrequencyanda xedpulsewidth.Other modulationstrategies(e.g.,frequencyorpulse-widthmodulation)couldhavealso beenimplemented(andappliedtootherskeletalmusclegroups)withoutlossof generality. TheexperimentalresultsfortheregulationscenarioaredescribedinSection 3.4.2,andthetrackingexperimentalresultsareprovidedinSection3.4.3.These preliminaryexperimentalresultsindicatethatthedesiredknee-jointanglecanbe regulatedwithin 0 5oferrorforthe xedangleexperiment,andwithin 3 5of steady-stateerrorforthetrackingexperiment.

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26 3.2MuscleActivationandLimbModel Thetotalknee-jointdynamicscanbemodeledas[5] + + + + = (31) In3, ( ) R denotestheinertiale ectsoftheshank-footcomplexabout theknee-joint, ( ) R denotestheelastice ectsduetojointsti ness, ( ) R denotesthegravitationalcomponent, ( ) R denotestheviscous e ectsduetodampinginthemusculotendoncomplex[33], ( ) R represents unknownunmodelledboundeddisturbances(e.g.,fatigue,signalandresponse delays,unmodelledphenomena),and ( ) R denotesthetorqueproducedatthe knee-joint. Theinertialandgravitationale ectsin(31)canbemodelledas ( ( ))= ( ) ( ( ))= sin( ( )) where ( ) ( ) ( ) R denotetheangular(Fig.3)position,velocity,and accelerationofthelowershankabouttheknee-joint,respectively, R denotes theunknowninertiaofthecombinedshankandfoot, R denotestheunknown combinedmassoftheshankandfoot, R istheunknowndistancebetweenthe knee-jointandthelumpedcenterofmassoftheshankandfoot,and R denotes thegravitationalacceleration. Theelastice ectsaremodelledontheempirical ndingsbyFerrarinand Pedottiin[33]as ( )= 1(exp( 2 ( )))( ( ) 3) (32) where 1, 2, 3 R areunknownpositivecoe cients.Asshownin[5],theviscous moment ( ) canbemodelledas ( ( ))= 1tanh( 2 ( )) 3 ( ) (33)

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27 Figure3:Knee-jointanglede nedby where 1, 2,and 3 R areunknownpositiveconstants. Thetorqueproducedaboutthekneeiscontrolledthroughmuscleforces thatareelicitedbyNMES.Forsimplicity(andwithoutlossofgenerality),the developmentinthispaperfocusesonproducingkneetorquethroughforces, denotedby ( ) R ,generatedbyelectricalstimulationofthequadriceps(i.e., wedonotconsiderantagonisticmuscleforces).Thekneetorqueisrelatedtothe quadricepsforceas ( )= ( ( )) ( ) (34) where ( ( )) R denotesapositivemomentarmthatchangeswiththeextension and exionofthelegasshowninstudiesby[34]and[35].Asindicatedin[34],the momentarm ( ( )) hasuniquevaluesforagivenrangeofmotion,whilein[35], themomentarmsuniquevaluesareobtainedfortheentirerangeofmotion. Themuscleforce ( ) isgeneratedbytheavailableactinandmyosin lament bindingsitesinthemuscle bers.Thevoltageappliedtothemusclealtersthe calciumionconcentrationwhichin uencestheactin-myosinbinding.Therelationshipbetweenthemuscleforceandtheappliedvoltageisdenotedbytheunknown function ( ) R as ( )= ( ) ( ) (35)

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28 where ( ) R isthevoltageappliedtothequadricepsmusclebyelectrical stimulation.Whileexactforceversusvoltagemodelsaredebatableandcontain parametricuncertainty,thegenerallyacceptedempiricalrelationshipbetweenthe appliedvoltage(orsimilarly,current,frequency,orpulsewidth)iswellestablished. Thefollowingpropertieshavebeenexploitedinsubsequentcontroldevelopment. Property1: Theunknowndisturbance ( ) isboundedandits rstand secondderivativeswithrespecttotimeexistandarebounded. Property2: Themomentarm ( ( )) isacontinuouslydi erentiable, positive,monotonic,boundedfunction[35],andempiricaldataindicatesthe function ( ) isalsoacontinuouslydi erentiable,positive,monotonic,andbounded function. 3.3ControlDevelopment TheobjectiveinthispaperistodevelopaNMEScontrollertoproduceaknee torquetrajectorythatwillenableahumanshanktotrackadesiredtrajectory, denotedby ( ) R .Withoutlossofgenerality,thedevelopedcontrolleris applicabletodi erentstimulationprotocols(i.e.,voltage,frequency,orpulsewidth modulation).Toquantifytheobjective,apositiontrackingerror,denotedby 1( ) R ,is 1( )= ( ) ( ) (36) where ( ) isanaprioritrajectorywhichisdesignedsuchthat ( ) ( ) L, where ( ) denotesthe derivativefor =1 2 3 4 .Tofacilitatethesubsequent analysis, lteredtrackingerrors,denotedby 2( ) and ( ) R ,arede nedas 2( )= 1( )+ 11( ) (37) ( )= 2( )+ 22( ) (38)

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29 where 1, 2 R denotepositiveconstants.The lteredtrackingerror ( ) is introducedtofacilitatetheclosed-looperrorsystemdevelopmentandstability analysisbutisnotusedinthecontrollerbecauseofadependenceonacceleration measurements. Aftermultiplying(38)by andutilizingtheexpressionsin(3)and(3) (37),thefollowingexpressioncanbeobtained: = + (39) where ( 12 ) R isanauxiliarysignalde nedas = ( + 1 1+ 22)+ + + (3) andthecontinuous,positive,monotonic,andbounded(seeProperty2)auxiliary function ( ) R isde nedas = (3) Aftermultiplying(3)by 1( ) R ,thefollowingexpressionisobtained: = + (3) where ( ) R ( 12 ) R ,and ( ) R arede nedas = 1= 1 = 1 Tofacilitatethesubsequentstabilityanalysis,theopen-looperrorsystemfor (3)canbedeterminedas = 1 2 + 2 (3) where ( 12 ) R denotestheunmeasurableauxiliaryterm = + 2 1 2 + ( ) (3)

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30 Tofurtherfacilitatetheanalysis,anotherunmeasurableauxiliaryterm, ( ... ) R ,isde nedas = ( ) + ( ) ... + ( ) + ( )+ ( )+ ( ) (3) Afteraddingandsubtracting(3)to(3 13),theopen-looperrorsystemcanbe expressedas = 2+ + (3) wheretheunmeasurableauxiliaryterm ( 12 ) R isde nedas ( )= (3) Using[36],theMeanValueTheoremisappliedtodevelopthefollowingupper bound ( k k ) k k (3) where ( ) R3isde nedas ( ) [ 1 2] (3) Basedon(3),andthefactthat ( ) ( ) L =1 2 3 4 ,thefollowing inequalitiescanbedeveloped k k (3) where and R areknownpositiveconstants. Thedevelopedopen-looperrorsystemin(316)isnowsimilartotheopenlooperrorsystemin[27,28,37,38].Basedonthedynamicsgiveninequations (3)(3)thefollowing RISEfeedbackcontroller ( ) isemployedasameans

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31 toachievethetrackingobjective: ( ) ( +1) 2( ) ( +1) 2( 0) (3) + Z 0[( +1) 22( )+ ( 2( ))] where R denotepositiveconstantadjustablecontrolgains,and ( ) denotesthesignumfunction. Theorem: Thecontrollergivenin(321)ensuresthatallsystemsignals areboundedunderclosed-loopoperationandthatthepositiontrackingerroris regulatedinthesensethat k 1( ) k 0 as (3) providedthecontrolgain ,introducedin(3)isselectedsu cientlylarge,and isselectedaccordingtothefollowingsu cientcondition: + 1 2 (3) where and areknownpositiveconstants. ThestabilityanalysisandcompletedevelopmentoftheRISEmethodcanbe foundin[27,28,37,38]. 3.4ExperimentalResults TwoexperimentswereperformedusingtheRISEcontrollergivenin(3). Thevoltagecontrollerwasimplementedthroughanamplitudemodulationscheme composedofavariableamplitudepositivesquarewavewitha xedpulsewidth of 100 sec and xedfrequencyof 100 Hz.Thefollowingresultsindicatethat theRISEalgorithmwasabletominimizethekneeangleerrorwhiledynamically trackingadesiredtrajectory.

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32 3.4.1ExperimentalSetup TheRISEexperimentwasperformedusingthetestbeddescribedinAppendix 2.3.1.Forbothexperimentsa 4 5 kg( 10 lb.)loadwasattachedtotheweightbarof theexercisemachine. Ineachexperiment,bipolarself-adhesiveneuromuscularstimulationelectrodes thatwereplacedoverthedistal-medialandproximal-lateralportionofthequadricepsfemorismusclegroupofeachsubjectandconnectedtothecustomstimulation circuitry.Priortoparticipatinginthestudy,writteninformedconsentwasobtainedfromallsubjects,asapprovedbytheInstitutionalReviewBoardatthe UniversityofFlorida.Testsubject 1 wasahealthy 25 yearoldmale,testsubject 2 wasahealthy 24 yearoldmale,andtestsubject 3 wasahealthy 50 yearold male.Eachtestsubjectwasinstructedto relaxasmuchaspossibleandtoallow thestimulationtocontrolthelimbmotion(i.e.,thesubjectswerenotsupposedto in uencethelegmotionvoluntarily). Todetermineboundsonthetestsubjectsresponsetostimulation,acalibrationprotocolwasperformedtodetermineappropriateupperandlowerstimulation bounds.Speci cally,aninitialstimulationvoltagewaschosenthatwouldgenerate aknee-jointangleof 25.Thepulsewidthwassetat 100 sec anddeliveredat 100 Hz.Stimulationvoltagewaslinearlyincreasedattherateof 2 voltspersecond untiltheknee-jointanglereached 45,atwhichpointthevoltagewouldlinearly decrease.Thisad-hocstrategyprovidessomeindicationofthemuscleresponseto stimulationforthedi erentsubjectssothatthevoltagelevelscouldbemaintained withinsaferegionsofoperation.Fig.3showsthetypicalmuscleexcursionofthe testsubjectsusedfortheregulationandtrackingexperiments. 3.4.2RegulationResults Theinitialstimulationvoltageforsubject 1 wasbasedonthelinearvoltage testdescribedpreviously(Fig.32)whichindicatedthatforsubject 1 25 volts

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33 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 55 Time (s)Knee Joint Angle (deg) Voltage (V)5 10 15 20 25 30 35 40 45 50 55 Knee Joint Angle Voltage Figure32:Typicalmuscleexcursionofthetestsubjectsusedfortheregulation andtrackingexperiments. generatedaknee-jointangleof 25.Thevoltagewasdeliveredasapositivesquare wavetrainwitha xed 100 sec pulsewidthat 100 Hz. Fortheregulationtest,thedesiredkneeangleshowninFig.33increases from 0to 45in 2 seconds,incontrasttosimplyassigningaset-pointof 45,for comfortandsafetyofthestudyparticipants.TheresultsobtainedbytheRISE methodareshowninFig.33whichindicatesthedesiredknee-jointangle(long dashedline)andtheactualknee-jointangle(solidline).Thecomputedoutput voltageisshowninFig.3andadetailoftheerror(Fig.3)showsthatafter 3 secondstheknee-jointanglewaswithin 4,andafter 3 8 secondstheerrornever exceeded 0 5.After 8 secondstheknee-jointanglewasapproximately 44 7.Using threetestsubjects,atotalofeightregulationexperimentswereperformed(the rst subjectwastestedontwoseparatedays).TheRMSerrors,standarddeviation,and steady-stateerrorsfortheeightexperimentsareshowninTable31.Thepointin timewhenthesystemachievedsteady-statewasestimatedtooccurat 2 3 ofthe

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34 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 35 40 45 50 Time (s)Knee Joint Angle (deg) 5 10 15 20 25 30 35 40 45 50 Knee Joint Angle Desired Angle Figure33:RegulationofkneejointangleusingtheRISEcontroller. totalexperimenttime,hencethesteady-stateerrorsusedatafromthe nal 1 3 of thestimulationperiod. 3.4.3TrackingResults Theinitialstimulationvoltageforsubject 2 wasbasedonthelinearvoltage testdescribedpreviously(Fig.32)whichindicatedforsubject 2 18 voltsgeneratedaknee-jointangleof 25.Thevoltagewasdeliveredasapositivesquare wavetrainwitha xed 100 sec pulsewidthat 100 Hz.Thesinusoidaltracking pro leinFig.36wasprogrammedforaminimumangleof 20andamaximum of 45.Toensureasmooth(andcomfortable)stimulationbehavior,twosinusoidal equationswereused:

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35 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 35 40 45 50 Time (s)Voltage (volts)5 10 15 20 25 30 35 40 45 50Figure3:RegulationvoltageusingtheRISEcontroller. Figure3:Regulationerrorofkneejoint angle(desiredangleminusactualangle).

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36 Table3:RMSandsteady-stateerrorforRISEregulationexperiments TestLegRMSErrorSteady-stateMax.Steady-state Subject Error(RMS)Error(deg.) 1Left17.0790.473 0.232 1Right17.9130.336 0.643 2Left17.9800.432 0.502 2Right17.9970.198 0.355 3Left18.6920.462 0.340 3Right18.7340.349 0.320 1Left18.6100.332 0.312 1Right18.5930.728 0.385 18.2000.414 0.386Mean Standard 0.5340.1450.121Deviation 1( )= 2 + 2 sin( + 3 2 ) (3) 2( )= 2 2 + (3) 2 2 sin( + 3 2 ) + where denotestheminimumknee-jointangle, representsthemaximumkneejointangle,and denotes 2 ,( equaledtheknee-jointperiod).Thedesired trajectoryin(3)wasuseduntil 1( )= ,andthenthedesiredtrajectorywas changedto 2( ) in(3). Arepresentativegraphofthetrackingexperiment(Fig.37).showsthedesiredkneeangle(longdashedline)andtheactualknee-jointangle(solidline).The computedoutputvoltageisshowninFig.3andadetailoftheerror(Fig.3) showsamaximumtransienterrorof 17 3at 1 secondwhichcorrespondstothe pointofmaximumvelocity.After 1 secondtheerrordecreasesuntilapproximately

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37 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 Time (s)Desired Knee Joint Angle (deg) 0 5 10 15 20 25 30 35 40 45 50Figure3:Desiredtrackingpro leextendedto20seconds. 2 secondswhentheerrorreachessteady-state,neverexceeding 3 5.Usingthree testsubjects,atotalofeighttrackingexperimentswereperformed(the rstsubject wastestedontwoseparatedays).TheRMSerrors,standarddeviation,andsteadystateerrorsfortheeightexperimentsareshowninTable32.Thepointintime whenthesystemachievedsteady-statewasestimatedtooccurat 2 seconds,hence thesteady-stateerrorsusedatastartingat 2 secondsandcontinuinguntiltheend ofthestimulationperiod. 3.5Discussion Resultsfrombothexperimentswerepromising.Speci cally,theexperimental resultsindicatedthatwithnomusclemodel(andonlyvoltageamplitudemodulation),theRISEalgorithmcoulddeterminetheappropriatestimulationvoltagefor bothregulationandtracking.TheRISEalgorithmobtainedaregulationerrorof lessthan 0 5andatrackingerrorofapproximately 3 5. Theprimaryobjectiveofthe rstexperimentwasregulatingtheknee-jointto adesired nalangle( 45).Theexperimentshowedawellbehavedtransientand

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38 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 Time (s)Knee Joint Angle (deg) 5 10 15 20 25 30 35 40 45 50 Knee Joint Angle Desired Angle Figure3:KneejointtrackingusingtheRISEcontroller. 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 Time (s)Voltage (volts)5 10 15 20 25 30 35 40 45 50Figure3:TrackingvoltageusingtheRISEcontroller.

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39 Figure3:Trackingerrorofkneejointangle(desiredangleminusactualangle). Table3:RMSandsteady-stateerrorforRISEtrackingexperiments TestLegRMSErrorSteady-stateMax.Steady-state Subject Error(RMS)Error(deg.) 1Left4.3832.276 8.940 1Right4.6212.299 7.438 2Left4.1931.950 5.907 2Right4.7212.315 6.830 3Left3.5611.311 3.841 3Right5.3624.796 5.928 1Left4.7014.383 4.866 1Right4.7514.621 5.082 4.5372.994 5.788Mean Standard 0.4861.2852.097Deviation

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40 thatwithinthreesecondstheerrorwaswithin 4.After 3 8 secondstheerrornever exceeded 0 5. Theobjectiveforthesecondexperimentrequiredtheknee-jointtotracka desiredsinusoidaltrajectorywithaperiodoffour-seconds.Theexperimentshowed thatatthepointofmaximumvelocity(one-second),thecontrollerhadatransient errorof 17 3.Afterapproximately2-seconds(thepointwherethevelocityiszero) theknee-jointtrackingerrorneverexceeded 3 5. 3.6ConcludingRemarks ARISEnonlinearcontrolalgorithmwasappliedtoNMEStoelicitnonisometriccontractionsofthehumanquadricepsmuscle.Twoexperimentswere performedtodeterminetheperformanceoftheRISEcontrolmethod. Futuree ortswillfocusonimplementingdi erentmodulationmethods, stimulatingforfunctionaltasks,examiningfatigueinducedbytheRISEcontroller, comparingtheRISEcontrolresultswithotherNMESmethods,andexperimental trialsonmorevolunteers,potentiallyincludingpersonswithneurologicaldisorders.

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CHAPTER4 CONCLUSIONSANDRECOMMENDATIONS Twononlinearcontrollerswereimplementedfornon-isometricexperiments thatcontrolledahumankneejointanglev iaNMESofthequadricepswithoutthe useofamusclemodel.The rstcontrollerwasanumericaliterativeextremum seekingroutine(BrentsMethod)thatwasimplementedfortworegulationexperiments;the rstexperimentusedvoltagemodulationforNMES,andthesecondwas aself-testthatusedfrequencymodulation.ThesecondcontrollerwasanimplementationofarecentlydevelopedschemecoinedRISE(RobustIntegralSignofthe Error)thatusedvoltagemodulationinaregulationexperimentandasinusoidal trackingexperiment.TheexperimentalresultsfromBrentsMethodshowedthat withbothmodulationschemesitwasabletopositionthekneeanglewithin 0 9its 45objectivein veiterations.TheresultsfortheRISEcontrollershowedthat regulationofthekneeanglewasaccuratetowithin 0 5ofits 45objective.For thesinusoidaltrackingexperimenttheRISEcontrollermaintainedasteady-state trackingerrorofapproximately 3 5. Theresultsshownbythisresearchindicatethatitispossibletoperform reasonableNMEStrackingandregulationcontrolofthehumanquadriceps musclegroupwithoutusingamusclemodel.BrentsMethodislimiteddue toitsdependenceonthekneejointreac hingarelativesteady-statecondition (whichmaytake veseconds)beforeperformingitsnextiterations.TheRISE controllerwasverypromisingwiththereg ulationexperiments.Minoradjustments tothegainsshowedthatitcaneasilyaccommodateavarietyoftestsubjects withexcellentresults.LimitationstotheRISEmethodwereapparentduringthe trackingexperimentwhereitshowedsensitivitytogainchanges. 41

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42 ExpandinguponBrentsextremumseekingroutinedoesnotseemfruitful. Afeatureandlimitationofextremumseekingroutinesistheiriterativenatureto convergeonasolution.Whilethisiterativebehaviorworkswellwhentimebetween iterationsisnotanissue,itseverelylimitsitsuseforNMES. TheRISEcontrollerslimitationispartiallyduetotimedelayswhichoccur betweenthestimulusandthemusclecontractionforcewithnon-isometricNMES. WhenusedoutsideoftheNMES eld,addinganeuralnetwork(NN)totheRISE controllershowedimprovedbehaviorastimeprogressedandtheNNupdated. TheinabilitytodevelopaconsistentmappingbetweentheNMESparameters andthemusclescontractionforcecouldbeinvestigatedwithaNNthatlearnsan individualsmusclemodelbyusinguselinearvoltagegradientswithisometricand non-isometricNMES. Inadditiontofuturecontrolresearchwithnon-isometricNMES,theisometric attachmenttotheLEMenablestheabilitytoperformuniqueback-to-back experimentsthatmayshowinsightintotimedelayissuesaswellascause-ande ectofmusclefatigue.Withtheultimatemotivationofthisresearchbeing therehabilitationandpotentialimprovementsinthedailyactivitiesforpeople a ictedwithneuromusculardisorderssuchasstrokeorspinalcordinjuries,future experimentsneedtoincludepeoplewithinthispopulationsegment.

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APPENDIX ELECTRICALDESIGNANDINTERFACING A.1CircuitDesignfor100 secWidePulse Generatingapulsewidthof 100 sec fromasystemthathasasampling rateof1000HzrequiredbuildingthecustomcircuitshowninFigureA.After buildingaprototypeofthecircuitandverifyingitsbehaviortheprintedcircuit board(PCB)wasdesignedandthenecessary lesweresenttoImagineeringInc.to bemanufactured(Fig.A).UponreturnofthebarePCBfromthemanufacturer itwaspopulated(Fig.A),andagainitsbehaviorwasveri ed,afterwhich,the PCBtoPCinterfacingcableswerebuilt. A.2CircuitDescription TheoutputrequirementsforthePCBwerethatitdelivera 100 secwide pulseatafrequencyandvoltagedictatedby thecontroller.Thestimulationvoltage andfrequencyarebetween 0 150 voltsand 10 1000 Hzrespectively.FigureA showsthestimulationpulseshapethatisdeliveredtothetestsubject. TheoutputdemandsforthePCB(Fig.A)requiredsixseparateDCinput voltagesasdescribedinTableA1.InterfacingthePCBwiththeServoToGoI/O card(STG)requiredfourSTGoutputs;twobetween 10 0 to +10 0 voltsDCand twobetween 0 0 10 0 voltsDC. Generatingthe 100 secwidepulseatvaryingfrequenciesandvoltageswas controlledbyavoltage-to-frequencyconverter(VFC)describedbelow.Apower op-amp(describedbelow)usesa 0 10 VDCinputthenoutputsapositivesquare pulsebetween 1 150 voltswhichisfedtotheelectrodesattachedtothetest subject. ThecompletepartsusedtobuildthePCBarelistedinTableA2. 43

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44 FigureA1:Schematicofcircuitryusedtodeliverthecomputedstimulationpulse train.

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45 FigureA2:PCBlayoutofcircuitryusedtodeliverthecomputedpulsetrain. A.2.1HighVoltagePowerOp-AmpPA866EU Thefunctionoftheop-ampistoamplifytheinputsignalcomingfromthe STG.Theampli edsignal(thestimulationvoltagecomputedbythesoftware)is feddirectlytotheelectrodesattachedtothetestsubjectsmuscles.Theop-ampis suppliedwiththreeDCvoltages.The + 175voltsand 25voltsaresuppliedviaa TableA:Inputvoltagestocircuit DCSignal(volts)Purpose +175.0HighvoltageinputtoOp-amp -25.0LowvoltageinputtoOp-amp +10.0PositivesupplyvoltageforVFC -10.0NegativesupplyvoltageforVFC 0.0-10.0Op-ampcontrolsig nalforstimulationvoltage 0.0-10.0VFCcontrolsignalforstimulationfrequency

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46 FigureA3:Circuitboardusedtogenerateandamplifya100 secpulse.

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47 100 sec 0 150 voltsFigureA4:Shapeofstimulationpulse. commercialpowersupplyandservetopowertheop-amp.ThethirdDCsignal(0.0 -10.0volts)iscomputedbythesoftwareanddeliveredtotheop-ampviatheSTG. Theop-ampiscon guredasnon-invertingwith+IN(thevoltagetobeampli ed) beingsuppliedviatheemitteroftransistorQ2(Fig.A).Thenegativefeedback loopoftheop-ampconsistsofresistorsR2(470k )andR3(47k ) ,whichare con guredtogiveagainofapproximately10:1.Note:resistorR8isnolongerused. A.2.2VoltagetoFrequencyConversion-VFC32 Generatingthedesiredstimulationfrequencyandthe100 secwidepulseis controlledbyavoltage-to-frequencycon verterintegratedcircuit(IC).TheVFC issuppliedwiththreeDCvoltagesshowninTableA1.The 10voltsand + 10 voltspowertheVFCandaresuppliedviatheSTG.ThethirdDCsignal(0.0-10.0 volts)iscomputedbythesoftwareanddeterminesthesignalfrequency.Generation ofthe100 secwidepulseiscontrolledbytheproperselectionofcapacitorsand

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48 TableA:Partslistforstimulatorcircuit QuantityPartNumberDescription 1APEXPA866EUHighvoltagepowerop-amp 1TexasInst.VFC32Voltagetofrequencyconverter 210k ResistorCurrentlimiter 14.7k ResistorPullup 2100k ResistorR1,R2-VFCcircuit 1100 ResistorQ1Bias 210 ResistorOp-ampcurrentlimiter 23k ResistorOp-ampcompensation 1470k ResistorOp-ampfeedback 147k ResistorOp-ampfeedback 32.7nFCapVFCcontrol 40.1 FCapBypasscapacitor 22nFCapVFCcontrol 22pFCap Op-ampcompensation 17805 5voltsupply-pullup&LED 1LED Stimulationstatus 22N3565 NPNtransistor remainsconstantthroughouttheVFCfrequencyrange.Thearrayofparallel capacitorsC2,C9-C10areusedbytheVFCinordertogeneratethe100 secpulse width.TheseriesresistorsR1&R9determinethefrequencyrangewhensupplied a 0 10 VDCoutputfromtheSTG.Thefrequencyrangeis 10 1000 Hz.The voltagesuppliedtopin1oftheVFC(labeledIN-inFigA1),(suppliedfromthe STG)determinestheVFCsoutputfrequency. A.2.3Transistors2N3565 Transistors2N3565aretypeNPNandareusedinswitchingmode.Transistor Q1suppliestheDCvoltageoutputfromtheSTGtopin1ofVFC(labeledIN-in Fig.A1).ThebaseoftransistorQ2isconnectedtotheoutputoftheVFC.The collectorofQ2isconnectedtoaDCvoltagefromtheSTG.TheemitterfromQ2is connectedtothe+INoftheop-ampPA866EU.

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49 A.2.4VoltageRegulator7805 The7805voltageregulatorsupplies5voltDCpowertotheLED(which indicatescommunicationwiththeSTG )andasapull-upsupplytothebaseof transistorQ2(enablingittofunctionasadigitalswitch),andtheoutputofthe VFC. A.2.5CircuitBehavior Theoveralloutlineofcircuitsbehaviorisasfollows: TheSTGsuppliesa 0 10 VDCsignaltoIN-oftheVFCwhichdetermines thefrequency. TheSTGsuppliesa 0 10 VDCsignaltocollectorofQ2whichisthescaled downstimulationvoltage. TheoutputsignalfromtheVFCisdigital( 0 or 5 volts)andisfedtothe baseofQ2.TheoutputsignaldevelopedbytheVFCcontainsthefrequency ( 10 100 Hz),andcontainedwithinthatsignalisthepulsewidth( 100 sec). WhenthebaseofQ2ishighitsemittersuppliesthescaled-downstimulation voltageto+INoftheop-amp. Theoutputoftheop-ampcarriestheentireampli edstimulationsignal:The stimulationvoltageisdeliveredtothetestsubjectwitha 100 secwidepulse thatoccursatthedesiredfrequency. A.3Interfacingthe 100 secWidePulsePCBtotheComputer Generatingthedesiredfrequencyandstimulationvoltagerequiredcorrelating thetwo 0 10 VDCoutputsignalsfromtheSTGtothePCB.Thecorrect mappingvoltageswereobtainedempiricallybyrecordingtwosetsofdata-pairs. Usingacommercialpowersupply,tenrandomlychosenDCvoltageswereinput tothePCBsfrequencyandstimulationvoltagepins.TheoutputfromthePCB wasobservedonanoscilloscopeandthecorrespondingdata-pairswererecorded andenteredintoMATLABwheretwopolynomialsweregeneratedbyusingthe

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50 POLYFITandPOLYVALroutines.Thepolynomialswerethenhard-codedinto aC++program.Theaccuracyofthepolynomialswasveri edbyinputtinga desiredfrequencyandstimulationvoltageswhileobservingthePCBsoutputonan oscilloscope. A.4CircuitDesignforMultiplesof 25 secWidePulse ThemethodthatallowstheNMEScontrollertoincreaseordecreasethepulse widthinstepsof 25 secisperformedwithsixrelayscontrolledbytwoanalog multiplexers(MUX).Thecircuit(Fig.A5)wasdevelopedasaseparateadd-on circuitboardinterfacedtothePCBandtheSTG. A.4.1CircuitDescription Theoutputofthecircuitisakintothatofavariablecapacitor.Theoutput(a capacitance)isplacedinparallelwithcapacitorsC2andC9-C10onthemainPCB. ThecombinedcapacitancedeterminesthepulsewidthgeneratedbytheVFC. InputtothecircuitconsistsofeightdigitalsignalssuppliedbytheSTGand asix-voltDCpowersupply.Theeightdigitalsignalscontroltheoutputofthetwo MUXesthatdeterminewhichrelaysareenergized.Therelaysarecon guredas two-setsofthreeandarelabelled Relay1Array (RA1)and Relay2Array (RA2) asshowninFig.A.ThepurposeofRA1istocontroltheportionofthepulse widthwhichisamultipleof 100 sec.ThepurposeofRA2istocontroltheportion ofthepulsewidthwhichisamultipleof 25 sec.Functioningtogether,RA1and RA2allowforapulsewidthof 100 675 sec. Thefunctioningofeachrelayarrayisidenticalsothefollowingdiscussionof RA1alsoappliestoRA2 RA1iscontrolledbytheanalogmultiplexerlabeledU2. ThemultiplexerU2receivesfourdigitalsignalsfromtheSTGlabeledA-I,B-I, C-I,andE-I(Fig.A5)whosestatesdeterminewhichrelay(s)toenergize.The signalsA-I,B-I,C-I,andE-IaredeterminedbytheNMEScontrollerandarebased onthedesiredpulsewidth.U2decodesthefourdigitalsignalsthenenergizesthe

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51 FigureA5:Circuitforadjustingthepulsewidthwithstepsof 25 sec .

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52 TableA3:Digitalinputstothecircuit DigitalInputsto 74HC405 31 2from Purpose ServoToGo D1 A1-Relay1 D2 B1-Relay1 D3 C1-Relay1 D4 E1-Relay1 D5 A2-Relay2 D6 B2-Relay2 D7 C2-Relay2 D8 E2-Relay2 TableA4:Thee ectoftherelayonthecorrespondingpulsewidth Relay1 6 Purpose 1 100 secstep 2 300 secstep 3 200 secstep 4 25 secstep 5 25 secstep 6 25 secstep correspondingrelaysinRA1.ThecombinedstatesoftheRA1determinewhich capacitorsareinthecircuit,hencetheoverallcapacitanceappearingat" Feeds backtomainboard "inFig.A5.ThecapacitanceofcircuitA5isinparallelwith capacitorsC2andC9-C10onthePCBwhichdeterminesthecapacitanceseenby theVFC(whichdeterminesthepulsewidth). TableAshowstheSTGconnectionstotheMUXesandwhichrelaysare usedtocontroltheirportionofthepulsewidth.VariouscombinationsofRelays1 3givemultiplesof 100 secpulsewidthsfrom100-600 sec.Relays4 6eachadd anadditional 25 sectothepulsewidth.TableAshowstherelayse ectonthe pulsewidthrangingbetween 100 675 secin 25 secincrements. ThelistofpartsforthepulsewidthcontrollerisshowninTableA5.

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53 TableA5:Partslistforpulsewidthcontroller QuantityPartNumberDescription 6OmronG5V-15voltPCBmountrelay 2Fairchild74HC4053AnalogMultiplex 12 0 10 FCapacitor 6 0 01 0 05 FTuningcapacitor A.5IsometricAttachment IsometricexperimentsareperformedbyattachingaS-beamloadcell(LC) betweentheswing-armandtheframeoftheLEM.TheLCisratedfor 300 lbs. intensionandcompression.OutputfromtheLCisampli ed1000timesbyan instrumentationop-amp.ThecircuitdiagramfortheLCcircuitisshowninFig. A6.TheinputtotheLCcircuitis 10 VDCforloadcellexcitation.Outputfrom theLCis 0 10 VDCwhichismonitoredbyananalog-to-digitalinputontheSTG. TheLCoutputwascalibratedbysuspendingitthenaddingknownweightsto thefreeendandrecordingtheoutputvoltages.Theoutputwaslinear,requiring onlyslopeandy-intercepto sets.

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54 FigureA6:CircuitdiagramforS-beamloadcell.

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BIOGRAPHICALSKETCH KeithStegathwasborninAnnArbor,Michigan.In2005,hereceivedaBachelorofSciencedegreeinelectricalengin eeringfromtheUniversityofFlorida.In 2002,hereceivedanAssociateofArtsdegreefromSantaFeCommunityCollege,in Gainesville,Florida.WhileattendingFerrisStateUniversityinBigRapids,Michigan,from1975to1978,hereceivedanAssociateofSciencedegreeinautomotive technologyandamachinistcerti cate. From1993to1999,hewassoleproprietorofStegathCoachcraftwhich performedcompleterestorationsonclassicautomobiles.Hewasasoftwareengineer withAppliconCAD/CAMfrom1988to 1993.From1986to1988,hewasthe soleproprietorofCADsultantswhereheworkedwiththeCAD/CAMindustry anddevelopedacustomautomobilebasedontheCorvettechassis.From1986to 1987,hewasaCAD/CAMCoordinatorwithTroyDesignwherehedevelopeda procedureforgeneratingCNCtoolpathsfromaCADsolidmodeldatabase.From 1980to1986,hewasanApplicationandProductEngineeratManufacturingData SystemInc.whereheprovidedtechnicalsupportforcustomersusingcomputer assistedprogrammingoftheirCNCmachines.From1978to1980,hewasa machinist.HeworkedatJasperAutoPartsandMachineShopmachiningand rebuildingautomobileengines,thenatElectroArcManufacturingsettingup manualmillsandlathesandmanualprogrammingaCNCmill.Forashortperiod oftimehewasanautomechanicatLongChevrolet. From1998to2000,hewrotescience ctiontechno-thrillers.Hewrotesixshort stories,completedonenovel,andin2000workedwithapublishedauthorona secondnovel. 59