<%BANNER%>

Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2009-12-31.

Permanent Link: http://ufdc.ufl.edu/UFE0021833/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2009-12-31.
Physical Description: Book
Language: english
Creator: Stegath, Keith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

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

Notes

Statement of Responsibility: by Keith Stegath.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Dixon, Warren E.
Electronic Access: INACCESSIBLE UNTIL 2009-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021833:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021833/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2009-12-31.
Physical Description: Book
Language: english
Creator: Stegath, Keith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

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

Notes

Statement of Responsibility: by Keith Stegath.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Dixon, Warren E.
Electronic Access: INACCESSIBLE UNTIL 2009-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021833:00001


This item has the following downloads:


Full Text

PAGE 1

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

PAGE 2

c 2007KeithStegath

PAGE 3

Tomywifewhosesupportenabledmetoreturntocollege,myfatherwhose integrityneverfaltered,andmymotherwhoseshortlifewas lledwithloveand compassion.

PAGE 4

ACKNOWLEDGMENTS Mygratitudegoestothenumerouspe oplewhoallowedmethefreedomto improveexistingskillsandguidanceindevelopingnewones. iv

PAGE 5

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

PAGE 6

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

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

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

PAGE 11

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

PAGE 12

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

PAGE 13

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.

PAGE 14

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

PAGE 15

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

PAGE 16

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

PAGE 17

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

PAGE 18

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.

PAGE 19

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

PAGE 20

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.

PAGE 21

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

PAGE 22

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

PAGE 23

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)

PAGE 24

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

PAGE 25

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].

PAGE 26

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

PAGE 27

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

PAGE 28

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

PAGE 29

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

PAGE 30

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

PAGE 31

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.

PAGE 32

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.

PAGE 33

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

PAGE 34

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

PAGE 35

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

PAGE 36

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.

PAGE 37

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)

PAGE 38

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)

PAGE 39

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)

PAGE 40

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)

PAGE 41

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

PAGE 42

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.

PAGE 43

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

PAGE 44

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

PAGE 45

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:

PAGE 46

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).

PAGE 47

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

PAGE 48

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

PAGE 49

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.

PAGE 50

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

PAGE 51

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.

PAGE 52

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

PAGE 53

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.

PAGE 54

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

PAGE 55

44 FigureA1:Schematicofcircuitryusedtodeliverthecomputedstimulationpulse train.

PAGE 56

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

PAGE 57

46 FigureA3:Circuitboardusedtogenerateandamplifya100 secpulse.

PAGE 58

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

PAGE 59

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.

PAGE 60

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

PAGE 61

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

PAGE 62

51 FigureA5:Circuitforadjustingthepulsewidthwithstepsof 25 sec .

PAGE 63

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.

PAGE 64

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.

PAGE 65

54 FigureA6:CircuitdiagramforS-beamloadcell.

PAGE 66

REFERENCES [1]P.H.PeckhamandD.B.Gray,Functionalneuromuscularstimulation, J. Rehab.ResearchDev. ,vol.33,pp.9,1996. [2]J.J.AbbasandH.J.Chizeck,Feedbackcontrolofcoronalplanehipanglein paraplegicsubjectsusingfunctionalneuromuscularstimulation, IEEETrans. BiomedEng. ,vol.38,no.7,pp.687,1991. [3]N.Lan,P.E.Crago,andH.J.Chizeck,Controlofend-pointforcesofa multijointlimbbyfunctionalneuromuscularstimulation, IEEETrans. Biomed.Eng. ,vol.38,no.10,pp.953,1991. [4],Feedbackcontrolmethodsfortaskregulationbyelectricalstimulation ofmuscles, IEEETrans.Biomed.Eng. ,vol.38,no.12,pp.1213,1991. [5]T.Schauer,N.O.Negard,F.Previdi,K.J.Hunt,M.H.Fraser,E.Ferchland, andJ.Raisch,Onlineidenti cationandnonlinearcontroloftheelectrically stimulatedquadricepsmuscle, ControlEngineeringPractice ,vol.13,pp. 1207,2005. [6]K.Stegath,N.Sharma,C.M.Gregory,andW.E.Dixon,Anextremum seekingmethodfornon-isometricneuromuscularelectricalstimulation,in IEEEInt.Conf.Syst.,Man,Cybern. ,2007,accepted,toappear. [7]M.Ferrarin,E.Pavan,R.Spadone, R.Cardini,andC.Frigo,Standingup exerciserbasedonfunctionalelectricalstimulationandbodyweightrelief, MedicalandBiologicalEngineeringandComputing ,vol.40,no.3,pp.282, 2002. [8]G.KhangandF.E.Zajac,Paraplegicstandingcontrolledbyfunctional neuromuscularstimulation:PartI-computermodelandcontrol-system design, IEEETrans.Biomed.Eng. ,vol.36,no.9,pp.873,1989. [9]F.Previdi,M.Ferrarin,S.Savaresi,andS.Bittanti,Gainschedulingcontrol offunctionalelectricalstimulationforassistedstandingupandsittingdownin paraplegia:asimulationstudy, InternationalJournalofAdaptiveandSignal Processing ,vol.19,pp.327,2005. [10]H.KordylewskiandD.Graupe,Controlofneuromuscularstimulationfor ambulationbycompleteparaplegicsviaarti cialneuralnetworks, Neurol Res. ,vol.23,no.5,pp.472,2001. 55

PAGE 67

56 [11]J.A.RiessandJ.J.Abbas,Adaptiveneuralnetworkcontrolofcyclic movementsusingfunctionalneuromuscularstimulation, IEEETrans.Rehab. Eng. ,vol.8,pp.42,2000. [12]E.C.StitesandJ.J.Abbas,Sensitivityandversatilityofanadaptivesystem forcontrollingcyclicmovementsusingfunctionalneuromuscularstimulation, IEEETrans.Biomed.Eng. ,vol.47,pp.1287 ,2000. [13]K.Y.TongandM.H.Granat,Gait-controlsystemforfunctionalelectrical stimulationusingneuralnetworks, Med.Bio.Eng.Comput. ,vol.37,pp. 35,1999. [14]F.Sepulveda,M.H.Granat,andA.Cliquet,Twoarti cialneuralsystems forgenerationofgaitswingbymeansofneuromuscularelectricalstimulation, Med.Eng.Phys. ,vol.19,pp.21,1997. [15]F.Sepulveda, ComputerTechniquesinMedicalandBiotechnologySystems. KluwerAcademicPub.,Amsterdam,2003. [16]D.G.ZhangandK.Y.Zhu,Simulationstudyoffes-assistedstandingup withneuralnetworkcontrol,in IEMBS26thAnnualInternationalIEEE Conf.EngineeringinMedicineandBiologySociety ,vol.6,2004,pp.4118 4121. [17]J.P.Giu ridaandP.E.Crago,Functionalr estorationofelbowextensionafterspinal-cordinjuryusinganeuralnetwork-basedsynergisticfescontroller, IEEETrans.NeuralSyst.Rehabil.Eng.,vol.13,no.2,pp.147,2005. [18]Y.Chen,W.Chen,C.Hsiao,T.Kuo,andJ.Lai,Developmentofthe fessystemwithneuralnetwork+PIDcontrollerforthestroke,in IEEE InternationalSymposiumonCircuitsandSystems,vol.5,2005,pp.5119. [19]P.F.Blackman, AnExpositionofAdaptiveControl .Macmillan,1962,ch. Extremum-seekingregulators,pp.36. [20]O.L.R.JacobsandG.C.Shering,Designofasingle-inputsinusoidal perturbationextremum-controlsystem, Proc.Inst.Elect.Eng. ,vol.115,pp. 212,1968. [21]V.V.Kazakevich,Extremumcontrolofobjectswithinertiaandofunstable objects, Sov.Phys.J. ,pp.658,1960. [22]I.S.Morosanov,Methodofextremumcontrol Automat.RemoteContr. vol.18,pp.1077,1957. [23]I.I.Ostrovskii,Extremumregulation, Automat.RemoteContr. ,vol.18,pp. 900,1957.

PAGE 68

57 [24]B.KartikandM.Krstic, Real-TimeOptimizationbyExtremumSeeking Feedback .Hoboken,NewJersey:Wiley,2003. [25]G.M.Graham,T.A.Thrasher,andM.R.Popovic,Thee ectofrandom modulationoffunctionalelectricalstimulationparametersonmusclefatigue, IEEETrans.NeuralSyst.Rehabil.Eng.,vol.14,no.1,pp.3845,2006. [26]C.M.Gregory,W.E.Dixon,andC.S.Bickel,Impactofvaryingpulse frequencyanddurationonmusclefunctionduringnmes, MuscleandNerve vol.35,no.4,pp.504,2007. [27]P.M.Patre,W.MacKunis,C.Makkar,andW.E.Dixon,Asymptotic trackingforsystemswithstructuredandunstructureduncertainties, IEEE Trans.Contr.Syst.Technol. ,2006,accepted,toappear. [28],Asymptotictrackingforsystemswithstructuredandunstructured uncertainties,in IEEEConferenceonDecisionandControl ,2006,pp.441 446. [29]W.H.Press,S.A.Teukosky,W.T.Vetterly,andB.P.Flamneny, Numerical RecipesinFortran,theArtofScienti cComputing ,2nded.Cambridge UniversityPress,1992. [30]M.KrsticandH.Deng, StabilizationofNonlinearUncertainSystem Springer-Verlag,1998. [31]X.T.Zhang,D.M.Dawson,W.E.Dixon,andB.Xian,Extremumseeking nonlinearcontrollersforahumanexercisemachine, IEEETransactionson Mechatronics ,vol.11,no.2,pp.233,2006. [32]M.Lo er,N.Costescu,andD.Dawson,Qmotor3.0andtheqmotorrobotic toolkit-anadvancedpc-basedreal-timecontrolplatform, IEEEControlSyst. Mag. ,vol.22,no.3,pp.12,2002. [33]M.FerrarinandA.Pedotti,Therelationshipbetweenelectricalstimulusand jointtorque:[a]dynamicmodel, IEEETrans.Rehabil.Eng. ,vol.8,no.3,pp. 342,2000. [34]J.L.Krevolin,M.G.Pandy,andJ.C.Pearce,Momentarmofthepatellar tendoninthehumanknee, JournalofBiomechanics ,vol.37,pp.785, 2004. [35]W.L.Buford,Jr.,F.M.Ivey,Jr.,J.D.Malone,R.M.Patterson,G.L.Peare, D.K.Nguyen,andA.A.Stewart,Musclebalanceattheknee-momentarms forthenormalkneeandtheacl-minusknee, IEEETrans.Rehabil.Eng vol.5,no.4,1997. [36]V.I.Utkin, SlidingModesinControlandOptimization .NewYord:SpringerVerlag,1992.

PAGE 69

58 [37]B.Xian,D.M.Dawson,M.S.deQueiroz,andJ.Chen,Acontinuous asymptotictrackingcontrolstrategyforuncertainmulti-inputnonlinear systems, IEEETrans.Autom.Control ,vol.49,no.7,pp.1206,2004. [38]C.Makkar,G.Hu,W.G.Sawyer,andW.E.Dixon,Lyapunov-based trackingcontrolinthepresenceofuncertainnonlinearparameterizable friction, IEEETrans.Autom.Control ,2007,accepted,toappearOct.2007.

PAGE 70

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