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Cadence And Position Tracking for Switched FES-Cycling Combined with Motor Assistance for Cycle with Split-Crank

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Cadence And Position Tracking for Switched FES-Cycling Combined with Motor Assistance for Cycle with Split-Crank
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Esquivel Estay, Fidel Ignacio
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English

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Abstract:
Functional Electrical Stimulation (FES) has proven to be an effective method to improve health and regain muscle for people with limited or reduced motor skills. FES combined with an electric motor can facilitate recovery through the implementation of a closed-loop control system. Many people with movement disorders (e.g., stroke) have asymmetries in their motor control, motivating the need for a closed-loop controller that can be implemented on a split crank cycle. In this thesis, a nonlinear sliding mode controller is designed for each side of a split crank cycle to maintain a desired cadence and a constant crank angle difference of 180 degrees, simulating standard pedaling conditions. A Lyapunov-like function is used to prove stability and tracking to the desired cadence and position for the combined FES-Motor system. Experimental results on two able-bodied individuals show the feasibility and stability of the closed-loop controller. ( en )
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Awarded Bachelor of Science in Aerospace Engineering, summa cum laude, on May 8, 2018. Major: Aerospace Engineering
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College or School: College of Engineering
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Advisor: Warren Dixon, Ph.D.. Advisor Department or School: Mechanical and Aerospace Engineering

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University of Florida
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University of Florida
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Copyright Fidel Ignacio Esquivel Estay. 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.

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1 CadenceAndPositionTrackingforSwitched FES-CyclingCombinedwithMotorAssistancefor CyclewithSplit-Crank FidelI.EsquivelEstay Abstract FunctionalElectricalStimulationFEShasproven tobeaneffectivemethodtoimprovehealthandregainmusclefor peoplewithlimitedorreducedmotorskills.FEScombinedwith anelectricmotorcanfacilitaterecoverythroughtheimplementationofaclosed-loopcontrolsystem.Manypeoplewithmovement disorderse.g.,strokehaveasymmetriesintheirmotorcontrol, motivatingtheneedforaclosed-loopcontrollerthatcanbe implementedonasplitcrankcycle.Inthisthesis,anonlinear slidingmodecontrollerisdesignedforeachsideofasplit crankcycletomaintainadesiredcadenceandaconstantcrank angledifferenceof180degrees,simulatingstandardpedaling conditions.ALyapunov-likefunctionisusedtoprovestability andtrackingtothedesiredcadenceandpositionforthecombined FES-Motorsystem.Experimentalresultsontwoable-bodied individualsshowthefeasibilityandstabilityoftheclosed-loop controller. I.I NTRODUCTION FunctionalElectricalStimulationFESistheapplicationof electricalimpulsesonmusclebertoinduceaninvoluntary muscleresponseorcontraction.Overthelastfewdecades, severalstudieshavebeenperformedanalyzingthefeasibility ofFESexercises,suchasFESinducedcycling[1].Numerous physiologicalandpsychologicalbenetssuchasimprovedcardiovascularhealth[2],gainofmusclemass[3],andimproved motorcoordination[4]havebeendemonstrated.Previous studiesperformedonconventionalcycleshaveshownhow, bycombiningFESwithanelectricmotor,bettertrajectory andcadencetrackingcanbeobtainedthanbyFES-cycling alone[5].Inaddition,toaccountforpeoplewithvarying degreesofimpairment,[6]appliedassistive,resistiveand passivemodestoFESinducedcyclingtoencouragevolitional pedaling.Althoughthesestudieshaveshownahighdegreeof relevanceinrehabilitationtreatments,thecoupleddynamicsof bothpedalsonthebikefailstoisolatetheindividualmuscle contributionofeachleg.Peoplewhosufferfromasymmetric neurologicalconditions,suchasastrokeorlocalizedmuscle atrophy,mightdependentirelyononelimbtoreachthedesired torqueorcadence,likelyneglectingcontributionsoftheless responsivelimb.Therefore,itisdesirabletostudytheeffects ofdecoupledbikepedals,enablingeachmuscleregionto performindependentlyoftheother,maximizingmuscleuse andrecovery. Inthisthesis,aswitchingsignalisimplementedwhere thecontrolsignalswitchesbetweenmusclegroupsandthe *DepartmentofMechanicalandAerospaceEngineering,University ofFlorida,GainesvilleFL32611-6250,USAEmail:{delesquivel, wdixon}@u.edu electricmotorasafunctionofeffectivecrankangle,asin[5] and[6].Twodynamicsystemsareanalyzed,corresponding toeachleg.Forthedominantside,theobjectiveistotrack adesiredcadence,whereasthenon-dominantsideistasked withapositiontrackingobjectivetomaintain180phaseshift fromthepositionofthepedalonthedominantside. Aswitched-systemanalysisisusedtodesigntheproportionalgainandslidingmodecontrollersforeachsystem. Globalexponentialstabilityisobtainedforbothsub-systems independently.Thetwo-legsystemachievesglobalexponential stability,providedsufcientgainconditionsaremet. II.M ODEL Theswitchedcycle-riderdynamicsareconsideredas e l q l ; q l ; q l ;t = c l q l t ; q l t ;t + r l q l t ; q l t ; q l t ;t ; where q l : R > 0 !Q denotesthemeasurablecrankanglefor thelegs,wherethesubscript l 2L = f dom;non g indicates thedominantandnon-dominantlegs,respectively,and Q R denotesthesetofallpossiblecrankangles.Thetorques appliedaboutthecrankaxisbytheelectricmotor,thecycle, andtherideraredenotedby e l : Q R R R 0 R c l : R R R 0 R ,and r l : Q R R R 0 R respectively. Thetorqueaboutthecrankaxisbythecyclecanbe expressedas c l q l t ; q l t ;t = J c l q l t + b c q l t + d c l t ; where J c l 2 R > 0 b c l 2 R > 0 ; and d c l : R 0 R denote inertialeffects,viscousdampingeffects,anddisturbances appliedbythecycle,respectively.Thetorqueappliedabout thecrankbytheridercanbeseparatedintopassivetorques, p l : Q R R R ; theFESinducedmusclecontribution, M l : Q R R 0 R ; andthedisturbancesintheload, d r l : R 0 R asfollows: r l q l t ; q l t ; q l t ;t = p l q l t ; q l t ; q l t )]TJ/F11 9.9626 Tf 7.749 0 Td [( M l q l t ; q l t ;t + d r l t : In,thepassivetorquesappliedbytheriderare

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2 p l q l t ; q l t ; q l t = M p l q l t q l t + V l q l t ; q l t q l t + G l q l t + P l q l t ; q l t ; where M p l : Q! R > 0 ;V l : Q R R ;G l : Q! R ; and P l : Q R R ; denotetheinertial,centripetalCoriolis,gravitational,andpassiveviscoelastictissueforces, respectively.Thetorquesappliedbythemusclesaredenoted asthesumofeachmuscle'sindividualcontributionbyFES as M l q l t ; q l t ;t = X m 2M B m l q l t ; q l t u m l t;q l t ; 8 m 2M8 l 2L ; where u m l : Q R 0 R is thedesignedmusclecontrolcurrentinput,andthesubscript m 2M = f Q;G;H g indicatesthequadricepsfemoris Q ; gluteal G ; andhamstring H musclegroups,respectively. Itisassumedthattheriderdoesnotcontributevolitionallyto thisexperiment,andallvolitionalcontributionsaretherefore includedintothedisturbanceterm d r l : R 0 R : The uncertainmusclecontroleffectivenessisdenotedby B m l : Q R R > 0 ; 8 m 2M ; andcanbeexpressedas B m l = m l q l m l q l ; q l cos m l q l T m l q l ; 8 m 2M ; 8 l 2L ; where m l : Q! R > 0 denotesthe uncertainmomentarmofeachmusclegroup'sforceaboutits respectivejoint, m l : Q R R > 0 denotestheuncertain nonlinearfunctionrelatingstimulationintensitytotheforce outputbythemuscle,and m l : Q! R denotestheuncertain muscleberpennationangle.Thefunction T m l : Q! R denotesthetorquetransferratiobetweeneachmusclegroup andthecrank[7].Denitionsfortheeffectivestimulation regionsandswitchinglawsemployedduringthecadenceand trajectorytrackingarebasedon[5],wheretheportionofthe crankcycleinwhichaparticularmusclegroupisstimulated isdenotedby Q m l Q .Inthismanner, Q m l isdenedfor eachmusclegroupas Q G l f q l 2Qj T G l q l >" G l g ; Q Q l f q l 2Qj)]TJ/F11 9.9626 Tf 32.489 0 Td [(T Q l q l >" Q l g ; Q H l f q l 2Qj T H l q l >" H l g ; where m l 2 0 ; max T m l i isthelowerthresholdforeach torquetransferratio,limitingthestimulationregionssothat eachmusclegroupisengagedonlywhenitcontributesinthe positivecrankmotion. Basedonthestimulationregionsin-,thepiece-wise switchingsignal m l q l 2f 0 ; 1 g isdenedforeachmuscle groupsuchthat m l q l =1 when q l 2Q m l and m l q l =0 when q l = 2Q m l 8 m 2M ; 8 l 2L .Theregionofthecrank cyclewhereFESproducesefcienttorques, Q FES l ; isdened as Q FES l [ m 2M fQ m l g ; 8 m 2M ; 8 l 2L asdenedin [1]. Tofacilitatetheanalysisofacombinationofposition-based andvelocity-basedswitching,switchingtimesaredenotedby n t i n;l o ;i 2f s;e g ;n 2f 0 ; 1 ; 2 ;::: g ; 8 l 2L ,whereeach t i n;l representstheNthtimethatthesystemswitchestoactivate stimulationdenotedby i = s ortheelectricmotordenoted by i = e .Alsoin, u m l : R > 0 R denotesthecontrol inputandtheelectricalstimulationintensityappliedtoeach muscle,denedas u m l m l k m l u s l t ; 8 m 2M ; wherethesubsequentlydesignedFEScontrolinputisdenoted by u s l t and k m l 2 R > 0 isaconstantcontrolgain, 8 l 2L : Additionally,thetorqueaboutthecrankaxisbytheelectric motorforeachlegcanbeexpressedas e l = B e l u e l ; where B e l 2 R > 0 relatestheelectricmotor'sinputcurrentto theresultingtorqueaboutthecrankaxis,and u e l : R > 0 R isthedesignedelectricmotorcontroller, 8 l 2L Substituting-intoyields B M l u s l + B e l u e l = M l q l + b c l q l + d cl + V l q l + G l + P l + d r l ; where B M l : Q R R isthecombinedswitchedFES controleffectiveness,denedas B M l q l ; q l = X m 2M B m l m l k m l ; 8 m 2M ; 8 l 2L ,and M l : Q! R isdenedasthe summation M l J c l + M p l Thesubsequentdevelopmentisbasedontheassumptionthat adominantlowerlimbisidentiedandsetforthecontroller. Therefore,aone-waysystemdependenceisestablished,in thesensethatthenon-dominantsubsystemdependsonthe dominantsubsystem,butthereverseisnotapplicable.Both subsystemsareautonomousandstate-dependent. Theswitchedsysteminhasthefollowingproperties andassumptions,aslistedin[8].Inaddition,bothlegsare boundedbythesameconstants.Inthecaseofseverephysical differences,i.e.severemuscleatrophyorhighspasticityon oneleg,thehigherboundisusedforbothlegs. Property:1 c m M l c M ; where c m ;c M 2 R > 0 are knownconstants. Property:2 j V l j c V j q l j ; where c V 2 R > 0 isaknown constant. Property:3 j G l j c G ; where c G 2 R > 0 isaknown constant. Property:4 j P l j c P 1 + c P 2 j q l j ; where c P 1 ;c P 2 2 R > 0 areknownconstants. Property:5 j b c j c b j q l j ; where c b 2 R > 0 isaknown constant. Property:6 j d r l + d c l j c d ; where c d 2 R > 0 isaknown constant. Property:7 Byskewsymmetry, 1 2 M l = V l .

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3 Property:8 Theunknownmomentarmofeachmuscle groupabouttheirrespectivejointisnon-zero,i.e., l 6 =0 [9]. Property:9 Theauxiliaryterm l independsonthe force-lengthandforce-velocityrelationshipsofthemuscle beingstimulatedandisupperandlowerboundedbyknown positiveconstants, c ;c 2 R > 0 ,respectively,providedthe muscleisnotfullyextended[10]orcontractingconcentrically atitsmaximumshorteningvelocity[8]. Property:10 Thefunctionrelatingtheunknownmuscleberpennationangletooutputtorqueisneverzero, i.e., cos m l 6 =0 [11]. Property:11 Byproperties8-10, B m l hasalowerbound forall m l ,andthus,when P m 2M m l > 0 ;c b M B M l ; where c b M 2 R > 0 Property:12 c b e B e c B e ; where c b e ;c B e 2 R > 0 III.C ONTROL D EVELOPMENT Thecontrolobjectiveforeachofthelegsubsystemsisdenedseparatelyandissplitintothedominantsubsystem,with acadencetrackingobjective,andthenon-dominantsubsystem, withapositiontrackingobjectivetomaintainaconstantphase shiftof180degreesfromthedominantleg.Switchingsignals fortheFESmusclecontrolandandtheelectricmotorcontrol forbothlegsaredenedas s l : R 0 !f 0 ; 1 g and e l : R 0 !f 0 ; 1 g ,respectively, 8 l 2L ; 8 m 2M : These signalsaredesignedas s l 1 0 if q l 2Q m l if q l = 2Q m l ; e l 1 0 if q l = 2Q m l if q l 2Q m l : A.DominantSystem Thecontrolobjectiveforthedominantsubsystemistotrack adesiredcadencewhichisquantiedbythecadenceerror e 1 : R 0 R ; denedas e 1 t q dom d t )]TJ/F8 9.9626 Tf 11.811 0 Td [(_ q dom t ; where q dom d ; q dom 2 R > 0 arethedesiredandmeasured cadencesforthedominantleg.Theswitchingsignalforthe FESmusclecontrolandtheelectricmotorcontrolforthe dominantlegaredenedinand,respectively,where l = dom and m 2M : Takingthetimederivativeof,multiplyingby M dom andsubstitutingintoyields M dom e 1 = dom )]TJ/F11 9.9626 Tf 9.963 0 Td [(V dom e 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(B e dom u e dom )]TJ/F11 9.9626 Tf 9.963 0 Td [(B M dom u s dom ; wheretheauxiliaryterm dom : Q R R 0 0 isdened as dom b c dom q dom + d c dom + G dom + P dom + d r dom + V dom q dom d + M dom q dom d ; andusingProperties1-6, dom canbeboundedas dom c 1 + c 2 j e 1 j ; where c 1 c 2 2 R > 0 areconstants,and j j denotestheabsolute value.Basedon,,,andthestabilityanalysisin SectionIV,themusclecontrolinputfortheFESactinginthe dominantsubsystemisdenedas u s dom = s dom k 1 e 1 + k 2 sgn e 1 ; where k 1 k 2 2 R > 0 areselectableconstantcontrolgains,and s dom isdenedin.Likewise,theswitchedcontrolsystem fortheelectricmotorisdenedas u e dom = e dom k 3 e 1 + k 4 sgn e 1 ; where k 3 k 4 2 R > 0 areconstantcontrolgains,and e dom is denedin.Substitutingandintoresultsin theclosed-looperrorsystemforthedominantsubsystem M dom e 1 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(B M dom s dom k 1 e 1 + k 2 sgn e 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(B e dom e dom k 3 e 1 + k 4 sgn e 1 + dom )]TJ/F11 9.9626 Tf 9.963 0 Td [(V dom e 1 : B.Non-dominantSubsystem Thecontrolobjectiveofthenon-dominantsubsystemisto trackthedesiredcrankangle,whichisaphaseshiftof180 degreestothedominantleg.Thisobjectiveisquantiedby thepositionerror e 2 : R 0 R andtheauxiliaryerror r : R 0 R ,denedas e 2 t q non d t )]TJ/F11 9.9626 Tf 9.962 0 Td [(q non t ; r t e 2 + e 2 ; where q non d ;q non ; 2 R > 0 ,arethedesiredcrankanglefor thenon-dominantleg,themeasuredcrankangleforthenondominantleg,andaknownconstant.Theswitchingsignalfor theFESmusclecontrolandtheelectricmotorcontrolforthe non-dominantlegaredenedinand,respectively, where l = non and m 2M : Takingthetimederivativeof,multiplyingby M non andsubstitutingintoyields M non r = non )]TJ/F11 9.9626 Tf 9.963 0 Td [(V non r )]TJ/F11 9.9626 Tf 9.963 0 Td [(B e non u e non )]TJ/F11 9.9626 Tf 9.962 0 Td [(B M non u s non )]TJ/F11 9.9626 Tf 9.962 0 Td [(e 2 ; wheretheauxiliaryterm non : Q R R 0 0 isdened as non M non q non d + e 2 + b c non q non + d c non + P non + d r non + V non q non d + e 2 + G non : Basedon,andProperties1-6, non canbeboundedas non c 3 + c 4 k z k + c 5 k z k 2 ;

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4 where c 3 c 4 ;c 5 2 R > 0 areknownconstants,andtheerror vector z 2 R 2 isdenedas z =[ e 2 r ] T : Basedon,,,andthestabilityanalysisin SectionIV,themusclecontrolinputfortheFESeffective regionactingonthenon-dominantsubsystemisdenedas u s non = s non k 5 r + sgn r [ k 6 + k 7 k z k + k 8 k z k 2 ] ; where k 5 k 6 k 7 and k 8 2 R > 0 areselectablecontrolgains, and s non wasdenedin.Likewise,theswitchedcontrol systemfortheelectricmotorisdenedas u e non = e non k 9 r + sgn r [ k 10 + k 11 k z k + k 12 k z k 2 ] ; where k 9 k 10 k 11 and k 12 2 R > 0 areselectableconstant controlgains,and e non wasdenedin.Substituting andintoresultsintheclosed-looperrorsystemfor thenon-dominantsubsystem M r = )]TJ/F11 9.9626 Tf 7.749 0 Td [(B M non s non k 5 r + sgn r [ k 6 + k 7 k z k + k 8 k z k 2 ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(B e non e non k 9 r + sgn r [ k 10 + k 11 k z k + k 12 k z k 2 ]+ non )]TJ/F11 9.9626 Tf 9.963 0 Td [(V non r )]TJ/F11 9.9626 Tf 9.963 0 Td [(e 2 : IV.S TABILITY A NALYSIS TheStabilityAnalysisisdividedintodominantSection IV,Aandnon-dominantSectionIV,Bsubsystems.Four theoremsarepresentedtoevaluatethestabilityofthemotor andFEScontrollersdevelopedinSectionIII. A.StabilityofDominantSubsystem Let V L 1 : R R beapositivedenite,continuouslydifferentiable,commonLyapunov-likefunctioncandidatedened as V L 1 = 1 2 M dom e 2 1 : TheLyapunov-likefunctionisradiallyunboundedandsatises thefollowinginequalities c m 2 e 2 1 V L 1 c M 2 e 2 1 ; where c m and c M aredenedinProperty1.Afterusing, thetimederivativeofis V L 1 = 1 2 M dom e 2 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(V dom e 2 1 + dom e 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(B M dom s dom k 1 e 1 + k 2 sgn e 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(B e dom e dom k 3 e 1 + k 4 sgn e 1 : UsingProperty7,becomes V L 1 = e 1 dom )]TJ/F11 9.9626 Tf 9.962 0 Td [(B M dom s dom k 1 e 1 + k 2 sgn e 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(B e dom e dom k 3 e 1 + k 4 sgn e 1 : Theorem1. For q dom 2Q FES dom ,thecadenceerrorsystem denedinisexponentiallystableinthesensethat j e 1 t j r c M c m j e 1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(t s n;dom j exp )]TJ/F11 9.9626 Tf 8.944 6.74 Td [( dom 1 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t s n;dom ; where dom 1 2 R > 0 isdenedas dom 1 2 c M c b M k 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 2 ; where c M wasdenedinProperty1, c b M wasdenedin Property11, k 1 wasintroducedin,and c 2 wasintroducedin.Theinequalityinholdsforalltime t 2 [ t s n;dom ;t e n +1 ;dom ] ,providedthefollowinggainconditions aresatised k 1 > c 2 c b M ;k 2 > c 1 c b M : Proof: When q dom 2Q FES dom s dom =1 and e dom = 0 .Therefore,theexpressionincanbewrittenas V L 1 = e 1 dom )]TJ/F11 9.9626 Tf 9.962 0 Td [(B M dom k 1 e 1 + k 2 sgn e 1 : Using,thefollowingupperboundoncanbedeveloped V L 1 j e 1 j c 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k 2 c b m + e 2 1 c 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k 1 c b m ; whichisnegativedenite,providedthesufcientgainconditionsinaresatised.From,theinequalityin canbeupperboundedas V L 1 )]TJ/F11 9.9626 Tf 18.265 0 Td [( dom 1 V L 1 ; where dom 1 isdenedin.Thedifferentialequationin canbesolvedas V L 1 t V L 1 t s n;dom exp [ )]TJ/F11 9.9626 Tf 7.749 0 Td [( dom 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t s n;dom ] : Rearrangingandusingyieldsforalltime t 2 [ t s n;dom ;t e n +1 ;dom n 2f 0 ; 1 ; 2 ::: g Theorem2. For q dom = 2Q FES dom ,thecadenceerrorsystem denedinisexponentiallystableinthesensethatitcan beboundedas j e 1 t j r c M c m j e 1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(t e n;dom j exp )]TJ/F11 9.9626 Tf 8.944 6.74 Td [( dom 2 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t e n;dom ; foralltime t 2 [ t e n;dom ;t e n +1 ;dom ] ,where dom 2 2 R > 0 is denedas dom 2 2 c M c b e k 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 2 ; providedthefollowinggainconditionsaresatised k 3 > c 2 c b e ;k 4 > c 1 c b e :

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5 Proof: When q dom = 2Q FES dom s dom =0 and e dom = 1 .Therefore,theexpressionincanbeexpressedas V L 1 = e 1 dom )]TJ/F11 9.9626 Tf 9.962 0 Td [(B e dom k 3 e 1 + k 4 sgn e 1 : UpperboundingusingandProperty12,yields V L 1 j e 1 j c 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k 4 c b e + e 2 1 c 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k 3 c b e ; whichisnegativedeniteprovidedthegainconditionsin aresatised.Usingtoupperboundyields V L 1 )]TJ/F11 9.9626 Tf 18.265 0 Td [( dom 2 V L 1 ; where dom 2 isdenedin.Theinequalityincanbe solvedtoyield V L 1 t V L 1 t e n;dom exp [ )]TJ/F11 9.9626 Tf 7.749 0 Td [( dom 2 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t e n;dom ] : Theresultincanbeobtainedfromandforall t 2 [ t n;dom ;t n +1 ;dom for n 2f 0 ; 1 ; 2 ::: g B.StabilityofNon-dominantSubsystem Let V L 2 : R 2 R beapositivedenite,continuouslydifferentiable,commonLyaponov-likefunctioncandidatedened as V L 2 = 1 2 M non r 2 + 1 2 e 2 2 : TheLyapunov-likefunctionisradiallyunboundedandsatises thefollowinginequalities i jj z jj 2 V L 2 j jj z jj 2 ; where i ; j 2 R > 0 areknownconstantsdenedas i min 1 2 ; c m 2 ; j max 1 2 ; c M 2 ; and jj jj istheeuclideannormofavector.Usingand Property7,thetimederivativeofcanbeexpressedas V L 2 = r non )]TJ/F11 9.9626 Tf 9.963 0 Td [(rB M non s non k 5 r + sgn r [ k 6 + k 7 k z k + k 8 k z k 2 ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(rB e non e non k 9 r + sgn r [ k 10 + k 11 k z k + k 12 k z k 2 ] )]TJ/F11 9.9626 Tf 9.963 0 Td [(e 2 2 : Theorem3. For q non = 2Q FES non ,theerrorsystemdened inandisexponentiallystableinthesensethatitis boundedas jj z t jj r j i jj z t e n;non jj exp )]TJ/F11 9.9626 Tf 8.944 6.739 Td [( non 1 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(t )]TJ/F11 9.9626 Tf 9.962 0 Td [(t e n;non ; foralltime t 2 [ t e n:non ;t s n +1 ;non ] ,where non 1 2 R > 0 is denedas non 1 min ;c b e k 9 j ; providedthefollowinggainconditionsaresatised: k 10 > c 3 c b e ;k 11 > c 4 c b e k 12 > c 5 c b e : Proof: When q non = 2Q FES non ; s non =0 and e non =1 Therefore,theexpressionincanbereducedto V L 2 = r non )]TJ/F11 9.9626 Tf 9.963 0 Td [(e 2 2 + r )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [(B e non )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k 9 r + sgn r k 10 + k 11 jj z jj + k 12 jj z jj 2 : UpperboundingusingandProperty12,yields V L 2 )]TJ/F11 9.9626 Tf 25.46 0 Td [(c b e k 9 r 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(e 2 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( c b e k 10 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 3 j r j )]TJ/F8 9.9626 Tf 7.749 0 Td [( c b e k 11 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 4 j r jjj z jj)]TJ/F8 9.9626 Tf 17.711 0 Td [( c b e k 12 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 5 j r jjj z jj 2 ; whichisnegativedenite,providedthegainconditionsin aresatised.Usingtoupperboundyields V L 2 )]TJ/F11 9.9626 Tf 18.264 0 Td [( non 1 V L 2 ; where non 1 isdenedin.Thesolutiontothedifferential equationinyields V L 2 V L 2 t e n;non exp [ )]TJ/F11 9.9626 Tf 7.749 0 Td [( non 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t e n;non ; foralltime t 2 [ t e n;non ;t s n +1 ;non ] n 2f 0 ; 1 ; 2 ::: g ; from whichcanbeobtained. Theorem4. For q non 2Q FES non ,theerrorsystemdened inandisexponentiallystableinthesensethatitis boundedas j z t j r j i jj z t s n;non jj exp )]TJ/F11 9.9626 Tf 8.944 6.74 Td [( non 2 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t s n;non ; foralltime t 2 [ t s n;non ;t e n +1 ;non for n 2f 0 ; 1 ; 2 ::: g ,where non 2 2 R > 0 isdenedas non 2 min ;c b M k 5 j ; providedthefollowinggainconditionsaresatised k 6 > c 3 c b M ;k 7 > c 4 c b M ;k 8 > c 5 c b M : Proof: When q non 2 Q FES non ; s non =1 and e non = 0 .Therefore,theexpressionincanbereducedto V L 2 = r )]TJ/F11 9.9626 Tf 7.749 0 Td [(B M non k 5 r + sgn r [ k 6 + k 7 jj z jj + k 8 jj z jj 2 ]+ r non )]TJ/F11 9.9626 Tf 9.963 0 Td [(e 2 2 : UpperboundingusingandProperty11,andrearranging,yields V L 2 )]TJ/F11 9.9626 Tf 25.46 0 Td [(c b M k 5 r 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(e 2 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( c b M k 6 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 3 j r j )]TJ/F8 9.9626 Tf 7.749 0 Td [( c b M k 7 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 4 j r jjj z jj)]TJ/F8 9.9626 Tf 17.711 0 Td [( c b M k 8 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 5 j r jjj z jj 2 ; whichisnegativedeniteprovidedthegainconditionsin aresatised.Usingtoupperboundyields

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6 V L 2 )]TJ/F11 9.9626 Tf 18.264 0 Td [( non 2 V L 2 ; where non 2 isdenedin.Theinequalityincanbe solvedtoyield V L 2 V L 2 t s n;non exp [ )]TJ/F11 9.9626 Tf 7.749 0 Td [( non 2 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t s n;non ; foralltime t 2 [ t s n;non ;t e n +1 ;non ] n 2f 0 ; 1 ; 2 ::: g ; from whichcanbeobtained. C.SwitchedSystemStability FromTheorems1and2,thedominantlegsubsystem achievesexponentialcadencetrackingasseeninand .Furthermore,fromTheorems3and4,thenon-dominant subsystemachievesexponentialpositiontrackingasindicated byand.Itisnotedthatbothautonomoussubsystems achieveexponentialstabilitywhenanalyzedindependently. Furthermore,giventhatthedesiredtrajectoryofthenondominantsubsystemdependsontheinstantaneouspositionof thedominantsubsystematanymomentintime,itcanbe assertedthatprovidedthedominantsystemisstableforall time,thenon-dominantsystemwillalsobestableforthegiven intervaloftime.Thus,thedynamicsystemoutlinedinis stable,giventhatthesufcientgainconditionsshownin, ,andaresatised. V.E XPERIMENTS ExperimentswereconductedwiththeobjectiveofevaluatingtheperformanceoftheFESandmotorcontrollers developedforthedominantandthenon-dominantsubsystems, shownin,,,and,respectively.Experiments wereperformedontwoable-bodied24yearoldindividuals, onemaleandonefemale,afterwrittenconsentapproved bytheUniversityofFloridaInstitutionalReviewBoardwas provided.Thesubjectswereinstructedtoprovidenovolitional contributiontothepedalingeffort,thusallowingtheresponse ofthesystemtobeadirectmeasureofthecontrollersposition andcadencetracking. A.Split-CrankMotorizedFES-CyclingTestBed Toconducttheexperiments,acommerciallyavailablerecumbenttricycleTerraTrikeRoverX8wasmodiedto providestationarycycling.Thecyclecrankwassplitinto decoupledrightandleftsectionssuchthattherewasno mechanicalengagementbetweeneachsideofthecycle.This testbedcanbeseeninFigure1.Custompedalsweremanufacturedtotorthoticbootsthatservedtottherider'sfeet, preventingdorsiexionandplantarexionoftheankleswhile maintainingsagittalalignmentofthelowerlegs.Anoptical encoderUSDigitalH1wasafxedtoeachcyclecrankto measureangularpositionandvelocityofthedominantand non-dominantsubsystems.Two250Watt,24DCmotors,one foreachdynamicsystem,wereusedtoassisttheforward pedalingmotionoftherider.Foreachmotor,anADVANCED MotionControlAMCPS300W24powersupply,linkedwith anAMCFC15030ltercard,wasusedtopowerthemotor Figure1.MotorizedFES-cyclingtestbedwithsplit-crankusedforexperiments.AElectricmotors,BFESstimulator,Corthoticpedals,andD split-cranksystem. andreduceelectricalnoise,respectively.Themotorswere controlledusingtwoAMCAB25A100motordrivers. Acurrent-controlledstimulatorHasomedRehaStimwas usedtodeliverbiphasic,symmetric,andrectangularpulsesto thesubject'sactivemusclegroupsviaself-adhesiveelectrodes. Thestimulationamplitudewasxedto90mAforthequadriceps.Thestimulationintensitywascontrolledbymodulation ofthepulsewidthaccordingtoforthedominantleg,and forthenon-dominantleg. AQuanserQ8-USBdataacquisitiondevicewasusedto collectsignaldatafromtheencodersanddelivermotor current.TheFESandmotorcontrollerswereimplemented onacomputerrunningrealtimecontrolsoftwareQUARC, MATLAB/Simulink,Windows10atasamplingrateof500 Hz. B.ExperimentalSetup Electrodeswereplacedoverthesubjects'quadriceps femorisaccordingtotheAxelgaardelectrodeplacementmanual 1 .Thesubjectswerethenseatedonthecycleandtheirlegs securedtotheorthoticboots.Theseatwasadjustedtoensure hyperextensionofthekneewouldnotoccuratanypointduring thepedalingcycle.Thelengthsofthelowerlimbsaswellas thedistancesbetweenthecrankandthehipsweremeasured andusedtocalculatetheeffectivetorquetransferratiosused todeterminethemuscle-motorswitchingsignals,asin[5]. Theriders'legswerethenapproximatelypositionedata180 degreephasedifferencetopreventthesubjectfromalarge responsecausedbyalargeinitialpositionerror. Forthedominantsubsystem,thecadenceobjectivewasset to50RPM.Theexperimentwasstartedfromrest.Forthenondominantsubsystem,thedesiredpositionwasdenedasa180 degreeoffsetfromtheinstantaneouspositionmeasuredforthe dominantleg.Thedesiredcrankvelocityforthedominantleg 1 http://www.palsclinicalsupport.com/videoElements/videoPage.php

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7 q dom d andthedesiredcrankpositionforthenon-dominantleg q non d aredenedas q dom d = 5 3 1 )]TJ/F119 9.9626 Tf 9.962 0 Td [(exp )]TJ/F8 9.9626 Tf 8.945 6.739 Td [(2 5 t )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 0 ; q non d = q dom + : Duringtherst10secondsafterthesimulationwasinitiated, noFESwasappliedforeitherleg,andthemotorsalone wereusedtobringthedominantlegtothedesiredcadence of50rpmwhilekeepingthenon-dominantlegataconstant phasedifferenceof180degrees.After10seconds,themuscle stimulationwasappliedaccordingtotheswitchingsignalsand controlmethodsdescribedin,,and,respectively, wherethetorquetransferratioswerecalculatedforevery subject. C.Results Figure2depictsthetrackingperformanceforboth,dominantandnon-dominantsubsystemsforonesubjectduring astandardexperimentwithadurationof150seconds.The trackingerrorisquantiedbytheerrorsignalsdenedin SectionIII.Alsodepictedisthemotor/FEScontrolinputsover thedurationofthetrial. D.Discussion Experimentswereperformedtodepictthesystemresponseofthedecoupleddynamicsystemsdominantand non-dominantlegsforcadenceandpositiontrackingona stationarybike.TheresultsshowninFigure2successfully demonstratetheimplementationoftheFES-muscleinputcontrollersinandaswellasthemotorcurrentcontroller inand,toachieveexponentiallystabletrackingof cadenceforthedominantlegandpositionforthenondominantleg.Theresultsindicateexponentialconvergence toanultimateboundonthetrackingerrors e 1 and e 2 .These werefoundtobe 2 : 397 RPMwithastandarddeviationof 3 : 384 RPMand 0 : 7179 degreeswithastandarddeviationof 8 : 546 degrees,respectively.Whenthecontrollersareactive, themotorpreventsthelegsfromgoingbackwards,ortoofast, whiletheFESstimulatestheminthepositivecrankmotion. Onastandardcoupledcycle,mechanicalengagementbetweenthecrankaxlesprovidesacounteractingforceabout thecrankaxiswheneveroneofthelegsmove,balancing forcesduetogravityactingonthecyclesystem.Onthesplitcranktestbedusedfortheseexperiments,whenthelegsare positionedinthekinematicdeadzonesi.e.directlyinfront orbehindthethecrankshaft,thegravitationalforcesacting oneachlegdonotcanceleachother,causingmoretorque deviationsinthisregionofthecrank.Therefore,whilethe controllersmaintainthedesiredtracking,theaddedcyclic disturbanceproduceaforwardincreaseinspeed,thatcan accountfortheoffsetinthecadenceerror. Thepositionerror e 2 wascenterednearzerodegreesand hadastandarddeviationof 8 : 546 degrees.Giventhenatureof theexperimentwherebothlegsubsystemsaremechanically independent,thephasedifferencebetweentheshaftsisfreeto pivotaround180degrees,creatingrapidlyoscillating,negative topositive,positiontrackingerror.Thisbehaviorcausesthe motortoprovidearapiductuationbetweenpositiveand negativecurrents,whichcanbeseeninthe u e vstimegraphs inFigure2. Thecycle-riderclosed-loopsystemsinandare intendedtobeusedbyriderswithasymmetricalneurological conditions.Duetothisconditions,onastandardmotorized FES-cyclingtestbedwheretheshaftsarecoupled,theworking legmaydominatethecycle,effectivelynullifyinganyinput fromthelesseffectivelimb.Thecontrollersin,,, andsplitthelegsintotwodecoupledsubsystemswhere eachlegcontributesonlyitsportionofthetorqueneeded tomaintainthedesiredpositionandcadence.Thisimplies thatonelegcanreceiveextendedmotorassistancewithout affectingtheotherleg'sbehaviororresponse,andthemuscles fromeachlegwillbesufcientlyexercised. VI.C ONCLUSION ThecombinedFES-cyclingandelectricmotorslidingmode controllersdevelopedinthispaperweredesignedtoenable aridertopedalatadesiredcadencewhilemaintaininga constantcrankangledifferenceof180degreeswhenpedaling onamechanicallydisengagedsplit-crankstationarycycle.A Lyapunov-likeanalysiswasusedtoprovethestabilityofthe controllersforthedominantandnon-dominantsubsystems, andshowedexponentialtrackingfortheerrorsignals.Two experimentsperformedontwoable-bodiedindividualsvalidatedthecontroleffectivenesstoreachthedesiredcadenceof 50RPMandpositionoffsetof180degrees,attainingglobal ultimatelyboundedtracking. Thedevelopedcontrolsystemforadecoupledcyclecrank hasthepotentialtoadvancemotorizedFESrehabilitation proceduresforpeoplewithmovementdisordersthatresult inasymmetries.Theintroductionofdecoupleddynamicsfor thelegsubsystemsallowsforindividuallegstocontribute proportionallytotheirabilities,preventingoneeffectiveleg fromnullifyingtheinputfromalesseffectivelegoramuscle group. Inthefuture,potentialprotocolsactingonthekinematic deadzoneswherelegsacceleratedownwardsordecelerate upwardscouldbeimplementedtoreducethepositionerror andimproveoveralltracking.Theauthoralsoplanstoconduct experimentsonsubjectswithdifferentdegreesofneurological conditionstotesttherobustnessandusefulnessofthecontrollerdesign. R EFERENCES [1]M.J.Bellman,T.H.Cheng,R.J.Downey,C.J.Hass,andW.E. Dixon,Switchedcontrolofcadenceduringstationarycyclinginduced byfunctionalelectricalstimulation, IEEETrans.NeuralSyst.Rehabil. Eng. ,vol.24,no.12,pp.1373,2016. [2]S.P.Hooker,S.F.Figoni,M.M.Rodgers,R.M.Glaser,T.Mathews, A.G.Suryaprasad,andS.C.Gupta,Physiologiceffectsofelectrical stimulationlegcycleexercisetraininginspinalcordinjuredpersons, Arch.Phys.Med.Rehabil. ,vol.73,no.5,pp.470,1992. [3]M.Blanger,R.B.Stein,G.D.Wheeler,T.Gordon,andB.Leduc, Electricalstimulation:canitincreasemusclestrengthandreverseosteopeniainspinalcordinjuredindividuals? Arch.Phys.Med.Rehabil. vol.81,no.8,pp.1090,2000.

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8 Figure2.Trackingperformancefordominantlimbleftandnon-dominantlimbright.Resultsforthedominantlimbarequantiedbythecadenceerror e 1 trajectorytracking q dom ,FESmuscleinput u s dom ,andmotorcurrentinput u e dom : Likewise,resultsforthenon-dominantlimbarequantiedbyposition error e 2 ,positiontracking q non ,FESmuscleinput u s non ,andthemotorcurrentinput u e non [4]S.Ferrante,A.Pedrocchi,G.Ferrigno,andF.Molteni,Cyclinginduced byfunctionalelectricalstimulationimprovesthemuscularstrengthand themotorcontrolofindividualswithpost-acutestroke, Eur.J.Phys. Rehabil.Med. ,vol.44,no.2,pp.159,2008. [5]M.J.Bellman,R.J.Downey,A.Parikh,andW.E.Dixon,Automatic controlofcyclinginducedbyfunctionalelectricalstimulationwith electricmotorassistance, IEEETrans.Autom.ScienceEng. ,vol.14, no.2,pp.1225,April2017. [6]C.Rouse,C.Cousin,V.H.Duenas,andW.E.Dixon,Cadencetracking forswitchedfescyclingcombinedwithvoluntarypedalingandmotor resistance,in Proc.Am.ControlConf. ,2018. [7]E.S.Ids,T.Johansen,andK.J.Hunt,Findingthemetabolically optimalstimulationpatternforFES-cycling,in Proc.Conf.oftheInt. Funct.ElectricalStimulationSoc. ,Bournemouth,UK,Sep.2004. [8]M.J.Bellman,T.-H.Cheng,R.J.Downey,andW.E.Dixon,Stationary cyclinginducedbyswitchedfunctionalelectricalstimulationcontrol, in Proc.Am.ControlConf. ,2014,pp.4802. [9]J.L.Krevolin,M.G.Pandy,andJ.C.Pearce,Momentarmofthe patellartendoninthehumanknee, J.Biomech. ,vol.37,no.5,pp. 785,2004. [10]D.A.Winter, BiomechanicsandMotorControlofHumanMovement NewYork:Wiley,1990. [11]R.L.LieberandJ.Friden,Functionalandclinicalsignicanceof skeletalmusclearchitecture, MuscleNerve ,vol.23,no.11,pp.1647 1666,Nov.2000.