TY - JOUR

T1 - Identification of Stretch Reflex Dynamics from Closed-Loop Data

AU - Karami, Kiana

AU - Westwick, David T.

N1 - Funding Information:
1. INTRODUCTION where F(s) and X(s) are the Laplace transforffis of the 1. INTRODUCTION where F(s) and X(s) are the Laplace transforffis of the 1.1. IINNTTRROODDUCTIOUCTIONN jowhwhinererteetooFrqueq(su)eanandd Xang(gsle)e,arereespthehecetiLvaeplalyl,acaenddtrarτa,nsBforffianfnfids Kof thearhree joint torque and angle, respectively, and τ, B and K are Joint dynaffiics describe the relationship between the jotheintinetortiarquel,daandffipingangleand, resspringpectivceolnsy,taandnts.τT,heB iffiandpeKdancaree Joint dynaffiics describe the relationship between the the inertial, daffiping and spring constants. The iffipedance Jtooriqnutesdyanpapflfiieicds tdoesacrjiobinet,thaendretlhaetiornessuhlitpingbecthwaenegnesthtoe tsheaniniefrfitpiarol,pdearffsiypsitnegffai,nsdinscperiintgccoonntastinanstosn.lTyhzeeirfofispaednadnncoe torques applied to a joint, and the resulting changes to is an iffiproper systeffi, since it contains only zeros and no tttoohrreqquujoeeissntaa’ppsppplliioeesdditittooon,aavjjooeliionnctt,,ityaannaddndtthheeacrrceeessluuerllttaiitnniggoncchh(aaKnneggaeerssnettyoo psoalens.iffIitpsroinpveerrssye,sttehffei,asidnfcfieitittancocen,taiisnsproonplyerz,earossitanids nano the joint’s position, velocity and acceleration (Kearney poles. Its inverse, the adffiittance, is proper, as it is an tahned jHouinntt’esr,p1o9si9t0io).n,Thveloffcieitcyhaannicdalaicfcfiepleerdaatniocne o(fKaeajorninety, all-pole systeffi: and Hunter, 1990). The ffiechanical iffipedance of a joint, all-pole systeffi: asonfdfieHtiuffnietser,re1f9e9rr0e)d. Tthoe affsieicthsandiycnalafiffifiicpesdtaifnfnceesos,f adejofiinets, X(s) = 1 F (s) (2) sthofefierteilfafiteisonrsehfieprrbedetwtoeenasanitsindpyuntadffiisipclascteiffffni ensts,adnedfinthees X(s) = 2 1 F (s) (2) sthofefierteilfafiteisonrsehfieprrbedetwtoeenasanitsindpyuntadffiisipclascteiffffnieensts,adnedfinthees X(s) = F(s) (2) trheseulrteilnagtiotnorsqhuipesbgeetwnereanteadnairnopuuntddtihspelajoceinffti.enIttsainndvetrhsee XX((ss))== ((τsτs22++BBss++KK))FF((ss)) ((2)2) trheseulrteilnagtiotnorsqhuipesbgeetwnereanteadnairnopuuntddtihspelajoceinffti.enIttsainndvetrhsee (τs2 + Bs + K) risestuhlteinagdftfioirtqtaunecse,gesnoeffriaetteifdfieasroreufnedrretdhetojoianst.coItffsipilniavnercsee, (τs + Bs + K) risestuhlteinagdftfioirtqtaunecse,gesnoeffriaetteifdfieasroreufnedrretdhetojoianst.coItffsipilniavnercsee, The siffiplest reflex to consider is the ffiono-synaptic iwshtehretahdeffiaitptpalniecde,fosrocffeieotrifftioesrqrueef,erXre(ds)t,oisatsrecaotfefidplaiasntchee, The siffiplest reflex to consider is the ffiono-synaptic iwshtehre tahdeffiaitptpalniecde,fosrocffeieotrifftioesrqrueef,erXre(ds)t,oisatsrecaotfefidplaiasntchee, Tthreetcshiffriepflleesxt (rKefelaexrnetyo acnodnsHiduenrteirs, t1h9e90f)f.ioTnoh-esyfnfiaupsctilce winhpeurte,tahnedatphpelireedsuflotrincegoffriottoiroqnu,eF,(Xs)(sis),tihsetroeuattpeudt.asthe stretch reflex (Kearney and Hunter, 1990). The ffiuscle winhpeurte,tahnedatphpelireedsuflotrincegoffriottoiroqnu,eF,(Xs)(sis),tihsetroeuattpeudt.asthe stprientdclhesrerfelsepxon(KdeaffrionsetysatnrodngHlyunttoer,ve1l9o9c0it)i.esTihnetfhfieuscdlie-input,input,aandndthethereressultingultingffiffiootiotion,n,FF((ss))isisthetheoouutput.tput. spindles respond ffiost strongly to velocities in the di-repcintidolnesthraestpsotnredtcfhfieoss thsetrfofinugslcylestoinvwelhoiccihtietsheiyn atrhee efdfi-If one ignores the effects of reflexes, a joint can be ffiodelled rection that stretches the ffiuscles in which they are effi-Iafsoaneseigcnoonrde-sortdheereffffeieccthsaonfirceafllesxyesst,eaffjiocinotfficparnisbinegffiaodffeiallsesd, rreeccdttiodioennd.tthaThahtte ssettrexrceitttccahhteeissontthhfeeroffffffiiiuutsshccelleesspiininndwwlehhsiiccthhetthenhecyyauaasrreeeseetfffihfie-- Iafsoaneseigcnoonrde-sortdheereffffeieccthsaonfirceafllesxyesst,eaffjiocinotfficparnisbinegffiaodffeiallsesd, bedded. The excitation froffi the spindles then causes the asaspsraaingseconseaconndddd-or-oarffddiperererff,ffiireeeccshhuanalntiiincalcaglinsysytstshteffieeffiincoffipctroifnfispirrciisisiifnnfiggpeaadaffiass,ffniacess:, ffieuddesdceleed..tooTcheoenterxcaccitat.t.atioAiosniffifffroirplifieolffiiffiiethedheffiofisopindledeienldolefsthishtheihsesnysscteaeusffifsietasathekhees aspsraingseaconnddd-oarffdipererf,fireecshualntiincaglinsytshteeffiinctroifnfispircisiifnfigpeadaffniacess:, ffiuscle to contract. A siffiplified ffiodel of this systeffi takes springanddaffiper,result2ingintheintrinsiciffipedance: tfffhiiuuessccjlleoeinttoto ccaoonnngttlrreaaccatt.s. AAitsssiiffififnipplifiepluiftie, ddafffionfidoddeheall loof-ffwtthishavisessyysresttecetfffiiffiiettasakkieetss spring and daffFip(esr),=res(uτslst2in+gBinst+heKin)Xtri(nss)ic iffipedanc(e1):) the joint angle as its input, and half-wave rectifies its F(s)=(τs2+Bs+K)X(s) (1) theerivjaotinivtea(inngpleraacsticietsthienpduert,ivatnidvehisaalfp-wpraovxeiffrieactteidfieussiintgs ★ Supported by Fthe(s)N=atural(τs Scie+ Bncses+andK)XE(ngineerings) Resea(r1)ch a high-pass filter). This signal is then delayed, and filtered ★ Supported by the Natural Sciences and Engineering Research daehhigriigvhh-at-ppaivaessss(iffiltenilpterrr).a)c.tTTichhiseitshsseiiggdnnaearlliviisastttheihveennisddaeepllaapyyreeodd,x,iffaainndatdedffilteilutesrreineddg C★oSuunpcpiloortfeCdanbaydathtehrNouagtuhrgarlanStciNenScEesRCan-RdPEGnINgin2e3e8r9in3g9-2R0e1s0earch ayhitghhe-peaxscsitfailttieorn).dTyhniasffsiiigcnsaolfisthtehefnfiudsecllaeyetod,parnodduficlteerthede Council of Canada through grant NSERC-RPGIN 238939-2010 by the excitation dynaffiics of the ffiuscle to produce the Council of Canada through grant NSERC-RPGIN 238939-2010 by the excitation dynaffiics of the ffiuscle to produce the
Publisher Copyright:
© 2015

PY - 2015

Y1 - 2015

N2 - The torque and angle data used to identify models of limb impedance are usually generated under closed-loop conditions. In many cases, the joint position is controlled using a powerful position servo, so that the joint interacts with a very stiff environment. Under these conditions, the data may be treated as if they had been obtained in open-loop conditions. When experiments are performed under compliant conditions, the effects of feedback can no longer be ignored. A closed-loop identification algorithm which identifies a finite impulse response model of joint admittance has previously been developed, and used to estimate the admittance of the elbow joint under a variety of experimental conditions. The approach was based on prediction error minimization, and identified the FIR system model with an ARMA noise model. The experimental input was chosen to minimize the effects of reexes, so that the system would be very nearly linear. Standard correlation based tests were used to validate the resulting models. In this paper, we use two simulation models of the human ankle joint together with a typical experimental apparatus. The actuator was configured so that it could create both stiff and compliant environments, which allowed for the simulation of a broad range of joint dynamics experiments. In one case, a linear model was used, while the other included a nonlinear model of the stretch reex. Data from both simulations was processed using the closed-loop linear identification methods described above. Significant differences were noted between the models obtained from the linear and nonlinear simulations. Thus, the presence of the reex pathway biased the linear identification. However, the standard correlation based model tests suggested that the linear model structures were appropriate in both cases. The residuals appeared to be white noise, and not causally correlated with the input torque. High-order correlation based tests, normally used in the validation of parametric nonlinear models, revealed the presence of unmodeled nonlinearities in the system. This analysis is then applied to experimental data from a study on elbow compliance.

AB - The torque and angle data used to identify models of limb impedance are usually generated under closed-loop conditions. In many cases, the joint position is controlled using a powerful position servo, so that the joint interacts with a very stiff environment. Under these conditions, the data may be treated as if they had been obtained in open-loop conditions. When experiments are performed under compliant conditions, the effects of feedback can no longer be ignored. A closed-loop identification algorithm which identifies a finite impulse response model of joint admittance has previously been developed, and used to estimate the admittance of the elbow joint under a variety of experimental conditions. The approach was based on prediction error minimization, and identified the FIR system model with an ARMA noise model. The experimental input was chosen to minimize the effects of reexes, so that the system would be very nearly linear. Standard correlation based tests were used to validate the resulting models. In this paper, we use two simulation models of the human ankle joint together with a typical experimental apparatus. The actuator was configured so that it could create both stiff and compliant environments, which allowed for the simulation of a broad range of joint dynamics experiments. In one case, a linear model was used, while the other included a nonlinear model of the stretch reex. Data from both simulations was processed using the closed-loop linear identification methods described above. Significant differences were noted between the models obtained from the linear and nonlinear simulations. Thus, the presence of the reex pathway biased the linear identification. However, the standard correlation based model tests suggested that the linear model structures were appropriate in both cases. The residuals appeared to be white noise, and not causally correlated with the input torque. High-order correlation based tests, normally used in the validation of parametric nonlinear models, revealed the presence of unmodeled nonlinearities in the system. This analysis is then applied to experimental data from a study on elbow compliance.

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U2 - 10.1016/j.ifacol.2015.12.328

DO - 10.1016/j.ifacol.2015.12.328

M3 - Article

AN - SCOPUS:84988527563

SN - 2405-8963

VL - 48

SP - 1397

EP - 1402

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

IS - 28

ER -